Interaction between Autocorrelation and Conditional Heteroscedasticity

Interaction between Autocorrelation and Conditional Heteroscedasticity: A Random-Coefficient Approach Anil K. Bera; Matthew L. Higgins; Sangkyu Lee Journal of Business & Economic Statistics, Vol. 10, No. 2. (Apr., 1992), pp. 133-142. Stable URL: http://links.jstor.org/sici?sici=0735-0015%28199204%2910%3A2%3C133%3AIBAACH%3E2.0.CO%3B2-E Journal of Business & Economic Statistics is currently published by American Statistical Association.

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0 1992 American Statistical Association

Journal of Business & Economic Statistics, April 1992, Vol. 10, No. 2

Interaction Between Autocorrelation and Conditional Heteroscedasticity: A Random-Coefficient Approach Anil K. Bera Department of Economics, University of Illinois at Urbana-Champaign, Champaign, IL 61820

Matthew L. Higgins Department of Economics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201

Sangkyu Lee CNB Economic Research Institute, 1536-7 Seoul, Korea In applied econometrics, we tend to tackle specification problems one at a time rather than considering them jointly. This has serious consequences for statistical inference. One example of this is considering autocorrelation and autoregressive conditional heteroscedasticity (ARCH) separately. In this article we consider a linear regression model with random coefficient autoregressive disturbances that provides a convenient framework to analyze autocorrelation and ARCH simultaneously. Our stationarity conditions and testing results reveal the strong interaction between ARCH and autocorrelation. An empirical example of testing the unbiasedness of experts' expectations of inflation demonstrates that neglecting conditional heteroscedasticity or misspecifying the autocorrelation structure might result in unreliable inference.

KEY WORDS: Lagrange multiplier test; Livingston biannual survey data; Price expectation, Stationarity condition; Unbiasedness hypothesis.

The autoregressive conditional heteroscedasticity (ARCH) model, introduced by Engle (1982), and its generalizations have recently become popular in modeling monetary and financial variables [for example, see Bera, Bubnys, and Park (1988), Bollerslev, Chou, and Kroner (in press), Engle and Bollerslev (1986), and the references cited therein]. Econometric tradition has been to incorporate autocorrelation in a model to capture the dynamics of economic time series (see Hendry and Mizon 1978). It is surprising, therefore, that most of the empirical works do not explicitly consider the presence of serial correlation simultaneously with ARCH. In some cases the absence of serial correlation is reported, as indicated by a Durbin-Watson, Durbin-h, or other tests, prior to finding a significant test for ARCH. If ARCH is found to be present, however, the presence of serial correlation must be reevaluated because the usual tests for autocorrelation are not valid in the presence of ARCH. Diebold (1986) demonstrated that the presence of ARCH invalidates the standard asymptotic distribution theory of the sample autocorrelations and hence of the Box-Pierce and Box-Ljung test statistics for serial correlation. Furthermore, as indicated by Engle, Hendry, and Trumble (1985, p. 75), the presence of autocorrelation can readily be mistaken for ARCH when, in fact, no ARCH is present. In light of the increasing use of ARCH models and the prevalence of serial correlation in economic data,

the purpose of this article is to investigate the theoretical and empirical consequences of the interrelationship between ARCH and autocorrelation in terms of stationarity of the time series process and inference. The presence of ARCH could be interpreted in several ways such as nonnormality of the disturbance term (Engle 1982, p. 992) and nonlinearity of the process (Higgins and Bera 1989, in press). In a recent paper, Bera and Lee (in press) established a close link between autocorrelation and ARCH through parameter heterogeneity. They applied White's (1982) information matrix (IM) test to the standard linear regression model with autocorrelated errors and found that Engle's (1982) Lagrange multiplier (LM) test for ARCH is a special case of one component of the IM test. Given Chesher's (1984) interpretation of the IM test as a test for parameter variation, it can be said that the presence of ARCH is "equivalent" to random variation in the autocorrelation coefficients (see also Tsay 1987). This leads us to consider the regression model with random autocorrelation parameters. Until recently, when testing economic theory emphasis was put on the systematic part of the model, not on the specification of the distribution of the disturbance term. Now it is recognized that the nature of the disturbance is quite crucial, and in applied work attempts are generally made to take account of the violation of the ideal iid situation. Adjustments are usually

