forecasting of seasonal cointegrated processes

ELSEVIER International Journal of Forecasting 13 (1997) 369-380

Forecasting of seasonal cointegrated processes

Hans-Eggert Reimers* Hochschule Wismar, Fachhochschule fiir Technik, Wirtschaft und Gestaltung, Postfach 12 10, D-23952 Wismar, Germany

Abstract

In this paper, forecasts of seasonally cointegrated models are analysed applying the ML-approach for seasonal cointegration suggested by Lee (1992). The forecasts of seasonally cointegrated models in fourth differences are compared with those in first differences, including seasonal dummies. The comparison is done by means of simulating various bivariate data generating processes containing seasonal cointegrating relationships. Forecasts are calculated for different specifications of lag length and cointegrating rank. One main result of the simulation study is that the models in first differences with seasonal dummies forecast smaller errors for short horizons than the seasonally cointegrated models. For longer forecast horizons models in fourth differences outperform the former. Furthermore, a model with more seasonal cointegrating relations than in the underlying model produces very high forecast errors. This is not affected by the choice of the lag length. Both modelling procedures are applied to the German income-consumption-wealth-process. The forecast results of the models confirm the evidence of the simulation study. © 1997 Elsevier Science B.V.

Keywords: Seasonal cointegration; VAR forecasts; Consumption process

1. Introduction

Since the publication of Engle and Granger's paper in 1987, the analysis of cointegrating relations has become common practice in applied work for macroeconomic data. Seasonally adjusted time series are often investigated. If seasonally unadjusted time series are examined, their possible seasonality is taken into account by including seasonal dummies (see for example Johansen and Juselius (1990)). Recently the cointegration analysis for the zero frequency is extended by investigating cointegration for seasonal unit roots. Hylleberg et al. (1990) define seasonal cointegration and propose a multi-step procedure, adopting the two-step procedure of Engle

*Tel: +49-3841-753601; fax: +49-3841-753383; e-mail: h.reimers @ wi.hs-wismar.de

and Granger (1987). Corresponding to Johansen's maximum likelihood estimator at zero frequency Lee (1992) suggests a maximum likelihood (ML) estimator for seasonal cointegrating relations. Ahn and Reinsel (1994) give evidence that the ML-procedure estimates the cointegration coefficients precisely.

However, not much is known about how good the forecasts of seasonally cointegrated models are. One exception is the work of Kunst (1993a). He analyses the forecasting performance of models with seasonal cointegrating vectors. In sum, he presents some evidence that models accounting for seasonal cointegration produce lower forecast errors than models with seasonal dummies. However, the individual forecasts of the series are not summarized by a forecast measure of the process, which accounts for different variances of the series. Furthermore, the effects of different specifications of models ~are not

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370 H. Reimers I International Journal of Forecasting 13 (1997) 369-380

investigated. These questions and more general, the forecasting performance of models with seasonal cointegrating vectors are examined in this paper by means of simulating different data generating processes with seasonal cointegrating relationships.

The models are estimated for varied lag lengths and alternative numbers of cointegrating relations. One main result of the simulation study is that the models with seasonal dummies, and restricted by cointegrating relations at zero frequency, forecast smaller errors for short horizons than the seasonally cointegrated models. The former are more robust against specification changes than the latter. For longer forecast horizons the latter outperform the former. Furthermore, models with more seasonal cointegrating relations than in the underlying model produce very high forecast errors. This is not affected by the choice of the lag length.

Both modelling procedures are applied to the German consumption process including consumption, disposable income and a wealth variable. Evidence is presented that the approach with seasonal dummies produces smaller forecast errors in the short-term than the model with seasonal cointegrating relations. The paper proceeds as follows. The next section gives the statistical framework. This section is divided into two subsections: forecasting and seasonal cointegration. Section 3 contains a simulation study to compare forecasts for different model specifications. In Section 4 the procedures are applied to forecast income and consumption in Germany. Concluding remarks follow in Section 5.

2. Statistical framework

2.1. Forecasting cointegrated processes

Assuming a vector autoregressive process of lag length p (VAR(p))

X , = AIX,_ l + • . . + ApX t_p q- Et, (1)

where X, is a (K× 1) vector, the coefficients Aj are (K X K) matrices, et is a K-dimensional white noise process with zero mean and nonsingular covariance matrix X, and p is the order of the process. For a VAR(p) process the optimal h-step forecast with

minimal mean squared errors (MSE) is given by the conditional expectation, even if det(l~c-AlV . . . . . Apv p) has roots on the unit circle (see for example Ltitkepohl (1991), Chapter 11). Thus, the optimal h-step forecast at origin t is

X t ( h ) = A i X , ( h - 1)+ • • • + A p X t ( h - p ) , (2)

where Xt( j )=X,+ j for j--<0. This equation relates the optimal h-step forecast to the optimal (h-l) forecast etc. It is possible to express the h-step forecast conditional on information available in period t. The forecast errors are

h - I

e(h) = Xt+ h - Xt(h) = ~ ~e,+ h_,, (3) I=0

where ~0 = Ix. The matrices ~ are obtained from Aj by the following recursions

l

= ~ ~_jAj, (4) j = l

where A j = 0 for j > p . The forecast MSE matrix becomes

h - I

MSE(h) = ~ ~ . , ~ : . (5) l = 0

If one or more eigenvalues of the determinantal polynomial det( IK-A~z . . . . . ApZ p) have a length of unity, the ~ do not converge to zero in this case. If a data generating process (DGP) contains cointegrating relationships, accounting for them reduces the unit roots of the process and should therefore decrease the MSE growth (see Engle and Yoo (1987)).

