Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

Abstract

A number of recent papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, taken the lag length in the unit root test regression to be a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. We investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent lag selection methods in augmented Dickey–Fuller type regressions and propose new lag selection criteria which allow for unconditional heteroskedasticity. Standard lag selection methods are shown to have a tendency to over-fit the lag order under heteroskedasticity, resulting in significant power losses in the (wild bootstrap implementation of the) augmented Dickey–Fuller tests under the alternative. The proposed new lag selection criteria are shown to avoid this problem yet deliver unit root tests with almost identical finite sample properties as the corresponding tests based on conventional lag selection when the shocks are homoskedastic.

Keywords:

• Information criteria
• Lag selection
• Nonstationary volatility
• Unit root test
• Wild bootstrap

JEL Classification:

• C22
• C15

ACKNOWLEDGMENT

We thank three anonymous referees for their helpful and constructive comments on an earlier draft of this paper.

Notes

¹The recent financial turmoil and disruption of economic activity associated with the onset of the 2008 credit crisis has undoubtedly reversed this decline and produced a corresponding rise in unconditional volatility. Such changes reinforce the need to allow for the possibility of non-constancy in unconditional volatility.

²This initialization ensures that we obtain T residuals and, hence, T bootstrap errors in Step 4. An asymptotically equivalent alternative is to omit this initialisation thereby yielding only T − q residuals, but to then initialize the recursion in Step 4 with the first q detrended sample values. We found virtually no differences between the two schemes for the sample sizes considered.

³In this paper, we take ξ _t to be standard normal. Other choices are also possible, although Cavaliere and Taylor (2008, Remark 6) mention that this has almost no impact on finite sample behavior.

⁴This differs from the approach taken by Smeekes and Taylor (2012, Assumption 5) for reasons explained in their Remark 15.

⁵Cavaliere and Taylor (2009a) assume that q ≤ p* for all T but this is not necessary for the validity of the bootstrap. By allowing p* to be smaller than q one can replicate the effect of underfitting the lag length in the bootstrap, which may improve finite sample performance (Richard, 2009, cf.][).

⁶We also considered the re-weighted local constant estimator proposed by Xu and Phillips (2011). As discussed by Xu and Phillips (2011, this estimator shares all the advantages of the local linear estimator. However, unlike the local linear estimator (but like the local constant estimator), it cannot be negative. The simulation results with this estimator were virtually identical to the results reported here with the local constant estimator and, hence, are omitted in the interests of space.

⁷Similarly, it is possible to use the residuals which are obtained when imposing the unit root null hypothesis. Unreported simulation results indicated that the results do not change in this case either.

⁸For smaller sample sizes the differences between the regular and re-scaled IC are less obviously seen, at least in part because the maximum lag length parameters will be smaller; see, for example, the additional results for T = 50 reported in Cavaliere et al. (2012).

⁹Different specifications again gave very similar results.

The DGP is (8a)–(8d) with β = 0 and c = 7, for each of the 14 ARMA models considered in Table 1 and where σ _t is constant for T = 150 and T = 250. The MAIC and RSMAIC lag selection criteria are as outlined in sections 3.2 and 3.3, respectively, with p _max = ⌊A(T/100)^1/4⌋. As in Perron and Qu (2007), the information criteria are applied to OLS demeaned data, for the case of z _t  = 1. Results are based on 5,000 replications.

As for Table 2 except that σ _t follows the smooth transition model in Volatility Model 1.

¹⁰Full results can be found in Cavaliere et al. (2012).

As for Table 2 except that σ _t follows the stochastic volatility model in Volatility Model 2.

See notes for Table 2. Bootstrap ADF tests constructed as detailed in Algorithm 1 for QD demeaned data (z _t  = 1) using either MAIC or RMAIC lag selection. Results are for the nominal 5% significance level and based on 5,000 simulations.

See notes for Tables 3 and 4. Wild bootstrap ADF tests constructed as detailed in Algorithm 1 for QD demeaned data (z _t  = 1) using either MAIC or RMAIC lag selection. Results are for the nominal 5% significance level and based on 5000 simulations.

¹¹Results for the remaining volatility models are available in Cavaliere et al. (2012).

¹²Notice that the local power curves for these models are quite different from the corresponding local power curves seen in Fig. 1 under homoskedasticity, even for T = 250. This is not an effect of the lag order selection method but rather a consequence of the result that if nonstationary volatility is present, then the limiting distributions of the ADF statistic, , under both the null hypothesis and local alternatives, and hence the asymptotic local power function of the associated bootstrap test, are functions of the underlying volatility process (Cavaliere and Taylor, 2008, cf.][p. 8).