134

Journal of Business & Economic Statistics, April 1992

made for one misspecification at a time, however and that might lead to misleading inference since the departure from the iid set up could be multidirectional. To illustrate this, we consider the problem of testing the unbiasedness of experts' expectations of inflation. We explored different distributional structure for the error, such as iid, either autocorrelated or conditionally heteroscedastic, and the presence of both autocorrelation and ARCH. Our empirical analysis shows that a combined model of autocorrelation and ARCH explains the data best. It also demonstrates that the test for unbiasedness crucially depends on the distributional assumption of the error term. The plan of the article is as follows. After describing the model in Section 1, in Section 2 we show how this formulation helps to study stationarity of the resulting process. In Section 3, we discuss how the tests for ARCH are affected by the presence of autocorrelation and vice versa. In Section 4, the empirical example is carried out using the Livingston survey data. Finally, Section 5 offers some concluding remarks. 1. THE MODEL

To analyze autocorrelation and ARCH simultaneously, we specify a general framework that encompasses models encountered in practice. For this purpose, we consider a static linear regression model with random coefficient autoregressive (AR) disturbances:

where y, is the tth observation on the dependent variable, x, is a k x 1 vector representing the tth observation on k regressors that are either stochastic or nonstochastic, and p is a k x 1vector of unknown constant coefficients. The disturbance E, follows a stochastic pth order autoregressive (AR) process with c$~ constant and rli, stochastic for all j. If the x, are stochastic, then they are assumed to be independent of E, for all t and s. This assumption can be relaxed to include lagged dependent variables within our framework. In an earlier expanded version of this article, we addressed this issue, which requires tedious algebra (see Bera, Lee, and Higgins 1990a). For the model (1.1), a number of other fairly general assumptions are made. Assumptions. (1) (7, = (qlt, . . . , t = 1, 2, . . . , N } , is a sequence of iid random vectors in which E(7,) = OPx1 and E(q,q;) = 2 , where 2 is a p x p positive-semidefinite matrix. (2) {u,, t = 1, 2, . . . , N) is a sequence of iid random variables with E(u,) = 0 and E(u:) = a2.(3) {q;, t = 1 , 2 , . . . , N} and {u,, t = 1, 2, . . . , N } are mutually independent. A wide range of models are encompassed as special %,)I,

cases of the model (1.1). Let be the information set available at time t. The series of past regression disturbances { E , - ~=J ~1, 2, . . . } is assumed to belong to q,-,. Then the conditional mean and variance of the disturbance E, under our assumptions can be represented as

where

4 = (4,, . . . , 4p)' and E , - ~ = and, letting 7, = (q,,, . . . , rl,,)',

...,

If 4 f 0, the error process is serially correlated, and if C # 0, the error process has ARCH. A diagonal C specifies the linear ARCH process proposed by Engle (1982), but a nondiagonal C specifies an ARCH process with additional cross-product terms between the past errors. We call the latter model the augmented autoregressive conditional heteroscedasticity (AARCH) to differentiate it from the standard ARCH. Unlike ARCH, AARCH is not symmetric in the sense that the conditional variance h, depends on the signs of the individual lagged E,'s. Therefore, AARCH displays the leverage effect of the asymmetric ARCH model suggested by Nelson (1991). Recent empirical applications have shown that an asymmetric ARCH model is particularly useful for stock-price data. We should note that there are other ways to introduce autocorrelation and ARCH jointly. An obvious alternative to our random coefficient formulation is to consider an A R disturbance with ARCH innovations. For observed time series data, as opposed to regression disturbances, Weiss (1984) considered such a model. The basic difference between these two approaches is that in Weiss's model the conditional variance h, is a function of both the past regression disturbances E, and the innovations u,, but in our model, as in that of Engle (1982), h, is a function of only the past E,'s. In that sense Weiss's model is more general. Weiss, however, did not consider the augmented ARCH terms, and our derivation of the stationarity conditions in the following section is more explicit and simpler. We should also note that if we start with an ARMA model with random coefficients, then h, will be a function of both E, and u, as in the work of Tsay (1987). In a follow-up paper (see Bera, Lee, and Higgins 1990b), we generalized the framework (1.1) so that the conditional variance of E, is also a function of lagged h,'s, thus obtaining a model

Bera, HigginIS, and Lee: ARCH and a Random-Coefficient Approach

that we termed generalized AARCH (GAARCH). As discussed by Bera et al. (1990b), derivation of the stationarity conditions for a combined A R and GAARCH model is very complicated. In Section 2, however, we use a known result from the time series literature to obtain the stationarity conditions for an A R model with AARCH errors. The condition is very simple and elegant, and no new derivation is required. Conditions of Engle (1982) and Milhoj (1985) can be derived as special cases. In this article, we emphasize the need for considering the presence of serial correlation and conditional heteroscedasticity simultaneously. In our empirical example, we find their joint presence and illustrate how considering only one of them can lead to misleading inference about the parameters of the systematic part of the model. Of course, for other data it is quite possible that only one of them might be present. As we have mentioned earlier, autocorrelation can be mistaken for conditional heteroscedasticity even when the latter is absent. It would be useful to have a test to distinguish pure serial correlation from A R C H or AARCH. Higgins and Bera (1990) suggested nonnested tests for ARCH and bilinearity, and their procedure can be easily modified for ARCH and AR. Another approach would be first to apply a joint test for ARCH and AR, as discussed by Bera et al. (1990a). If the joint test is significant, then we can use the tests of Section 3 to identify the departure(s). 2.

