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The wild bootstrap and heteroskedasticity-robust tests for serial correlation in dynamic regression models
Computational Statistics & Data Analysis 49 (2005) 377 – 395 www.elsevier.com/locate/csda
The wild bootstrap and heteroskedasticity-robust tests for serial correlation in dynamic regression models L.G. Godfreya,∗ , A.R. Tremaynea, b a Department of Economics, University of York, Heslington, York YO10 5DD, UK b School of Economics and Political Science, University of Sydney, Sydney NSW 2006, Australia
Available online 19 June 2004
Abstract Conditional heteroskedasticity is a common feature of financial and macroeconomic time series data. When such heteroskedasticity is present, standard checks for serial correlation in dynamic regression models are inappropriate. In such circumstances, it is obviously important to have asymptotically valid tests that are reliable in finite samples. Monte Carlo evidence reported in this paper indicates that asymptotic critical values fail to give good control of finite sample significance levels of heteroskedasticity-robust versions of the standard Lagrange multiplier test, a Hausman-type check, and a new procedure. The application of computer-intensive methods to removing size distortion is, therefore, examined. It is found that a particularly simple form of the wild bootstrap leads to well-behaved tests. Some simulation evidence on power is also given. © 2004 Elsevier B.V. All rights reserved. Keywords: Heteroskedasticity; Serial correlation; Wild bootstrap
1. Introduction The importance of testing for serial correlation after the least-squares estimation of a dynamic regression model has been understood for many years. Lagrange multiplier (LM) tests allow for flexibility in the choice of alternative hypothesis, are easily implemented, and are now used as a matter of routine in estimation programmes. A standard means ∗ Corresponding author. Tel.: +44-1904-433754; fax: +44-1904-433759.
E-mail addresses: [email protected] (L.G. Godfrey), [email protected] (A.R. Tremayne). 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.05.020
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of implementing such tests is to carry out an artificial regression and base the test on the significance of coefficient(s) in that regression. It is shown in Belsley (1997, 2000) that misleading inferences can result from using such an approach in static regression with small and moderate samples. In the earlier of these two papers, an adjustment to the t-statistic for testing the null of no serial correlation against the alternative of a simple scheme of specified order is proposed. The companion paper extends the approach to an F-test for joint autoregressive schemes and the efficacy of both suggestions is demonstrated by reporting Monte Carlo results based upon independent and identically distributed (iid) regression errors. The present paper is concerned with the application of such tests when the regressors include lagged dependent variables, as well as exogenous variables. In the presence of both types of regressors, tests based on the popular portmanteau-type approach are, of course, invalid and so LM tests may be particularly attractive to empirical workers. However, standard LM-type tests (and the modifications suggested by Belsley), along with many others, are predicated on an assumption of homoskedasticity. It is now recognized that conditional heteroskedasticity may be common, especially (but certainly not exclusively) when the data relate to financial variables. In the presence of conditional heteroskedasticity, standard checks for serial correlation cannot be assumed to be asymptotically valid and may lead to misleading inferences. There is therefore a need to examine heteroskedasticity-robust (HR) procedures. An obvious way to derive LM-type checks that are asymptotically HR is to employ heteroskedasticity-consistent covariance matrix estimates (HCCME) as discussed in White (1980). The basic idea of using HCCME to obtain diagnostic tests is long established; see, for example, Pagan and Hall (1983). Details of a specific application of HCCME to testing for serial correlation are given in Godfrey (1994). However, there is a considerable body of evidence indicating that tests derived from the HCCME can have finite sample distributions that are quite unlike those predicted by asymptotic theory. In particular, finite sample significance levels can be far from what is desired. The main purpose of this paper is to report Monte Carlo results on the wild bootstrap approach when it is used to obtain critical values or p-values for serial correlation tests based upon HCCME. Given the significant increases in the availability of cheap and powerful computers in the last few years, this approach is attractive. Once implemented in standard programmes, it could be used not only for serial correlation tests, but also for other variable-addition diagnostic checks, e.g. the RESET test. Thus the wild bootstrap described below provides a way for applied workers to obtain misspecification tests that are asymptotically robust to heteroskedasticity (and nonnormality) with the bootstrap being used to improve upon the approximation provided by asymptotic theory. This combination of features matches recommendations in Hansen (1999) for good econometric practice. In order to assess the added value of the results given below, reference should be made to the existing body of published work pertaining to serial correlation tests and conditional heteroskedasticity. There are relatively few contributions. Robinson (1991) considers a static linear regression model and derives an LM test for serial correlation that is asymptotically valid under dynamic conditional heteroskedasticity and nonnormality. No evidence on finite sample performance is provided and, as noted by Robinson, the test is inappropriate in the presence of lagged dependent variables. Whang (1998) derives tests for serial correlation in dynamic models that are asymptotically valid in the presence of heteroskedasticity of
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unknown form. Whang (1998) does not use the HCCME approach. Instead his test statistic involves a nonparametric kernel estimate of the unknown variance function, and his technique requires the specification of kernel and trimming functions. He reports Monte Carlo results for his asymptotic tests based upon a static simple regression. His estimates show that asymptotically valid approximations cannot be relied upon. For example, with 50 observations and a nominal significance level of 5%, the majority of his estimates are between 2% and 3.5%. In addition to these published articles, there is an unpublished study by Binbin Guo and Peter Phillips. In Guo and Phillips (2000), they provide HR tests that are asymptotically valid either when there are only exogenous regressors or when there are only lagged dependent variables as regressors. They report Monte Carlo results for a model of the latter type. The use of asymptotic critical values produces mixed results with some evidence for tests being undersized. The use of HCCME can also be avoided if there is precise information about the error variances, for then specific adjustments can be made to obtain HR checks for serial correlation. For example, ARCH-corrected tests are discussed in Silvapulle and Evans (1998). However, such an approach suffers from the general drawback that applied researchers are not likely to have accurate information about the precise form of heteroskedasticity. Mistakes made in specifying the variance model will, in general, lead to asymptotically invalid inferences; see Belsley (2002). Moreover, as noted above and contrary to what is asserted in Silvapulle and Evans (1998, p. 35), the Box–Pierce and Ljung–Box portmanteau tests are not asymptotically valid when exogenous variables and lagged values of the dependent variable are both included amongst the regressors. Consequently their Monte Carlo analysis includes inappropriate tests and the corresponding results must be treated with caution. In view of the above, the novel features of our study are: the use of a wild bootstrap to gain better control over significance levels than is provided by asymptotic theory; Monte Carlo designs using regression models with regressor sets having both exogenous variables and lagged dependent variables; and a new test derived as a modification of the HR version of the LM statistic. The paper consists of six sections. Section 2 contains descriptions of models and tests, including the new variant of the LM test. Bootstrap methods are explained in Section 3. The designs of the Monte Carlo experiments are provided in Section 4. Results from these experiments are summarized in Section 5. Section 6 concludes. 2. Models and test procedures The dynamic regression model is written as yt = Yt + Xt + t = Wt + t ,
(1)
where: Yt = (yt−1 , . . . , yt−P ) with P 1; = (1 , . . . , P ) having values such that the roots of zP − 1 zP −1 − · · · − P = 0 are all strictly inside the unit circle; Xt is an L-vector of regressors that are strictly exogenous; Wt = (Yt , Xt ); and = ( , ). Let K = P + L denote the number of regression
