special topics in applied econometrics 1) linear panel models · m. bluhm special topics in applied...

Special Topics in Applied Econometrics1) LINEAR PANEL MODELS

Marcel Bluhm

Wang Yanan Institute for Studies in EconomicsXiamen University

Antwerp University, 13 - 17 February 2012

1) Linear Panel Models:→ Agenda

⇒ Introduction

The Pooled Model

Unobserved Effects Panel Data Models

M. Bluhm Special Topics in Applied Econometrics: 1) LINEAR PANEL MODELS 2/51

1.1) Introduction

General linear model

yit = αit + xitβit + uit (1)

where i = 1, 2, · · · ,N and t = 1, 2, · · · ,T

Coefficients and slope vary over individual and time

Model too general and not estimable



Introduction

⇒ The Pooled Model

Unobserved Effects Panel Data Models


1.2) The Pooled Model

yit = α + xitβ + uit (2)

Assumptions:

E (x′tut) = 0,t = 1, 2, · · · ,T

’Regressors are contemporaneously uncorrelated with the error term’

rank[∑T

t=1 E (x′

txt)] = k’Regressors are not perfectly linear dependent’

E (u2t x′txt) = σ2E (x

′

txt), t = 1, 2, · · · ,T ; σ2 = E (u2t )’Homoscedasticity’

E (utusx′txs) = 0, t 6= s; t, s = 1, 2, · · · ,T

’Conditional covariance of errors across time periods are zero’



The pooled ordinary least squre (OLS) estimator is given by

θ =(

X′X)−1

X′y (3)

where X and y are the stacked explanatory variables xkit and dependendvariable yit over time, respectively:



y =

y11...

y1Ty21

...y2T

...yN1

...yNT

NTx1

θ =

α

β1β2...

βK

K+1x1

X =

1 x111 · · · xK11...

......

...1 x11T · · · xK1T1 x121 · · · xK21...

......

...1 x12T · · · xK2T...

......

...1 x1N1 · · · xKN1...

......

...1 x1NT · · · xKNT

NTxK+1

wherek = 1, · · · ,K denote different regressorsi = 1, · · · ,N denote the cross-section dimensiont = 1, · · · ,T denote the time dimension



The estimator for the asymptotic variance of β is given by

Avar(θ)

= σ2(

X′X)−1

(4)

where σ2 is the usual OLS variance estimator from the pooled regression1

NT−K−1 u′u with

u =

u11...

u1T...

uN1...

uNT



A heteroscedasticity and serial correlation robust estimate of β can beobtained as

Avar(θ) = (N∑i=1

X′iX)−1(N∑i=1

X′i ui u′iX)(

N∑i=1

X′iX)−1 (5)

where the residuals ui are the T × 1 pooled OLS residuals forcross-section observation i :

ui =

u11...

u1T

NTx1

Whether a robust variance estimator is necessary can be infered viatesting for serial correlation and heteroscedasticity


1.2) The Pooled Model⇒ Testing for Serial Correlation

Estimate Equation (2) by pooled OLS and obtain the errors uit

Estimate Equation (2), including the lagged errors as additionalexplanatory variable in xit

yit = α + xit β + uit (6)

where xit contains the lagged error term

Compute heteroscedasticity robust variance for θ =

[α

β

]

Avar(ˆθ) = (X′X)−1X′ΨX(X′X)−1 (7)

where Ψ = diag(û211,û212, ...,

û21T ,û221, ..., ...,

û2NT )

If the usual t-statistic for the coefficient of the lagged uit issignificant, serial correlation is present


1.2) The Pooled Model⇒ Testing for Heteroscedasticity

The null hypothesis (H0) can be stated asE (u2t |xt) = σ2, t = 1, 2, ...,T

Obtain R2c from the POLS regression of u2it on a constant and

hit ; t = 1, ...,T ; i = 1, ...,N:

u2it = const + hitτ + εit (8)

where:

hit contains elements of xit , as well as squares and cross-products ofelements of xitR2c = 1− ε′ ε