CONDITIONS FOR STATIONARITY

In this section we examine the necessary and sufficient conditions for the random coefficient A R disturbances to be covariance stationary. It is shown that the stationarity conditions for ARCH and A R are related when both are present. In the statistics literature, the stationarity conditions for A R series with random parameters has been extensively studied. Andel (1976) derived the necessary and sufficient conditions for stationarity of a univariate A R series with stochastic coefficients. Nicholls and Quinn (1982) extended the results of Andel to the multivariate A R model with random coefficients. Based on Andel (1976) and Nicholls and Quinn (1982), Nicholls (1986) stated simple conditions that are easy to check for second-order stationarity of the univariate A R series with random coefficients (see also Ray 1983). We now give the stationarity conditions for the error process of the model (1.1) as Proposition 1. In addition to Assumptions 1-3, let -n,Proposition be the field generated by {us, 7,; s t} and let 1.

CT

I

where O ~ p ~ , , x is, the ( p - 1) x 1 null vector, I,-, is the ( p - 1) x ( p - 1) identity matrix, and 4-, = . . . , 4,)'. Then the random coefficient AR(p)

135

disturbance process (1.1) has a unique =,-measurable second-order stationary solution iff (a) the eigenvalues of M lie within the unit circle and (b) vec(C)'a < 1, where a is the last column of the matrix ( I - M @ M)-I. Proof. See theorem 2 and lemma 3 of Andel (1976) and corollary 2.3.2 of Nicholls and Quinn (1982). To investigate the validity of the stationarity conditions for Engle7sARCH process in the presence of autocorrelation, we first consider the pure ARCH(p) process specified as h, = a2 + y,~:-, + . . . + y,~:-,. In this case, C = diag(y,, y2, . . . , yp) and a is a p 2 x 1vector with one as elements corresponding to y,, j = 1, 2, . . . , p. Given the regularity conditions y, r 0, j = 1, 2, . . . ,p , the required stationarity condition is summarized as Zj'=, y, < 1. Proposition 1, however, can be easily used to show that this condition is not sufficient to ensure stationarity in the presence of autocorrelation. We now consider an AR(p) process combined with ARCH(p). For this special case, after some simplifications we can state the two conditions for stationarity as (1) the roots of the polynomial zp - 4,zp-' - 42zp-2 - . . - 4p = 0 lie inside the unit circle, and (2) w(4) X?=, yj < 1, where w(.) is a function of (4,, 4 2 , . . . , &). It is clear that the influence of autocorrelation exercised through w(4) may invalidate the stationarity conditions for the ARCH process. This point will be further stressed in our empirical example. For the inflation model errors an AR(2) model with second order ARCH is found to be appropriate, and the stationarity condition involves both A R and ARCH parameters. In Section 3, we investigate how the tests for AARCH and ARCH are affected by the presence of autocorrelation and vice versa. 3. TEST STATISTICS

Before proceeding to derive the test statistics, some discussion of the likelihood function and the information matrix is provided because the tests will be based on the score or LM principle of Rao (1948). To simplify the derivation of the tests, we reparameterize the conditional variance as follows. Let y = vech[2C - diag(C)], where vech(.) is the vector-half operator, which stacks the lower triangular elements of a matrix, and diag(C) is a diagonal matrix whose diagonal elements equal the main diagonal of C. We then rewrite the conditional variance (1.3) as

. write the likelihood for where Z, = ~ e c h ( s , - + - ~ )To the model (1.1) we make the assumption that the elements of the sequences (7, = (vlr, . . . , qpr)', t = 1, 2, . . . , N), and {u,, t = 1, 2, . . . , N ) are normally distributed. Let L,(8) denote the likelihood function, where 8 = ( P ' , +', y', a 2 ) ' is the [k + p + p ( p + 1)/ 2 + 11 vector of all the parameters in the model and N is the effective number of observations. The likeli-

Journal of Business & Economic Statistics, April 1992

136

hood function LN(8) is assumed to be the true one and to satisfy the regularity conditions given by Weiss (1986). We also assume that the stationarity conditions for the model are satisfied. Under the normality assumption, the log-likelihood function is I,(@)

=

1 const - 2

z

ln(h,)

