tesis doctoral de la universidad de alicante. tesi doctoral...

Three essays on specification testing in econometric models. Alicia Pérez Alonso.

Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d´Alacant. 2006.

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Three Essays on Specification Testing in Econometric Models

Alicia Pérez Alonso

.^i^-//-'^' ^- ' '•''^^\ Supervisor: Juan Mora López

Quantitative Economics Doctorate Departamento de Fundamentos del Análisis Económico

Universidad de Alicante

July 2006



Os meus avós, ós meus pais e a miña irmá:

Por darme ás para voar e facerme sentir que sempre terei un fogar ó que voltar.



Agradecimientos

Quiero agradecer a mi director de tesis Juan Mora la disposición, paciencia y apoyo mostrados a lo largo de estos años, en los que ha compartido conmigo sus conocimientos y su tiempo. Para mí serás siempre un referente a seguir tanto en lo profesional como en lo personal. Muchas gracias por todo lo que me has enseñado.

Quiero expresar mi gratitud también a los miembros del Departamento de Fundamentos del Análisis Económico de la Universidad de Alicante, desde mis profesores en los cursos del doctorado hasta aquellos que han participado en el Workshop de Econometria. En concreto me gustaría mencionar a Lola Collado por la lectura de mi trabajo y su ayuda.

Gracias también a mis profesores de Vigo, y en especial a Eduardo Giménez, María Jesús Freiré, Consuelo Pazo, Manuel Besada, Gustavo Bergantiños y José María da Rocha por haberbe animado a emprender esta locura y confiar en mis posibilidades. Me gustaría darle las gracias también a Daniel Miles por su hospitalidad y por abrirme de nuevo las puertas de Vigo.

Agradezco también a los miembros del CAM (Universidad de Copenhague) su hospitalidad durante mi estancia allí. Gracias muy especialmente a Sergio y a Mirtha por vuestra amistad y hacerme sentir como en casa.

Durante estos años en Alicante, son tantos los buenos amigos que he hecho que no sé muy bien por dónde empezar a dar las gracias. Sin duda, pese a todos los sacrificios que conlleva estar lejos de la familia, la morriña por mi tierra y los momentos de desánimo vividos durante los cursos y la tesis, empezar este doctorado es la mejor decisión que he tomado en mi vida. Por este motivo, Alicante siempre será para mi un punto de referencia.

Muchas gracias especialmente a Patricia Castromán y Pilar Castillo por compartir conmigo risas, llantos y charlas interminables durante los cursos. Por toda la confianza que habéis depositado en mí desde el principio, vuestro ánimo y amistad siento que esta tesis también es vuestra.

Para mis niñas, Laura y Marisa, un gracias enorme por todo, que es muchísimo. La verdad es que no me hago a la idea de que no volvamos a compartir despacho. Os voy a echar mucho de menos. Gracias también a mis queridos compañeros de piso Jorge, Gabriel, Leonora y Fiorenzo. También a Patricia Restrepo y Monica Contestabile que vinieron de su mano. No puedo olvidar a todos los compañeros del doctorado con quienes he compartido muchos buenos momentos. Me gustaría mencionar especialmente a Antonio, Chony, Miguel, Rebeca, Dunia, Juandi, Bea, Ricardo Alberola, Lore, Arantxa, Szabi, José María, Paco, Ricardo Martínez, Patri (te has ganado ha pulso el diminutivo), Aida, Jaromir, Nataliya, Aitor, Lari y Frede. Muchas gracias a todos por vuestra amistad.

Gracias muy especialmente a Silvio, porque sin él no habría podido concluir esta tesis. Llegaste en el momento más oportuno. Trabajar contigo ha significado



recobrar de nuevo las ganas de dedicarme a esta profesión. Gracias por todo lo que me has enseñado, admiro tu fuerza y coraje para enfrentarte a las adversidades.

Finalmente me gustaría mencionar a mi familia, mi hermana Romi y mis padres. Sin su cariño, paciencia y apoyo incondicional no podría haber concluido con éxito esta tesis.



Contents

Agradecimientos 2 Introduction and Summary 6 Introducción y Resumen en Español 9

A Bootstrap Approach to Test the Conditional Symmetry in Time Series Models 13 1.1 Introduction 13 1.2 The nonlinear dynamic model 16 1.3 Tests for conditional symmetry 20 1.4 Symmetric bootstrap 21

1.4.1 Asymptotic properties 23 1.5 A Monte Carlo study 26

1.5.1 Experimental design 26 1.5.2 Simulation results: a comparative study of symmetry tests . . 29

1.6 Conclusions 37 Appendix 39 References 42 Tables 45

Unemployment and Hysteresis: A Nonlinear Unobserved Components Approach 56 2.1 Introduction 56 2.2 An extension of Jaeger and Parkinson's model 61 2.3 Experimental design for computing the bootstrap p-value for the lin

earity hypothesis test 65 2.3.1 The state-space model 66 2.3.2 Homoskedastic bootstrap 66 2.3.3 Heteroskedastic bootstrap 69 2.3.4 Monte Carlo evidence 70

2.4 Empirical results 72 2.5 Conclusions 74 References 75



Tables and Figures 78

Specification Tests for the Distribution of Errors in Nonparametric Regression: a Martingale Approach 85 3.1 Introduction 85 3.2 Statistics based on the estimated empirical process 88 3.3 Statistics based on a martingale-transformed process 92 3.4 Simulations 94 3.5 Concluding Remarks 95 Appendix 97 References 107 Tables 109



Introduction and Summary

This thesis is composed of three chapters, in which we focus on three related, but

different, issues regarding specification testing^. In particular, in the first chapter,

we analyze how to test for conditional symmetry in time series data. In the second

chapter, we focus on an empirical application of an example of a nested hypothesis

test; this simply means that the null hypothesis that is tested is a special case of the

alternative model. Finally, in the third chapter, we describe how to test parametric

assumptions about the distribution of a regression error, making no parametric

assumption about the conditional mean or the conditional variance in the regression

model.

More precisely, in Chapter 1, "A bootstrap approach to test the conditional sym

metry in time series models ", we focus on the evaluation of several statistical testing

procedures that can be used to test for conditional symmetry. In particular, we con

sider the nonparametric test for conditional symmetry of Bai and Ng [2001, Journal

of Econometrics 103, 225-258]. The test, which is based on martingale transfor

mations, does not require the data to be stationary or independent and identically

distributed (i.i.d.), and the dimension of the conditional variables can be infinite.

The test is shown to be consistent and asymptotically distribution-free, but its com

putation is rather intensive. The literature on the closely related problem of testing

for (unconditional) symmetry is large. A classical test of symmetry is the test of

skewness. We show that under standard regularity conditions that ensure asymp

totic normality of parameter estimators, the asymptotic null distribution of this test

does not change when replacing the unknown errors by well-behaved residuals. In

addition, commonly used nonparametric tests are the Wilcoxon signed-rank test,

the Runs test and the Triples test, among others. These test are asymptotically

distribution-free for i.i.d. observations. It is not clear whether these tests can be

extended to testing for conditional symmetry, since it has not yet been rigorously

proved that statistics computed by using regression residuals instead of the true

^ Chapter 2 is a joint work with Silvestre Di Sanzo and Chapter 3 is a joint work with Juan Mora.



errors have approximately the same distribution as those based on the errors. It is

by no means obvious that this is so. Another problem encountered when using real

data is that, for finite samples, the distributions of the symmetry tests included in

this study are still unknown. As a consequence, the true size of these tests often

differs to a large extent form its nominal size based on asymptotic critical values.

The main purpose of this chapter is to show if the bootstrap can be used to ob

tain finite-sample critical values. We describe the bootstrap method and establish

a consistency property of the bootstrap for a nonlinear dynamic model. We also

perform, for a wide variety of alternative symmetric and asymmetric distributions,

Monte Carlo simulations to compare the finite-sample size and power of the tests

when critical values are obtained using a bootstrap procedure with that we could

achieve using the asymptotic theory when available. The results of Monte Carlo ex

periments show that for the cases investigated, the bootstrap methodology proposed

performs reasonably well.

In Chapter 2, "Unemployment and hysteresis: a nonlinear unobserved compo

nents approach", we propose a definition of hysteresis taken from Physics which

allows for nonlinearities. To provide an operational statistical framework for our

concept of hysteresis we use the unobserved components approach, which decom

poses unemployment rate into a non-stationary natural component and a stationary

cyclical component, which are both treated as latent variables. We extend the model

of Jaeger and Parkinson [1994, European Economic Review 38, 329-42] by introduc

ing nonlinearities in the specification of the natural rate component. In particular,

we allow past cyclical unemployment to have a difi erent impact on the current nat

ural rate depending on the regime of the economy. The estimation methodology used

can be assimilated into a threshold autoregressive representation in the framework

of a Kalman filter. Under this new framework, the problem of testing for hysteresis

becomes a problem of testing for linearity. When we implement a test for linearity a

problem of unidentified nuisance parameters under the null hypothesis arises. As a

result, conventional statistics do not have an asymptotic standard distribution. To

circumvent this problem and derive an appropriate p-value for a test for hysteresis we

7



propose two alternative bootstrap procedures: the first is vaHd under homoskedastic

errors and the second allows for general heteroskedasticity. We investigate the per

formance of both bootstrap procedures using Monte Carlo simulations. Our study

concerns Italy, France and the United States. For European countries, we reject the

null of linearity. This is related to the presence of hysteresis. On the other hand,

for the United States, we reject the hysteresis hypothesis, but we find evidence in

favour of persistence.

In Chapter 3, "Specification tests for the distribution of errors in nonparametric

regression: a martingale approach", we focus on the so important problem of test

ing whether the distribution of a regression error belongs to a parametric family of

continuous distribution functions, without assuming any specific parametric form

for the conditional mean function or the conditional variance function. If errors

were observable and the parameter were known, the problem we consider is usually

referred to as the one-sample problem. Suitable statistics for testing the hypothesis

of interest in this case are, for example, the Kolmogorov-Smirnov or Cramer-von

Mises type test statistics. In our context, we do not observe the error term, but we

can construct residuals using nonparametric estimators of the conditional mean and

the conditional variance. Additionally, we have to replace the unknown parame

ter by a well-behaved estimator. The consequences of replacing errors by residuals

and unknown parameters by estimators is that, in general, the empirical process

in which test statistics are based converges to a process that depends on unknown

quantities (the underlying true distribution and the true parameters). Therefore,

these test statistics are no longer asymptotically distribution-free; hence, asymptotic

critical values cannot be tabulated. To circumvent this problem, we propose test

statistics based on a martingale transformation of the estimated empirical process;

this martingale-transformed process is asymptotically distribution free and, hence,

asymptotic critical values can be obtained without bootstrap or simulation meth

ods. We also perform a Monte Carlo experiment to check the behaviour of the

asymptotically distribution-free test statistics for small and moderate sample sizes.



Introducción y Resumen en Español

Esta tesis consta de tres capítulos, en los que se abordan diferentes aspectos de los

contrastes de especificación^. Así, en el primero se estudia cómo contrastar simetría

condicional en series temporales. El segundo se centra en una aplicación empírica

de un ejemplo de un contraste de hipótesis anidado; esto quiere decir que la hipóte

sis nula a contrastar es un caso particular del modelo bajo la hipótesis alternativa.

Finalmente, el tercero analiza cómo contrastar supuestos paramétricos sobre la dis

tribución de los errores de regresión sin hacer ningún supuesto paramétrico sobre la

media condicional o la varianza condicional del modelo de regresión.

Más concretamente, en el Capítulo 1, "Un enfoque bootstrap para contrastar

simetría condicional en modelos de series temporales", se evalúan varios proced

imientos estadísticos de contraste que pueden utilizarse para contrastar simetría

condicional. En particular, estudiamos el contraste no paramétrico para simetría

condicional de Bai y Ng [2001, Journal of Econometrics 103, 225-258]. Este con

traste, basado en una transformación de martingala, no exige que los datos sean

estacionarios o independientes e idénticamente distribuidos (i.i.d.), y la dimensión

del conjunto de variables condicionales puede ser infinita. Además, éste es consis

tente y tiene distribución asintótica libre, pero su cálculo es bastante intensivo. La

literatura sobre contrastes para la simetría no condicional es enorme. Un contraste

clásico es el del coeficiente de asimetría. Bajo condiciones de regularidad estándares

que aseguren la normalidad asintótica de los estimadores de los parámetros, la dis

tribución asintótica de este contraste bajo la hipótesis nula no cambia cuando se

sustituyen los errores no conocidos por residuos que se comportan correctamente.

Por otra parte, otros contrastes no paramétricos comúnmente utilizados son el con

traste de rangos-signos de Wilcoxon, el contraste de las rachas y el contraste de

los tripletes, entre otros. Estos contrastes tienen distribución asintótica libre con

observaciones i.i.d. No sabemos si estos contrastes pueden utilizarse para simetría

condicional, ya que no se ha demostrado de forma rigurosa que estos estadísticos cal-

^El Capítulo 2 es un trabajo conjunto con Silvestro Di Sanzo y el Capítulo 3 es un trabajo conjunto con Juan Mora.

9



culados con los residuos tengan aproximadamente la misma distribución que cuándo

se calculan usando los verdaderos errores de regresión. Además, la demostración

no es trivial. Otro problema que aparece cuando usamos datos reales es que, para

muestras finitas, la distribución de los contrastes de simetría incluidos en este es

tudio es todavía desconocida. Por tanto, el tamaño real de los contrastes difiere

con frecuencia de su tamaño nominal, basado en valores críticos asintóticos. El

principal objetivo de este artículo es estudiar si el bootstrap se puede usar para

obtener valores críticos para muestras finitas. Se describe el método bootstrap y se

estable la consistencia del bootstrap para un modelo dinámico no lineal. También se

realizan, para una amplia variedad de distribuciones simétricas y asimétricas, simu

laciones de Monte Cario para comparar el tamaño y la potencia en muestras finitas

de los contrastes cuando los valores críticos se calculan utilizando un procedimiento

bootstrap con los que se obtienen cuando utilizamos la teoría asintótica, en caso

de que ésta exista. Los resultados de los experimentos de Monte Cario muestran

que, para los casos investigados, la metodología bootstrap que se propone funciona

razonablemente bien.

En el Capítulo 2, "Desempleo e histéresis: un enfoque no lineal de componentes

no observables", se propone una definición de histéresis extraída de la Física que

permite no linealidades. Para proporcionarle a este concepto de histéresis un marco

estadístico en el que operar se usa el enfoque de componentes no observables, el

cuál descompone el ratio de desempleo en una componente natural no estacionaria

y en una componente cíclica estacionaria, ambas tratadas como variables latentes.

El modelo de Jaeger y Parkinson [1994, European Economic Review 38, 329-42]

se amplía introduciendo no linealidad en la especificación de la componente ratio

natural. En concreto, se permite que la componente cíclica pasada tenga un impacto

diferente en el ratio natural de desempleo actual dependiendo del régimen de la

economía en el que nos encontremos. La metodología utilizada se puede asimilar a

la de los modelos umbral autoregresivos en el marco de un filtro de Kalman. En este

contexto, el contraste de histéresis se transforma en un contraste sobre la linealidad

del modelo. Al realizar el contraste de linealidad surge el problema de que hay

10



parmámetros que no están identificados bajo la hipótesis nula. Como consecuencia,

los estadísticos convencionales no tienen una distribución asintótica estándar. Para

solucionar este problema y poder obtener p-valores adecuados para un contraste

de histéresis se proponen dos procedimientos bootstrap alternativos: el primero es

válido bajo homocedasticidad de los errores y el segundo permite formas generales de

heterocedasticidad. El funcionamiento de ambos procedimientos boost rap se estudia

mediante simulaciones de Monte Cario. Los países incluidos en el estudio son Italia,

Francia y Estados Unidos. Para los países europeos, se rechaza la hipótesis nula

de linealidad, lo cuál se asocia a la presencia de histéresis. Por el contrario, para

Estados Unidos, se rechaza la hipótesis de histéresis, pero hay evidencia sobre la

existencia de persistencia.

Finalmente, el Capítulo tercero "Contrastes de especificación para la distribución

de los errores en regresiones no paramétricas: un enfoque martingala", se centra en el

problema de cómo contrastar si la distribución de los errores de regresión pertenece a

una familia paramétrica de funciones de distribución continuas, sin suponer ninguna

forma paramétrica específica para las funciones media condicional y varianza condi

cional. Si los errores fuesen observables y los parámetros conocidos, el problema

estudiado se conoce en la literatura como el problema de una muestra. Estadísti

cos adecuados para contrastar la hipótesis relevante son, por ejemplo, estadísticos

del tipo Kolmogorov-Smirnov y Cráter-von Mises. En este contexto, el término de

error no es observable, pero podemos construir residuos utilizando estimadores no

paramétricos de la media condicional y la varianza condicional. Además, es necesario

reemplazar los parámetros desconocidos por estimadores adecuados de los mismos.

Las consecuencias de reemplazar los errores por residuos y los parámetros descono

cidos por estimadores son que, en general, el proceso empírico en el que se basan

estos estadísticos converge a un proceso que depende de variables desconocidas (la

verdadera distribución subyacente y los verdaderos parámetros). Por tanto, estos

estadísticos de contraste dejan de tener una distribución asintótica libre; como con

secuencia, los valores críticos asintóticos no pueden ser tabulados. Para solucionar

este problema, en este capítulo se proponen estadísticos de contraste basados un

11



transformación martingala del proceso empírico estimado. El proceso que resulta de

esta transformación tiene distribución asintótica libre y, por tanto, se pueden obtener

valores críticos sin necesidad de utilizar el bootstrap o métodos de simulación. Un

experimento de Monte Cario estudia el comportamiento de estos estadísticos con

distribución asintótica libre para muestras de tamaño pequeño y tamaño moderado.

12



Chapter 1

A Bootstrap Approach to Test the Conditional Symmetry in Time Series Models

1.1 Introduction

The problem of testing conditional symmetry in time series data is fundamental in

both theoretical and empirical research. In the last few years considerable research

has been devoted to model and forecast the conditional mean and the conditional

variance of financial time series, that is, the return and risk of financial assets, re

spectively. The class of (Generalized) Autoregressive Conditional Heteroskedasticity

((G)ARCH) models, introduced by Engle (1982) and BoUerslev (1986), is the most

widely used among economists and other applied practitioners to model time vary

ing conditional variances. In essence, all empirical studies that assume conditional

heteroskedasticity also use a quasi-maximum likelihood estimator (QMLE). If the

likelihood is assumed to be Gaussian, the QMLE is known to be consistent if the

conditional mean and the conditional variance are correctly specified. However, nor

mality of innovations is frequently not a very realistic assumption for high-frequency

financial time series because the resulting model fails to capture the kurtosis in the

data. Alternative distributions for innovations are considered in the literature. For

example, following BoUerslev (1987), a popular choice is the standardized Student-í

distribution. If the likelihood is assumed to be non-Gaussian, Newey and Steigerwald

13



(1997) show that consistency of a QMLE requires that both the assumed innovation

density and the true innovation density are unimodal and symmetric around zero.

