GLS detrending and unit root testing

Transcription

1 Economics Letters 97 (2007) GLS detrending and unit root testing Dimitrios V. Vougas School of Business and Economics, Department of Economics, Richard Price Building, Swansea University, Singleton Park, Swansea SA2 8PP, UK Received 7 May 2006; received in revised form 11 February 2007; accepted 19 March 2007 Available online 29 June 2007 Abstract This paper simulates power of unit root tests based on alternative procedures for undertaking GLS detrending in a linear trend model. Many of the proposed methods produce improvements (over the original approach) for small samples and autoregressive parameter near unity Elsevier B.V. All rights reserved. Keywords: Autoregressive estimator; GLS detrending; Unit root test; Power JEL classification: C12; C15; C22 1. Introduction For regressions with first-order (or higher-order) serial correlation, efficient two-step generalized least squares (GLS) approaches utilize various initial estimates (or preset values) for the autoregressive parameter. In this paper, interest focuses on linear trend as a regressor and subsequent unit root testing. 1 Early literature of GLS use in this context is given by Canjels and Watson (1997) who analyze relative efficiency of the resulting GLS estimators under a root near unity. 2 The GLS approach is used in efficient unit root testing. Elliott et al. (1996) (ERS, hereafter) propose local-to-unity GLS detrending (using Tel.: ; fax: address: D.V.Vougas@swan.ac.uk. URL: 1 With intercept-only as a regressor (not examined here), there is still room for finite-sample (but not asymptotic) improvement, using similar GLS approaches. 2 Spurious detrending of a random walk is examined by Durlauf and Phillips (1988) /$ - see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.econlet

2 D.V. Vougas / Economics Letters 97 (2007) specific sample-dependent autoregressive values). This is the so-called DF-GLS unit root test, that is a DF test (see Dickey and Fuller, 1979) applied (with no deterministic component) to the regression residual, which arises from the GLS estimators employed in the original regression. It is of interest to examine power of unit root tests based on alternative GLS detrending procedures with various initial autoregressive estimators (rather than specific values). 3 To this end, this paper employs various autoregressive estimators for GLS detrending that have been proposed in the literature. It examines finite-sample sizeadjusted power of the resulting GLS-type unit root tests. In addition, a newly proposed estimator for the autoregressive parameter is employed, along with a new DF-GLS test based on (explosive) local-to-unity values. Power of resulting unit root tests is compared to the benchmark power of both the original and new DF-GLS unit root tests in a model with linear trend and AR(1) error. Provided simulation evidence, based on half-a-million replications, favors the new approaches. The paper is organized as follows: Section 2 discusses GLS estimation and related unit root testing, while Section 3 provides Monte Carlo design, results and discussion. Finally, Section 4 concludes. 2. GLS estimation and unit root testing Let y t be generated by a k 1 vector of variables w t and the AR(1) process u t, y t ¼ dvw t þ u t ; t ¼ 1; N ; T; ð1þ u t ¼ qu t 1 þ e t ; t ¼ 2; N ; T: ð2þ The initial error u 1 needs special treatment, depending on the value of ρ. In full information GLS detrending, inherent is the assumption that u 1 is independent from ε 2,, ε T, see Dufour (1990) and Dufour and King (1991). This assumption is fully employed by ERS and Elliott (1999). 4 For given ρ, y 1 ¼ dvw 1 þ u 1 ð3þ is employed along with ð1 qlþy t ¼ð1 qlþd V w t þ e t ; t ¼ 2; N ; T: ð4þ In matrix form y ¼ W d þ u ; ð5þ with y ={y 1,(1 ρl)y 2,,(1 ρl)y T }, u ={u 1, ε 2,, ε T }, w 1 =w1 and w j =(1 ρl)wj ( j=2,, T ); w j is the j-th row of W. 5 The GLS estimator of δ, denoted δˆgls, is the OLS estimator of Eq. (5). The resulting series y d ¼ y Wˆd GLS ð6þ 3 Note that the GLS class of unit root tests includes the Lagrange Multiplier (LM) test, i.e., the Schmidt and Lee (1991) (see also Schmidt and Phillips, 1992) version. These LM tests are examined by this author elsewhere. 4 Note the assumption of Berenblut and Webb (1973) that u 1 =ε 1 with ε 1 iid(0, σ 2 ). 5 In fact Pu = ε, withε ={ε 1,, ε T } and P is the T T GLS transformation matrix defined as P[1,1] =1, P[ j,j]=1, P[ j, j 1] = ρ for j =2, 3,,T and 0 elsewhere. In effect, it is assumed that u 1 = ε 1.