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coefficients in (1) and T denote the number of observations available for estimation. The null hypothesis to be tested is that the errors t are serially uncorrelated. It will be argued that the use of a wild bootstrap is very important to the reliable assessment of the statistical significance of HR variants of the LM tests of Breusch (1978) and Godfrey (1978), and the Hausman-type test of Godfrey (1997). Consequently the regularity conditions must not only support the usual HCCME and associated asymptotic tests, but also validate the bootstrap procedure. All tests are constructed using the results of ordinary least-squares (OLS) estimation of (1). It is essential that the OLS estimators be consistent under the null hypothesis. Consequently it is assumed that RSS j , the residual sum of squares from the artificial regression of the j th variable of Wt on all other regressors, is at least Op (T ), j = 1, . . . , K, i.e., all regressors are asymptotically cooperative. Restrictions on the coefficients of the lagged dependent variables of Yt have been given above. The exogenous variables Xt are not restricted to be nonstochastic or covariance-stationary I(0) terms; see Wooldridge (1999) for a discussion of nonstationary regressors in the context of testing for serial correlation. Let = ( , ) denote the OLS coefficient estimator for (1) and the terms et = yt − Wt be the residuals derived from this estimator. Turning to assumptions that are made, under H0 , about the errors of (1), the following, provided by Gonçalves and Kilian (2002) for wild bootstraps, are adopted for the case of conditional heteroskedasticity: (a) E(t | Ft−1 ) = 0, almost surely, where Ft−1 is the sigma-field generated by (t−1 , t−2 , . . .). (b) E(2t ) = 2 , 0 < 2 < ∞. (c) plimT →∞ T −1 Tt=1 E(2t | Ft−1 ) = 2 . (d) E(2t t−r t−s ) = 0 for all r = s, for all t, r 1, s 1. (e) plimT →∞ T −1 Tt=1 t−r t−s E(2t | Ft−1 ) = 0 for any r 1, s 1. (f) E(| t |4r ) is uniformly bounded for some r 2 and all t. These assumptions are somewhat stronger than those required for the asymptotic validity of the HCCME and, in particular, require that at least eight, rather than at least four, moments exist. Also Gonçalves and Kilian (2002) remark that assumption (d) is satisfied for ARCH and GARCH processes with symmetric innovation processes but is ruled out in some asymmetric cases. The model (1) differs from that used by Gonçalves and Kilian (2002) to discuss regularity conditions because the latter is a pure autoregression, i.e., = 0. Provided they are asymptotically cooperative, it seems reasonable to assume that the inclusion of strictly exogenous regressors is not problematic. The case of predetermined, but not strictly exogenous, regressors is discussed below. Suppose that the null hypothesis is to be tested against the alternative that the errors are generated by a Qth-order autoregression, denoted AR(Q). [Using a Qth-order moving average, denoted MA(Q), as the alternative leads to the same test statistic.] A convenient variable-addition form of the LM test is a test of = (1 , . . . , Q ) = 0 in the augmented model yt = Yt + Xt + Et + t = Wt + Et + t ,
(2)
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in which Et = (et−1 , . . . , et−Q ) and et−q is set equal to zero when t − q 0. Denote the OLS estimator of the regression coefficients for (2) by ˆ and ˆ . The Breusch–Godfrey test is then a check of the joint significance of the elements of ˆ . Although Breusch (1978) and Godfrey (1978) propose a large-sample 2 form of the test, subsequent investigations by Kiviet (1986) revealed that better finite sample behaviour is achieved using an approximate F statistic. Kiviet’s version of the test is denoted by LM F , and it is this form that will be shown to be vulnerable to dynamic heteroskedasticity. If the errors were iid under the null hypothesis, one simple possibility for potentially improving the small-sample behaviour of a test like LM F would be to use a standard bootstrap procedure based on a residualresampling scheme to estimate critical values. We experiment with such an approach; the test is termed LM B in the experiments reported below and it is seen to suffer from the same deficiencies as LM F . An alternative to LM procedures is derived in Godfrey (1997) as an extension of Dezhbakhsh and Thursby (1994). The test is a check of the significance of the difference between the estimators of from (1) and (2), i.e., ( − ˆ ). Godfrey (1997) gives a formula based upon a direct application of Hausman’s (1978) general result for the variance–covariance matrix of estimator contrasts. However, a variable-addition form is also available. Let A denote the cross-product moment matrix for the regressors of (2), i.e., A=T
−1
Y Y X Y E Y
Y X X X E X
Y E XE , E E
in which Y is T × P with rows Yt , X is T × L with rows Xt , and E is T × Q with rows Et . The submatrices of the inverse of A partitioned as lagged endogenous regressors, exogenous regressors, and lagged residuals are denoted by AYY , AYX , AYE , etc. The variable-addition form of Godfrey’s (1997, p. 201, Eq. (2.6)) test is then obtained by testing = (1 , . . . , P ) = 0 in the extended model yt = Yt + Xt + (AYE Et ) + t = Wt + Jt + t ,
(3)
i.e., the test variables are the P linear combinations of the terms in Et given by Jt = AYE Et . A proof that (3) is appropriate can be derived by modification of Theorem 2 in Davidson et al. (1985). If P Q with rank(AYE ) = Q, the test variables of (3) are (omitting any redundant terms) equivalent to those of (2). If P < Q with rank(AYE )=P , the Hausman and LM tests are not asymptotically equivalent and there is no generally valid ranking of these procedures by asymptotic local power; see, e.g., Holly (1982). Using White’s (1980) general formula for the HCCME, the HR version of the LM statistic can be shown to be M(W)E}−1 E e, LM HR = e E{E M(W)
(4)
is diag(e2 , . . . , e2 ), where e is the T-dimensional OLS residual vector with elements et , 1 T and M(W) = IT − P(W) = IT − W(W W)−1 W ,
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where W is the T × K matrix with rows Wt . The calculation of the HR-Hausman test of = 0 in (3), when P < Q, is carried out in a similar fashion. We denote the test statistic for this procedure by HAHR , which is given by M(W)J}−1 J e, HAHR = e J{J M(W)
(5)
where J is the T × P matrix with rows Jt . It should be noted that the et2 derived from OLS estimation of (1) are used to compute the HCCME for both robust serial correlation tests. This corresponds to using the squared residuals from restricted (under the null hypothesis) estimation, which, compared to using the unrestricted squared residuals, has been found to give superior finite-sample performance; see Davidson and MacKinnon (1985) and Godfrey and Orme (2002). As in Davidson and MacKinnon (1985), HCCME are sometimes derived by multiplying squared residuals by terms that tend to unity. It is reported in Davidson and MacKinnon (1985) that, when restricted residuals are used, such adjustments lead only to slight changes. The simplest version is, therefore, used. Consideration of finite-sample performance suggests that, in addition to using et2 in expressions for HCCME, it may be useful to examine devices that reduce the sampling variability of the HCCME. One such device is used here to derive a modified version of LM HR . The basic idea is to eliminate asymptotically negligible terms. To identify and remove such terms, partition W as (W1 , W2 ), where, under the null hypothesis, p lim T −1 W1 E = 0
and p lim T −1 W2 E = 0,
so that W1 contains terms like yt−j , j = 1, . . . , min(P , Q), and W2 contains the exogenous regressors Xt and any terms yt−j with j > Q. Standard results on projection matrices imply that 1 ) − P(W2 ), M(W) = M(W 1 = M(W2 )W1 . Hence where W M(W)E = 1 + 2 + + 3 , E M(W) 2 where E, ˇ 1 = Eˇ
P(W2 )E, 2 = −Eˇ
P(W2 )E 3 = E P(W2 )
1 )E. Since p lim T −1 W E = 0, 2 and 3 are both op (T ) and are and Eˇ denotes M(W 2 asymptotically negligible relative to 1 , which is Op (T ). Hence the HR-variant of the LM test given by (4) is asymptotically equivalent to the modified procedure based upon E) ˇ −1 E e. MLM HR = e E(Eˇ
(6)
Under the null hypothesis, LM HR and MLM HR are asymptotically distributed as 2 (Q). When P < Q, the asymptotic null distribution of HAHR is 2 (P ). (If P Q, all three tests are asymptotically equivalent.) Asymptotically valid tests are obtained by comparing sample values of test statistics with critical values from the right-hand tail of the appropriate 2 distributions. However, as illustrated by the results provided in Godfrey and Orme (2002), the finite sample distributions of HR statistics may differ appreciably from those predicted
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by asymptotic theory when several restrictions are being tested. Their results also indicate the potential usefulness of wild bootstrap methods, which will be discussed in the next section.
3. Wild bootstrap methods To describe the wild bootstrap, it is useful to write the OLS estimated version of (1) as yt = Wt + et ,
t = 1, . . . , T .
Let Z be the discrete random variable with the two-point probability distribution √ √ √ Z= − ( 5 − 1)/2 with probability ( 5 + 1)/(2 5) √ = ( 5 + 1)/2, otherwise,
(7)
(8)
so that E(Z) = 0, E(Z 2 ) = 1, and E(Z 3 )=1. The conditions that E(Z)=0 and E(Z 2 )=1 are essential for the validity of the bootstrap procedure. The additional restriction pertaining to the third moment of (8) is suggested in Liu (1988). After examination of the distribution of a single linear combination of OLS estimators, it is shown in Liu (1988) that, if E(Z 3 ) = 1, the wild bootstrap enjoys second-order properties with the first three moments of the relevant test statistic being estimated correctly to O(T −1 ). The probability distribution (8) is denoted PD1 in the discussion in Section 5. The wild bootstrap is then implemented as follows. (i) For t = 1, . . . , T , observations are generated by ∗ ∗ yt∗ = ˜ 1 yt−1 + · · · + ˜ P yt−P + Xt + ∗t ,
(9)
where bootstrap sample starting values are set equal to actual estimation sample starting values (see Li and Maddala, 1996, Section 2.3), and the errors ∗t are given by ∗t = et Zt , Zt being a drawing from (8). Note that, in contrast to the standard residual resampling scheme for iid errors, it is not necessary to centre the OLS residuals if the null model does not contain an intercept when using the wild bootstrap; see Liu (1988, pp. 1706–1707). (ii) Given the bootstrap data, the regression model is estimated and the associated values of the test statistics HA∗HR , LM ∗HR , and MLM ∗HR are calculated. (iii) Repeat (i) and (ii) B times in order to estimate the p-values of the observed statistics HAHR , LM HR , and MLM HR . The null hypothesis of serial independence is rejected for p-values that are sufficiently small. Several other distributions have been suggested as alternatives to (8); see Davidson and Flachaire (2001), henceforth DF, Liu (1988), and Mammen (1993). However, with the exception of the DF scheme, using these alternative wild bootstraps produces Monte Carlo results here that are markedly inferior to those produced by (8). Consequently, detailed results are provided below only for (8) and the DF scheme given by the simple two-point distribution Z=1 with probability 0.5, = − 1, otherwise.
