The Japanese consumption function

Journal of Econometrics 55 (1993) 275-298. North-Holland

Seasonal cointegration

The Japanese consumption function*

R.F. Engle and C.W.J. Granger University of California at San Diego, La Jolla, CA 92037, USA

S. Hylleberg University of Aarhus, DK-8000 Aarhus C. Denmark

H.S. Lee Tulane University, New Orleans, LA 70118, USA

A theory of seasonal cointegration and integration is discussed in Hylleberg, Engle, Granger, and Yoo (1990) and tests for seasonal unit roots are also developed. To estimate and test for seasonal cointegration at each frequency, a two-step procedure similar to the one suggested by Engle and Granger (1987) is investigated in this paper. Using Japanese data on consumption and income, evidence in favour of seasonal cointegration at frequency l/4 is found. An economic interpretation of this cointegrating relation is presented using the notion of a slightly impatient borrowing-constrained utility-maximizing consumer. While the test statistics for noncointegration occurring at the frequency l/2 of a cycle have the same distribution as the test statistic obtained for the zero frequency case by Engle and Granger (1987) and Engle and Yoo (19871, the distribution of the test statistics for noncointegration at the frequency l/4 (and 3/4) is derived based on the asymptotic distribution theory for testing a pair of complex roots on the unit circle [Ahtola and Tiao (19871, Chan and Wei (1988)]. The critical values and evidence on the power of the test are obtained through Monte Carlo simulations.

1. Introduction

The theory and application of integration and cointegration have been of major interest in economics for a number of years. Recently the theory was

*We gratefully acknowledge help and comments from Palle Schelde Andersen (BIS), Erich Ghysels (CRDE, Montreal), Denise Osborn (University of Manchester), Franz Palm (University of Limburg), Kirsten Stentoft (University of Aarhus), and an anonymous referee. An earlier version was also presented at the World Meeting of the Econometric Society of Barcelona, August 1900, and at the NBER/NSF Time Series Conference in San Diego, October 1990. The Danish Social Science Research Council supported the project financially.

0304-4076/93/$05.00 0 1993-Elsevier Science Publishers B.V. All rights reserved

216 R.F. Engle et al., Seasonal cointegration

extended to cover aspects of economic time series other than the long-run or the zero frequency characteristics. Especially Hylleberg, Engle, Granger, and Yoo [HEGYI (1990) consider the seasonal frequency in quarterly time series, while Beaulieu and Miron (1990) and Franses (19901 extend their testing procedure to the monthly case.’

In previous work [HEGY (199011 it was found for UK data that there were seasonal unit roots in consumption but not in income, presumably because of different tastes for consumption at different seasons. In this paper Japanese data is examined because of the strong seasonality in income generated by the end-of-year bonus system and the increases in bonus payments in the second quarter. Hence we examine whether the seasonality in income is cointegrated with that in consumption and find some empirical support of a cointegrating relation at the annual frequency.’ The economic interpretation of such a cointegrating relation is that of a slightly impatient utility-maximizing borrowing-constrained consumer who uses his bonus payments to replace his worn out clothes, furniture, etc.

An estimation and testing procedure similar to the one suggested by Engle and Granger (19871 is developed, and it is argued that the test statistics for noncointegration occurring at the frequency $ of a cycle have the same distribution as the test statistic obtained for the zero frequency case by Engle and Granger (1987) and Engle and Yoo (1987).

The distribution of the test statistics for noncointegration at the frequency f (and +) is obtained using the asymptotic distribution theory for a pair of complex roots on the unit circle [see Chan and Wei (1988) or Ahtola and Tiao (198711, and the small sample distribution and critical values are found by help of Monte Carlo simulations. The power of the test is also analyzed via Monte Carlo experiments. They indicate that the power is quite high at least for moderate sample sizes and when compared to the power of other tests for cointegration and integration.

2. Tests for seasonal integration

In HEGY (1990) an integrated series is defined as a series that has an infinite value in its spectrum at a particular frequency 8. For quarterly data the frequencies of interest correspond to 8 = 0, $, i, i of a cycle (2~r). In the following we will concentrate on series integrated of order 1 implying that the frequency 8 = 0 corresponds to a component of the form

‘See Hylleberg (1992) and Ghysels (1990) for a survey.

2Cointegration at seasonal frequencies are also found by Kunst (1990) using macro series for four European countries.

R. F. Engle et al., Seasonal cointegration 277

(1 - B)x, = Et, while 8 = i corresponds to (1 +B)x, = .st and 0 = b, i to (1 + B%, = (1 + iBX1 - iB)X, = E,, .st being a stationary process and B the lag operator. Notice that 1 - B4 = (1 - BXl + BXl + iBX1 - iB) = (l-B)(l+BXl+B*)=(l-B)(1+B+B2+B3)=(1+B2X1-B*).

Due to habit and convenience E, is often assumed to be i.i.d. or n.i.d. It is quite easy to show that a process such as (1 - B4h, = E, or (1 + B)x, = E* has the unlikely property that a shock to the process may alter the seasonal pattern completely. Just consider X, = -x,-i + E, and a draw of .st dominat- ing x,-i. In such cases x, may have the same sign as x,-i and the seasonal pattern has been turned around. The definition of integration does not require at to be anything else than stationary, in fact E, can be bounded, heteroskedastic, autocorrelated conditional heteroskedastic, autocorrelated, nonsymmetric, etc. Due to the complex nature of the economic system, i.e., of the data-generating process Ed, a simple univariate representation such as (1 - B4)x, = Ed may be expected to display such behaviour. Such a univariate representation is therefore also best seen as an approximation that should be interpreted with care. Here an integrated seasonal model is applied to data with a varying seasonal pattern and the choice between a deterministic

0.20 -

0 2 0.10.

’ -0.00

- -0.10

- 0.20 1 , 0 20 40 60 80 100 121

1961.1-1987.4

-0.61 0 20 40 60 80 100 12(

1961.1-1987.4

Fig. 1. Quarterly change in the log of real total consumption and in the log of real disposable income.

278 RF. Engle et al., Seasonal cointegration

seasonal model and integrated model depends on the degree of variation in the seasonal pattern.

Based on an expansion of an autoregressive representation such as +(B)x, = E, around the points 8, = f 1, f i, a test procedure extending the well-known Dickey-Fuller test for integration at frequency 0 = 0 is developed. The test is based on the auxiliary regression,

Y4t = (1 - B4)X,

= Tlyl,t-l + T2Y2,r-1 + T3y3,t-2 + r4y3,t-1 +&t, (1)

where yrt = (1 +I3 + B2 + B3)x, is the observed series adjusted for the seasonal unit roots at 8 = +, +, +, y,, = - (1 - I3 + B2 - B3)x, is the observed series adjusted for the unit roots at 8 = 0, i, f, while y,, = - (1 - B2)x, is the observed series adjusted for the unit roots at 0 = 0, $. The tests for a unit root at frequency 0, i, and $ are then based on the ‘t’ values of rTTI and rTT2 both of which are distributed as a Dickey-Fuller distribution and an ‘F’ test

0.20

0.16

0.12

0.08

0.04

0.W

-0.04

- 0.06

-0.12

0.4

4 -0.3 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28

1961-1987 1961-1987

Fig. 2. Log of real total consumption [qc(i), i = 1,2,3,4] and log of real disposable income [qy(i), i = 1,2,3,4], ith quarter value minus the average of the calendar year.

