Band Covariance Matrix Estimation Using Restricted Residuals: A Monte Carlo Analysis

Economics Department of the University of Pennsylvania

Institute of Social and Economic Research -- Osaka University

Band Covariance Matrix Estimation Using Restricted Residuals: A Monte Carlo AnalysisAuthor(s): Antonio V. Ligeralde and Bryan W. BrownSource: International Economic Review, Vol. 36, No. 3 (Aug., 1995), pp. 751-767Published by: Wiley for the Economics Department of the University of Pennsylvania and Institute of Socialand Economic Research -- Osaka UniversityStable URL: http://www.jstor.org/stable/2527369 .

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INTERNATIONAL ECONOMIC REVIEW Vol. 36, No. 3, August 1995

BAND COVARIANCE MATRIX ESTIMATION USING RESTRICTED RESIDUALS: A MONTE CARLO ANALYSIS*

BY ANTONIO V. LIGERALDE AND BRYAN W. BROWN1

Using Monte Carlo simulations, we examine the performance of Wald-type

test statistics based on alternative versions of a heteroskedasticity consistent band covariance matrix estimator which is algorithmically constrained to be

positive definite in finite samples. We find that the test statistic based on the

originally proposed estimator tends to result in excessive Type I errors. This

problem can be alleviated to some extent by employing a quasi-maximum likelihood procedure. However, by simply using restricted, as opposed to the usual OLS residuals when constructing the band covariance matrix estimator, excessive Type I errors can be substantially reduced, if not eliminated.

1. INTRODUCTION

The question of conducting inferences when the disturbance term in a linear regression model exhibits both heteroskedasticity and serial correlation of unknown forms can generally be handled by employing one of several heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators discussed in An- drews (1991) and Andrews and Monahan (1992). Because these estimators depend on the choice of weighting schemes and lag truncation parameters and not on any specific structure that the covariance matrix may take, HAC covariance estimators are especially versatile.

In some applications, however, economic theory may imply a specific covariance structure which, by construction, the above HAC covariance estimators do not impose. Sometimes, it may therefore be desirable to employ an estimator which is less versatile but more appropriate from the standpoint of economic theory. This is the case, for instance, when testing for market efficiency.

To illustrate, market efficiency implies that forecast errors must be orthogonal to information available at the time the forecasts are made. More precisely, E(yt+d I0)d

0, where the expectation of the d-period ahead prediction error, Yt+d, is condi- tioned on Ot, the information set at time t. On the basis of this orthogonality condition and an appropriate definition of the forecast error, we can test the statistical adequacy of our model of market efficiency using a simple regression framework. In particular, consider

(1) Yt+d XIt + Ut+d

* Manuscript received October 1992; revised July 1994. 1 Financial support from the Committee on Faculty Research, Miami University, is gratefully

acknowledged by the first author. We thank Adrian Pagan and the referees for helpful comments.

751



752 ANTONIO V. LIGERALDE AND BRYAN W. BROWN

where xt are elements of the information set at time t. By specifying the null hypothesis /3 = 0, we can test whether or not the orthogonality condition implied by market efficiency can be rejected.

When testing market efficiency using the above procedure, we can often increase the size of our sample by considering a sampling interval that is shorter than the forecast interval. However, this leads to serially correlated forecast errors. The overlap in the forecast errors occurs because the orthogonality condition only guarantees that E(ut+d Iut, ..1 ) = 0. Thus, we must entertain the possibility that E(ut+dut+d,) # 0 for 0 ? i < d. Consequently, the covariance matrix will not be diagonal but instead have a band structure similar to what we might obtain if the disturbances followed a moving average process of order (d - 1). Since generalized least squares can lead to inconsistent parameter estimates (Hansen and Hodrick 1980, Brown and Maital 1981), a tractable alternative is to use ordinary least squares to obtain consistent parameter estimates but employ a covariance estimator that accounts for the band structure of the covariance matrix for inferences.

A heteroskedasticity consistent estimator that imposes a band structure on the covariance matrix has been proposed by Eichenbaum, Hansen, and Singleton (1988). Unlike the usual band covariance matrix estimators that have been previously employed, notably Hansen's generalized method of moments (GMM) covariance estimator (see Hodrick 1987, pp. 41-46), the Eichenbaum, Hansen, and Singleton (EHS) estimator offers the distinct advantage of being algorithmically constrained to be positive definite in finite samples. However, while it has been demonstrated elsewhere that in finite samples, test statistics based on Hansen's GMM covariance estimator results in excessive Type I errors (Ligeralde 1989, Mishkin 1990), the accuracy of inferences based on the EHS estimator has not yet been assessed.

