The power of autocorrelation tests near the unit root in models with possibly mis-specified linear restrictions

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1 Economics Letters 94 (2007) The power of autocorrelation tests near the unit root in models with possibly mis-specified linear restrictions Alan T.K. Wan a,, Guohua Zou b, Anurag Banerjee c a Department of Management Sciences, City University of Hong Kong, Tat Chee Ave., Kowloon, Hong Kong b Chinese Academy of Sciences, Beijing, China c University of Southampton, U.K. Received 17 November 2005; received in revised form 22 May 2006; accepted 29 June 2006 Available online 8 December 2006 Abstract It is well known that the Durbin Watson and several other tests for first-order autocorrelation have limiting power of either zero or one in a linear regression model without an intercept, and a constant lying strictly between these values when an intercept term is present. This paper considers the limiting power of these tests in models with possibly incorrect restrictions on the coefficients. It is found that with linear restrictions on the coefficients, the limiting power can still drop to zero even with the inclusion of an intercept in the regression. Our results also accommodate the situation of a possibly mis-specified linear model Elsevier B.V. All rights reserved. Keywords: Autocorrelation test; Linear restrictions; Mis-specified models; Power JEL classification: C12; C13 1. Introduction The Durbin Watson (DW) test is by far the most common test for first-order autocorrelated (AR(1)) errors. Over the past three decades there has been a sizable literature on the power properties of the DW test and its power comparisons with various other testing procedures under differing circumstances. See, for example, Blattberg (1973), Schmidt and Guilkey (1975), Bartels and Goodhew (1981), Bartels (1992), White (1992), among others. For a survey of the literature on testing for autocorrelation, see King (1987). A number Corresponding author. Tel.: ; fax: addresses: msawan@cityu.edu.hk (A.T.K. Wan), ghzou@iss.ac.cn (G. Zou), A.N.Banerjee@soton.ac.uk (A. Banerjee) /$ - see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.econlet

2 214 A.T.K. Wan et al. / Economics Letters 94 (2007) of authors have shown that the power of the DW test can drop to zero when the autocorrelation coefficient in the AR(1) process is close to one. Krämer (1985) found that in a linear regression without an intercept, the limiting power of the DW test is either zero or one, and whether the limit is zero or one is determined by the sign of an eigenvalue that depends on the underlying regressor matrix. Krämer and Zeisel (1990) provided further analytical evidence suggesting that in addition to the DW test, the alternative DW (ADW) (King, 1981), the Berenblut Webb (BW) (Berenblut and Webb, 1973), and King's point optimal (King, 1985) tests can all have zero power when the correlation between neighbouring errors is close to one. The latter two tests are most powerful invariant in a given neighbourhood of the alternative hypothesis' parameter space. So it may very well happen that all of these tests do not reject the null hypothesis of no autocorrelation, but nevertheless the neighbouring disturbances are nearly perfectly correlated. On the other hand, Zeisel (1989) proved that in the model with an intercept, the DW test has limiting power that never reaches unity or drops to zero. Small (1993) showed that Zeisel's (1989) results apply also to the ADW, BW and King's point optimal tests. It appears therefore that, by adding a constant to the model, one can avoid the worst scenario of the autocorrelation tests having no power at all as one approaches the unit root. One purpose of this paper is to extend these results to the models with restrictions on coefficients. There has been some indirect evidence suggesting that the DW test applied to the restricted regression has somewhat similar properties to that applied to the unrestricted model. Giles and Lieberman (1992) studied the size and power of the DW test when its application is preceded by a preliminary t-test on the significance of a coefficient. Their Monte- Carlo results indicate no size distortion for the DW test applied without pre-testing to the restricted model when the restriction is true; but if the false restriction is imposed without pre-testing, then the subsequent DW test generally yields sizes far greater than the nominal size. The restricted version of the DW test is generally more powerful than the corresponding unrestricted and pre-test versions of the test when the restriction is true; if the restriction is false, the power of the restricted DW test can be vastly inferior to the other two. Giles and Lieberman (1992) also reported the possibility of very poor power properties of the pre-test DW test when highly correlated errors are combined with a grossly incorrect restriction. Dufour (1990), on the other hand, suggested a joint procedure for testing linear restrictions on the coefficients and autocorrelated errors. This procedure, however, fails when applied to the testing of a zero intercept at the unit-root. In this paper we present analytical results on the limiting power of the DW, ADW, BW and King's point optimal tests when applied to a model with (possibly) incorrect linear restrictions on the coefficients. For a restricted regression with no intercept, it is found that the limiting power of all four tests tends to either one or zero as in the case of an unrestricted model; but with restrictions on the coefficients, these limiting power characteristics can still apply even with the inclusion of an intercept. This is true even when the restrictions are wrongly imposed. The latter finding provides an interesting contrast with the existing results on the power properties of these tests in the unrestricted model. Whether or not the restrictions are in fact valid has no effect on the limiting power of the tests. Applications of our findings to possibly misspecified linear models will also be discussed. 2. Model framework, test statistics and notations We consider the following linear regression with possibly AR(1) errors: y ¼ X b þ u; u t ¼ qu t 1 þ e t ; jqjb1; e t fnidð0;r 2 Þ ð1þ

