Wild-bootstrapped variance-ratio test for autocorrelation in the presence of heteroskedasticity

Abstract

The Breusch–Godfrey LM test is one of the most popular tests for autocorrelation. However, it has been shown that the LM test may be erroneous when there exist heteroskedastic errors in a regression model. Recently, remedies have been proposed by Godfrey and Tremayne [9] and Shim et al. [21]. This paper suggests three wild-bootstrapped variance-ratio (WB-VR) tests for autocorrelation in the presence of heteroskedasticity. We show through a Monte Carlo simulation that our WB-VR tests have better small sample properties and are robust to the structure of heteroskedasticity.

Keywords:

• variance-ratio test
• Breusch–Godfrey LM test
• autocorrelation
• heteroskedasticity
• wild bootstrap

JEL Classifications:

• C12
• C15

Acknowledgements

We are grateful to Douglas Barthold, Hidehiko Ichimura, Dong H. Kim, Tae H. Kim, the participants of 2009 WEAI conference, and the participants of the seminar at the University of Tokyo for their helpful comments. We are also indebted to the anonymous referee and the associate editor. The authors are members of the ‘Brain Korea 21’ Research Group of Yonsei University.

Notes

For example, see Mammen 18, Davidson and Flachaire 5, and Flachaire 6, among others.

The modified VR test proposed by Shim et al. 21 also employs such a pre-correction for heteroskedasticity.

Pagan 19 20 elaborated this general problem of a two-step estimation.

Say, when , , and .

For a review, see 13.

See [12, pp. 3215–3217], for an introduction to wild bootstrap method.

In addition to the uniform distribution, we also tried a normal distribution for the regression errors, η _t . As the simulation results are not much different, we only report the uniform distribution case. It is noteworthy that the WB-VR tests work well for a flat distribution (i.e. extremely fat tails) as well as for a usual bell-shaped distribution. The simulation results for a normal distribution case are available from the authors upon request.

We experimented with two alternative types of heteroskedasticity: a decreasing multiplicative heteroskedasticity case and a variance break case. The results are not qualitatively different from those of the positive heteroskedasticity case presented here. The simulation results are available from the authors upon request.

In addition to the 5% nominal size, we experimented with 1% and 10% significance levels, too. The results are consistent with the 5% ones reported in the paper. The simulation results are available from the authors upon request.

Thus, as long as the true lag length of the serial correlation is smaller than 12, the MVR test is robust to the preset p.

As the MVR test statistic is identical regardless of the leg length, p, we present just one rejection rate for the MVR.

Godfrey and Tremayne [10, p. 392], case (21) versus case (25) presented in Table 6.

Of course, a special case like case (22) presented in Table 6 of Godfrey and Tremayne 10 is an exception, as the lower order terms have zero coefficients.

We also examined a single variance break case. The simulation results are available from the authors upon request.