On Perron s Unit Root Tests in the Presence. of an Innovation Variance Break

Transcription

1 Applied Mathematical Sciences, Vol. 3, 2009, no. 27, On Perron s Unit Root ests in the Presence of an Innovation Variance Break Amit Sen Department of Economics, 3800 Victory Parkway Xavier University, Cincinnati, OH , USA sen@xavier.edu Abstract he unit root tests of Perron 1989, Econometrica were designed to have power against the stationary alternative characterized by a break in the trend function. We show that all versions of Perron s 1989 tests can be over-sized when there is a break in the innovation variance. We propose modified Perron statistics based on the GLS transformation proposed by Kim, Leybourne, and Newbold 2002, Journal of Econometrics that maintain size and have power against the trend-break stationary alternative. he modified Perron statistics weakens evidence against the unit root null for the Nelson-Plosser macroeconomic series. Mathematics Subject Classification: 62F03, 62M10, 62P20, 91B84 Keywords: Unit Root; rend Break; Innovation Break, Break Date 1. Introduction Perron s 1989 seminal paper demonstrated that the conventional Dickey-Fuller 1979 test will spuriously accept the unit root null hypothesis if the trend-stationary alternative did not allow for a break in the trend function. herefore, Perron 1989 developed unit root tests that are designed to have power against the trend-break stationary alternative. 11 hree different characterizations of the trend-break were considered, namely, a the Crash model that allows for a break in the intercept; b the Changing Growth model that allows for a break in the time trend with the two segments joined at the time of the break; and c the Mixed model that allows for a simultaneous break in the intercept and the time trend. he location of the break or break-date is assumed to be known to the practitioner. Perron s 1989 tests are based on a regression that nests the unit root 1 We consider the Innovation Outlier model in which any change in the trend function evolves in the same manner as any other shock, see Perron 1989 for further details.

2 1342 Amit Sen null and the appropriate trend-break stationary alternative hypothesis. he unit root test, denoted by t i P for i=a,b,c corresponding to the Crash, the Changing Growth, and the Mixed model respectively, is the t-test for the hypothesis that the coefficient on the first lag of the dependent variable is equal to one. Perron 1989 provides empirical evidence regarding the Nelson and Plosser 1982 macroeconomic series. In a recent paper, Kim, Leybourne, and Newbold 2002 considered the problem of testing for the presence of a unit root when there is a break in the innovation variance. 2 2 Given that Perron s 1989 statistics do not allow for a break in the innovation variance, we, in this paper, consider the limiting null behaviour of Perron s 1989 unit root tests in the presence of a break in the innovation variance. Our results indicate that the limiting null distributions of t i P i=a,b,c depend on both the break-fraction τ c, the pre-break variance σ1 2, and the post-break variance σ2 2. Simulation evidence reveals that the Perron statistics are over-sized if there is either a fall in the variance relatively early in the sample or an increase in the variance relatively late in the sample. We adapt Kim, Leybourne, and Newbold s 2002 modified GLS strategy to correct for the size distortions in t i P i=a,b,c. he limiting null distribution of our modified Perron unit root statistics, denoted by t i P i=a,b,c, in the presence of an innovation variance break are the same as those described by Perron Our simulation evidence confirms that the modified Perron statistics, t i P finite samples. i=a,b,c, maintain their size in Finally, we illustrate the use of the modified Perron statistics by testing for the presence of a unit root in the thirteen macroeconomic series contained in the Nelson and Plosser 1982 data set. Our evidence shows that, for most series, there can be substantial changes in the innovation variance across the pre-break and post-break sub-samples. Given that Perron s 1989 statistics can be over-sized in the presence of a break in the innovation variance, we re-evaluate the empirical evidence by formally incorporating this potential break in the innovation variance. Our empirical results weaken the evidence against the unit root null hypothesis considerably. Specifically, while Perron 1989 rejected the unit root null hypothesis for ten series, we are only able to reject the unit root null for four series: Real GNP, Nominal GNP, Industrial Production, and Employment. Our analysis, therefore, suggests that the practitioner should use the modified Perron statistics if a break in the innovation variance is suspected. he remainder of the paper is organized as follows. In Section 2, we discuss the data generating process, the null and alternative hypothesis, Perron s 1989 unit root tests, and their limiting null behaviour in the presence of a break in the innovation variance. 2 Kim, Leybourne, and Newbold 2002 show that the conventional Dickey-Fuller 1979 pseudo t- statistics suffer from severe size distortions. 3 For further details on the limiting distribution of the Perron 1989 statistics, see his heorem 2 on pp