3 A.T.K. Wan et al. / Economics Letters 94 (2007) where y and u are n 1 random vectors, X is an n k non-stochastic matrix of full column rank and β is a k 1 vector of unknown coefficients. In addition to sample information, there exists prior information in the form of the following set of m independent linear restrictions on the coefficients: Rb ¼ r; ð2þ where R is an m k known prior information design matrix of rank m and r is a m 1 vector of known elements. We are interested in testing the hypothesis of H 0 : ρ=0 of no first-order autocorrelation versus H 1 : ρn0 It is assumed that tests of H 0 are preceded by fitting the regression (1) with the linear restrictions (2) incorporated. Let E(uu )=σ 2 V(ρ). Clearly, V(0)=I is an identity matrix. Imposing the restrictions in (1) gives rise to the restricted least squares (RLS) estimator bðqþ ¼ˆbðqÞþS 1 ðqþr V ðrs 1 ðqþr V Þ 1 ðr RˆbðqÞÞ; ð3þ where βˆ(ρ)=s 1 (ρ)x V 1 (ρ)y and S(ρ)=X V 1 (ρ)x. In practice, prior to testing H 0, the estimator used is the one in (3) with ρ=0, i.e., bð0þ ¼ˆbð0ÞþS 1 ð0þr V ðrs 1 ð0þr V Þ 1 ðr Rˆbð0ÞÞ: ð4þ Let MðqÞ ¼I XS 1 ðqþx V V 1 ðqþ; BðqÞ ¼XS 1 ðqþr V ðrs 1 ðqþr V Þ 1 RS 1 ðqþx V V 1 ðqþ; ð5þ ð6þ and dðqþ ¼XS 1 ðqþr V ðrs 1 ðqþr V Þ 1 ðrb rþ=r: If the restrictions are correct then δ(ρ)=0 for which y X bðqþ ¼ðMðqÞþBðqÞÞu ¼ MðqÞu: On the other hand, if the restrictions are incorrect, then δ(ρ) 0. Thus, y X bðqþ ¼ MðqÞu þ rdðqþ: Eq. (9) is the vector of residuals corresponding to the RLS estimator given in (3). We consider the test statistics which can be expressed as ratio of the following quadratic forms: ð7þ ð8þ ð9þ Gðq o ;QÞ ¼ ðy X bðq o ÞÞ V Qðy X bðq o ÞÞ ðy X bð0þþ V ðy X bð0þþ ¼ ðu þ rdðq oþþ V M Vðq o ÞQ Mðq o Þðu þ rdðq o ÞÞ ðu þ rdð0þþ V ; Mð0Þðu þ rdð0þþ where ρ o is a known constant. Note that (10) is in fact the generalized likelihood ratio test for testing the hypothesis of the autoregressive parameter being equal to ρ o and Q=V 1 (ρ o ) under the assumption that Rβ=r. ð10þ

4 216 A.T.K. Wan et al. / Economics Letters 94 (2007) Below we give the restricted regression analogues of four of the autocorrelation test statistics considered in Krämer and Zeisel (1990): (i) The DW test: The restricted DW test statistic is given by G(0, A) where A ¼ : ð11þ B A (ii) The ADW test: For this test, the test statistic has the form G(0, A 1 ), where A 1 =A+C, and C is an n n matrix whose top left and bottom right elements are one and all remaining elements are zero. For regressions without restrictions, this test is found to be more powerful than the DW test against negative autocorrelation, with the reverse being the case against positive autocorrelation. See King (1981). (iii) The BW test: Here ρ o =1 and Q=W in (10) (i.e., V 1 (1) is replaced by W ), where W ¼ : ð12þ B A Empirical evidence has suggested that this test has a very clear power advantage over the DW test for large values of ρ in the unrestricted regression. (iv) King's point optimal (s(ρ 1 ))test: This test corresponds to setting ρ o =ρ 1 and Q=V 1 (ρ 1 ) in (10) for some known constant ρ 1. For the unrestricted model, King (1985) showed that this test is most powerful invariant in a given region of the alternative hypothesis parameter space. By construction, this test yields a different test statistic for each choice of ρ 1. The BW and the s(ρ 1 ) tests are approximately identical as ρ Main results In this section, we consider the limiting power of the test statistics based on the quadratic form G(ρ o, Q) given in (10). Using results from Banerjee and Magnus (1999), we can write V ðqþ ¼ 1 1 q 2 LL V ð13þ p where L ¼ L o þ ul 1 þ Oðu 2 Þ; u ¼ ffiffiffiffiffiffiffiffiffiffi 1 q 2 ; L o ¼ði; 0 nðn 1Þ Þ; L 1 ¼ 0 0V, and Φ is an (n 1) 0 U