In this paper, we therefore examine the performance of Wald-type test statistics based on the EHS estimator through Monte Carlo simulations. It turns out that the EHS estimator, based on Durbin's (1960) two-step procedure, yield test statistics that are equally disappointing in finite samples. Thus, we also consider three other ways of constructing a feasible EHS estimator. The first employs a quasi-maximum likelihood estimation procedure, the second utilizes restricted, as opposed to unrestricted OLS residuals and the third combines these two procedures. We find that significant reductions in excessive Type I errors can be achieved primarily through the use of restricted residuals in forming the EHS estimator.

The remainder of the paper is divided into 3 parts. Part 2 describes alternative ways of constructing the EHS estimator. Part 3 presents Monte Carlo results and part 4 concludes.

2. THE EHS COVARIANCE MATRIX ESTIMATOR

In this section, we describe how inferences regarding /3 in eq. (1) may be carried out using the EHS band covariance estimator that is not only robust to general and unspecified forms of heteroskedasticity but is also algorithmically constrained to be positive definite in finite samples. Stacking all n observations, we can rewrite



BAND COVARIANCE ESTIMATION 753

eq. (1) as

y =Xf3+u

where y is [n X 1], X is [n X k], /3 is [k x 1] and u is [n X 1]. Under appropriate assumptions (as in White 1984, p. 125), the least squares estimator /3 will have the following asymptotic distribution:

nl/2( (-) d N(OM-1VM-1)

where

fl

M = plim n-1 E xi t=1

(n d-I Iz Vplim fl-1{[ZZ*] + K4 Z z/+

Zt -Xtut+d -

Given a known [j x k] matrix R and a known j-vector q, an appropriate statistic for testing a set of linear restrictions in the null hypothesis Ho: R /3 = q is then given by

W--n(R -q)'(RM-1VM-1R') 1(Rt3-q)

which asymptotically will follow Xi2 under the null hypothesis and be large under the alternative hypothesis.

Assuming zt is covariance stationary, the estimator proposed by Eichenbaum, Hansen and Singleton (1988) is based on the Wold decomposition of the process zt. Imposing the orthogonality condition implied by market efficiency, we can write zt in the form of a (d - 1) vector moving average process.

zt =At + BlAt-1 + +Bdl At-d+l

where Bi is a [kxk] coefficient matrix. The covariance matrix V can then be expressed as

d-1

V=E(ztz) + E E(ztztT + zt-Tz z) T= I

or

wh(Ie+eB+ A E +Bd(A)A(I+)B + --- +B.-1),

where A = E( At At).




If ut+d were observable, an estimator for V can be calculated using a two-step procedure due to Durbin (1960). First, we estimate the [k X k] matrices A1 .I. AL in the regression

Zt =Alzt-1 + +A-- L +A+t

Since we are essentially approximating an infinite order vector autoregressive representation of zt, a sufficiently large lag length L must be chosen.

Using the residuals At from the previous regression, we then estimate the square matrices Bi in the regression

A A

Zt = B1Atl + *** +Bd- IAt-d+l + Ut.

Using the residuals Dt, we then form itA==(n-L-d+1)l d+LDvJ and obtain the estimator

A A A A A A

V (+ B,+ *-- +Bd- )A(I+ B'+ --- +Bd -l)-

Note that generally, in finite samples, V will be nonsingular with probability one as long as A = E(At A') is nonsingular and n > k.

Since ut+d is unobservable, we can instead apply Durbin's two-step procedure to Zt -XtUt+d where ut+d =Yt+d -t x/. We shall refer to the resulting feasible estimator, VE, as the unrestricted EHS estimator. Tests of linear restrictions on ,3 can then be carried out using

W-(R ,B -q )[ [R(X' X/n) V(XX/) WR'] (R13-

Alternatively, we can directly estimate the coefficients of the vector moving average process Zt using quasi-maximum likelihood estimation (QMLE) (see Judge et al. 1985, p. 679). Note that although this nonnormality robust procedure entails nonlinear optimization, it is likely to yield a better approximation than Durbin's two-step procedure insofar as the lag length of the autoregressive representation of Zt iS not arbitrarily truncated at L. We shall refer to the resulting estimatorVM, as the unrestricted QMLE. The corresponding Wald-type statistic is then given by

WM-n(R 3-q)[R(X'X/n) VM(XX/n) R](R q).

A third way of constructing the EHS estimator is obtained by taking note of the fact that if we are interested in the probability distribution of W when the null




hypothesis (R,3 = q) is true, it would be equally valid to base our feasible test statistic on an estimate of V that assumes our specified null hypothesis is true. Judge, Griffiths, Hill, Lutkepohl, and Lee (1985, pp. 472-475) point out that in the context of generalized least squares estimation, this approach would be equivalent to using the Lagrange multiplier test as opposed to the Wald test in conducting inferences. They further document research suggesting that a test statistic based on restricted residuals performs better in finite samples than a test statistic based on unrestricted residuals. As we mentioned in the introduction, however, generalized least squares can lead to inconsistent parameter estimates and might therefore be inappropriate for the model we have in mind. We could, however, form Wald-type statistics using restricted residuals and expect better finite sample results.