1

Note that u, = E, - 4'_~,-, = (y, - y:-,$) - (x, x_,- ,4)'P is a function of /3 and 4 , where y,-, - = (y,. . . , yt -p)l and x_, - = (x, - , . . . , xi-,)I. The LM test is based on the score vector d = alN(8)la6, which can be partitioned as d = (db, d i , dC , dm*)'to conform with 8 = (PI, +I, yl, u2)'. Let d be the score vector evaluated at the restricted maximum likelihood estimator (MLE) of 8 and let 9 be a consistent estimator of the information matrix. Then the LM statistic is and has, under the null hypothesis, a limiting 1 1d'f-la ~ x2 distribution with m df, where m is the number of restrictions imposed by the null hypothesis. A derivation of the elements of and 1is given in the Appendix. The expression for I++, depends on the estimate of y (through h,), and, therefore, standard tests for autocorrelation will be misleading if the presence of ARCH is neglected. Next we discuss the tests for AARCH or ARCH in the presence of A R and vice versa. In our model, to test for the AARCH (or ARCH) is to test whether or not the coefficients of autoregressive disturbances are varying. Obviously, the hypothesis to be tested is H,: y = 0 and the LM statistic for testing this hypothesis in the presence of autocorrelation will be denoted by LMAARcHIAR. Under the null hypothesis, our model is reduced to a linear regression model with AR(p) disturbances. The restricted MLE's of P , 4 , and u 2 are easily obtained from most computer packages. We define the two residuals i = y, and lit = 6, - 4'!,-,. Here we should note that li, should be viewed as an estimate of u, + C$ in (1.1); We further denote E = (if-,, . . . , if-,,)', 2, = vech(E,-,E~;,), W, = (1,Z:)', f, = (liflB2) - 1, Z = (z,, . . . , Z,)', i as an N x 1 unit vector, W = (i, Z), and f = (f,, . . . , f,)'. Then the LM statistic for testing H,: y = 0 in the presence of autocorrelation can be expressed as

,

,,

,

a

$6

, - , A

where R2 is the square of the multiple correlation in the regression off on W. This statistic has an asymptotic x2 distribution with p ( p + 1)/2 df when the null hypothesis is true. The proof of this result follows from Engle's (1982) derivation of the LM test for ARCH with straightforward modifications. Since multiplying by a scalar and adding a constant to the dependent variable of a linear regression that includes an intercept

does not alter the squared multiple correlation, the R2 can be easily calculated from the regression of uf on and (1, 2;,, Zit)', where^,, = (if-,, i f - 2 , . . . , Note 2 2 , = (E,-,E,-~,~ , - 1 ~ , - 3. ,. . , i,-p+lit-p)l. that z,, is a vector of p lagged squared residuals and z2,is a p ( p - 1)/2 x 1vector of distinct cross products between p lagged residuals. Considering the argument of Koenker (1981), it is expected that, when u, is not normally distributed, LMAARcHIAR is still an appropriate statistic as long as the fourth moment is finite and constant. We can derive the LM test for ARCH in the presence of autocorrelation (LMARcHIAR) simply by assuming that the covariance matrix C is diagonal. When C is a diagonal matrix, Z, reduces to z,,, a vector of p lagged values of if.Then the LM statistic LMARcHIAR is easily calculated from the regression of lif on (1, 6:..., if-,). The statistic will asymptotically follow a x2 distribution with p df. In Section 1, we noted that models of Weiss (1984) and Tsay (1987) also involve lagged innovations u,. Presence of that form of heteroscedasticity can be tested by running a similar regression. The only difference is that now the regressors should be (1, Lif-,, . . . , All of these tests will be carried out for our empirical application. Note that the validity of these tests depends on the correct specification of the A R process. Finally, we discuss a simple LM test tor autocorrelation, which takes account of a specified form of AARCH or ARCH disturbances. Since a specific parametric form of conditional heteroscedasticity is assumed, it is obvious that the performance of the test will depend on how accurately the AARCH or ARCH represents the conditional variance. The null hypothesis is formulated as H,: 4 = 0. Under the null hypothesis, our model is reduced to a linear regression model with the AARCH disturbances. Let 6 be the MLE's for the AARCH (or ARCH if 2 is diagonal) regression model. Then we can define 6, = y, - xi6 and h, = ?'z, + G2. The derivation of the tests is very simple since under the null hypothesis u, = E, and hence E(u,lq,-,) = 0. Then the appropriate score vector and corresponding part of the information matrix is given by A

A

A

A

,,

and (see the Appendix). Therefore, the LM statistic for testing H,: 4 = 0 in the presence of AARCH disturbances is

The statistic has an asymptotic x2 distribution with p df when the null hypothesis is true. The statistic LMARIAARcH is based on transformed

