The Size and Power of Four Tests for Detecting Autoregressive Conditional Heteroskedasticity in the Presence of Serial Correlation

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1 The Size and Power of Four s for Detecting Conditional Heteroskedasticity in the Presence of Serial Correlation A. Stan Hurn Department of Economics Unversity of Melbourne Australia and A. David McDonald * CSIRO Division of Fisheries Castray Esplanade Hobart 7000 Australia Abstract Recently Bera et al. proposed a test for time-varying variance of regression errors, known as autoregressive conditional heteroskedasticity (ARCH), in the presence of serial correlation. We examine the empirical size and power of this and three other tests. The primary purpose is to provide information to applied researchers who might wish to choose selectively from a range of diagnostic tests when ARCH and/or serial correlation are likely to feature in the data. Key Words size, power, autoregressive conditional heteroskedasticity, serial correlation. * The authors would like to thank Adam Davidson for excellent research assistance.

2 . Introduction Univariate regression models often yield errors with nonconstant variance when fitted to economic time-series data. This renders implausible the standard assumption of constant one-period forecast variance that underpins most econometric models. A more plausible assumption, introduced by Engle (98), is that economic time-series regression errors are mean-zero, seriallyuncorrelated processes with nonconstant variances conditional on the past but with constant unconditional variance. Engle's initial specification was for the conditional regression-error variance to take the form σt = αo + αεt +... αp εt p where αo, α,..., αp are fixed coefficients and ε t iis the i-th lagged regression error. To describe such an error process Engle referred to this as an autoregressive conditional heteroskedastic (ARCH) process. The ARCH model and its generalizations, particularly the generalised ARCH (GARCH) model (Bollerslev, 986) and the ARCH-in-mean (ARCH-M) model (Engle et al. 987), have become increasingly popular in econometric modeling because of the large number of economic time series that exhibit nonconstant one-period forecast variance. This paper is focused on methods for identifying ARCH when the variable being studied exhibits serial correlation. In such a situation, it is easy to confuse the effects of ARCH and serial correlation, and the empirical performance of tests to detect ARCH may be impaired. Of particular relevance is the work of Bera et al. (99) which concentrates on the simultaneous detection of ARCH and serial correlation. Using simulated data we compare the size and power of the LM test for ARCH proposed by Bera et al. with the size and power of other tests, including Engle's standard test for ARCH.. The Bera et al. Random Coefficient Framework for ing for ARCH in the Presence of Serial Correlation Consider the static linear regression model y = X β+ ε () t t t One way of dealing with serial correlation and ARCH simultaneously is to specify ε t as a random-coefficient autoregressive process ε = ( φ + η ) ε + ζ () t p j= i j jt t j t As pointed out by the Bera and Higgins (99), Bollerslev et al. (99) cited several hundred papers involving application of ARCH processes to financial markets alone.

3 with φ j constant and η jt stochastic for all j. η = ( η η η t it, t... pt ) is a sequence of i.i.d. random vectors and ζ t is a sequence of i.i.d. random variables. η and ζ t t are independent and have the properties [ η ] 0 [ η η '] E t = ( px ) = Σ t t ( pxp ) [ ζ ] = 0 [ ζ ] E t t E () E = σ () To demonstrate how the random-coefficient specification encompasses a wide range of models as special cases, it is necessary to consider the mean and variance of the disturbances of the linear regression model, ε t. If Ψ t is the set of known past regressive errors at time t, then p µ = E( ε Ψ ) = E ( φ + η ) ε + ζ j= t t t j jt t j t t Ψ () p ht = var( εt Ψt = E ) ηjtεt j + ζt Ψ j = t (6) It is convenient to express the results by rewriting equations () and (6) in vector notation. Let φ= ( φ,..., φp ) and ε = ( ε ε ) t t,..., t-p. Then, because ε t is known at time t and in light of the properties given in equation () and (), µ φ ' ε (7) t = t h = ε 'Σ ε + σ = γz + σ (8) t t t t = Σ Σ t = t ' t and vech is the operator that vectorizes the lower triangular portion of a symmetric matrix. where γ vech [ - diag( )] and Z vech ( ε ε ) It follows that if φ 0 the error process is serially correlated and to test for ARCH is to test whether or not the coefficients of the autoregressive disturbances vary. The hypothesis to be tested is H 0 : γ =0 and under this null hypothesis the model is a homoskedastic linear regression model with AR(p) disturbances. It is useful to note that if is a nondiagonal matrix then the ARCH process incorporates cross product terms between past errors of different lag lengths, a generalization of the linear ARCH process proposed by Engle (98) and named augmented ARCH (AARCH) by Bera et al. (99). In Under this hypothesis h t =σ, implying that all elements of are zero.

