American Journal of Theoretical and Applied Statistics 2013; 2(6): 293-298 Published online January 30, 2014 (http://www.sciencepublishinggroup.com/j/ajtas) doi: 10.11648/j.ajtas.20130206.28 A new criterion for lag-length selection in unit root tests Agunloye, Oluokun Kasali*; Arnab, Raghunath; Shangodoyin, Dahud Kehinde Department of Statistics, University of Botswana, Gaborone, Botswana Email address: [email protected] (Agunloye O. K.) To cite this article: Agunloye, Oluokun Kasali; Arnab, Raghunath; Shangodoyin, Dahud Kehinde. A New Criterion for Lag-Length Selection in Unit Root Tests. American Journal of Theoretical and Applied Statistics. Vol. 2, No. 6, 2013, pp. 293-298. doi: 10.11648/j.ajtas.20130206.28 Abstract: This paper examines lag selection problem in unit root tests which has become a major specification problem in empirical analysis of non-stationary time series data. It is known that the implementation of unit root tests requires the choice of optimal truncation lag for good power proper ties and it is equally unrealistic to assume that the true optimal truncation lag is known a prior to the practitioners and other applied researchers. Consequently, these users rely largely on the use of standard information criteria for selection of truncation lag in unit root tests. A number of previous studies have shown that these criteria have problem of over-specification of truncation lag-length leading to the well-known low power problem that is commonly associated with most unit root tests in the literature. This paper focuses on the problem of over-specification of truncation lag-length within the context of augmented Dickey-Fuller (ADF) and generalized least squares Dickey-Fuller (DFGLS)unit root tests. In an attempt to address this lag selection problem, we propose a new criterion for the selection of truncation lag in unit root tests based on Koyck distributed lag model and we show that this new criterion avoids the problem of over-specification of truncationlag-length that is commonly associated with standard information criteria. Keywords: Truncation Lag, Information Criteria, Koyck Distributed Lag Model, Unit Root Test, Low Power, Partial Correlation Coefficient 1. Introduction Testing for the presence of unit root in time series data is a major precondition in any cointegration analysis and other empirical research using time series data. Determination of appropriate truncation lag is quite a challenging aspect of unit root testing. A number of previous studies such as [1], [2], [3],[4],[5] and [6] have shown that there is a strong connection between truncation lag and the empirical power of unit root tests. Among numerous unit root tests proposed in the literature, the augmented Dickey–Fuller (ADF) test introduced by [7] and the generalized least squares Dickey-Fuller (DF-GLS) test introduced by [8] appears to be the most popular unit root tests among applied researchers. Both ADF and DF-GLS unit root tests are well-formulated to handle any possible serial correlations in the error terms of the Dickey–Fuller regressions by augmenting the regressions with lagged differences of the original series. However, the empirical implementation of the ADF and DF-GLS unit root tests requires the inclusion of appropriate number of lagged differences in the Dickey–Fuller regression, which is commonly referred to as lag selection problem in the literature. In practice, lag-lengths are commonly selected by two different lag selection techniques such as general- to-specific criterion proposed by [9] which is based on some sequential t-test procedure and standard information criteria such as Akaike information criterion (AIC), Final Prediction Error (FPE),Bayesian information criterion(BIC) and Hannan-Quinn information criterion (HQIC) to mention only a few but these criteria are generally wellknown to have problem of over-fitting the truncation lags in unit root tests leading to low power in unit root tests.