3 224 D.V. Vougas / Economics Letters 97 (2007) is used in (efficient) unit root testing. Note that W has typical row w t (t=1,, T). ERS employ this unit root testing method and recommend using ρ=1 7/T when w t ={1} (intercept-only) and ρ=1 13.5/T when w t ={1,t} (linear trend). 6 On the other hand, estimators for ρ (rather than specific preset values) 7 can be used. Such an approach is less important for a model with intercept-only and more important when there are regressors, for example a linear trend. The asymptotic results of ERS point towards this direction. This paper uses various estimators for ñ in GLS detrending and examines power of the resulting unit root tests. From OLS residuals, û t ¼ y t ˆd V w t ; ð7þ where δˆ is the OLS estimator of Eq. (1), various estimators for ρ arise. The Durbin Watson (DW) (see Durbin and Watson, 1950) statistic ˆ d ¼ ðû t û t 1 Þ 2 N0 t¼1 results in the associated estimator ˆq d ¼ 1 ˆ d 2 : ð9þ (Note that dˆ is a unit root test statistic relating to the coefficient unit root test, since T(ρˆd 1)= Tdˆ/2. If one uses a local-to-unity specification, ρˆd=1 γˆd/t with γˆd=tdˆ /2.) Also ρˆd is always in the ( 1, 1) interval and used by Durlauf and Phillips (1988) to alleviate the effect of spurious detrending. A corrected estimator based on ρˆd, proposed by Theil and Nagar, is ˆq TN ¼ T 2ˆq d þ k 2 T 2 k 2 : ð10þ (T is the sample size and k the number of regressors.) An alternative estimator, also bounded in ( 1, 1), is the Cauchy Swartz estimator (a term used here) ˆq CS ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð11þ ux t T ux t T 1 ð8þ 6 Note that setting ρ=1, for any w t, results in the LM unit root test. 7 In addition, new preset sample-dependent values may be employed.

4 D.V. Vougas / Economics Letters 97 (2007) The OLS estimator ˆq LS ¼ 1 ð12þ is calculated from the autoregression of û t on û t 1. On the other hand, the autocorrelation coefficient is ˆq C ¼ t¼1 : ð13þ An alternative Theil Nagar-type estimator is proposed here for the first time and defined as ˆq ATN ¼ Tˆq d þ k T k : ð14þ In addition, a set of non-iterative estimators is discussed in Park and Mitchell (1980). The estimator used in the Cochrane Orcutt (CO) iterative procedure is ˆq CO ¼ : ð15þ 1 t¼1 A second estimator used in the Prais Winsten (PW) iterative procedure is ˆq PW ¼ : ð16þ 1 Finally, a third estimator (used in Monte Carlo studies) is ˆq MC ¼ : ð17þ

5 226 D.V. Vougas / Economics Letters 97 (2007) Table 1 Finite-sample power of various DF-GLS unit root tests DF-GLS ρˆd ρˆ TN ρˆ LS ρˆ C ρˆatn ρˆ PW ρˆ M ρˆ CS ρˆ CO This paper also proposes a new bias-corrected estimator for ρ. The bias correction is via asymptotically negligible sample statistics that appear to be very successful in reducing finite-sample bias under a unit root. For the model y ¼ qy 1 þ e ð18þ with y={y 2,, y T }, y 1 ={y 1,, y T 1 } and ε={ε 2,, ε T }. The proposed estimator is ˆq 1 ¼ yv 1 y y V 1 y þ p ffiffi DyV Dy 3 y V 1 y : ð19þ 1 Under a unit root, the additional term is O p (T 3/2 ) (that is asymptotically negligible, even after multiplication by T). This estimator appears virtually bias-free for T N 15. Similarly, the model with deterministic component is y ¼ qy 1 þ W s d s þ e; ð20þ W τ is W with its first row omitted. The proposed bias-corrected estimator for this model is ˆq M ¼ yv 1 My y V 1 My þ p ffiffi DyV MDy 3 y V 1 My ; ð21þ 1

6 D.V. Vougas / Economics Letters 97 (2007) Table 2 Finite-sample power of corrected DF-GLS unit root tests New DF-GLS ρˆd ρˆ TN ρˆ LS ρˆ C ρˆatn ρˆ PW ρˆ M ρˆ CS ρˆ CO M=I T 1 W τ (W τ W τ ) 1 W τ. Unreported Monte Carlo evidence indicates that both ad hoc estimators of Eqs. (19) and (21) have better bias properties than the corrected estimators of Ullah (2004, Chaper 6) (for the model with no deterministic component) and Kiviet and Phillips (1993) (for the model with deterministic component), respectively. The bias-reduction comparison is based on the large-sample (not the small-sigma) first-order bias correction of these authors. 3. Numerical issues: design and results From asymptotic results in ERS, p. 825, it appears that the alternative GLS methods of this paper, for intercept-only, must result in finite-sample improvements and not asymptotic ones. On the other hand, for linear trend, both finite-sample and asymptotic (although bounded) improvements are to be expected. Nevertheless, there is scope for finite-sample size-adjusted power evaluation of unit root tests based on the GLS detrending methods above. A Monte Carlo simulation study is undertaken for the linear trend model. Following Elliott (1999), the generating mechanism for u t is u t ¼ qu t 1 þ e t e t fiidnð0;1þ; t ¼ 2; N ; T; ð22þ

