ESSAYS IN TIME SERIES ECONOMETRICS By Nasreen Nawaz A DISSERTATION Submitted to Michigan State University in partial ful…llment of the requirements for the degree of Economics –Doctor of Philosophy 2015 ABSTRACT ESSAYS IN TIME SERIES ECONOMETRICS By Nasreen Nawaz In the …rst chapter, We focus on the estimation of the ratio of trend slopes between two time series where is it reasonable to assume that the trending behavior of each series can be well approximated by a simple linear time trend. We obtain results under the assumption that the stochastic parts of the two time series comprise a zero mean time series vector that has su¢ cient stationarity and has dependence that is weak enough so that scaled partial sums of the vector satisfy a functional central limit theorem (FCLT). We compare two obvious estimators of the trend slope ratio and propose a third biascorrected estimator. We show how to use these three estimators to carry out inference about the trend slope ratio. When trend slopes are small in magnitude relative to the variation in the stochastic components (the trend slopes are small relative to the noise), we …nd that inference using any of the three estimators is compromised and potentially misleading. We propose an alternative inference procedure that remains valid when trend slopes are small or even zero. We carry out an extensive theoretical analysis of the estimators and inference procedures with positive …ndings. First, the theory points to one of the three estimators as being preferred in terms of bias. Second, the theory unambiguously suggests that our alternative inference procedure is superior both under the null and under the alternative with respect to the magnitudes of the trend slopes. Finite sample simulations indicate that the predictions made by the asymptotic theory are relevant in practice. We give concrete and speci…c advice to empirical practitioners on how to estimate a ratio of trend slopes and how to carry out tests of hypotheses about the ratio. The second chapter is an extension of the …rst, where the stationarity assumption in the analysis is relaxed. It is assumed that the stochastic parts of the trending series follow an I(1) process. We consider the case of unit root in the noise term in the IV regression equation. We also consider the case of cointegration between the two series. The theory explicitly captures the impact of the magnitude of the trend slopes on the estimation and inference about the trend slopes ratio. If the trend slopes are relatively large in magnitude, the IV estimator is consistent for both I(1) and I(0) regression errors. For medium and small trend slopes, the IV estimator is inconsistent for I(1) case, but consistent for I(0) regression error. For inference, the test based on IV estimator has been compared with the alternative testing approach. Asymptotic theory and …nite sample simulations suggest that the alternative testing approach is superior both under the null and under the alternative with respect to the magnitudes of the trend slopes. Whether the noise term in the IV regression equation is I(0) or I(1) has an impact on the power performance of the test for the trend slopes ratio. The third chapter is an empirical application of the methodology developed in the …rst and the second chapters. The empirical …ndings on convergence of per capita income across regions in convergence literature are mixed. There is evidence of convergence in a substantial number of cases, whereas evidence contrary to convergence has also been found. Where there is -convergence found, it is interesting to come up with a measure of speed of convergence and estimate it. The speed of convergence has been shown to be proportional to a ratio of trend slopes, and using the methodology developed in the …rst and the second chapters, we estimate this ratio for all US regions which are converging. The higher the ratio, the greater is the speed of convergence. TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii 1 ESTIMATION AND INFERENCE OF LINEAR TREND SLOPE RATIOS WITH I(0) ERRORS (with Timothy J. Vogelsang). . . . . . . .1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Model and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 1.2.1 Model and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Estimation of the Trend Slope Ratio . . . . . . . . . . . . . . . . . . . . . . . .5 1.2.3 Asymptotic Properties of OLS and IV . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Bias Corrected OLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 1.2.5 Asymptotic Properties of Estimators for Small Trend Slopes . . . . . . . . . 