Macroeconomic forecasting with a vector arima model

143

international Journal of Forecasting 1 (1985) 143-150 North-Holland

MACROECONOMIC FORECASTING A case study of the finnish economy

WITH

A VECTOR

ARIMA

MODEL

*

Lars-Erik GLLER Ministry

of Finance,

SF-001

71 Helsinki.

F!inland

The vector ARIMA (VARIMA) model is a multivariate generalization of the univariate ARIMA model. VARIMA can accomodate assumptions on exogeneity and on contemporaneous relationships. Exogeneous forecasts and non-zero future shocks make it possible to generate alternative forecasts. In a case study VARIMA well describes developments in the 1970’s and successfully competes with judgemental methods and ARIMA in providing a general outlook of the early 1980’s. Keywords:

VARIMA, Macroeconomic forecasts, Causal assumptions, Adjustment of forecasts.

1. Introduction The vector ARIMA (VARIMA) model [Tiao and Box (1981)] is a new method that allows for modeling lagged correlations between variables. It will be shown here that this generalization of univariate ARIMA [Box and Jenkins (1976)] can be brought closer to econometric forecasting techniques [cf. Granger and Newbold (1977, ch. 7)]. In section 2 an exogeneous variable is introduced through a transfer function [cf. Koreisha (1983)]. This makes it possible both to generate alternative forecasts and to define contemporaneous relationships. More generally, contemporaneous relationships can be introduced when unidirectional causality can be assumed at lag zero. It is also shown how the MA form of the VARIMA model lends itself to economic interpretation. A method for adjusting forecasts according to knowledge, or assumptions concerning the forecasting period, is discussed. In section 3 a case study is presented with macroeconomic data from Finland. The interim multipliers show that the VARIMA model makes economic sense. Faced with a real forecasting situation it performs quite well compared to univariate ARIMA and available judgemental forecasts. The results are summarized in section 4.

2. The VARIMA

model

The VARIMA model [Tiao and Box (1981) Hillmer et al. (1983)] is a generalization of the univariate case to consideration of multiple time series that are both auto- and cross-correlated. The model can be written:

* This research was supported by the U.S. Educational Foundation in Finland and by the Ministry of Finance. Finland. I am also indebted to the University of Wisconsin-Madison. Statistics Department. for offering me the use of the WMTS-1 program as well as working facilities during the 1980-1981 academic year. Special thanks go to Professor Timo L. Terasvirta for valuable remarks in connection with the introduction of contemporaneous relationships in VARIMA models. 0169-2070/85/$3.30

0 1985, Elsevier Science Publishers B.V. (North-Holland)

L.-E. iiller / Mocroeconomi~/orecostinR with LI vector ARIMA

144

model

where stationary (or stationarized) Z, and a, are column vectors containing k variables each and polynomials in the backshift operator B have k X k matrices +,, i = 1,. . ,p and 6,, j = 1,. . . , ( coefficients, a, is white noise (0,Z) and both cp and 8 have all their roots outside the unit circle. Suppose a model (1) has been adequately specified and estimated from data on two stationary t series z,, and z2, (k = 2) and that at lag zero there is strong correlation between the two serie residuals. (Note that only lagged correlations between the original variables have been taken car by the model.) This information can be exploited if one is prepared to make assumptions at causality. A simple two variable case arises if model (1) is AR(l) with +,2 = 0. Assume that a,, a,, are strongly correlated and that it is known in advance that causality for lag zero must run fl z,, to z2, but not the other way round. Then z,, is exogeneous for z2, and z,, can be introduced lag zero in a transfer function for z2 [cf. Koreisha (1983)J. A way of introducing z,, for lag zen equation for z2 even if C#Q~ # 0 is the following [cf. Hillmer et al. (1983)]. First estimate model containing only lagged relationships. Still assuming the same direction of causality for lag zero reg the estimated residuals ci,, on ci,, ( a = estimated). Let the regression coefficient be &!,. Multiply estimated VARIMA model from the left by

1 0 [-+;,11’

yielding

where

This result is easily generalized to any internally consistent causal chain at lag zero among obser variables in more general VARIMA models. Although the VARIMA model is primarily designed for forecasting one would like the relati ships in it to make economic sense [cf. Klein and Moore (1983)]. The MA form of model (1)

where JI( B) = +-I( B)8( B), shows, through the entries in $4 B), how the system reacts to a shock in any of the variables z,,, i = 1,2,. . .,k. The cumulative sums of the (i,j)th elements of the matrices are called ‘interim multipliers’ [see, e.g., Harvey (1981), p. 231)]. They give an idea of t the model describes the data. The starting point of a VARIMA forecast [cf. Tiao and Box (1981)] is the latest camp observation vector. Now, the publication speed of economic time series varies. A way to help VARIMA model to pick up the right course by introducing all information at the time of being is following: (i) (ii)