Moreover, if conditional symmetry fails, an additional parameter is needed to ensure

consistency of a non-Gaussian QMLE. The additional parameter accounts for the

location of the innovation density. The reader may refer to the work of Franses and

van Dijk (2000) for an extensive survey of the recent developments of modelling,

estimation and hypothesis testing for time-varying conditional variance models.

Whether or not conditional symmetry holds is also an issue of interest for adap

tive estimation. An adaptive estimator shares the asymptotic optimality properties

of the maximum likelihood estimator, differing from it in that a nonparametric es

timator of the score function of the log likelihood replaces the analytic expression

that would be used if the actual functional form of the disturbance distribution was

known. Bickel (1982) shows that if the density function of the disturbance is sym

metric about the origin, then the parameters of a linear regression model can be

estimated adaptively. Newey (1988) constructs adaptive estimators of linear regres

sion parameters by a generalized method of moments (GMM) when the foregoing

is true. The above results are extended to stationary autoregressive moving average

(ARMA) process by Kreiss (1987) and reduced-rank vector error correction models

by Hodgson (1998). In the case of testing, the efficiency of the methods can be

improved under the additional assumption of a symmetric error distribution, see for

example Azzalini and Bowman (1993) or Kulasekera and Wang (2001). Further,

conditional symmetry is part of the stochastic restrictions on unobservable errors

used in semiparametric modelling (see Powel (1994) and references therein). The

conditional symmetry restriction implies constant conditional mean and median,

which is quite familiar in econometric theory and practice.

The conventional asymptotic theory of the bootstrap relies on Edgeworth ex

pansions in order to prove the existence of asymptotic refinements. In many cases

the efficiency of this method can be improved under the additional assumption of

symmetry. Davidson and Flachaire (2001) study various versions of the wild boot

strap applied to a linear regression model with heteroskedastic errors. They show

14



that when the error terms are symmetrically distributed about the origin, the wild

bootstrap applied to statistics based on heteroskedasticity-conistent standard errors

benefits from better asymptotic refinements than when errors are asymmetrically

distributed. In particular, they found that the error in rejection probability (ERP)

is at most of order T~ / with symmetric errors and at most of order T^^/^ with

asymmetric errors, where T denotes the sample size. Comparable results are ob

tained by Hall (1992) for the case of homoskedastic regression models. He shows

that bootstrap tests on the slope parameters benefit from refinements in the case of

unskewed error terms.

There is also a growing literature addressing the problem of conditional symmetry

of macroeconomic time series related to asymmetries in business cycles. As discussed

in Brunner (1992), the assumption of Gaussian shocks places strong restrictions on

the time series behaviour of economic fluctuations. Since the Gaussian distribution

is symmetric about zero, the conditional density is symmetric about its conditional

mean. Our notion of conditional symmetry is that, in an expansion (contraction),

the probability of further expansion (contraction), relative to the conditional mean,

is equal to the probability of a contraction (expansion). That is, positive shocks

to the conditional mean are as likely as negative shocks. There is a substantial

body of empirical evidence that suggests that business cycles expansions appear

to be more persistent and less volatile than contractions. That is, economic time

series behave asymmetrically over the business cycle; see e.g., DeLong and Summers

(1986), Hussey (1992), Verbrugge (1997) and Belaire-Franch and Contreras (2002).

Thus, symmetry tests are an essential first step in practical model-building exercises

since it is desirable to establish the validity or otherwise of the symmetry assumption

before exploring more complicated business cycle structures.

Tests for symmetry have a long tradition in both Statistics and Econometrics. In

this paper, we focus on the evaluation of several statistical testing procedures that

can be used to test for conditional symmetry. In particular, we consider the non-

parametric test for conditional symmetry of Bai and Ng (2001). The closely related

problem of testing for (unconditional) symmetry was investigated by Wilcoxon (see

15



Gibbons and Chakraborti, 1992), Gupta (1967), McWilliams (1990) and Randies

et al. (1980) among others. It is not clear whether these tests can be extended to

testing for conditional symmetry, since it has not yet been rigorously demonstrated

that statistics computed by using regression residuals instead of the true errors have

approximately the same distribution as those based on the errors. It is by no means

obvious that this is so. However, for the case of tests of symmetry based on sample

moments, we show that under standard regularity conditions that ensure asymptotic

normality of moment estimators, the asymptotic null distributions of the tests do

not change when replacing the unknown errors by well-behaved residuals. Another

problem encountered when using real data is that, for finite samples, the distribution

of the symmetry tests included in this study is still unknown. As a consequence,

the true size of these tests often differs to a large extent from its nominal size when

asymptotic critical values are used. The main purpose of this paper is to investigate

whether the bootstrap can be used to obtain improved finite-sample critical values.

The remainder of the paper is organized as follows. Section 2 details the class

of nonlinear dynamic processes under which we will work. In Section 3, we briefly

review all the tests for conditional symmetry used in this paper. Section 4 describes

the bootstrap method and establishes a consistency property of the bootstrap for

nonlinear regression models. Section 5 performs a wide variety of Monte Carlo

simulations to compare the finite-sample size and power of the tests when critical

values are obtained using a bootstrap procedure with the size and power that we

could achieve using the asymptotic theory. Concluding comments are presented in

Section 6. Technical proofs of all results are deferred to an Appendix.

1.2 The nonlinear dynamic model

Suppose that { (^ , Xt)} is a strictly stationary discrete-time stochastic process with

y e M and Xt G W^, defined on some probability space {il, T, P). Here, Xt is

a vector containing both explanatory variables and lagged values of Yt. That is,

Xt = {Zt, Yt-^i,..., Yt^p)', where Zt G M.'^~P is a vector of some explanatory variables.

16



Let Yt and Xt be both defined based on a stationary process {Vt} by

Yt = M/y(T4,yi_i,y,_2,...),

Xt = {Xn,...,Xuy^^x{VuV,^j,Vt-^2,...),

where ^ y : K°° ^- E and ^ x : K°° -^ M.^ are two Borel measurable functions,

respectively, and {Vt} may be vector-valued. We see that {{Yt,Xt)} depends upon

the infinite history of {Vt} . Let r > 0 be a positive real number. Following Gal

lant and White (1988), we define {{Yt,Xt)} to be Lr-near epoch dependent {Lr-

NED) with respect to a stationary process {Vt}, provided -E | l i f < oo and Vr{m) =

E Yt-Yt (m)

E Xt - x't^ ^ 0 as TO oo, where |-| and ||-|| are the absolute

value and the Euclidean norm of W^, respectively, Yt = '^Y.miyt, Vt-i,..., Vt-m+i),

^(m) ^ (x(™)^...^x,(-)y ^ ^xAyuyi-i.-.Vt-m+i). and ^ y „ and v^xm are R-

and K"^-valued Borel measurable functions with m arguments involved, respectively.

In particular, if Vr{m) = 0(TO"""^'*') for some A > 0 we say {(It, Xt)} is L^—NED of

size —a. The more negative —a is, the more quickly the dependence of {{Yt-,^t)}

on past values of Vt dies out. We will call Vrim) the stability coefficients of order r

of the process {(Yt.,Xt)}. Since NED is only a measure of how {{Yt.Xt)} depends

on {Vt}, we place no conditions here on the dependence properties of {V^}.

We are interested in the conditional distribution of Yt conditional on Xt. Condi

tional symmetry implies that the distribution of Yt, given Xt, has a symmetric form

about its conditional mean. That is to say, ft{y + Ht/^t) = ft{—y + l^'t/^t), where

ft{-/Xt) is the density of Yj conditional on Xt, and fj,^ = E [Yt/Xt] is the conditional

mean. We assume that the dynamic behaviour of Yt is given by the general nonlinear

time series regression model:

Yt = fi{Xt, e) + a{Xt, d)ut, t = l,2,...,T (LI)

where fi{Xt,, 9) and cr^{Xt, 0) are the conditional mean and the conditional variance

of Yt, respectively. The functional forms of /i : M"* x M'' ^ M and cr : E"* x E'' ^

E are known except for ^ E 0 C E^, where G is the parameter space. {ut}t^i

17

are assumed to be independent and identically distributed (i.i.d.) zero-mean unit-

variance unknown errors with Ut being independent of Xt for all t. Fy(-) is the

cumulative distribution function (cdf) of Ut with density function /«(•). Let ^ be a

root-T consistent estimator of the parameter vector 9. The estimated residuals are

computed from the the estimated parameters. Then Ut = {Yt — ii{Xt,9))/a{Xt,6).

Unless otherwise stated, all summations considered here are taken from 1 to T,

where T denotes the number of observations. Note that the general framework (1.1)

encompasses linear regression models as a particular case.

Under model (1.1), conditional symmetry of Yt is equivalent to the symmetry

of Ut about zero, that is, fu{u) = fu{—u) for all u. Therefore, the null hypothesis

under test is that "HQ: Ut is symmetric about 0", versus the general alternative "Hi:

Ut is not symmetric about 0". It is pointed out that conditional symmetry does not,

in general, imply unconditional symmetry^.

An example of a NED process less trivial than a finite moving average process is

a simple AR(1) process (see Gallant and White, 1988, pp. 27-28). ARMA models

of finite order with zeros lying outside the unit circle can be shown to be NED of

arbitrarily large size, provided the parameters are chosen such that the stationarity

as well as the invertibility condition is fulfilled and the innovations satisfy appropri

ate moment conditions. Infinite MA processes can also be shown to be NED under

mild conditions on the moving average weights (see Wooldridge and White, 1988,

example 3.3). As Hansen (1991) has shown, strictly stationary GARCH processes

are NED under mild regularity conditions. This framework also includes the AR

process with ARCH/GARCH errors, discussed in Engle (1982), which is widely ap

plied in financial econometrics. Consider the AR(1)-GARCH(1,1) process, in which

observed data are generated as a realization of a stochastic compound process

Yt^'j + OYt.i + et, 1/2 u _ X , Ru , „,„2 Ct = Utht , ht = \ + (3ht-i + ae i - l ;

^To illustrate this, consider a MA{1) process Yt = Ut ~ Out~i with Ut i.i.d. and 6=1. The unconditional distribution of Yt is always symmetric with independence of whether or not /„(•) is symmetric, since Yj and —Yt have exactly the same distribution. However, the conditional distribution of Yt on Xt (which inlcudes Ut-^i) will be asymmetric in case Ut is asymmetric.

18



with {ut} being i.i.d., so ht is strictly stationary. If |^| < 1, it is well-known that

this model can be expressed as

oo

Yt = 7/(1 -0)+E dêt^r. T = 0

It can be shown that Yt is NED of order r on the stationary process {ct}, if for

some r > 2 , £'|ei|'^ < A < o o , with stable coefficients

Vr{m) = E Yt~Y, (m) = E

oo

E Oê,. T=m

r < \6r E i r E ie,_^^.r < \er A / ( I - I D, T = 0

decaying at a geometric rate. The conditions to ensure that E \etf < oo are E \ut\^ <

oo for some r > 2, and p + a < 1.

We next show that e = Uth^ is NED of order r on the stationary process {ut}. oo k oo fe

By repeated substitution we have /it = A + A E n( /^ + '^^í-¿) = - + ' E H ^Í-ÍJ k=li=l k=li=l

where zt — (3 + aef. Because under f5 + a < 1, sup¿>; E\zt\^ < c < 1 for some r > 2,

it follows that

Elht]'' = X + XE oo fe

En *- <X + XJ2E k=l fc=l 1 = 1

by the Minkowski's inequality for infinite sums

1 = 1

< A(l + c / ( l - c ) ) < oo,

, M m~l k To see that ht is NED on {ut}, let h^"^' = A + A E 0 zt-^• By Minkowski's

fc=l i=l

inequality

v'^{m) E ht- ht (m)

XE oo k

E U^t-k=m i= l

< Ac" -' E E fc=i

11 ^t~{m-l)-i ¿ = 1

< c ™ A / ( l - c ) .

Thus ht is NED of order r on {«t}. By Theorem 4.2 of Gallant and White (1988),

et = iti/it is L^-NED on {î}. This is also true for ARCH errors (/3 = 0).

19

1.3 Tests for conditional symmetry

We next describe the tests for symmetry considered in our Monte Carlo study.

To test for conditional symmetry, tests are applied to regression residuals. Since

these tests have been discussed extensively in the literature, their description here

is relatively brief.

A classical test of symmetry is the test of skewness (see Gupta, 1967, for a review

of this test). This test is developed for demeaned data, but the statistic has the same

limiting distribution when applied to residuals from a simple linear regression model.

It might be of interest to compare this test with a joint test of the third and fifth

central moments. In principle, a joint test of more moments is possible, but the

higher order-moments are difficult to estimate precisely. The potential advantage is

to be more powerful than a test based on the third moment in isolation. A practical

strategy would be to start with the skewness coefficient and consider joint tests of

higher moments only if we do not reject HQ. Standard asymptotic results will lead

to the derivation of both a joint test of the third and fifth central moments and

the skewness coefficient. The proof is omitted here in order to save space, but is

available upon request. An advantage of these tests is that they are intuitive and

easy to compute. However, they present a number of limitations. First, the limiting

distributions of the estimators are known and have a simple form for the case of

ordinary least squares. Different estimation methods may yield different limiting

distributions. Second, they are moment-based tests, which require the existence of

the sixth and tenth moments, respectively. This is not satisfied by many useful

distributions such as the student-Í5 or GARCH process. Finally, these tests are not

consistent against alternatives which are asymmetric and yet have the third moment

and/or the fifth moment equal to zero.

Bai and Ng (2001) discuss how to test whether the regression residuals from

a nonlinear time series regression model are symmetrically distributed. The test,

which is based on martingale transformations, does not require the data to be sta

tionary or i.i.d., and the dimension of the conditional variables can be infinite. The

20



test is shown to be consistent and asymptotically distribution free, but its compu

tation is rather intensive.

The literature on symmetry is large, an commonly used nonparametric tests are

the Wilcoxon signed-rank test (for further details, see Gibbons and Chakraborti

(1992), the Runs test of McWiUiams (1990), and the Triples test of Randies et al

(1980)). These test are asymptotically distribution-free for i.i.d. observations. In

the present setting, we replace the unobservable errors by well-behaved residuals.

Thus, the asymptotic distribution of these statistics is unknown.

1.4 Symmetric bootstrap

We consider the nonlinear regression model (1.1). Under the null hypothesis, we

know that the population {ui^ ...,UT} is symmetric about zero. The tests under

consideration were computed with estimated regression residuals when testing for

symmetry of regression errors. Let TV = Triui, ...^UT) denote the test statistic of

interest, which is a function of the standardized residuals. By using standardized

residuals, we are guaranteed that all model residuals have, at least, the same two

first moments.

In this section, we consider a bootstrap procedure for approximating the distrib

ution of the test statistic of interest, which is a function of the residuals, for testing

on the symmetry about the mean of the underlying distribution of the errors. When

bootstrapping any test statistic, our aim is to find a bootstrap distribution that

mimics the null distribution of the data, even though the data may be generated

by an alternative distribution. We propose a resampling scheme so that the null

hypothesis is respected in the bootstrap data-generating process. That is, a re

sampling method that ensures the bootstrap distribution to be symmetric. To be

precise, we define the bootstrap sample by T^ = {(y¿*,X¿*) : t = 1,2, . . . .T}, where

y / = /i(X;,?) + a{X;,e)u¡ and X; = {Zt,Y^*^^, ...,Y^*^p)'. Note that the exogenous

explanatory variables are fixed in repeated samples, and 9 is some estimate of the

parameter vector 9. Bootstrap residuals u* = (UJ , . . . ,M^) ' were constructed by a

21



two-stage procedure:

Stage 1: Construct recentred versions of the residuals Ut = Ut — T~^Ylt^t-

Random signs are assigned to the centered residuals u^ according to independent

realizations of a Rademacher random variable St, independent of ut, which takes

values +1 and —1 with probability 1/2 each. By doing that, we obtain a set of

symmetrized residuals {sitfi, ...JSTUT}-

Stage 2: A random number device independently selects integers ij, ...îr, each of

which equals any value between 1 and T with probability 1/T. We allow a single unit

StUt to appear more than once in the sample, that is to sample with replacement.

Therefore, the bootstrap data set {u^, ...,u^} consists of members of the original

data set {siUi, ...JSÛT}, some appearing zero times, some appearing once, some

appearing twice, etc.

Each bootstrap sample T^ is then used to re-estimate the parameter vector

9. Let 6 denote the bootstrap estimator of 9. The estimated residuals from the

bootstrap sample are

{u¡ - y; - /i(x;,r))/<T(x;,r): t = i,2, ...,T} .

Using bootstrap residuals, we compute the bootstrap test statistic TJ. = TT{UI, ..., u^).

Repeating this procedure B times gives a sample I T^^ : b = 1, ...,B> of TV val

ues. This sample mimics a random sample of draws of Tx under the null hypothesis.

In particular, we consider the problem of estimating the a-level critical value of the

TT test from its empirical distribution. Let c^^ denote the bootstrap estimate of

the a-level critical value. Let T^H\ < " (2) ^ ••• ^ ^T(B) denote the B realizations

of TT arranged in order of increasing size, and suppose we choose B and u such that

iz/B = 1 — a. Since the B values of T^¿ divide the real line into 5-1-1 parts, not B,

then it makes sense to select c^^ = Tî^î)- For example, in the case of o; = 0.05

and B = 1000, this would involve taking c^^ = T^,QQ\-

It is convenient to choose a single value of B at which to monitor the performance

of all the tests. In this study, this is not possible since there are large differences

22

between the run times of the tests. Different values of B were chosen, so as to make

the run time of each test approximately the same. We set B ~ 1999 for the moment-

based tests, the Wilcoxon signed-rank test and the Runs test, while B = 999 for the

Bai and Ng test. For the Triples test, we carry out i? = 99 bootstrap replications,

which is the smallest value of B that is commonly suggested. The processing time

becomes excessive when greater values are used, especially for T > 100. We will

illustrate the finite sample performance of the bootstrap proposal of the paper by

means of simulation in Section 4.

1.4.1 Asymptotic properties

To study the asymptotic properties of the proposed bootstrap, we need to state the

underlying assumptions.

(B l ) For some small 5 > 0 and some r > 2, the data generating process (DGP)

(1.1) is L2+¿-NED on {Zt, Ut} of size —2(r — l ) / ( r — 2). The constant S is specified

in A2 below.

(B2) E \Ytf+^ < oo for some S > 0.