(10)
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(The Monte Carlo results for all wild bootstrap schemes are available on request.) The distribution of (10) satisfies the essential requirements that E(Z) = 0 and E(Z 2 ) = 1, but has E(Z 3 ) = 0, rather than E(Z 3 ) = 1 as in (8). The probability distribution of (10) recommended by DF is denoted PD2 below. Provided the regularity conditions mentioned in Section 2 are satisfied, the wild bootstrap can be used for either conditional or unconditional heteroskedasticity by combining (9) with either PD1 or PD2 . The bootstrap scheme (9) assumes that the variables of Xt are strictly exogenous. If the Xt were only predetermined, it could be argued that (9) is inappropriate and that joint simulation of conditional and marginal models is required to approximate the actual data generation process. However, Giersbergen and Kiviet (1996) argue that joint modelling introduces the possibility of misspecifying the marginal model for Xt and find Monte Carlo evidence that joint modelling fails to deliver useful improvements, a finding corroborated in Godfrey and Orme (2000). The wild bootstrap is not the only way to produce HR tests. If the model is static with all the regressors being strictly exogenous, and the variances are functions of regressor values, a paired bootstrap can be used; see Flachaire (1999). Brownstone andValletta (2001) discusses why the wild bootstrap is likely to be more accurate than the paired bootstrap when both are appropriate. Additional evidence in favour of using the wild bootstrap method recommended below, rather than the paired bootstrap, is to be found in Flachaire (2004). 4. Monte Carlo design The number of Monte Carlo replications for each experiment is R = 25000, with the number of bootstraps being B = 399. MacKinnon (2002) remarks that, while he would not recommend using B = 399 in genuine applications, sampling errors associated with this value tend to cancel out in Monte Carlo experiments. In empirical work, B = 999 could be used without incurring important waiting times, given the power of modern personal computers. The dynamic regression model upon which experiments are based is yt = 1 yt−1 + 2 yt−2 + 1 + 2 xt + t ,
t = 1, . . . , T ,
(11)
in which xt is a scalar variable and T equals 40 or 80. This model corresponds to (7) of Dezhbakhsh (1990). The values used for (1 , 2 ) are (0.5, 0.3), (0.7, −0.2), (1.0, −0.2), (1.3, −0.5), (0.9, −0.3), and (0.6, 0.2) which are regarded in Dezhbakhsh and Thursby (1995) as typical of values observed in applied work. In all experiments, 1 = 2 = 1. The exogenous variable xt is constructed in two ways. First, it is generated as an artificial variable according to the first order autoregression xt = xt−1 + vt ,
(12)
with vt being NID(0, 2v ), = 0.5 or 0.9, and 2v selected, given the value of , so that Var(xt ) = 1. The vt and all other pseudo-random numbers are generated using subroutines from the NAG Library. The starting value x0 is a drawing from a standard normal distribution. Second, xt is obtained by standardizing the logs of quarterly observations for GDP in the UK, with the basic data being taken from the file GDP95.FIT in the Microfit 4.0 package (Pesaran and Pesaran, 1996). The transformed data appear to exhibit deterministic and stochastic
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trends, as might be expected for this type of macro-economic variable. The starting values y−1 and y0 are set equal to their unconditional mean with x = 0, i.e., 1 /(1 − 1 − 2 ). The effect of using these starting values is reduced by discarding the first 40 of any sequence of (T + 40) generated values. The finite sample significance levels of versions of the tests described in Section 2 are estimated for nominal levels of 1% and 5%. As a useful example of testing using quarterly data, HR-tests LM HR and MLM HR of 1 = 2 = 3 = 4 = 0 in yt = 1 yt−1 + 2 yt−2 + 1 + 2 xt +
4
j et−j + error,
(13)
j =1
are used, with the et−j being lagged residuals from the OLS estimation of (11). The Hausman-type test HAHR is a test of the joint significance of the estimator contrasts for 1 and 2 arising from a comparison of OLS estimates of (11) and (13), i.e., it is a special case of the test associated with (3). With conditional heteroskedasticity, the error terms t of (11) can be written as t = h t t , (14) where ht denotes a conditional variance and, under the null hypothesis, the terms t are iid(0,1). Various distributions for t are used. The normal distribution serves as a benchmark and standardized forms of the t (5) and 2 (8) distributions are also employed. The t (5) distribution is used, following Gonçalves and Kilian (2002), to investigate robustness of the wild bootstrap methods to departures from condition (f) of Section 2. The 2 (8) distribution is used to provide evidence on the effects of skewness. This distribution, with a coefficient of skewness equal to 1, is heavily skewed, according to the arguments of Ramberg et al. (1979). The final component required to derive a typical error t , after drawing t , is the con√ ditional standard deviation ht . Since the HR-tests for serial correlation are intended for general use, it is important to obtain evidence not only for several forms of heteroskedasticity, but also for the case of homoskedastic errors. The following five specifications for variance schemes are used. First, the ht are observation-invariant; that is, the errors are homoskedastic with ht = 2 ,
t = 1, . . . , T ,
(15)
2 being set equal to 1 or 10. Second, the ARCH(1) process provides an important example of conditional heteroskedasticity and is used in the form ht = 0 + 1 2t−1 ,
(16)
where 0 = 2 /(1 − 1 ), 1 = 0.4 or 0.8, and 2 is defined as for (15). The GARCH(1, 1) model is also frequently used in applied studies. This variance scheme is written as ht = 0 + 1 2t−1 + 2 ht−1 ,
(17)
where 0 = 1, 1 = 0.1, and 2 = 0.8; see Bollerslev (1986). The values of 1 and 2 are similar to those reported in empirical work. The average R 2 varies between 0.4 and 0.9
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across the five variance schemes, which seems consistent with values observed in applied work. Given the widespread use of quarterly data in applied work, it seems apposite to examine seasonal schemes of heteroskedasticity. Consideration is, therefore, given to a fourth-order model taken from Engle’s (1982) classic article. As in (38) of Engle (1982, p. 1002), conditional variances ht are written as ht = 0 + 1 (0.42t−1 + 0.32t−2 + 0.22t−3 + 0.12t−4 ),
(18)
in which 0 and 1 are as defined for (16). The fifth and final model to be adopted has unconditional quarterly heteroskedasticity and can be written as 22 22 2c2 2c2 2 2 2 2 , , , , (19) ( 1 , 2 , 3 , 4 ) = (1 + c) (1 + c) (1 + c) (1 + c) in which 2j denotes the variance in quarter j, the average of these terms is the 2 specified in (15), and c equals 4 or 9. Model (19) is similar in spirit to that used in Burridge and Taylor (2001, p. 104). Note that, although (19) generates unconditional heteroskedasticity, its effects on serial-correlation tests cannot be ignored. In the estimation of the covariance matrix of the OLS estimators for (13), the second moments of the errors are not asymp2 , j = 1, . . . , 4, so that the conventional totically orthogonal to the squared regressors et−j (homoskedasticity-based) estimator is not consistent; see White (1980, p. 826). In addition to significance levels, the sensitivity of tests to serial correlation is clearly of interest. For experiments designed to provide evidence on power, various alternatives are used, some of which involve a combination of conditional heteroskedasticity and serial correlation. Five alternatives with homoskedastic errors are obtained as special cases of the stationary fifth-order autoregression
t =
5 j =1
j t−j + t
(20)
which replaces (14). The coefficient vectors = ( 1 , 2 , 3 , 4 , 5 ) of these special cases are as follows:
(1) = (0.7, 0.0, 0.0, 0.0, 0.0),
(21)
(2) = (0.0, 0.0, 0.0, 0.7, 0.0),
(22)
(3) = (1.4, −0.71, 0.15, −0.1, 0.0),
(23)
(4) = (0.8, 0.0, −0.2, 0.06, 0.0) and
(24)
(5) = (0.5, 0.0, 0.0, 0.5, −0.25).
(25)
Thus the alternative hypothesis of the tests LM HR and MLM HR from (13) is overspecified for the first two cases and underspecified for the last. The roots of the polynomial equations z4 − 1 z3 − · · · − 4 = 0 implied by the coefficients of (23) and (24) are (0.5, 0.4, 0.3, 0.2) and (0.5, −0.5, 0.4 ± 0.3i), respectively.
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Table 1 Sample of cases used in Tables 2–5
Case 1 Case 2 Case 3 Case 4 Case 5
Variance model
Regressor
(15) with 2 = 1 (16) with 1 = 0.8 (17) with 1 = 0.1, and 2 = 0.8 (18) with 0 = 0.1, and 1 = 0.8 (19) with c = 9
log(UK GDP)a (12) with = 0.5 log(UK GDP)a (12) with = 0.5 log(UK GDP)a
a These data are standardized before OLS estimation.
To derive power estimates in the presence of ARCH errors, the general random-coefficient specification discussed in Bera and Higgins (1993) is adopted. The two specific error models used in the Monte Carlo experiments are
t = (0.7 + 0.1t )t−1 + t
(26)
t = (0.7 + 0.1t )t−4 + t ,
(27)
and
in which the t are NID(0, 1) and are independent of the s for all s and t. Clearly the probability that the random coefficient of the serial correlation process is strictly between 0 and 1 is high for Eqs. (26) and (27). 5. Monte Carlo results The estimates of significance levels, as sampled in Tables 2–5, provide clear evidence of the following main features: (i) As expected from Kiviet (1986), the standard LM F test is well-behaved when there is neither conditional nor unconditional heteroskedasticity. However, in the presence of either of these forms of heteroskedasticity, LM F rejects too frequently relative to the nominal value. The extent of the problem usually increases with the sample size, markedly so in the case of some forms of conditional heteroskedasticity. The error distribution does not appear to have any important effects on rejection rates. (This feature is illustrated by Table 2, part (a).) (ii) The standard Lagrange Multiplier test based on an iid bootstrap, LM B , gives estimated significance levels very similar to LM F . Since LM B suffers from the same unacceptable size distortions as the standard test using critical values based on the F-distribution when there is heteroskedasticity and offers no noticeable advantages, it is not considered further. (Examples of the similarity of estimates for LM B and LM F are provided in Table 2.) (iii) The estimates for LM HR and those for MLM HR are similar, whichever of the three methods is used to obtain the critical value. This can be seen by comparing Tables 3
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Table 2 Estimated significance levels of (a) LM F and (b) LM B with T = 40, 80
Sample size
Nominal significance level 1%
Nominal significance level 5%
T = 40
T = 40
(a) LM F Distribution for t of (14) is N (0, 1) Case 1 0.92 Case 2 2.26 Case 3 1.34 Case 4 2.35 Case 5 1.99
T = 80
4.61 8.29 5.86 8.95 7.81
4.27 15.94 6.38 16.22 8.31
Distribution for t of (14) is standardized t (5) Case 1 1.00 1.04 Case 2 2.37 6.18 Case 3 1.87 2.14 Case 4 2.65 7.22 Case 5 2.77 2.78
4.54 7.95 6.44 9.22 8.07
4.64 15.05 7.57 17.51 8.23
Distribution for t of (14) is standardized 2 (8) Case 1 1.06 0.97 Case 2 2.34 6.07 Case 3 1.66 2.14 Case 4 2.61 6.42 Case 5 2.89 2.80
4.72 8.17 6.52 8.93 9.00
4.33 15.44 7.50 16.68 8.45
4.72 9.76 5.23 10.50 7.66
4.62 16.69 5.27 16.90 8.16
Distribution for t of (14) is standardized t (5) Case 1 0.98 1.06 Case 2 2.94 6.10 Case 3 1.36 1.20 Case 4 3.30 7.22 Case 5 2.36 2.01
4.63 9.61 5.78 10.92 7.82
4.83 15.92 6.04 18.36 7.80
Distribution for t of (14) is standardized 2 (8) Case 1 1.05 0.92 Case 2 2.94 6.02 Case 3 1.20 1.21 Case 4 3.20 6.58 Case 5 2.50 2.16
4.72 9.80 5.65 10.57 8.58
4.69 16.05 5.83 17.34 8.03
(b) LM B Distribution for t of (14) is N (0, 1) Case 1 0.93 Case 2 2.68 Case 3 1.14 Case 4 2.80 Case 5 1.71
0.90 6.48 1.59 6.14 2.26
T = 80
0.96 6.48 1.13 6.23 2.07
Notes: Cases are as specified in Table 1. All estimates are given as percentages.
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Table 3 Estimated significance levels of LM HR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (4)
PD1
PD2
2 (4)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.30 1.33 0.17 1.36 0.21 1.43 0.18 1.27 0.16 1.56
0.98 1.00 1.14 1.02 1.29
3.69 3.41 3.34 3.52 3.36
5.98 5.99 5.86 6.26 6.62
5.04 5.08 5.07 5.20 5.91
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.17 1.12 0.85 0.14 1.23 0.94 0.14 1.20 1.00 0.15 1.25 0.90 0.12 1.22 1.10
2.88 2.90 2.98 3.04 2.62
5.48 5.82 5.92 5.78 6.08
4.41 4.78 5.14 4.87 5.47
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.22 1.15 0.90 0.16 1.15 0.96 0.17 1.23 0.90 0.15 1.20 0.87 0.17 1.28 1.18
3.24 3.18 3.09 3.20 2.99
5.55 6.01 5.75 5.88 6.40
4.61 4.92 4.94 4.96 5.76
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (4) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
and 5 below. When the asymptotic 2 (4) distribution is used, LM HR and MLM HR are both markedly undersized at 1% and 5% levels. The problem is especially severe when the nominal size is 1%, as might be appropriate when a check for serial correlation is one of a battery of misspecification tests. When PD1 is used in the wild bootstrap, LM HR and MLM HR have estimates that are slightly greater than required values. Despite what might be expected from the asymptotic theory for a quasi-t test in Liu (1988), the use of PD2 leads to better agreement betweenestimates and nominal values than is observed with PD1 . Indeed all the evidence suggests that using PD2 leads to very good control of finite sample significance levels. (iv) The Hausman-type test HAHR is, like LM HR and MLM HR , undersized at the nominal 1% level when asymptotic critical values are used. However, asymptotic critical values produce estimates, a sample of which is reported in Table 4 below, that are close to the nominal significance level of 5%. It might be conjectured that the improved performance of asymptotic theory reflects the reduction in the number of restrictions being tested, i.e., two rather than four. The use of PD1 again leads to slightly oversized tests and good control is derived when PD2 is employed.