Consequently, we also consider carrying out Durbin's two-step procedure on z2 -XtUt+d, where U+d -Yt+d -X, h, and /3* is the restricted least squares estimator given by /3 8 = /3- (X' X)1 R'[R(X' X)- 1 R']1(R / - q). We shall refer to the resulting feasible estimator, VET, as the restricted EHS estimator. We could then calculate

A

(R A

q) [R(X X/n) AV (XXn-R'] -I(R A

q)

to carry out tests of linear restrictions on /3.

Finally, we take note that we can also carry out the QMLE procedure on Zt which is generated from restricted residuals, to obtain an analogous feasible estimator VM and Wald-type statistic, WM. We shall refer to VM as the restricted QMLE.

3. MONTE CARLO EXPERIMENTS

3.1. Experimental Set-Up. To evaluate the finite sample performance of the feasible EHS estimators, we considered the following basic regression specifications:

Yt+d =0.200 + 0.237xt+ Ut+d for d = 1, 2,3

Yt+d = 0.200 + 0.362yt + 0.237xt + Ut+d for d = 1, 2,3.

The intercept term and slope coefficients are based on empirical results from Hansen and Hodrick's (1980) tests of the simple foreign exchange market efficiency hypothesis using the regression framework discussed in Section 1. For x,, we used monthly observations on the French franc-US dollar exchange rate forecast error formed by taking the difference between the natural log of the two-month ahead forward rate at time t - 2 and the natural log of the corresponding spot rate at time t. We obtained the data from the International Monetary Market Yearbook from




February 7, 1983 to March 6, 1987, yielding 50 observations. If we view the regressors as information available at the time the forward rates are set, then Yt+d can be viewed as the d-period ahead forecast error between the US dollar and a foreign currency. Following MacKinnon and White (1985), we replicated xt the appropriate number of times in order to meet the sample sizes of 50, 100 and 200 considered in our experiments. We utilized this technique to introduce a dose of realism into our experiments and to obtain a set of results based on the first specification where we could control for (X'X/n)- 1 regardless of sample size. In the second specification, we included a lagged endogenous regressor since many tests of market efficiency typically include lagged forecast errors as an obvious element of the information set.

For each of the above regressions, we specified Ut+d to be a moving average of order d - 1. In particular,

Ut+d = Et+d for d = 1,

Ut+d = 0.873(Et+d + 0.560Et+d-I) for d = 2,

Ut+d = 0.857(at+d + 0.560 Et+d-1 + 0.220Et+d-2) for d = 3.

The moving average coefficients, primarily based on values obtained by Brown and Maital (1981), were adjusted so that the variance of ut+d would be equal in each case.

Heteroskedasticity was introduced into the experiments as follows. Letting vt+d

N(0, 1), we considered the following cases:

Cl: Et+d =t+d

C2: it+d = Vt+d [exp (-0.701 + 20x,)] 1/2

C3: Et+d = Vt+d(0.634 + 12.680x,).

C1 represents the base case where the disturbances are homoskedastic, whereas C2 and C3 represent cases where the disturbances are heteroskedastic. In particular, C2 portrays a multiplicative heteroskedasticity model whereas C3 allows the standard deviation of Et+d to be a function of xt. As before, the parameters have been set such that the variance of ut+d would, on average, be equal in each case.

In view of the empirical findings that financial data appear to be well described by autoregressive conditional heteroskedasticity models, we included the following constant correlation ARCH(1) and GARCH(1, 1) processes (Baillie and Bollerslev 1989, Bollerslev, Chou, and Kroner 1992) in our experiments.




C4: Ut+d = ut+d(0.654 + 0.346ut)1 2 ford=1,

Ut+d = (ut+d +0.560Vt+d1I )(0.415 + 0.346u )1/2 for d = 2,

Ut+d (vt+d +0.56OVt+d I + 0.220vt+d-2 )(0.388 + 0.346u2)1/2 for d = 3.

C5: ut+d = 1.146vt+d(ht+1)1/2 for d = 1,

Ut+d = (Vt+d+ 0.56OVt+d1 )(ht+l ) / for d = 2,

Ut+d =0.982(vt + 0.560vt+d- I+ 0.220 t+dO

2 )(ht+ )1/2 for d = 3,

ht+1 = 0.295 + 0.249u 2 + 0.285hz.

The ARCH and GARCH parameters are based on results obtained by Baillie and Bollerslev (1989) adjusted to let the variance of ut+d be equal in each case. Note that although the covariance matrix estimators being investigated are not specifically tailored to handle autoregressive conditional heteroskedasticity, C4 and C5 are nonetheless relevant insofar as OLS estimation and inference in the presence of lagged dependent variables as regressors necessitates the use of a heteroskedasticity consistent covariance matrix estimator. This is so because the cross products of the regressors would be correlated to the squares of the disturbances (Engle 1982).