(B3) The errors ut are i.i.d. random variables with zero mean, unit variance

and E' |iii| < oo. The density of Ut is /„(•) and the cdf Fu{-). Furthermore, Ut is

independent of Xt.

(B4) /i(-, •) and a{-, •) are twice continuously differentiable with respect to the

second argument with bounded derivatives. Additionally, there exists CTQ > 0 such

that (T(-, •) > (To-

(B5) The estimator ? satisfies VT0 -9) = Op(l).

(B6) /i.(-, •) and a(-, •) are Lipschitz continuous with respect to the first argument,

i.e., there exist a constant L^ such that \fj-{u, 9) — ¡i{v, 9)\ < L^ \\u — 7;||, and a[-, •)

satisfies a similar inequality for a certain constant L^.

(B7) '^x.m is continuously difi^erentiable with respect to the m arguments with

bounded derivatives.

(B8) m-^oo with m = o{T).

Assumptions Bl and B2 are related to the nonlinear process itself. Assumption

23

B3 is concerned with the behaviour of the errors. The differentiabihty condition

required in B4 is relatively standard in nonlinear estimations. B5 is a standard

assumption, which ensures that the estimators are root-T consistent. Conditions

B6-B8 are required for purely technical reasons.

We next provide a little theory for the convergence of the empirical distribution

of standardized residuals under the symmetric bootstrap proposed above. The idea

behind the bootstrap is to replace the true distribution function of the error term

Ut by its empirical estimate. Let FT be the empirical distribution function of the

recentred standardized residuals, putting mass 1/T on each Uj, t — 1, ...,T. That is,

the centered residuals are equally likely to appear in the bootstrap sample. Following

Efron and Tibshirani (1993), a bootstrap sample is defined to be a random sample

of size T drawn from FT-, say u* = (Si,..., u^)'- The start notation indicates that u*

is not the actual data set u, but rather a randomized, or resampled, version of it.

We can construct the distribution GT, which places mass 1/T at StUt, t =

1, 2,..., T, where Sj is a Rademacher random variable, independent oiut- We use GT

as the basis for our bootstrap resampling scheme. It is straightforward to prove that

the distribution of the random variable StUt is symmetric about zero under both HQ

and Hi. Let G„ be its distribution function defined by

G„(x) = ^ ( l - F „ ( - 2 ; ) + F„(x))

It is pointed out that Gu{x) = F„(a;) for every given x under the null hypothesis.

Note that the symmetry of the bootstrap errors does not depend on whether the null

hypothesis holds or not, although Ut does. That is, our bootstrap approximation to

the null hypothesis is always valid even the data { ( ^ , - ^ Í ) } Í =X were drawn from a

population under which the null hypothesis does not hold. Therefore, the derived

bootstrap tests automatically follow the first guideline set by Hall and Wilson (1991).

Namely resampling should be done in a way that reflects the null hypothesis, even

when the true hypothesis is distant from the null. As they pointed out, this ensures

the reasonable power of the bootstrap test against the departure from the null

hypothesis.

24



In order to investigate the asymptotic behaviour of the symmetric bootstrap,

we use the Mallows metric? c?2 to show that the bootstrap errors u^ approximate

the true errors Ut under HQ. There is one key result that make this metric a useful

tool in proving asymptotic results for regression models. From Bickel and Preedman

(1981; Lemma 8.3), given distributions F, Fi, F2,..._ the condition (Í2(-PT, F ) —> 0 as

T —> oo implies that the probability measures corresponding to Fx converge weakly

to the measure corresponding to F.

Proposition 1: Suppose that assumptions B1-B5 hold. Then, under HQ,

d2{ut,ul)-^0 as T —> oo.

As next step, we show that T^ replicates the structure of (1.1), given the original

data TT ~ {{Yt,Xt),t = 1,...,T}. For this purpose, we define TT = { (^ , Xt) ,

t = 1,...,T} as

Yt = ^iiXt,9) + a{Xt,e)et, i = l , 2 , . . . , T ,

where Xt = {Zt,l^„i,..., Yi„p} and {£t}t=i ^^^ conditionally i.i.d. random variables

with the following properties. Given T^, (i) £t has conditional distribution F^, (ii)

d2{£t,Uf) = d2{ut,u*), (iii) TT is L2+<5-NED on {Zt,et} for some 5 > 0. Here and

in the following, a star appearing in E denotes expectation with respect to T^

conditional on the data T7-.

Proposition 2: Suppose that assumptions B1-B8 hold. Then, under HQ,

sup E* \<t<T

y I - Y: = Op(l) jor T ^ 00.

The following corollary, which show that

given T7-, follows immediately from Proposition 2.

Yt ~ y: 0 in mean for T ^ 00

^The Mallows metric is defined by dl{X.Y) = dl{G,H) = inf {£;[i|X - Flpj : X''G,Y~H} , where the infimum is over all joint distributions of {X, Y) whose fixed marginal distributions are G and H respectively and where ||.|| denotes the Euchdean norm on R. See Bickel and Freedman (1981; Section 8) for a detailed discussion of this metric.

25

Corollary 2: Suppose that assumptions B1-B8 hold. Then, under HQ,

E*{T-^ T. Yt-Y; }^0 forT ^ oo. t

Note that we do not prove that the conditional distribution of T^ given Ty is

asymptotically equal to the null-hypothesis distribution of TT since the asymptotic

distribution of TV is unknown for some of the statistics under consideration.

1.5 A Monte Carlo study

In this section, we investigate the finite-sample properties of the symmetry tests

of Section 2 by means of Monte Carlo simulation^. The aim of the experiments is

two-fold. First, to investigate whether the bootstrap procedure proposed in Section

3 can be used to obtain improved finite-sample critical values with respect to the

asymptotic theory, whenever this is available. Second, to identify the size and power

properties of the test statistics under various scenarios, including linear, AR, MA

and GARCH models. We first describe the data-generating processes (DGP) and

the experimental design that is used in our simulations. A discussion of the results

obtained in these simulation experiments follows.

1.5.1 Experimental design

The time series considered in our study are generated according to model (1.1),

where functions fJ.{-,-) •iid cr(-, •) are generated according to four basic types of

DGPs:

DGPi: ii{Xt,e) = f3, + J:Zu(3,, {Z^t,Z,u-.ZH)' '•- N{0,h), and a{Xt,e) =

a = l;

DGP2: fiiXt, 9) = c + pYt.j and a{Xt, 9) = a = 1;

DGP3: ^(Xi, #) = /i + (j)Ut-i and a{Xt, 6) = a = 1;

DGP4: ii{Xt,e)=^ and a{Xt, 9) = {ao + aMXt~i,9Y+ a2a{Xt^i,9ful^y/\

^All the procedures for estimating the models described in this section were written in GAUSS programming language. Programs are available from the author upon request.

26



DGPi is a linear regression model with an intercept component and k i.i.d. variables

as regressors. Data are generated setting /3o = ... = / ^ = 1 and k — 1,4. The reason

for increasing the number of regressors is to observe the sensitiveness of the size and

the power of the tests to the additional regressors. For an AR(1) specification,

our simulation experiment is based on DGP2. We set c = 0 and p = 0.5,0.8.

We denote by DGP3 the MA(1) design. We set the constant regressor /i equal

to zero and 0 = 0.5,0.8. Finally, DGP4 corresponds to a GARCH(1,1) model. In

this framework, we set 7 = 1 and {ao,ai,a2) = (2,0.5,0.3). Also, we consider the

model with (tto, «i, 02) = (2, 0.9, 0.05), which is close to being an IGARCH(1,1). All

parameter combinations considered were selected to make the results of our study

comparable with those obtained by Bai and Ng (2001), whenever this is possible.

For each DGP, we draw Ut from symmetric and asymmetric distributions to

derive conditionally symmetric and asymmetric distributions for Y¿. To asses the

size of the tests, we first generate Ut from the standard normal distribution and

the student-i distribution with 5 degrees of freedom. To evaluate the power of

the tests, we draw random variables from the exponential distribution and the chi-

square with two degrees of freedom. We then consider another ten distributions,

four symmetric and six asymmetric, from the generalized lambda family (GLF)

discussed in Ramberg and Schmeiser (1974). The choice of all these distributions is

motivated by the fact they are used in previous studies of testing symmetry and in

consequence provide a benchmark for comparing size and power. In addition, they

cover a wide range of values of third and fourth standardized moments. The GLF

is easily generated since it is defined in terms of the inverse cumulative distribution

function F^^{u) = Ai + [u'^^ + (1 — u)" *] /A2, 0 < tí < 1, with mean and variance

given by:

// = Ai + [(l + A 3 ) " ' - ( l + A4)-V-^2,

a^ = [(l + 2A3)- ' -2/?(l + A3,l + A4) + (l + 2A4)"'

27

where (3{-, •) denotes the beta function. The A parameters defining the ten selected

distributions are taken from Randies et al. (1980) and are listed in Table 1, together

with the associated skewness (773) and kurtosis (774) values. The distributions are

arranged in ascending order of departure from symmetry'*. To be under the assump

tions of the regression model, all error distributions are standardized to have zero

mean and unit variance. Among these distributions, the Student-í distribution with

5 degrees of freedom has finite variance, but does not have finite sixth and tenth

moments. The generalized lambda distributions have finite gth moment if, and only

if, —1/g < min{\z, A4). All other distributions have finite sixth and tenth moments.

This is aimed at checking how moment-based tests behave when data do not possess

proper moments.

The experiments proceed by generating artificial time series of length T from

(1.1) with T G {50,100,200}. We have to estimate k + 2 parameters in DGPi.

The parameters of interest are estimated using ordinary least squares. Next, in

DGP¿ {i = 2,3) and DGP4, we have three and four parameters to be estimated,

respectively. In order to do that, we use maximum likelihood (ML) estimation. In

the context of DGP4, as Fiorentini et al. (1996) proposed, for estimation purposes

we employ the analytic first and second derivatives of the log-likelihood instead

of numerical approximations in order to benefit for computational reductions and

avoid convergence problems. Finally, we compute the relevant test statistic TT =

Tx{ui, ...,UT), which is based on the standardized residuals from estimation of (1.1).

Due to the computational demand required by some of the tests included in

this study is very high, experiments were conducted using 500 replications for the

Triples test, 1000 for the Bai and Ng test, and 2000 for the remaining tests. For each

replication, we reject the null being tested at the nominal a-level, based on both

bootstrap and asymptotic critical values, if the observed test TT is above c^ ^ and

exceeds the (l-a) quantile of the corresponding asymptotic distribution, respectively.

We finally count the proportion of times that the null hypothesis is rejected for

each test statistic using bootstrap- and asymptotic-based critical values. For non-

'^The shapes of the GLF density functions are shown in McWilHams (1990).

28

symmetric alternatives, this proportion yields an estimate of the power of the test.

In all cases power is not size-adjusted. On the other hand, the proportions from

the symmetric distributions imply estimates of the Type I error. Since the results

for tests performed at the 0.01, 0.05, and 0.10 significance levels are qualitatively

similar and lead to the same conclusions about the relative merits of different tests,

we focus on 5%-significance level tests^.

1.5.2 Simulation results: a comparative study of symmetry tests

The reader has to consider that the nonparametric tests included in this study, that

is, the Wilcoxon Signed-Rank test (WSRT), the Runs test {RT) and the Triples test

{TRT), are originally constructed for the problem of testing the unconditional sym

metry of an i.i.d. sample of observations. We investigate the performance of these

tests when testing for conditional symmetry. Under (1.1), conditional symmetry is

equivalent to the symmetry of the error term about zero. Furthermore, at this point

we do not provide an asymptotic distribution theory for these tests when unknown

errors are replaced by well-behaved residuals. This is not the case of Bai and Ng

test (CST) and moment-based tests {S^ and Sj¡ ), whose corresponding asymptotic

distributions are completely known. We implement a bootstrap version of all the

tests. Tables 2 to 9 show the empirical size and empirical power of the various tests

obtained using artificial time series generated according to DGPi, DGP2, DPG3

and DGP4. It should be pointed out that moment-based tests are only computed

when the process is uncorrelated (DGPi). We report empirical rejection rates (%)

under the null and the alternative based on both asymptotic critical values as well

as bootstrap critical values obtained from Monte Carlo trials. To establish heuristic

comparisons, for the set of nonparametric tests we use the tabulated asymptotic

critical values that will correspond to tests statistics computed with "observable"

errors^. This should be borne in mind when assessing the results. Based on the

^Results at the 1% and 10% levels of significance are available upon request. ^All the tests conducted in this simulation study are one-tailed tests at the 0.05 level. The

asymptotic critical values for the tests S^^^ and CST sxe 5.99 and 2.20 respectively. For the

29



selected 5% nominal level (or size) of the tests, the empirical rejection frequencies

should be around 5% under the null, while they should be around 100% under the

alternative.

Table 2 presents the empirical size of the symmetry tests for DGPi. Let us

initially consider the Si^ test. In the six symmetric cases investigated, this test

performs similarly under bootstrap and asymptotic procedures. It is notably un

dersized (around 3%) in distributions S4 to S6, but only slightly undersized for S3.

In contrast, it has the correct size in Si and S2 cases, even when T = 50. For a

sample size of 50 observations, it is also worth noting that increasing the number

of regressors leads to a significant^ reduction in the empirical size for SI and S2

distributions. For larger sample sizes, test performance is not affected by increasing

the number of regressors.

A feature of the size properties of the S^^: statistic with asymptotic critical values

is that it is consistently undersized with actual size about 2% for most of the cases

investigated, but not all. This result is in stark contrast to the size properties of the

test with bootstrap-based critical values. The bootstrap does bring the empirical

size of the test closer to its nominal level, a sample size of T = 50 is large enough

for distributions SI and S2, while it is necessary T = 200 for distributions S3 to

S6, which are far from a mesokurtic distribution. The empirical size of this test is

rather stable to an increase in the number of regressors. The performance of both

moment-based tests under distributions S4 to S6 deserves further analysis, since

sixth and tenth moments of these distributions do not exist. The empirical size

of S^ test in S3, for which sixth and tenth moments exist, is comparable to its

size under distributions S4 to S6. The same applies to S^ when comparing S3 and

S5. Interestingly, when bootstrap critical values are used, the empirical size of the

remaining tests, the asymptotic critical value is 3.84. ^Because the results depicted here present numerous opportunities for comparing empirical sizes

and powers between symmetry tests, and for fixed test, between parameterizations of the DGP, it is difficult to assign a threshold percentage to determine a difference as statistically significant. We use the 1% when comparing sizes and the 10% when comparing powers. Despite being subjective choices, these values provide a good indication of whether differences are larger than can reasonably be explained by random simulations.

30



S^^ test in S4 is comparable to that of CST, which is not a moment-based test,

for any values of T and k. This is not the case for thicker-tailed distributions such

as S5 and S6, where 5^' is clearly more conservative than CST- Comparing the

size of both moment-based statistics across distributions, shows that the Sj^ test

statistic rejects less often than the skewness coefficient under the null when using

asymptotic critical values. The results are the reverse with bootstrap-based critical

values except for distributions with kurtosis equal to 3.

We next consider the Bai and Ng test. Note that for all DGPs considered in

BN, the estimated regression model imposes a^ = 1. Under this circumstance, the

variance is not treated as a parameter to be estimated, as it is assumed in (1.1),

but a constant to be specify, and for this reason it is not possible to establish direct

comparisons between their results and those of this paper^. Fixing fc = 1, the size of

the test based on asymptotic critical values is largely satisfactory for distributions

SI to S4 when T — 50. However, this result should be interpreted with caution,

since when the sample increases to T = 100 empirical sizes fall drastically. This

may suggest that this test presents inflated sizes for small samples under these

distributions. Turning to fat-tailed distributions, we may see that S5 is slightly

oversized for T = 200. The S6 case is more seriously oversized, since at the 5% level

the rate of wrong rejections is 8.5% for T = 50. This distortion increases with T,

being the percentage of over-rejections of the empirical size above 5% of its nominal

size for T = 200. This reflects an efficiency loss in conducting the CST test based

on asymptotic critical values in distributions with high kurtosis, since it tends to

reject a true null too often. It should be pointed out, however, that oversizing is not

so large as to render the test unattractive for applications. These results appear to

be robust to increase the number of regressors to fc = 4. On the other hand, the

results from bootstrap critical values show that, in S5 and S6 cases, the bootstrap

performs well, with sizes close to the nominal level for T > 50. In the remaining

cases, the bootstrap test yields low sizes for values of T up to 100, with sizes mostly

^Models 1, 2, 4, 5 and 6 in Bai and Ng (2001) correspond to distributions SI, S4, A5, A7 and A8 in this paper, respectively.

31



between 2% and 4%. It is also shown that this undersizing is corrected for large T.

Increasing the number of regressors moves bootstrap-based empirical sizes upward

in small samples. Since size distortions may lead to misleading inference, to guard

against possible over-rejection of the symmetry hypothesis, it is advisable to compute

this test statistic with bootstrap-based critical values.

For both values of k considered, the actual sizes of the Runs test based on asymp

totic critical values are close to the 5% nominal level even for T = 50. Increasing

the number of regressors increases the empirical size for Si, while it decreases for

S5. This is mainly a small sample effect, since if we compute the test with large T,

size is not affected by the number of regressors. When A; = 1, the Runs test with

bootstrap-based critical values has poor size. Interestingly, the performance of the

test steadily deteriorates as T increases, its rejection frequencies being around 1%

for T = 200. In unreported simulations, we found that the size distortions of the

bootstrap test do disappear slowly as T increases further. The size properties of RT

for fc = 4, on the other hand, seem largely satisfactory, being the performance of

the bootstrap comparable to asymptotic values.

In the case of the Wilcoxon Signed-Rank test, the striking feature of the results

is the quite severe size distortion of the test when asymptotic critical values are

used for any of the distributions considered. In the S6 case, a 1% size is reached,

while in the remaining cases rejection frequencies are equal or close to zero, which

illustrates the conservative nature of the test. These size distortions do not disappear

as T increases from 50 to 200. These results should be interpreted with caution.

The poor size properties of WSRT statistic might stem from incorrectly assuming

the asymptotic distribution of the test is invariant to the replacement of errors by

residuals. The use of bootstrap-based critical values instead of asymptotic ones

corrects the differences between the empirical and nominal sizes. For fixed k = 1

and sample sizes of T = 50 and above, the size of the WSRT test is fair in cases S3

to S6, with sizes above 4% and below 6%. In contrast, in cases SI and S2, this test

tends to be slightly undersized with actual sizes between 3% and 4% for T = 50.

A sample size of T = 200 observations appears large enough to ensure good size

32



properties of WSRT test for these two distributions, which are between 4% and 5%.