The VAR representation (auto) may be reparametrized to obtain vector error correction representations in first differences and in fourth differences. They are considered to characterize the cointegrating properties of a process.

2.2. Seasonal cointegration

If, for instance, two variables are seasonally integrated it is possible that a linear combination among them eliminates some seasonal unit roots. Hence, the variables are seasonally cointegrated (see Lee (1992)), To investigate this property (1) is reparametrized to obtain a so-called seasonal error

H. Reimers / International Journal of Forecasting 13 (1997) 369-380 371

correction representation. The process X, may have unit roots at seasonal frequencies as well as at the zero frequency. Thus, the determinant IA(v)l of the matrix polynomial (I r - A i v . . . . . Ap V p) may have roots on the unit circle. Precisely, it has one at - 1 and at least one at + 1. Assuming that the fourth differences of X, (AaX ,) are stationary the process can be written as

A 4 X ' = H I Z I , t _ I -[- H2Z2. t_I -~ H3Z3. ,_ 2 -F HnZ3. , 1

-F F I A 4 X t I + " " " -F j~p_4A4Xt_p+4 "F ~t"

(6)

This form is referred to Lee (1992) as seasonal error correction representation. The vectors Z/., ( i= 1,2,3) are defined as Z I . t = ( I + L + L 2 + L 3 ) X t , Z2.t=(1 - L + L 2 - L 3 ) X t and Z 3 . t = ( 1 - L 2 ) X t , where L is the backshift operator (LX, = X,_ l ). If H 2 = H 3 = H a = 0 (6) reduces to a cointegrated vector autoregressive (VAR-) model with the long-run matrix H~. The matrix H 1 has rank r~. Thus there are r I cointegrating relations which have the usual long-run interpre- tation. If the matrices H l , H 2, or H 3 and H a have full rank, then the series do not contain unit roots at the corresponding frequency. If their ranks are zero, seasonal cointegrating relationships at that frequency do not exist. In the intermediate case where 0 < r a n k ( H , ) = r i < K it can be shown that H~=BiC~ for suitable K X r i matrices Bi and Cg such that C~Zg.,_~ is stationary even though Zi, , itself is nonstationary. The matrices C i (B~) are called cointegration matrices (loading matrices).

Lee (1992) shows that ML-estimates of C~ (C 2, C 3, and Ca) are obtained from the canonical vectors

on Zl,,_ j (Z2,t_l, Z3,t_2, and Z3,,_ l, respectively) with respect to A4X , corresponding to the non-zero canonical roots under the assumption C 3 = C 4. Canonical analysis is conducted conditional on the other nonstationary parts and on p - 4 lags of A4X r For instance the ZL, ~ to A a X t correlations are calculated conditional on Zz.t_ l, Z3.,_ 2, Z3.t_ l, and on p - 4 lags of A 4 X t using the OLS approach. Model (6) contains as specification parameters the lag length p and the cointegrating ranks at seasonal and nonseasonal frequencies (rl, r 2, r3). It is rearranged to determine the forecast of the model in the VAR representation.

The above analysis is done without a deterministic

component. Kunst (1993b) and Lee and Siklos (1995) extend the basic model of Lee with an intercept:

X , = v + A , X , l + " ' ' + A p X t p + E t, ( 7 )

where v is a (K× 1) intercept term. Applying Lee 's estimation procedure the intercept is included in the dynamic component.

The analysis of the seasonal cointegration processes in fourth differences is accompanied by examining models in first differences and investigating only the cointegrating relations at the zero frequency using Johansen's framework (see Johansen (Johansen, 1988, 1991)). The seasonality is taken into account by seasonal dummies. Thus the following model is analysed:

A I X t = ld + ~SD, + qb, A ,X ,_ , + • • .

-1- I~l,_ l A i X t _ p + 1 -- H X t _ p -1- Et, ( 8 )

where v is a (K x 1) intercept term. The factor ( is an (K × 3) coefficient matrix and SD, the appropriate vector of the (3 × T) seasonal dummy matrix for the T observations. The rank s of the matrix H corre- sponds to the independent cointegrating relations at zero frequency. This model contains as specification parameters the lag length p and the number of cointegrating relations s. Eq. (8) is reparametrized to obtain a VAR to forecast the process.