3 On Perron s unit root tests 1343 In Section 3, we discuss modifications of the Perron s 1989 tests that incorporate the innovation variance break. We illustrate the use of the suggested modified Perron s 1989 tests within the context of the Nelson-Plosser series in Section 4. Section 5 contains some concluding remarks, and all proofs are relegated to an appendix. 2. Behaviour of Perron s Unit Root Statistics Perron 1989 developed unit root tests that are designed to have power against a trend-break stationary alternative. he location of the break is assumed to be known a priori. For the asymptotic results, we assume that the break-date is a constant fraction of the sample size, that is, b c = τ c with the break-fraction τ c 0,1. Perron 1989 considers the following three different characterizations of the break under the stationary alternative: Model A: y t = μ 0 + μ 1 DU c t + γd c t + μ 2 t + ρy t 1 + ɛ t 1 Model B: Model C: y t = μ 0 + μ 2 t + μ 3 D c t + ρy t 1 + ɛ t 2 y t = μ 0 + μ 1 DU c t + γdc t + μ 2 t + μ 3 D c t + ρy t 1 + ɛ t 3 where DUt c =1 t> c b is an intercept dummy, Dt c =1 t = c b +1, Dt c =t b c1 t>b c, Dt c = t 1 t> c b, and ɛ t is the error term. Model A or the Crash model includes the intercept dummy DUt c to allow for a break in the intercept, Model B or the Changing Growth model includes the trend dummy Dt c to allow for a break in the time trend with the two segments joined at b c, and Model C or the Mixed model includes an intercept dummy DUt c and a trend dummy D c t to allow for a simultaneous break in the intercept and the time trend. In the general case, k additional regressors {Δy t j } k j=1 are included in regressions 1-3 to eliminate correlation in the disturbance term. ypically, the value of the lag-truncation parameter k is unknown, and so a data-dependent method for choosing the appropriate value of k is used. For example, one may use Perron and Vogelsang s 1992 data-dependent method kt sig for selecting the lag-truncation parameter which is described in what follows. Specify an upper bound kmax for the lag-truncation parameter. he chosen value of the lag-truncation parameter k is determined according to the following general to specific procedure: the last lag in an autoregression of order k is significant, but the last lag in an autoregression of order greater than k is insignificant. he significance of the coefficient is assessed using the 10% critical values based on a standard normal distribution. Under the null hypothesis, the data generating process contains a unit root, that is: y t = μ 0 + γdt c + y t 1 + ɛ t 4 Perron 1989 presents the limiting null distribution of the unit root tests based on

4 1344 Amit Sen regressions 1-3, denoted by t i P for i=a,b,c corresponding to the Crash, the Changing Growth, and the Mixed model respectively, see heorem 2 in Perron he limiting distribution of t i P i=a,b,c are non-standard and depend on the break-date. herefore, Perron 1989 tabulates the critical values for a selection of break-dates to facilitate their use in empirical applications, see ables IV.B, V.B, and VI.B on pp in Perron We study the behaviour of t i P i=a,b,c in the presence of a break in the innovation variance which may occur under the unit root null hypothesis or under the trend-break stationary alternative. We assume that the break in the innovation variance coincides with the break in the trend-function if it exists. Following Kim, Leybourne, and Newbold 2002, we model the break in the innovation variance as: ɛ t = σ t η t with σt 2 = σ1 2 1 [t τ c ] + σ2 2 1 [t>τ c ], and η t is a martingale difference sequence with Eηt 2 η t 1,... = 1 and E η t 4+β η t 1,... =κ< for some β>0. So, the innovation variance in the pre-break sample is σ1 2 and that in the post-break sample is σ2 2. he following results characterize the limiting null distribution of t i P i=a,b,c when there is a break in the innovation variance. Let B 1 = τ c 0 W r dr, B 2 = 1 τ c W r dr, B 3 = τ c 0 rwr dr, B 4 = 1 τ c rwr dr, B 5 = τ c 0 W r 2 dr, and B 6 = 1 τ c W r2 dr. 5 heorem 1: Suppose the true data generating process for the time series {y t } t=0 is given by equations 4 and 5. he limiting distribution of t-statistic for H o : ρ = 1, denoted by t A P, based on the OLS regression 1 is given by: t A P 1 {σ1 2 τ c + σ2 2 1 τ c } VA 1 W A [2,1] VA 1 where V A is a 2 2 symmetric matrix, and W A is a 2 1 matrix given by: V A [1, 1] = 1 [1 3 τ c +3τ c 2] 12 [2,2] V A [1, 2] = σ 1 B 3 + σ 2 B τ c σ 1 B τ c σ 2 B 2 [ ] [ ] V A [2, 2] = σ1 2 B 5 τ c 1 B1 2 + σ2 2 B 6 1 τ c 1 B2 2 W A [1, 1] = 1 2 σ 1 τ c W τ c σ 2 1 τ c {W 1 W τ c } σ 1 B 1 σ 2 B 2 W A [2, 1] = 1 { 2 σ2 1 W τ c 2 τ c} τ c 1 σ1 2 W τ c B { } 2 σ2 2 W 1 2 W τ c 2 1 τ c 1 τ c 1 σ2 2 {W 1 W τ c } B 2

