The Performance of Lag Selection and Detrending Methods for HEGY Seasonal Unit Root Tests

Abstract

This paper analyzes two key issues for the empirical implementation of parametric seasonal unit root tests, namely generalized least squares (GLS) versus ordinary least squares (OLS) detrending and the selection of the lag augmentation polynomial. Through an extensive Monte Carlo analysis, the performance of a battery of lag selection techniques is analyzed, including a new extension of modified information criteria for the seasonal unit root context. All procedures are applied for both OLS and GLS detrending for a range of data generating processes, also including an examination of hybrid OLS-GLS detrending in conjunction with (seasonal) modified AIC lag selection. An application to quarterly U.S. industrial production indices illustrates the practical implications of choices made.

Keywords:

• Data-based lag selection
• HEGY tests
• OLS and GLS detrending
• Seasonal unit root

JEL Classification:

• C22
• C52

Notes

¹The test regression and HEGY-type tests corresponding to (2.5) for a general seasonal aspect S are presented by Smith and Taylor (1999) and COT, among others.

²For this procedure, and also those suggested by Beaulieu and Miron (1993) and Rodrigues and Taylor (2004), results were also obtained for a significance level of 15%. These are excluded to conserve space, but exhibit qualitatively similar patterns to the corresponding 10% ones.

³We are grateful to a referee who suggested the inclusion of some DGPs where unit roots were present at some but not all (zero and seasonal) frequencies of interest. However, an extension of the analysis here found that DGPs with local departures from the unit root null at some frequencies yielded very similar results overall to those reported.

⁴Applied in the context of increasing sample size, this k _max satisfies k = o(T ^1/3) and hence also yields valid asymptotic inference in (2.5) when the innovation process is conditionally heteroscedastic. Although we experimented with a variety of non-IID martingale difference specifications for ϵ_4t+s in the context of the conventional HEGY test regression, the results were almost identical to those reported.

The DGP is (3.1) with c = 0 and u _4t+s  = ϵ_4t+s  ∼ IIDN(0, 1), for quarterly data over N = 60 years. Tests and lag selection criteria as in Section 2, with k _max = int[ℓ(4N/100)]^1/4 for ℓ = 4 or 12. All tests allow for seasonal means and a zero frequency trend: OLS and GLS indicates OLS-detrending and GLS-detrending, with PQ indicating that the latter uses the OLS-GLS method of Perron and Qu (2007). The statistics are t-type tests for unit roots at the zero and π frequencies (t ₀, t ₂) and joint F-type statistics for unit roots at the π/2 frequency. (F ₁), all seasonal frequencies (F ₁₂) and the zero and all seasonal frequencies (F ₀₁₂). Results are based on 5,000 replications for a nominal 5% level of significance.

As for Table 1, except that the DGP has moving average disturbances, with u _4t+s  = (1 − θL)(1 − ΘL ⁴)ϵ_4t+s and maximum lag given by k _max = int[ℓ(4N/100)^1/4] with ℓ = 12.

As for Table 1, except that the DGP has seasonal autoregressive disturbances with (1 − ΦL ⁴)u _4t+s  = ϵ_4t+s and maximum lag given by k _max = int[ℓ(4N/100)^1/4] with ℓ = 12.

As for Table 1, except that the DGP is (3.1) with c = 5, 10, and 20.

As for Table 2, except that the DGP is (3.1) with c = 5, 10, and 20.

As for Table 3, except that the DGP is (3.1) with c = 5, 10, and 20.

⁵These critical values were obtained by direct simulation using 100, 000 replications and T = 2, 000.

⁶We also investigated the performance of these lag selection criteria in the context of the HEGY test regression (2.5) for the MA(1) case of θ = 0.8, which interchanges the roles of the zero and Nyquist frequencies. As anticipated, this leads to oversizing for t ₀, as found by Hall (1994) and Ng and Perron (2001) for the Dickey–Fuller test. Analogous results were also found for the MA(2) u _4t+s  = (1 − 0.64L ²)ϵ_4t+s , where near-cancellation applies at the Nyquist frequency, and hence the oversizing relates to F ₁.

⁷The results for other cases are available on request.

⁸For both N = 60 and N = 100, the corresponding results with MBIC lag selection and GLS detrending exhibit similar patterns to those shown with OLS-GLS detrending.

⁹All data are from the U.S. Federal Reserve website http://www.federalreserve.gov/releases/g17/ table1_2.htm.

ARMA models are estimated for component quartely U.S. industrial production growth series, computed as 100 times the first difference of the logarithm of the corresponding index. Estimated models have the form (1 − φL)(1 − ΦL ⁴)Δx _4t+s  = Δμ_4t+s  + (1 − θ₁ L − θ₂ L ²)(1 − ΘL ⁴)ϵ_4t+s , where x _4t+s is the observed series, the deterministic component Δμ_4t+s includes a constant and three seasonal dummy variables, ϵ _t is white noise; values in parentheses are standard errors; R ² and s are the conventional coefficient of determination and the residual standard error, respectively; LM(4) is the p-value for the F-test version of the Lagrange–Multipler test for autocorrelation to 4 lags.

¹⁰This general specification was adopted based on the serial correlation properties of the first differenced series. The ARMA models were estimated in the program EViews using conditional least squares.

^{*, **} and *** indicate statistically sigficant at 10%, 5%, and 1% levels, respectively. All tests allow for seasonal intercepts and a zero frequency trend, and are computed over the sample 1947Q1 to 2010Q4, with PQ indicating use of the OLS-GLS method of Perron and Qu (2007). The maximum lag length considered is given by k _max = ⌊12(4N/100)^1/4⌋ = 15. Critical values have been obtained by simulation based on 100,000 replications for a sample of 64 years of quarterly data.

See notes for Table 1, except that N = 100.

See notes for Table 2, except that N = 100.

See notes for Table 3, except that N = 100.

See notes for Table 4, except that N = 100.

See notes for Table 5, except that N = 100.

See notes Table 6, except that N = 100.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lecr.