Finally, we considered the following constant correlation integrated GARCH(1, 1) process that appears to characterize financial data sampled at very short intervals (Engle and Bollerslev 1986).

C6: Ut+d = 1.146vt+d(ht+1)1/2 for d = 1,

Ut+d = (vt+d+ 0.56Ot+d1 )(ht+l) for d = 2,

Ut+d =0.982(vt+d+ 0.560vt+d-1 + 0.220vt+d-2 )(ht+1) for d = 3,

h t+1 = 0.295 + 0.249u72 + 0.751h

C6 is interesting insofar as the 2nd and 4th moments of an IGARCH(1, 1) process do not exist (Bollerslev 1986), suggesting that sample statistics based on the covariance matrix estimators under study could lead to erroneous inferences. Consequently, C6 should give us an indication of the extent to which this is true.




TABLE 1

BASIC EXPERIMENT

MA(O): Y. + d = 0.200 + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 8.6* 6.8 4.4 9.0* 5.8 5.8 7.5* 6.1 5.7 restricted EHS 4.6 4.3 3.6 4.7 4.0 4.6 4.5 4.8 4.8 true cov. 5.5 4.6 3.3 5.0 4.8 5.0 6.3 5.6 4.9

False null unrestricted EHS 30.3* 44.5* 74.4* 44.9* 74.3* 96.5* 63.1* 90.1* 99.3* restricted EHS 21.9* 40.3* 72.2* 36.7* 69.7* 95.9* 54.7* 88.4* 99.1* true cov. 24.4* 41.7* 73.0* 41.2* 72.9" 96.2* 59.7* 90.5* 99.2*

C4 CS C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 9.4* 5.9 5.8 8.3* 6.3 5.3 8.5* 5.4 6.4 restricted EHS 6.2 4.1 5.1 4.6 4.3 4.8 4.7 3.9 4.8

False null unrestricted EHS 30.7* 50.1* 73.1* 28.5* 48.9* 74.9* 8.4* 6.2 5.8 restricted EHS 21.8* 45.7* 71.4* 20.9* 43.7* 73.2* 5.1 4.1 4.9

C1: Et+d = Ut+d; Ut+d is Niid(0, 1) C2: Et +d = vt+d [exp( -0.701 + 20x, )]1/2

C3: Et+d = U.t+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH(1, 1) C6: IGARCH (1, 1) True null: Ho: f3' = [0.200,0.237] False null: Ho 3 = [0, 0] '-Significantly different from 5 percent at the 1 percent significance level.

The simulations were carried out on an IBM PS/ValuePoint 433DX/D personal computer using the GAUSS programming language (Aptech Systems 1992).

3.2. Note on Tables. The results of our experiments are reported in Tables 1 to 9. All entries represent percentage of rejections out of 1000 replications using a 5 percent significace level. In general, we would like to see rejection rates close to 5 percent when a true null hypothesis, such as /3 = [0.200,0.362,0.237] is specified. On the other hand, we would like a false null hypothesis, such as /3 = [0,0,0], to be rejected close to 100 percent. Asterisks indicate cases where the fraction of rejections differs from 0.05 by more than + 2.576[(0.05 0.95)/1000]1/2. This essentially tells us whether or not the actual rejection rate is significantly different from a 5 percent rejection rate at the 1 percent significance level.

For cases C1 to C3, we also tested the null hypothesis using a Wald-type test statistic based on the true covariance matrix V. We could not do the same for the autoregressive conditional heteroskedasticity cases C4 to C6 as calculating V proved intractable.




TABLE 2 BASIC EXPERIMENT

MA(1): Yt + d = 0.200 + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 12.6* 9.0* 6.8 14.3* 8.4* 6.2 10.7* 7.2* 7.0* restricted EHS 4.5 4.6 4.5 3.5 4.1. 4.2 2.3* 3.3 5.1 true cot). 5.1 5.2 5.4 3.9 5.1 3.7 3.7 4.1 4.5

False null unrestricted EHS 23.4* 31.2* 48.9 32.1* 43.9* 71.3* 40.4* 59.9* 87.5* restricted EHS 9.6* 21.1* 43.6* 15.9* 35.3* 67.4* 21.4* 50.9* 84.7* true coy. 13.3* 25.4* 47.9* 16.6* 31.7* 61.1* 15.6* 35.6* 72.8*

C4 CS C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 13.5* 7.9* 7.3* 13.4* 7.4* 7.3* 12.4* 8.0* 5.6 restricted EHS 6.5 4.9 5.6 5.1 2.7* 5.0 4.5 3.9 4.0

False null unrestricted EHS 27.6B 30.4* 50.7* 26.5 * 30.0* 49.6B 12.8* 8.1 ' 6.1 restricted EHS 13.9* 21.0* 45.7* 12.5* 22.5* 43.3* 5.0 4.3 4.5

Cl: Et+d = Vt+d; ,t+d is Niid(O, 1) C2: Et+d = )t+d[exp(- 0.701 + 20x,)]1/2 C3: et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) CS: GARCH (1,1) C6: IGARCH(1, 1) True null: Ho: /3' = [0.200,0.237] False null: Ho: 3' = [0,0 ] *Significantly different from 5 percent at the 1 percent significance level.