When T = 50, as the number of regressors increases, size significantly decreases in

S2, while increasing in S6. In both cases, this is a small sample phenomenon.

Focusing on TRT, for series of length T = 50 and fc = 1, we may see that the

empirical size of the test with asymptotic critical values tends to be much smaller

than the nominal size, although the size distortions for SI and S2 are considerably

less than they are for distributions with kurtosis higher than normal (S3 to S6). In

particular, the empirical sizes of the test range between 2% and 3%, and between 1%

and 2%, respectively. In all the cases, sample sizes of at least 200 observations are

needed to avoid significant undersizing. When considering leptokurtic distributions

(S3 to S6) and fixing T = 50, size increases substantially with k. A similar result

holds for all the distributions when bootstrap critical values are used. Again, this

is a small sample effect. For fixed T and k, the performance of the test improves

with bootstrap critical values. For /c = 1, the test is slightly undersized, with sizes

between 3% and 4%. But, as T becomes larger, these size distortions disappear. For

k = 4, bootstrap-based empirical sizes are accurate throughout, even when T = 50.

The size properties of the Bai and Ng test together with the nonparametric tests

in DGP2, DGP3 and DGP4 are reported, respectively, in Tables 3 to 5. Overall,

the evidence from our simulations suggests that the relevant test statistics replicate

the same patterns found for DGPi. Fourth points are worth making regarding

the differences with respect to the discussion above. First, with bootstrap critical

values and T = 50, the TRT test is the most accurate for all three DGPs, and

also DGPi when fe = 4. Second, the asymptotic-based empirical size of CST under

S5 is slightly oversized under these three DGPs even when T = 50. Third, when

data come from a GARCH model (DGP4), the bootstrap-based empirical sizes of

WSRT under distributions SI and S2 are slightly oversized for T > 100. Moreover,

the conservative nature of RT when using bootstrap critical values decreases as T

increases, with sizes about 2% for T = 200. Fourth, focusing on DGP3 when the

moving average coefliicient is large, the performance of TRT under S4 is considerably

worse with both bootstrap and asymptotic critical values. Surprisingly, the CST

33



test under S5 turns out to perform correctly with asymptotic critical values, while

becoming undersized with bootstrap ones. For both tests, this power reduction is

a small sample effect that vanishes when T = 100. On the contrary, the results

of CST and the nonparametric tests under DGP2 are quite robust to an increasing

level of autocorrelation. It is the same for DGP4 when the error process is close to

being an IGARCH.

We now briefly review the performance of the tests under the eight alternatives

of asymmetry. The results for each one of the four DGPs considered here are dis

played in Tables 6 to 9, respectively. The Monte Carlo simulations reveal that the

bootstrap performance is better or at worst equal to that of asymptotic critical val

ues for all the tests under any DGP, with the exception of RT- AS intuition would

suggest, this test is more successful with gisymptotic critical values, given the size

of the test based on bootstrap critical values is too small. When T = 50, the more

asymmetric are the distributions (A4, A5, A7 and A8), the greater are the difl er-

ences between asymptotic and bootstrap critical values (around 20%) for all the

DGPs. Conversely, when T = 200, power differences are meaningful for the most

asymmetric distributions, whereas reaching about 20% for A2 and A3. One might

have anticipated this result in view of the conservative nature of the bootstrap RT

test for distribution under the null hypothesis. Since CST does not hold its 5% level

very well with asymptotic critical values, it is difficult to include it in any asymp

totic power comparisons, since it high power might easily arise out of these inflated

levels. Note also that WSRT has disappointing power properties with asymptotic

critical values. The reason for such poor performance, when compared to bootstrap

critical values, is that the asymptotic-based empirical size of this test is extremely

conservative (bearing in mind that the under-size in the WSRT test computed with

residuals is attributable to the use of critical values for the corresponding WSRT

test computed with errors). The distributions for which this test is effective appear

ing to be rather small. When T = 200, it has enough power to reject A4, A5, A7

and A8 distributions. The range is reduced to A7 and A8 for T = 100. Therefore,

unless stated otherwise, all power comparisons between tests reported hereafter are

34



based on bootstrap critical values, which are more reliable.

Turning attention to comparisons between DGPs, results seem not to be affected

by the DGP where the data come from. There are two exceptions to this rule in

the case of DGP4, for which the bootstrap-based empirical power of WSRT and

iiy.tests undergoes a signiñcant reduction for most of the distributions, no matter

the sample size considered. It should be pointed out that this power distortion does

not affect comparisons between tests described below. For a fixed DGP, the results

of all the tests are quite robust across parameterizations for AR, MA and GARCH

models. On the contrary, under DGPj with A; = 4, for the sample size 50, there is

a reduction in the power of CST and RT tests with asymptotic critical values for

distributions A4, A5, A7 and A8. For CST, when the number of observations is 100,

the increase in the number of regressors does not alter the power of this test. At

least 200 observations are needed to avoid this reduction in the case of RT- We do

not observe power reductions when the number of regressors is large with bootstrap

critical values.

Non-symmetry is detected with reasonable frequencies in nearly all cases. For

a fixed distribution, power increases with T with both bootstrap and asymptotic

critical values. For a fixed T, Bai and Ng test and the nonparametric tests exhibit

monotonic power with the power of the tests increasing for increasing levels of asym

metry, except for A6, which is analyzed in depth below. For the moment-based tests,

this monotonic behaviour is interrupted in A7 and A8, being their powers lower than

in A5. This may reflect the sensitivity of these tests to the high kurtosis displayed

by these distributions. The alternative of non-symmetry is detected with the lowest

probability in the case Al, as we would expected given this distribution is rather

close to symmetry. When T = 50, the WSRT test is the only one with power above

the nominal level under Al. This scenario improves for T = 200, with only RT and

TRT having power around 5%. Distribution A6 is introduced to show the sensitivity

analysis of the power to the kurtosis of the underlying distribution by comparing

the behaviour of each particular test under A6 against alternative A5, which has the

35



same asymmetry as A6 but a lower level of kurtosis^. From the empirical results,

we can assert that RT test is the most affected by the kurtosis of the underlying

distribution. For example, non-symmetry under A5 is more than 8 times as likely

to detect as under A6 by RT test, when T = 50 and we consider DGPi with 1 re-

gressor. The number of times increases to 15 for T = 200. For the remaining tests,

the differences in power between these two distributions decrease as T increases. In

particular, when T = 50, the power of CST under A5 is approximately 5.5 times

the power of A6, while this number reduces to 1.5 for T = 200. Moment-based tests

have a similar behaviour to Bai and Ng test. Although also sensitive, WSRT and

TRT show lower differences than CST- It is noteworthy that A6 only outperforms,

in terms of power, Al.

Overall, the WSRT test clearly dominates the others on power for all the DGPs,

being the differences in performance more remarkable when T < 100. It is followed

by the CST test for DGPa to DGP4. Under DGPi, CST and S^^ statistics are the

best performing competitors. Furthermore, these tests have complementary power,

since Sj! performs better than CST for distributions A2 and A3, while CST appears

to be more preferable for A7 and A8, which are thick-tailed distributions. For the

remaining distributions, the power of both tests is comparable. The TRT test on the

other hand, which has good size properties for all the DGPs, has disappointingly

low power, i.e, less than 50% even for AS with T = 200. It's worth noting that

the bootstrap power of RT is not affected by the same erratic behaviour of the size.

Its power increases substantially with T, although it remains low even for T = 200

when data come from Al, A2, A3 or A6. In fact, under these distributions RT is

slightly dominated by the TRT test. Turning to the properties of moment-based

tests in DGPi, we may see that S^: is more competitive in power than ST in all

cases, especially when the sample size is moderate to large (T = 100,200), which

corresponds with our intuition.

To summarize, the following conclusions emerge when size and power perfor-

^To establisli comparisons with A6, it would be also possible to use distribution A4, which has the same levels of asymmetry and kurtosis as A5.

36

manee are jointly considered. First, the simulations provide a strong case for the

use of bootstrap critical values, especially in sample sizes as small as 50. The Runs

test is the only one for which asymptotic-based critical values outperform bootstrap

ones. For this test, the use of bootstrap critical values provides additional protection

of the 5% level, but at a nontrivial cost in terms of power. Second, S' dominates

S^, since both moment-based tests have similar size properties, but a joint test of

the third and fifth moments is more competitive in power. The obvious simplicity

of this test and its robustness to the number of regressors in the model make it use

attractive despite of its sensitivity to thick-tailed distributions. Third, considering

the Monte Carlo results and the fact that WSRT is easy to calculate, we recommend

it over any of the competitors included in this study, being aware of the fact that its

asymptotic distribution when replacing regression errors by residuals is unknown.

The level of protection of the nominal level is higher with CST, but at a no minor

reduction in power, especially for distributions with low skewness. Finally, note that

the size and power advantages held by WSRT using bootstrap critical values are

limited to the specific class of null and alternative distributions considered in this

study.

1.6 Conclusions

This paper investigates the finite sample properties of the Bai and Ng test commonly

employed to detect conditional symmetry. We also explore the possibility of evalu

ating conditional symmetry by using some widely used tests for the unconditional

symmetry of observations when the tests are applied to regression residuals. The

tests investigated included the coefficient of skewness, a joint tests of the third and

fifth moments, the Runs test, the Wilcoxon signed-rank tests and the Triples test.

The limiting distribution of the conditional tests is only provided for moment-based

tests. For this reason, the performance of a symmetric bootstrap to compute critical

values for all the tests is discussed. The proposed symmetric bootstrap is easy to

implement and is flexible enough to be adapted to a variety of nonlinear regression

37



models. The potential of the methodology is illustrated using Monte Carlo simula

tion. The following general conclusions can be drawn from the results. First, the

ability of the bootstrap to overcome the problem of oversizing observed for the Bai

and Ng test when asymptotic critical values are used. Second, the size and power

properties of the tests do not appear to be affected by the data-generating process

for time series of relatively large length. Finally, the evidence from our simulations

suggests that the Wilcoxon signed-rank test dominates the others in terms os size

accuracy and power to detect non-symmetry.

38



APPENDIX

Proof of Proposi t ion 1: To proceed, corresponding to FT and GT, let FT and GT

denote the empirical distributions of {«ijiî and {stUt}^^^, respectively. By

the triangular inequality,

4{ut,u¡) = dl{F^,GT) < dl{Fu,GT) + dl{GT,GT).

Under HQ, the first term converges to 0 by Lemma 8.4 in Bickel and Freedman

(1981). In order to obtain an upper bound for the second term, we consider

particular random variables UT and VT, where UT = {SÍ^Í} and VT = {stUt} ,

t = 1 T. Hence,

dliGT.GT) < E*{UT--VTf = T~^Y.i^t-ut-T~'Y.Ut? t t

Í Í Í

2 _, if,{X„e)-i,{Xj) ^ ^^â{X,,e)~a{xJ)

t \ a{Xj) * a[Xj)

{î{Xue) - ^x{X^,'6)f

i a{Xt,eY

t a{Xt,9y t

For the first term on the right hand side (r.h.s.), we make use of the lower

bound for o'{-,-) and a Taylor expansion for the numerator. By adding and

subtracting terms, it is bounded by

2

4(70'T-1E {di^{Xt, e)/de)\e -e) + o,(i)

Op{T-^) + Op(l) = Op(l) as r -> oo,

39

which is obtained applying assumptions B4-B5.

Exactly along the same line, for the second term on the r.h.s,

Í a{Xt,6y t

X (T-^/^E y,^{a{X^,9)-a{X„9)f

t a{Xt,ey

Under assumption B3, by the Central Limit Theorem,

T-'/'Zu't=0,{l).

The remainder is treated as follows,

^_i/2 ^ {a{X^,9) a{Xt,0)f^ ^^,^_, ^ ^^^^^^^ e)/d9yVf0 -9) + 0^(1) i a{Xt,9y t

which is of order Op(l).

By the law of large numbers

{T-'Eutf = o,{l),

which completes the proof of Proposition 1.

Proof of Proposition 2: By the definition of TT and T ^

E* Y, - Y; E*\fi{Xt,9)-fx{X:,9)

+ {a{X,, 9) - a{X:,0))u: + a{Xt, 9){u^ - u¡]

< E

+E

l,{Xt,9)--li{X:,9) + E* i,{x:,9)-f,{x;,9)

a{Xt,9)-a{X:,9) \u¡\ + E* a{X:,9)~a{X:,9) u.

+E*a{Xt,9)\{et~u:)\.

Following the same reasoning as in the proof of Proposition 1, the second and

the fourth term converge to zero in probability. For the first term, we have

from B6

E* fi{X,,9)-ii{X:,9) <L,E* Xt ~ x;

40

Exactly along the same lines we obtain

E* a{Xue)-a{X:,e) \u;\<E*\ul\L„E* Xt-X¡

Finally, by Proposition 1,

E*a{Xt, 9) \Í6t - u¡)\ = E*a{Xt, 9)E* \{e, - w*)| = Op(l).

Thus,

E* Yt - Y: <{L^ + E*\ul\L,)E* Xt-X; +Op(l).

Under Bl, we use the following approximation to the NED process Xt by the

stationary process Xi''. That is,

;(m; (m)N Xt=xr + {Xt - xr) = xr+VT,. n* r^('^)\2 (m)

(1.2)

M-.2 where E*{r]flY = £{77^/1X7}^ = 0{v2{m)) asm ^00. Note that E*{rj'-^¡}

will never increase as m —> 00. Thus, by (B8)

Letting £i = 0 for í < 0, we have from (1.2) that for a given t

(1.3)

E* Xt- ~x: < E*

< {E*

Xt-

Xt

-xP -xP

+ E* 2 ,

Nl/2

X, (T)

x:

+E* | |*xr(£i , et-i, -,£1) - ^ x , r « , «i-i, •••> «í

< 0 ( v ' ^ ; ¡ M ) + E* mxAx)/dx\\ \\et - u*t\\ = Op(l),

where the second inequality follows by the Liapounov's inequality, and the last

inequality follows from (1.3), (B7) and Proposition 1. •

41

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44



TABLES

Table 1: Distributions used in the Monte Carlo study

Distribution Al A2 A3 A4 V3 V4 Symmetric distributions

SI S2 S3 S4 S5 S6

0 0

0 0

N{0,1) 0.197454 0.134915

-1 -0.080000 h

-0.397912 -0.160000 -1 -0.240000

0.134915 -0.080000

-0.160000 -0.240000

0 0 0 0 0 0

3.0 3.0 6.0 9.0 11.6 126.0

Asymmetric distributions Al A2 A3 A4 A5 A6 A7 A8

-0.116734 3.586508

0

-0.351663 -0.130000 0.043060 0.025213

-1 -0.007500

-0.160000 0.094029 -0.030000

exponential: —ln{e), e ~ U{0,1)

0 0 0

xi -1 -0.100000 -1 -0.001000 -1 -0.000100

-0.180000 -0.130000 -0.170000

0.8 0.9 1.5 2.0 2.0 2.0 3.16 3.88

11.4 4.2 7.5 9.0 9.0 21.2 23.8 40.7

45



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Chapter 2

Unemployment and Hysteresis: A Nonlinear Unobserved Components Approach

2.1 Introduction

The business press make continual references to the high unemployment that char

acterizes European countries. For example, in France and Italy unemployment since

the mid 1970s has steadily increased without any significant decrease or evident ten

dency to revert to a stable underlying unemployment rate. It remains very high at

around 10% (see Figure 2.1). Many theories have emerged to provide an economic

explanation which could account for this observed unemployment persistence. Most

of the work in the relevant literature assumes that it can be attributed to changes in

the natural rate of unemployment and/or changes in the cyclical rate of unemploy

ment. Based on this framework, two main approaches are the natural rate theory

and the unemployment hysteresis theory.

The first approach supposes that output fluctuations generate cyclical move

ments in the unemployment rate, which in the long run, will tend to revert to its

equilibrium. The crux of the natural rate hypothesis is that cyclical unemployment

and natural unemployment evolve independently. Hence, the tendency of the nat

ural rate to remain at a high level is the result of permanent shocks on the structure

of the labour market such as increased unemployment benefits, strong trade unions,

56



legislative restrictions on dismissal and minimum wage laws (see Friedman, 1968).

The second approach supposes that cyclical unemployment rate and natural rate

do not evolve independently. The basic idea of the hysteresis hypothesis is that a

change in the cyclical component of the unemployment rate may be permanently

propagated to the natural rate. Based on this idea, an increase in the cyclical

unemployment rate may lead to an increase, over time, in the level of the natural rate

(see Phelps, 1972). A direct corollary of hysteresis is that short-run adjustment of

the economy can take place over a very long period. Then, aggregate demand policy,

traditionally considered as ineffective in changing the natural rate of unemployment,

can have a permanent effect on it.

In this paper, we focus on this second approach. The word hysteresis derives

from the Greek varepeuj, which means to come later. The physicist James Alfred

Ewing was the first to introduce this term into the scientific literature to explain the

behaviour of electromagnetic fields in ferric metals. As pointed out in Amable et

al. (1995), a mathematical modelling of hysteresis requires us to consider a system

subject to an external action, that is an input-output system. Hysteresis is defined

as a particular type of response of the system when one modifies the value of the

input: the system is said to exhibit som,e rem,anence when there is a permanent effect

on output after the value of the input has been modified and brought back to its initial

position. This formal definition implies that a hysteretic process is characterized by

the following properties:

1. It is necessary to know the history of the system in order to assess its position.

Hence, the history of the system matters. This implies the presence of a unit root

in the process.

2. There is a remanence effect. If one transitory shock is followed by a second of

the same intensity in the opposite direction the system does not revert to its former

equilibrium. Hence, a transitory shock has a permanent efi ect on the system's

equilibrium, since the system retains traces of past shocks on it even after those

influences have ceased to apply. It must be noted that this property is only present

in nonlinear systems.

57



To sum up, hysteresis occurs in nonlinear systems that exhibit multipHcity of

equihbria and the remanence property.

Many mechanisms are hsted in economic hterature as giving rise to hysteresis (see

R0ed, 1997, for an extensive survey). For instance, after a negative shock, firms may

reduce capital stock along with employment. The latter will cause unemployment

to persist because the firm may not re-open its plant once the shock is removed.

Second, long periods of unemployment may cause workers to lose skill, which could

lead to long-term unemployed workers losing the possibility of returning to the

labour market. Moreover, long term unemployment may have a demoralising eflFect

on search behaviour, contributing to a less efficient matching process. Third, after a

negative shock, the insider (currently employed worker), has the power to push up

wages due to the cost to the firm of labour turnover and this increase in wages may

permanently raise the unemployment rate (insider-outsider theory, see Blanchard

and Summers, 1987). Therefore, a cyclical shock that reduces the number of insiders

leads to a permanent change in the natural rate.