It is worth noting that model (8) is misspecified under the assumptions of model (6), since it does not exactly account for seasonal unit roots. It contains less unit roots than (6). Furthermore, it neglects cointegrating relations at seasonal frequencies and focuses only on the cointegrating vectors at zero frequency. But the model includes seasonal dummies and is often used in empirical work. Thus it seems worthwhile to mention it as second approach.

3. Simulation study

A bivariate VAR(4) is used by Ahn and Reinsel (1994). None of their variables is an independent driving force. Each influences the other. A more simple framework is analysed in this study. The bivariate DGP X, = E4= l AjX,_j + ~, is considered for


X, = [x , , x2,]' which has the following seasonal error correction representation:

A 4 X t = B l C l Z l , t _ 1 -.}- B2C2Z2,t_ 1 q- B3C2g3,t_ 2

+ B4C2Z3. t - I + ~:t'

where

B~ = [bl,0.0]', C I -- [1.0, - 1.2]

B 2 = [b2,0.0], C 2 = [1.0, - 0.4]

B 3 = [b3 ,0 .0] , B4 = [b4,0.0 ]

0.3 10) The values of the elements b i are given in Table 1. The second variable (x2,) drives the process. The cointegrating vectors are clearly different. In this way possibility is avoided that an identical vector eliminates unit roots at all seasonal and nonseasonal frequencies. Processes with these vectors may be well approximated by models in first differences which include this cointegrating vector for s = 1. To describe the dynamic structure of each process the eigenvalues of the characteristic equation de t {A(L)}=O are determined. They are ordered ac- cording to their length. Table 1 gives the values of different DGP, the implied number of unit roots and the length of the next eigenvalue. For example, the first item (0.2; 0.2; - 0 . 4 ) reads as: The process parameters a re b 2 = 0 . 2 , b 3 = 0 . 2 , b 4 = - 0 . 4 , and imply 4 unit roots; the next eigenvalue has the length 0.821. Often its follower has the same length. They are a conjugate complex pair.

Samples of series length of T= 100 and T=200 are generated from the above processes using the RNDNS subroutine in GAUSS to create the bivariate normal distribution with zero mean vector and covariance matrix ~7,. Furthermore, 40 presample values and 20 postsample for ex-ante forecasting are used. The starting condition is set to X 0 = X _ , = X_ 2 =X_ 3 =0. 2000 replications (N) of the sample series are generated.

The rank of the cointegrating vectors at seasonal frequencies as well as of the corresponding loading matrices are restricted to be identical. This limits the specification alternatives. Estimates of the B~ and the C i are calculated for each series for varied lag lengths ( p = 4 , 5, 6, 7, 8) and different seasonal cointegration ranks r I =0, 1, 2 as well as r 2 =0, 1, 2. Hence, 5 × 3 × 3 = 45 systems were determined to forecast the series in the fourth differences framework. Each system generates only one h-step forecast for h = 1 . . . . . 20 at origin T= 100 (T=200). Applying Johansen's procedure with seasonal dummies (in first differences) lag lengths of p = 3, 4, 5, 6 and cointegrating ranks of s = 0, 1, 2 are specified. Hence, 4 × 3 = 12 alternative systems are estimated.

The forecast errors of the different models are compared by normalized forecast variances using MSE matrices. Conducting a simulation study with N replications a measure Mj of the forecasting performance is defined as

1 N 1 K (ejk)~

Mj = ~ n~=l -~ k~=l MSEk(j ) for j = 1 . . . . . h.

The square of the j-step forecast error of the k-th

Table 1 Values and implied eigenvalues of the simulated processes

b~ b2; b3; b 4 b2; b3; b 4 b2; b3; b 4 b2; ba; b 4

-0 .75 a) 0.2; 0.2; b) 0.4; 0.2; c) 0.2; 0.2; d) 0.0; 0.0; - 0 . 4 4; 0.821 - 0 . 2 4; 0.878 - 0 . 2 4; 0.677 0.0 7; 0.250

-0 .50 e) 0.2; 0.2; f) 0.4; 0.2; g) 0.2; 0.2; h) 0.0; 0.0; - 0 . 4 4; 0.805 - 0 . 2 4; 0.836 - 0 . 2 4; 0.777 0.0 7; 0.500

-0 .25 i) 0.2; 0.2; j) 0.4; 0.2; k) 0.2; 0.2; 1) 0.0; 0.0; - 0 . 4 4; 0.878 - 0 . 2 4; 0.884 - 0 . 2 4; 0.857 0.0 7; 0.750

0.0 m) 0.2; 0.2; n) 0.4; 0.2; o) 0.2; 0.2; p) 0.0; 0.0; - 0 . 4 5; 0.922 - 0 . 2 5; 0.919 - 0 . 2 5; 0.899 0.0 8;

Each cell contains first the denomination of the process and the values of the DGP, second the number of unit roots and third the length of the next eigenvalue.


component of the process (ejk) ~ is divided by the corresponding diagonal element of the MSE matrix MSEk(j). The sum of K terms divided by the process dimension K indicates the performance of a model.