Bera, Higgins, and Lee: ARCH and a Random-Coefficient Approach

residuals obtained by dividing each residual by esti- mated conditional standard deviation. Allowing that " 6, = h;1122t and 61-1 = h;112((2t-l, 21-2, . . . &t-p) r LMARIAARcH can be expressed as

PI + P2xr +

Et,

137

(4.1) where yt is the actual rate of inflation and xt is the expected rate. Forecasts are said to be unbiased if Ho: pl = 0, P2 = 1 is true. To test H,, a proper specification of the distribution of E, is crucial, and we will emphasize LMAR~AARCH this point rather than the economic implications of Ho -1 in the following discussion. The first row of Table 1 gives the ordinary least squares (OLS) estimates. LM statistics for testing the presence L " ~ ~ have ~ the ~ same ~ c ~ as L " ~ ~ l of ARCH ~ ~ and ~ autocorrelation ~ ~ based on the OLS reexcept that C is a diagonal matrix and all terms with siduals are given in rows 1 and 4 of Table 2. These tests at the MLE parameters for Engle's hats are indicate the presence of autocorrelation and ARCH up ARCH regression model. to the second order. We also draw the same conclusion In the preceding discussion, we have shown that from Table 3, which presents the autocorrelations of L M ~ and ~ L l M~ ~~can ~ b~e constructed ~~ ~~ ~ On the c ~ OLS residuals and the squared OLS residuals. As embasis of the transformed residuals. In this sense, our phasized previously, these individual tests for AR results can be contrasted with the ARCH-corrected tests (ARCH) are not valid in the presence of ARCH (AR). for autocorrelation suggested by Diebold (1986) in the of ARCH and The MLE9s for different time series framework that are based on the sample AR through order 2 are also given in ~~b~~ We autocorrelations. Generalizations of the Diebold (1986) obtained these estimates using I M ~ L subroutine Z X ~ I N . type the regressi0n were s"gFirst we check whether we should consider the auggested WOO1dridge and be discussed in mented ARCH terms. In the context of Equation (l.3), the context of our empirical illustration. for p = 2, the augmented ARCH term is E,- I ~ t - 2 with coefficient y3, which is related to E(qtq:) = 2 by-y3 = 4. ILLUSTRATIVE EXAMPLE 2 cov(qlt, q2J. From Table 1, we note that the augT o illustrate our methodology, we consider the probmented ARCH parameter y, in Specification (2) is not lem of testing whether experts' forecasts are unbiased significant. Moreover, when a second-order ARCH specification is estimated, the first-order ARCH coefestimates of the actual inflation rate using the Livingston biannual survey data from June 1952 to December ficient is also not significant. With only an AR model, 1985. Since July 1, 1946, this survey has reported exboth first and second-order A R coefficients are found perts' semi-annual predictions of a set of key economic to be significant. The first-order AR coefficient, howvariables including the consumer price index. Starting ever, becomes insignificant when A R is combined with with Turnovsky (1970), the Livingston data have been ARCH as seen from the estimates of Specification (7). Here we should note that the reported t statistic is based used extensively to test hypotheses of the formation of price expectations [for example, see Brown and Maital on standard errors calculated from the Hessian matrix. (1981); Chan-Lee (1980); Figlewski and Wachtel (1981)l. Therefore, they will not be very reliable when the models are misspecified. A better approach would be to use The underlying model for testing the unbiasedness of Weiss's (1986) standard errors, which are robust to nonforecasts is Yt

=

I

3

Table 1 . Parameter Estimates of ARCH and AR Models Model number

(1)

Error process

81

8 2

White noise

(2) AARCH ( ~ 1 9 Y21

~ 3 )



(3) ARCH

(71372)



(4) ARCH (72)

(5) AR

(41)

(6) AR (

4

3

42)

(7) ARCH ( ~ 2 ) + AR (6, 42)



NOTE. Values in parentheses are asymptotic t statistics.

41

4 2

3

?7

72

?3

AIC

138

Journal of Business & Economic Statistics, April 1992 Table 2. LM Statistics for ARCH and Serial Correlation

LM test

1

3

2

6

5

4

ARCH

ARCHlAR(4,)

ARCHIAR(4,, 42)

AR

ARIARCH(Y2)

ARIARCH(Y,, Y,)

R-AR

R-ARIARCH(y2)

R-ARIARCH(y,, Y,)

NOTE: Values are the LM statistics for individual lag components and follow an asymptotic crit~calvalues at the 5% and 1% significance levels are 3.84 and 6.63.