The first attempt to introduce a measure of hysteresis into unemployment theory

was made by Blanchard and Summers (1986). They argue that unemployment

exhibits hysteresis when current unemployment depends on past values with the

sum of their coefficients equal to or very close to unity. That is, hysteresis in

unemployment arises when unemployment series has a unit root. The presence

of a unit root in the process means it is path dependent. That is, any shock is

entirely incorporated into the series level. Therefore, hysteresis is assimilated into

the concept of "full persistence". Based on this framework, a great number of

studies have investigated whether unemployment series, which is modelled as an

ARMA process, exhibits a unit root (see, for example, Brunello, 1990, León-Ledesma

and McAdams, 2004, Mitchell, 1993, Papell et al, 2000, and Song and Wu, 1997).

Therefore, the dominant approach in the empirical literature to determine whether

hysteresis exists focuses on testing for the existence of a unit root in a linear process.

Two problems arise with this kind of model. The first is that natural and cyclical

shocks are summarized in the innovation with no distinction. As pointed out above.

58



hysteresis in unemployment arises when a change in cycHcal unemployment induces

a permanent change in the natural rate. Having said that, the presence of a unit

root in the unemployment rate is a necessary condition for the existence of hysteresis

but not a sufficient one since the unit root could be generated by accumulation of

natural shocks and be completely independent of whether there is hysteresis. Hence,

separating the respective effects of transitory and permanent shocks on the natural

rate of unemployment is the only way to assess if changes in it are due to cyclical

(this is the case of hysteresis) or natural shocks or both. The second problem refers

to the practice of checking for the presence of hysteresis using a linear model. A

linear model lacks the property of remanence. Under linearity, a shock on the system

followed by a second one of the same intensity in the opposite direction will bring

the system back to its initial position. In this context, it is incorrect to use the term

hysteresis, and we should refer rather to persistence. There is a major difference

between persistence and hysteresis. In a system exhibiting persistence the response

to impulses is a linear function, which is not the case for a system with hysteresis.

A number of papers have studied methods for checking for the presence of hys

teresis in a nonlinear framework. They employ a battery of unit root tests that

control for the possible existence of nonlinear behaviour in unemployment series.

Papell et al. (2000) test for unit roots in autoregressive models with structural

changes and León-Ledesma (2002) implements a unit root test in a threshold au

toregressive model. Though these models incorporate nonlinearites to model the

behaviour of unemployment rate series they have the same weak point as the linear

models described above: it is not possible to tell whether a change in the natural

rate is due to transitory or permanent shocks.

So, if our goal is to check for the presence of a hysteresis effect on the unem

ployment rate we need a nonlinear econometric model that discriminates between

natural and cyclical sources of influence on the unemployment rate.

Jaeger and Parkinson (1994, henceforth JP) put this idea into perspective and

adopt an unobserved components (UC) modeP to test the validity of the hysteresis

^See Harvey, 1989, for a detailed description of the Unobserved Component models.

59



hypothesis. They generate a pure statistical decomposition of the actual unemploy

ment rate into a natural rate component and a cyclical component, which are both

treated as latent variables. They also assume a particular structure to describe the

variation over time of these latent variables. The hysteresis effect is introduced by

allowing cyclical unemployment to have a lagged effect on the natural rate, which is

assumed to contain a unit root. They only consider symmetric responses of the nat

ural rate as regards cyclical unemployment fluctuations. In this way, they implicitly

assume hysteresis is a linear phenomenon. This approach is insufficient to correctly

identify hysteresis. Though it allows the difí"erent sources of shocks (i.e., cyclical or

permanent) to be identified, it does not take into account the existence of nonlin

ear dynamics in unemployment series: this is necessary to capture the remanence

property of a hysteretic process. Under JP's model it is only possible to establish

whether delayed cyclical unemployment has a significant impact on the natural rate.

This describes persistence but it does not correspond to hysteresis.

In order to take into account this nonlinear feature of a hysteretic process, we

propose an extended version of JP's model. In particular, we allow past cyclical

unemployment to have a different effect on the natural rate, which depends on

the regime of the economy. It is thus possible to capture the stylized fact that

natural rate does not decrease in cyclical expansion periods as much as it increases

in cyclical recession periods. This provides a plausible explanation for the tendency

of the natural rate to remain at a high level. The parameters of the model are

estimated by maximum likelihood using a modified Kalman filter that incorporates

the methodology implemented for the estimation of the threshold autoregressive

(TAR) models (see Tong, 1990) in order to split the sample into two groups, which

we may call regimes.

Under this new framework, the problem of testing for hysteresis becomes a prob

lem of testing for linearity. The relevant null hypothesis is a one-regime model

(i.e. the non presence of hysteresis) against the alternative of two regimes (i.e. the

presence of hysteresis). The absence of a body of finite sample theory for nonlin

ear models means that empirical research must rely either on asymptotic theory

60



or bootstrapping for inference. Testing for the econometric hypothesis of interest

in the context of nonhnear models raises a particular problem known in statistics

literature as hypothesis testing when a nuisance parameter is not identified under

the null hypothesis (see, among others, Andrews and Ploberger, 1994, Chan, 1990,

and Hansen, 1996). In particular, the threshold parameter and the delay lag of the

threshold variable are not identified under the null of linearity. If the model is not

identified, the asymptotic distributions of standard tests are unknown, which means

that tabulation of critical values is not possible. There is no shortcut solution for this

problem. Hansen (1996), derives an asymptotic distribution free of nuisance para

meters that is useful for testing and inference in TAR models. He shows that critical

values are easily approximated via Monte-Carlo simulation. As far as we know, no

distributional theory is available to implement a linearity test in the framework of a

UC model with nonlinear dynamics described by the TAR methodology. With this

in mind, we extend the bootstrap proposed by Stofi er and Wall (1991) for assessing

the precision of Gaussian likelihood estimates of the parameters of linear state-space

models to the context of performing a test of linearity for a nonlinear state-space

model. Then we use this method to check for the presence of hysteresis in Italy,

Prance and the United States.

The paper proceeds as follows. Section 2 brieñy describes JP's model and intro

duces an extended version that accounts for nonlinearity. Section 3 proposes two

alternative bootstrap procedures to compute the p-value for a linearity test under

the framework of interest. It also discusses the design of the Monte Carlo experi

ments that are used to investigate the small sample performance of the bootstrap

version of the test and presents the results of the experiments. Empirical results for

Italy, France and the United States are presented in Section 4. Section 5 concludes.

2.2 An extension of Jaeger and Parkinson's model

JP propose a pure statistical decomposition of the unemployment rate to evaluate

the data for evidence of hysteresis effects. They assume the actual unemployment

61



rate to be the sum of two unobservable components: a non-stationary natural rate

component, U^, and a stationary cyclical component, t/*",

u, = ur+i/f. (2.1)

In order to contemplate the necessary condition for hysteresis, i.e. the presence

of a unit root in the process, they first test for the presence of a unit root in the

unemployment series and then impose it in the model. Having said that, the natural

rate component is defined as a random walk plus a term capturing possible hysteresis

eñ"ects,

[/f = [/, , + «í/f^,+ef. (2.2)

Coefficient a measures, in percentage points, how much the natural rate increases

if the economy experiences a cyclical unemployment rate of 1.0 percent. The size of

this coefficient is their measure of hysteresis.

The cyclical^ component of the unemployment rate is defined as a stationary

second-order autoregressive process,

Ul" = cj>,U^.^ + AUÍL2 + ^?. (2.3)

The system is completed by augmenting the model with a version of Okun's law,

which relates cyclical unemployment and output growth,

A = /3iA~i+<5f/f+ ef, (2.4)

where Dt stands for the output growth rate at date t. Equation (2.4) defines the

output growth rate as an autoregressive process of order one plus a term capturing

the influence of the cyclical rate of unemployment^. Since the cyclical component is

^As in JP. we find that AR(2) processes for the cychcal component fit the data well for all the countries under study.

•^Jaeger and Parkinson (1994) introduce this equation to identify the model. However. Proietti (2004) points out that these authors fail to recognize that the model is just identify. Nevertheless, we also use this additional series to estimate the model in a more efScent way.

62

assumed to be stationary, we consider Uf instead of AUf as in JP's model in order

to avoid a problem of over-differentiation.

The disturbances e^, ef and ef* are assumed to be mutually uncorrelated shocks

which are normally distributed with variances a'j^, a^ and cr|,, respectively.

In order to test the hysteresis hypothesis, i.e. past cyclical movements on unem

ployment have a permanent impact on the natural rate, JP perform a significance

test on parameter a,

Ho:a = 0.

If parameter a is significantly different from zero, they argue there exists a

hysteresis effect on the unemployment rate. It is important to note that JP's model

is linear, since it implies that past cyclical unemployment changes have the same

impact, in absolute terms, on the natural unemployment rate. For example, a

variation in the cyclical component of (1%) or (—1%) causes a variation in the

natural rate of («%) or {—a%) respectively. Again, we remark that this linear

context lacks the property of remanence, so it is not possible to observe hysteresis.

We would do better to refer to persistence rather than hysteresis.

At this point, we want to relax the assumption of linearity and we introduce

nonlinearities into JP's model. This extension allows us to detect whether hysteresis

is present in unemployment series. Nonlinearites are introduced by allowing past

cyclical unemployment to have a different impact on the natural rate, which depends

on the regime of the economy. To that end, equation (2.2) becomes

where qt is the threshold variable and 7 stands for the threshold parameter. Equa

tions (2.1), (2.3) and (2.4) remain the same together with assumptions about shocks.

This kind of model is estimated via maximum likelihood in the framework of

the Kalman filter'^. We employ a modified Kalman filter in order to incorporate a

'See Hamilton (1994, Chapter 13) and Harvey (1989, Chapter 3) for a more detailed description

63



deterministic cut-off of the sample that corresponds to a raw indicator for favor

able and unfavorable periods, which is based on the methodology implemented for

the estimation of TAR models^. We choose the long difference Ut^i — Ut^d, with

d G {2,3}, as our threshold variable. This variable is an indicator of the state of

the economy to identify the regimes. The integer d is called the threshold delay

lag. Whether this variable is lower or higher than the threshold parameter 7 de

termines whether an observation belongs to one regime or the other. We consider

an economy with two regimes, one related to high long differences (regime 1), i.e.

an unfavorable regime, and the other with low long differences (regime 2), i.e. a

favorable regime. Parameters d and 7 are unknown so they must be estimated along

with the other parameters. The maximization is best solved through a grid search

over two-dimensional space (7, c¿). To execute a grid search we need to fix a region

over which to search. It is important to restrict the set of threshold candidates a

priori so that each regime contains a minimal number of observations. We restrict

the search to values of 7 lying on a grid between r th and (1 — r)th quantiles of gt_i

for each value of d. In our applications we choose r = 0.30. Then we estimate the

model for each pair (7, d) belonging to this grid and retain the one that provides

the highest log-likelihood value.

As mentioned in the previous section, in this context a test for hysteresis becomes

a test for linearity, i.e. a test for a single regime against the alternative of two

regimes. The null hypothesis we are interested in is

Ho : ai = «2.

At this point, a remark is needed. If the unemployment rate displays a nonlinear

behaviour, JP's model is misspecified and any inference based on the parameters

of this model may lead us to wrong conclusions. This reflection suggests that the

following testing strategy should be implemented.

The starting point is the extended JP model, where we test the null hypothesis

«1 = «2- If we reject it, we are accepting the presence of hysteresis in unemployment

of the Kalman filter. ^See Harvey (1989, Section 3.7) for a more detailed description of the Threshold Kalman filter.

64



series. If it is not rejected, hysteresis is not present in the unemployment rate but

there is still a place for the presence of persistence. Once this point is reached, the

next step is to estimate the linear model proposed by JP and test for persistence

following the strategy they propose.

As pointed out in the previous section, when we perform the test of linearity a

problem of unidentified nuisance parameters under the null hypothesis arises. That

is, there exists a set of parameters that are not restricted under the null hypothesis.

In particular, the null hypothesis aj = «2 does not restrict the threshold parameter

7 and the delay d. As a result, conventional statistics do not have an asymptotic

standard distribution. In order to circumvent this problem, we employ a bootstrap

technique to compute the p-value associated with the test of interest.

2.3 Experimental design for computing the bootstrap p-value for the linearity hypothesis test

Our aim in this section is to approximate the distribution of the test statistic of

interest by a consistent bootstrap procedure. In particular, we implement a Wald

test statistic. The difficulty is that there is no well-accepted bootstrap method that

is appropriate in the present framework. Stoffer and Wall (1991) propose a bootstrap

method to asses the precision of the Gaussian maximum likelihood estimates of the

parameters of linear state-space models. They also prove the asymptotic validity

of this bootstrap under appropriate conditions. These conditions have been not

verified for nonlinear state-space models, and may in fact not hold. However, the

result they obtain is sufficient to justify using the bootstrap for the statistic supW.

Following their resampling mechanism, we propose two bootstrap procedures: the

first is valid if the errors in our model are homoskedastic and the second allows for

the presence of general heteroskedasticity. We check that both bootstrap procedures

work well in our framework by means of Monte Carlo simulations, of course, we have

no guarantee that they work in general.

65



2.3.1 The state-space model

The state-space model is defined by the equations

Si = F{qt-i)stî+wt

Vt = Hst + Dyt-i + vt

(2.5)

(2.6)

where St = {U^.UfMî)' i^ ^ vector of unobserved state variables and y =

{Ut^DtY is a vector of observed variables. Equation (2.5) is known as the tran

sition equation and equation (2.6) is known as the measurement equation. The

coefficients of the model are stored in the constant matrices

F(gt_i) = Fi/(5i_i > 7) + F2l{qt-î < 7) , Qtî = Ut-i - Ut-d,

H 1 1 0 0 (5 0 F

1 0 0

« i

01 1

0 02 0

, ¿ = 1,2; D = 0 0 0 p.

where I{B) = 1 when B holds. The vectors Wt = (e¿ , e f , 0 ) ' and Vt = (0, e D\

represent white noise processes with E{'Wtw'^ = Q, E{vtv[)

where

R and E['Wtv[) = 0,

Q = a N 0 0

0 0 0 cr\ c

0 0

and R = 0 0 0 al

Note that under the null hypothesis of linearity F{qt-.i) = F. To simplify the

notation, let ^0 = {(^N-, CC , c p , a, 0^, 02i <) /î)' be the vector with the model

coefficients and the correlation structure under the null hypothesis, and 0i = (CTJV,

o"c cz), «1, «2; 01, 02) <) /î)' be the vector of parameters under the alternative of

nonlinearity.

2.3.2 Homoskedastic bootstrap

We propose the following algorithm for the homoskedastic bootstrap:

66

STEP 1: We estimate the supW test. To compute this test we need only to

estimate the model under the alternative hypothesis of nonlinearity. The parame

ters of interest are {9i, 7, d}. For each given value of (7,6?) belonging to the grid

A described in the previous section, we obtain the maximum likelihood estimates

(MLE) 9i{-y,d) and compute the pointwise Wald test statistic,

W{j,d) = RA{l,d){RV^r0,{j,d))R')-\ROih.d)y,

where R = (0 0 0 1 — 1 0 0 0 0 0) and Var{9i) is a heteroskedasticity-robust

maximum likelihood estimator of the variance-covariance matrix. Then, arguing

from the union-intersection principle, Davies (1987) propose the statistic

supW = sup VF(7, (i). (7,rf)eA

STEP 2: Using the Kalman filter we obtain linear forecasts of the state vector

at time t based on all the available information up to time í — 1, S Í | Í - I , and the

mean square error matrix associated with each of these forecasts, Pt\t-i- We also

obtain from the Kalman filter the innovations, the innovations covariance matrix,

the Kalman gain matrix and the updating of the state variable,

€t = yt- Hst\t~-i- Dyt^i,

E, = HPt\t-,H' + R,

Kt = Pt\,_^H'T.-\

st\t = st\t-i + Kttt.

Following Anderson and Moore (1979), we derive the innovations form represen

tation of the observations,

st\t^i = Fst_,\t_i + FKtet, (2.7)

yt = Hst\t_^ + Dyt-i + et. (2.8)

67



Let 9Q denote the maximum likelihood estimated parameters under the null.

Evaluating ej, Ej, Kt and St\t at OQ, we obtain'ej, T,t, Kt and 'st\t. We next construct

the standardized residuals by setting

STEP 3: To construct the bootstrap data set, we use equations (2.7) and (2.8).

Let F , H and D be the matrices of coefficients evaluated at 6*0. Ki is the Kalman gain

matrix obtained in step 2, and the bootstrap errors {e¿,í = 1, ...,T} are indepen

dent values obtained by resampling, with replacement, from the set of standardized

residuals {etA = 1, . . . ,T}. so|o = 0 contains the first 3 values of the state variables

(thus, these are prespecified and set equal to the initial conditions for the Kalman

filter). The remaining elements of the vector St\t-^i are constructed by computing a

first-order autoregressive given by (2.7):

The vector yt is constructed by computing a first-order autoregressive process, with

initial conditions fixed at the observed values, and then by adding the results to the

corresponding elements of St\t-i- That is, the row ith of y is given by (2.8):

y; = Hsl^,_,+DyU+t]"el

All initial conditions are kept fixed throughout the bootstrap replications.

STEP 4: The bootstrap sample {yl,t — 1, ...,T} is then used to re-estimate the

parameters under Hi. The algorithm employed to estimate the bootstrap threshold

parameter 7* and the delay lag d* proceeds as in Step 1, where the threshold variable

is given by U^^-^ — U^_¿*. Let 6j{'y*,d*) denote the estimator of 61 when using the

bootstrap sample. We then compute the pointwise Wald test statistic associated

with the bootstrap sample as

68



* — ^ - ^ * . T -—v+

W*{^*,d*) = Re,{Y,d*){RVar{e,{Y,d*))R')-\Rej{-f*,d*)y,

and the supW* test as

supW* = sup W*{Y,d* (7»,d*)eA*

STEP 5: Repeating steps 3 and 4 for 6 = 1,..., 5 , gives a sample {supW* : 6 =

1,..., B} of s«pVF values. This sample mimics a random sample of draws of supW

under the null hypothesis. We compute the bootstrap p-value as PB =card(s«pVF* >

supW)/B, that is the fraction of supW* values that are greater than the observed

value supW.

We carry out B = 1000 bootstrap replications.