R.F. E&e et al., Seasonal cointegration 219

on rr3 n r4 = 0 or in case of rd = 0 on the ‘t’ values of ~a, respectively. The critical values can be found in HEGY (1990) and also in Fuller (1976) for tTl and t_ and in Dickey, Hasza, and Fuller (1984) for t_.

The auxiliary regression in (1) may be augmented by lagged values of the dependent variable (1 -B4)x, without any effect on the distribution under the null as is the case with the Dickey-Fuller procedure. However, the power and size of the test may depend critically on the ‘right’ augmentation being used. From Monte Carlo experiments we know [see Haldrup (1989) among others] that the power of the test suffers if too many auxiliary parameters are applied to render the errors white noise, while the size may be far greater than the chosen level of significance if we use too few parameters [see also Hall (199011. In addition one may add deterministic terms like an intercept, seasonal dummies, and a trend, but this will change the distribution.

The Japanese data used are total real consumption, cI, and real disposable income, yr, from 1961.1 to 1987.4 in 1980 prices and in logs. Both series have a distinct seasonal pattern. The income series have strong seasonal peaks

x Q 0.20 I u 0.20

& 0.10 %. G , 0.10

z , -0.00 ’ -0.00

d II 11 -0.10 z u” -0.10

-0.2oI 0 20 40 60 80 100 120

1961.1-1987.4

3 0.6 f 0.4 Q I 0.4 3 0.3

N k 0.2

’ + 0.1

‘p -0.0 B -0.0

- -0.2 II

7 z -0.1

“1 1’ -0.4

.‘I‘ “II -0.2 ^, -0.3 j: -O.,+ 80 100 120

1961.1-1987.4

-0.201 I

0 20 40 60 80 100 120 1961.1-1987.4

-0.41 0 20 40 60 80 100 120

1961.1-1987.4

Fig. 3. Unit root transformations of the log of real total consumption and the log of real disposable income.

280 R.F. En& et al., Seasonal cointegration

Table 1

Tests for seasonal integration in the log of total real consumption, cc, in the log of real disposable income, y,, and in the log of the consumption-income relation, c, -y,, in Japan,

1961.1 to 1987.4.

Auxiliary regression

Variable

Deter- ministic parta Augmentation

‘t’ ‘I’ ‘1’ ‘t’ ‘F’

Tl =2 773 r4 773n574

I,TR 1,2,4,5,6,8,10 - 1.49 - 1.71 - 1.67 0.94 1.81 cr I, SD, TR 1,4,5,6,8,10 - 1.41 - 0.94 - 4.23b 1.03 lO.llb

I,TR 1,3,4,7,8 - 1.14 0.62 - 1.58 - 0.46 1.48 Yf I, SD, TR 1,3,4,7,8 - 1.11 - 0.67 - 2.00 - 0.55 2.24

I, TR 1,3,4,5,7,8 - 0.78 1.74 - 1.77 0.26 1.60 ct - Yr I,SD,TR 1,4 - 0.60 - 1.57 - 2.50 - 0.45 3.22

aI = intercept, SD = seasonal dummies, TR = trend. bWe reject a unit root at a 5% significance level.

with the fourth-quarter peak being caused by massive bonus payments. In addition, a growing number of firms have engaged in bonus payments in the second quarter as well.

The first differences of the two series appearing in fig. 1 clearly show that the seasonal pattern changes over the period in both series. This is even more evident from fig. 2, which shows the four quarters of each of the series, and from fig. 3, which shows the transformations -(l - B + B2 - B3) and --Cl- B2) of both c, and yr. Especially the income series has a changing seasonal pattern where ‘spring’ becomes ‘summer’ as the qy2 series crosses qy3. Graphs like these were advocated by Franses (1991).

The outcomes of the formal test are shown in table 1. The results indicate that the income series is integrated of order 1 at all frequencies 0, f, 3, while the consumption series is integrated of order 1 at the frequencies 0 and 4, but the test has some difficulty in separating a seasonal unit root at frequency i from a deterministic seasonal pattern. Notice that the order of augmentation is presented and that ‘holes’ are allowed in the lag distribution. This augmentation is chosen in order to whiten the residuals at the cost of the minimum number of parameters. Too many parameters will decrease the power of the tests, while too few may render the size far greater than the level of significance.

3. Tests for seasonal cointegration

Based on the definition of integration at a specific frequency, HEGY (1990) extend the theory of cointegrated systems to cover cointegration at

R.F. Engle et al., Seasonal cointegration 281

frequencies other than the long-run frequency. Let us consider an N X 1 vector of zero mean variables x, which are all Z,(l) at the frequencies 0 = 0, a, i, i. The Wold representation can then be written as

(1 -B4)Xt = C(B)Ef, (2)

where E, is an N X 1 vector of NID(O,n) variables and C(B) and N X N matrix of lag polynomials.

Cointegration at the zero frequency then depends on the existence of an N x r matrix c+ N > rr 2 0 such that a;C(l> = 0, while cointegration at the frequency i requires the existence of an N X rl matrix (~z such that &C(- 1) = 0. The columns in (or and (~z are called the cointegrating vectors at the frequencies 0 and i, respectively, while rr and r2 are called the cointegrating ranks. Cointegration at the frequencies a and + corresponding to the root i and its complex conjugate -i is most elegantly handled by extending the notion of a cointegrating vector to that of a cointegrating polynomial vector in a(B) = (Ye + a4B such that a’(i>C(i) = 0, where cy3 and a4 are N X r3 vectors, N > r3 2 0.

The Wold representation is, however, not the most useful in applied work, and instead we will consider one of the alternatives the so-called Error Correction Model (ECM) which, in accordance with HEGY (1990), may be written in two forms:

or

A*(B)A,xt = Y,~,y,,t-1 + Y2a3y2,t-l - (7’3a3 - 3/4a4)y3,t-2

+ (Y4Ly3 + Y3’Y4)Y3,t-l+ Et (3)

A*(B)A,x, = ~I~,YI,,-I + ~2a2~2,t-1

-(73 + 74B)(a3 +a4B)y3,t-2 +&t, (4)

where yl, y2, y3, and y4 are N x rl, NX r2, N X r3, and NXr, matrices containing the weights to the cointegrating relations for the different frequencies in each of the N equations. yl,, yt2, and yt3 are N X 1 vectors containing the transformed observations in x,, i.e., y,, = (1 + B + B2 + B2)x,,

y2t = -(l -B + B2 - B3)_v,, and y3t = -(l - B2)xt. The N X N polynomial matrices A*(B) and A;“(B) have all their roots outside the unit circle, while A*(O) =A;“(01 = I,,,. This implies that the left-hand side is a stationary multivariate autoregression in the fourth differences, while the cointegrating relation on the right-hand side represents the long-run cointegrating relations a;~,,, the cointegrating relations 4y,, at the frequency i, and the cointegrating polynomial relations ((Y; + a6B)yt at the frequency $ (and $1.


In our case, where we consider a system of two variables consisting of consumption and income where yr and c, N 1,(l), 8 = 0, i, i, i, there may exist one or no cointegrating vector at each frequency. If the cointegrating rank for c, and yt is one at all frequencies, 8 = 0, f, 4, f, the first equation in the two-equation ECM systems corresponding to (4) becomes

+ Y&l,,-1 - (Y12Y1,t-1) + Y12@2,t-1 - a2*y2,*-1)

-(?I3 + Y14B)(C3,t-2 - a32Y3,t-2 - a41C3,t-3 - a42Y3,t-3)

+s*, (5)

and a similar form may be obtained for yt. Also, in cases where yt is weakly exogenous for the parameters in the model for c, (the parameters of interest), only the model for c, will have the error-correcting terms and the equation will look like (5) with j allowed to run from zero and not from one, i.e., A, y, is allowed.