normality. To rectify this, we base our model selection on the Akaike information criterion (AIC) and a number of specification tests reported in Tables 2 and 3. Looking at the AIC values reported in Table 1 for each model, we note that there is no improvement over model (1) when AR or ARCH is individually incorporated. When both A R and ARCH are included simultaneously, as in model (7), the AIC decreases from -420.9 to -427.4. The likelihood ratio statistic, with 3 df, for the hypothesis H,,: 4, = 4, = y2 = 0 is 12.4, which is significant at the 1% level. Note that the models in Table 1 were selected based on the LM test results in Table 2. This was a trial-anderror approach. First looking at Table 2, we look for those significant ARCH lags that are fairly robust to the different specifications of the A R part. It appears that the second-lag ARCH component is significant no matter how we specify the AR process. A similar approach is followed in selecting the A R specification. The significant first two orders of A R do not change over different choices of the ARCH model. We also present three versions of Wooldridge's (1990) robust LM test for serial correlation in Table 2. LMR-,, is constructed as N-SSR from regressing a vector of ones on 2, . where &,-,is 4,-, purged from its linear projection on the regressors in (4.1). This test will have a x2 distribution even if the conditional variance of the error is misspecified. LMR.ARIARcH(y,, and LMR-ARIARCH(yl,y2) are constructed in a similar fashion except that all quantities are weighted by the inverse of the estimated conditional standard deviations from Table 3. Autocorrelations of OLS and Standardized Residuals, and of Their Squares

NOTE: it and 4 denote OLS and standardized res~duals,respectively. The approximate

= ,121 standard error for these autocorrelations is N - '

'

2 distribution with 1 dl. The asymptotic

specification (3) and (4). (For a full discussion of the tests, see Wooldridge [1990].) The robust tests for autocorrelation, like our tests for autocorrelation in the presence of ARCH, indicate that autocorrelation through the second order is significant. These considerations lead us to select model (7). Further support for model (7) is indicated by the autocorrelation of the standardized residuals

and the squares of the standardized residuals reported in Table 3. Unlike the OLS residuals, the standardized residuals do not indicate any kind of dependence. We also computed White's (1980) test for static heteroscedasticity for the standardized residuals of model (7) and the OLS residuals of model (1). The computed value of the statistic for the standardized residuals is 2.821, which is insignificant for a x2 with 2 df. In contrast, the computed value for the OLS residuals is 17.28, which is highly significant. Finally, the standardized residuals were subjected to Jarque and Bera's (1987) LM test for normality, and we did not find any existence of nonnormality. As we discussed in Section 1, the difference between model (7) and the models of Tsay (1987) and Weiss (1984) is that lagged squared innovations do not appear in the equation for the conditional heteroscedasticity. Their model does represent a viable alternative to our formulation (1.3) of h,. In Section 3, we indicated the procedures to carry out tests for the presence of lagged innovations squares in h,. For this we need to run regressions of ri: on its lags and consider the N . R2 values. We ran three separate regressions, and the results are as follows:

Li:

=

.00006 (2.80)

+

.02 (.16)

Li:-,

+

.20 Li:-,, (1.55)

R2

=

.039,

Bera, Higgins, and Lee: ARCH and a Random-Coefficient Approach

where fit is the estimated innovations from our AR(2) model (6). Therefore, the tests were not significant. Nevertheless, we attempted to augment models (6) and (7) by adding the lagged innovation squares and u:-, to the h, formulations. This resulted in lower AIC values and negative estimates for a'. At this point it would be interesting to consider whether our ARCH(y,) + AR(4,, 4,) formulation would satisfy the stationarity condition stated in Section 1. For this special case, the stationarity condition is w(4) . y, < 1, where

Evaluating w(4) at 6, = .27 and 6, = .33, we find w(6) = 1.34, and since 9, = .51, model (7) satisfies the stationarity condition. Had we obtained 9, > .75, the process would have been nonstationary, even though -7.5value is within the stationary region of a pure ARCH process. Taking all of the evidence as presented previously, model (7) can be taken as a reasonable description of our data. It would be interesting to consider the pattern of the forecast errors of inflation and the conditional variance of the forecast errors over the sample period. In Figure 1 we plot the actual and expected rates of inflation, and in Figure 2 we plot the estimated conditional variance &. Prior to 1973, the ht do not vary much. Then there are three distinct peaks at 1975,1980,

1958

1962

1966

139

and 1983. The first two are possibly due to the oil price hike and interest rate deregulation, respectively. As we note from Figure 1, exactly during these periods experts systematically underpredicted the inflation rate. The last peak is more interesting because during this period we observe overprediction of the inflation rate. Therefore, it was not the level of inflation that led to higher conditional variance but rather the prediction error produced the higher variance. Returning to unbiasedness of experts' expectations, in Table 4 (p. 140) we present Wald tests of hypothesis H,: p, = 0, p2 = 1 for selected models from Table 1. The numerical magnitudes of the statistics are quite different. Except for the AR(4, , 4,) model, in all cases we reject the unbiasedness hypothesis. The white-noise specification gives a similar result to our preferred model (7). As also seen from Table 1, separately ARCH and A R have opposite effects on the estimates of p, and p,. Possibly due to this interaction, the results for OLS regression in which both A R and ARCH are ignored are similar to those for a model that incorporates A R and ARCH. Our findings can be summarized as follows. AIC and our specification tests clearly indicate model (7) as preferred, and the hypothesized constraints on the betas are rejected for this choice. On the other hand, the simpler model (6) leads to the opposite conclusion. This finding strengthens our main theme, that interaction between ARCH and autocorrelation could be crucial for economic time series analysis.