2.3.3 Heteroskedastic bootstrap

Our aim here is to calculate a bootstrap distribution of the Wald test allowing

for the possibility of general heteroskedasticity. The algorithm is similar to the

one described above, but replacing the resampling scheme in step 3. In particular,

the resampling we propose is based on the idea of the wild bootstrap, which was

studied for the first time by Wu (1986) in the context of variance estimation in

heteroskedastic linear models. In our context, it looks like this:

Step 3': We propose the following algorithm to generate the bootstrap sample

{y¡,t = l,...,T}:

I. Generate r]^ independent and identically distributed variables from a fixed

distribution®, such that E{r]f) = 0 and E[{r]^y] = E[{r]^)^] = 1. Define e¿ = €¿77 ,

where et is the tth residual calculated in step 2. The bootstrap errors e¿ satisfy

E*{e^) = 0 and E*[{e'¡y] = (et^, where E* denotes the expectation under bootstrap

distribution.

^In particular, the variable rj^ was sample from Mammen's (1993. p.257) two-point distribution attaching masses (5 + \/5)/10 and (5 - \/5)/10 at the points ~(VE - l ) /2 and (v^S -|- l ) /2 , respectively.

69



II. We set the initial condition so|o = 0 and, for t = 2,...,T, set sl_ûi =

St_i|i„i, that is, unobserved bootstrap components are generated with conditionally

set design on the estimated unobserved components in step 2,

s*t\t-i ^Fst-i\t-i + FKtel

III. To define the bootstrap observations, we use a conditional resampling on

{y~i,yo,yi,---,yTî),

y¡ = Hst\t-i + Dytî + e*.

2.3.4 Monte Carlo evidence

In this section we report on a Monte Carlo simulation study designed to evaluate

the small sample performance of both homoskedastic and heteroskedastic bootstrap

procedures in the problem of testing for linearity. We start with a brief description

of the design of the experiment, then proceed with the discussion of the results.

Design of the experiment

The time series considered in our analysis are generated according to the state-

space model given by equations (2.5) and (2.6), under the null and the alternative

hypotheses. Let MQ and Mi denote the class of linear and nonlinear state-space

models, respectively. In our experiments, we use MQ and Mi with (e^,ef,ef) '

iidA^(0,O), where

n al 0 0 0 al 0 0 0 (7¿

as the data-generating process (DGP). We test the null hypothesis of linearity. As

discussed at the end of Section 2, the null hypothesis is true if and only if aj = «2.

Hence, MQ is nested in Mi. We use the statistic supW based on an estimated

Ml setting d = 2, and compute the p-value using both the homoskedastic and the

heteroskedastic bootstrap procedures. The size of the test is investigated when the

70



data are generated according to MQ, while turning to the power properties of the

test under Mi.

To ensure the relevance of the simulations, the parameter values are chosen to

correspond to models that have been fitted successfully to real-world time series.

More specifically, the selected parameters are chosen according to the corresponding

estimated model for U.S. under the null, and for France under the alternative:

DGPo : 00 = (0.049, 0.246,0.842,1.472, 0.285, 0.162, -1.498,0.506)'.

DGPi : 01 = (0.436,0.157,0.5,4.1,1.106,0.117,0.337,-0.460,0.688)'; 7 = 0.16.

To study the effect of the size of the difference «i — «2 on the performance of

the test, we vary «i between (1.306,1.506,1.706,2.106), while a2 remains constant

at its fixed value. Each of these values gives rise to DGP2~DGP^, respectively.

The experiments proceed by generating artificial series of length T+50 according

to Mo or Ml with T = 150, and initial values set to zero. We then discard the

first 50 pseudo-data points in order to attenuate the effect of initial conditions and

the remaining T points are used to compute the test statistic. We simulate the

proportion of rejections of the test at the 5%, 10% and 20% significance levels.

The estimation of the rejection probabilities is calculated from B = 99 bootstrap

replications and R = 500 simulation runs. The processing time becomes excessive

when greater values of B or R are used.

Simulation results

In Table 1 we present simulation evidence concerning the empirical size and power

of the test. We observe a reasonable approximation of the nominal level at all signif

icance levels considered. Deviations from the null hypothesis are detected with high

probability across the various parameterizations. We observe that in all cases under

consideration the test based on the homoskedastic bootstrap approach yields slightly

lower rejection probabilities than the heteroskedastic bootstrap test. It should be

emphasized that this happens even though the model generated is homoskedastic.

71



As expected, the performance of both bootstrap procedures improves as the differ

ence between the values of parameters in the two regimes increases.

2.4 Empirical results

Our study concerns Italy, France and the United States. The economic series em

ployed are the quarterly unemployment rate (U) and real gross domestic product

(GDP). Data for Italy (running from 1970:1 through 2002:1), France (1978:1-2002:1)

and U.S. (1965:1-2002:1) come from OECD Main Economic Indicators. All data are

obtained as seasonally adjusted and all the variables except the unemployment rate

are in natural logs.

We have decomposed the unemployment rate assuming that the natural rate

contains a unit root. This assumption must be tested. To do this, we employ the

Phillips-Perron test for unit roots. We obtain that unemployment rates display non-

stationary behaviour for all countries. We also perform the unit root test for the

GDP series, which also displays non-stationary behaviour for all countries. Results

are presented in Table 2.

Tests for hysteresis are reported in Table 3. The p-values presented in Table 3

are calculated following the bootstrap technique described in Section 3. For compar

ison reasons, we also report the ;?-values obtained with the linear model. Diagnosis

checking of the residuals of the linear modeF leads us to implement a heteroskedastic

bootstrap for the U.S. and a homoskedastic bootstrap for France and Italy. Accord

ing to bootstrap p-values, the hysteresis effect is significant at the 5% level for

Italy and France. As argued in Section 2, under the presence of nonlinearity, JP's

model may lead to obtain spuriously good inference results. In fact, note that JP's

methodology fails to detect hysteresis for the case of France. This result stresses the

importance of testing linearity before fitting any statistical model.

'^The assumptions underlying the errors of the linear model are tested via appropriate autocorrelation, heteroskedaticity and normality test statistcs, which are available from the authors upon request. We find evidence in favour of non-autocorrelation in all countries. Evidence against homoskedasticity is only found in the U.S..

72



Results concerning the estimated models for Italy and Prance are available in Ta

bles 4 and 5. For the case of Italy, the maximum likelihood estimate of the threshold

parameter is 7 = 0.1, with a 90% bootstrap confidence interval [0.015,0.266]. Our

estimate of the delay parameter is d = 2. Hence, the threshold model splits the

regression into two regimes depending on whether or not the threshold variable is

higher than this threshold parameter. That is, we consider we are in regime 1 when

Ut-i — f/i-2 ^ 0.1 and in regime 2 when Ut^i — Ut-2 < 0.1 (see Figure 2.2). For

Italy, there are less observations in regime 1 (41%) than in regime 2 (59%), which

means that this country spent more periods of time in the favorable regime. This is

also the case for France. Analyzing the estimated hysteresis parameter, we observe

a point of great interest. Both parameters are positive and the one associated with

Regime 1 is greater than that of Regime 2. This points to asymmetric responses

of the natural rate as regards cyclical unemployment movements in the following

direction: the natural rate does not decrease in favorable cyclical periods as much

as it increases in unfavorable cyclical periods. The size of the coeflacients suggests

that this mechanism is more pronounced in France than in Italy. In fact, for Italy,

the natural rate decreases (2.76%) in unfavorable periods, while a cyclical shocks

have an impact of (1.85%) in favorable periods. For France, these values are (4.1%)

and (1.11%), respectively.

It is worth analyzing the U.S. separately. The information concerning the model

estimated is provided in Table 6. According to the hysteresis test, we can not reject

the null of linearity. However, as we mentioned in Sections 1 and 2, there is still a

place for persistence. In fact, we find evidence in favour of it, given that parameter

a is significant at 5%. Hence, though there is no hysteresis, cyclical shocks have a

significant impact on the natural rate. In particular, if the economy experiences a

cyclical unemployment rate of 1% the natural rate increases by a 1.472%.

73

2.5 Conclusions

The aim of this paper is twofold. First, to take into account the nonhnear feature

of a hysteretic process we propose a definition of hysteresis taken from Physics. To

provide an operational statistical framework for our concept of hysteresis we use

the unobserved components approach, which decomposes unemployment rate into a

non-stationary natural component and a stationary cyclical component. We extend

the model of Jaeger and Parkinson (1994) by introducing nonlinearities into the

specification of the natural rate component. We do this by allowing past cyclical

unemployment to have a different eñ'ect on the current natural rate depending on

the regime of the economy. To estimate the model we use a modified Kalman filter

that incorporates a sample partition that corresponds to two difi^erent regimes. The

procedure for identifying these regimes is related to the TAR methodology. Under

this framework, a test for hysteresis becomes a test for linearity. Second, when we

implement a test for linearity a problem of unidentified nuisance parameters arises

since the threshold parameter and the delay lag of the threshold variable are only

identified under the alternative hypothesis of hysteresis. As a result, the standard

asymptotic distributions of the classical tests are unknown under the null. Our

objective is to implement a correct test for the relevant null hypothesis of a one-

regime model. We rely on bootstrapping techniques to calculate an appropriate

p-value for the decision rule. We propose two bootstrap procedures: the first is

valid if the errors in our model are homoskedastic and the second allows for general

forms of heteroskedasticity. In a Monte Carlo simulation study, both bootstrap

approximations of the linearity test are investigated in greater detail, and we find

that they work quite well. Our study concerns Italy, France and the United States.

For European countries, we reject the null of linearity. This is related to the presence

of hysteresis. On the other hand, for the United States we reject the hysteresis

hypothesis. We find symmetric responses of the natural rate as regards to cyclical

fluctuations in unemployment.

74



REFERENCES

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Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter

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Blanchard, O.J., Summers, L.H., 1986. Hysteresis and the European Unemployment

Problem. NBER Macroeconomics Annual, Vol. I, MIT Press, Cambridge MA, pp.

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Blanchard, O.J., Summers, L.H., 1987. Hysteresis in Unemployment. European

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Davies, R.B., 1987. Hypothesis testing when a nuisance parameter is present only

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Hall, P., 1992. The Bootstrap and Edgeworth expansion. Springer Series in Statis

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Hamilton, J.D., 1994. Time Series Analysis. Princeton.

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the null hypothesis. Econometrica 64, 413-30.

Harvey, A.C., 1989. Forecasting, structural time series models and the Kalman

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Jaeger, A., Parkinson, M., 1994. Some evidence on Hysteresis in Unemployment

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Papell, D.H., Murray, C.J., Ghiblawi, H., 2000. The structure of unemployment.

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Proietti T., 2004. Unobserved Components Models with Correlated Disturbances.

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R0ed, K., 1997. Hysteresis in Unemployment. Journal of Economic Surveys 11,

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Song, F.M., Wu, Y., 1997. Hysteresis in unemployment: evidence from 48 U.S.

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Stoffer, D.S., Wall, K.D., 1991. Bootstrapping state-space models: Gaussian maxi

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76



Wu, C.F.J., 1986. Jackknife, Bootstrap and Other Resampling Methods in Regres

sion Analysis. The Annals of Statistics 14, 1261-95.

77



TABLES AND FIGURES

Table 1: Monte Carlo Results Nominal size Simulated size DGPo Homoskedastic bootstrap Heteroskedastic bootstrap

Simulated power DGPi : «1 — Q;2 = 3 Homoskedastic bootstrap Heteroskedastic bootstrap

DGP2 : «1 - «2 = 0.2 Homoskedastic bootstrap Heteroskedastic bootstrap

DGP3 : cti - «2 = 0.4 Homoskedastic bootstrap Heteroskedastic bootstrap

DGPi : «1 - «2 = 0.6 Homoskedastic bootstrap Heteroskedastic bootstrap

DGF5 : «x - «2 = 1 Homoskedastic bootstrap Heteroskedastic bootstrap

5%

0.042 0.045

0.739 0.757

0.089 0.126

0.155 0.164

0.279 0.216

0.495 0.499

10%

0.092 0.091

0.800 0.856

0.111 0.121

0.158 0.177

0.292 0.316

0.498 0.543

20%

0.203 0.195

0.826 0.879

0.136 0.150

0.169 0.183

0.333 0.351

0.505 0.587

78



Table 2: Unit Root Tests

Phillips-Perron test on GDP series Italy -2.743 ~ Prance 0.888 U.S. -0.365

Phillips-Perron test on Unemployment series

Italy -1.716 Prance -2.593 U.S. -2.089

Note 1: For the Phillips-Perron test, we use Mackinnon critical values for rejecting the hy

pothesis of a unit root. We do not reject the null hypothesis of a unit root at 1%, 5% or 10%.

Table

Italy France U.S.

3 : Tests for the Hysteresis Assumption Nonlinear Model HQ : ai = Q!2

Bootstrap j9-value=0.0375* Bootstrap p-value=0.033* Bootstrap p-value=0.890

Linear Model iio : a = 0

p-value=0.015* j9-value=0.204 j9-value=0.000*

"Significant at 5%

79



Table 4 : Estimation Results^ ITALY

NONLINEAR MODEL

Percentage of observations

Natural Rate Equation ai

Cyclical Rate Equation </>!

02

Identification Equation

/5i 5

O-D

Threshold 90% confidence^

Delay lag

1=1 1=2

41% 59% i

2.762 = 1 i = 2

(0.134) 1.846 (0.066) 0.481 (0.057)

0.047 (0.001) 0.896 (0.049) 0.020 (0.006)

0.509 (0.011) 6.401 (0.735) 0.700 (0.055)

7 = 0.1 [0.015,0.266]

d = 2

* Following StofFer and Wall (1991), standard errors are calculated from B = 1000 runs of the

bootstrap and provided in brackets. These standard errors are the square root of ^ {O^j^ — Oi) /{B —

1), where 9i represents the ith parameter of the vector 0i, i = 1,..., 10, and 9i is the MLE of 0¿. "We compute the confidence interval based on the bootstrap percentiles described by Hall

(1992).

80

Table 5 : Estimation Results^" FRANCE

NONLINEAR MODEL

Percentage of observations

Natural Rate Equation Oii

Cyclical Rate Equation

01 02 0-c


Pi 5

0"D

Threshold 90% confidence"

Delay lag

i=l i=2 44% 56%

i = 4.100

= 1 z = 2 (0.213) 1.106 (0.022) 0.436 (0.035)

0.117 (0.041) 0.337 (0.056) 0.157 (0.074)

0.688 (0.069) -0.460 (0.052) 0.500 (0.033)

7 = 0.16 [0.123,0.532]

d = 3

^''Following Stoifer and Wall (1994). standard errors are calculated from B = 1000 runs of the

bootstrap and provided in brackets. These standard errors are the square root of "^ {(^ib~()i) /{B —

1), where 9i represents the ¿th parameter of the vector 9i, i = 1...., 10, and 9i is the MLE of 9i. ^HVe compute the confidence interval based on the bootstrap percentiles described by Hall

(1992).

81



Table 6 : Estimation Results^^ U.S.

LINEAR MODEL Natural Rate Equation

«(«)

Cyclical Rate Equation

01 02


Pi 5

C^D

1.472 0.049

0.285 0.162 0.246

0.506 -1.498 0.842

(0.297) (0.165)

(0.200) (0.130) (0.061)

(0.059) (0.660)

(0.077)

(a) The Wald test statistic for the null hypothesis ( a = 0) is distributed chi-square with 1

degree of freedom under the null. It is significant at 5%.

^ Standard errors are calculated from a consistent MLE of the variance-covariance matrix and provided in brackets.

82



Figure 2.1: Unemployment Rate

83



Figure 2.2: Threshold variable: Ut^i — Í7._T and 7.

84



Chapter 3

Specification Tests for the Distribution of Errors in Nonparametric Regression: a Martingale Approach

3.1 Introduction

Specification tests for the distribution of an observable random variable has a long

tradition in Statistics. However, there are many situations in which the random vari

able of interest for the researcher is a non-observable regression error. For example,

in Economics, the productivity of a firm is defined as the error term of a regression

model whose dependent variable is firm profits; and, in Finance, the return of an

asset in a period is usually defined as the error term of a dynamic regression model.

In contexts such as these, knowing whether the distribution of the error term belongs

to a specified parametric family or not may be crucial to achieve efficient estima

tion, to determine certain characteristics of interest (such as percentiles or number

of modes) of the error term, or to design an efficient boostrap procedure. This is

the problem that we study in this paper.

Let us describe the specific framework that we consider. Let {X,Y) be a bivariate

continuous random vector such that E{Y'^) is finite, and denote m{x) = E{Y\X =

x) and a'^{x) =Var(y|X = x). We can consider then the error term e = {¥ —

85



Chapter 3 Specification Tests for the Distribution of Errors in Nonparametric Regression

m{X)}/a{X), which is, by definition, a zero-mean unit-variance random variable.

The objective of this paper is to describe how to test a parametric specification about

the cumulative distribution function (c.d.f.) of e, making no parametric assumptions

about the conditional mean function m(-) or the conditional variance function a^{-).

Specifically, if F(-) denotes the c.d.f. of £ and JP" = {F{-, 9), 9 e Q C M™} denotes

a parametric family of zero-mean unit-variance continuous c.d.f.'s, each of them

known except for the parameter vector 9, we propose a testing procedure to face the

hypotheses Ho : 3 ^0 e e such that F(-) - F{-. 9o)

vs. Hi : F(-) ^ T,

assuming that independent and identically distributed observations {(Xi,l¿)}^^j,

with the same distribution as {X,Y), are available. The testing procedure that we

propose here could also be used, with appropriate changes, if the family JF reduces

to one known c.d.f. (i.e. when there is no unknown parameter 9), or if the error

term that is to be analyzed is defined by removing only the conditional mean (i.e.

when we consider the error term Y — m{X)). The specific test statistics that should

be used in these more simple contexts are discussed below.

The testing problem that we study in this paper can also be considered as an

extension of the classical goodness-of-fit problem. Suppose that a parametric speci

fication for the c.d.f. of an observable continuous variable Y is rejected using a tradi

tional nonparametric goodness-of-fit statistic, such as the Kolmogorov-Smirnov one;

one of the drawbacks of these statistics is that the rejection of the null hypothesis

gives no intuition about the cause of the rejection. In this situation, it would be of

interest to examine if the only reason why the null hypothesis has been rejected is

because the parametric family fails to capture appropriately the behaviour in mean

of y ; if we want to check whether this is the case, then we would have to analyze

if the parametric specification is appropriate for Y — m{X). li the null hypothesis

were rejected again, we might be interested in going one step further and testing

whether the parametric family fails to capture appropriately the behaviour in mean

and variance of Y; thus, we would have to analyze if the parametric specification is

appropriate for {Y — m{X)}/a{X), and this is precisely the testing problem that

86




we consider here.