It is worth noting that the forecast results of models in fourth differences differ extremely by varying the cointegrating ranks. Contrary to what would be expected a change of the seasonal cointegrating rank affects dramatically the values of the

H~-matrices. If a seasonal cointegrating rank of r 2 = 2 is specified, it results in a VAR model, where some eigenvalues have a length greater than one. This length indicates an explosive process. The portions of models with explosive roots are given in Table 2 for T=100. The table contains the portions of models with lag order p = 4 ( p = 6 for b 1 =0.75) and p = 8. All considered seasonal cointegration ranks are presented. If the cointegration ranks are set to zero

Table 2 Portions of models with explosive roots

bt P

0 1 2

0 1 2 0 1 2 0 1 2

- 0.75 a) 4 0.00 1.00 1.00 6 0.00 0.964 1.00 8 0.00 0.245 0.999

b) 4 0.00 1.00 1.00 6 0.00 0.982 1.00 8 0.00 0.092 0.997

c) 4 0.00 1.00 1.00 6 0.00 0.937 1.00 8 0.00 0.247 0.999

d) 4 0.00 0.944 1.00 6 0.00 0.982 1.00 8 0.00 0.982 1.00

- 0.50 e) 4 0.00 1.00 1.00 8 0.00 0.010 0.994

f) 4 0.00 1.00 1.00 8 0.00 0.062 0.995

g) 4 0.00 0.997 1.00 8 0.00 0.108 0.994

h) 4 0.00 0.881 1.00 8 0.00 0.947 0.999

-0.25 i) 4 0.00 1.00 1.00 8 0.00 0.054 0.985

j ) 4 0.00 0.996 1.00 8 0.00 0.052 0.992

k) 4 0.00 0.971 1.00 8 0.00 0.058 0.993

1) 4 0.00 0.819 0.999 8 0.00 0.905 1.00

0.0 m) 4 0.00 1.00 1.00 8 0.00 0.043 0.981

n) 4 0.00 0.971 1.00 8 0.00 0.068 0.992

o) 4 0.00 0.859 1.00 8 0.00 0.041 0.984

p) 4 0.00 0.760 0.996 8 0.00 0.854 1.00

0.013 1.00 1.00 0.833 1.00 1.00 0.000 0.971 1.00 0.691 0.998 1.00 0.001 0.264 0.999 0.531 0.636 0.999 0.211 1.00 1.00 0.955 1.00 1.00 0.000 0.989 1.00 0.804 0.997 1.00 0.000 0.128 0.999 0.576 0.631 0.999 0.045 1.00 1.00 0.898 1.00 1.00 0.000 0.960 1.00 0.730 0.997 1.00 0.000 0.276 0.999 0.560 0.646 1.00 0.001 0.993 1.00 0.769 1.00 1.00 0.000 0.985 1.00 0.619 0.997 1.00 0.000 0.981 1.00 0.496 0.995 1.00 0.001 1.00 1.00 0.677 1.00 1.00 0.000 0.119 0.996 0.541 0.588 0.998 0.008 1.00 1.00 0.859 1.00 1.00 0.000 0.092 0.998 0.616 0.648 1.00 0.001 1.00 1.00 0.770 1.00 1.00 0.000 0.128 0.995 0.577 0.617 0.998 0.001 0.984 1.00 0.601 0.999 1.00 0.000 0.951 0.999 0.510 0.985 1.00 0.003 1.00 1.00 0.533 1.00 1.00 0.005 0.092 0.990 0.554 0.588 0.997 0.002 1.00 1.00 0.714 1.00 1.00 0.003 0.164 0.995 0.650 0.714 0.999 0.002 0.999 1.00 0.607 1.00 1.00 0.005 0.107 1.00 0.584 0.630 0.998 0.002 0.956 1.00 0.487 0.993 1.00 0.006 0.920 1.00 0.512 0.974 1.00 0.318 1.00 1.00 0.631 1.00 1.00 0.334 0.436 0.997 0.687 0.779 0.999 0.390 0.996 1.00 0.749 1.00 1.00 0.418 0.597 0.998 0.801 0.890 1.00 0.350 0.973 1.00 0.680 0.996 1.00 0.366 0.493 0.998 0.739 0.813 1.00 0.325 0.910 1.00 0.630 0.976 1.00 0.337 0.941 1.00 0.687 0.982 1.00

p: lag length; b~: process parameter; a)-p): DGP; r~, r 2 cointegrating ranks.

3 7 4 H. Reimers / International Journal of Forecasting 13 (1997) 369-380

(r~ = r z =0) the models have more or less no explosive roots. This result changes if r~ = 1 or r~ = 2 are considered. These latter specifications result in many models with explosive roots. Especially, all models with r I --2 and r2=2 have this property.

The portion of models with explosive roots decreases, if longer lags are specified. The large portions of such models imply that their forecast errors grow with increasing forecast horizon faster than the diagonal elements of MSE-matrices. High normalized forecast errors are found for all examined processes. However, these models can be easily eliminated in an empirical application, if the eigenvalues of the AR representation are investigated. Obtaining eigenvalues with a length greater than one, the conditions of a unit root process are not fulfilled. Therefore, the corresponding specification is not taken into account and less cointegrating relations should be considered. Moreover, a seasonal cointegrating test indicates seasonal cointegrating ranks and its application may avoid to select an inappropriate rank.