Make a one step ahead forecast from the latest complete observation vector. Substitute model forecasts for available observed figures. The difference A I a,+, =z,+1 -Z,+

Ill7

where i,, ,,, is the model forecast for t + 1 made at t, can be interpreted as a non-zero future shock/residual. (iii) Repeat (i). then (ii) until all observations have been used after which forecasting continues according to the standard VARIMA technique. More generally, the shock interpretation allows for subjective adjustment of forecasts, common among practical econometricians [cf. Klein (1984)]. A forecaster may for good reasons disagree with the model forecast for some future time point, or perhaps he/she wants to see, as an alternative. what happens if a plausible shock hits a variable at a certain point of time in the future. For another way of introducing prior information in forecasts using mixed estimation [Theil and Goldberger (1961 )I. see Cholette (1982).

3. A case study of the Finnish

economy

The main purpose of this study was to use a VARIMA model as a complement to other techniques in short-term macroeconomic forecasting. The problems of specification of VARIMA models increase with the number of variables being modeled jointly. Consequently the set of variables should be as small as possible but still reflect the most interesting features of the economy, from a policy point of view. Any forecasting model of a small economy, one third of whose production is exported, can be expected to profit from a relationship between foreign demand and exports. Here a one-sided relationship is reasonable since the impact of a small economy on foreign demand is negligible [cf. oiler (1978)]. Thus the first two variables to be included in the model were foreign demand and exports and the prior assumption was made that foreign demand is exogeneous. Another foreign factor that has a major influence on economic development is the price of oil. Again an exogeneity assumption is natural: oil prices affect the Finnish economy but not the other way round. The domestic economy is described by unemployment, wages and investment. Employment is generally seen as one of the most important target variables of economic policy and the relationship between wages and unemployment has been a hot topic in economics for a very long time. Both in economic theory and in policy investment is regarded as crucial for present and especially future production and employment. The model thus contains six variables, two of which (foreign demand and oil prices) are regarded as exogeneous to the rest. In order to simplify specification two further assumptions were made: foreign demand directly affects only exports and oil prices affect only foreign demand, exports and wages (not investment or unemployment). Quarterly observations for the period 1970:lQ through I980:2Q were used for estimation, leaving observations 1980:3Q through 1982:4Q for post sample checking. All variables were first transformed into logarithms, then differenced once to remove trends and level shifts (cf. fig. 2). All but the oil price and the wage variables contain seasonal variations. For the short estimation period the logarithms of the seasonals can be considered as nearly constant and were removed by changing to deviations from (logarithmic) seasonal averages. The following model was obtained, where lower case letters signify transformed variables: Foreign

demund

Y, = 0.009 - O.l03z,-, (0.007) (0.037)

+ b,,

(6.1)

L. -E. Oiler / Macroe~o~~omic/orecasrin~

146

wifh o uector A RI MA model

Exports x, = - 0.270x,-, (0.119)

-0.2ooz,_, (0.046)

+1.417y, +0.215u,-, (0.412) (0.064)

(6.2

+ 81,

Relative wages w, = O.l04z, 5 -O.l73u,-, (0.027) (0.038)

(6.:

+0.323a, ,,-, +0.249ri,,,mz + ii,, (0.069) (0.072)

Investment i, =0.112x,-, (0.081)

-O.O84u,_, + ci,, -0.4336,,,-, (0.150) (0.054)

(6.

Unemployment u, = 0.541u,-, -0.260x,-, (0.131) (0.181)

+0.754w,-, (0.363)

(6.t

+- ci,,,

where Z Y X W

= = = =

the price of oil in the Persian Gulf, foreign demand = industrial production in European OECD ’ countries, exports = volume of exports to OECD countries, relative wages = (Finnish wage index)/(export price index of industrialized index of Finnish currency), I = investment, volume, U = unemployment rate.

countries)

x

(pric

The numbers in parentheses are estimated standard errors. The model is adequate since there ar no statistically significant auto- or cross-correlations between residuals. The standard deviations o residuals (s) are given in the first row of table 2. Table 1 shows contemporaneous correlation between model residuals. Only the negative correlation between unemployment and investmen residuals is statistically significant. Could this information be exploited as outlined in section 2, b; arguing that the relationship is one-sided at lag zero? There seems to be little risk in assuming that at investment project is launched or discontinued irrespective of the current employment situatior Previous employment, as an indicator of capacity utilization, may according to the model have som impact but for the current situation time is too short to alter decisions. On the other hand, starting o finishing a project can be expected to have an immediate employment effect. Proceeding as outlined in (2). (3) and (4) the two last equations (6.4) and (6.5) were multiplied b, the matrix

’ Organisation

for Economic

Co-operation

and Development

147

L.-E. 6llkr / Macroeconomic forecasting with a vector AROMA model Table I Coefficients of contemporaneous residual correlation in the VARIMA Relative wages

Exports Exports Relalive wages Investment Unemployment

1 0.02 0.10 0.15

1 - 0.26 -0.10

model. Investment

Unemployment

1 - 0.37

1

where & = -0.775 is the estimate of the regression coefficient when regressing unemployment ^ It is easy to verify that only (6.5) changes: residuals ci,, on investment residuals u4,. u,=O.606u,-,

- 0.775i,-

0.260x,-,

- 0.087x,-,

+ 0.754w,-,

- 0.336ri,,,-,

+ ci,,.