5 On Perron s unit root tests 1345 heorem 2: Suppose the true data generating process for the time series {y t } t=0 is given by equations 4 and 5. he limiting distribution of t-statistic for H o : ρ = 1, denoted by t B P, based on the OLS regression 2 is given by: t B P 1 {σ1 2 τ c + σ2 2 1 τ c } VB 1 W B [3,1] VB 1 where V B is a 3 3 symmetric matrix, and W B is a 3 1 matrix given by: V B [1, 1] = 1 12 V B [1, 2] = 1 [1 12 τ c +9τ c 2 +2τ c 3] 12 V B [1, 3] = σ 1 [B 3 1 ] [ 2 B 1 + σ 2 B 4 1 ] 2 B σ 1 σ 2 τ c 1 τ c W τ c V B [2, 2] = 1 12 [1 6τ c 2 +8τ c 3 3τ c 4] V B [2, 3] = σ 2 B V B [3, 3] = σ1 2 B 5 + σ2 2 B 6 σ 1 B 1 + σ 2 B 2 2 +σ 1 σ 2 2 τ c 1 τ c W τ c 2 [3,3] [ 1 τ c 2] σ 1 B 1 + σ 2 B 2 τ c σ 2 B τ c 1 τ c 2 σ 1 σ 2 W τ c +2 σ 1 σ 2 W τ c {τ c σ 2 B 2 1 τ c σ 1 B 1 } W B [1, 1] = σ 1 {τ c W τ c B 1 } 1 2 σ 1 W τ c +σ 2 {W 1 τ c W τ c B 2 } 1 2 σ 2 {W 1 W τ c } W B [2, 1] = σ 2 {W 1 τ c W τ c B 2 } 1 2 σ 1 1 τ c 2 W τ c 1 [ 2 σ 2 1+τ c 2] {W 1 W τ c } W B [3, 1] = 1 { 2 σ2 1 W τ c 2 τ c} + 1 { } 2 σ2 2 W 1 2 W τ c 2 1 τ c [σ 1 W τ c +σ 2 {W 1 W τ c }] {σ 1 B 1 + σ 2 B 2 } σ 1 σ 1 σ 2 1 τ c W τ c 2 + σ 2 σ 1 σ 2 τ c W τ c {W 1 W τ c } heorem 3: Suppose the true data generating process for the time series {y t } t=0 is given by equations 4 and 5. he limiting distribution of t-statistic for H o : ρ = 1, denoted by t C P, based on the OLS regression 3 is given by: t C P 1 {σ1 2 τ c + σ2 2 1 τ c } VC 1 W C [3,1] VC 1 [3,3]