The main purpose of the paper is to compare forecasting performance between the two approaches. Therefore, the model specification is selected which generates the smallest forecast errors for a given DGP. The specification is easily chosen for models in first differences. The specification p = 4 and s = 1 dominates all others, if the DGP includes a cointegrating relation at zero frequency. Without this relation models with p =4 and s =0 generate the smallest forecast errors independent of the forecast horizon. Models with lag length p = 3 obtain higher forecast errors independent on the specified cointegration rank. The coefficient matrix of the fourth lag contains relative high values, which are neces- sary to describe the process. If the DGP contains a nonseasonal cointegation relation, models with s = 2 or s = 0 produce greater forecast errors than those with s = 1 for a given lag length. This evidence confirms former results of Engle and Yoo (1987).

For the processes in fourth differences the selection is not as easy as before. In some cases one specification is better than all others. A model with a higher lag length often dominates one with a lower lag. This is confirmed by a vector portmanteau test. It is an extension of the Box-Pierce (BP-) test (see for example Liitkepohl (1991), Chapter 4). Its statis-

tic is asymptotically x 2 ( K 2 ( q - m ) ) distributed, where q denotes the number of considered autocorrelations and K2m the estimated parameters.

Table 3 gives results of the BP-test. Each cell contains the portion of the rejections of the null hypothesis of no autocorrelation in the residuals up to lags q = 2 0 at the 5% significance level. The results depend on the specification of the cointegration ranks. The null hypothesis of no autocorrelation in the residuals is mostly rejected for a cointegration rank of r2=2 at seasonal frequencies and/or of r I = 2 at zero frequency. The null hypothesis is not often rejected for models with a high lag order and no cointegrating relations at seasonal frequency. These models contain residuals without significant vector autocorrelations. Only these models generate reliable forecasts.

In the simulation study frequently different specifications generate the best forecasts for different forecast horizons. Up to a horizon h = 3 models without any cointegration relation are superior to the other. In the medium-term and long-run models accounting for cointegration are better. To save space only the specification is selected which generates more often best results. The results of both approaches are given in Fig. 1. Each plot includes a solid line which presents the normalized forecast errors of the best model in fourth differences and a dashed line which gives the normalized forecast errors of the best model in first differences. The horizontal axis shows the forecast horizon from 1 to 20, the vertical axis takes up the forecast errors measured by Mj. The specification of the model in fourth differences is followed by the specification of the model in first differences. For example in plot a) of Fig. 1, p = 8 r~--1 r2=0 presents a model in fourth differences where the lag order is 8, the cointegration rank of H 1 is r~ = 1 and the cointegration ranks of H 2, H a and H 4 are r 2=0. Then the specification of the model in first differences follows. p = 4 s = l reads as: the lag order is p = 4 and the cointegration rank of H equals s = 1.

The best forecast results are obtained by models in fourth differences, where the cointegration rank is rt =1 or rl =0. The specification is often accompanied by a specification of r 2 = 0. In Fig. 1 the order p = 8 is mostly selected. The differences between theoretical and empirical errors are explained as

H. Reimers I International Journal of Forecasting 13 (1997) 369-380

Table 3 Results of the Box-Pierce vector autocorrelation test for T= 100

375

r, 0 1 2

r 2 0 1 2 0 1 2 0 1 2

bl P

- 0.75 a) 4 1.00 1.00 1.00 0.900 1.00 1.00 0.999 1.00 1.00 6 0.773 0.993 1.00 0.732 0.997 1.00 0.997 1.00 1.00 8 0.247 0.957 1.00 0.290 0.971 1.00 0.955 0.999 1.00

b) 4 0.999 1.00 1.00 0.907 1.00 1.00 0.999 1.00 1.00 6 0.842 0.999 1.00 0.816 1.00 1.00 0.999 1.00 1.00 8 0.241 0.971 1.00 0.282 0.984 1.00 0.965 1.00 1.00

c) 4 0.986 0.994 1.00 0.691 0.997 1.00 0.996 1.00 1.00 6 0.526 0.994 1.00 0.465 0.997 1.00 0.992 1.00 1.00 8 0.152 0.969 1.00 0.176 0.983 1.00 0.953 1.00 1.00

d) 4 0.481 1.00 1.00 0.502 1.00 1.00 0.989 1.00 1.00 6 0.079 1.00 1.00 0.045 1.00 1.00 0.972 1.00 1.00 8 0.028 1.00 1.00 0.027 1.00 1.00 0.912 1.00 1.00