5 A.T.K. Wan et al. / Economics Letters 94 (2007) (n 1) lower triangular matrix with ones on and below the diagonal and zeros elsewhere. Write u¼ r u L, where ξ is a scalar, η is an (n 1) 1 vector, and n f Nð0;I n Þ. Then we have n g u ¼ r u ðni þ u Pg þ O p ðu 2 ÞÞ; where P =JΦ and J =(0, I n 1 ). Upon substituting (14) in (10), we obtain Gðq o ;QÞ ¼ ni þ u Pg þ O p ðu 2 Þþudðq o Þ V M V ðq o ÞQ Mðq o Þ ni þ u Pg þ O p ðu 2 Þudðq o Þ ni þ u Pg þ O p ðu 2 : ð15þ Þþudð0Þ V Mð0Þ ni þ u Pg þ O p ðu 2 Þþudð0Þ So when M (0)i 0, Gðq o ;QÞ Y p i V M Vðq o ÞQ Mðq o Þi ug : i V Mð0Þi g ð14þ ð16þ On the other hand, if M (0)i=0, then M(ρ o )i=0, B(ρ o )i=0 and hence M (ρ o )i=0. Therefore, we have Gðq o ;QÞ Y p ð Pg þ dðq o ÞÞ V M V ðq o ÞQ Mðq o Þð Pg þ dðq o ÞÞ ð Pg þ dð0þþ V ugðþ; g Mð0Þð P þ dð0þþ ð17þ when M (0)i=0. Note that (16) and (17) hold regardless of the validity of the linear restrictions Rβ=r. Combining (16) and (17), Theorem 1 follows at once. Theorem 1. Consider model (1) subject to the linear restrictions Rβ = r. The following results apply irrespective of whether r Rβ is a zero vector: 1) if M (0)i 0, then lim PrðGðq o;qþncþ ¼ 1 if g Nc qy1 0 if g bc ; 2) if M (0)i=0, then lim PrðGðq o;qþncþ ¼PrðgðgÞNcÞ; qy1 ð18þ ð19þ where c is an arbitrary constant and g and g(η) are defined in (16) and (17), respectively. Let us interpret Theorem 1. The case of M (0)i 0 arises if and only if M(0)i 0 orm(0)i=0 and B(0) i 0. Observe that if M(0)i 0 then the regression contains no intercept; if M(0)i=0 then the regression contains an intercept (or at least i lies in the column space of X). Also, B(0)i 0 if at least one restriction in Rβ=r involves the intercept. Hence, part 1) of Theorem 1, which states that the limiting power will be either one or zero, holds for both regressions with and without an intercept, 1 and in the case of the former, 1 Note that zero restriction on the intercept would imply that the restricted model is identical to an unrestricted model with no intercept, and the corresponding results are well-known from the work of Krämer (1985) and Krämer and Zeisel (1990).