4 what follows we concentrate on the restricted case with constrained to be diagonal. In these circumstances Z t is a vector of p lagged squared residuals. Maximum likelihood estimates of β, φand σ are obtained to enable the Define ( Ζ t ) ( ) construction of $ $ εand ζ. W t =, $ $ and ft = ζ / $ t σ or, given N observations, W=(i, Ζ), $ Z $ = ( Z $,..., Z $ N ) and f=(f,...,f N ) with i a unit vector of dimension N. The LM test statistic is then fwww Wf LM ARCH AR = ' ( ' ) ' N NR ff ' =. (9) where R can be obtained as the coefficient of determination for the regression of ζt ( εt εt p) $ on, $... $. Under the null hypothesis this statistic is distributed asymptotically as χ with p degrees of freedom.. Alternatives to Bera et al.'s for ARCH In the absence of serial correlation in regression errors one obtains a special case of equation (). In this case the test statistic proposed by Engle (98) is obtained as a special case of equation (8) but where R is the coefficient of determination for the regression of ( ) $ $ εt εt,...$ on, εt p. That is, Engle's test statistic is obtained if one lets ε t in equation () be an uncorrelated, but possibly heteroskedastic, error process. Z t and W t are defined as above and f t = ( ε t ) $ / σ $. Equation (9) then defines the Engle test statistic which is distributed asypmptotically under the null hypothesis, H 0 : γ = 0, as χ with p degrees of freedom. Another variant of this test arises when serial correlation is modeled explicitly. The conditional variance of ε t is given by equation (8). Under the null hypothesis of no ARCH this conditional variance will obviously be constant, as will the variance of ζ t. If, however, the null hypothesis is false the restricted estimates of the autoregressive parameters will lead to estimates of µ t that are subject to bias which may result in the appearance of heteroskedasticity. In this case the statistic given by equation (9) is obtained $ $ via the regression of ζt ( ζt,..., ζ ) $ on, t p. This test statistic, which will be referred to as the alternative Engle statistic, is distributed asymptotically under the null hypothesis as χ with p degrees of freedom. The remaining test statistic examined in this paper is the commonly-used Q statistic proposed by McLeod and Li (98) for testing nonlinear dependence Standard errors of the estimates can be obtained via the information matrix or via the robust estimates put forward by Weiss (986).

5 in regression residuals. This is a modification of the statistic put forward by Ljung and Box (979) in the context of fitting ARIMA models to time-series data. The Q test statistic is based on the squared residuals ζ $ t and has the form Q p - = N(N + ) (N - τ) τ = r( τ) (0) r( τ ) = N t=+ τ ( ζ $ σ$ )( ζ $ σ$ ) t t τ N t= ( ζ $ σ$ ) t () where r(τ) is the serial correlation coefficient of the squared residuals ζ $ t, p is the highest order serial correlation calculated and N is the sample size. Under the null hypothesis Q is distributed asymptotically as χ with p degrees of freedom.. Comparison of Size and Power Among ARCH s Empirical size and power estimates have been obtained for tests appropriate to each of the above statistics. Size estimates were obtained for tests of up to fifth-order ARCH for data generated using models with one through three autoregressive terms, under the null hypothesis of homoskedasticity. These estimates were obtained for samples of 0, 00, 00 and 00 observations using 000 simulated replications and nominal size of %. All random components included in the simulated data are Gaussian. The models used to generate the data are given by y t = β + 0 ε t () where β 0 is a constant and ε t follows the AR process presented in equation (), with η jt set to zero. For the AR() model φ = 0. ; for the AR() model φ= 0. and φ = 0. ; and for the AR() model φ =0., φ = 0. and φ = 0.. The size estimates, based on LM statistics for individual lag components which follow an asymptotic χ distribution with one degree of freedom, appear in Tables, and. All the test statistics and model estimates reported in this paper were obtained using program written by the authors in the GAUSS programming language (Aptech Systems, 99). Results for samples of size 00 are omitted because they are very similar to those for samples of size 00.