- 0.50 e) 4 0.998 0.994 1.00 0.880 0.997 1.00 0.998 1.00 1.00 8 0.211 0.849 1.00 0.220 0.889 1.00 0.954 0.994 1.00

f) 4 0.992 0.993 1.00 0.900 0.998 1.00 0.999 1.00 1.00 8 0.195 0.931 1.00 0.193 0.956 1.00 0.970 0.999 1.00

g) 4 0.856 0.998 1.00 0.803 0.973 1.00 0.997 1.00 1.00 8 0.114 0.901 1.00 0.107 0.937 1.00 0.960 0.999 1.00

h) 4 0.228 1.00 1.00 0.753 1.00 1.00 0.995 1.00 1.00 8 0.018 1.00 1.00 0.013 1.00 1.00 0.922 1.00 1.00

-0.25 i) 4 0.991 0.992 1.00 0.977 0.994 1.00 1.00 1.00 1.00 8 0.163 0.703 1.00 0.233 0.814 1.00 0.962 0.990 1.00

j) 4 0.914 0.994 1.00 0.984 0.998 1.00 1.00 1.00 1.00 8 0.138 0.884 1.00 0.248 0.944 1.00 0.976 0.999 1.00

k) 4 0.599 0.987 1.00 0.960 0.973 1.00 0.999 1.00 1.00 8 0.075 0.781 1.00 0.205 0.884 1.00 0.967 1.00 1.00

1) 4 0.090 1.00 1.00 0.946 1.00 1.00 0.999 1.00 1.00 8 0.013 1.00 1.00 0.177 1.00 1.00 0.952 1.00 1.00

0.0 m) 4 0.967 0.997 1.00 0.992 1.00 1.00 1.00 1.00 1.00 8 0.096 0.584 1.00 0.905 0.971 1.00 0.993 0.998 1.00

n) 4 0.753 0.986 1.00 0.996 1.00 1.00 1.00 1.00 1.00 8 0.076 0.833 1.00 0.970 0.996 1.00 0.999 1.00 1.00

o) 4 0.386 0.985 1.00 0.990 1.00 1.00 1.00 1.00 1.00 8 0.038 0.688 1.00 0.935 0.990 1.00 0.997 0.999 1.00

p) 4 0.020 1.00 1.00 0.969 1.00 1.00 0.999 1.00 1.00 8 0.005 1.00 1.00 0.906 1.00 1.00 0.993 1.00 1.00

p: lag length; bt: process parameter; a)-p): DGP; r], r 2 cointegrating ranks, significance level is 5%, Box-Pierce-test up to 20 lags.

fo l lows . T h e theore t ica l va lue s reflect the unce r t a in ty

o f the fo recas t hor izon , w h e r e a s the empi r i ca l er rors

i nc lude add i t iona l ly the unce r t a in ty o f the e s t i m a t e d

coef f i c ien t s o f the p rocess . T h e s e coef f ic ien t s are not

k n o w n , but e s t ima ted . E x t e n d i n g the obse rva t i on

pe r iod r e d u c e s the d i f f e rence b e t w e e n them. T h e

fo recas t er rors o f the bes t m o d e l in four th d i f f e r ences

are c o n s i d e r a b l y h i g h e r for the ho r i zon h = 1 than for

h = 2. M o r e o v e r , s o m e l ines s h o w a s ea sona l pat tern.

Th i s resu l t s f r o m the fact that more seasona l uni t

roots are speci f ied than the u n d e r l y i n g p r o c e s s e s

conta in .

T h e c o m p a r i s o n o f the two a p p r o a c h e s s h o w s that

m o d e l s in first d i f f e r ences a lw a ys o u t p e r f o r m m o d e l s

in four th d i f f e rences for sma l l f o r ecas t i ng hor izons ,

as long as co in t eg ra t i ng re la t ions at s ea sona l fre-

q u e n c i e s are i nc lude d in the D G E Th i s is a lso valid,

i f m o d e l s in four th d i f f e r ences wi th r I = 0 and r 2 = 0


o) p=8 r1=1 r2=O; p=4 s=l b) p=8 r1=1 r 2 = O ; p = 4 s=l

e) p=8 r1=1 r2=O; p=4 s=l

- t = = 7 I I t ~ Is I I 111

• f) p--8 r 1=1 r2=O: p=¢ s=l

i) p=8 rt=O r2=O: p=4 s=l

- - / \ / ~,

j) p=8 r 1 =0 r2=O: p=¢ s=l

_ ~ . _ _ \ / I - - \ /

Jl m) p=7 r1=0 r2=O; p=4 s=O n) p=7 rl=O r2=O; p=4 s=O

Fig. 1. Forecast results for T= 100. Vertical axis: normalized forecast errors, horizontal axis: forecasting horizon, solid line: normalized forecast errors of best model in fourth differences, dashed line: normalized forecast errors of best model in first differences.

H. Reimers I International Journal of Forecasting 13 (1997) 369-380 377

c) p=8 r1=1 r2=O; p=4 s=l d) p=5 r1=1 r2=O; p=4- s=l

I \

,-" - k I k / / \ , . . / \ /

! \ / _ / -

1 | . . . . . . . . - 1 1 • 7 ii II la 1• 17

~ g) p=8 r1=1 r2=O: p=4 s=l

!