1970

197q

1978

1982

Figure 7. Biannual Actual and Expected Rate of Inflation, June 1952-December 1985: Actual, -; Expected, - - -.

140

Journal of Business & Economic Statistics, April 1992

5.

Figure 2. Estimates of the Conditional Variances of Forecast Errors.

5. CONCLUDING REMARKS

It is clear that if we misspecify the conditional first moment (mean), inferences on higher order conditional moments will be very misleading. Within our framework, several interesting problems could be investigated. First, as we have done in the context of hypothesis testing, it would be interesting to study analytically the effect on estimation if we ignore (or misspecify) the autocorrelation or conditional heteroscedastic structure. Second, it appears that with our random coefficient approach we can express the ARCH-type models in a Kalman-filter framework. This might facilitate the estimation procedure, particularly under a multivariate setup. Diebold and Nerlove (1987,1989) exploited this approach to obtain maximum likelihood estimates of a dynamic factor model with ARCH errors. Finally, we have not addressed in detail the issue of modeling the conditional variance as a function of lagged disturbances or innovations or both. This problem is currently Table 4. ~odel number (1 (3) (6)

(7)

Tests for Unbiased Forecasts Error process White noise ARCH(Y,, r2) AR(dJl342) ARCH(y2) + AR(4,, 4,)

under investigation using the nonnested test procedures of Cox. ACKNOWLEDGMENTS

We thank three anonymous referees and especially an associate editor and the editor for detailed comments and numerous suggestions. We are also grateful to Paul Newbold, Yuk Tse, and the participants of the 1988 North American Econometric Society Summer Meetings and econometrics seminars at the University of Indiana, Queen's University, University of Sydney, and McMaster University for their helpful comments. Yoon Dokko graciously provided us'with the Livingston data. This article is based on research funded by the Bureau of Economic and Business Research and the Research Board of the University of Illinois. APPENDIX: DERIVATION OF THE SCORE AND INFORMATION MATRIX

Differentiating the log-likelihood (3.2), the general form of the score vector is

Wald statistic 16.90 30.69 3.58 13.50

NOTE: Each statistic has 2 df. The asymptotic critical values at 5% and 1% significance levels are 5.99 and 9.21.

The componenents of the score, evaluated at the MLE7s, are easily shown to be

Bera, Higgins, and Lee: ARCH and a Random-Coefficient Approach

141

A typical expression for ah,lay, is E,-, . E,-, for r = 1,

. . . , p and s = r, . . . , p. And ah,la~,-, can be

u,E,where the w,'s are appropriate

expressed as choices from y,, y2, . . . , yp(p+ l)12. Therefore, the

expression in the expectation in (A.2) is the product of

two symmetric and an antisymmetric function of _ E , - ~ ,

making the whole expression antisymmetric in 5,Hence, the expectation in (A.2) is 0, giving I,,. = 0.

A similar argument gives IpUz = 0.

The nonzero elements of the information matrix can be consistently estimated by

,,

where d ~ , l d pis a k x p ( p + 1)/2 matrix with columns - ~ , - ~ t ,-- ,~ , - , 2 , - ~for , i = 1, . . . , p and j = i,

..

.,P.

Differentiating the log-likelihood twice, the Hessian is

,.

*,-

Iterating the expectation on the information set ,, the general form of the information matrix simplifies to

The I,,, , and elements of I(8) are immediately seen to be 0 because ah,la$ = 0, au,/ay = 0, and a u , l a ~=~ ~ 0. To show that Ip,., Ipyf and IPu2 are also 0, we employ the symmetry argument of Engle's (1982) theorem 4. First consider

-

-

Earlier we noted that AARCH does not satisfy Engle's (1982) condition of symmetry. By considering h, = h,(g,- ,) in Equation (1.3), however, AARCH is symmetric in the sense that h,( -g,-,) = h(gt-l). Therefore, the second expectation in (A.l) is of the product of a symmetric and an antisymmetric function of This implies that the product is an antisymmetric function of g,- the symmetric unconditional distribution of g,-, will ensure that the expectation is 0, and hence Ip+,= 0. Similarly, consider the (i, j) element of I,,,