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Table 4 Estimated significance levels of HAHR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (2)
PD1
PD2
2 (2)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.50 1.29 0.56 1.35 0.48 1.60 0.57 1.37 0.30 1.38
1.06 0.99 1.30 1.03 1.20
4.77 5.62 4.34 5.60 3.64
5.69 5.96 6.30 5.95 6.24
4.95 5.02 5.75 4.93 5.80
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.40 1.12 0.84 0.48 1.31 0.96 0.39 1.43 1.22 0.50 1.35 0.95 0.23 1.16 1.09
4.32 5.04 4.29 5.33 3.22
5.31 5.92 6.30 6.00 5.82
4.68 4.89 5.71 5.08 5.53
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.41 1.20 0.94 0.51 1.29 1.06 0.34 1.38 1.11 0.57 1.35 1.01 0.28 1.41 1.16
4.40 5.70 4.32 5.77 3.77
5.39 6.26 6.38 6.30 6.26
4.57 5.31 5.89 5.34 6.01
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (2) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
The full set of Monte Carlo results is available on request, but, in order to save space, only a representative sample is provided below. This sample is obtained from five cases. Details of these cases are given in Table 1. The cases cover all variance models and both types of exogenous regressor sequence. For variance models (16), (18) and (19), there is a choice between two levels of heteroskedasticity. The stronger of the two forms has been used in all cases in order to obtain stringent checks of the efficacy of the wild bootstrap. For the artificial AR(1) regressor of (12), the smaller value of , i.e., = 0.5, is used so that the time-series behaviour of this variable is not similar to that of the standardized log(UK GDP) series, which can be modelled by an I(1) process. All the cases in Table 1 are for (11) with (1 = 0.5, 2 = 0.3). Different choices of (1 , 2 ) from the pairs of values given in Section 4 do not lead to important changes in results; see Godfrey and Tremayne (2002). In Table 2, parts (a) and (b), when the non-robust forms of tests are under consideration, results for both values of T are given to show that distortions due to heteroskedasticity are not reduced (and can get much worse) when the sample size is increased. The departures from nominal significance levels observed when LM F is asymptotically inappropriate indicate the vulnerability of this well-known standard check to ARCH and GARCH forms of
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Table 5 Estimated significance levels of MLM HR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (4)
PD1
PD2
2 (4)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.24 1.30 0.14 1.25 0.17 1.26 0.13 1.24 0.13 1.43
0.97 1.00 1.05 0.99 1.20
3.25 2.86 2.71 2.98 2.70
5.68 5.78 5.51 5.90 6.22
4.82 4.98 4.84 5.12 5.65
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.11 1.12 0.86 0.12 1.14 1.02 0.13 1.08 0.90 0.14 1.16 0.90 0.11 1.18 1.10
2.65 2.64 2.50 2.64 2.24
5.25 5.63 5.55 5.57 5.80
4.48 4.74 4.96 4.81 5.45
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.18 1.02 0.83 0.15 1.07 0.90 0.13 1.23 0.96 0.12 1.18 0.86 0.14 1.25 1.16
2.74 2.72 2.63 2.80 2.54
5.12 5.78 5.58 5.72 6.02
4.33 4.92 4.89 4.97 5.53
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (4) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
conditional heteroskedasticity and to seasonal unconditional heteroskedasticity. The contents of Table 2, part (a), demonstrate this clearly, while Table 2, part (b), supports our contention that LM B and LM F behave similarly. The information in Tables 3–5 shows very clearly that asymptotic critical values cannot be relied upon when using HR tests. However, the combination of restricted residuals in the HCCME and the very simple PD2 wild bootstrap gives tests with good finite sample behaviour. The results for the t (5) error distribution indicate that the wild bootstrap can work well even when regularity condition (f) of Section 2 is not satisfied. The results for the 2 (8) error distribution show that symmetry of innovations is not essential for the wild bootstrap to yield accurate finite sample rejection rates. Turning to estimates of power derived from (20) to (27), results for the artificial regressor sequence of (12) with =0.5 using PD2 are given in Table 6. The following features emerge from consideration of the contents of Table 6. (v) When LM F is asymptotically valid, i.e., in the cases of panel (a), it is usually slightly more powerful than the HR variants LM HR and MLM HR . However, the cost of the ‘insurance premium’ for robustness to unspecified forms of heteroskedasticity is not
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Table 6 Estimates of power with T = 40 and T = 80. Model is (11) with (1 , 2 , 1 , 2 ) = (0.5, 0.3, 1.0, 1.0). AR(1) regressor with = 0.5 (a) Errors t generated by stationary AR(5) model (20) with perturbations t being NID(0,1) T = 40 T = 80
(j ) of
LM F
LM HR
HAHR
MLM HR
LM F
LM HR
HAHR
MLM HR
(21) (22) (23) (24) (25)
36.7 82.2 14.7 30.3 47.6
31.5 71.9 17.8 28.2 39.5
33.4 9.0 21.7 24.5 27.2
30.2 72.9 16.5 26.9 39.6
80.6 99.7 69.1 91.5 95.8
75.1 99.3 65.5 86.6 92.3
74.9 18.0 72.0 75.5 76.5
74.5 99.3 64.5 86.3 92.1
(b) Errors t generated by ARCH models (26) and (27) with perturbations t being NID(0,1) T = 40 T = 80 ARCH model LM F LM HR HAHR MLM HR LM F LM HR HAHR MLM HR (26) (27)
n/a n/a
30.9 67.3
32.2 9.7
29.5 67.6
n/a n/a
74.3 99.0
74.7 18.9
73.9 99.0
Notes: All critical values estimated using PD2 wild bootstrap, with nominal significance level of 5%. n/a denotes that the test LM F is inappropriate.
large. Estimates for LM F are not given for the ARCH models in panel (b) because the test is asymptotically inappropriate; see Table 2, part (a). (vi) There is no generally valid ranking by power that applies to HAHR , on the one hand, and the asymptotically equivalent LM HR and MLM HR , on the other; see Holly (1982). Differences between Hausman-type and LM-type tests vary in sign and are usually small. However, an important difference is observed in the cases of the simple AR(4) model defined by (22) and the corresponding ARCH process (27). For these two cases, HAHR has manifestly inferior power. The insensitivity of HAHR to serial correlation is worrying even if the implied inconsistencies of the OLS estimators are small. The usual HCCME is inappropriate in the presence of serial correlation, so failure to detect serial correlation may lead to misleading inferences. Overall it may be safer to use either LM HR or MLM HR , the differences between the power estimates of these asymptotically equivalent tests being small. (vii) Finally, comparison of the first and second rows of results in panel (a) of Table 6 with the corresponding results in panel (b) suggests that power estimates are not greatly affected by the presence of random variation in AR coefficients.
6. Conclusions Time-varying heteroskedasticity is recognized as an important phenomenon in applied studies; see, e.g., the discussions in Bollerslev et al. (1994). It is shown that misleading inferences may be made when standard checks for serial correlation are applied to dynamic
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regression models in the presence of either conditional or seasonal heteroskedasticity. To be more precise, simulation evidence presented in this paper illustrates a tendency for the wellknown Lagrange multiplier test to be oversized, sometimes by a substantial amount. This evidence is derived from Monte Carlo experiments designed to mimic quarterly empirical studies. In these experiments, heteroskedasticity-robust versions of the usual Lagrange multiplier test and a Hausman-type test are also examined, along with a modification of the former procedure. The modification is designed to improve performance by using a less variable estimate of the covariance matrix. All tests use restricted residuals to calculate the HCCME proposed in White (1980). It is clear from the Monte Carlo results that the use of asymptotic critical values cannot be recommended. Heteroskedasticity-robust tests conducted on this basis are found to be undersized. However, a wild bootstrap approach proves to be reliable, with good control over finite sample significance levels being provided by the simple twopoint probability distribution discussed by Davidson and Flachaire (2001). This result gives supporting evidence to Flachaire (2004) in which exactly the same combination of restricted residuals in the HCCME and form of wild bootstrap is recommended. The experiments described above, being, for the most part, based upon dynamic regressions with conditional heteroskedasticity, are complementary to those in Flachaire (2004), which are for static regressions with unconditional heteroskedasticity. Results on power of serial correlation tests are also reported. The original and modified forms of the Lagrange multiplier test are very similar in performance. Under several alternatives, the Hausman test rejects a false null with a frequency similar to the other two tests, but evidence is found that it has relatively weak power against simple fourth-order autocorrelation. Consequently it may be safer to use the Lagrange multiplier test in the form in which it is robust to heteroskedasticity of unspecified form, or the modified version of this test, rather than the Hausman test. Acknowledgements We are grateful to Peter N. Smith, two anonymous referees and David Belsley for helpful comments and suggestions. References Belsley, D.A., 1997. A small-sample correction for testing for gth-order serial correlation with artificial regressions. Comput. Econom. 10, 197–229. Belsley, D.A., 2000. A small-sample correction for testing for joint serial correlation with artificial regressions. Comput. Econom. 16, 5–45. Belsley, D.A., 2002. An investigation of an unbiased correction for heteroskedasticity and the effects of misspecifying the skedastic function. J. Econom. Dynam. Control 26, 1379–1396. Bera, A.K., Higgins, M.L., 1993. ARCH models: properties, estimation and testing. J. Econom. Surveys 7, 305–366. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327. Bollerslev, T., Engle, R.F., Nelson, D.B., 1994. ARCH Models. in: Engle,R.F.,McFadden,D.L. (Eds.), Handbook of Econometrics, Vol. IV. North-Holland, Amsterdam. Breusch, T.S., 1978. Testing for autocorrelation in dynamic linear models. Austral. Econom. Papers 17, 334–355.