Finally, we take note that for the set of experiments where d= 1 (i.e. no serial correlation), VE and VE* were calculated using

n1 EZZt and n1 Zt Z* t=1 t=1

since none of the moving average coefficients Bi had to be estimated using the two-step Durbin procedure.

3.3. Summary of Results. From Tables 1 to 6 we see that the Wald-type test statistic W, which is based on the unrestricted EHS estimator, generally resulted in excessive Type I errors. This problem became more severe with the inclusion of an additional regressor and with the increase in the order of the moving average process.

Given the results using the true covariance matrix V, it appears that the dismal performance of W was due to the fact that VE itself performed poorly in finite





MA(2): Yt + d = 0.200 + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 19.2* 10.2* 8.0* 17.0* 11.5* 9.4* 15.5* 7.7* 6.5 restricted EHS 4.8 3.2 4.2 4.7 3.7 5.0 4.0 2.6* 3.8 true cov. 4.6 4.4 4.9 4.2 4.5 6.1 3.2 2.7" 3.4

False null unrestricted EHS 27.3* 27.9* 42.3* 33.3* 42.5* 63.1* 36.9* 25.9* 79.3* restricted EHS 7.6* 14.2* 34.3* 10.7- 27.0* 56.1* 14.4* 39.2* 71.6* true cov. 12.0* 19.2* 38.1* 13.0* 26.3* 47.7* 11.2* 24.2* 55.2

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 16.8* 11.3* 6.9* 20.2* 11.2* 8.2* 18.1* 9.8* 6.3 restricted EHS 4.6 4.3 3.4 7.5* 4.1 4.3 6.1 3.8 3.8

False null unrestricted EHS 29.4* 31.9* 47.4* 30.5* 31.7* 42.0* 17.8* 10.1* 6.9* restricted EHS 9.5* 18.8* 38.2* 11.7* 17.8* 34.0* 6.0 4.0 3.7

Cl: Et+d = Vt+d; Vt+d is Niid(0, 1) C2: Et+d = ,t+d[exp(-0.701 + 20x,)]1/2 C3: et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1,1) C6: IGARCH (1, 1) True null: Ho: / = [0.200,0.237] False null: Ho: / = [0, ] *Significantly different from 5 percent at the 1 percent significance level.

samples and not simply because it was used as a component of a test statistic. In particular, the superior results from the MA(O) specifications where we did not have to estimate any moving average parameter suggest that the approximation provided by the two-step Durbin procedure may not be adequate in small samples.

To find out if a more efficient method of estimating the vector moving average coefficients of zi leads to an improvement in the finite sample performance of the unrestricted estimator VE, we ran experiments where the moving average coefficients Bi were estimated using unrestricted QMLE. We also ran experiments using restricted QMLE. The negative log likelihood function was minimized using the BFGS algorithm provided in the GAUSS OPTMUM package with starting values for Bi set to zero.

The results, given in Table 7, suggest that indeed, excessive Type I errors can be reduced by using quasi-maximum likelihood estimates for the moving average coefficients Bi. However, it is clear from Tables 1 to 7 that dramatic improvement in the actual size of the Wald-type test statistic can be achieved mainly by using restricted residuals. This result held true for all sample sizes and specifications considered.





MA(O): Yt + d = 0.200 + 0.362Yt + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 11.2* 10.2* 5.8 12.2* 8.5* 7.5* 9.7* 8.4* 5.7 restricted EHS 3.9 4.9 3.8 2.9* 3.7 3.9 2.9* 4.3 4.1 true cov. 5.5 4.4 4.6 4.1 4.2 5.9 4.6 5.5 4.9

False null unrestricted EHS 81.9* 98.5* 100* 88.2* 99.6* 100* 93.9* 99.5* 100* restricted EHS 63.8* 96.2* 100* 71.8* 98.3* 100* 82.9* 98.6* 100* true cov. 77.9* 97.6* 100* 84.7* 99.1* 100* 91.6* 99.6* 100*

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 12.0* 9.8* 7.6* 15.1 * 8.2* 7.6* 14.7* 10.4* 7.8* restricted EHS 3.8 4.4 5.6 3.3 3.9 5.1 4.7 4.8 4.1

