Bootstrapping unit root tests with covariates

ABSTRACT

We consider the bootstrap method for the covariates augmented Dickey–Fuller (CADF) unit root test suggested in Hansen (1995) which uses related variables to improve the power of univariate unit root tests. It is shown that there are substantial power gains from including correlated covariates. The limit distribution of the CADF test, however, depends on the nuisance parameter that represents the correlation between the equation error and the covariates. Hence, inference based directly on the CADF test is not possible. To provide a valid inferential basis for the CADF test, we propose to use the parametric bootstrap procedure to obtain critical values, and establish the asymptotic validity of the bootstrap CADF test. Simulations show that the bootstrap CADF test significantly improves the asymptotic and the finite sample size performances of the CADF test, especially when the covariates are highly correlated with the error. Indeed, the bootstrap CADF test offers drastic power gains over the conventional unit root tests. Our testing procedures are applied to the extended Nelson and Plosser data set.

KEYWORDS:

• Bootstrap consistency
• covariates
• unit root tests

JEL CLASSIFICATION:

• C12
• C15
• C22

Acknowledgements

We are grateful to Joon Park, Bill Brown, and David Papell for helpful discussions and comments.

Notes

Table 1 in Hansen (1995) provides asymptotic critical values for the CADF t-statistic for values of the nuisance parameter in steps of 0.1 via simulations. For intermediate values of the nuisance parameter, critical values are selected by interpolation.

We start with the simple model without the deterministic components to effectively deliver the essence of the theory. The models with the deterministic components will be considered at the end of this section.

Noting that the null limit distribution of the CADF t-test depends only on the correlation coefficient ρ², Hansen (1995, Table 1, p. 1155) provides the asymptotic critical values for the CADF t-test for values of ρ² from 0.1 to 1 in steps of 0.1. The estimate for ρ² is constructed as , where v_t = β(L)′w_t + ε_t, and , and are consistent nonparametric estimators of the corresponding parameters.

Here we use the simple terms “size” and “power” to mean “Type I error” and “rejection probability under the alternative hypothesis,” respectively.

To find σ_ηε yielding σ_ηε = 0.4 as in the iid simulations, we use the relationship and simulate using our specification of (ε_t) in (11).

In the tables below, we provide the results only for ϕ = 0.5 and 0.8 to save space because we have lower ρ² for these parameters.

The simulation results where ρ² is estimated using the true model in a parametric way are available from the authors. In this case, the estimation of ρ² becomes more precise, and the size distortions coming from the imprecise estimates of ρ² disappear as n grows, as expected (for example, in the cases of (β, ϕ)=(−0.5,0.5),(−0.8,0.5)). However, even with the more precise estimates of ρ², the size distortions observed in the cases of (β, ϕ)=(−0.5,0.8),(−0.8,0.8), where ρ² are very low, still remain, and they are 1.8% and 0.7%, respectively. Therefore, even with the more precise estimates of ρ², the CADF test still suffers from the size distortions problem, especially for low values of ρ².

We thank Elliott and Jansson, and Ng and Perron for sharing their codes with us. We do not provide the results of the Z_α, MZ_t, MSB, and P_t tests because their performances are very similar to that of the MZ_α test.

As the lag truncation parameter, eight was used following the suggestion in Kwiatkowski et al. (1992).

The simulation results for n = 25 and for the cases of ϕ = −0.5 and −0.8 are omitted to save space. Full results are available from the authors upon request.

Stock and Watson (1999) note that current theoretical literatures in macroeconomics provide neither intuition nor guidance on which covariates are candidates for our CADF and bootstrap CADF tests other than on the basis of stationarity.

BIC gives almost similar results.

is low when ϕ is positive according to the simulation results. When we calculate the estimates ϕ of AR(1) lags of potential covariates, they all take large positive numbers.