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The wild bootstrap and heteroskedasticity-robust tests for serial correlation in dynamic regression models L.G. Godfreya,∗ , A.R. Tremaynea, b a Department of Economics, University of York, Heslington, York YO10 5DD, UK b School of Economics and Political Science, University of Sydney, Sydney NSW 2006, Australia
Available online 19 June 2004
Abstract Conditional heteroskedasticity is a common feature of financial and macroeconomic time series data. When such heteroskedasticity is present, standard checks for serial correlation in dynamic regression models are inappropriate. In such circumstances, it is obviously important to have asymptotically valid tests that are reliable in finite samples. Monte Carlo evidence reported in this paper indicates that asymptotic critical values fail to give good control of finite sample significance levels of heteroskedasticity-robust versions of the standard Lagrange multiplier test, a Hausman-type check, and a new procedure. The application of computer-intensive methods to removing size distortion is, therefore, examined. It is found that a particularly simple form of the wild bootstrap leads to well-behaved tests. Some simulation evidence on power is also given. © 2004 Elsevier B.V. All rights reserved. Keywords: Heteroskedasticity; Serial correlation; Wild bootstrap
1. Introduction The importance of testing for serial correlation after the least-squares estimation of a dynamic regression model has been understood for many years. Lagrange multiplier (LM) tests allow for flexibility in the choice of alternative hypothesis, are easily implemented, and are now used as a matter of routine in estimation programmes. A standard means ∗ Corresponding author. Tel.: +44-1904-433754; fax: +44-1904-433759.
E-mail addresses: [email protected] (L.G. Godfrey), [email protected] (A.R. Tremayne). 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.05.020
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of implementing such tests is to carry out an artificial regression and base the test on the significance of coefficient(s) in that regression. It is shown in Belsley (1997, 2000) that misleading inferences can result from using such an approach in static regression with small and moderate samples. In the earlier of these two papers, an adjustment to the t-statistic for testing the null of no serial correlation against the alternative of a simple scheme of specified order is proposed. The companion paper extends the approach to an F-test for joint autoregressive schemes and the efficacy of both suggestions is demonstrated by reporting Monte Carlo results based upon independent and identically distributed (iid) regression errors. The present paper is concerned with the application of such tests when the regressors include lagged dependent variables, as well as exogenous variables. In the presence of both types of regressors, tests based on the popular portmanteau-type approach are, of course, invalid and so LM tests may be particularly attractive to empirical workers. However, standard LM-type tests (and the modifications suggested by Belsley), along with many others, are predicated on an assumption of homoskedasticity. It is now recognized that conditional heteroskedasticity may be common, especially (but certainly not exclusively) when the data relate to financial variables. In the presence of conditional heteroskedasticity, standard checks for serial correlation cannot be assumed to be asymptotically valid and may lead to misleading inferences. There is therefore a need to examine heteroskedasticity-robust (HR) procedures. An obvious way to derive LM-type checks that are asymptotically HR is to employ heteroskedasticity-consistent covariance matrix estimates (HCCME) as discussed in White (1980). The basic idea of using HCCME to obtain diagnostic tests is long established; see, for example, Pagan and Hall (1983). Details of a specific application of HCCME to testing for serial correlation are given in Godfrey (1994). However, there is a considerable body of evidence indicating that tests derived from the HCCME can have finite sample distributions that are quite unlike those predicted by asymptotic theory. In particular, finite sample significance levels can be far from what is desired. The main purpose of this paper is to report Monte Carlo results on the wild bootstrap approach when it is used to obtain critical values or p-values for serial correlation tests based upon HCCME. Given the significant increases in the availability of cheap and powerful computers in the last few years, this approach is attractive. Once implemented in standard programmes, it could be used not only for serial correlation tests, but also for other variable-addition diagnostic checks, e.g. the RESET test. Thus the wild bootstrap described below provides a way for applied workers to obtain misspecification tests that are asymptotically robust to heteroskedasticity (and nonnormality) with the bootstrap being used to improve upon the approximation provided by asymptotic theory. This combination of features matches recommendations in Hansen (1999) for good econometric practice. In order to assess the added value of the results given below, reference should be made to the existing body of published work pertaining to serial correlation tests and conditional heteroskedasticity. There are relatively few contributions. Robinson (1991) considers a static linear regression model and derives an LM test for serial correlation that is asymptotically valid under dynamic conditional heteroskedasticity and nonnormality. No evidence on finite sample performance is provided and, as noted by Robinson, the test is inappropriate in the presence of lagged dependent variables. Whang (1998) derives tests for serial correlation in dynamic models that are asymptotically valid in the presence of heteroskedasticity of
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unknown form. Whang (1998) does not use the HCCME approach. Instead his test statistic involves a nonparametric kernel estimate of the unknown variance function, and his technique requires the specification of kernel and trimming functions. He reports Monte Carlo results for his asymptotic tests based upon a static simple regression. His estimates show that asymptotically valid approximations cannot be relied upon. For example, with 50 observations and a nominal significance level of 5%, the majority of his estimates are between 2% and 3.5%. In addition to these published articles, there is an unpublished study by Binbin Guo and Peter Phillips. In Guo and Phillips (2000), they provide HR tests that are asymptotically valid either when there are only exogenous regressors or when there are only lagged dependent variables as regressors. They report Monte Carlo results for a model of the latter type. The use of asymptotic critical values produces mixed results with some evidence for tests being undersized. The use of HCCME can also be avoided if there is precise information about the error variances, for then specific adjustments can be made to obtain HR checks for serial correlation. For example, ARCH-corrected tests are discussed in Silvapulle and Evans (1998). However, such an approach suffers from the general drawback that applied researchers are not likely to have accurate information about the precise form of heteroskedasticity. Mistakes made in specifying the variance model will, in general, lead to asymptotically invalid inferences; see Belsley (2002). Moreover, as noted above and contrary to what is asserted in Silvapulle and Evans (1998, p. 35), the Box–Pierce and Ljung–Box portmanteau tests are not asymptotically valid when exogenous variables and lagged values of the dependent variable are both included amongst the regressors. Consequently their Monte Carlo analysis includes inappropriate tests and the corresponding results must be treated with caution. In view of the above, the novel features of our study are: the use of a wild bootstrap to gain better control over significance levels than is provided by asymptotic theory; Monte Carlo designs using regression models with regressor sets having both exogenous variables and lagged dependent variables; and a new test derived as a modification of the HR version of the LM statistic. The paper consists of six sections. Section 2 contains descriptions of models and tests, including the new variant of the LM test. Bootstrap methods are explained in Section 3. The designs of the Monte Carlo experiments are provided in Section 4. Results from these experiments are summarized in Section 5. Section 6 concludes. 2. Models and test procedures The dynamic regression model is written as yt = Yt + Xt + t = Wt + t ,
(1)
where: Yt = (yt−1 , . . . , yt−P ) with P 1; = (1 , . . . , P ) having values such that the roots of zP − 1 zP −1 − · · · − P = 0 are all strictly inside the unit circle; Xt is an L-vector of regressors that are strictly exogenous; Wt = (Yt , Xt ); and = ( , ). Let K = P + L denote the number of regression
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coefficients in (1) and T denote the number of observations available for estimation. The null hypothesis to be tested is that the errors t are serially uncorrelated. It will be argued that the use of a wild bootstrap is very important to the reliable assessment of the statistical significance of HR variants of the LM tests of Breusch (1978) and Godfrey (1978), and the Hausman-type test of Godfrey (1997). Consequently the regularity conditions must not only support the usual HCCME and associated asymptotic tests, but also validate the bootstrap procedure. All tests are constructed using the results of ordinary least-squares (OLS) estimation of (1). It is essential that the OLS estimators be consistent under the null hypothesis. Consequently it is assumed that RSS j , the residual sum of squares from the artificial regression of the j th variable of Wt on all other regressors, is at least Op (T ), j = 1, . . . , K, i.e., all regressors are asymptotically cooperative. Restrictions on the coefficients of the lagged dependent variables of Yt have been given above. The exogenous variables Xt are not restricted to be nonstochastic or covariance-stationary I(0) terms; see Wooldridge (1999) for a discussion of nonstationary regressors in the context of testing for serial correlation. Let = ( , ) denote the OLS coefficient estimator for (1) and the terms et = yt − Wt be the residuals derived from this estimator. Turning to assumptions that are made, under H0 , about the errors of (1), the following, provided by Gonçalves and Kilian (2002) for wild bootstraps, are adopted for the case of conditional heteroskedasticity: (a) E(t | Ft−1 ) = 0, almost surely, where Ft−1 is the sigma-field generated by (t−1 , t−2 , . . .). (b) E(2t ) = 2 , 0 < 2 < ∞. (c) plimT →∞ T −1 Tt=1 E(2t | Ft−1 ) = 2 . (d) E(2t t−r t−s ) = 0 for all r = s, for all t, r 1, s 1. (e) plimT →∞ T −1 Tt=1 t−r t−s E(2t | Ft−1 ) = 0 for any r 1, s 1. (f) E(| t |4r ) is uniformly bounded for some r 2 and all t. These assumptions are somewhat stronger than those required for the asymptotic validity of the HCCME and, in particular, require that at least eight, rather than at least four, moments exist. Also Gonçalves and Kilian (2002) remark that assumption (d) is satisfied for ARCH and GARCH processes with symmetric innovation processes but is ruled out in some asymmetric cases. The model (1) differs from that used by Gonçalves and Kilian (2002) to discuss regularity conditions because the latter is a pure autoregression, i.e., = 0. Provided they are asymptotically cooperative, it seems reasonable to assume that the inclusion of strictly exogenous regressors is not problematic. The case of predetermined, but not strictly exogenous, regressors is discussed below. Suppose that the null hypothesis is to be tested against the alternative that the errors are generated by a Qth-order autoregression, denoted AR(Q). [Using a Qth-order moving average, denoted MA(Q), as the alternative leads to the same test statistic.] A convenient variable-addition form of the LM test is a test of = (1 , . . . , Q ) = 0 in the augmented model yt = Yt + Xt + Et + t = Wt + Et + t ,