6 1346 Amit Sen where V C is a 3 3 symmetric matrix, and W C is a 3 1 matrix given by: V C [1, 1] = 1 [1 3 τ c +3τ c 2] 12 V C [1, 2] = τ c 3 V C [1, 3] = σ 1 B 3 + σ 2 B τ c σ 1 B τ c σ 2 B 2 V C [2, 2] = τ c 3 V C [2, 3] = σ 2 B τ c σ 2 B 2 V C [3, 3] = σ 2 1 B 5 τ c 1 σ 2 1 B2 1 + σ2 2 B 6 1 τ c 1 σ 2 2 B2 2 W C [1, 1] = 1 2 σ 1 τ c W τ c σ 2 1 τ c {W 1 W τ c } σ 1 B 1 σ 2 B 2 1 W C [2, 1] 2 σ 2 1 τ c {W 1 W τ c } σ 2 B 2 W C [3, 1] = 1 { 2 σ2 1 W τ c 2 τ c} τ c 1 σ1 2 W τ c B { } 2 σ2 2 W 1 2 W τ c 2 1 τ c 1 τ c 1 σ2 2 {W 1 W τ c } B 2 he limiting null distributions of t i P i=a,b,c depends on the break-fraction τ c, the pre-break variance σ1 2, and the post-break variance σ2 2. herefore, we assess the size of t i P i=a,b,c using finite sample simulations based on the data generating process described by 4 and { 5. We set γ = 0, and μ 0 = 0. he error term is generated as follows: ɛ t = 1 [t τ c ] + σ } 2 σ 1 1 [t>τ c ] η t, and η t i.i.d.n0, 1, and used all combinations arising from τ c = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} and 2 σ 1 = {4, 2.5, 1.67, 1.25, 1, 0.8, 0.6, 0.4, 0.25}. We used 10,000 replications for two different sample sizes, namely, = {100, 200}. he empirical size of all statistics is calculated based on the critical values tabulated in Perron 1989 at the 5% significance level. he empirical size of t A P are presented in ables 1 and 2, of tb P in ables 3 and 4, and of tc P in ables 5 and 6. A general pattern emerges from these simulations which shows that the statistics are over-sized if either there is a fall in the innovation variance relatively early in the sample or if there is an increase in the innovation variance relatively late in the sample. 3. Modified Perron Unit Root ests Given that Perron s 1989 unit root statistics can be over-sized in the presence of an innovation break, we adapt the modified GLS strategy proposed by Kim, Leybourne, and Newbold 2002 to develop modified Perron tests that do not suffer such size distortions. First, we estimate the pre-break variance and the post-break variance based on the estimated residuals {ˆɛ i t} i=a,b,c from regressions 1-3 respectively as follows: ˆσ 1 i 2 = 1 τ c τ c ˆɛi 2 t 6

7 On Perron s unit root tests 1347 ˆσ 2 i 2 = 1 t=τ τ c c +1 ˆɛi 2 t Next, we apply the following transformation to the dependent variable: ỹt i = y t ˆσ 1 i 1 1 τ c + y t ˆσ 2 i 1 1 τ c +1 8 he GLS transformation based on 8 when applied to regressions 1-3 yields: μ 0 ˆσ A + μ 2 1 ˆσ A t + ρ ỹt 1 A + ɛ t 1 ˆσ 1 A ỹt A μ 0 + μ 1 + γ = ˆσ A + μ 2 2 ˆσ 2 A μ 0 + μ 1 ˆσ 2 A + μ 2 ˆσ 2 A t + ρ ỹt 1 A + ɛ t ˆσ 2 A + μ 2 ˆσ 1 B t + ρ ỹt 1 B + ɛ t ˆσ 1 B ỹt B = μ 0 ˆσ 1 B μ 0 ˆσ 2 B μ 0 ˆσ B 2 + μ 2 ˆσ B 2 + μ 2 ˆσ B 2 t + μ 3 ˆσ B 2 t + μ 3 ˆσ B 2 μ 0 ˆσ C + μ 2 1 ˆσ C t + ρ ỹt 1 C + ɛ t 1 ˆσ 1 C ỹt C μ 0 + μ 1 + γ = + μ 2 + μ 3 ˆσ 2 c ˆσ C 2 μ 0 + μ 1 ˆσ C 2 + μ 2 + μ 3 ˆσ C 2 t + ρ ỹ A t 1 + ˆσ A 1 ˆσ A 2 ˆσ A 2 + ρ ỹ B t 1 + ˆσ B 1 ˆσ B 2 ˆσ B 2 t τ c +ρ ỹ B t 1 + ɛ t ˆσ B 2 ỹ A t 1 + ɛ t ˆσ A 1 ỹ B t 1 + ɛ t ˆσ B 1 t + ρ ỹ C t 1 + ˆσ C 1 ˆσ C 2 ˆσ C 2 t + ρ ỹ C t 1 + ɛ t ˆσ C 2 ỹ C t 1 + ɛ t ˆσ C 1 if t τ c 7 if t = τ c +1 if t τ c +2 if t τ c if t = τ c +1 if t τ c +2 if t τ c if t = τ c +1 if t τ c +2 he modified unit root statistics, denoted by t i P i=a,b,c, are defined as the t-statistic for the null hypothesis H o : ρ = 1 in the following OLS regressions: ỹ i t = α 0 + α 1 DU c t + βd c t + α 2 t + α 3 D c t + ρ ỹ i t 1 + η t 9 i=a,b,c with ỹt i defined in 8. he dummy variable Dc t removes the observation corresponding to t = τ c + 1 as it does not belong to either the pre-break or post-break regimes. If k additional lagged first differences {Δy t j } k j=1 are included in regression 1-3, then additional dummy variables need to be included in regression 9 to remove the middle k +1 observations corresponding to t = {τ c +1,τ c +2,...,τ c +k +1}. Regression 9 is the same as the Mixed model regression of Perron 1989, and so the critical values of t i P i=a,b,c are the same as those of tc P given in able VI.B of Perron i=a,b,c are shown in ables are shown in ables 11 and We assessed the empirical size of the modified Perron 1989 tests, t i P using the simulation design described in Section 2. he results for t A P 7 and 8, for t B P are shown in ables 9 and 10, and for tc P