! \ / . . . , " ~. ~ \ 1

h) p=5 r1=1 r2=O; p=4 s = l

_ / \

/ \ / s \ / \ /

/ \ /

\ /

\

. k) p=8 rl=O r2=O: p=4 s=l

! _ ~" \ / / N / ! t*

s ~- / a

I

I) p = 5 r 1 = 0 r2=O; p=¢ s = l

/-.. \ /

I " / \ -

I \ \ /

, , ; ~ ,' ~, - , ~ ;, - , ~ '. ~ : ,, . . . . . . , ; , , ; ~', - I a •

o) p=7 rl=O r2=O; p=4 s=O

\ \ i I

p) p=4 rl=O r2=O; p=4 s=O

i 1 \ \

,~ i I , ' ,~ \

/ _ ~ \ / \ / / /

\ \ /

F i g . 1 . (continued)

378 H. Reimers / International Journal o f Forecasting 13 (1997) 3 6 9 - 3 8 0

are considered. The forecast errors of the former are

20-30% lower than these of the latter. The forecast errors of the former are in general more smooth than those of the latter. For DGP with no cointegrating

relation at Zero frequency models in fourth differ-

ences dominates the models in first differences. In other words, the former seems to fit strong and

changing seasonality better than the latter. Extending the sample size (T=200) decreases the

normalized forecast errors. The selected lag order for

models based on fourth differences is not reduced. The best forecast results are more often obtained by specifications of the fourth differences approach

which include seasonal cointegrating relationships.

The main difference of the results between the two approaches does not change.

It is worth noting that only processes including seasonal cointegrating vectors and seasonal unit roots are used in this simulation. Thus the simulation

design contains changing seasonality and is in favour of models in fourth differences. Moreover, only bivariate DGP are considered. Nevertheless, there are

examples presented, where models in first differences stand comparison with the models in fourth differences. These results confirm the results of

Kunst (1993a). He concludes that substantial gains in forecasting precision seem unlikely if seasonal cointegration is taken into account.

tests based on the maximum eigenvalue and the trace

statistics are reported in Table 4. Given these results it seems sensible to concluded that there exist cointegrating vectors at the zero and seasonal fre-

quencies. ( r t = r 2 = r 3 = 1). This model specification is used for forecasting in the following. The Box-

Pierce-test confirms the lag length selection, because

the value of the test statistic is not significant (see Table 4 last row).

Reserving twelve observations to determine ex- ante forecast errors the estimation period is accord-

ingly shortened. The estimation period covers the period 1962:1-1990:4. The model in fourth differences (6) with intercepts is estimated to determine

the coefficients of the seasonal error correction representation. Then it is rearranged to get the VAR

representation and to forecast the process six steps

ahead. Next the sample period is extended by one quarter. The model is reestimated and rearranged.

Table 4 Seasonal cointegration tests

Null Fourth differences First differences hypothesis Different frequencies:

0 1/2 1/4

Trace test: - T 3" In (1 -h , ) r i =0 30.96** 22.40* 37.84*** 34.23** r i : 1 11.80 6.01 6.74 12.99 r i = 2 2.42 1.00 0.54 2.74

4. F o r e c a s t for the c o n s u m p t i o n - p r o c e s s

Strong seasonality in consumption and income data is found for various countries (see for example

Ahn and Reinsel (1994), Engle et al. (1993), Lee and Siklos (1993), Reimers (1994)). For western Ger- many Reimers (1994) investigates the consumption- process explained by real consumption, real dispos-

able income deflated by the price index of private consumption and net financial wealth of private households deflated by the price index of private consumption. All series are from the data bank of the Deutsche Bundesbank. All variables are in logarithms. Reimers shows that the variables are seasonally cointegrated. Using order selection criteria the lag length of the unrestricted process is determined to be p = 5 for the period 1962:1 to 1993:4. The results of the seasonal cointegration

A .... test: - T In (1-Ai) ri=O 19.16" 16.39" 31.10'** 21.24"* r i = 1 9.38 5.01 6.20 10.25 ri = 2 2.42 1.00 0.54 2.74

Decision r~ = 1 r 2 = 1 r 3 = 1 s = 1

Box-Pierce- test (28) 247.27 (0.127) 218.16 (0.184)

Sample period 1962:1-1993:4. *** (**, *): at the 1% (5%, 10%)-level significant. Lee (1992) gives critical values up to a process of three variables and no deterministic component. Franses and Kunst (1995) give evidence that the critical values from Table 1 of Osterwald-Lenum (1992) should be used for the zero frequency. Furthermore, they notice that an intercept do not change the critical values for the other frequencies, hence the tables of Lee (1992) are used. Box-Pierce-test up to 28 lags, p-values in parentheses. The Box-Pierce-test statistic has an asymptotic x:-distribution, where the degrees of freedom depend on the number of considered autocorrelations and the number of estimated parameters of the model.