6 218 A.T.K. Wan et al. / Economics Letters 94 (2007) at least one restriction must involve the intercept 2. Hence, it is possible for the tests to have zero power even with the inclusion of an intercept in the model if at least one of the linear restrictions involves the intercept, irrespective of whether the restrictions are in fact valid. Thus the existing results, which in particular have shown that the limiting power of the test cannot drop to zero or reach one when the (unrestricted) regression contains an intercept, do not apply to regression models with linear restrictions. The limiting power in part 1) is determined by the non-zero eigenvalues of the matrix [M (ρ o )QM (ρ o ) cm (0)]ii which has rank of at most one. When M (0)i 0andg,[M (ρ o )QM (ρ o ) cm (0)]ii has a unique nonzero eigenvalue whose sign determines the limiting power of G(ρ o,q); if g Nc this eigenvalue is positive and the limiting power takes the value of 1; alternatively, if g bc, this eigenvalue is negative and the limiting power is 0. In part 2) of Theorem 1, the limiting power is a constant lying strictly between zero and one. This occurs when M (0)i=0 or equivalently, M(0)i=0 and B(0)i=0, i.e., the model contains an intercept but the restrictions do not involve the intercept. Theorem 1 yields the corresponding results for the unrestricted model when m=0. In this case, M (0)=M(0) and the parts 1) and 2) of Theorem 1 give the limiting power characteristics of the tests for the regressions with no intercept and with an intercept, respectively. 4. The case of mis-specified linear models It is clear from the foregoing discussion that the four test statistics in restricted regressions do not possess the same properties as their unrestricted counterparts. Now we consider the case of mis-specified linear models. It is easily seen that a correct model with a false constraint on coefficients is equivalent to a misspecified model with a correct constraint. More precisely, the expression of G(ρ o,q) in model (1) with an incorrect constraint is algebraically equivalent to the corresponding quadratic form for the model y=xβ+δ (ρ o )+u with the correct restriction Rβ=r where δ (ρ o )=XS 1 (ρ o )R (RS 1 (ρ o )R) 1 δ 0 and δ 0 represents the degree of departure from the true constraint. More generally, the conclusions similar to Theorem 1 apply to the over-fitted linear model y=xβ+uγ+u (the true model is y=xβ+u) with the restrictions Rβ=r, or the under-fitted linear model y=xβ+u (the true model is y=xβ+uγ+u)) with the restrictions Rβ = r. In particular, our results can also be applied to the mis-specified unrestricted models. For example, our result can be applied to the following economic growth model with a structural break at t o : y t =β 1 +β 2 t+(β 3 +β 4 t)i(tnt o )+u t. The practitioner often assumes there is no break, that is, β 3 =β 4 =0 when the break is in fact there. Applying our results, when β 1 =0, the limiting power of test of the form G (ρ o,q) will be one or zero depending on g 0 u i M V ðq V o ÞQMðq o Þi Nc or bc (where X=(t imð0þi V 1,, t n ) ). On the other hand, when B 0, the limiting power will lie between one and zero. 5. Conclusions This paper has provided new insights into the limiting power of four well-known tests for AR(1) errors. We demonstrate in particular that with restricted coefficients, the probability of detecting autocorrelation 2 The case of models testing for non-zero restrictions involving the intercept often comes up in the context of the Solow growth models. These models look at the Solow residuals between countries and test for the differences in productivity growth represented by the intercept term, and then compare them with the productivity growth of the benchmark region (see Bernard and Jones, 1996). Our results in Theorem 1 suggest that diagnostic checking for AR(1) errors in such models may be prone to errors because the power of the autocorrelation tests may be low.

7 A.T.K. Wan et al. / Economics Letters 94 (2007) can fall to zero as autocorrelation becomes very severe, even when the model includes an intercept term. So the well-known results regarding the limiting power of these tests do not apply to regressions with restricted coefficients, even though the restrictions are incorrectly imposed. Our result also applies to the mis-specified linear models. Altogether, given the widespread use of these tests, in particular the Durbin Watson test, our findings should be of interest to both theorists and applied researchers. Acknowledgements The first author was supported by a competitive earmarked research grant from the Hong Kong Research Grant Council, and the second author was supported by the National Natural Science Foundation of China (Grant Nos , ). References Banerjee, A., Magnus, J.R., The sensitivity of OLS when the variance matrix is (partially) known. Journal of Econometrics 92, Bartels, R., On the power function of the Durbin Watson test. Journal of Econometrics 51, Bartels, R., Goodhew, J., The robustness of the Durbin Watson test. Review of Economics and Statistics 63, Berenblut, I.I., Webb, G.I., A new test for autocorrelated errors in the linear regression model. Journal of the Royal Statistical Society B 35, Bernard, A.B., Jones, C.I., Comparing apples to oranges: productivity convergence and measurement across industries and countries. American Economic Review 86, Blattberg, R.C., Evaluation of the power of the Durbin Watson statistic for non-first order serial correlation alternatives. Review of Economics and Statistics 55, Dufour, J.M., Exact tests and confidence sets in linear regressions with autocorrelated errors. Econometrica 58, Giles, D.E.A., Lieberman, O., Some properties of the Durbin Watson test after a preliminary t-test. Journal of Statistical Computation and Simulation 41, King, M.L., The alternative Durbin Watson test: an assessment of Durbin and Watson's choice of test statistic. Journal of Econometrics 17, King, M.L., A point optimal test for autoregressive disturbances. Journal of Econometrics 27, King, M.L., Chapter 3. Testing for autocorrelation in linear regression models: a survey. In: King, M.L., Giles, D.E.A. (Eds.), Specification Analysis in the Linear Models: Essay in honour of Donald Cochrane. Routledge, London. Krämer, W., The power of the Durbin Watson test for regressions without an intercept. Journal of Econometrics 28, Krämer, W., Zeisel, H., Finite sample power of linear regression autocorrelation tests. Journal of Econometrics 43, Schmidt, P., Guilkey, Some further evidence on the power of the Durbin Watson and Geary tests. Review of Economics and Statistics 57, Small, J.P., The limiting power of point optimal autocorrelation tests. Communications in Statistics. Theory and Methods 22, White, K.J., The Durbin Watson test for autocorrelation in nonlinear models. Review of Economics and Statistics 74, Zeisel, H., On the power of the Durbin Watson test under high autocorrelation. Communications in Statistics. Theory and Methods 18,