False null unrestricted EHS 78.5* 93.3* 99.8* 77.3* 94.8* 99.7* 55.9* 68.8* 75.2* restricted EHS 60.2* 90.1* 99.6* 61.7* 91.4* 99.9* 30.0* 53.5* 61.8*

Cl: Et+d = vt+d; Vt+d is Niid(0, 1) C2: Et+d = -u+d[exp(-0.701 + 20x,)]1/2 C3: Et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1, 1) C6: IGARCH(1, 1) True null: Ho: i' = [0.200,0.362,0.237] False null: Ho: ' = [0, 0, 0] *Significantly different from 5 percent at the 1 percent significance level.

How might we motivate the substantial reduction in Type I errors from using restricted residuals? The following analytical result for the simplest case where ut is serially uncorrelated, homoskedastic and independent of x, is suggestive. In this

case, the feasible estimator VE =n E i2 t can be written as t=1

n fl

VE = n EUt+dXtXt- 2n E U+ dXt /3 _

) T t=1 t=1

n

+ n - 1 E 3 ( 3 ,B- * ) 3 x, x' B, *)x ' t=1

In the special case where the restriction matrix R is equal to the identity matrix, we have Ut+d = Ut+d and f3* =13 so that VE can also be expressed as

A A

VE = VE* -2F1 + F2





MA(1): Yt + d = 0.200 + 0.362Yt + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 25.7* 14.4* 8.7* 25.1* 12.5* 8.8* 23.0* 11.6* 8.8* restricted EHS 8.9* 6.1 4.9 5.7 4.0 4.3 6.2 2.8* 3.9 true cov. 6.6 6.6 5.0 6.5 4.6 4.5 4.7 3.3 4.5

False null unrestricted EHS 71.5* 89.2* 99.9* 76.6* 93.7* 99.9* 83.1* 94.0* 99.8* restricted EHS 34.1* 68.9* 98.5* 37.8* 73.8* 98.8* 45.0* 77.9* 99.1* true coy. 54.2* 84.9* 99.4* 55.9* 87.4* 99.9* 60.8* 88.7* 99.3*

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 26.8* 16.6* 10.7* 30.3* 15.6* 9.5* 28.3* 14.6* 11.0* restricted EHS 10.5* 6.8 4.8 11.3* 6.9* 4.9 12.4* 5.3 5.4


Cl: et+d = Vt+d; Vt+d is Niid(0, 1) C2: Et+d = ,t+d[exp ( -0.701 + 20x,)]1/2 C3: 'et+d = V,+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1, 1) C6: IGARCH (1, 1) True null: Ho :' = [0.200,0.362,0.237] False null: Ho: [0, 0,0 ] *Significantly different from 5 percent at the 1 percent significance level.

where

F1 = n - E ut+df x,( fB-B3 )xtx', and F2 =n1 E( 3-f3 )'x xx( 3- )xtx'. t=1 t=1

Since E(F1 Ix) = E(F2lx,) we get

E(VEIxt) = E(VE*Ixt)-a positive semidefinite matrix.

This means that in finite samples, unrestricted residuals would yield a test statistic W that would be greater than or equal to the statistic W* based on restricted residuals. The result for C1, n = 50 in Table 1 is therefore not surprising. While it is not clear how this analytical result extends to cases where ut is nonspherical, we suspect that similar considerations apply. Our conjecture is that although W and W* have the same asymptotic distribution, in finite samples VE* exceeds VE by a positive semidefinte matrix so that W ? W*. Thus, the probability of a Type I error associated with W will be bigger than that associated with W*. We leave this conjecture open for future investigation.





MA(2): Yt + d = 0.200 + 0.362Yt + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 37.8* 21.1* 11.4* 36.6* 17.3* 10.5* 37.1* 17.9* 11.4* restricted EHS 12.4* 5.5 4.9 9.7* 3.7 4.1 9.7* 2.5" 3.9 true cov. 7.0* 6.2 5.4 4.9 5.7 4.3 4.6 5.0 4.0

False null unrestricted EHS 75.2* 88.0* 99.0* 79.1* 91.0* 98.9* 83.5* 92.5 99.6* restricted EHS 40.1* 58.8* 96.3* 45.6* 64.7* 95.7* 46.6* 68.4* 97.7* true cov. 47.6* 77.6* 98.4* 51.5* 81.1* 97.7* 51.0* 81.7* 98.4*

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 41.0* 17.7* 14.2* 20.9* 20.9* 11.5* 19.4* 19.6* 17.4* restricted EHS 14.7* 5.6 6.2 9.9* 6.4 4.8 10.1 B 6.4 6.8


Cl: Et+d = Vt+d; Vt+d iS Niid(0, 1) C2: Et+d = u,+d[exp( -0.701 + 20xt)]1/ C3: 'et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1, 1) C6: IGARCH(1, 1) True null: Ho: /' = [0.200,0.362,0.237] False null: Ho: [0, 0,0 ] *Significantly different from 5 percent at the 1 percent significance level.