8 1348 Amit Sen 12. Based on the corresponding 5% level critical values tabulated in Perron 1989, the empirical size of t i P i=a,b,c are close to the nominal size in all cases. herefore, the modified Perron tests correct the size distortions present in the Perron tests. 4. Application to Nelson-Plosser Macroeconomic Series Perron 1989 used his statistics, t i P i=a,c, to test for the presence of a unit root in the thirteen macroeconomic series contained in the Nelson and Plosser 1982 data set. Perron 1989 specified the Crash model for all series except the Common Stock Price series and the Real Wages series. For the latter two series, Perron 1989 used the Mixed model. he empirical evidence, based on Perron s 1989 testing strategy, reveals substantial evidence in favour of the trend-break stationary alternative. Specifically, Perron 1989 rejected the unit root null hypothesis for all series except the Consumer Price Index, the Velocity, and the Interest Rate series. 4 4 However, Perron s 1989 analysis did not take into account the potential presence of a break in the innovation variance. he asymptotic distribution of Perron s 1989 statistics, together with their finite sample simulation performance, show that these statistics can be over-sized when there is a break in the innovation variance under the unit root null hypothesis. herefore, we re-evaluate the empirical evidence by allowing for the presence of a break in the innovation variance. We begin by estimating the pre-break standard deviation of the innovation term ˆσ 1 i, i=a,c, and the post-break standard deviation of the innovation term ˆσi 2, i=a,c based on regressions 1 and 3 for the Crash model and Mixed model specifications respectively. he results, shown in able 13, indicate that there may be substantial changes in the innovation variance across sub-samples as measured by the ratio of the estimated post-break innovation standard deviation to the estimated pre-break innovation standard deviation ˆσ 2 i /ˆσi 1. While the innovation variance for the Employment series and the Interest Rate series are fairly constant across sub-samples, there is an increase in the innovation variance for the Common Stock Price series and the Velocity series. For all other series, there is a decrease in the innovation variance. herefore, our preliminary evidence suggests that one should incorporate a potential break in the innovation variance when testing for the presence of a unit root. We calculate the modified Perron statistic, t i P i=a,c based on regression 9, see able 13 below. Since the asymptotic null distribution of the modified Perron statistics are the same as that of Perron s 1989 Mixed model statistic t C P, we extrapolated the critical values for t i P i=a,c based on the tabulated critical values in able VI.B of Perron We reject the unit root null hypothesis for only four series: Real GNP at the 1% level, Nominal GNP at the 5% level, Industrial Production at the 2.5% level, and Employment at the 10% level. While Perron 1989 rejected the unit root null hypothesis for ten out of the thirteen series, our evidence against the unit root null hypothesis, in comparison, is 4 he reader is referred to able VII of Perron 1989, pp for further details.