Notice, that as cointegration at the long-run frequency is interpreted as indication of a ‘parallel’ long-run movement in the nonstationary series c, and Y*, cointegration at a particular seasonal frequency is interpreted as evidence for a ‘parallel’ movement in the corresponding seasonal component of the two series which both exhibit a varying seasonal pattern.

The terms in eq. (5) must be stationary since components with different periodicities must be uncorrelated. Although yir and tit, i = 1,2,3, have asymptotically infinite variance and almost all linear combinations will have infinite variance, the following particular linear combinations will be I(O) at all frequencies:

Z1t =c1t - al2Ylt,

z2t = C2t - ff22Y2ty

Z3t = C3t - (Y32Y3t - a41c3,t-1 - (Y42Y3,t-1. (6)

For the first two a least squares regression will give a superconsistent estimate of the cointegration parameters as in the Engle-Granger two-step method. Furthermore, these estimates can be used directly in specifying and estimating the error correction model, and tests for cointegration at these frequencies can be carried out by testing the residuals from such cointegrating regressions for any remaining unit roots at the particular frequencies 0 and a.


In the case of .z~~, this is a dynamic relation and hence is potentially more complicated. First note that y,, +Y~,~_~ is Z,,,(O) as is cg, + Q_~. Since linear combinations of Z(O) series must be at most Z(O), zqr = z~,~_ I +

(1 + m&,+1 - ‘~~~y~,~+~) will again be at most Z(0). It will also be a linear combination of the same variables as in zgt, hence these parameters are not identified simply by the restriction that the linear combination is Z(O). A simple solution is to eliminate one of the variables, and in our case the lagged value of c3r has been eliminated. The cointegrating relation between c31 and y,, is estimated by regressing cjl on yjr and ~~,~_i. As usual this will be superconsistent for the true parameters and can be used in a two-step estimation procedure of the error correction model. Again, the residuals from this model will be used to test for seasonal cointegration at the annual frequency f. Although the estimates obtained are superconsistent, the small sample biases can be expected to be large unless the R*, i.e., the fit, is high.3

Let us denote the residuals obtained from regressing cl1 on yi,, c2, on y2*, and cgl on Y,, and ~~,~-i, u,, u,, and w,, respectively. The test of noncointegration at the zero frequency can then be performed by an auxiliary regression on Au, on u,_, with or without deterministic parts and augmented by the necessary lagged values of Au,. However, due to the estimation of the cointegrating vector the ‘t’ ratio for the coefficient to u,_, is not distributed as the usual Dickey-Fuller ‘t’. Instead, one may use the fatter-tailed distributions obtained by Engle and Granger (1987) and Engle and Yoo (1987) for exactly such cases.

The same distributions can be applied when testing for noncointegration at the biannual frequency 4. Here the basic part of the auxiliary regression becomes (u, + u,_ i) on - (u,_ i), where the minus sign is used in order for the distribution to be the same as for Au, on u,_i. Otherwise we would have to use the mirror image of that distribution.

Thus, while the distributions of test statistics for cointegration at the frequencies 0 and i may be found in the literature, this is not the case for tests of cointegration at the frequency $. Let us therefore discuss the theoretical results which are available for testing complex unit roots, and develop the asymptotic distributions of test statistics for seasonal cointegration at frequency $ (and i).

Suppose (c3,, y3r) is generated by cgt + ~3,~-2 = .scl and y3r +y3,,_2 = &yf

with the starting values c 3,0=c3,;1=0 and y,,,=y,,_,=O, while &CI and E,,~ are independent identically distributed with mean 0, variances CT?* and a,*, and COV(F,,~, E,~) = 0.

Similarly to the above two cases, the tests for seasonal cointegration at frequency $ can be performed by testing for the existence of a pair of

‘In cases where there are seasonal unit roots in the series and the cointegrating relation is thought to be a long-run relation between c, and y,, the cointegrating regression c,-on yt gives inconsistent estimates [see HEGY (1990)].


complex roots on the unit circle in the residuals from a regression which here is

Cgt=P11Y3r+P^2Y3,t-1+W,. (7)

In analogy with the HEGY test, the null hypothesis of no cointegration at frequency $ would imply that both rTT3 and r4 were zero in the auxiliary regression of the form

(wt + w,-~) = 7r3( -w,_*) + 7r4( -w,_~) + error. (8)

If & and & are known, the problem becomes simply a test for a pair of complex unit roots in a univariate process, since this model implies that w, is equal to c, for the true values of pi and & being zero.

The distributional results for this case are presented in Theorem 1.

Theorem 1. PI and & known. Consider the regression (8) where & and & are assumed known in (7). The least squares estimator of n3 and rr, converges weakly in distribution4 as

T7j3 -% 2$J$, and T7;, 3 21,4/14~, (9)

where 3 means weak convergence in distribution, while the t values are

The F statistic on r3 n T, = 0 has the same limiting distribution as +[t& + ti4], i.e.,

F .+i,n?i, -s +(rcr: + J11)/43, (11)

where

RF. E&e et al., Seasonal cointegration 285

W,(r) and W,(r) are independent standard Brownian motions on the unit interval [0, 11.

Proof: Follows directly from Ahtola and Tiao (1987) and Chan and Wei (1988).

Notice that the asymptotic distribution of te3 is independent of the value of r,. This result, observed first by Ahtola and Tiao (19871, is very useful in practice since the test for the existence of a pair of complex unit roots can be undertaken without any a priori information on the value of TV. This explains the similarity of the quantiles in the distribution of ts3 in HEGY (1990) to those for 7^’ in Dickey, Hasza, and Fuller (1984).

When PI and & are unknown they must be estimated and the distributional results are presented in Theorem 2.

Theorem 2. PI and & unknown. Again consider (7) and (8) and let &and & be unknown but estimated by least squares. The estimates PI and pz then converge weakly in distribution as

(12)

while the least squares estimates of r3 and r, converge weakly in distribution as

T 7j3 5 28,/4 and T G4 -% 2e,/$.

Furthermore, the t statistics converge weakly in distribution as

tsqJ+ e4/[4(1 +s:+rS)]“‘. (14)

The F statistic on r3 n rd = 0 has the same limiting distribution as +[t& + ti,], i.e.,

F =3nw4 -S ;((e; + Q/4(1 + s: + G)),

J.Econ-K


where

e,=j’(WldWl+W,dW,)+(5:+S:)j1(W,dW,+KdK) 0 0

- &/ol( WI dW, + W, dW, + W, dW_, + W, dW,)

+ 52j1( W, dW, - WI dW, + W, dW, - W, dW,), 0

e,=/‘(w,dW,- W,dW,) + (5:+5:)/1(w,dW3-WsdW,) 0 0

-@W,dW,- W,dW,+ W,dW,- W,dW,)

+ &/1(W3 dW, - W, dW, + W, dW, - W, dW,), 0

d, = /‘(W;” + W;) dr 0

-(tl + C2)(‘(W,W, + W2K + W2W3 - J+‘,W,) dr. 0

Proof. The proof is presented in the accompanying working paper [Engle, Granger, Hylleberg, and Lee [EGHL] (1991, app. C).