The test statistics that we propose in this paper can be motivated by studying

the relationship between our problem and the classical goodness-of-fit problem. If

the error term e were observable and parameter 9Q were known, our test would be

the classical goodness-of-fit test. In our context, the unobservable errors must be re

placed by residuals, which must be derived using nonparametric estimations of m(-)

and (j'^{-) since no parametric form for these functions is assumed, and parameter

6Q must be replaced by an appropriate estimator, say 0. Thus, we could think of

using as a statistic for our test any of the traditional nonparametric goodness-of-fit

statistics, but computing it with nonparametric residuals and the estimator 6. How

ever, it is well-known in the literature that the consequence of replacing errors by

parametric residuals and parameters by estimators in goodness-of-fit tests is that the

resulting statistics are no longer asymptotically distribution-free (see e.g. Durbin,

1973 or Loynes, 1980); furthermore, the asymptotic null distributions usually depend

on unknown quantities and, hence, asymptotic critical values cannot be tabulated.

In this paper we prove that this is also the case when nonparametric residuals are

used, and we discuss how this problem can be circumvented in our testing problem.

Specifically, using the results derived in Akritas and Van Keilegom (2001), we derive

the asymptotic behaviour of goodness-of-fit statistics based on nonparametric resid

uals and estimators; and then, following the methodology introduced in Khmaladze

(1993), we derive the martingale-transformed test statistics that are appropriate in

our context.

The rest of the paper is organized as follows. In Section 2 we introduce the em

pirical process on which our statistics are based and derive its asymptotic properties.

In Section 3 we describe the martingale transformation that leads to asymptotically

distribution-free test statistics. In Section 4 we report the results of a set of Monte

Carlo experiments that illustrate the performance of the statistics with moderate

sample sizes. Some concluding remarks are provided in Section 5. All proofs are

relegated to an Appendix.

87



3.2 Statistics based on the estimated empirical process

If we had observations of the error term {£¿}f ^ and parameter o were known, we

could use as a statistic for our test the asymptotic Kolmogorov-Smirnov statistic

Kn or the Cramer-von Mises statistic Cm which are defined by

K^ = n i /2sup |F„(2)-F(^ ,^o) | , zeM

n Cn = 5]{i;(£,)-F(£,,^0)f,

¿=1

where -F„(-) denotes the empirical c.d.f. based on {ei}^^^. Both Kn and C„, are

functionals of the so-called empirical process V„(-), defined for 2; G M by

n

V„(^) = n-1/2 5^{/(£, <z)- F{z, Oo)},

where /(•) is the indicator function. Hence, the asymptotic properties of K^ and Cn

can be derived by studying the weak convergence of the empirical process V„(-). In

our context, the test statistics must be constructed replacing errors by residuals and

the unknown parameter by an estimator. Since no parametric assumption about the

conditional mean m(-) or the conditional variance cP'i^ is made, the residuals {EÍYÍ=\

must be constructed using nonparametric estimates of these functions. Specifically,

we consider Nadaraya-Watson estimators, i.e.

n

¿=1

1=1

where Wi{x, hn) = K{{x — Xi)/hn}/ Yll=i -^{(^ ~ ^j)/hn}, K{-) is a known kernel

function and {hn} is a sequence of positive smoothing values. With these estimates

we construct the nonparametric residuals £j = {Yi — m{Xi)}/a{Xi). On the other

hand, the unknown parameter must be replaced by an appropriate estimator 0; we

discuss below the asymptotic properties that 0 must satisfy. Using this estimator

and the nonparametric residuals, we can define now the statistics

zeiR I n

where -F„(-) denotes the empirical c.d.f. based on {ej}"^^. Both Kn and Cn are

functionals of the process V„(-), defined for 2; G M by

n

This process will be referred to as the "estimated empirical process". First of all

we discuss the asymptotic relationship between the empirical process V„(-) and the

estimated empirical process V„(-), since this relationship will be crucial to establish

the asymptotic behaviour of íí„ and Cn- The following assumptions will be required:

Assumption 1: The support oí X, hereafter denoted Sx, is bounded, convex and

has non-empty interior.

Assumption 2: The c.d.f. of X, denoted Fx{-)i admits a density function fx{-)

that is twice continuously differentiable and strictly positive in Sx-

Assumption 3: The conditional c.f.d. oiY\X = x, hereafter denoted F(-|x),

admits a density function f{-\x). Additionally, both F{y\x) and f{y\x) are

continuous in [x, y), the partial derivatives ^f{y\x), •^F{y\x), ^F{y\x) exist d and are continuous in {x, y), and sup . , \yf{y\^)\ < 00, sup^, ^ \y-^F{y\x)\ < 00,

sup^.y \y^-§^f{y\^)\ < oo> s^Px,y \y^£2F{y\x)\ < oo.

Assumption 4: The functions m{-) and o" (-) are twice continuously differentiable.

Additionally, there exists C > 0 such that infa;gs^ a^{x) > C.

Assumption 5: The kernel function K{-) is a symmetric and twice continuously

differentiable probability density function with compact support and J uK(u)áu

0.

89

Assumption 6: The smoothing value hn satisfies that n/i^ = o(l), nh^^/log h~^ =

0(1) and log h-y{nhl+^^) = o(l) for some 5 > 0.

Assumption 7: The c.d.f F{-, 9) admits a density function /(•, 9) which is positive

and uniformly continuous in M. Additionally, /(•,•) is twice differentiable with

respect to both arguments, F{-,-) has bounded derivative with respect to the

second argument and sup^gu \zf{z,9)\ < oo for every 9 E Q.

Assumption 8: If HQ holds, then there exists a function •0(-, •, •) such that •n}l'^{9 —

Oo) = n-^/^J2'^^,i^{X,,£„9o) + Op{l). Additionally, E{i;{X,e,9o)} = 0, O =

E{ip{X, e, 9o)tp{X, e, 9Q)'} is finite, IIJ{-, •, •) is twice continuously differentiable

with respect to the second argument.and sup^gjj \-^'ip{x, z,9)\ < co.

Assumption 9: If Hi holds, then there exists 9^ G M" such that 'n}^^{9 ~ 9^) =

0,(1).

Assumptions 1-6, which are similar to those introduced in Akritas and Van Kei-

legom (2001), guarantee that the nonparametric estimators of the conditional mean

and variance behave properly. Assumption 7 allows us to use mean-value arguments

to analyze the effect of introducing the parametric estimator 9. Assumptions 8-9

ensure that the parametric estimator behaves properly both under HQ and Hi.

Our first proposition states an "oscillation-like" result between the empirical

process and the estimated empirical process in our context.

Proposition 1: If HQ holds and assumptions 1-8 are satisfied then

sup zeR

V „ ( z ) - { V n ( z ) + A i „ ( z ) + A2n{z) - A 3 „ ( z ) } = 0 , ( 1 )

where AU^) = f[zMn-'^'T.U{ivÁX^^y^)+^ln},

A2n{z) = zf{z, 9o)n-'/' i:tiW2{X..Y^ + P^J, Asn{z)~Feiz,9oyn'/\d-9o),

and Fe{z, 9) = §^F{z, 9), ip^{x, y) = -a{x)-^ ¡{I{y < v)-'F{v\x)}dv, ip^ix, y)

—a{x)~~^f{v — m{x)}{I{y < v) — F{v\x)}dv, and, for j = 1,2, (3j^ =

lhl{Ju^K{u)du}E{cpj^^{X,Y)}, iPj^^{x,y) = ¿ipj{x,y).

90

Note that processes Ain(-) and A2n(-) arise as a consequence of the nonpara

metric estimation of the conditional mean and variance, respectively, whereas A3„(-)

reflects the effect of estimating ô- The following theorem states the asymptotic be

haviour of Kn and Cn-

Theorem 1: Suppose that assumptions 1-1 hold. Then:

a) If Ho holds and assumption 8 is satisfied then:

K„^sup|D(i)| and C^ ^ í{D{t)Ydt, Í6K J

where D{-) is a zero-mean Gaussian process on M with covariance structure

Cov{D{s), D{t)} = F(min(s, i), ô) - F{s, eo)F{t Bo) + H{s, t, 9o),

and

H{s, t, Oo) = f{s, 9o)[E{I{e < t)e} + f £;{J(e < t){e^ - 1)}] +f{t,9o)[E{I{6 < s)e} + \E{I{e < s){e'' - 1)}] + / (5 , ô)/(i, ô)[l + '-Ê{e^) -h î{E{é) - 1}] -Fe{s,9oyE{I{e <t)iP{X,e,eQ)} -Fe{t,9o)'E{I{s < s)i;{X,s,9o)} -f{s, 9o)Fe{t, 9oy[E{i;{X, e, 9o)e} + fi?{^(X, e, 9o){e^ - 1)}] -fit,9o)Fe{s,9oy[E{,k{X,e, 0o)e} + ¡E{^ÁX,e,9o){e^ - 1)}] +Fe{s,9o)'^Fe{t,9o).

b) If Hi holds and assumption 9 is satisfied then, 'i c EM.,

P{kn > c) ^ 1 and P{dn > c) ^ 1.

Since the covariance structure of the limiting process depends on the underly

ing distribution of the errors and the true parameter, it is not possible to obtain

asymptotic critical values valid for any situation. To overcome this problem, in the

next section we propose to consider test statistics that are based on a martingale

transform of the estimated empirical process, in the spirit of Khmaladze (1993), Bai

(2003) and Khmaladze and Koul (2004).

91


3.3 Statistics based on a martingale-transformed process

As Proposition 1 states, three new processes appear in the relationship between the

estimated empirical process V„(-) and the true empirical process V„(-). These three

additional processes stem from the estimation of the conditional mean, the condi

tional variance and the unknown parameter. If we follow the methodology described

in Bai (2003), this relationship leads us to consider the martingale-transformed

process

Wn{z) = n^'^{Fn{z)~ f q{u)'C{uY^dn{u)f{u,eo)áu), J —oo

where

q{u) = (1, Uu, eo)/fiu, 9o),l + ufuiu, 9o)/f{u, Oo), fe{u, OoY/fiu, Oo))', C{u)^J^°°q{r)q{ryf{r,9o)dr,

4(«) ^ /;°° q{r)dF^{r) = n^' Y.U ( ^ > ^W^)-^

and fu{u,6) = •§^f{u,9), fe{u,9) = -§^f{u,9). Since process W„(-) depends on

the unknown parameter 9Q, we cannot use it to construct test statistics; obviously,

the natural solution is to replace again 6*0 by 9. Thus, we consider the estimated

martingale-transformed process W„(-), defined in the same way as W„(-), but re

placing ^0 by 9. With this estimated process we can construct the Kolmogorov-

Smirnov and Cramér-von-Mises martingale-transformed statistics

Kn = sup W„(2)

n

£i}^.

i=l

The asymptotic behavior of these statistics can be derived studying the weak con

vergence of W„(-). The following additional assumptions, which ensure that the

martingale transformation can be performed and behaves properly, are required.

Assumption 10: C(M) is a non-singular matrix for every u G [—oo, -|-oo).

+00

Assumption 11: If HQ holds, then J \\q0{u)\\'^f{u,9o)du = Op{l), where qg{-) ~oo

denotes the derivative oí q{-) with respect to 9.

92



Chapter 3 Specification Tests for the Distribution of Errors in Nonpararaetric Regression

+ 0 0

A s s u m p t i o n 12: If Hi holds, then J \\qe{u)\\'^f{u,6^:)du = Op{l). —00

T h e o r e m 2: Suppose that assumptions 1-7 and 10 hold. Then:

a) If Ho holds and assumptions 8 and 11 are satisfied then:

Kn^ sup lW(i ) | and Cn^ Í {W(í)}2dt, te[o,il J[o,i]

where W(- ) is a Brownian motion.

b) If Hi holds and assumptions 9 and 12 are satisfied then, V c G M;

P{Kn> c) ^ l and P (Ü„ > c) -^ 1.

It follows from this theorem that a consistent asymptotically valid testing pro

cedure with significance level a is to reject HQ if Kn > k^, or to reject HQ if

Cn > Ca, where ka and c^ denote appropriate critical values derived from the c.d.f.'s

of supjgjoij | W ( i ) | and L j ,{W(í)}^dí . Specifically, the critical values for Kn with

the most usual significance levels are /ÍQ.IO = 1-96, fco.os = 2.24, fco.oi = 2.81 (see e.g.

Shorack and Wellner, 1986, p.34), and the critical values for C„ with the most usual

significance levels are CQ.IO = 1.196, CQ.OS = 1.656, CQ.OI = 2.787 (see e.g. Rothman

and Woodroofe, 1972).

The statistics Kn and Cn are designed to test if the c.d.f. of the error term

£ = {Y — m{X)}/a{X) belongs to a parametrically specified family of zero-mean

unit-variance continuous c.d.f.'s. If we were interested in testing if the c.d.f. of the

the error term Y — m{X) belongs to a parametrically specified family of zero-mean

continuous c.d.f.'s, then the statistics that we would use are defined in the same way

as Kn and C^, but considering q{u) = (1, /„(n, 6 'O) / / (M, 6*0), fe{u,0o)'/ f{u,0(¡))'. If

we were interested in testing if the c.d.f. of the error term e = {Y — m{X)}/a{X)

is a known zero-mean unit-variance c.d.f. FQ{-), then the statistics that we would

use are sup_jgjj |W„(2) | and ^ " ^ 1 W„(ej)^, where W„(-) is defined as above but now

considering q{u) = (1, fo.u{u)/fo{u), 1 + ufo^^iu)/fo{u)y, where /o(-) and /o,u(-)

denote the first and second derivative of Fo(-). Finally, if we were interested in

93




testing if the c.d.f. of the error term Y — m{X) is a known zero-mean c.d.f. Fo{-),

then the statistics that we would use are again sup^g^ |W„(^)| and Yl^=i W„(ei)^,

but now W„(-) is defined as above but considering q{u) = (1, /O,«('W)//O(M))'-

3.4 Simulations

In order to check the behaviour of the statistics, we perform a set of Monte Carlo ex

periments. In each experiment, independent and identically distributed {(X¿, ^)}"=i

are generated as follows: Xi has uniform distribution on [0,1] and Y^ = 1 + Xi + EÍ,

where Xi and e are independent, and e has a standardized Student's t distribu

tion with 1/6 degrees of freedom. The value of 5 varies from one experiment to

another; specifically, we consider ¡5 = 0, 1/12, 1/9, 1/7, 1/5 and 1/3 (when 5 = 0,

the distribution of Si is generated from a standard normal distribution). Using the

generated data set {( -^Í ,^Í )}¿LI cis observations, we test the null hypothesis that

the distribution of the error term {Y — m{X)}/a{X) is standard normal. Observe

that, according to the data generation mechanism, the null hypothesis is true if and

only if 5 = 0; thus the experiment with 5 = 0 allows us to examine the empirical

size of the test, and the experiments with 5 > 0 allow us to examine the ability of

the testing procedure to detect deviations from the null hypothesis caused by thick

tails.

The test is performed using the statistics described at the end of the previous

section, i.e. the Kolmogorov-Smirnov type statistic sup^g^ |W„(2;)| and the Cramér-

von Mises type statistic XliLi Wn(ei)^, where Wn(-) is defined as above. Note that

in the specific test that we are considering in this set of experiments, the function

q{-) that appears in the definition of W„(-) proves to be q{u) = (1, ~u, 1 — u^)'. The

computation of the statistics requires the use of Nadaraya-Watson estimates of the

conditional mean and variance functions. We have used the standard normal density

function as a kernel function K{-), and various smoothing values to analyze how

the selection of the smoothing value influences the results; specifically, we consider

/iü) = C^^'^axn^^^^, for j = 1, ...,4, where ax is the sample standard deviation of

94




{Xi}2^j and C^^^ = j/2. The integrals within the martingale-transformed process

have been approximated numerically. We only discuss the results for the Cramér-

von Mises type statistic, since the results that are obtained with the Kolmogorov-

Smirnov type statistic are quite similar. In Table 1, we report the proportion of

rejections of the null hypothesis for n = 100 and n = 500 with various significance

levels; these results are based on 1000 replications. The results that we obtain show

that the statistic works reasonably well for these sample sizes, and its performance

is not very sensitive to the choice of the smoothing value.

3.5 Concluding Remarks

In this paper we discuss how to test if the distribution of errors from a nonparametric

regression model belongs to a parametric family of continuous distribution functions.

We propose using test statistics that are based on a martingale transform of the

estimated empirical process. These test statistics are asymptotically distribution-

free, and our Monte Carlo results suggest that they work reasonably well in practice.

The present research could be extended in several directions. First of all, it seems

interesting to extend our results to the case of symmetry tests. Under a nonlinear

regression model, conditional symmetry is equivalent to the symmetry of the error

term about zero. This is the null hypothesis we are interested in. Symmetry and

conditional symmetry play an important role in many situations. The following ex

amples may illustrate the relevance of constructing consistent tests of symmetry and

conditional symmetry. Conditional symmetry is part of the stochastic restrictions

on unobservable errors used in semiparametric modelling (Powell, 1994). Adaptive

estimation relies on the assumption of conditional symmetry (Bickel, 1982; Newey,

1988). In macroeconomics, the symmetry of innovations also plays an important role

(Campbell and Henstchel, 1992). In Finance, knowing whether returns or risks ex

hibit symmetry may help in the choice of an adequate risk measure for portfolio risk

management (Gouriérox, Laurent and Scaillet, 2000). Knowledge of the properties

of the error term in a regression model has efficiency implications for bootstrapping

95




(Davidson and Flachaire, 2001).

In addition to this, it is also interesting to extend the results we have already

obtained to dynamic models. The main point here is to extend Theorem 1 of Akritas

and Van Keilegom (2001), which proposed a consistent estimator of the distribution

of the error term based on nonparametric regression residuals for iid observations, to

a context with dependent observations. This would allow us to apply a martingale

transform to the nonparametric-oscillation like results derived.

96



APPENDIX

Proof of Proposition 1: Assume that Ho holds and let 9 be an appropriate esti

mator of 6o. If we add and substract F{z, 9o) to V„(-), we obtain

n

Vn{z) = n-'/'J2Î{e,<z)-F{z,eo)]^n'/'[F{z,e)-F{z,9o)]{3.1)

By Taylor expansion, the second term admits the approximation

(//) = Fe{z,9oyn'/^d-9o) + Fee{z,9)'n^'-'0 - 9^?/2, (3.2)

where Fee denotes the second partial derivative of F(-, •) with respect to the

second argument and 9 denotes a mean value between 9 and ô- Apply As

sumption 8 to show that the last term is Op{n~^/'^).