7 228 D.V. Vougas / Economics Letters 97 (2007) p with u 1 =0 for ρ =1 and u 1 ¼ e 1 = ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q 2 for ρ b1 (stationarity). 8 Only w t ={1,t} is considered with (true) δ =0 with no loss of generality. In the first step, GLS estimators for δ are derived from Eq. (5), using various estimators for ρ that arise from OLS residuals (Eq. (7)) or preset values. Corresponding DF-type t-ratio tests are employed to the resulting GLS residual series from Eq. (6) with no deterministic component. For T {25, 50, 100, 200}, critical values are calculated via simulation and then used in power calculation for ρ {.95,.9,.8,.7} via further simulation. Both critical values and power calculations are based on half-a-million replications. In total, ten estimators are considered, including the ERSvalue /T. Thislattertestis termeddf-glshere. Poweris presentedintable 1. Note that each test is signified by the underlying estimator for ρ. (The results based on ρˆ MC are not reported, since they are identical to the ones based on ρˆ CO.) Table 1 shows that, although power differences of various tests are relatively small, the original DF-GLS test has power which is matched or beaten by the power of the other tests for all sample sizes and alternative values. It seems that the ERS autoregressive value does not uniformly generate extra power, relative to other GLS-type unit root tests. To complete the power picture, the paper employs bias-corrected versions of the autoregressive estimators (including ρˆ M). The correction of Judge et al. (1988, p. 393) is followed. For an autoregressive estimator ρ (say) (from residuals obtained using detrending with a time polynomial of order b), the used bias-corrected estimator is q c ¼ q þ ðb þ 1Þð1 þ qþþ2 q : ð23þ T Taking b= 1 to denote no deterministic (b=0 intercept-only, b=1 linear trend, etc.), the correcting terms are familiar for b= 1, 0. This correction is employed here for every one of the estimators in the previous section, with b=1. In an attempt to enhance power of the original DF-GLS test, this paper also proposes detrending with alternative (explosive) autoregressive values. Scores of unreported Monte Carlo experiments have been employed to find explosive autoregressive values that maximize power for T=50 at ρ=.8. The recommended (explosive) preset values are: ρ μ =1+11.5/T (intercept-only) and ρ τ =1+17/T (linear trend). Table 2 reports new power results based on new critical values, along with power of the new DF-GLS test. Finite-sample critical values for the new coefficient and t-ratio DF-GLS tests, both intercept-only (b=0) and linear trend (b=1), are provided by the author upon request. 9 From Table 2, it is now more pronounced that the original DF-GLS test has power which is lower than the power of other GLS-type tests that are based on corrected estimators. Power of the original DF-GLS test is inferior to the power of many alternative tests, including the new DF-GLS which is based on ρ τ. However, there is no uniformly most powerful test. 4. Conclusions This paper examines finite-sample size-adjusted power of various GLS-type unit root tests (t-ratios). It is found that most tests have improved power, although no one is uniformly better. Two proposed tests 8 All unit root tests of this paper are invariant with respect to u 1 under the null, hence there is no loss of generality by setting u 1 =0. 9 All GLS-type unit root tests of this paper require new finite-sample critical values.

8 D.V. Vougas / Economics Letters 97 (2007) appear to have better power near unity and for smaller sample sizes. The power is generally low, but in line with (if not better than) other unit root tests in the literature, see Elliott (1999). However, power alone cannot help in judging which GLS test to use in practice. Acknowledgement The author thanks an anonymous referee for comments that helped improving exposition and clarity of the paper. References Berenblut, I.I., Webb, G.I., A new test for autocorrelated errors in the linear regression model. Journal of the Royal Statistical Society, Series B 35, Canjels, E., Watson, M.W., Estimating deterministic trends in the presence of serially correlated errors. Review of Economics and Statistics 79, Dickey, D.A., Fuller, W.A., Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, Dufour, J.M., Exact tests and confidence sets in linear regressions with autocorrelated errors. Econometrica 58, Dufour, J.M., King, M.L., Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or non-stationary AR(1) errors. Journal of Econometrics 47, Durbin, J., Watson, G., Testing for serial correlation in least squares regression I. Biometrika 37, Durlauf, S.N., Phillips, P.C.B., Trends versus random walks in time series analysis. Econometrica 56, Elliott, G., Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review 40, Elliott, G., Rothenberg, T.J., Stock, J.H., Efficient tests for an autoregressive unit root. Econometrica 64, Judge, G.G., Hill, R.C., Griffiths, W.E., Lutkepohl, H., Lee, T.-C., Introduction to the Theory and Practice of Econometrics, Second edition. Wiley. Kiviet, J.F., Phillips, G.D.A., Alternative bias approximations in regressions with a lagged dependent variable. Econometric Theory 9, Park, R.E., Mitchell, B.M., Estimating the autocorrelated error model with trended data. Journal of Econometrics 13, Schmidt, P., Lee, J., A modification of the Schmidt Phillips unit root test. Economics Letters 36, Schmidt, P., Phillips, P.C.B., LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, Ullah, A., Finite Sample Econometrics. Oxford University Press, Oxford.