ˆε′ˆεwith ε the stacked error terms from Equation (8) and ˆε

the stacked error terms from regressing u2it on a constant only

Under H0 the test statistic NTR2c has a χ2

Q distribution, with Q thenumber of elements in hit


1.2) The Pooled Model⇒ Digression: The Solow Growth Model

Solow (1956) obtained Nobel prize for model

Simple framework that helps investigating the proximate causes andmechanics of the process of economic growth

Model contains abstract representation of a complex economy.Supply of goods based on classical production function: Y = F (K , L)

One theoretical implication of the model: ’convergence’

Solow model can be used to investigate economic growth over time

In particular, regression analyses have been extensively used toconfront theory with empirics

Barro-type regressions to investigate ’convergence’ and the causes ofeconomic growth


1.2) The Pooled Model⇒ Digression: The Solow Growth Model (ctd.)

The basic Solow model with constant population growth and laboraugmenting technological change with a Cobb-Douglas productionfunction can be used to derive the following equation [see, forexample, Acemoglu (2008)]

yit − yi ,t−1yi ,t−1

≈ g − τ(log yi ,t−1 − log y∗i ,t−1)

where

y is outputy∗ is equilibrium output to which economies convergeg is technological progressτ = (1− α)(δ + g + n), with α a parameter from the Cobb-Douglasproduction function, δ the depreciation of capital, and n the populationgrowth



Assuming y∗ constant, the equation can be put into the followingestimable framework

yit − yi ,t−1yi ,t−1

= yit ≈ c + τ log yi ,t−1 + εit

where c = g + log τy∗

In case of convergence, τ should be smaller than zero

This model implication can be empirically tested via estimating theabove equation


1.2) The Pooled Model⇒ Digression: Introduction to Matlab

Introduction to Matlab


1.2) The Pooled Model⇒ HANDS-ON

Hands-On 1



The estimated growth equation is usually refered to as ’unconditionalconvergence’ (Sala-i-Martin, 1992), that is, countries convergeregardless of differences in individual characteristics and policies

This may be too demanding, since it implies that the income gapbetween any two countries irrespective of their institutionalenvironment, investment behavior, policies etc. should shrink overtime

If countries differ with respect to these factors, the Solow modelpredicts that they converge to different levels y∗

A more appropriate regression equation may thus take the form

yit ≈ ci + τ log yi ,t−1 + εit



Introduction

The Pooled Model

⇒ Unobserved Effects Panel Data Models


1.3) Unobserved Effects Panel Data Models⇒ Introduction

Key assumption in previous sub-section: E (x′tut) = 0; t = 1, 2, ...,T

This assumption is very likely to be violated in the model fromEquation (2)

Panel data models explicitely containing a time constant and anunobserved effect can solve/alleviate the omitted variables problem

Key issue: is the unobserved effect uncorrelated with the explanatoryvariables?


1.3) Unobserved Effects Panel Data Models⇒ Introduction (ctd.)

yit = ci + xitβ + uit (9)

Implicit assumption in Equation (9):Each cross-section features a time constant unobserved variable(’unobserved effect’)

The unobserved effect captures features of individual cross-sectionsthat are given and do not change over time (e.g. socio-culturalfactors such as religion)

To consistently estimate the parameter vector of interest, β, it iscrucial whether the unobserved effect, ci , is (Fixed Effects model) oris not (Random Effects model) correlated with the vector ofexplanatory variables, xit for all t


1.3) Unobserved Effects Panel Data Models⇒ Random Effects (RE) Methods

yit = x′itβ + vit (10)

where vit = uit + ci

Under the following assumptions the coefficient vector of interest can beestimated consistently and efficiently by feasible generalized least squares(FGLS) within an RE framework:

E (uit |xi , ci ) = 0 for t = 1, ...,T ; with xi = (xi1, xi2, ...xiT )’Strict exogeneity of the explanatory variables and the unobservedeffect’

E (ci |xi ) = E (ci ) = 0’Orthogonality between ci and Xi ’. The second part of thisassumption is without loss of generality if xit contains an intercept



rank E (X′iΩ−1Xi ) = K

’Regressors are not perfectly linear dependent’where the elements of Ω are determined by two further assumptionson the idiosyncratic errors:

E (u2it) = σ2u; t = 1, 2, ...,T

’Constant unconditional variance across t’E (uituis) = 0; ∀t 6= s’Idiosyncratic errors are serially uncorrelated’

E (uiu′i |xi , ci ) = σ2uIT

Orthogonality of outer product of idiosyncratic errors with respect toxi and ci implies the previous assumptions (constant variance andserial uncorrelatedness)

E (c2i |xi ) = σ2c’Homoscedasticity of the unobserved effect’



For the FGLS procedure, define σ2v = σ2c + σ2uIf we have consistent estimators of σ2u and σ2c we can form

Ω = σ2uIT + σ2c ιT ι′T

where ιT is a vector of ones of dimension T × 1

The FGLS-RE estimator is given by

βRE = (N∑i=1

X′iΩXi )−1(

N∑i=1

X′iΩyi ) (11)

The asymptotic variance Avar(βRE ) is given by

(N∑i=1

X′iΩ−1Xi )

−1N∑i=1

X′iΩ−1vi v

′iΩ−1Xi (

N∑i=1

X′iΩ−1Xi )

−1 (12)



To implement the RE procedure, we need σ2c and σ2u

Given the previous assumptions we have σ2v = σ2c + σ2u withσ2v = T−1

∑Tt=1 E (v2it) ∀i

Estimation procedure:1 Estimate Equation (10) using pooled OLS2 Obtain the pooled OLS residuals, ˆvit3 Compute consistent estimator of σ2

v : σ2v = 1

NT−K

∑Ni=1

∑Tt=1

ˆv2it

4 Compute consistent estimator of σ2c :

σ2c = 1

NT (T−1)/2−K

∑Ni=1

∑T−1t=1

∑Ts=t+1

ˆvit ˆvis

5 Form Ω and obtain the parameter vector of interest using Equation(11)

6 Use Equation (12) to compute Avar(βRE )



If the idiosyncratic errors are heteroscedastic and serially correlatedacross t, a more general estimator Ω can be used:

Ω = N−1N∑i=1

ˆvi ˆv′i

where the ˆvi are the pooled OLS residuals

If N is not several times larger than T , an unrestricted FGLS featuresundesireable finite sample properties whereas the RE aproach onlyrequires estimation of two variance parameters



If the RE assumptions hold but the model does actually not containan unobserved effect, pooled OLS is efficient

To test H0 : σ2c = 0, the following asymptotically standard normallydistributed test statistic can be used:

TS =

∑Ni=1

∑T−1t=1

∑Ts=t+1 vit vis

[∑N

i=1(∑T−1

t=1

∑Ts=t+1 vit vis)2]

12

(13)


1.3) Unobserved Effects Panel Data Models⇒ Digression: Manifesto for a Growth Econometrics

DISCUSSION


1.3) Unobserved Effects Panel Data Models⇒ HANDS-ON

Hands-On 2


1.3) Unobserved Effects Panel Data Models⇒ Fixed effects (FE) Methods

yit = xitβ + ci + uit (14)

Assuming that ci and xit are uncorrelated, the RE approach puts ciinto the error term

FE methods allow for ci to be arbitrarily correlated with xit

This robustness comes at a price: we cannot include time constantfactors in xit


1.3) Unobserved Effects Panel Data Models⇒ Fixed effects (FE) Methods (ctd.)

To allow for estimation within an FE framework the following assumptionsneed to hold:

E (uit |xi , ci ) = 0, for t = 1, 2, ...,T ; with xi = (xi1, xi2, ...xiT )’Strict exogeneity of the explanatory variables and the unobservedeffect’

rank(∑T

t=1 E (x′it xit)) = rank(∑T

t=1 E (X′i Xi )) = K’Regressors are not perfectly linear dependent’with xit the time demeaned explanatory variables: xit = xit − xi

E (uiu′i |xi , ci ) = σ2uIT

’Homoscedasticity and serial uncorrelatedness of the idiosyncraticerrors’



Under the given assumptions the FE model can be estimatedconsistently and efficiently by applying pooled OLS after a fixedeffects transformation of Equation (14)

The fixed effects transformation removes the unobserved effect fromEquation (14)

Alternative transformations are available, for example, afirst-differencing transformation



Estimation of the FE model proceeds in three steps:

1 Average Equation (14) over t = 1, 2, ...,T

yi = xiβ + ci + ui (15)

with yi = T−1∑T

t=1 yit2 Subtract Equation (15) from Equation (14) on the FE transformed

equation to obtain the FE transformed equation

yit − yi = (xit − xi )β + uit − ui

oryit = xitβ + uit ; t = 1, 2, ...,T

3 Use the pooled OLS estimator from Equation (3) on the transformedequation to obtain βFE



To obtain the asymptotic variance of the estimated parameter vector,Avar(βFE ) one proceed in two steps:

Estimate the variance of the idiosyncratic errors

σ2u =1

N(T − 1)− K

N∑i=1

T∑t=1

u2it

where uit = yit − x′itβFE ; t = 1, 2, ...,T ; i = 1, 2, ...,N

The variance-covariance matrix is given by

Avar(βFE ) = σ2u(N∑i=1

X′i Xi )−1 (16)



If serial correlation is present in the idiosyncratic errors the variance matrixestimator in Equation (16) will be improper.To test the idiosyncratic errors for serial correlation one proceeds asfollows:

Run the regression uiT on ui ,T−1; i = 1, 2, ...,Nwith uiT , ui ,T−1 the last two time periods of cross-section i

Test H0 : ’No serial correlation:’ δ = − 1T−1 (see Wooldridge (2004),

p. 275) where δ is the coefficient of ui ,T−1. Under the previousassumptions the usual t-statistic is asymptotically normally distributed



If serial correlation is present, the asymptotic variance estimator needsto be adjusted. The robust variance estimator of βFE is

Avar(βFE ) = (X′X)−1(N∑i=1

X′i ui u′i Xi )(X′X)−1 (17)

where ui = yi − X′βFE



Alternatively to computing a robust variance estimator of βFE we can relaxtwo previous assumptions:

Instead of E (uiu′i |Xi , ci ) = σ2uIT we assume E (uiu

′i |Xi , ci ) = Λ, with

Λa T × T positive definite matrixan unrestricted, constant covariance matrix

Instead of rank(E (X′i Xi )) = K we assume rank(E (X′iΩXi )) = K ,with Ω a (T − 1)× (T − 1) unrestricted, positive definite matrix

Note:

Due to a rank deficiency (from first differencing) under the adjustedassumption (see Wooldridge (2002) p.276f) one time period needs tobe dropped from the observationsThe assumption made on Λ carries through to Ω



Fixed effects generalized least squares is carried out in the following steps:

Estimate βFE

Drop the last time period for each i and compute the N residualvectors ûi = yi − Xi β with dim(ûi ) = (T − 1)× 1

A consistent estimator for Ω is Ω = N−1∑N

i=1ûi

û′iThe FEGLS estimator is given by

βFEGLS = (N∑i=1

X′iΩ−1Xi )

−1(N∑i=1

X′iΩ−1yi ) (18)

The asymptotic variance is then given by

Avar(βFEGLS) = (N∑i=1

X′iΩ−1Xi )



Hands-On 3


1.3) Unobserved Effects Panel Data Models⇒ First Differencing (FD) Methods

yit = xitβ + ci + uit (19)

Instead of using the FE transformation one can also estimate β inEquation (19) by FD methods

Model and interpretation of β are exactly as in the previoussub-section


1.3) Unobserved Effects Panel Data Models⇒ First Differencing (FD) Methods (ctd.)

To allow estimation within an FD framework, the following assumptionsneed to hold:

E (uit |xi , ci ) = 0; t = 1, 2, ...,T ; with xi = (xi1, xi2, ...xiT )’Strict exogeneity of the explanatory variables and the unobservedeffect’

rank(∑T

t=2 E (∆x′it∆xit)) = K’Regressors are not perfectly linear dependent’with ∆xit = xit − xi ,t−1

E (eie′i |xi ,1, ..., xi ,T , ci ) = σ2e

’The first differenced errors are serially uncorrelated andhomoscedastic’with eit = ∆uit ; t = 2, ...,T



Assuming eit to be serially uncorrelated implies that uit follows arandom walk: uit = ui ,t−1 + eit