[2],[9] and [10]compared the performance of ADF test under these two lag-selection techniques and showed that sequential ttest outperformed standard information criteria.[11] compared the performance of DF-GLS test under modified Akaike information criteria (MAIC) and sequential t-testing procedure and show that the latter perform better than former. In an attempt to address the well-known problem of low power of ADF unit root test occasioned by overspecification of truncation lag-length, [8] introduced a class of efficient unit root tests called generalized least squares Dickey-Fuller(DF-GLS) test using ADF test based on GLSdetrending. Although, DF-GLS test was shown to be more powerful than the ADF test but it also inherits the lag length selection problem since it is a modification of the original ADF test. Hence, DF-GLS test is also characterized by low power problem due to over-specification of truncation lag-length. Another noticeable problem of 294 Agunloye, Oluokun Kasali et al.: A New Criterion for Lag-Length Selection in Unit Root Tests standard information criteria is the fact that different lag selection criteria choose different optimal truncation laglengths for the same dataset. This multiple suggestions of possible truncation lags by different information criteria raises a fundamental question as to on which particular information criterion could we rely upon for the choice of optimal truncation lag-length. To date, there exists no operational procedure for selecting optimal truncation laglength for unit root tests that gives the best results. Hence, this paper focuses on the choice of optimal truncation laglength for the ADF family of unit root tests and DF-GLS test since these are the most widely used unit root tests in empirical analysis. yt( 2) yt( 3) yt( 4) 2. Standard Information Criteria In this study, we considered four standard information criteria for lag selection in unit root testing. The procedure is to fit autoregressive model on yt( ) ,i = 1,....,4 for order i ranging from 1 to 9 and subsequently obtaining the quantities to compute the values of the criteria. The following lag selection methods are considered: i.[12]: Final Prediction Error (FPE) ( ) FPE ( k ) = ln σˆ k2 + 2 (1 + k ) N −k ii.[13]: Akaike Information Criterion(AIC) AIC ( k ) = ln σˆ k2 + 2 (1 + k ) N iii.[14]:Bayesian Information Criterion (BIC) ( ) BIC ( k ) = ln σˆ k2 + (1 + k ) ln N N iv.[15]: Hannan-Quinn Information Criteria (HQIC) ( )+ HQIC ( k ) = ln σˆ 2 k j =0 k = ∆yt − α − ρ yt −1 = ∑ γ j ∆yt − j + ε 2t j =0 k = ∆yt − α − β t − ρ yt −1 = ∑ γ j ∆yt − j + ε 3t j =0 k = ∆ytd − α − γ t − ρ ytd−1 = ∑ δ j ∆ytd− j + ε 4t j =0 k yt(1) = ∆yt − ρ yt −1 = ∑ γ j ∆yt − j + ε1t Where yt(1) is ADF Model I with no constant and no trend yt( 2) is ADF Model IIwith constant but no trend yt( ) is ADF Model III with both constant and trend 3 yt( 4) is DF-GLS with both constant and trend ε ti , i = 1,..., 4 are white noise error terms, k is the truncation lag-length to be determined empirically, γ j and δ j are coefficients of differenced lagged values. The R.H.S of equation (1) is similar to typical distributed lag models used in econometric modeling. [16] has proposed an ingenious method of specifying the lag in distributed lag model by assuming that the coefficients of γ ’s and i ’s on the R.H.S of equation (1) are all of the same sign and declines geometrically, then without loss of generality we can assume Koyck postulations as follows: Let λ k = δ k N λk = λ0 R(ki ) We consider a distributed lag representation of ADF and DF-GLS regression models in the form: k (2) k = 0,1, i = 1,....,4 (3) Where R is the partial correlation coefficient between i yt( ) /i = 1,..., 4 and ∆yt − j or ∆ytd− j at which 3. A New Approach to Selection of Truncation Lag in Unit Root Tests or γ then we have k to ascertain if these conventional lag selection criteria either over-fit or under-fit the truncation lags by comparing the