9 On Perron s unit root tests 1349 much weaker. It follows that substantially strong empirical evidence against the unit root null found by Perron 1989 is a result of his statistic being over-sized owing to a break in the innovation variance. 5. Conclusion Perron 1989 developed unit root tests that are specifically designed to have power against the stationary alternative characterized by a break in the trend function. Following Perron 1989, we consider three different characterizations of the break under the stationary alternative: the Crash model that allows for a break in the mean; the Changing Growth model that allows for a break in the time trend with the two segments joined at the time of the break; and the Mixed model that allows for a simultaneous break in the mean and the time trend parameters. We assume that the break in the innovation variance, if it occurs, coincides with the known break-date for the trend function parameters. Our assessment of the size of Perron s 1989 unit root tests show that they can suffer from serious size distortions. We show that the limiting null distribution of Perron s statistics depend on the pre-break variance, the post-break variance, and the breakfraction. Our simulation evidence reveals that, in the presence of a break in the innovation variance, the Perron tests are over-sized when there is a decrease in the innovation variance relatively early in the sample or when there is an increase in the innovation variance relatively late in the sample. herefore, we propose modified Perron tests that are based on the GLS transformation proposed by Kim, Leybourne, and Newbold he modified Perron tests have the same limiting null distribution as Perron s Mixed model statistic, and so the practitioner can use the critical values tabulated in Perron 1989 for empirical applications. Simulation evidence confirms that the modified Perron tests maintain their size in the presence of a break in the innovation variance. We illustrate the use of the modified Perron test with an application to the Nelson and Plosser 1982 macroeconomic series. Our empirical evidence shows that most series can be characterized by a break in the innovation variance, and incorporating this innovation break substantially weakens the evidence against the unit root null hypothesis compared to the evidence presented in Perron herefore, the practitioner should evaluate the innovation variance before testing for the presence of a unit root. In the eventuality that a break in the innovation variance is suspected, the practitioner should use the modified Perron test given that it maintains its size and continues to have power against the trend-break stationary alternative. Acknowledgements his research was partially supported by the D. J. O Conor Professorship grant at Xavier University.

10 1350 Amit Sen References [1] D.A. Dickey and W.A. Fuller, Distribution of the Estimator for Autoregressive ime Series With a Unit Root, Journal of the American Statistical Association, , [2].-H. Kim, S.J. Leybourne, and P. Newbold, Unit Root ests With a Break in Innovation Variance, Journal of Econometrics, , [3] C.R. Nelson and C.I. Plosser, rends and Random Walks in Macroeconomic ime Series, Journal of Monetary Economics, , [4] P. Perron, he Great Crash, the Oil Shock and the Unit Root Hypothesis, Econometrica, , [5] P. Perron and.j. Vogelsang, Nonstationarity and Level Shifts With an Application to Purchasing Power Parity, Journal of Business and Economic Statistics, , Appendix In what follows, we outline the proofs of heorems 1-3. he location of the breakdate is b c =[τc ]. All summations are taken over the sample, that is, from to unless otherwise specified. he results are based on the functional weak convergence result 1/2 [r] η t W r r [0, 1] where is W r is the Wiener Process defined on the unit interval, and denotes weak convergence. Based on the data generating process given in 1, and assuming without loss of generality that μ 0 =0,γ = 0 and y 0 =0,we can show that: c b A.1 1/2 ɛ t σ 1 W τ c A.2 1/2 t= c b +2 ɛ t σ 2 {W 1 W τ c } c b A.3 1 y t 1 ɛ t 1 2 σ2 1 {W τ c 2 τ c} A.4 1 t= c b +2 y t 1 ɛ t 1 2 σ2 2 { } W σ2 2 {W τ c 2 τ c} + σ 2 σ 1 σ 2 W τ c {W 1 W τ c }