Notice: Arguments similar to those used in Stock (1987) yield @i and & would converge to constants when c, and y, were seasonally cointegrated at frequency $.

The test statistics have so far been constructed using the residuals w, from the regression model (7). If the statistics are based upon the least squares regression


where SDj, is a quarterly seasonal dummy, the limiting distributions of the test statistics have the same form as in (9)-(16), except that the W,(r)‘s are now replaced by the demeaned standard Brownian motions, say, q(r) = q::(r) - /,‘B$~,(r)dr (i = 1,2,3,4). In this case the asymptotic distribution of tg, in (9) is the same as that of tG, in Dickey, Hasza, and Fuller (1984).

Notice that our discussion has been confined to the case of i.i.d. error terms. Following the work of Phillips (1987) we can show that the asymptotic distributions are also valid for heterogeneous errors under a mixing condition. When the error terms are intertemporally dependent, however, the limiting distributions depend on nuisance parameters, i.e., the variance of E,~ and the spectral density of E,~ at the zero frequency in our case. When E, = (E&, E,,~)’ follows a stationary and invertible ARMA process, we can correct the dependence on nuisance parameters by augmenting lagged dependent variables in the regression model (8) or by using the asymptotically similar tests suggested by Phillips and Ouliaris (1990, thms. 4.1, 4.2). Again the power and size of the test may be expected to depend critically on the right augmentation being used.

To actually obtain the critical values for the test of no cointegration at the frequencies $ and f, i.e., a test of a complex pair of roots with module 1 at these frequencies, a series of Monte Carlo experiments have been conducted.5 A cointegrating regression of cjt on y,, and y,, f _ 1 with or without deterministic parts were performed. Based on the residuals from this regression, wt, an auxiliary regression (wt + w,_~) = ~~J-Jv,_~) + rJ-~~_i) + deterministic parts + error is performed and the ‘F’ value of the joint test rr3 n r4 = 0 calculated together with the ‘t’ values for rg = 0 and “4 = 0.

Comparison of the critical values of the ‘F’ statistics in the appendix with the critical values obtained by HEGY (1990) reveals that the use of estimated residuals implies a fatter-tailed distribution. In fact, the 0.95 fractiles presented in the appendix range from 7.0 in the case where seasonal dummies are not in the cointegrating regression to 10.0 when seasonal dummies are present. This must be compared to 95% fractiles ranging from 3.0 to 6.6 when the series are observed. In addition, the distribution is only affected by the inclusion of deterministic seasonal terms in the cointegrating regression, while the inclusion of just an intercept does not change the distributions. The inclusion of seasonal dummies in the auxiliary regression also fattens the tail of the distribution although to a lesser extent. A similar result is obtained for the critical values of the ‘t’ statistics on TV, while the distribution of te4 has somewhat fatter tails than the one obtained in HEGY (1990).

‘The design of the experiments all made in GAUSS 2.0 are the following. The number of replications for each distribution is 30,000, and there were two noncointegrating variables with unit roots at frequencies l/4 and 3/4. The random number generator used is RNDN supplied by GAUSS 2.0. The two variables y, and cl are generated by processes such as (1 + B*)ys, = .zr,, where aCI - NID(0, 1) and (1 + B2)c3, = eCI, where eCl - NID(0, 1). This implies that car and y3, are both Z,,,(l), but not cointegrated at the frequencies l/4 and 3/4.

288 R.F. Engie et al., Seasonal cointegration

.-_ - _

______-_------c,-“‘~~~-~==, ---__ -_ ’ \

-----____ --._ 7=136\ . \ -_ \ ’

\ --_ 1’ ’ \ , ’ I,7 = 200

\ \ ’ I \ \’ \ \ \ ’ \ 1’ ’ \ \ ’

\ \

T= loo’! ‘) I \ \ \

\ \ ‘\ , \

T= 48’, \ ’ \

\ \ ’ \ \ \ ’

\ \ \ ’ \ 1

\ \

\ ‘:I \ ‘\ \’

\ \’ \ ‘.\’

Nominal size 0.05. \

Number of replications 3000. h 0.0 ’ ’ ’ ’ ’ ’ ’ ’ s I ’ ’ ’ ’ n I ’ ’ ’

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

The error in the cointegration relation is (1 + qB*)e. q is on the axes.

Fig. 4. Power of the cointegration F test at frequency l/4. Cointegration relation with intercept and seasonal dummies; auxiliary regression with no deterministic parts.

Notice, that the small sample critical values were obtained by use of the cointegrating regression csr on yj, and ~s,~_r and not from the asymptotically equivalent regression with cs,+r included. In order to investigate the effect of this on the critical values, Monte Carlo experiments were conducted based on the cointegrating regression of cjt on ys,, cj, ,_ r, and ys, ,_ i. We found that the effect on the critical values was negligible.

The power of the test was evaluated by 3000 Monte Carlo replications with a basic model such as (71, where y, is generated as (1 + B’)y,, -

NID(0, I>, ~3.0, ~3, -, = 0, while the error term was generated according to (1 + qP)w, - NID(O,‘l) for 0 I q 5 1. The limiting case of 4 = 1 is the null hypothesis of no cointegration and the power is calculated for q = 0.98, 0.96, 0.90, 0.80, and 0.50, T = 48, 100, 136, and 200, and for different combinations of deterministic components. In general the power of tests is quite impres- sive, especially for sample sizes of 100 and over. An illustration of a representative case is depicted in fig. 4, where it is shown that no cointegration is rejected in more than 80% of the cases if the error in the cointegrating relation follows an AR process such as w, = O.~W,_, + e, where T = 100 and the power is very close to one for sample size of 136 and 200.


Table 2

Test for cointegration at frequency 0: the long run.a

Cointegrating regression

Coefficient Tests for unit (cointegrating Deter-

roots in vector) ministic Auxiliary regression residuals

Regres- Regressor components Determin- Augmen- ‘DF sand YlI included R2 istic part tation DW to,

Cl1 0.772 I,TR 0.998 - 1,475 0.08 - 0.59 (0.015)

Cl1 0.959 I 0.995 - 1,4,5 0.04 - 0.83 (0.007)

Cl1 0.970 - 1.00 - 1,4,5 0.04 - 0.99 fO.OOlJ

Cl1 1 (fixed) - I, TR 1,435 - 0.86 I, SD, TR 1,495 - 0.88

aThe tests are based on the ordinary augmented Dickey-Fuller regression Au, = w,u,_ 1 + Z,k=!b, Au,_~ + e,, where u, is the residual from the cointegrating regression. The ‘t’ statistic is distrtbuted as described in Engel and &anger (1987) and Engle and Yoo (1987).

At last in this section let us consider the possible cointegrating relations for the Japanese data.‘j A likely candidate for a cointegrating vector common to all frequencies is (1, - 11, which implies that the log of the consumption income relation, i.e., c, - y,, is Z,(O) at the frequencies 8 = 0, f, i, %. As the cointegrating vector is not estimated, the distributions given in HEGY (1990) apply, but the results shown in table 1 clearly cannot reject a unit root at any of the frequencies 0, $, i, and $.

Table 2 contains the results for the zero frequency case where the cointegrating regression is run (a) with an intercept and a trend, (b) with an intercept only, and (c) without any deterministic part. The ‘Dickey-Fuller’ tests based on the residuals all indicate that a unit root cannot be rejected implying noncointegration at the long-run frequency.’