lag-length suggested by these information criteria with lag-length suggested by the new lag selection criterion. δ i 2 (1 + k ) ln ( ln ( N ) ) Where k is the lag-length selected, N is the sample size and σˆ 2 is the MLE of the model residual. Our objective is (1) yt(i ) depends the rate of decline of on either λk ’s and measures the rate ∆yt − j or ∆ytd− j and for we take R i such that 0 < R( i ) < 1 () is the indicator of decay of the distributed lag and 1 − R( i) is the speed of adjustment[17]. Equation(2) fundamentally postulates that each λ coefficient is a measure of dependence of y (i ) on either t American Journal of Theoretical and Applied Statistics 2013; 2(6): 293-298 ∆yt − j or λ and each successive ∆ytd− j coefficient is numerically less than each preceding λ ,this implies that as one goes back into the distant past the effect of that lag on y (i ) becomes progressively smaller. t To obtain the mean lag, we ensure that the long-run multiplier is finite That is the sum of λ is defined as k ∞ ∑ k =0 295 interval [ 0.5, 0.999 ) (ii)The simple linear regression model will be fitted to the L.H.S of equation (1) to generate the parameters needed and to compute yt(i ) , ∀ i = 1,.........,4 (iii)If R(i ) < 0.3 , then the mean lag will be assumed to be R(i) zero since 1− R for R(i ) < 0.3 is a fraction not up to 0.5. (i ) (iv)The partial correlation coefficient λk = λ0 ∑ R (ki ) (4) R(i) should be computed such that we include all variables and adjusted for maximum lag until it gives a value less than 0.3 but ∑ R (ki ) = 4. Power of Test for R 1 1 − R i) ( (5) We define power of test for R as follows: (i ) is the inverse of speed of adjustment. Therefore, the Mean lag is L i = ∑ k λk k ∑ λk = ∑ λk λ 0 1 − ∑ k λ 0 Rk i = λ0 −1 1 − R i R i −1 (6) i define the test statistic as k =0 i We reject the null hypothesis of R = 1 and hence leads to (i ) undefined mean lag L at a specified level of significance if Z < − Zα .The Power of possible mean lag is the α 2 (7) R(i) < 1 = R∗ Thus we have So using (7) in (6) gives R L(i) = (i) ,i =1,.....,5 1− R(i) The notation R(i) P R ∗ = P Z < − Zα / R(i ) = R ∗ (8) k decline in λ .This ensures that, to have reasonable mean k L( i ) we expect the absolute value of R to be in the (10) This reduces to indicates that we generate different R’s for each of the models described in equation (1) and we shall demonstrate the power of test for R for different (i ) values of ρ = 0.001,0.5, and 0.999 . The following are the immediate consequences of equation (8) : (i).The closer R is to +1 ,the slower the rate of decline in λ ,whereas the closer it is to zero, the more rapid the lag ( probability of rejecting H when H is true i.e when 0 1 R(i ) 1 − R(i ) (9) V Rˆ i ) (i ) ∞ Rˆ(i ) − 1 Z= ∑ kR ki Applying arithmetic-geometric progression rule to (6) we obtain the sum to infinity for the term kR k to be ∑ kR(k) = R(i) is normally distributed, then we Assuming that = 1 − R H0 : R(i) =1 against H1 : R(i) <1 P( = P ∗ R = P ) Rˆ i () V Rˆ i ( ) Rˆ(i ) −1 V Rˆ(i ) < −Zα + < − Zα / R(i ) = 1 V Rˆ i ( ) ∗ R / R i = R∗ () (11) 296 Agunloye, Oluokun Kasali et al.: A New Criterion for Lag-Length Selection in Unit Root Tests = P Rˆ i − R∗ 1 R∗ () ∗ < −Zα + − / Ri = R () V Rˆ i V Rˆ i V Rˆ i ( ) ( ) ( ) yt( 2) = P Z < −Zα + = P Z 1− R = P Z ˆ V Ri ( ) ( ∗ ) < −Zα − R −1 V Rˆ i ( ) ( ∗ where α1 , α 2 regression model values of and ( 2) yt ρ .With α3 are obtained by fitting of a equals to a constant for different values of α1 , α 2 and α3 therefore proceed to generate the series for ∗ R −1 V Rˆ i ( ) ( ) ∆yt − α1 − 0.001* yt −1 = b1 , ρ = 0.001 ρ = 0.5 = ∆yt − α 2 − 0.5* yt −1 = b2 , ∆yt − α 3 − 0.999* yt −1 = b3 , ρ = 0.999 ) known ,we b1 , b2 and (12) b3 .Thereafter, we compute partial correlation coefficient between : b1 and different choices of variables from the set After some algebraic manipulation of Equation (11) we have Equation (12) which defines the power of test for R while controlling