(2)
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in which Et = (et−1 , . . . , et−Q ) and et−q is set equal to zero when t − q 0. Denote the OLS estimator of the regression coefficients for (2) by ˆ and ˆ . The Breusch–Godfrey test is then a check of the joint significance of the elements of ˆ . Although Breusch (1978) and Godfrey (1978) propose a large-sample 2 form of the test, subsequent investigations by Kiviet (1986) revealed that better finite sample behaviour is achieved using an approximate F statistic. Kiviet’s version of the test is denoted by LM F , and it is this form that will be shown to be vulnerable to dynamic heteroskedasticity. If the errors were iid under the null hypothesis, one simple possibility for potentially improving the small-sample behaviour of a test like LM F would be to use a standard bootstrap procedure based on a residualresampling scheme to estimate critical values. We experiment with such an approach; the test is termed LM B in the experiments reported below and it is seen to suffer from the same deficiencies as LM F . An alternative to LM procedures is derived in Godfrey (1997) as an extension of Dezhbakhsh and Thursby (1994). The test is a check of the significance of the difference between the estimators of from (1) and (2), i.e., ( − ˆ ). Godfrey (1997) gives a formula based upon a direct application of Hausman’s (1978) general result for the variance–covariance matrix of estimator contrasts. However, a variable-addition form is also available. Let A denote the cross-product moment matrix for the regressors of (2), i.e., A=T
−1
Y Y X Y E Y
Y X X X E X
Y E XE , E E
in which Y is T × P with rows Yt , X is T × L with rows Xt , and E is T × Q with rows Et . The submatrices of the inverse of A partitioned as lagged endogenous regressors, exogenous regressors, and lagged residuals are denoted by AYY , AYX , AYE , etc. The variable-addition form of Godfrey’s (1997, p. 201, Eq. (2.6)) test is then obtained by testing = (1 , . . . , P ) = 0 in the extended model yt = Yt + Xt + (AYE Et ) + t = Wt + Jt + t ,
(3)
i.e., the test variables are the P linear combinations of the terms in Et given by Jt = AYE Et . A proof that (3) is appropriate can be derived by modification of Theorem 2 in Davidson et al. (1985). If P Q with rank(AYE ) = Q, the test variables of (3) are (omitting any redundant terms) equivalent to those of (2). If P < Q with rank(AYE )=P , the Hausman and LM tests are not asymptotically equivalent and there is no generally valid ranking of these procedures by asymptotic local power; see, e.g., Holly (1982). Using White’s (1980) general formula for the HCCME, the HR version of the LM statistic can be shown to be M(W)E}−1 E e, LM HR = e E{E M(W)
(4)
is diag(e2 , . . . , e2 ), where e is the T-dimensional OLS residual vector with elements et , 1 T and M(W) = IT − P(W) = IT − W(W W)−1 W ,
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where W is the T × K matrix with rows Wt . The calculation of the HR-Hausman test of = 0 in (3), when P < Q, is carried out in a similar fashion. We denote the test statistic for this procedure by HAHR , which is given by M(W)J}−1 J e, HAHR = e J{J M(W)
(5)
where J is the T × P matrix with rows Jt . It should be noted that the et2 derived from OLS estimation of (1) are used to compute the HCCME for both robust serial correlation tests. This corresponds to using the squared residuals from restricted (under the null hypothesis) estimation, which, compared to using the unrestricted squared residuals, has been found to give superior finite-sample performance; see Davidson and MacKinnon (1985) and Godfrey and Orme (2002). As in Davidson and MacKinnon (1985), HCCME are sometimes derived by multiplying squared residuals by terms that tend to unity. It is reported in Davidson and MacKinnon (1985) that, when restricted residuals are used, such adjustments lead only to slight changes. The simplest version is, therefore, used. Consideration of finite-sample performance suggests that, in addition to using et2 in expressions for HCCME, it may be useful to examine devices that reduce the sampling variability of the HCCME. One such device is used here to derive a modified version of LM HR . The basic idea is to eliminate asymptotically negligible terms. To identify and remove such terms, partition W as (W1 , W2 ), where, under the null hypothesis, p lim T −1 W1 E = 0
and p lim T −1 W2 E = 0,
so that W1 contains terms like yt−j , j = 1, . . . , min(P , Q), and W2 contains the exogenous regressors Xt and any terms yt−j with j > Q. Standard results on projection matrices imply that 1 ) − P(W2 ), M(W) = M(W 1 = M(W2 )W1 . Hence where W M(W)E = 1 + 2 + + 3 , E M(W) 2 where E, ˇ 1 = Eˇ
P(W2 )E, 2 = −Eˇ
P(W2 )E 3 = E P(W2 )
1 )E. Since p lim T −1 W E = 0, 2 and 3 are both op (T ) and are and Eˇ denotes M(W 2 asymptotically negligible relative to 1 , which is Op (T ). Hence the HR-variant of the LM test given by (4) is asymptotically equivalent to the modified procedure based upon E) ˇ −1 E e. MLM HR = e E(Eˇ
(6)
Under the null hypothesis, LM HR and MLM HR are asymptotically distributed as 2 (Q). When P < Q, the asymptotic null distribution of HAHR is 2 (P ). (If P Q, all three tests are asymptotically equivalent.) Asymptotically valid tests are obtained by comparing sample values of test statistics with critical values from the right-hand tail of the appropriate 2 distributions. However, as illustrated by the results provided in Godfrey and Orme (2002), the finite sample distributions of HR statistics may differ appreciably from those predicted
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by asymptotic theory when several restrictions are being tested. Their results also indicate the potential usefulness of wild bootstrap methods, which will be discussed in the next section.
3. Wild bootstrap methods To describe the wild bootstrap, it is useful to write the OLS estimated version of (1) as yt = Wt + et ,
t = 1, . . . , T .
Let Z be the discrete random variable with the two-point probability distribution √ √ √ Z= − ( 5 − 1)/2 with probability ( 5 + 1)/(2 5) √ = ( 5 + 1)/2, otherwise,
(7)
(8)
so that E(Z) = 0, E(Z 2 ) = 1, and E(Z 3 )=1. The conditions that E(Z)=0 and E(Z 2 )=1 are essential for the validity of the bootstrap procedure. The additional restriction pertaining to the third moment of (8) is suggested in Liu (1988). After examination of the distribution of a single linear combination of OLS estimators, it is shown in Liu (1988) that, if E(Z 3 ) = 1, the wild bootstrap enjoys second-order properties with the first three moments of the relevant test statistic being estimated correctly to O(T −1 ). The probability distribution (8) is denoted PD1 in the discussion in Section 5. The wild bootstrap is then implemented as follows. (i) For t = 1, . . . , T , observations are generated by ∗ ∗ yt∗ = ˜ 1 yt−1 + · · · + ˜ P yt−P + Xt + ∗t ,
(9)
where bootstrap sample starting values are set equal to actual estimation sample starting values (see Li and Maddala, 1996, Section 2.3), and the errors ∗t are given by ∗t = et Zt , Zt being a drawing from (8). Note that, in contrast to the standard residual resampling scheme for iid errors, it is not necessary to centre the OLS residuals if the null model does not contain an intercept when using the wild bootstrap; see Liu (1988, pp. 1706–1707). (ii) Given the bootstrap data, the regression model is estimated and the associated values of the test statistics HA∗HR , LM ∗HR , and MLM ∗HR are calculated. (iii) Repeat (i) and (ii) B times in order to estimate the p-values of the observed statistics HAHR , LM HR , and MLM HR . The null hypothesis of serial independence is rejected for p-values that are sufficiently small. Several other distributions have been suggested as alternatives to (8); see Davidson and Flachaire (2001), henceforth DF, Liu (1988), and Mammen (1993). However, with the exception of the DF scheme, using these alternative wild bootstraps produces Monte Carlo results here that are markedly inferior to those produced by (8). Consequently, detailed results are provided below only for (8) and the DF scheme given by the simple two-point distribution Z=1 with probability 0.5, = − 1, otherwise.
(10)
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(The Monte Carlo results for all wild bootstrap schemes are available on request.) The distribution of (10) satisfies the essential requirements that E(Z) = 0 and E(Z 2 ) = 1, but has E(Z 3 ) = 0, rather than E(Z 3 ) = 1 as in (8). The probability distribution of (10) recommended by DF is denoted PD2 below. Provided the regularity conditions mentioned in Section 2 are satisfied, the wild bootstrap can be used for either conditional or unconditional heteroskedasticity by combining (9) with either PD1 or PD2 . The bootstrap scheme (9) assumes that the variables of Xt are strictly exogenous. If the Xt were only predetermined, it could be argued that (9) is inappropriate and that joint simulation of conditional and marginal models is required to approximate the actual data generation process. However, Giersbergen and Kiviet (1996) argue that joint modelling introduces the possibility of misspecifying the marginal model for Xt and find Monte Carlo evidence that joint modelling fails to deliver useful improvements, a finding corroborated in Godfrey and Orme (2000). The wild bootstrap is not the only way to produce HR tests. If the model is static with all the regressors being strictly exogenous, and the variances are functions of regressor values, a paired bootstrap can be used; see Flachaire (1999). Brownstone andValletta (2001) discusses why the wild bootstrap is likely to be more accurate than the paired bootstrap when both are appropriate. Additional evidence in favour of using the wild bootstrap method recommended below, rather than the paired bootstrap, is to be found in Flachaire (2004). 4. Monte Carlo design The number of Monte Carlo replications for each experiment is R = 25000, with the number of bootstraps being B = 399. MacKinnon (2002) remarks that, while he would not recommend using B = 399 in genuine applications, sampling errors associated with this value tend to cancel out in Monte Carlo experiments. In empirical work, B = 999 could be used without incurring important waiting times, given the power of modern personal computers. The dynamic regression model upon which experiments are based is yt = 1 yt−1 + 2 yt−2 + 1 + 2 xt + t ,
t = 1, . . . , T ,
(11)
in which xt is a scalar variable and T equals 40 or 80. This model corresponds to (7) of Dezhbakhsh (1990). The values used for (1 , 2 ) are (0.5, 0.3), (0.7, −0.2), (1.0, −0.2), (1.3, −0.5), (0.9, −0.3), and (0.6, 0.2) which are regarded in Dezhbakhsh and Thursby (1995) as typical of values observed in applied work. In all experiments, 1 = 2 = 1. The exogenous variable xt is constructed in two ways. First, it is generated as an artificial variable according to the first order autoregression xt = xt−1 + vt ,
