forecasting growth with time series...

Journal of Forecasting, Vol. 14, 97-105 (1995)

FORECASTING GROWTH WITH TIMESERIES MODELS

DANIEL PENAUniversidad Carlos III de Madrid, Spain

ABSTRACT

This paper compares the structure of three models for estimating fumregrowth in a time series. It is shown that a regression model gives minimumweight to the last observed growth and maximum weight to the observedgrowth in the middle of the sample period. A first-order integrated ARIMAmodel, or 1(1) model, gives unifonn weights to all observed growths.Finally, a second-order integrated ARIMA model gives maximum weightsto the last observed growth and minimum weights to the observed growthsat the beginning of the sample period. The forecasting performance ofthese models is compared using annual output growth rates for sevencountries.

KEYWORDS ARIMA models; integrated processes; regression;stationary processes

INTRODUCTION

An important problem in modeling economic time series is forecasting the future growth of agiven time series. Assuming that a linear model is appropriate for the data, the procedures mostoften used are as follows:

(1) Detrend the observed data by regressing the observations on time, and use the residualsfrom this regression to build a stationary time series model. The series is forecasted byadding the values of the deterministic future trend and the forecast of the stationaryresidual.

(2) Differentiate the series, test for unit roots and if the series is assumed to be integrated oforder one (1(1)) build a stationary ARMA model in the first difference of the series.Typically models built in this way include a constant for many economic time series.

(3) Differentiate twice the series and build the ARMA model on the second difference of theprocess that is assumed to be 1(2). Then in most cases the 1(2) model does not include aconstant term.

The decision between these three procedures should be made by testing the number of unit rootsin the time series model. However, the available tests are not very powerful, specially for shorttime series, and therefore it is important to understand the consequences of using these models.CCC 0277-6693/95/020097-09 Received January 1994© 1995 by John Wiley & Sons, Ltd. Revised August 1994

98 Journal of Forecasting Vol. 14, Iss. No. 2

Let Z,, be the time-series data and let us call b,-Z,- Z,_, the observed growth at time /. It isshown in this paper that the estimate of future growth by the three procedures can be written as

where the coefficients to, are a weighting function, that is, w, > 0 and I w = 1. The next sectionof this paper proves that linear regression gives minimum weights to the last observed growthand maximum weights to the observed growth in the middle of the sample. The third sectionshows that an 1(1) model with a constant term gives a unifonn weight throughout the sample,that is, a>, = n'\ The fourth section shows that an 1(2) model gives maximum weight to the lastobserved growth and minimum to the oldest values. The fifth section compares these models inforecasting annual output growth for seven countries in the period 1960-91. The final sectioncontains some conclusions.

REGRESSION ON TIME

Let us call Z, the observed time series and let us assume for simplicity that the sample size isn = 2 m + l . Let r= {-m,..., 0, . . . ,+m}. Then the least squares estimator of the slope in theregression on time

Z,=ô+ît+u (1)

is given by

Z_,) (2)

Calling bi = Z,-Z,^^, for t=-m + l,...,m, the observed growth at each period, we note that

Z,-Z_,=

and, after some straightforward manipulations that are shown in Appendix 1, the estimate of theslope can be written as

' y( > + ^ - y ) (3)7=1

where the weights are given by

and add up to one. Therefore the estimated growth ^^ is a weighted mean of all the observedgrowths bj, such that the maximum weights are given to fo, and b^ that correspond to theobserved growth in the middle of the sample period, and the minimum weights are given to bând foi_™, the first and last observed growth.

The estimator (3) has an interesting interpretation. On the assumption that the linear model(1) holds, the 2m values fo, {t= -m+ 1, ...m) are unbiased estimates for^. These estimates are

Daniel Pena Forecasting Growth with Time-series Models 99

correlated and have covariances

,b,,,) = E[{b, -

Therefore, the covariance matrix of these 2m estimates is the Toeplitz matrix:

0

V =-a" la" .

0 -CT

-CT2 «_2

(5)

It is easy to show (Newbold and Granger, 1974) that given a vector B of unbiased estimators ofa parameter Q with covariance matrix V, the best (in the mean squared sense) linear unbiasedestimator of d given by

where 1' = (1 1... 1). Now, the inverse of the Toeplitz matrix (5) has been studied by Shaman(1969), who obtained the exact inverse of a first-order moving average process. As V can beinterpreted as the covariance matrix of a non-invertible (0=1) first-order moving averageprocess, then V"' = \v^\ will be given by

_'7

2m+11 = 1,..., 2m

and Vy = Vy,. Therefore

(2m+ 1)

2m 2m - 1 2m - 22 m - 1 2 ( 2 m - l ) 2(2m-2)2m - 2 2(2m - 2) 3(2m - 2)

123

2 m - 12m

(7)

It is proved in Appendix 1 that the estimator (3) can also be obtained by applying (6) to theunbiased but correlated estimates b,.