for the effects of other remaining independent variables. We also repeat the same procedure for b2 and b3 > Zα + of independent variables (i ) ∆yt , ∆yt −1 , ∆yt −2 ,......, ∆yt −12 For ADF MODEL III, k yt(3) = ∆yt − α − β t − ρ yt −1 = ∑ γ j ∆yt − j + ε t 3 , 5. Empirical Illustration j =0 For empirical illustration, we fit simulated data to the distributed lag specifications of Dickey-Fuller regression models of ADF and DF-GLS unit root tests as follows: we set ρ = 0.001, 0.5 and 0.999 in following representations: k For ADF MODEL I , yt(1) = ∆yt − ρ yt −1 = ∑ γ j ∆yt − j + ε t1 j =0 We set ρ = 0.001,0.5 and 0.999 in following representations: yt(1) ∆yt − 0.001* yt −1 = a1 , = ∆yt − 0.5* yt −1 = a2 , ∆yt − 0.999* yt −1 = a3 , We generate series of yt( ) to have the 1 ρ = 0.001 3) We ( 3) ∆yt − α1 − β1t − 0.001* yt −1 = d1 , = ∆yt − α 2 − β 2t − 0.5* yt −1 = d 2 , ∆yt − α 3 − β 3t − 0.999* yt −1 = d3 , estimate the 3) to have the ρ = 0.001 ρ = 0.5 ρ = 0.999 regression equation , for values of yt = α + β * t + ∆yt + ρ * yt −1 ρ = 0.001,0.5 and 0.999 .Thereafter, ρ = 0.5 we proceed to generate series for d1 , d 2 and d3 .We compute partial ρ = 0.999 correlation coefficients between: d1 and different choices a1 , a2 and a3 for different values ρ = 0.001, 0.5 and 0.999 .We compute partial correlation coefficients between : a1 and different choices of variables from the set of independent variables ∆yt , ∆yt −1 , ∆yt − 2 ,......, ∆yt −12 while controlling for the effects of other remaining independent variables. We also repeat the same procedure for a2 and a3 k For ADF MODEL II , y ( 2) = ∆y − α − ρ y = γ ∆y + ε , ∑ j t− j t2 t t t −1 j =0 we set ρ = 0.001,0.5 and 0.999 in following representations: yt( yt( ( 2) yt of variables from the set of independent variables ∆yt , ∆yt −1 , ∆yt −2 ,......, ∆yt −12 while controlling for the effects of other remaining independent variables. We also repeat the same procedure for d 2 and d 3 For DF-GLS unit root test, k yt( 4) = ∆ytd − α − γ t − ρ ytd−1 = ∑ δ j ∆ytd− j + ε t 4 , j =0 we set ρ = 0.001,0.5 and 0.999 in following representations: to have the yt( 4) ∆ytd − α1 − γ 1t − 0.001* ytd−1 = e1 , = ∆ytd − α 2 − γ 2t − 0.5* ytd−1 = e2 , ∆ytd − α 3 − γ 3t − 0.999* ytd−1 = e3 , yt( 4) to have ρ = 0.001 ρ = 0.5 ρ = 0.999 the American Journal of Theoretical and Applied Statistics 2013; 2(6): 293-298 We ( 4) estimate the regression equation for values of yt = α + γ * t + ∆y + ρ * y ρ = 0.001, 0.5 and 0.999 .Thereafter, we d t d t −1 , is higher compared with ADF tests at lag 0 and lag 1 respectively. Table 2.POWER OF TEST FOR R WHEN ρ proceed to generate series for e1 , e2 and e3 .We compute partial correlation coefficients between: e1 and different choices of variables from the set of independent variables ∆yt , ∆yt −1 , ∆yt −2 ,......, ∆yt −12 while controlling for the effects of other remaining independent variables. We also repeat the same procedure for e2 and e3 6. Power of Test for ADF and DF-GLS Unit Root Tests under the New Lag Selection Criterion In this section, we discuss the power of test for R for the ADF and DF-GLS unit root tests based on the optimal truncation lag selected by the new criterion. TABLE 2A Lag R p-value Power of Test ADF MODEL I 0 0.785 0.000 0.9535 ADF MODELII 0 0.785 0.000 0.9535 ADF MODEL III 0 0.785 0.000 0.9535 DF-GLS 0 0.815 0.000 0.9607 TABLE 2B Unit Root Test Lag R p-value Power of Test ADF MODEL I 1 0.062 0.051 0.3771 ADF MODELII 1 0.062 0.051 0.3771 ADF MODEL III 1 0.062 0.051 0.3771 DF-GLS 1 0.412 0.022 0.6237 Table 3.POWER OF TESTFOR R WHEN ρ = 0.999 TABLE 3A TABLE 1A Lag R p-value Power of Test ADF MODEL I 0 0.652 0.000 0.6118 ADF MODELII 0 0.652 0.000 0.6118 ADF MODEL III 0 0.652 0.000 0.6118 DF-GLS 0 0.752 = 0.5 Unit Root Test Table 1.Power of Test for R When ρ = 0.001 Unit Root Test 297 0.000 0.7563 Unit Root Test Lag R p-value Power of Test ADF MODEL I 0 0.850 0.000 0.9871 ADF MODELII 0 0.850 0.000 0.9871 ADF MODEL III 0 