A similar result is obtained if the cointegrating vector is hxed at (1, - l), in which case the ordinary Dickey-Fuller distribution is applicable as the test is used on the observed although seasonally adjusted consumption-income ratio, cIt - yr,. Notice that the results cl, -yt, shown in table 2 and the t,, column for c, -yI in table 1 are quite similar. This should be no surprise as the regression in table 2 adjusts for the seasonal unit roots by preadjusting, while the regression in table 1 adjusts by including the proper variables in the regression.

‘The data used are from the National Accounts Japan. They are available from the authors and can be found in EGHL (1991).

‘The critical values used are found in Engle and Yoo (1987).


Table 3

Test for cointegration at frequency l/2: biannual.”


Regres- sand

Coefficient (cointegrating

vector)

Regressor

Y2r

Deter- ministic

components included R2

Auxiliary regression

Augmentation

Tests for unit roots in

residuals

‘DF DW r+?,

c2r 0.236 I, SD 0.939 L&3,4 2.72 - 1.81 (0.067)

c2t 0.340 I 0.893 1,4,5 3.65 - 1.23 (0.012)

C2r 0.343 - 0.881 1,4,5 3.15 - 1.27 (0.001)

aThe tests are based on the auxiliary regression (u, + uI _ I) = P,( -u, _ II + x5= ,b,(u, _j t

Ut_j_l)+e,, where U, is the residual from the cointegrating regression. The ‘t’ statistic IS distributed as the ‘Dickey-Fuller’ described by Engle and Granger (1987) and Engle and Yoo (1987).

Based on the residuals from the cointegrating regression of czI = -(l - B +B2-B3)ct on yZt= -Cl -B + B2 - B3)y,, cointegration at the biannual frequency i is also clearly rejected as will be seen from the results in table 3. Hence the series doesn’t seem to be cointegrated at the biannual frequency either.

Table 4

Test for cointegration at frequency l/4 (and 3/4): annual.”


Coefficient Tests for unit roots

(cointegrating in residuals

vector) Deter-

ministic ‘HEGY’ test

Regres- Regressors components Auxiliary regression t t F

sand Y3rr Y3,r-1 included R2 Augmentation is GA +,n+4

c3t 0.264 0.109 I, SD 0.974 1,2,3,4,5 -3.60 -2.17 8.90 (0.033) (0.034)

C3r 0.434 0.144 I 0.967 1,4,5 - 2.41 - 1.62 3.89 (0.008) (0.008)

C3l 0.441 0.151 - 0.956 1,4,5 - 2.31 - 1.55 3.59 (0.030) (0.010)

“The tests are based on the auxiliary regression (w, + w,_~) = P,(-w,_~) + rrJ- w,~,) + cf,lbj(w,_j + w~_,_~) + e,, where w, is the residual from the cointegrating regression. The distribution of the joint ‘F’ test for rr3 f~ rr4 = 0 and the ‘t’ test on z-3 and rre are given below.


A somewhat different picture emerges from the results shown in table 4. Here we depict the polynomial cointegrating vector between cat and y,,, i.e., a regression of cgi on ySt and Y~,~._.~. An asymptotically equivalent test could be obtained by running the regression with c3,(_ i. The regression is run with an intercept and seasonal dummies, with just an intercept, and without any deterministic components at all.

Notice that the results of the test for integration for c, at the frequency $ was inconclusive as the test was unable to separate between a deterministic seasonal component and a unit root component at the annual frequency. Based on the graphical evidence a varying seasonal pattern seems not unlikely, But of course in case c, is I(O) at frequency $ while yt clearly is f(l), cointegration is ruled out.

When applying the distributions given in the appendix, the ‘F’ value lies in between the 90% and 95% fractiles, while the results for the cointegrating regression with no seasonal dummies indicate noncointegration. A similar result is obtained by use of the ‘t’ statistics on r3 and rd. The results are thus indicating a weak ‘maybe’ when the question is asked whether there is cointegration between c, and yt at the seasonal frequencies f and a, while noncointegration at the long-run frequency and at the frequency i cannot be rejected.

4. Conclusions and economic interpretations

In this paper we extend the theory of seasonal integration and cointegration of Hylleberg, Engle, Granger, and Yoo (19901.s The extension consists of a test for seasonal non~integration at the annual frequency as the distribution obtained for the test of noncointegration at the frequencies 0 and l/2 are in Engle and Granger (1987). Besides providing the distribution of the test statistics, asymptotic results are presented both for the test for seasonal integration and the tests for seasonal cointegration. The critical values for the tests for seasonal cointegration at the annual frequency were found by Monte Carlo simulations, and the power of the test is evaluated in a similar way, The power of the tests seems quite satisfactory at least for sample sizes of 100 and over.

The tests were applied to the tota consumption and disposable income in Japan from 1961.1 to 1987.4. The results indicate that the log of the income series is integrated of order 1 at both the Iong-run frequency and at the seasonal frequencies. This result indicates that income is nonstationary and that the seasonal pattern has significant variation over the period. A similar result is obtained for consumption, although the seasonal pattern is more regular, and it is in fact a question whether a deterministic annual seasonai pattern is to be preferred.

*For an alternative approach see Engle and Yoo (1989) and Lee (1989).


The results of the cointegration analysis show that the log of the two series is not cointegrated at any frequency with cointegrating vector (1, - 11, and it was also shown that the seasonally adjusted data, i.e., the data adjusted for seasonal unit roots by summing over four consecutive quarters, is not cointegrated at the long-run frequency, neither with cointegrating vector (1, - 1) nor with an estimated cointegrating vector. The tests for seasonal cointegration indicates no cointegration at the bi-annual frequency, while there may be signs of cointegration at the annual frequency. This indicates that the annual seasonal components of consumption and income are similar.

Since Hall (1978) theories of consumption have been a major research area. In a recent paper by Winder and Palm (1990) on the life cycle consumption model it is shown that under certain (strong) assumptions the first-order condition of the constrained maximization of the representative agents life time utility implies 4(Z3Xl -B)c,+i = ca + ~~+r, where c, is consumption, 4(B) is a lag polynomial with no roots inside the unit circle, cc a constant, and E,+ 1 = c,+~ - E(c,+,lZ,) = r(1 + r)-i&(1 + r>-’ x E[~(B)Y t+l+i)Zt+l] - E(~#@3)y,+~+ilZ,), where Z, is the information set at time 1. The lag polynomial stems from the utility being a function of x, = @I)c,, and the closed form solution is obtained by assuming that the functional form of the utility function is V(x): -~-‘e-~~, y > 0. In case c$(B) includes a component with unit roots such as (1 + BXl + 8’) = (1 + B + B2 + Z33) the consumer cares about last year’s consumption, while a component counting only (1 + B2> implies that the consumer cares about the last half year’s consumption. The integration results obtained above for consumption may be interpreted correspondingly.

However, interpreting the finding of seasonal integration in the consumption series as an indication of a varying seasonal pattern does not correspond to the habit persistence model of Osborn (1988) who models UK consumption as an Z,(O), 8 = $, a, f, process with seasonally varying coefficients and seasonally varying intercepts.

Deaton (1989) presents a model based on a slightly impatient utility-maximizing consumer who faces borrowing constraints. In case of an integrated income series the optimal strategy for this consumer is to spend all his income in each period. Although further conditions have to be set up for this result to aggregate, for instance that the share of impatient consumers with a borrowing constraint is constant, cointegration between income and consumption may follow. This is of course not supported by the data either, but may give some hints of a theory model where seasonal cointegration exists between the consumption and income relation. In short, if a slightly impatient borrowing-constrained consumer has the habit of using his bonus payments to replace his worn out clothes, furniture, etc. when the payment occurs, one may expect cointegration at the annual frequency.