6 Power estimates were obtained for tests of ARCH for data generated with one through three autoregressive terms and ARCH The models used to generate the data are as above but where the η jt for each model are distributed as Gaussian with mean zero and variance φ j /. The power estimates are displayed in Tables through 6 for samples of 0, 00 and 00 observations, again for 000 simulated replications and nominal size of %. It is clear that in the presence of serial correlation, the size for Engle's test is much greater than the nominal size, especially for the first two lags. For small samples the other tests have empirical size below the nominal size. For larger samples the point estimates of size are higher for all tests: in many cases the empirical size is significantly greater than %. Engle's test reveals the confusion between ARCH and serial correlation emphasized by Bera et al. The Bera et al. test statistic offers a clear improvement in size for the simulated data used here, particularly at the first lag. As a consequence of the excessive empirical sizes appearing in Tables through the power estimates reported in Tables through 6 were obtained using the empirical critical value (based on our 000 simulations) appropriate for an empirical size of %. These size-corrected powers are necessary for meaningful comparisons whenever the empirical size diverges significantly from the nominal size. 6 Table indicates that all of the tests fail to detect ARCH when the sample is restricted to 0 observations. The results reported in Tables and 6 indicate that power improves as sample size increase but that Engle's test performs significantly worse than the other tests. With regard to unadjusted power, not reported here, the disparity between Engle's test on ε t and its alternative on ζ t is particularly notable because it illustrates that the presence of serial correlation not only increases the size but also reduces the power of Engle's original test. In response to the low powers reported in Tables through 6, the random components of the AR coefficients, η jt, were generated with the higher variance: an increase from φj / to φj. These higher variances strengthened the ARCH, resulting in improved power for each of the tests. This highlights the sensitivity of this representation of ARCH to the degree of randomness in the AR coefficients. The size-corrected powers reported in Table 7 through 9 repeat the earlier pattern of increased power with sample size. Again the Engle test on ε t performs poorly. For the sample size of 00, the Bera et al. test arguably provides the greatest power. Three points of practical relevance emerge from the reported results. First, even correctly-specified tests have very low power for small samples. It is therefore, very difficult to test successfully for ARCH in small samples of data. 6 Size-corrected powers are reported for all test statistics, even if the empirical size is not significantly different from the nominal size, not only for ease of comparison but also because asymptotic properties may not hold for samples of up to 00 observations.

7 Second, before testing for ARCH it is important to implement heteroskedasticity-robust tests for serial correlation 7 because i. Engle's test, which is used most widely in practice, is unreliable in the presence of serial correlation; and ii. there is no serial-correlation-robust ARCH test available so LM tests for ARCH (conditional on the presence of serial correlation) must be used and these require correct modelling of serial correlation. Finally, once serial correlation is modelled adequately the Bera et al. test appears to be reliable in testing for ARCH for samples of at least 00 observations.. Concluding Remarks Empirical size and power estimates have been generated for four ARCH tests in the presence of serial correlation. The test suggested by Bera et al., in particular, seems to offer improvement in size and power over Engle's test, especially for larger samples. At least for the data generated in the present study, the results lend support to the arguments put forward by Bera et al. in favor of their test. Economists are therefore likely to gain from implementing the Bera et al. test in applied work in which ARCH and/or serial correlation are likely to be present. 7 See MacKinnon (99), Diebold (986), Wooldridge (99) and Silvapulle and Evans (99), for example. 6