Suppose now that an ARMA model is fitted to the residuals of the regression model (1). Thenthe equation for the /i-steps-ahead forecast is

where n,[K) is the forecast of the zero mean stationary process fitted to the residuals. As for astationary process the long-run forecast converges to the mean, rf,(/i) -» 0, and the parameter jOjis the long-run estimated growth of the time series.

FORECASTING GROWTH WITH AN 1(1) MODEL

The ARIMA approach in modelling time series with trend is to differentiate the data and then fita stationary ARMA process. Assuming that a difference is enough to obtain a stationary series,


that is, the series is integrated of order one or 1(1), the fitted model is

VZ,=^ + n, (8)where n, follows an ARMA model

n, = IV,a,-i (9)

The process [a,] is a Gaussian white-noise process and the series {V,l converge, so that n,, is azero mean stationary process. Calling V to the covariance matrix of n,, the estimate of ^ inmodel (8) is given by the generalized least squares estimator

^=( l 'V- ' l ) - ' ( l 'V- ' f t ) (10)

where the vector b has components b, = VZ,. Assuming that n, is stationary and invertible it iswell known (see Fuller, 1976) that 6= \fn S fc, is asymptotically unbiased for ^ with variance

When n is large, the expected growth h periods ahead is given by

and it will be estimated by

where n,{h) is the /i-steps-ahead forecast of the stationary process n,. As for h large the rf,will go to zero, the mean value forecast, the long-run growth will be estimated by p . As

the long-run growth will be estimated simply by using the first and last observed values. Also,this estimate can be interpreted as a weighted average with unifonn weighting of the observedgrowths h,.

FORECASTING GROWTH WITH AN 1(2) MODEL

Some economic time series require differencing twice to obtain a stationary model. Then theseries is called integrated of order two or 1(2), and the model used is

V'Z, = «, (12)where

n,=IV,a,-, (13)

and the process [a,] is a Gaussian white-noise process. The series (V,} converge so that n, is azero mean stationary and invertible process. The /i-steps-ahead forecast from model (12) can bewritten

Z,(h) = $^" + l"h+n,(h) (14)

where ^Q'" and 4*'' depend on the origin of the forecast and n,{h) is the /i-steps-ahead forecastof the zero mean stationary process. Again, as the forecast n,(h) will go to zero, the long-rungrowth will be estimated by ^/". To understand the structure of ;3/" let us consider first thesimplest case in which n, follows an MA(1) process, n, = (1 -6B)a,. Then the forecast for anylag h is given by

Uh) = M + M"h (15)


because n,(l) is a constant. Let us obtain the form of j8f" as a function of the observed growthsVZ,. Assuming that the origin is t=T-l, then we can obtain y3o* "'* in model (15) using the twoforecasts Zj-.i (1) and Z^.i (2) as follows:

and substracting the first equation from the second.

which leads to

, + ebr.2 + 0' b,_3...] (16)

that is, the forecasted future growth is an exponentially weighted average of past observedgrowths.

In general, it is easy to show that

where the a^ coefficients depend on the moving-average structure of the process and behave like' structure of weights.

FORECASTING INTERNATIONAL GROWTH RATES

In order to illustrate the performance of the three models compared in this paper we haveapplied them to forecast gross national product for seven countries in the period 1960-91. Manysophisticated models have been used to forecast intemational growth rates and tuming points.See, for instance, Garcia-Ferrer et al. (1987), Zellner and Hong (1989), Min and Zellner(1993), and the references in these papers. Our objective here is not to build the 'best' model toforecast annual output growth, but to illustrate the forecast accuracy for different forecasthorizons of the three models analysed in the paper.

The data we use are given in Appendix 2 and represent gross national product in the period1960-91 for the United States, Japan, and the five largest countries of the European Union(France, Germany, Italy. Spain, and the United Kingdom).

Four models are used for the logarithm of the gross nationat product. The first (Ml) is thelinear regression on time given by equation (1). The second (M2) is a random walk with drift,that is model (8) with n, = a,. The third is an IMA (2, 1) model, that is, model (12) withn, = (l-6B)a,. The fourth is a random walk without drift on the rate of growth In Y,(Y,_i,where Y, is gross national product, and it is equivalent to the third model with ^ = 0. Note thatthe forecast from any of these four models is a linear trend. In the first the slope is obtained by agiven maximum weight to the observed growth in the middle of the sample; in the second theslope gives equal weight to all observed growths; in the third the weights decreaseexponentially; and in the fourth only the last observed growth is taken into account to build theforecast. Therefore, Ml gives, relatively, minimum weight to the more recent data whereas M4gives them the maximum weight.