[Received April 1989. Revised September 1990.1

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References Information Matrix Test, Parameter Heterogeneity and ARCH: A Synthesis Anil K. Bera; Sangkyu Lee The Review of Economic Studies, Vol. 60, No. 1. (Jan., 1993), pp. 229-240. Stable URL: http://links.jstor.org/sici?sici=0034-6527%28199301%2960%3A1%3C229%3AIMTPHA%3E2.0.CO%3B2-S

What do Economists Know? An Empirical Study of Experts' Expectations Bryan W. Brown; Shlomo Maital Econometrica, Vol. 49, No. 2. (Mar., 1981), pp. 491-504. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198103%2949%3A2%3C491%3AWDEKAE%3E2.0.CO%3B2-X

Testing for Neglected Heterogeneity Andrew Chesher Econometrica, Vol. 52, No. 4. (Jul., 1984), pp. 865-872. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198407%2952%3A4%3C865%3ATFNH%3E2.0.CO%3B2-R

The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor Arch Model Francis X. Diebold; Marc Nerlove Journal of Applied Econometrics, Vol. 4, No. 1. (Jan. - Mar., 1989), pp. 1-21. Stable URL: http://links.jstor.org/sici?sici=0883-7252%28198901%2F03%294%3A1%3C1%3ATDOERV%3E2.0.CO%3B2-T

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Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation Robert F. Engle Econometrica, Vol. 50, No. 4. (Jul., 1982), pp. 987-1007. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198207%2950%3A4%3C987%3AACHWEO%3E2.0.CO%3B2-3

Small-Sample Properties of ARCH Estimators and Tests Robert F. Engle; David F. Hendry; David Trumble The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 18, No. 1, Econometrics Special. (Feb., 1985), pp. 66-93. Stable URL: http://links.jstor.org/sici?sici=0008-4085%28198502%2918%3A1%3C66%3ASPOAEA%3E2.0.CO%3B2-T

The Formation of Inflationary Expectations Stephen Figlewski; Paul Wachtel The Review of Economics and Statistics, Vol. 63, No. 1. (Feb., 1981), pp. 1-10. Stable URL: http://links.jstor.org/sici?sici=0034-6535%28198102%2963%3A1%3C1%3ATFOIE%3E2.0.CO%3B2-C

Serial Correlation as a Convenient Simplification, Not a Nuisance: A Comment on a Study of the Demand for Money by the Bank of England David F. Hendry; Grayham E. Mizon The Economic Journal, Vol. 88, No. 351. (Sep., 1978), pp. 549-563. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28197809%2988%3A351%3C549%3ASCAACS%3E2.0.CO%3B2-D

A Class of Nonlinear Arch Models M. L. Higgins; A. K. Bera International Economic Review, Vol. 33, No. 1. (Feb., 1992), pp. 137-158. Stable URL: http://links.jstor.org/sici?sici=0020-6598%28199202%2933%3A1%3C137%3AACONAM%3E2.0.CO%3B2-8

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A Test for Normality of Observations and Regression Residuals Carlos M. Jarque; Anil K. Bera International Statistical Review / Revue Internationale de Statistique, Vol. 55, No. 2. (Aug., 1987), pp. 163-172. Stable URL: http://links.jstor.org/sici?sici=0306-7734%28198708%2955%3A2%3C163%3AATFNOO%3E2.0.CO%3B2-3

Conditional Heteroskedasticity in Asset Returns: A New Approach Daniel B. Nelson Econometrica, Vol. 59, No. 2. (Mar., 1991), pp. 347-370. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28199103%2959%3A2%3C347%3ACHIARA%3E2.0.CO%3B2-V

Conditional Heteroscedastic Time Series Models Ruey S. Tsay Journal of the American Statistical Association, Vol. 82, No. 398. (Jun., 1987), pp. 590-604. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198706%2982%3A398%3C590%3ACHTSM%3E2.0.CO%3B2-3

Empirical Evidence on the Formation of Price Expectations Stephen J. Turnovsky Journal of the American Statistical Association, Vol. 65, No. 332. (Dec., 1970), pp. 1441-1454. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197012%2965%3A332%3C1441%3AEEOTFO%3E2.0.CO%3B2-H

A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity Halbert White Econometrica, Vol. 48, No. 4. (May, 1980), pp. 817-838. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198005%2948%3A4%3C817%3AAHCMEA%3E2.0.CO%3B2-K

Maximum Likelihood Estimation of Misspecified Models Halbert White Econometrica, Vol. 50, No. 1. (Jan., 1982), pp. 1-25. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198201%2950%3A1%3C1%3AMLEOMM%3E2.0.CO%3B2-C