Going back to the results of the Monte Carlo experiments we further take note that the power to reject an incorrect null hypothesis exhibited by the test statistic based on the restricted EHS estimator is comparable to that of the test statistic based on the true covariance matrix (see cases C1 to C3). Not surprisingly, the only specification where the test statistic based on the restricted EHS estimator showed little or no power to reject an incorrect null hypothesis even at n = 200 was the IGARCH(1, 1) case for which the unconditional 2nd and 4th moments of the disturbance term do not exist. For this specification, however, even the test statistic based on the unrestricted EHS estimator worked just as poorly. This is most readily seen from Tables 1 to 3.

Finally, insofar as we may not always be interested in testing hypotheses which involve all regression coefficients, we also ran experiments where we tested restrictions on only one and two out of three regression coefficients. The results are given in Tables 8 and 9. The Wald-type test statistic based on the restricted EHS estimator continued to yield actual rejection rates that were closer to the nominal size of the test than those from the test statistic based on the unrestricted EHS estimator. Moreover, as can be seen from Tables 6, 8 and 9, the improvement in the




TABLE 7 QUASI-MAXIMUM LIKELIHOOD EXPERIMENT

MA(1): Yt + d = 0.200 + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 13.2* 7.3* 6.1 12.0* 7.2* 7.1* 10.4* 8.4* 6.9* restricted EHS 4.6 3.1* 4.7 3.5 3.9 4.5 2.3* 5.1 3.7 unrestricted QMLE 10.5* 5.9 5.3 9.4* 6.5 5.7 9.7* 7.7* 6.6 restricted QMLE 3.0* 3.3 3.8 2.5* 3.3 3.8 2.8* 4.8 3.4

False null unrestricted EHS 23.2* 31.2* 48.8* 32.0* 43.5* 71.1* 42.8* 62.1* 64.4* restricted EHS 10.3* 22.3* 42.9* 16.9* 33.6* 65.8* 24.4* 49.5* 51.2* unrestricted QMLE 19.4* 27.4* 43.1* 26.6* 37.4* 65.7* 38.4* 55.9* 57.9* restricted QMLE 7.5* 21.0* 39.4* 13.8* 30.0* 61.1* 23.9* 46.0* 46.5"

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 13.9* 10.0* 6.4 13.0* 7.8* 5.7 13.9* 7.7* 5.1 restricted EHS 4.6 5.2 4.7 6.4 4.4 4.4 5.7 4.3 3.5 unrestricted QMLE 10.1* 8.2* 6.1 11.2* 6.5 5.2 11.3* 6.9* 5.3 restricted QMLE 3.0* 4.9 4.6 4.4 3.9 3.9 4.4 4.0 3.5

False null unrestricted EHS 27.5* 31.9* 51.4* 25.8* 32.3* 46.2* 13.2* 8.2* 5.8 restricted EHS 12.9* 22.3* 45.3* 11.2* 23.6* 40.9* 5.5 4.3 3.8 unrestricted QMLE 22.8* 27.6* 47.6* 22.0* 28.6* 41.7* 11.6" 6.9* 5.5 restricted QMLE 11.3* 20.3* 44.0* 9.1* 21.9* 37.8* 4.2 3.7 3.9

Cl: Et+d = Vt+d; Vt+d is Niid(0, 1) C2: Et+d = ut+d[exp(-0.701 + 20x,)]1/2 C3: et+d = Vt+d[0.634 + 12.68x,] C4: ARCH (1) C5: GARCH (1, 1) C6: IGARCH (1, 1) True null: Ho: /' = [0.200,0.237] False null: Ho: /' = [0, 0] * Significantly different from 5 percent at the 1 percent significance level. Note. This table is comparable to Table 2.

actual size of the test statistic became more pronounced as we increased the number of restrictions being jointly tested.

4. CONCLUSION

When dealing with models with an intertemporal dimension, one has to allow for the possibility that the regression disturbances may be serially correlated such that the covariance matrix has a band, instead of the usual diagonal structure. This covariance structure appears prominently, for instance, in econometric tests of market efficiency (Hodrick 1987).