We have used the period 1960-79 to fit the models and 1980-91 to check their forecastingperformance. Ml has been fitted by least squares to the 20 points, the parameter fi in M2 (seeequation (8)) has been estimated with equation (11), and the parameter 0 in M3 has been


estimated by maximum likelihood for the seven countries with the results given in Table L M4does not require any parameter estimation.

Table n presents the forecasting accuracy of the four models for three forecast horizons: one,two, and three steps ahead. The procedure used to build this table is as follows:

(1) The fitted models were employed to generate twelve one-step-ahead forecasts, eleven two-step-ahead forecasts, and ten three-step-ahead forecasts for the years 1980-91. ModelsMl and M2 were re-estimated to include all past data prior to the forecast origin. Theparameter d in M3 was always kept fixed to 0.7, the mean value for the seven countries(see Table I). We have checked that re-estimating the parameter 0 with each new dataimproves very few results, but makes model M3 more expensive in computing time. In thisway the updating of the forecast equation is very simple in all the models used. Finally,M4 does not need any parameter estimation.

(2) The error of the three types of forecasts were computed for each of the seven countries. Asan overall measure of accuracy we have used the mean squared error of the forecast. Thismeasure has been computed for the three forecast horizons: one (SI), two (S2) and three(S3) steps-ahead.

It can be seen that for the one-step-ahead forecast 1(2) models are the best in six of the sevencountries. Only for the United States are the 1(1) forecasts slightly better than those generatedby the 1(2) models. The 1(2) models are also the best for two and three-steps-ahead forecasts for

Table I. Maximun likelihood estimation of the moving average parameter

France Germany Italy Japan Spain UK USA

0.62 0.84 0.83 0.52 0.3 0.95 0.93

Table II. Mean squared error of the one, (SI), two, (S2), and three, (S3), steps-ahead forecasts for thefour models

SIS2S3

SIS2S3

SIS2S3

SIS2S3

Ml0.01200.01660.0220

Ml0.00840.01200.0162

Ml0.02520.03400.0435

Ml0.00320.00430.0054

]

M20.00030.00120.0026

M20.00030.00140.0031

M20.00060.00250.0051

M20.00060.00150.0024

FranceM3

0.00010.00050.0010

ItalyM3

0.00020.00110.0025

SpainM3

0.00020.00090.0022

USAM3

0.00070.00190.0033

M40.00010.00060.0011

M40.00020.00130.0037

M40.00020.00070.0019

M40.00070.00290.0065

SIS2S3

SIS2S3

SIS2S3

Ml0.00660.00930.0125

Ml0.03200.04320.0569

Ml0.00220.00290.0030

GermanyM2

0.00040.00150.0031

M30.0003O.OOU0.0024

JapanM2

0.00050.00190.0043

M30.00010.00040.0008

United KingdomM2

0.00070.00190.0031

M30.00070.00230.0043

M40.00020.00110.0031

M40.00020.00070.0014

M40.00050.00240.0061

Daniel Fena Forecasting Growth with Time-series Models 103

five of the seven countries. However, for the United States and the UK the 1(1) model providesbetter forecasts than the best 1(2) model. As can be expected, the forecasting performance of thedeterministic linear trend is very poor for all horizons.

The main conclusion from this exercise is that, on average, 1(2) models, even the simplestones that do not require any estimation, seem to be best for forecasting intemational growth inthe countries considered in this example.

CONCLUSION

We have compared three time-series models in this paper. The three models forecast futuregrowth by using a weighted average of the observed growths in the sample. Linear regressiongives minimum weight to the last observed growth and maximum weight to the centre of thesample period. This implies that, for instance, if we use this method to forecast next year'sgross national product (GNP) with a sample of 40 data, we are saying that the most informativeitem to forecast 1994 growth is the growth in 1974, whereas the last observed growth in 1993receives a weight equal to the one in 1954. If we use an 1(1) model, the growth is forecast byusing a uniform weighting in all the years in the sample. In the GNP example the observed 1993growth is as relevant as the one observed in 1960 or 1965 for forecasting 1994 growth. Thelogical requirement that the most relevant year to forecast GNP growth are the last observedgrowth is only accomplished by using an 1(2) model. In particular, an ARIMA (0, 2,1) modelleads to an exponentially weighting of last observed growths.

Many econometric papers and some well-known books on time series (see, for instance,Brockwell and Davis, 1987, p. 25) use least squares regression on time as an altemative todiflFerencing for removing a trend in a time series. However, the logical implications of bothprocedures are seldom analysed. It is important to stress that if a series follows an 1(2) modelbut we detrend it by least squares regression on time, the residuals from this fit do not provide,in general, a sound basis for fitting an ARMA model, and the forecast perfomance of theprocedure may be poor.