11 On Perron s unit root tests 1351 A.5 c b 3/2 τ c y t 1 σ 1 0 W r dr A.6 3/2 t= c b +2 y t 1 σ 2 1 τ c W r dr +σ 1 σ 2 1 τ c W τ c A.7 c b 2 y 2 t 1 σ 2 1 τ c 0 W r 2 dr A.8 2 A.9 t= c b +2 y 2 t 1 σ 2 2 3/2 c b A.10 3/2 A.11 A.12 5/2 t= c b +2 te t σ 2 1 τ c W r 2 dr +σ 1 σ τ c W τ c 2 + 2σ 1 σ 2 σ 2 W τ c τ c te t σ 1 [τ c W τ c 0 c b 5/2 t= c b +2 ty t 1 σ 2 1 Regressions 1-3 can be written as: 1 τ c W r dr W r dr [ 1 ] W 1 τ c W τ c W r dr τ c ty t 1 σ 1 τ c 0 rwr dr τ c rwr dr σ 1 σ 2 ] 1 τ c 2 W τ c A.13 Y = X i θ i + ɛ with i=a,b,c corresponding to regressions 1, 2, and 3 respectively, and Y = [y t ], ɛ =[ɛ t ], X A =[1,DU t,d t,t,y t 1 ], θ A =μ 0,μ 1,γ,μ 2,ρ A, X B =[1,t,Dt,y t 1 ], θ B =μ 0,μ 2,μ 3,ρ B, X C =[1,DU t,d t,t,d t,y t 1 ], and θ C =μ 0,μ 1,γ,μ 2,μ 3,ρ C. We partition the matrix X i as [X i,1 X i,2 ] for i=a,b,c with X A,1 =[1,DU t,d t ], X A,2 = [t, y t 1 ], X B,1 = [1], X B,2 =[t, Dt,y t 1], X C,1 =[1,DU t,d t ], and X C,2 =[t, D t,y t 1 ]. It follows that regression A.13 will yield numerically equivalent results as: A.14 Y i = X i,2 θ i,2 + ɛ i for i=a,b,c where Yi, X i,2, ɛ i are respectively the projections of Y, X i,2, and ɛ on the space spanned by the columns of X i,1, θ A,2 =μ 2,ρ A, θ B,2 =μ 2,μ 3,ρ B, θ C,2 =

12 1352 Amit Sen μ 2,μ 3,ρ C. he OLS estimator of θ i,2 is equal to ˆθ i,2 = also be written as: A.15 D i, ˆθ i,2 θ i,2 = [ ] Di, 1 1 [ ] X i,2 X i,2 D 1 i, Di, 1 X i,2 ɛ i 1 Xi,2 X i,2 X i,2 Y i which can where D A, = diag 3/2,, D B, = diag 3/2, 3/2,, and D C, = diag 3/2, 3/2,. Also, the estimated error variance from regression 1-3 are given by: A.16 ˆσ 2 i = 1 n i for i=a,b,c, n 1 = 2, and n 2 = n 3 =3. { [ ] ɛ i ɛ i Di, 1 [ ] X i,2 ɛ i Di, 1 1 [ ] } X i,2 X i,2 D 1 i, Di, 1 X i,2 ɛ i Define the n i n i symmetric matrix V i, = Di, 1 X i,2 X i,2 D 1 i,, and the n i 1 matrix W i, = Di, 1 X i,2 ɛ i for i=a,b,c. We can show that: τ c V A, [1, 1] = 3 t t t t 2 2 τ c V A, [1, 2] = 5/2 ty t 1 τ c 1 t 1 3/2 y t 1 + 5/2 ty t 1 1 t 2 3/2 y t 1 τ c V A, [2, 2] = 2 τ c 2 yt τ c 1 y t yt τ c 1 1 y t 1 W A, [1, 1] = W A, [2, 1] = τ c 3/2 tɛ t τ c 1 t 1 1/2 ɛ t + 3/2 tɛ t 1 t 2 1/2 ɛ t τ c 1 τ c y t 1 ɛ t τ c 1 3/2 τ c y t 1 1/2 ɛ t + 1 y t 1 ɛ t 1 τ c 1 3/2 y t 1 1/2 ɛ t

13 On Perron s unit root tests 1353 V B, [1, 1] = 3 t t 2 τ c V B, [1, 2] = 3 t t0 t t tt τ c t 3 t=τ c +1 V B, [1, 3] = 5/2 ty t /2 y t 1 2 τ c V B, [2, 2] = 3 0 t t τ c t 3 2 t=τ c +1 V B, [2, 3] = 5/2 ty t τ c τ c +1 3/2 y t t=τ c +1 τ c 3/2 y t 1 + τ c 1 τ c 3/2 y t 1 t=τ c +1 V B, [3, 3] = 2 2 yt 1 2 3/2 y t 1 W B, [1, 1] = 5/2 tɛ t /2 ɛ t 2 W B, [2, 1] = 3/2 τ c tɛ t + 1/2 [ ɛ t τ c 2 1 ] 2 1 τ c W B, [3, 1] = t=τ c /2 ɛ t [{ τ c 1 } 2 1 τ c 2 t=τ c +1 1 y t 1 ɛ t 1/2 ɛ t 3/2 1 ] 2 1 τ c y t 1 τ c V C, [1, 1] = 3 V C, [1, 2] = 3 t t t t 2 2 t t 2 2 V C, [1, 3] = τ c 5/2 ty t 1 τ c 1 t 1 3/2 y t 1