App

endi

x: C

riti

cal

valu

es f

or t

he t

est

of c

oint

egra

tifm

at

freq

uenc

y 3

30,0

00 M

onte

C

arlo

re

plic

atio

ns

wer

e ap

plie

d in

the

com

puta

tions

. T

he

stan

dard

er

rors

on

the

es

timat

es

vari

ed,

but

mos

t w

ere

fess

tha

n 0.

015.

Tab

le A

. 1

?J

Cri

tical

va

lues

fo

r th

e ‘t

’ an

d ‘F

’ st

atis

tics

on

r3

and

v.+

C

oint

egra

ting

regr

essi

on:

c, =

i,

y, +

& y

,_ , +

det

er

+ w

,. A

uxili

ary

regr

essi

on:

3 W

? + w

rcz

=7i

3(-W

t_2f

+rr

q(-W

1__,

)+“r

‘ !?

00

C

oint

. re

gr.

__

_.__

~”

____

__..

__.“

““.

s

with

‘I

’ ?r

3 ‘t

’ T

r,

‘F’

“3nv

4 E

~

~...

.-

_~.

p

Det

er.

T

0.01

0.

025

0.05

0.

1 -0

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0.02

5 0.

05

0.95

0.

975

0.99

0.

50

O.Y

O

0.95

-0

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0.

99

~~.

.._

~--

$

48

- 4.

04

-3.6

6 -

3.34

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.00

-2.9

9 -2

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5 2.

05

2.42

2.90

2.59

6.

01

7.46

8.

81

10.8

0 10

0 -

3.94

-

359

-3.3

0 -3

.00

-3.0

1 -2

.54

-2.1

2 2.

10

2.50

2.

94

2.61

5.

91

7.21

8.

63

10.2

4 P E

-

I36

- -

- 3.

90

3.57

3.

28

-2.9

8 -3

.01

2.53

-2

.15

2.13

2.

51

- 2.

92

2.60

5.

83

7.11

8.

39

IO.1

4 2

200

- 3.

89

- 3.

56

-3.2

9 -2

.98

-3.0

4 -2

.56

-2.1

3 2.

13

2.52

2.

99

2.63

5.

84

7.11

8.

35

10.1

0 $

.-__

___

. .._ -

____

_ ~.

.. .._

......

~

“.

..^_ _

..... ~

w

48

-

3.96

-

3.57

-

3.27

-

2.93

-

2.93

-2

.44

-2.0

3 2.

12

2.54

2.

96

2.59

5.

96

7.35

8.

77

10.5

1 N

O

- 3.

86

- 3.

54

-327

-2

.95

-2.9

5 -2

.49

-208

2.

12

2.52

29

.5

2.59

5.

83

7.10

8.

42

10.1

5 i

136

-3.8

4 -3

.54

- 3.

26

-2.9

6 -2

.99

-2.5

2 -2

.10

2.14

2.

52

2.98

2.

60

5.83

7.

13

8.40

10

.09

g.

8

200

- 3.

86

- 3.

52

- 3.

26

- 2.

96

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95

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53

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3 2.

15

2.55

3.

00

2.61

5.

79

7.01

8.

26

10.0

2 _-

..--

... -

_....

__

__

____

_ __

____

_-_,

- __

_ __

__...

--

48

-

4.87

-

4.49

-4

.18

-3.8

4 -

2.97

-

2.48

-2

.07

2.08

2.

47

2.95

4.

71

d:O

O

10.6

5 12

.18

14.1

1 10

0 -4

.77

- 4.

40

-4.1

2 -

3.81

-

3.02

-

2.56

-2

.14

2.10

2.

50

2.98

4.

70

8.66

10

.12

If.4

8 13

.26

I, S

D

136

- 4.

77

- 4.

42

-4.1

4 -

3.81

-

2.99

-

2.55

-2

.14

2.13

2.

50

2.97

4.

71

8.57

9.

99

11.4

1 13

.25

200

- 4.

76

- 4.

40

-4.1

2 -3

.81

- 2.

96

- 2.

52

-2.1

3 2.

12

2.52

2.

97

4.71

8.

57

9.99

11

.41

13.2

5 --

.-

~__

_ __

____

Tab

le A

.2

Cri

tica

l va

lues

fo

r th

e ‘f’

an

d ‘F

’ st

atis

tics

on

‘T

T~

and

rd.

Coi

nte

grat

ing

regr

essi

on:

c, =

i,y

I +

& y

r_ 1

+ d

eter

+ w

,. A

uxi

liar

y re

gres

sion

: w

,+w

,+2=

“3(-

Wr~

2)+

~~

(-W

,_l)

+I+

e,.

Coi

nt.

reg

r.

wit

h

‘t’

iT3

‘t’

r‘j

‘F’

rrg

n i-

r‘,

Det

er.

T 0.

01

0.02

5 0.

05

0.1

0.01

0.

025

0.05

0.

95

0.97

5 0.

99

0.50

0.

90

0.95

0.

975

0.99

48

- 3.

89

- 3.

52

-3.2

1 -

2.89

-

2.84

-

2.41

-2

.00

2.08

2.

46

2.90

2.

49

5.77

7.

13

8.51

10

.22

100

- 3.

86

- 3.

53

- 3.

24

- 2.

93

- 2.

91

- 2.

50

-2.1

0 2.

11

2.51

2.

94

2.56

5.

84

7.08

8.

33

9.88

-

136

- 3.

87

-3.5

3 -3

.26

- 2.

96

- 2.

96

- 2.

49

-2.1

0 2.

13

2.50

2.

93

2.58

5.

83

7.06

8.

20

9.76

20

0 -

3.86

-

3.52

-

3.26

-

2.95

-

2.97

-

2.52

-2

.12

2.13

2.

53

3.02

2.

59

5.79

7.

05

8.33

10

.07

48

- 3.

90

- 3.

52

-3.2

3 -

2.90

-

2.89

-

2.40

-2

.00

2.07

2.

47

2.93

2.

50

5.83

7.

22

8.59

10

.19

100

- 3.

87

- 3.

53

- 3.

26

- 2.

93

- 2.

98

- 2.

52

-2.1

0 2.

09

2.46

2.

93

2.57

5.

80

7.09

8.

31

10.0

2 I

136

-3.8

3 -3

.50

- 3.

24

- 2.

93

-3.0

0 -2

.51

-2.1

0 2.

14

2.53

3.

01

2.59

5.

73

7.07

8.

32

9.96

20

0 -

3.87

-

3.52

-

3.25

-

2.95

-

3.03

-

2.54

-2

.12

2.12

2.

52

2.98

2.

58

5.78

7.

10

8.27

9.

95

48

-4.8

1 -

4.44

-4

.11

-3.7

9 -2

.95

- 2.

46

-2.0

4 2.

08

2.48

2.

93

4.61

8.

79

10.4

2 11

.92

13.8

6 10

0 -

4.74

-

4.38

-4

.10

- 3.

78

- 2.

94

- 2.

49

-2.1

1 2.

09

2.46

2.

92

4.66

8.

56

10.0

3 11

.41

13.1

2 I,

SD

13

6 -

4.72

-

4.38

-4

.10

- 3.

79

- 2.

98

- 2.

51

-2.1

1 2.

12

2.50

2.

95

4.65

8.