ACKNOWLEDGEMENTS

The author is very grateful to a referee for useful comments and to Fernando Lorenzo forproviding me with the data in Appendix 2. This research has been supported by GrantPB93-0232, DGICYT, Spain.

REFERENCES

Box, G. E. P. and Jenkins, G. M., Time Series Analysis, Forecasting and Control, San Francisco, CA:Holden-Day, 1976.

Brockwell, P. J. and Davis, R. A., Time Series: Theory and Methods, New York: Springer-Verlag, 1987.Fuller, W. A., Introduction to Statistical Time Series, New York: John Wiley, 1976.Garcia-Ferrer, A. et al., 'Macroeconomic forecasting using pooled intemational data', Joumal of Business

and Economic Statistics, 5 (1987), 53-67.Min. C. and Zellner, A., 'Bayesian and non-Bayesian methods for combining models and forcasts with

applications to forecasting intemational growth rates'. Journal of Econometrics, 56 (1993), 89-118.Newbold, P. and Granger, C. W. J., 'Experience with forecasting univariate time series and the

combination of forecasts', Joumal of the Royal Statistical Society A, (1974), 131-46.


Shaman, P., 'On the inverse of the covariance matrix of a first order moving average', Biometrika, 56(1969), 595-600.

Zellner, A. and Hong, C, 'Forecasting intemational growth rates using Bayesian shrinkage and otherprocedures', Joumal of Econometrics, 40 (1989), 183-202.

Author's biography:Daniel Pena is Professor of Statistics at the Universidad Carlos III de Madrid. He has held visitingpositions at the University of Wisconsin-Madison and the University of Chicago. His papers have appearedin JASA. Technometries, JBES, Biometrika and JRSSB, among other joumals. His research interests aretime series, Bayesian statistics and robust methods.

Author's address:Daniel Pena, Departamento de Estadistica y Econometria, Universidad Carlos in de Madrid, c/Madrid,126, 28903 Getafe, Spain.

APPENDIX 1

Using

andm - f+1

1=1 j = t i = l 2

we havem

0:=J]wj(bj + b,_j) (A2);=i

where

and the sum of all the weights tOj adds up to one:

- l m m

On the other hand, let 1' = (1 , . . . , 1) be a vector of 2m ones. Then using equation (7)

l'V"' = (m, (2m - 1), - (2m - 2),..., - (2m - / + 1),..., m)

andlm .

Therefore the estimate is given bylm

3f(2m-/+l) ,. i — m


However,

2m3(m - j + l)(m + j) ,

1) '"^

y 3(m+;)(m-y+l)jtt 2m(2m+l)(m+l)

in agreement with equation (A.2).

APPENDIX 2

Year

19601961196219631964196519661967196819691970197119721973197419751976197719781979198019811982198319841985198619871988198919901991

11222222223333333444444444445555

France

843 797945 321075 117186 077328 572439 832567 050687 399801 853997 718169 517320 319455 696642 994741 918729 687893 818031 470167 773300 036359 708411 784513 213548 895616 372700 143813 350917 827126 894320 876440 386501 546

Germany

856 480891 267933 451962 240

1 026 3401 081 4501 111 9601 108 7501 169 9901 257 0901 321 4001 361 1601 419 1201 488 1901 492 0801 471 2201 549 8001 593 9101 641 6401 709 1701 727 5101 730 5201 714 1401 740 9001 789 3501 823 1801 863 7701 890 2801 960 5102 027 3302 130 5002 209 640

Italy

299 823322 014341 991363 885374 066386 290409 409438 795467 514496 023522 366530 750545 077583 826615 520599 197638 619660 174684 480725 461756 197760 366761 991769 370790 036810 580834 262860 422895 397921 714942 271955 817

Japan

63 53068 75874 88384 09193 91099 376109 947122 132137 866155 068171 674178 993193 712208 484207 197213 123222 098232 566243 891257 390266 741276 287285 022292 721305 208320 419328 839342 340363 592380 735400 627418 347

911121314151616181920212325262627282828292829303031323435373839

Spain

997 409268 160388 150577 180255 234146 669244 187949 235067 357676 720512 121465 753215 030023 181429 127572 397450 289229 612642 478654 512027 187975 987429 760082 958524 354321 697323 992147 515910 027611 409973 365839 726

UK

201 042207 630209 774217 831229 627235 432239 850245 261255 247260 529266 433271 734281 202301 667296 503294 236302 201309 155320 439329 601322 432318 195323 576335 406343 232356 096370 860388 750405 512414 053416 206406 842

11122222222223333333333334444444

USA

864 700854 445952 558096 600213 900336 100473 400536 400641 400715 800714 400791 800934 400086 600068 400043 500193 800336 400490 000579 200563 800632 900551 800675 000900 700016 649119 600243 300410 600525 300555 000496 100

forecasting growth with time series...

Documents