14 1354 Amit Sen and 5/2 V C, [2, 2] = 3 V C, [2, 3] = V C, [3, 3] = ty t 1 t t t 2 3/2 y t 1 5/2 ty t 1 1 t 2 3/2 y t 1 τ c 2 τ c 2 yt 1 2 τ c 1 3/2 y t 1 2 y 2 t 1 1 τ c 1 2 3/2 y t 1 W C, [1, 1] = W C, [2, 1] = W C, [3, 1] = τ c 3/2 tɛ t 1 t τ c 1 1/2 ɛ t 3/2 tɛ t 1 t 2 1/2 ɛ t 3/2 tɛ t 1 t 2 1/2 ɛ t τ c 1 τ c y t 1 ɛ t τ c 1 3/2 τ c y t 1 1/2 ɛ t 1 y t 1 ɛ t 1 τ c 1 3/2 y t 1 1/2 ɛ t where t 1 =τ c 1 τ c t, t 2 = τ c 1 1 t, t = 1 t, and t 3 = 1 τ c t. Based on the limiting behaviour of the moments in A.1-A.12, tedious calculations imply that V i, V i, and W i, W i for i=a,b,c as described in heorems 1-3, and so expression A.15 implies that: A.17 D i, ˆθ i,2 θ i,2 Vi 1 W i Also, equation A.16 implies that ˆσ i 2 is asymptotically equivalent to 1 ɛ i ɛ i = 1 τ c ɛ t ɛ ɛ t ɛ 2 2 where ɛ 1 =τ c 1 τ c ɛ t and ɛ 2 = τ c 1 1 ɛ t. herefore,

15 On Perron s unit root tests 1355 A.18 ˆσ 2 i τ c σ τ c σ 2 2 for i=a,b,c. Finally, the limiting distribution of expressed as: varˆθ ˆ i,2 =ˆσ i 2 X i,2 X i,2 1 can be [ ] A.19 D i, varˆθ ˆ i,2 D i, { τ c σ τc σ2 2 } V 1 i i=a,b,c based on re- herefore, the limiting null distribution of the t-statistic t i P gressions 1-3 respectively, namely: t i P = ˆρ i 1 ˆσ i 2 2 Xi,2 X i,2 1 [n i,n i ] can be obtained by combining expressions A.17, A.18, and A.19.

16 1356 Amit Sen able 1. Size of t A P with sample size = 100, μ 0 = γ = able 2. Size of t A P with sample size = 200, μ 0 = γ = able 3. Size of t B P with sample size = 100, μ 0 = γ =

17 On Perron s unit root tests 1357 able 4. Size of t B P with sample size = 200, μ 0 = γ = able 5. Size of t C P with sample size = 100, μ 0 = γ = able 6. Size of t C P with sample size = 200, μ 0 = γ =

18 1358 Amit Sen able 7. Size of t A P with sample size = 100, μ 0 = γ = able 8. Size of t A P with sample size = 200, μ 0 = γ = able 9. Size of t B P with sample size = 100, μ 0 = γ =

19 On Perron s unit root tests 1359 able 10. Size of t B P with sample size = 200, μ 0 = γ = able 11. Size of t C P with sample size = 100, μ 0 = γ = able 12. Size of t C P with sample size = 200, μ 0 = γ =

20 1360 Amit Sen able 13. Modified Perron Unit Root ests for the Nelson-Plosser Data Series Sample Form-of Series Period -Break τ c k ˆσ 1 i ˆσ 2 i ˆσ 2 i /ˆσi 1 t i P Real GNP A a Nominal GNP A c Real Per Capita GNP A Industrial Production A b Employment A d GNP Deflator A Consumer Prices A Nominal Wages A Money Stock A Velocity A Interest Rate A Common Stock Prices C Real Wages C Note: he small letters in parenthesis that appear as superscript indicate the significance of these statistics. he letters a, b, c, and d indicate significance with respect to the asymptotic critical values at the 1%, 2.5%, 5%, and 10% significance level respectively. he asymptotic critical values of t i P were extrapolated from able VI.B of Perron 1989 based on the break-fraction. Received: November, 2008