In this paper, we examined the question of choosing an appropriate covariance estimator when the regression disturbances are both serially correlated and possibly heteroskedastic. Special attention was given to the band covariance matrix estimator




TABLE 8 TEST OF RESTRICTION ON THE COEFFICIENT OF Yt MA(0): Yt + d = 0.200 + 0.362Yt + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 20.5* 14.1* 8.6* 19.0* 12.7 8.9 19.6 14.5* 9.5* restricted EHS 9.7* 8.8* 6.0 6.6 4.6 5.5 5.1 5.8 5.6 true cov. 6.1 6.5 5.7 6.2 4.5 6.1 5.0 6.6 6.0

False null unrestricted EHS 49.6* 78.2* 97.2* 47.5 74.2* 95.0* 45.0* 69.6* 93.7* restricted EHS 27.8* 64.8* 96.3* 20.7* 55.8* 92.3* 19.5* 51.3* 89.7* trie cov. 31.3* 68.0* 96.6" 27.6" 62.5* 93.2" 24.0* 59.2* 91.9*

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 23.0* 14.8* 11.0* 20.2* 13.5* 10.4* 21.5* 14.8* 11.9* restricted EHS 10.4* 8.5* 6.6 10.0* 7.3* 6.9* 7.6* 6.4 6.0


Cl: Et+d = Vt+d; Vt+d is Niid(0, 1) C2: Et+d = t+d[exp( -0.701 + 20X,)]1/ C3: Et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1, 1) C6: IGARCH (1, 1) True null: Ho: 1 = [0.362] False null: Ho: 1 = [0] *Significantly different from 5 percent at the 1 percent significance level.

proposed by Eichenbaum, Hansen, and Singleton (1988) that is not only robust to heteroskedasticity but is also algorithmically constrained to be positive definite in finite samples. We evaluated the finite sample performance of Wald-type test statistics based on four alternative versions of the EHS estimator through Monte Carlo simulations. In particular, we considered a feasible estimator based on Durbin's two-step procedure using unrestricted residuals (unrestricted EHS); a quasi-maximum likelihood estimation procedure using unrestricted residuals (unrestricted QMLE); Durbin's two-step procedure using restricted residuals (restricted EHS); and a quasi-maximum likelihood estimation procedure using restricted residuals (restricted QMLE).

Our experiments indicate that in finite samples, a Wald-type test statistic based on the unrestricted EHS estimator tends to result in excessive Type I errors. This problem can be alleviated to some extent by employing a quasi-maximum likelihood procedure which avoids the lag length approximation of Durbin's two-step procedure. The most interesting finding of this paper, however, is that using restricted residuals to form the EHS estimator can dramatically improve the finite sample performance of the resulting test statistic. For all specifications considered in the study, test statistics based on feasible EHS estimators using restricted residuals




TABLE 9

TEST OF RESTRICTIONS ON THE COEFFICIENTS OF Yt AND X,

MA(2): Yt + d = 0.200 + 0.362yt + 0.237x, + Ut+d

C1 C2 C3

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 29.0* 16.4* 8.8* 30.0* 15.9* 11.3* 29.5* 15.4* 10.3* restricted EHS 11.3* 6.8 4.9 10.4* 4.9 5.8 8.5* 5.0 5.3 true cov. 5.6 5.1 5.1 7.1* 6.4 6.8 6.6 6.3 5.6

False null unrestricted EHS 56.1* 72.5* 95.7* 59.4* 72.2* 91.1* 57.0* 70.9* 93.3* restricted EHS 25.6* 47.0* 91.9* 22.6* 40.4* 80.7* 19.9* 35.8* 78.5* true cov. 26.6* 62.3* 94.1* 29.8* 56.9* 87.5* 27.4* 53.9* 87.3*

C4 C5 C6

Sample size 50 100 200 50 100 200 50 100 200

True null unrestricted EHS 31.8* 18.0* 10.9* 30.5* 17.2* 10.4* 31.6* 19.2* 13.3* restricted EHS 11.5* 6.3 5.7 11.4* 6.6 5.9 11.1* 7.5 6.3

False null unrestricted EHS 50.0* 56.6* 80.4* 47.9* 63.9* 84.1* 47.2* 54.2* 69.6" restricted EHS 23.0* 35.2* 66.3* 22.8* 39.4* 74.4* 19.8* 29.2* 49.3

Cl: Et+d = Vt+d; Vt+d is Niid(0, 1) C2: Et+d = vt+d[exp(-0.701 + 20x,)]1/2 C3: Et+d = Vt+d[0.634 + 12.68x,] C4: ARCH(1) C5: GARCH (1,1) C6: IGARCH(1,1) True null: Ho: [I 1, 0B2] = [0.362,0.237] False null: Ho: [I 3,3 2]=[0, ] *Significantly different from 5 percent at the 1 percent significance level.

resulted in actual rejection rates that came closer to the nominal size of the test than the test statistic based on the unrestricted EHS estimator. When the restrictions on each of the regression coefficients and the intercept term were tested jointly, as is usually the case when testing market efficiency, the improvement in the finite sample performance of the Wald-type test statistic based on restricted residuals was especially noteworthy.

Miami University, U.S.A Rice University, U.S.A

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