Forecasts are calculated for one to six quarters

ahead. These steps are repeated up to the third quarter of 1993 to determine the last 1-step forecast. Hence in total, there are twelve 1-step forecast

errors, eleven 2-step forecast errors . . . . . seven 6-

step forecast errors. To sum up the forecast results root mean squared

forecast errors (RMSF) and mean absolute forecast errors (MAF) are determined (see Table 5). The

forecast errors of income and consumption are higher

than those of wealth. Alternative forecasts are determined by a model in

first differences using Johansen's procedure, where the model includes seasonal dummies. For the period 1962:1 to 1993:4 lag length and cointegration rank

are specified. Choosing a lag length of p = 5 likelihood ratio tests are conducted to select the cointegrating rank. Only the null hypothesis of s = 0 is

rejected at the 5% level (see Table 4 right part).

Hence a cointegrating rank of s = 1 is chosen. The value of the BP-test statistic is 218.16 which is not

significant at the 10% level (see Table 4). Following the previous approach the estimation period is shortened to determine the coefficients of the model and to forecast the process up to six-step ahead. The forecast errors of the model in first differences have the same magnitude as the errors of the previous

procedure. The forecast results are summarized by the RMSF and MAF statistics. Their values are given

relatively to the results of the model in fourth

differences. Analysing the summary statistics, the model in

first differences yields more often lower forecast

errors than the model in fourth differences for forecast horizons up to 4 quarters (see Table 5). For 5 or 6 quarters ahead the latter approach outperforms

the former. These results confirm the findings of the simulation study. In the short-term models in first

differences forecast a process with lower forecast

errors than a model in fourth differences. The latter dominates the former in the medium-term. Using the

estimates of the fourth differences approach as a

DGP, the simulation experiment of the former section is repeated. The best forecast results are obtained by the model based on first differences and a

long-run relationship (s = 1 and p = 5). It outperforms the models of the fourth differences approach. Thus

the main results of the former section are confirmed.

5. Conclusion

This paper focuses on forecasting seasonally co-

integrated processes. They are analysed by the ML- approach suggested by Lee, where the forecasts are determined by their VAR representation. The forecasts of seasonally cointegrated models in fourth differences are compared with forecasts of cointegrated models in first differences including seasonal

dummies. The comparison is done by means of simulating different bivariate seasonal cointegrating

models. In the simulation the forecasting performance of

models in fourth differences are strongly affected by

Table 5 Forecast errors for consumption, income and wealth

First differences Model Fourth differences

Variable Consumption Income Wealth Consumption Income Wealth

Results relative to competitor Horizon R M S F MAF R M S F MAF R M S F MAF RMSF MAF RMSF MAF R M S F MAF

1 0.0143 0.0108 0.0208 0.0179 0.0072 0.0061 1.098 1.046 0.938 0.821 0.917 0.902 2 0.0167 0.0147 0.0242 0.0200 0.0083 0.0069 0.994 0.973 0.975 0.960 0.964 0.957 3 0.0170 0.0156 0.0247 0.0214 0.0064 0.0053 0.947 0.897 0.915 0.907 1.016 0.926 4 0.0189 0.0164 0.0258 0.0221 0.0081 0.0067 1.021 0.970 0.988 0.991 0.951 0.955 5 0.0268 0.0192 0.0327 0.0267 0.0128 0.0109 1.175 1.214 1.055 0.959 0.875 0.826 6 0.0261 0.0185 0.0375 0.0291 0.0100 0.0083 1.077 1.119 1.011 0.852 0.900 0.771

RMSF: Root mean squared forecast error; MAF: Mean absolute forecast error. The measures of forecast errors of the model in first difference are divided by the corresponding measure of the model in fourth differences. Values less than unity indicate that the forecast error of the model in frist differences is smaller than the error of the competitor.

380 H. Reimers / International Journal of Forecasting 13 (1997) 369-380

changes in the specification of the parameters. If more seasonal cointegrating relations are specified than in the DGP, a high portion of models has explosive roots. This implies large forecast errors. Moreover, the choice of the lag length affects the forecasting performance. The models are not robust against specification changes. On the other hand, models in first differences with seasonal dummies forecast smaller errors for short horizons than the seasonally cointegrated models. For longer forecast horizons models in fourth differences outperform the former. Given the available sample size in empirical work and the importance of forecasts for short-term horizons a concentration on models based on first differences with seasonal dummies seems sensible.

Acknowledgments

Earlier versions of this paper were presented at the 15th International Symposium on Forecasting in Toronto, in seminars at the Institute for Statistic and Econometrics of the universities in Kiel and in Nrnberg. The author is grateful to Philip Hans Franses, Jan G. de Gooijer, the participants of the seminars and three anonymous referees for their helpful comments.

References

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Biography: Hans-Eggert REIMERS received his Ph.D. in Econ- omics at the Christian-Albrechts-University of Kiel in 1991. In 1991-1994 he worked at the Research Department of the Deutsche Bundesbank. He is Professor in the Economics Depart- ment of the Hochschule Wismar. His research interests include seasonal cointegration analysis, monetary policy analysis, and applied econometrics.

forecasting of seasonal cointegrated processes

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