Accurately sized test statistics with misspecified conditional homoskedasticity

Abstract

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.

Keywords:

• robust testing
• test size
• confidence interval estimation
• heteroskedasticity
• autocorrelation

2000 Mathematics Subject Classification :

• 62F03
• 62F35
• 62J05
• 62M10
• 91B84

Acknowledgements

We thank the members of the Econometrics Research Group at the University of California, Santa Barbara for helpful comments.

Notes

Other variance estimators have also been proposed such as the vector autoregression heteroskedasticity and autocorrelation consistent (VARHAC) estimator 6 or the bootstrap estimator 7 8 9 10. The simulation results of Den Haan and Levin 6 11 show that the VARHAC estimator performs comparably to the Andrews and Monahan prewhitened estimator, and Kiefer and Vogelsang 2 show that the block bootstrap performs comparably to tests employing the fixed-bandwidth asymptotic critical values.

Rothenberg assumes that the regressors are strictly exogenous, while Newey–West, Andrews–Monahan and Kiefer–Vogelsang assume that the regressors are only weakly exogenous. The simulations in Erb and Steigerwald 13 study the relative performance of the Rothenberg correction under both strict and weak exogeneity.

We set and η _k0 equal to zero in the simulations and discard the first 50 observations to remove any influence from initial values.

Data were also simulated according to a moving-average specification, under which the serial correlation is of limited duration and conditional heteroskedasticity plays a correspondingly larger role. The results were similar to those found in the autoregressive specifications and can be found in Erb and Steigerwald 13.

The values of ζ have been chosen to ensure that the unconditional variance of U _t is the same in all specifications. Andrews and Monahan also study this specification, albeit without reference to the findings in Rothenberg.

To see that this model brings conditional heteroskedasticity, consider the case in which . Then

and the covariance conditional on X is

Not considered here are the classic OLS variance estimator under the assumption of i.i.d. errors and the parametric, AR(1) variance estimator. Simulation results for these estimators can be found in Erb and Steigerwald 13.

We use n−5, rather than n, as the divisor because the degrees-of-freedom calculation is likely to be used when the sample size is small, as is the case in our simulations where n=50.

To ensure the matrix I−∑ _s A _s is not too close to singularity, we restrict the eigenvalues of to be no larger than 0.97 in absolute value. See Andrews and Monahan \cite[p. 957]{5} for the details.

This estimator, which includes prewhitening, retains many of the asymptotic properties of including the rate of convergence.

Our finite-sample results are designed to guide researchers with moderate sample sizes, in which size inflation is known to be a problem. Although is not a consistent estimator of J under conditional heteroskedasticity, a consistent estimator is easily obtained by switching from to for large sample sizes (say n>1000).

A similar limit theory is developed by Phillips et al. 14.

Our goal is to evaluate the accuracy of the Kiefer–Vogelsang limit theory under both ‘large’ and ‘small’ specifications of the bandwidth parameter, and we use the Andrews method to select the ‘small’ bandwidth. Technically, the limit theory will not hold if the Andrews method is employed, as the Andrews procedure gives a bandwidth value that is o(n) and Kiefer and Vogelsang assume that the bandwidth parameter is O(n). However, we find in our simulations that the Kiefer–Vogelsang testing procedure performs substantially better (in terms of test size) when the Andrews procedure is implemented for each replication, as opposed to using some fixed value for the bandwidth, say m/n=0.2, for all replications. For this reason, we report the simulation results using the Andrews method.

Unfortunately, it is not clear whether the asymptotic theory of Kiefer and Vogelsang extends to tests employing the prewhitened HAC estimator. The Kiefer–Vogelsang critical values depend on the ratio of the bandwidth to the sample size (m/n) in the HAC estimator. Although the prewhitened variance estimator also requires a specified bandwidth in the second stage, it is typically much smaller than the bandwidth without the first-stage prewhitening (as much of the dependence has already been removed). Although the empirical size may be improved in some cases by prewhitening, the limit theory may not hold in all cases. For example, Vogelsang and Franses 15 employ prewhitening with the alternative critical values and find (at least in one simulation specification) that, ‘over-rejections can be a problem with moderately persistent data and prewhitening may not improve matters’ (p. 15).

The precise form of f is detailed in the appendix.

Rothenberg considers covariance estimators of the form , where j corrects for the number of observations lost due to the lag length. To ensure a positive semi-definite estimator, we replace the factor 1/(n−j) with 1/(n−5) 16.

As the table makes clear, the conditionally homoskedastic variance estimator can only be recommended in conjunction with Rothenberg's second-order critical value adjustment when the data are heteroskedastic. Although the cho test has better size than the hac test if the serial correlation is high, it rarely exhibits smaller size than the pw or tests.

Intuition would suggest that for a fixed value of ρ _U the performance of the cho and tests should deteriorate as ρ _X increases. However, ρ _X also adds to the overall pattern of serial correlation in the regressors as well as the errors. For this reason, the cho and tests show size gains over other HAC tests when ρ _U is large, even when ρ _X is large.

For example, if the upper bound is set at 6, then the distance between each element of ψ is 6/10, and .

Note that the range of values along the β₂-axis differs for alternative values of the serial correlation parameters. All tests are showing substantial increases in power as the correlation in the data falls.

Researchers often prefer ‘automatic’ bandwidth procedures as they simplify the estimation process and provide a level of uniformity across different projects. However, researchers must still make decisions regarding the specifications which underpin all mechanical bandwidth methods, such as objective functions and data dependence. For example, the Andrews bandwidth procedure is a function of J, and therefore, a preliminary estimate of J is necessary to compute the bandwidth to be used in the estimation of . Typically, the AR(1) model is used as a preliminary estimate of J, but alternative models will lead to alternative bandwidths.

There is no data-dependent method of choosing an ‘optimal’ bandwidth for the test. Phillips, Sun, and Jin propose a data dependent rule for their test that minimizes a weighted sum of type I and type II errors, for which Kiefer and Vogelsang conjecture can be extended to their test. However, in place of selecting the bandwidth, the researcher is now left to choose the proper weights for the type I and type II errors.

To ensure the estimator accurately accounts for the serial correlation, we impose a minimum bandwidth of 5. Allowing for bandwidths less than 5 further inflates the rejection probabilities.

Not surprisingly, when the errors truly are homoskedastic, the test outperforms all other robust tests in terms of finite-sample empirical size, as it exploits the homogeneity in the data. These results can be found in Erb and Steigerwald 13.

A heteroskedasticity pretest would undoubtedly alter the test size and we do not recommend choosing among the testing procedures in this manner.