8 References Aptech Systems Inc. GAUSS, Maple Valley. Bera, A.K. and Higgins, M.L. (99) "ARCH Models: Properties, Estimation and ing" Journal of Economic Surveys, 7 (), Bera, A.K., Higgins, M.L. and Lee, S. (99), "Interaction Between Autocorrelation and Conditional Heteroskedasticity: A Random-Coefficient Approach", Journal of Business and Economic Statistics, 0(), -. Bollerslev, T. Chou, R.Y. and Kroner, K.F. (99) "ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence", Journal of Econometrics,, - 9. Bollerslev, T. (986), "Generalized Conditional Heroskedasticity", Journal of Econometrics,, Box, G.E.P. and Pierce, D.A. (970) "Distribution of Residual Autocorrelations in Integrated Moving Average Time Series Models', Journal of the American Statistical Assosication, 6, Diebold, F.X. 986, ing for Serial Correlation in the Presence of ARCH, Proceedings of the American Statistical Association, -8. Engle, R.F. (98), " Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica, 0, Engle, R.F., Lilien, D. and Robins, R. (987), "Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model", Econometrica,, Harvey, A.C. (989), Forecasting Structural Time Series Models and the Kalman Filter, Cambridge University Press. Ljung, G.M. and Box, G.E.P. (978) " On a Measure of Lack of Fit in Time Series Models', Biometrika, 6, MacKinnon, J.G. (99) "Model Specification s and Artificial Regressions" Journal of Economic Literature, 0, 0-6. McLeod, A.T. and Li, W.K. (98) Diagonistic checking ARIMA time series models using squared-residual autocorrelations, The Journal of Time Series Analysis,, 69-7 Silvapulle, P. and Evans, M. (99) "ing for Serial Correlation in the Presence of Conditional Heteroskedasticity" Discussion Paper, Monash University, Australia. 7

9 Weiss, A.A. (986) "Asymptotic Theory for ARCH Models: Estimation and ing", Econometric Theory,, 07-. Wooldridge, J.M. 99, On the Application of Robust, Regression-based Diagnostics to Models of Conditional Means and Conditional Variances, Journal of Econometrics, 7, -6. 8

10 Table : Size Results for Samples of 0 Observations Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error of % size for 000 replications = 0.00 Estimated standard errors based on empirical size are bounded above by

11 Table : Size Results for Samples of 00 Observations Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error of % size for 000 replications = 0.00 Estimated standard errors based on empirical size are bounded above

12 Table : Size Results for Samples of 00 Observations Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error of % size for 000 replications = 0.00 Estimated standard errors based on empirical size are bounded above by

13 . Table : Size-corrected Power Results for Samples of 0 Observations Generated with ARCH Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error estimates are bounded above by

14 Table : Size-corrected Power Results for Samples of 00 Observations Generated with ARCH Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error estimates are bounded above by 0.00.

15 Table 6: Size-corrected Power Results for Samples of 00 Observations Generated with ARCH Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals ) Alternative Engle for (ζ t only) Standard error estimates arebounded above by

16 Table 7: Size-corrected Power Results for Samples of 0 Observations Generated with ARCH (higher-variance η jt ) Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error estimates are bounded above by

17 Table 8: Size-corrected Power Results for Samples of 00 Observations Generated with ARCH (higher-variance η jt ) Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error estimates are bounded above by

18 Table 9: Size-corrected Power Results for Samples of 00 Observations Generated with ARCH (higher-variance η jt ) Engle's for ARCH on ε t Residuals Lag for LM Term in Model Bera et al. for ( ζt and εt ) Q for ARCH on ζ t Residuals Alternative Engle for (ζ t only) Standard error estimates are bounded above by