(12)
with vt being NID(0, 2v ), = 0.5 or 0.9, and 2v selected, given the value of , so that Var(xt ) = 1. The vt and all other pseudo-random numbers are generated using subroutines from the NAG Library. The starting value x0 is a drawing from a standard normal distribution. Second, xt is obtained by standardizing the logs of quarterly observations for GDP in the UK, with the basic data being taken from the file GDP95.FIT in the Microfit 4.0 package (Pesaran and Pesaran, 1996). The transformed data appear to exhibit deterministic and stochastic
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trends, as might be expected for this type of macro-economic variable. The starting values y−1 and y0 are set equal to their unconditional mean with x = 0, i.e., 1 /(1 − 1 − 2 ). The effect of using these starting values is reduced by discarding the first 40 of any sequence of (T + 40) generated values. The finite sample significance levels of versions of the tests described in Section 2 are estimated for nominal levels of 1% and 5%. As a useful example of testing using quarterly data, HR-tests LM HR and MLM HR of 1 = 2 = 3 = 4 = 0 in yt = 1 yt−1 + 2 yt−2 + 1 + 2 xt +
4
j et−j + error,
(13)
j =1
are used, with the et−j being lagged residuals from the OLS estimation of (11). The Hausman-type test HAHR is a test of the joint significance of the estimator contrasts for 1 and 2 arising from a comparison of OLS estimates of (11) and (13), i.e., it is a special case of the test associated with (3). With conditional heteroskedasticity, the error terms t of (11) can be written as t = h t t , (14) where ht denotes a conditional variance and, under the null hypothesis, the terms t are iid(0,1). Various distributions for t are used. The normal distribution serves as a benchmark and standardized forms of the t (5) and 2 (8) distributions are also employed. The t (5) distribution is used, following Gonçalves and Kilian (2002), to investigate robustness of the wild bootstrap methods to departures from condition (f) of Section 2. The 2 (8) distribution is used to provide evidence on the effects of skewness. This distribution, with a coefficient of skewness equal to 1, is heavily skewed, according to the arguments of Ramberg et al. (1979). The final component required to derive a typical error t , after drawing t , is the con√ ditional standard deviation ht . Since the HR-tests for serial correlation are intended for general use, it is important to obtain evidence not only for several forms of heteroskedasticity, but also for the case of homoskedastic errors. The following five specifications for variance schemes are used. First, the ht are observation-invariant; that is, the errors are homoskedastic with ht = 2 ,
t = 1, . . . , T ,
(15)
2 being set equal to 1 or 10. Second, the ARCH(1) process provides an important example of conditional heteroskedasticity and is used in the form ht = 0 + 1 2t−1 ,
(16)
where 0 = 2 /(1 − 1 ), 1 = 0.4 or 0.8, and 2 is defined as for (15). The GARCH(1, 1) model is also frequently used in applied studies. This variance scheme is written as ht = 0 + 1 2t−1 + 2 ht−1 ,
(17)
where 0 = 1, 1 = 0.1, and 2 = 0.8; see Bollerslev (1986). The values of 1 and 2 are similar to those reported in empirical work. The average R 2 varies between 0.4 and 0.9
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across the five variance schemes, which seems consistent with values observed in applied work. Given the widespread use of quarterly data in applied work, it seems apposite to examine seasonal schemes of heteroskedasticity. Consideration is, therefore, given to a fourth-order model taken from Engle’s (1982) classic article. As in (38) of Engle (1982, p. 1002), conditional variances ht are written as ht = 0 + 1 (0.42t−1 + 0.32t−2 + 0.22t−3 + 0.12t−4 ),
(18)
in which 0 and 1 are as defined for (16). The fifth and final model to be adopted has unconditional quarterly heteroskedasticity and can be written as 22 22 2c2 2c2 2 2 2 2 , , , , (19) ( 1 , 2 , 3 , 4 ) = (1 + c) (1 + c) (1 + c) (1 + c) in which 2j denotes the variance in quarter j, the average of these terms is the 2 specified in (15), and c equals 4 or 9. Model (19) is similar in spirit to that used in Burridge and Taylor (2001, p. 104). Note that, although (19) generates unconditional heteroskedasticity, its effects on serial-correlation tests cannot be ignored. In the estimation of the covariance matrix of the OLS estimators for (13), the second moments of the errors are not asymp2 , j = 1, . . . , 4, so that the conventional totically orthogonal to the squared regressors et−j (homoskedasticity-based) estimator is not consistent; see White (1980, p. 826). In addition to significance levels, the sensitivity of tests to serial correlation is clearly of interest. For experiments designed to provide evidence on power, various alternatives are used, some of which involve a combination of conditional heteroskedasticity and serial correlation. Five alternatives with homoskedastic errors are obtained as special cases of the stationary fifth-order autoregression
t =
5 j =1
j t−j + t
(20)
which replaces (14). The coefficient vectors = ( 1 , 2 , 3 , 4 , 5 ) of these special cases are as follows:
(1) = (0.7, 0.0, 0.0, 0.0, 0.0),
(21)
(2) = (0.0, 0.0, 0.0, 0.7, 0.0),
(22)
(3) = (1.4, −0.71, 0.15, −0.1, 0.0),
(23)
(4) = (0.8, 0.0, −0.2, 0.06, 0.0) and
(24)
(5) = (0.5, 0.0, 0.0, 0.5, −0.25).
(25)
Thus the alternative hypothesis of the tests LM HR and MLM HR from (13) is overspecified for the first two cases and underspecified for the last. The roots of the polynomial equations z4 − 1 z3 − · · · − 4 = 0 implied by the coefficients of (23) and (24) are (0.5, 0.4, 0.3, 0.2) and (0.5, −0.5, 0.4 ± 0.3i), respectively.
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Table 1 Sample of cases used in Tables 2–5
Case 1 Case 2 Case 3 Case 4 Case 5
Variance model
Regressor
(15) with 2 = 1 (16) with 1 = 0.8 (17) with 1 = 0.1, and 2 = 0.8 (18) with 0 = 0.1, and 1 = 0.8 (19) with c = 9
log(UK GDP)a (12) with = 0.5 log(UK GDP)a (12) with = 0.5 log(UK GDP)a
a These data are standardized before OLS estimation.
To derive power estimates in the presence of ARCH errors, the general random-coefficient specification discussed in Bera and Higgins (1993) is adopted. The two specific error models used in the Monte Carlo experiments are
t = (0.7 + 0.1t )t−1 + t
(26)
t = (0.7 + 0.1t )t−4 + t ,
(27)
and
in which the t are NID(0, 1) and are independent of the s for all s and t. Clearly the probability that the random coefficient of the serial correlation process is strictly between 0 and 1 is high for Eqs. (26) and (27). 5. Monte Carlo results The estimates of significance levels, as sampled in Tables 2–5, provide clear evidence of the following main features: (i) As expected from Kiviet (1986), the standard LM F test is well-behaved when there is neither conditional nor unconditional heteroskedasticity. However, in the presence of either of these forms of heteroskedasticity, LM F rejects too frequently relative to the nominal value. The extent of the problem usually increases with the sample size, markedly so in the case of some forms of conditional heteroskedasticity. The error distribution does not appear to have any important effects on rejection rates. (This feature is illustrated by Table 2, part (a).) (ii) The standard Lagrange Multiplier test based on an iid bootstrap, LM B , gives estimated significance levels very similar to LM F . Since LM B suffers from the same unacceptable size distortions as the standard test using critical values based on the F-distribution when there is heteroskedasticity and offers no noticeable advantages, it is not considered further. (Examples of the similarity of estimates for LM B and LM F are provided in Table 2.) (iii) The estimates for LM HR and those for MLM HR are similar, whichever of the three methods is used to obtain the critical value. This can be seen by comparing Tables 3
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Table 2 Estimated significance levels of (a) LM F and (b) LM B with T = 40, 80
Sample size
Nominal significance level 1%
Nominal significance level 5%
T = 40
T = 40
(a) LM F Distribution for t of (14) is N (0, 1) Case 1 0.92 Case 2 2.26 Case 3 1.34 Case 4 2.35 Case 5 1.99
T = 80
4.61 8.29 5.86 8.95 7.81
4.27 15.94 6.38 16.22 8.31
Distribution for t of (14) is standardized t (5) Case 1 1.00 1.04 Case 2 2.37 6.18 Case 3 1.87 2.14 Case 4 2.65 7.22 Case 5 2.77 2.78
4.54 7.95 6.44 9.22 8.07
4.64 15.05 7.57 17.51 8.23
Distribution for t of (14) is standardized 2 (8) Case 1 1.06 0.97 Case 2 2.34 6.07 Case 3 1.66 2.14 Case 4 2.61 6.42 Case 5 2.89 2.80
4.72 8.17 6.52 8.93 9.00
4.33 15.44 7.50 16.68 8.45
4.72 9.76 5.23 10.50 7.66
4.62 16.69 5.27 16.90 8.16
Distribution for t of (14) is standardized t (5) Case 1 0.98 1.06 Case 2 2.94 6.10 Case 3 1.36 1.20 Case 4 3.30 7.22 Case 5 2.36 2.01
4.63 9.61 5.78 10.92 7.82
4.83 15.92 6.04 18.36 7.80
Distribution for t of (14) is standardized 2 (8) Case 1 1.05 0.92 Case 2 2.94 6.02 Case 3 1.20 1.21 Case 4 3.20 6.58 Case 5 2.50 2.16
4.72 9.80 5.65 10.57 8.58
4.69 16.05 5.83 17.34 8.03
(b) LM B Distribution for t of (14) is N (0, 1) Case 1 0.93 Case 2 2.68 Case 3 1.14 Case 4 2.80 Case 5 1.71
0.90 6.48 1.59 6.14 2.26
T = 80
0.96 6.48 1.13 6.23 2.07
Notes: Cases are as specified in Table 1. All estimates are given as percentages.
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Table 3 Estimated significance levels of LM HR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (4)
PD1
PD2
2 (4)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.30 1.33 0.17 1.36 0.21 1.43 0.18 1.27 0.16 1.56
0.98 1.00 1.14 1.02 1.29
3.69 3.41 3.34 3.52 3.36
5.98 5.99 5.86 6.26 6.62
5.04 5.08 5.07 5.20 5.91
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.17 1.12 0.85 0.14 1.23 0.94 0.14 1.20 1.00 0.15 1.25 0.90 0.12 1.22 1.10
2.88 2.90 2.98 3.04 2.62
5.48 5.82 5.92 5.78 6.08
4.41 4.78 5.14 4.87 5.47
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.22 1.15 0.90 0.16 1.15 0.96 0.17 1.23 0.90 0.15 1.20 0.87 0.17 1.28 1.18
3.24 3.18 3.09 3.20 2.99
5.55 6.01 5.75 5.88 6.40
4.61 4.92 4.94 4.96 5.76
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (4) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
and 5 below. When the asymptotic 2 (4) distribution is used, LM HR and MLM HR are both markedly undersized at 1% and 5% levels. The problem is especially severe when the nominal size is 1%, as might be appropriate when a check for serial correlation is one of a battery of misspecification tests. When PD1 is used in the wild bootstrap, LM HR and MLM HR have estimates that are slightly greater than required values. Despite what might be expected from the asymptotic theory for a quasi-t test in Liu (1988), the use of PD2 leads to better agreement betweenestimates and nominal values than is observed with PD1 . Indeed all the evidence suggests that using PD2 leads to very good control of finite sample significance levels. (iv) The Hausman-type test HAHR is, like LM HR and MLM HR , undersized at the nominal 1% level when asymptotic critical values are used. However, asymptotic critical values produce estimates, a sample of which is reported in Table 4 below, that are close to the nominal significance level of 5%. It might be conjectured that the improved performance of asymptotic theory reflects the reduction in the number of restrictions being tested, i.e., two rather than four. The use of PD1 again leads to slightly oversized tests and good control is derived when PD2 is employed.