55

9.94

11

.21

12.9

5 20

0 -

4.68

-

4.37

-

4.10

-3

.79

- 3.

01

- 2.

56

-2.1

3 2.

14

2.53

2.

99

4.67

8.

53

9.90

11

.22

12.8

7

Tab

le A

.3

Cri

tica

l va

lues

fo

r th

e ‘t

’ an

d ‘F

’ st

atis

tics

on

rT

T3 a

nd

r,.

Coi

nte

grat

ing

regr

essi

on:

c, =

bly

t +

& y

, L

+ d

eter

+ w

,. A

uxi

liar

y re

gres

sion

: W

,+W

,+*=

~3(

-WI_

2)+

~~

(-W

t_,)

+I+

SD

+e,

.

Coi

nt.

reg

r.

Fe

?Y

wit

h

‘t’

773

‘t’

7rd

‘F’

rr3

I-I z

-z,

,h

Det

er.

T

0.01

0.

025

0.05

0.

1 0.

01

0.02

5 0.

05

0.95

0.

975

0.99

0.

50

0.90

0.

95

0.97

5 0.

99

3

48

- 3.

92

- 3.

60

- 3.

34

- 3.

02

-3.1

4 -

2.63

-2

.18

2.18

2.

59

3.11

2.

95

6.44

7.

89

9.36

P

11

.39

100

- 4.

02

-3.6

8 -3

.43

-3.1

4 -

3.24

-

2.73

p

-2.2

8 2.

24

2.61

3.

11

3.18

6.

69

8.10

9.

50

11.4

6 -

136

- 4.

05

- 3.

71

- 3.

44

-3.1

6 -

3.28

-

2.77

-2

.31

2.31

2.

74

3.22

3.

24

6.73

8.

14

9.64

11

.64

$ 20

0 -

4.04

-3

.74

-3.4

9 -3

.19

-3.2

8 -2

.78

-2.3

3 2.

31

2.75

3.

23

3.30

6.

85

8.32

9.

77

11.6

0 8

48

- 3.

97

- 3.

63

- 3.

34

- 3.

04

-3.0

8 -2

.60

-2.1

6 2.

15

2.56

3.

01

2.97

6.

47

7.94

9.

37

11.4

8 ;

100

- 4.

05

- 3.

70

-3.4

3 -3

.13

- 3.

23

- 2.

71

-2.2

7 2.

25

2.71

3.

20

3.19

6.

70

8.14

9.

63

11.6

2 s’

I

136

- 4.

04

- 3.

72

- 3.

46

-3.1

7 -

3.29

-

2.76

-2

.30

2.27

2.

69

3.16

3.

26

6.76

8.

14

9.44

11

.49

g 20

0 -

4.08

-3

.76

-3.5

0 -

3.21

-

3.24

-

2.76

-2

.31

2.31

2.

74

3.21

3.

28

6.88

8.

29

9.59

11

.44

2

48

- 4.

69

- 4.

30

- 4.

00

- 3.

67

- 2.

90

- 2.

42

-2.0

1 2.

00

2.39

2.

85

4.26

8.

24

9.78

11

.26

13.3

8 8

100

- 4.

68

- 4.

33

- 4.

05

- 3.

74

- 2.

92

- 2.

47

-2.0

8 2.

10

2.48

2.

88

4.54

8.

33

9.74

11

.04

12.6

2 I,

SD

13

6 -

4.66

-4

.31

- 4.

05

- 3.

75

- 2.

99

- 2.

49

-2.1

0 2.

12

2.50

2.

95

4.55

8.

34

9.67

11

.05

12.7

9 20

0 -

4.64

-

4.33

-

4.08

-

3.77

-3

.00

- 2.

53

-2.1

2 2.

12

2.51

2.

94

4.58

8.

41

9.72

11

.00

12.7

0

Tab

le A

.4

Cri

tica

l va

lues

fo

r th

e ‘t

’ an

d ‘F

’ st

atis

tics

on

rr

r3 a

nd

T+

~

ifft~

grat

i~g

regr

essi

on:

c, -

&y,

+

&t-

t t

dete

r f

w,.

Au

xili

ary

regr

essi

on:

w,+

w,,,

=7C

3(-W

~_Z

)+rr

4(-W

r__!

)+I+

TR

+et

.

Coi

nt.

rep

. -

_.

ff

wit

h

‘?’

7r3

‘t’

7r.j

“F’rr

,nq

d;:

&

Det

er.

T

0.01

0.

025

0.05

0.

1 P

.--P

__

__.

0.01

i?

.o25

0.

05

0.95

0.

975

0.99

0.

50

0.90

0.

95

0.97

5 0.

99

2 .-

~__

_-

48

- 3.

77

-3.4

1 -3

.12

-2.8

1 -

2.82

-

2.35

-

1.96

2.

12

2.49

2.

93

2.40

5.

55

6.91

8.

22

9.87

.!

100

- 3.

81

- 3.

48

- 3.

22

- 2.

YO

-

2.93

-

2.45

-2

.06

2.12

2.

52

2.95

2.

49

5.6R

6.

93

8.16

10

.01

$J

- 13

6 -

3.82

-

3.48

-

3.22

-

2.92

-

2.96

-

2.49

-2

.08

2.12

2.

50

2.93

2.

53

5.71

6.

96

8.14

9.

51

8 20

0 -

3.84

-

3.50

-

3.22

-

2.93

-2

.99

-2.5

2 -2

.13

2.13

2.

53

3.04

2.

54

5.77

7.

01

8.29

9.

90

3

48’.

-3.7

9 -3

.41

-3.1

4 -

2.82

-2

.80

-2.3

1 -1

.92

2.14

2.

52

2.96

2.

39

5.52

6.

80

8.04

9.

72

100

- 3.

81

- 3.

49

- 3.

21

- 2.

90

- 2.

92

- 2.

43

-2.0

4 2.

11

2.50

2.

96

2.51

5.

66

6.89

8.

17

9.86

8.

I 13

6 -

3.84

-

3.50

-

3.23

-

2.92

-3

.02

- 2.

54

-2.1

2 2.

10

2.50

2.

90

2.55

5.

72

6.98

8.

20

9.81

3

200

- 3.

81

- 3.

51

- 3.

24

- 2.

93

- 2.

99

- 2.

50

-2.0

8 2.

14

2.52

2.

96

2.55

5.

73

I.00

8.

20

9.90

-_

..II

-___

g

48

- 4.

67 -

-

4.30

-

4.01

-

3.69

-2

.90

- 2.

39

- 1.

98

2.13

2.

53

3.01

4.

43

Sk

- 9.

96

11.4

2 13

.40

100

- 4.

65

- 4.

34

- 4.

06

- 3.

76

- 2.

98

- 2.

49

-2.0

7 2.

11

2.51

2.

96

4.58

8.

44

9.89

11

,22

12.9

1 I,

SD

13

6 -

4.65

-4

.31

- 4.

06

- 3.

16

- 2.

95

- 2.

45

-2.0

8 2.

14

2.53

2.

99

4.58

8.

39

9.78

11

.12

12.5

9 20

0 -

4.65

-4

.32

-4.0

5 -

3.78

-

2.99

-

2.49

-2

.11

2.14

2.

53

2.95

4.

62

8.36

9.

71

11.0

2 12

.72

Tab

le A

.5

Cri

tical

val

ues

for

the

‘t’

and

‘F’

stat

istic

s on

~‘s

and

?r,

. C

oint

egra

ting

regr

essi

on:

cI =

it

yI +

jay

,_

f + d

eter

+

w,.