From Theorem 1 in Akritas and Van Keilegom (2001), we obtain the following

expansion of the empirical c.d.f. based on the estimated residuals e¿:

n

F„{z) = n~^J]/(£, <z) i=l

n n

Y. I[e, <z)+ n-' Y^ ifiXi, y„ z) + P^{z) + Rn{z), (3.3) = n ¿=i ¿=i

where ^{x,y,z) = ~f{z,eo)a~\x) ¡[I{y < v) - F{v\x,9om + z^-^)dv,

^n(^) = IÎU u^K{u)du}E{ip^^{X,Y, z)}, ip^^{x,y,z) = ^^{x,y,z) and

sup gjK \Rn{z)\ = Oj,{rr^/'^) + Op{hD = Op(n^^/^). Note that

(p{x,y,z) = f{z,9o)(p^^{x,y) + zf{z,9o)ip2n{x,y),

(3^{z) = f{z,9o)(3,^ + zf{z,9o)(3,^

where ip-^^{-,-), <P2n{'i')-- fîn ^^d l^2n ^^6 as defined above. The proposition

follows immediately by appeling to (3.2) and (3.3) in (3.1). •

97

Chapter 3 Specification Tests for the Distribution of Errors in Nonparametr ic Regression

Proof of Theorem 1: First we prove the theorem for Kn- Note that, under Hi 0,

Kn = sup zei . where we define Dn(2)

n

D„(^) = n-'/'J2{I{ei <z)- F{z,d) - P^{z)}, (3.4)

and /3„(-) is defined above. To derive the asymptotic distribution of Kn, it

suffices to prove that D„(-) converges weakly to D(-), and then apply the

continuous mapping theorem. Prom Proposition 1 and (3.4), it follows that

D„(-) has the same asymptotic behaviour as D„(z) = n~^^^ Yl^=i[H^i ^ ^) ~

F{z,9o) + (p{Xi,Yi,z)] ~ Fe{z,0oyn^/'^(9-9o), where the function ip{-,-, •) is

defined above.

To analyze the process D„(-), we follow a similar approach to that used in the

proof of Theorem 3.1 in Dette and Neumeyer (2003), though now an additional

term turns up due to the estimation of parameter ^o- We can rewrite ¡f{-,\ •)

as follows: oo y

^{x,y,z) = -f^{l-^-^){J{l-Fiv\x))dv- J F{v\x)dv} y - o o

oo y zf(z,9o)

y - c »

{J v{l — F{v\x))dv — J vF{v\x)dv}

= - ^ ( 1 - T f )( ( ) -y)- f f f ( ( ) + 'i^) - y'y

For y = m(x) + a{x)e, we have

ip{x, y, z) = ip{x, m{x) + a{x)e, z) = f{z, 0o){e + -(e^ - 1)). (3.5)

We also have for the bias part

/5n(^) = -hl{Jk{uydu}x{f{z,eo)j~-r[{m'afx){x)

+2{m'af'^){x) - 2{a'm'fx)ix)]dx + zf{z,eo) J ~-r[2{a'af'x)ix)

+ {a"afx){x) - {m'{x)ffx{x) - 3{a'{x)ffx{x)]dx}/2,

98

where we use the prime and the double prime to denote the first and second

order derivatives of the corresponding function, respectively. Observe that the

bias can be omitted if n/i^ = o(l).

By Assumption 8 and replacing (3.5) in D„(2;), we obtain

n

D„(z) = n-'/'Y.[I{ei<z)-F{z,eo) + f{z,9o){ei + ^{e¡-l))

= ñn{z)+Op{l),

where the last line defines the process D„(-). Obviously, under our assump

tions, E[Ún{z)] = 0. For s,i G M, straightforward calculation of the covari-

ances yields that Cov{D„(s),D„(i)} = F(min(s,i), ^o) - F{s,eo)F{t,eo) +

H{s,t,9o), where H{-,-,-) is defined in Theorem 1. Hence, the covariance

function of D„(-) converges to that of D(-).

To prove weak convergence of process D„(-), it suffices to prove weak conver

gence of D„(-). Let £°°{Q) denote the space of all bounded functions from a

set ^ to R equipped with the supremum norm \\v\\g = sup^gg N(p)l) and define

^ = {5^(-), 2:eM}as the collection of functions of the form

¿.(e) = I{e <z) + f{z, eo){e + ^{e'- 1)) - Fe{z, ^o)V(X e, Oo). (3.6)

With this notation, observe that

n

DM = n-'/'J2^6{e,)-E[5{e,)]) i=\

is an ^-indexed empirical process in £°°(^). Proving weak convergence of D„(-)

in £°°(^) entails that the class Q is Donsker. Following Theorem 2.6.8 of van

der Vaart and Wellner (1996, p. 142), we have to check that Q is pointwise

separable, is a Vapnik-Cervonenkis class of sets, or simply VC-class and has

99

an envelope function A(-) with weak second moment^. Using the remark in the

proof of the aforementioned theorem, the latter condition on the envelope can

be promoted to the stronger condition that the envelope has a finite second

moment.

Pointwise separability of G follows from p. 116 in van der Vaart and Well-

ner (1996). More precisely, define the class Qi = {Sz{-), z G Q}, which is a

countable dense subset of Q (dense in terms of pointwise convergence). For

every sequence z^ G Q with Zm \ z as m —> oo, which means that z^

decreasingly approaches z as m —> oo, and 6z{-) G G, we consider the se

quence Szmi') ^ Oi- First, for each e G M, the sequence 5^^(-) fulfils that

^zmi^) —^ ^zi^) pointwise as m —> oo, since Sî-) is right continuos for every

e G M. Second, ¿z^(-) —> ¿z(-) in L2(P)-norm, where P is the probability

measure corresponding to the distribution of e,

\\5.Js) - 5z{e)\\%, EE J \5zje) - 5Ae)\'f{v,eo)dv

< 3[F{zrn, eo) ~ F{z, Oo) + {f{z^, 9o) - f{z, eo)yE{s^)

H^mf{zm,eo) - zf{z,eo)fE{e^ ~ if/A]

+{Fe{^m,Oo) - F0{z,9o))'n{Fe{zm,eQ) - Fg{z,9o))

-2{Fe{zm, Oo) - Fe{z, 9o)yE{{I{e < z^) - I{e < z))i;{X, e, ô)}

- 2 ( / ( 2 ^ , ô) - f{z, 9o)){Fe{z^, 9o) - Fe{z, 9o))'E{Íj{X, e, 9o)e}

-2{z^f{zm,9o) - zf{z,9o)){Fg{z,n, 9o) - Fe{z,9o)yE{i:{X,e, 9o){e' - 1)}

—> 0 as 771 —> oo.

For 2; G M, we may rewrite (3.6) as Sz{e) = gi{e)+g{e), where gi{e) = I{e < z)

and f{z, 9o){e + |(e^ — 1)) — Fg{z, 9o)'ip{X, e, 9o). Let us now define the class

^Consider an arbitrary collection Xn = {xj, ..-.Xn} of n points in a set X and a collection C of subsets of X. We say that C picks out a certain subset A of Xn if ^ = C Pi X„ for some C € C. Additionally, we say that C shatters Xn if all of the 2" subsets of X„ are picked out by the sets in C. The VC-index V{C) of the class C is the smallest n for which no set Xn C A" is shattered by C. We say that C is a VC-class if V{C) is finite. Finally, a collection 5 is a VC-class of functions if the collection of all subgraphs {{x,t), g{x) < t}, where g ranges over Q, forms a VC-class of sets in A' X M. See van der Vaart and Wellner (1996, chapter 2.6) for further details.

100

of all indicator functions of the form C^ = {e i—^ I{e < d), d EW} such that

gi{-) G Ci. Consider any two point sets {ei, £2} C R and assume, without

loss of generality, that £1 < £2- It is easy to verify that Ci can pick out

the null set and the sets {ei} and {£1, £2} but can't pick out {£2}- Thus,

the VC-index V{Ci) of the class Cj is equal to 2; and hence Ci is a VC-

class. Note that •)/;(•,•,•) = ('i/'i(-, •, •); •••; V^mi';'; •))• We define the class of

functions C2 = {£ 1—> as + 6(£^ - 1) + Ci'ip'¡^{X,e,9o) + ... + Cmtp^{X,e,9o)\

a, b, Ci, ...,Cm G R} such that g2{-) G €2- By Lemma 2.6.15 of van der Vaart

and Wellner (1996) and Assumption 8, for fixed X G E and 60 E Q, the class

of functions C2 is a VC-class with ¥{62) < dim{C2) + 2. Finally, by Lemma

2.6.18 of van der Vaart and Wellner (1996), the sum of VC-classes builds out

a new VC-class. This yields the VC property of Q.

Recall that an envelope function of a class Q is any function a; 1—> A{x) such

that [¿^(a;)! < A{x) for every x and Sz{-). Using that f{-,0) is bounded away

from zero, sup^gjj |£/(£, ^)| < 00 and that F{-,-) has bounded derivative with

respect to the second argument, it follows that Q has an envelope function of

the form

A(£) = 1 + «!£ + Q;2(£^ - 1) - «S^ i^ , ^, 0),

where a = (1, «i, a2, «3)' is a (3 + m) x 1 vector of constants. Finally, note

that our assumption 8 readily implies that this envelope has a finite second

moment.

Additionally, under our assumptions it is readily checked that Q is pointwise

separable and is a VC-class, which completes the proof of part a. On the

other hand, under our assumptions sup^gjj \Fn{z) — FE{Z)\ = Op(l). Also, by

applying the mean-value theorem, F{z,9) = F{z,9) -f- F0{z,6**){6 — 9) for

some ^ 6 0 and 9** a mean value between 9 and 9. Clearly, under Ho,

9 = ^0) ciiid the last term is Op{n~^/'^) from assumption 8. Analogously,

under Hi, ^ = 6' , and the last term is Op(n~^/^) from assumption 9. Thus,

irrespective of whether HQ hold true or not, sup^gjj \F{z, 9) — F{z, 9)\ = Op(l).

101

Therefore sup^^^\Fn{z) - F{z,9)\ -^ sup^^^\Fe{z) - F{z,9)\. Under Hi,

sup^gjj \Fe{z) — F{z, ^*)| > 0 and this concludes the proof of part b.

For the second test statistic observe that C„ = J{Fn{v) — F{v,0)}'^dFn{v).

As before, the asymptotic distribution of this statistic can be obtained from

Proposition 1 and the uniform convergence of F„(-). •

Let us define q{-) in the same way as q{-) but replacing 6*0 by 9. The following

two propositions are required in the proof of Theorem 2.

Proposition A l : Suppose that assumptions 1-7 hold. Then:

a) If Ho holds and assumptions 8 and 11 are satisfied then:

\\q{u)-q{u)\\''f{u,9o)áu^o,{l).

b) If Hi holds and assumption 9 and 12 are satisfied then: +00

\\q{u)-q{u)\\^f{u,9Mu = 0p{l).

Proof of Proposition A l : Under assumption 7, g(-) is continuously difí'erentiable

with respect to 9. Thus, by a Taylor expansion we obtain

q{-) = q{-) + qe{-,9*)0-9^)/2,

where qe{-,9*) denotes the derivative of g(-) with respect to 9, evaluated at 9*,

and 9* lies between 9 and ^o- Observe that +00

\q{u) -q{u)\\'^f{u,9o)du

+ 00

1 " ^ " '12 / \\„ / n*\\\2 < ^ l l ^ - ^ o i r / \\qe{;9*)\\'f{u^9o)du

102

where the first inequality follows using \\q{-)-q{-W < lke(^ ^*)lñl^-ô| |V4,

and the last equality follows using assumptions 8 and 11. More precisely, under

assumption 8, it is straightforward to show that {9 — OQ) = Op{n~^^'^). Then,

11 — ô|P = Op{n~'^). This completes the proof of part a. The result of part

b is obtained along the same line of argument using assumptions 9 and 12. •

Proposition A2: Suppose that assumptions 1-7 hold. Then:

a) If Ho holds and assumption 8 is satisfied then:

n

sup|ln-^/^5^[I(e. > z){q{e,) ~ q{e,)} z6lR " ^

1=1

~ j{q{u)-q{u)}f{u,eo)du]\\ = 0,(1).

2

b) If Hi holds and assumption 9 is satisfied then:

n

snY>\\n'^'^y\I{e, > z){q{e,) ~ q{e,)}

+ 00

~ j{q{u)-q{u)}f{u,9^)du]\\ = 0^(1).

Proof of Proposition A2: As above, under assumption 7, q{-) is continuously

differentiable with respect to 6. Thus, by a Taylor expansion we obtain q{-) =

q{') + qe{',9*){6 — 6Q)/2, where qe{-,9*) denotes the derivative of g(-) with

respect to 9, evaluated at 9*, and 9* lies between 9 and ô- Thus, under HQ,

observe that

+ 00

n-"' T.tlV{^^ > ^mXî) - g(£.)} - / {Q{U) - q{u)}f{u, 9o)du] z

= n-'l^ Er=i[^(£. > A{n{e.) - g(£0} - m{e > z){q{e) ~ q{e)})]

103


n ' Er=i[^(^^ > ^^)Qo{ei,9*) - E{I{e > z)qe{e,e*)]n^l''0- eo)/2.

Under assumption 8, it is straightforward to show that v}/'^[6 — OQ) = Op{l).

On the other hand, the first term on the right hand side is Op(l) using some

uniform strong law of large numbers. This completes the proof of part a. The

result of part b is obtained along the same line using assumptions 9. •

Proof of Theorem 2: In the following reasoning we assume that the null hypoth

esis holds. Interchanging the variables, setting t = F{z,9o), we shall first

show that W„(-) = W„(-P^-'^(-,^o)) converges weakly to a standard Brown-

ian motion. Let D[0, b] {b > 0) denote the space of cadlag functions on [0, b]

endowed with the Skorohod metric^. Furthermore, define the linear mapping

r : £)[0,1] ^ ^[0,1] as follows

Í 1

T{ai-)){t)^ jq{F-\s,eo))'C{F-\s,eo))-'[lq{F-\r,9o))da{r)]ds.

0 s

Let

Q{t) = {Qi{t),Q2{t),Q,{t),Q4{t)y

= (Í, f{F'Hu 0o)), f{F-\t, 0o))F-'{t, 9o), Fe{F-\t, ^o))')',

so that q{F~^{-, OQ)) is the derivative of Q{-). It is easy to check that

nQii-)) = Qi{-), for / = 1,2,3,4. (3.7)

Prom C{F-'^{s,eo)y^C{F-\sJo)) = I4 we have C(F~i(s, ^0))^^ x

{}Q{r)dQ,{r)} = (1,0,0,0)'. Thus T{Q,{-)){t) = /Q(s) ' ( l ,0 ,0,0) 'ds = s 0

Qi{t). A parallel analysis establishes similar results for the remaining compo

nents of Q{-). ^See Section 14 of Billingsley (1968).

104




Let t = F{F-'{t),d). Thus V„(i) =. nV2[F„(i) - i] + ^V^fi _ ¿j. Note that

V„(-) can be rewritten as follows

V„(-) = n'^'[Fn{F~'{;9o))~Q^{-)]+ny'[Qi{-)~F{F~-\Q,{.),0,),9)]. (3.8)

Using the linearity of r(-), (3.4) and (3.5), routine calculations yield that

w„(-) = v„(-)-r(v„(-)).

Using Proposition 1, the linearity of r(-) and (3.4), it follows that

r(V„(z)) = r(V„(^)) + n-V2 n^^ [/(^^ 9,){^,^{X,,Y^ + Í3,J

Notice that the bias term /?„(•) = f{z, 9o)P-in + ^fi^^ ^o)P2n can be omitted if

nh^ = o(l). Using Proposition 1 again, we have

w„(-) = v„(-) - r(v„(0) + op(i) + 0(1).

Thus, as V„(-) converges weakly to a standard Brownian bridge B{-) on [0,1],

Wn(-) converges weakly to B{-) — T{B{-)), which is a standard Brownian

motion on [0,1] (see Khamaladze, 1981 or Bai, 2003, p. 543).

Let us now define W„(-) = W„(F"^(-,^o))- Under assumptions 7 and 8,

/(•, 9) = f{-,9o) + Op(l) (this follows applying a Taylor expansion). Addition

ally, propositions Al and A2 imply that Assumption Dl of Bai (2003) holds.

Hence, to prove that W„(-) = Wn(-) + Op(l), we follow exactly the lines of the

proof of Theorem 4 of Bai (2003), what completes the proof of a.

On the other hand, under Hi, the assertation can be deduced from the prob

ability limit of n~^/^W„(2;), which is

H(z) = F{z) ~ I q{u)C{ur'dn{u)f{u, 9,)du},

105




where

q{u) = (1, Uu, 9,)/f{u, e,), 1 + ufuiu, 9,)/f{u, 9,), fe{u, 9,y/f{u, 6,))',

C(«)EEXrg( r )g ( r )7 ( r , ^ . )d r , dn{u) = ¡^°°q{T)f{T)dT,

It can be easily checked that S{z) ^ 0 under Hi. The result of part b follows

from here. •

106




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Annals of Statistics 10, 647-671.

Billingsley, P., 1968. Convergence of Probability Measures. John Wiley, New York.

Campbell, J.Y. and Hentschel, L., 1992. No news is good news. Journal of Financial

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Davidson, R. and Flachaire, E., 2001. The Wild Bootstrap, Tamed at last. Queen's

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108




TABLES

TABLE 1; Propor t ion of Rejections of H, 0

5

0 1/12

1/9 1/7 1/5 1/3

0 1/12

1/9 1/7 1/5 1/3

0 1/12

1/9 1/7 1/5 1/3

/i(i)

0.005 0.015 0.049 0.072 0.144 0.369

0.015 0.044 0.079 0.126 0.216 0.484

0.041 0.074 0.110 0.174 0.269 0.569

/i(2)

n =

0.004 0.032 0.037 0.069 0.132 0.376

0.018 0.060 0.090 0.127 0.202 0.494

0.053 0.086 0.122 0.165 0.266 0.578

/i(3)

100

0.003 0.016 0.027 0.064 0.130 0.371

0.013 0.049 0.074 0.111 0.201 0.493

0.041 0.075 0.109 0.161 0.250 0.568

h^^^

a = 0.003 0.026 0.034 0.064 0.151 0.376

a = 0.015 0.059 0.075 0.105 0.238 0.481

a = 0.044 0.081 0.114 0.150 0.288 0.554

/i^i)

0.01 0.007 0.095 0.254 0.419 0.712 0.988

0.05 0.045 0.177 0.378 0.555 0.821 0.996

:0.10 0.083 0.237 0.460 0.629 0.873 0.998

/i(2)

n =

0.006 0.126 0.307 0.491 0.769 0.994

0.037 0.232 0.455 0.618 0.865 0.998

0.076 0.308 0.527 0.690 0.902 1.000

/i(3)

500

0.008 0.149 0.339 0.521 0.792 0.995

0.038 0.258 0.486 0.650 0.884 0.998

0.077 0.344 0.556 0.719 0.909 1.000

h^'^

0.006 0.162 0.357 0.535 0.803 0.996

0.040 0.280 0.499 0.663 0.892 0.998

0.079 0.359 0.570 0.736 0.917 1.000

109



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