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Table 4 Estimated significance levels of HAHR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (2)
PD1
PD2
2 (2)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.50 1.29 0.56 1.35 0.48 1.60 0.57 1.37 0.30 1.38
1.06 0.99 1.30 1.03 1.20
4.77 5.62 4.34 5.60 3.64
5.69 5.96 6.30 5.95 6.24
4.95 5.02 5.75 4.93 5.80
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.40 1.12 0.84 0.48 1.31 0.96 0.39 1.43 1.22 0.50 1.35 0.95 0.23 1.16 1.09
4.32 5.04 4.29 5.33 3.22
5.31 5.92 6.30 6.00 5.82
4.68 4.89 5.71 5.08 5.53
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.41 1.20 0.94 0.51 1.29 1.06 0.34 1.38 1.11 0.57 1.35 1.01 0.28 1.41 1.16
4.40 5.70 4.32 5.77 3.77
5.39 6.26 6.38 6.30 6.26
4.57 5.31 5.89 5.34 6.01
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (2) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
The full set of Monte Carlo results is available on request, but, in order to save space, only a representative sample is provided below. This sample is obtained from five cases. Details of these cases are given in Table 1. The cases cover all variance models and both types of exogenous regressor sequence. For variance models (16), (18) and (19), there is a choice between two levels of heteroskedasticity. The stronger of the two forms has been used in all cases in order to obtain stringent checks of the efficacy of the wild bootstrap. For the artificial AR(1) regressor of (12), the smaller value of , i.e., = 0.5, is used so that the time-series behaviour of this variable is not similar to that of the standardized log(UK GDP) series, which can be modelled by an I(1) process. All the cases in Table 1 are for (11) with (1 = 0.5, 2 = 0.3). Different choices of (1 , 2 ) from the pairs of values given in Section 4 do not lead to important changes in results; see Godfrey and Tremayne (2002). In Table 2, parts (a) and (b), when the non-robust forms of tests are under consideration, results for both values of T are given to show that distortions due to heteroskedasticity are not reduced (and can get much worse) when the sample size is increased. The departures from nominal significance levels observed when LM F is asymptotically inappropriate indicate the vulnerability of this well-known standard check to ARCH and GARCH forms of
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Table 5 Estimated significance levels of MLM HR with T = 40 Nominal significance level 1%
Nominal significance level 5%
Critical value
2 (4)
PD1
PD2
2 (4)
PD1
PD2
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is N (0, 1) 0.24 1.30 0.14 1.25 0.17 1.26 0.13 1.24 0.13 1.43
0.97 1.00 1.05 0.99 1.20
3.25 2.86 2.71 2.98 2.70
5.68 5.78 5.51 5.90 6.22
4.82 4.98 4.84 5.12 5.65
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized t (5) 0.11 1.12 0.86 0.12 1.14 1.02 0.13 1.08 0.90 0.14 1.16 0.90 0.11 1.18 1.10
2.65 2.64 2.50 2.64 2.24
5.25 5.63 5.55 5.57 5.80
4.48 4.74 4.96 4.81 5.45
Distribution for t Case 1 Case 2 Case 3 Case 4 Case 5
of (14) is standardized 2 (8) 0.18 1.02 0.83 0.15 1.07 0.90 0.13 1.23 0.96 0.12 1.18 0.86 0.14 1.25 1.16
2.74 2.72 2.63 2.80 2.54
5.12 5.78 5.58 5.72 6.02
4.33 4.92 4.89 4.97 5.53
Notes: Cases are as specified in Table 1. All estimates are given as percentages. Columns headed 2 (4) are based on asymptotic critical values. Columns headed PD1 /PD2 are based on the wild bootstrap schemes.
conditional heteroskedasticity and to seasonal unconditional heteroskedasticity. The contents of Table 2, part (a), demonstrate this clearly, while Table 2, part (b), supports our contention that LM B and LM F behave similarly. The information in Tables 3–5 shows very clearly that asymptotic critical values cannot be relied upon when using HR tests. However, the combination of restricted residuals in the HCCME and the very simple PD2 wild bootstrap gives tests with good finite sample behaviour. The results for the t (5) error distribution indicate that the wild bootstrap can work well even when regularity condition (f) of Section 2 is not satisfied. The results for the 2 (8) error distribution show that symmetry of innovations is not essential for the wild bootstrap to yield accurate finite sample rejection rates. Turning to estimates of power derived from (20) to (27), results for the artificial regressor sequence of (12) with =0.5 using PD2 are given in Table 6. The following features emerge from consideration of the contents of Table 6. (v) When LM F is asymptotically valid, i.e., in the cases of panel (a), it is usually slightly more powerful than the HR variants LM HR and MLM HR . However, the cost of the ‘insurance premium’ for robustness to unspecified forms of heteroskedasticity is not
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Table 6 Estimates of power with T = 40 and T = 80. Model is (11) with (1 , 2 , 1 , 2 ) = (0.5, 0.3, 1.0, 1.0). AR(1) regressor with = 0.5 (a) Errors t generated by stationary AR(5) model (20) with perturbations t being NID(0,1) T = 40 T = 80
(j ) of
LM F
LM HR
HAHR
MLM HR
LM F
LM HR
HAHR
MLM HR
(21) (22) (23) (24) (25)
36.7 82.2 14.7 30.3 47.6
31.5 71.9 17.8 28.2 39.5
33.4 9.0 21.7 24.5 27.2
30.2 72.9 16.5 26.9 39.6
80.6 99.7 69.1 91.5 95.8
75.1 99.3 65.5 86.6 92.3
74.9 18.0 72.0 75.5 76.5
74.5 99.3 64.5 86.3 92.1
(b) Errors t generated by ARCH models (26) and (27) with perturbations t being NID(0,1) T = 40 T = 80 ARCH model LM F LM HR HAHR MLM HR LM F LM HR HAHR MLM HR (26) (27)
n/a n/a
30.9 67.3
32.2 9.7
29.5 67.6
n/a n/a
74.3 99.0
74.7 18.9
73.9 99.0
Notes: All critical values estimated using PD2 wild bootstrap, with nominal significance level of 5%. n/a denotes that the test LM F is inappropriate.
large. Estimates for LM F are not given for the ARCH models in panel (b) because the test is asymptotically inappropriate; see Table 2, part (a). (vi) There is no generally valid ranking by power that applies to HAHR , on the one hand, and the asymptotically equivalent LM HR and MLM HR , on the other; see Holly (1982). Differences between Hausman-type and LM-type tests vary in sign and are usually small. However, an important difference is observed in the cases of the simple AR(4) model defined by (22) and the corresponding ARCH process (27). For these two cases, HAHR has manifestly inferior power. The insensitivity of HAHR to serial correlation is worrying even if the implied inconsistencies of the OLS estimators are small. The usual HCCME is inappropriate in the presence of serial correlation, so failure to detect serial correlation may lead to misleading inferences. Overall it may be safer to use either LM HR or MLM HR , the differences between the power estimates of these asymptotically equivalent tests being small. (vii) Finally, comparison of the first and second rows of results in panel (a) of Table 6 with the corresponding results in panel (b) suggests that power estimates are not greatly affected by the presence of random variation in AR coefficients.
6. Conclusions Time-varying heteroskedasticity is recognized as an important phenomenon in applied studies; see, e.g., the discussions in Bollerslev et al. (1994). It is shown that misleading inferences may be made when standard checks for serial correlation are applied to dynamic
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regression models in the presence of either conditional or seasonal heteroskedasticity. To be more precise, simulation evidence presented in this paper illustrates a tendency for the wellknown Lagrange multiplier test to be oversized, sometimes by a substantial amount. This evidence is derived from Monte Carlo experiments designed to mimic quarterly empirical studies. In these experiments, heteroskedasticity-robust versions of the usual Lagrange multiplier test and a Hausman-type test are also examined, along with a modification of the former procedure. The modification is designed to improve performance by using a less variable estimate of the covariance matrix. All tests use restricted residuals to calculate the HCCME proposed in White (1980). It is clear from the Monte Carlo results that the use of asymptotic critical values cannot be recommended. Heteroskedasticity-robust tests conducted on this basis are found to be undersized. However, a wild bootstrap approach proves to be reliable, with good control over finite sample significance levels being provided by the simple twopoint probability distribution discussed by Davidson and Flachaire (2001). This result gives supporting evidence to Flachaire (2004) in which exactly the same combination of restricted residuals in the HCCME and form of wild bootstrap is recommended. The experiments described above, being, for the most part, based upon dynamic regressions with conditional heteroskedasticity, are complementary to those in Flachaire (2004), which are for static regressions with unconditional heteroskedasticity. Results on power of serial correlation tests are also reported. The original and modified forms of the Lagrange multiplier test are very similar in performance. Under several alternatives, the Hausman test rejects a false null with a frequency similar to the other two tests, but evidence is found that it has relatively weak power against simple fourth-order autocorrelation. Consequently it may be safer to use the Lagrange multiplier test in the form in which it is robust to heteroskedasticity of unspecified form, or the modified version of this test, rather than the Hausman test. Acknowledgements We are grateful to Peter N. Smith, two anonymous referees and David Belsley for helpful comments and suggestions. References Belsley, D.A., 1997. A small-sample correction for testing for gth-order serial correlation with artificial regressions. Comput. Econom. 10, 197–229. Belsley, D.A., 2000. A small-sample correction for testing for joint serial correlation with artificial regressions. Comput. Econom. 16, 5–45. Belsley, D.A., 2002. An investigation of an unbiased correction for heteroskedasticity and the effects of misspecifying the skedastic function. J. Econom. Dynam. Control 26, 1379–1396. Bera, A.K., Higgins, M.L., 1993. ARCH models: properties, estimation and testing. J. Econom. Surveys 7, 305–366. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327. Bollerslev, T., Engle, R.F., Nelson, D.B., 1994. ARCH Models. in: Engle,R.F.,McFadden,D.L. (Eds.), Handbook of Econometrics, Vol. IV. North-Holland, Amsterdam. Breusch, T.S., 1978. Testing for autocorrelation in dynamic linear models. Austral. Econom. Papers 17, 334–355.
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