0.850 0.000 0.9871 DF-GLS 0 0.883 0.000 0.9924 TABLE 3B TABLE 1B Unit Root Test Lag R p-value Power of Test ADF MODEL I 1 0.054 0.094 0.3423 ADF MODELII 1 0.054 0.094 0.3423 ADF MODEL III 1 0.054 0.094 0.3423 DF-GLS 1 0.374 0.014 0.4475 From Tables 1A and 1B above, it is obvious that the partial correlation coefficient denoted by R is significant at zero lag but not significant at lag 1.Hence,for ρ = 0.001 , lag 1may be pre-specified as maximum lag for these unit root tests but optimal truncation lag is 0 because the power of test for R is higher at lag 0 than at lag 1.From both tables it is also seen that the power of test for R under DF-GLS test is higher compared with ADF tests at lag 0 and lag 1 respectively. From Tables 2A and 2B above, it is obvious that the partial correlation coefficient denoted by R is significant at zero lag but not significant at lag 1.Hence,for ρ = 0.5 , lag 1may be pre-specified as maximum lag for these unit root tests but optimal truncation lag is 0 because the power of test for R is higher at lag 0 than at lag 1.From both tables it is also seen that the power of test for R under DF-GLS test Unit Root Test Lag R p-value Power of Test ADF MODEL I 1 0.066 0.038 0.4317 ADF MODELII 1 0.066 0.038 0.4317 ADF MODEL III 1 0.066 0.038 0.4317 DF-GLS 1 0.312 0.013 O.5978 From Tables 3A and 3B above, it is obvious that the partial correlation coefficient denoted by R is significant at zero lagbut not significant at lag 1.Hence,for ρ = 0.999 , lag 1may be pre-specified as maximum lag for these unit root tests but optimal truncation lag is 0 because the power of test for R is higher at lag 0 than at lag 1.From both tables it is also seen that the power of test for R under DF-GLS test is higher compared with ADF tests at lag 0 and lag 1 respectively. 7. Lag-Selection by Conventional Information Criteria In order to evaluate the performance of standard information criteria, we fitted auto regression of order 1 to 9 to our simulated data and these information criteria select the following truncation lag as represented in table 4 below: 298 Agunloye, Oluokun Kasali et al.: A New Criterion for Lag-Length Selection in Unit Root Tests Table 4. LAG-SELECTION BY CONVENTIONAL INFORMATION CRITERIA MODEL AR (1) AR ( 2 ) AR ( 3 ) AR ( 4 ) AR ( 5 ) AR ( 6 ) AR ( 7 ) AR ( 8 ) AR ( 9 ) RMSE σˆ k2 AIC FPE BIC HQIC 2.38 2.36 2.34 2.34 2.35 2.32 2.33 2.33 2.33 5.6787 5.5649 5.4943 5.4897 5.5037 5.4242 5.4382 5.4010 5.4103 1.7496 1.7358 1.7296 1.7353 1.7445 1.7366 1.7459 1.7458 1.7543 1.7496 1.7360 1.7299 1.7358 1.7452 1.7375 1.7471 1.7474 1.7563 1.7736 1.7720 1.7779 1.7959 1.8174 1.8218 1.8435 1.8558 1.8769 1.7592 1.7503 1.7489 1.7595 1.7736 1.7706 1.7849 1.7898 1.8033 Table 4 has autoregressive (AR) models of order 1 to 9 in the first column and root mean square error (RMSE) for each model in the second column. The optimal truncation lag for a particular criterion is the lag that minimizes the value of that criterion. Hence, the optimal truncation lag chosen by AIC, FPE and HQIC is 3 respectively whilst BIC picked 2.This is clearly an over-specification of truncation lag when compared with truncation lag suggested by our new criterion. 8. Conclusion In this paper, we have highlighted the lag selection problem in the context of ADF and DF-GLS tests based on standard information criteria which we have shown to overestimate the truncation lag leading to the well-known problem of low-power associated with unit root tests. Given the persistent problem of over-specification of truncation lag by data-dependent standard information criteria, we introduced a new lag selection criterion based on Koyck distributed lag model where truncation lag is specified as a deterministic function of the partial correlation coefficient between dependent variable and different choices of independent variables of distributed lag specifications of ADF and DF-GLS unit root tests. 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