A random walk has strong serial dependence. This assumption is thusopposite to the assumption on the error terms under FE methods

Under the previous assumptions βFD can be estimated consistentlyand efficiently by the pooled OLS estimator from the regression

∆yit on ∆xit ; t = 2, ...,T ; i = 1, 2, ...,N (20)

with ∆yit = yit − yi ,t−1

The asymptotic variance of the FD estimator is given by

Avar(βFD) = σ2e (∆X′∆X)−1

with σ2e = (N(T − 1)− K )−1∑N

i=1

∑Tt=2 e

2it ; eit = ∆yit −∆xit βFD



Under the previous assumption the errors eit = ∆uit should be seriallyuncorrelated

To test whether this assumption holds, the pooled OLS residuals fromthe regression in Equation (20) can be employed for the followingregression:

eit = ρei ,t−1 + εit ; t = 3, 4, ...,T ; i = 1, 2, ...,N

The null hypothesis H0 : ρ = 0 can be tested with the usualt-statistic



If the assumption of serially uncorrelated error terms is violated, thefollowing robust asymptotic variance estimator can be applied:

Avar(βFD) = (∆X′∆X)−1(N∑i=1

∆X′i ei e′i∆Xi )(∆X′∆X)−1

with ∆X the N(T − 1)× K matrix of stacked first differences of xit

When T = 2, FE and FD are identical

FD easier to implement

When T > 2 the choice between FE and FD hinges on theassumption about the idiosyncratic error terms


1.3) Unobserved Effects Panel Data Models⇒ Hausman Test

Since FE is consistent when ci and xit are correlated, but RE isinconsistent, a statistically significant difference is interpreted asevidence against the RE model

To test the null hypothesis H0 : ′βRE is the correct specification’, thefollowing test statistic is asymptotically χ2

M distributed, with M thefirst dimension of the vector of parameter estimates

H = (βFE − βRE )′[Avar(βFE )− Avar(βRE )]−1(βFE − βRE )

Note that βRE does not contain the coefficients on time-constantvariables or aggregate time variables



Hands-On 4


1.3) Unobserved Effects Panel Data Models⇒ Additional Remarks

To implement the notion of ’conditional convergence’ in a moresophisticated empirical way, Barro (1991) and Sala-I-Martin (1997)model ci as a function of, among other things, the schooling rate,fertility rate, investment, government consumption, inflation,openness, and democracy.

In regression form this can be written as

yit = xitβ + τ logyi ,t−1 + εit

where x contains the variables mentioned above

In a latter part of the lecture we will try to explain countries’ outputgrowth with some of these variables

This kind of equations has been used to estimate the ’determinants ofgrowth’


1.3) Unobserved Effects Panel Data Models⇒ Additional Remarks (ctd.)

With all models seen so far care needs to be taken to distinguishcorrelation from causationExample: schooling and fertility rate

Besides the simplicity of regression framework and attractiveness as abridge between theory and data, several problematic features thusneed to be taken into consideration:

Endogeneity of variablesMeasurement error of dataIn the Solow model, investment is key; if controlled for, other variablessuch as schooling should have no explanatory powerThe growth regression framework is derived from a closed-economySolow model. In reality, economies trade and are no islands.


1.3) Unobserved Effects Panel Data Models⇒ Additional Remarks (ctd.)

Furthermore, noting that yit ≈ log yit − log yi ,t−1, growth regressionsmay be written more naturally in an autoregressive framework as

log yit = ci + (1 + τ)log yi ,t−1 + εit

We will investigate this equation at a later point in the lecture

Including the x variables in the equation above can uncover thecorrelations between these variables and countries’ level of output


SUMMARY

The pooled panel model

yit = α + xitβ + uit

The unobserved effects panel model

yit = αi + xitβ + uit

Fixed effects?Random effects?


SUMMARY (ctd.)

The Solow growth model and Barro-type growth regressions

yit = αi (Xitβ) + τyi ,t−1 + εit

Pitfalls:

endogeneitypoor data qualityparameter heterogeneity


Digression: The Growth of Growth Theory

DISCUSSION


special topics in applied econometrics 1) linear panel models · m. bluhm special topics in applied...

Documents