Aux

iliar

y re

gres

sion

: W

,iW1;

2=~3

(-W

,_2)

+~~

(-W

,_If

+zS

+tf

+T

R+

e,.

Coi

nt.r

egr.

--

~...~

~~

..-.~

...-~

...-~

...-~

F

with

‘t

’ 8s

‘1

’ T‘

$ ‘F

’ vr

3 n

r4

h

Det

er.

T

____

_ -.

.___

__...

_ 0.

01

0.02

5 0.

05

0.1

0.01

0.

025

0.05

0.

95

0,97

5 0.

99

6.50

0.

90--

&G

----

--~

0975

0.

99

$

41,

- 3.

89

- 3.

52

-3.2

1 -2

.89

-2.8

4 --

-:2.

41

-2.0

0 2.

08

2.46

2.

90

2.49

5.

77

7.13

??

8.

51

100

-3.8

3 -

3.49

10

.22

-3.1

9 -

2.89

-

2.87

-

2.42

-2

.04

2.13

2.

52

2.96

2.

52

5.68

6.

93

8.15

8.

81

F -

136

- 3.

80

-3.5

0 -3

.21

- 2.

91

- 2.

94

- 2.

45

-2.0

6 2.

13

2.53

3.

00

2.52

5.

67

6.99

8.

25

9.86

$

2co

- 3.

84

- 3.

52

-3.2

3 -

2.92

-

3.05

-

2.55

-2

.10

2.14

2.

54

2.98

2.

57

5.72

6.

99

8.28

9.

87

..__I

_.

._~_

_...

!!_

48

- 3.

90

- 3.

52

- 3.

23

- 2.

90

- 2.

89

- 2.

40

-2.0

0 2.

07

2.47

2.

93

2.50

5.

83

7.22

8.

59

100

10.1

9 r;

-

3.87

-3

.53

-3.2

6 -

2.93

-

2.98

I

- 2.

52

-2.1

0 2.

09

2.46

13

6 -

3.83

2.

93

2.57

5.

80

7.09

8.

31

10.0

2 s

- 3.

50

- 3.

24

- 2.

93

-3.0

0 -2

.51

-2.1

0 2.

14

2.53

3.

01

2.59

5.

73

7.07

20

0 8.

32

9.96

2

- 3.

87

- 3.

52

-3.2

5 -

2.95

-

3.03

-

2.54

-2

.12

2.12

2.

52

2.98

2.

58

5.78

7.

10

8.27

9.

95

9 -_

___.

. 48

-

4.67

-

4.32

-4

.02

-3.6

9 -

2.89

-

2.40

-1

.99

2.14

2.

53

2.96

4.

43

8.46

9.

98

11.4

5 13

.46

g

100

- 4.

67

- 4.

30

- 4.

03

- 3.

72

- 2.

94

- 2.

47

-2.0

7 2.

11

2.49

2.

91

4.43

8.

26

9.64

10

.99

I, S

D

136

- 4.

62

12.6

1 -

4.30

-

4.03

-

3.73

-

2.93

-

2.47

-2

.09

2.14

2.

52

2.99

4.

53

8.28

9.

65

10.9

0 20

0 12

.51

- 4.

64

- 4.

32

- 4.

07

-3.7

8 -3

.04

- 2.

56

-2.1

2 2.

15

2.51

2.

96

4.60

8.

38

9.70

11

.07

12.6

9 -..

..~

__

.- __

__-.

..

298 R.F. Engle et al,, Seasonal cointegration

References

Ahtola, J. and G.C. Tiao, 1987, Distributions of least squares estimators of autoregressive parameters for a process with complex roots on the unit circle, Journal of Time Series Analysis 8, l-14.

Beaulieu, J. and J.A. Miron, 1990, Seasonal unit roots and deterministic seasonals in aggregate U.S. data, Manuscript (Boston University, Boston, MA).

Chan, N.H. and C.Z. Wei, 1988, Limiting distributions of least squares estimates of unstable autoregressive processes, Annals of Statistics 16, 367-401.

Deaton, A., 1989, Saving and liquidity constraints, Paper presented to the Munich Meeting of the Econometric Society.

Dickey, D.A., H.P. Hasza, and W.A. Fuller, 1984, Testing for unit roots in seasonal time series, Journal of the American Statistical Association 79, 355-367.

Engle, R.F. and C.W.J. Granger, 1987, Cointegration and error correction: Representation, estimation, and testing, Econometrica 55, 251-276.

Engle, R.F. and B.S. Yoo, 1987, Forecasting and testing in cointegrated systems, Journal of Econometrics 35, 143-159.

Engle, R.F. and B.S. Yoo, 1989, Cointegrated economic time series: A survey with new results, in: R.F. Engle and C.W.J. Granger, eds., Long run economic relationship: Readings in cointegration (Oxford University Press, Oxford) 237-266.

Engle, R.F., C.W.J. Granger, S. Hylleberg, and H.S. Lee, 1991, Seasonal cointegration: The Japanese consumption function, Discussion paper (University of California, San Diego, CA).

Frames, PH., 1990, Testing for seasonal unit roots in monthly data, Report no. 9032/A (Erasmus University, Rotterdam).

Franses, P.H., 1991, Multivariate approach to modeling univariate seasonal tune series, Report no. 9301A (Erasmus University, Rotterdam).

Fuller, W.A., 1976, Introduction to statistical time series (Wiley, New York, NY). Ghysels, E., 1990, On the economics and econometrics of seasonahty, Invited paper presented at

the Sixth World Congress of the Econometric Society, Barcelona. Hall, A.R., 1990, Testing for unit root in time series with pretest data based model selection,

Mimeo. (Economics Department, North Carolina State University, Raleigh, NC). Hall, R.E., 1978, Stochastic implications of the life cycle-permanent income hypothesis, Journal

of Political Economy 86, 971-987. Haldrup, N., 1989, Least squares biases in autoregression with testing for unit roots, Manuscript

(Institute of Economics, University of Aarhus, Aarhus). Hylleberg, S., ed., 1992, M~elling seasonahty (oxford University Press, oxford). Hyheberg, S., R.F. Engle, C.W.J. Granger, and B.S. Yoo, 1990, Seasonal integration and

cointegration, Journal of Econometrics 44, 215-238. Kunst, R.M., 1990, Seasonal cointegration in macroeconomic systems: Case studies for small and

large European countries, Research memorandum no. 271 (Institute for Advanced Studies, Vienna).

Lee, H.S., 1989, Maximum likelihood inference on cointegration and seasonal cointegration, Discussion paper no. 89-19 (Department of Economics, University of California, San Diego, CA).

Osborn, D.R., 1988, Seasonal&y and habit persistence in a life cycle model of consumption, Journal of Applied Econometrics 3, 255-266.

Phihips, P.C.B., 1987, Time series regression with a unit root, Econometrica .55,277-301. Phillius, P.C.B. and S. Ouliaris. 1990, Asymptotic properties of residual based tests for cointegra-

t&n; Econometrica 58, 165-193. _ _ _ _ Stock, J.H., 1987, Asymptotic properties of least squares estimators of cointegrating vectors,

Econometrica 5.5, 1035-1056. Winder, C.C.A. and F.C. Palm, 1990, Stochastic implications of the life cycle ~nsumption model

under rational habit formation, Mimeo. (University of Limburg, Maastricht).

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The Japanese consumption function