(6.5’)

Substituting (6.5’) for (6.5) we computed the MA form (5) of model (6). For lack of space we present in fig. 1 only the interim multipliers (cumultive responses) to shocks in oil prices. In order to avoid further causal assumptions no overall orthogonalization of residuals was attempted as in Sims (1980). The curves in fig. 1 show how rising oil prices resulted in compensatory wage drift which in turn led to unemployment. Forecasts were generated both with model (6.1-5) and with the modified model (6.1-5’). The former was slightly more accurate (ex post) and was chosen for closer study (fig. 2). This does not necessarily mean that the causal assumption in (6.5’) is wrong. Rather, it shows the sensitivity of joint forecasting. Investment should have declined in a slump year such as 1982, instead it went up. The contemporaneous link to unemployment in (6.5’) transmitted the error to unemployment and from there to wages and exports. Forecasts of the two exogeneous variables: foreign demand and oil prices can be chosen subjectively. The OECD’s own forecast of industrial production is taken as given in judgmental official forecasts of domestic developments in Finland. In order to get a fair comparison the OECD forecast was used as an input in the VARIMA forecast. Another exogeneous input was provided by the OPEC meeting in December 1980, where the price of oil was set for the next year. What would happen after that would not, according to the model, affect a forecast that ends in the last quarter of 1982. The non-zero shock technique presented in section 2 was not applied here since it would have complicated the definition of forecasting period and the checking of forecasting accuracy. In practical forecasting with model (6) the technique is applied regularly. All four jointly modeled variables were

OIL PRICESHOCK InLault

1

d

i

1

i

i2

i6

io

quarters

Fig. 1. Interim multipliers of the VARIMA

model.

EXPORTS

130

lllll.Fltl,75p

1975=‘100 ’

REM1IUE IHGES

80 IfWESlllENT

UNEtlPLOYtlENl

%

10

+ + 5

I Fig. 2. VARIMA

and ARIMA

forecasts

(- = outcome.

+ = VARIMA.

0 = ARIMA).

also modeled univariately. For lack of space the models are not given here but are supplied request. ARIMA and VARIMA forecasts were compared using the same summary accuracy measu as in Makridakis et al. (1982): Mean Square Error (MSE), Mean Average Percentage Error (MAI and Median of Absolute Percentage Error (Md). Summary measures as well as standard deviati of sample one step ahead forecast errors (s) are given in table 2. The better score is marked with z asterisk (*). Except for the Md measure for exports, VARIMA beats ARIMA. ’ Summary measures are good for comparing different forecasts for a single variable. Gem economic outlooks, on the other hand, are usually given in annual percentage numbers or in sim graphs. Table 3 compares VARIMA, ARIMA and available official annual forecasts with act outcome. The latter are judgmentally pooled from different sources of information: order stoc Table 2 Forecasting

accuracy

measures

of VARIMA

Exports VARIMA

s \IMSE

MAPEI MdS

and univariate Relative

ARIMA

0.064 *

0.082

0.454 * - 3.7 * 7.4

0.467 5.9 4.3 *

wages

VARIMA 0.026 * 11.3 * - 3.9 * 6.6 *

ARIMA

models. Investment

ARIMA 0.031 12.4 - 5.7 8.2

VARIMA 0.048 * 493 * - 3.4 * 4.6 *

Unemployment ARIMA 0.057 513 - 5.2 5.0

VARIMA 0.098 * 0.79 * 9.6 * 9.9 *

’ The exogeneous forecast of foreign demand 1s partly to blame for the poor VARIMA export forecast. to change by -3% in 1981 and by 3% in 1982. The outcome was -2% both years. For problems forecasts of exogeneous variables, see Ashley (1983).

ARIMA 0.113 1.16 - 17.1 20.1

Foreign demand when using avail,

L.-E.

iill&

/ Macroeconomic/orecasring

wirh a vector AROMA

149

model

Table 3 Annual percentage change and annual rate of unemployment, forecasts and outcome. ’ Relative wages

Exports

(2)

(1) 1980 1981 1982

2 -6 -5

(3) 31

-16 4

4 -5* 1*

(4) 4 -4 3-4

a Co/umns: (1) Outcome, (2) VARIMA.

Investment

(1)

(2)

(3)

1 17 12

6 14” -3

5*9 1*

(3) ARIMA.

(4)

(1)

(2)

-

10 1 3

10*7 1’ -3

Unemployment (3)

(4)

(1)

(2)

(3)

(4)

0 0’

13 15 -

4.8 5.3 6.2

4.8 * 6.0 7.0 *

4.7 4.6 4.6

4.9 4.7 * -

(4) Official.

UARIIIA .4

0 -.4 198013198111 Fig. 3. Forecast errors of VARIMA

198Ol3198111

198211 and ARIMA

198211

models.

interviews with business executives, leading indicators, consistency checks, etc. The first remark to be made is that the official forecast is a shade better than model forecasts only in one case, unemployment 1981, but for the wrong reason. A decrease is forecasted instead of an increase as correctly predicted by VARIMA. Table 3 may give the impression that VARIMA and ARIMA are approximately equally accurate. However, the general outlooks provided by the two models are different. The ARIMA forecasts are not far from ‘no change’ whereas VARIMA reproduces what happened after the first oil shock. Exports reacted slower and unemployment more modestly to the 1979 shock and this resulted in large VARIMA forecast errors for 1981 : 14-1982: 2Q (fig. 3). ARIMA, however, completely misses the continuous rise in unemployment and generates serious errors for exports at the end of the period.

4. Discussion This study shows that the VARIMA model can be fit into a macroeconomic context. As in econometrics variables can be classified into exogeneous and endogeneous. Relationships with exogeneous variables can be modeled through transfer functions, where contemporaneous relationships are allowed. Such relationships can also be introduced between jointly modeled variables if a one sided relationship can be assumed at lag zero. However, the case study is an example of the fact that the more variables are related through the model the greater the risk that if the forecast of one variable goes wrong all will go wrong. Exogeneity assumptions make it possible to generate alternative forecasts. Another way to achieve this is by adjustment of forecasts based on non-zero future shocks. The same technique can be applied for exploiting all observations when time series are updated at different times. Most decision makers will be happier with both a model forecast and a judgmental one if the former proves to be at

150

L. -E. tiller / Macroeconomicjorecasring

with a ueclor A RJMA model

least as accurate as the latter. Both the VARIMA and the ARIMA models satisfy this criterion in the case study. At the point of time a macroeconomic forecast is made the latest reasonably accurate observation is at least one quarter old. If it is enough to get an idea of the position of the economy right now and of where it is heading during the coming 1-2 quarters ARIMA does the job in this case study. However, economic policy often works with a longer perspective. In this case the usefulness of ARIMA is more doubtful, whereas a VARIMA model that also makes economic sense can, as in this case study, be of some help.

References Ashley, R., 1983. On the usefulness of macroeconomic forecasts as inputs to forecasting models, Journal of Forecasting 2, 211-223. Box, G.E.P. and G.M.J. Jenkins. 1976, Time series analysis, forecasting and control (Holden-Day. San Francisco, CA). Cholette, P.A.. 1982, Prior information and ARIMA forecasting, Journal of Forecasting 4, 375-383. Granger. C.W.J. and P. Newbold. 1977, Forecasting economic time series (Academic Press. New York). Harvey, A.C.. 1981, The econometric analysis of time series (Philip Allan. Oxford). Hillmer, SC.. D.F. Larcker and D.A. Schroeder. 1983. Forecasting accounting data: A multiple time-series analysis. Journal of Forecasting 2, 389-404. Koreisha. SC.. 1983, Causal implications: The linkage between time series and econometric modeling. Journal of Forecasting 2. 151-168. Klein. L.R.. 1984. The importance of the forecast. Journal of Forecasting 3, l-19. Klein. P.A. and G.H. Moore. 1983, The leading indicator approach to economic forecasting - Retrospect and prospect. Journal of Forecasting 2. 119-135. Makridakis. S., A. Andersen. R. Carbone. R. Fildes, M. Hibon. R. Lewandowski. J. Newton E. Parzen and R. Winkler, 1982. The accuracy of extrapolation (time series) methods: Results of a forecasting competition, Journal of Forecasting 1. 111-153. Gller, L.-E.. 1978, Time series analysis of Finnish foreign trade. Ph.D. thesis (The Finnish Statistical Society, Helsinki). Sims, C.A.. 1980, Macroeconomics and reality, Econometrica 48, l-48. Theil, H. and A.S. Goldberger. 1961. On pure and mixed estimation in economics, International Economic Review 2, 65-78. Tiao. G.C. and G.E.P. Box. 1981, Modeling multiple time series with applications. Journal of the American Statistical Association 76. 802-816.