8 1.2.6 Implications (Predictions) of Asymptotics for Finite Samples . . . . . . . . 10 1.3 Finite Sample Means and Standard Deviations of Estimators . . . . . . . . . 10 1.4 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Linear in Slopes Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Con…dence Intervals Using t 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.3 Asymptotic Results for t-statistics . . . . . . . . . . . . . . . . . . . . . . . . .18 1.5 Finite Sample Null Rejection Probabilities and Power . . . . . . . . . . . . . .24 1.6 Practical Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7 Conclusion and Directions for Future Research . . . . . . . . . . . . . . . . . . 27 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 ESTIMATION AND INFERENCE OF LINEAR TREND SLOPE RATIOS WITH I(1) ERRORS . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2 The Model and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Model and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.2 Estimation of the Trend Slope Ratio . . . . . . . . . . . . . . . . . . . . . . . .52 2.2.3 Asymptotic Properties of IV when t ( ) is an I(1) Process . . . . . . . . . . 53 2.2.4 Asymptotic Properties of IV when t ( ) is an I(0) Process . . . . . . . . . . 54 2.2.5 Implications (Predictions) of Asymptotics for Finite Samples . . . . . . . . 55 2.3 Finite Sample Means and Standard Deviations of IV . . . . . . . . . . . . . . 55 2.4 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.1 Linear in Slopes Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.2 Asymptotic Results for t-statistics . . . . . . . . . . . . . . . . . . . . . . . . .58 2.5 Finite Sample Null Rejection Probabilities and Power . . . . . . . . . . . . . .62 2.6 Unit Root Tests for t ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 2.7 Finite Sample Null Rejection Probabilities and Power of Unit Root Tests . . 70 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 iv 3 SPEED OF ECONOMIC CONVERGENCE OF U.S. REGIONS. . . . . . . . 110 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 3.3 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 v LIST OF TABLES Table 1.1: Finite Sample Means and Standard Deviations. . . . . . . . . . . . . . 43 Table 1.2: Empirical Null Rejection Probabilities, 5% Nominal Level. . . . . . . 44 Table 1.3: Finite Sample Proportions of Con…dence Interval Shapes Based on t 0 46 Table 1.4: Finite Sample Power, 5% Nominal Level, T=100, Two-sided Tests. . .48 Table 2.1a: Finite Sample Mean and Standard Deviation, t I(1) . . . . . . . 102 Table 2.1b: Finite Sample Mean and Standard Deviation, t I(0) . . . . . . . 103 Table 2.2a: Empirical Null Rejection Probabilities, 5% Nominal Level, t I(1)104 Table 2.2b: Empirical Null Rejection Probabilities, 5% Nominal Level, t I(0)105 Table 2.3a: Finite Sample Power, 5% Nominal Level, Table 2.3b: Finite Sample Power, 5% Nominal Level, I(1) . . . . . . . . . . 106 t t I(0) . . . . . . . . . .107 Table 2.4: Finite Sample Performance of ADF and ADF-GLS Test for bt . . . . 108 Table 2.5: Finite Sample Power of ADF and ADF-GLS Unit Root Test for bt . .109 Table 3.1: Regression Results for Regions against Time. . . . . . . . . . . . . . .120 Table 3.2a: IV Point Estimates and the 95% Con…dence Intervals. . . . . . . . .121 Table 3.2b: IV Point Estimates and the 95% Con…dence Intervals (Post 1946). .121 Table 3.2c: IV Point Estimates and the 95% Con…dence Intervals (Post 1973). .122 Table 3.3a: Pairwise IV Point Estimates and the 95% CIs. . . . . . . . . . . . . .123 Table 3.3b: Pairwise IV Point Estimates and the 95% CIs (P1946). . . . . . . .124 Table 3.3c: Pairwise IV Point Estimates and the 95% CIs (P1973). . . . . . . . 125 vi LIST OF FIGURES Figure 1: Natural Log of Per Capita Income of US Regions. . . . . . . . . . . . .119 vii 1 ESTIMATION AND INFERENCE OF LINEAR TREND SLOPE RATIOS WITH I(0) ERRORS (with Timothy J. Vogelsang) 1.1 Introduction A time trend refers to systematic behavior of a time series that can approximated by a function of time. Often when plotting macroeconomic or climate time series data, one notices a tendency for the series to increase (or decrease) over time. In some cases it is immediately apparent from the time series plot that the trend is approximately linear. In the econometrics literature there is a well developed literature on estimation and robust inference of deterministic trend functions with a focus on the case of the simple linear trend model. See for example, Canjels and Watson (1997), Vogelsang (1998), Bunzel and Vogelsang (2005), Harvey, Leybourne and Taylor (2009), and Perron and Yabu (2009). When analyzing more than one time series with trending behavior, it may be interesting to compare the trending behavior across series as in Vogelsang and Franses (2005). Empirically, comparisons across trends are often made in the economic convergence literature where growth rates of gross domestic products (GDPs), i.e. trend slopes of DGPs, are compared across regions or countries. See for example, Fagerberg (1994), Evans (1997) and Tomljanovich and Vogelsang (2002). Often empirical work in the economic convergence literature seeks to determine whether countries or regions have growth rates that are consistent with convergence that has either occurred or is occurring. There is little, if any, focus on estimating and quantifying the relative speed by which convergence is occurring. In simple settings, quantifying the relative speed of economic convergence amounts to estimating the ratio of gross domestic product growth rates, i.e. estimating relative growth rates. In the empirical climate literature there is a recent literature documenting the relative warming rates between surface temperatures and lower troposphere temperatures. See Santer, Wigley, Mears, Wentz, Klein, Seidel, Taylor, Thorne, Wehner, Gleckler et al. (2005), Klotzbach, Pielke, Christy and McNider (2009) and the references cited 1 therein. In this literature there is an explicit interest in estimating trend slope ratios and reporting con…dence intervals for them. As far as we know, there are no formal statistical methodological papers in the econometrics or climate literatures focusing on estimation and inference of trend slope ratios. This paper …lls that methodological hole in the literature. We focus on estimation of the ratio of trend slopes between two time series where is it reasonable to assume that the trending behavior of each series can be well approximated by a simple linear time trend. We obtain results under the assumption that the stochastic parts of the two time series comprise a zero mean time series vector that has su¢ cient stationarity and has dependence that is weak enough so that scaled partial sums of the vector satisfy a functional central limit theorem (FCLT). We compare two obvious estimators of the trend slope ratio and propose a third bias-corrected estimator. We show how to use these three estimators to carry out inference about the trend slope ratio. When trend slopes are small in magnitude relative to the variation in the stochastic components (the trend slopes are small relative to the noise), we …nd that inference using any of the three estimators is compromised and potentially misleading. We propose an alternative inference procedure that remains valid when trend slopes are small or even zero. We carry out an extensive theoretical analysis of the estimators and inference procedures. Our theoretical framework explicitly captures the impact of the magnitude of the trend slopes on the estimation and inference about the trend slope ratio. Our theoretical results are constructive in two important ways. First, the theory points to one of the three estimators as being preferred in terms of bias. Second, the theory strongly suggests that our alternative inference procedure is superior both for robustness under the null and power under the alternative with respect to the magnitudes of the trend slopes. Finite sample simulations indicate that the predictions made by the asymptotic theory are relevant in practice. Therefore, we are able to give concrete and speci…c advice to empirical practitioners on how to estimate a ratio of trend slopes and how to carry out tests of hypotheses about the ratio. 2 The remainder of this chapter is organized as follows: Section 1.2 describes the model and analyzes the asymptotic properties of the three estimators of the trend slope ratio. Section 1.3 provides some …nite sample evidence on the relative performance of the three estimators. Section 1.4 investigates inference regarding the trend slope ratio. We show how to construct heteroskedasticity autocorrelation (HAC) robust tests using each of the three estimators. We propose an alternative testing approach and show how to compute con…dence intervals for this approach. We derive asymptotic results of the tests under the null and under local alternatives. The asymptotic theory clearly shows that our alternative testing approach is superior under both the null and local alternatives. Additional …nite sample simulation results reported in Section 1.5 indicate that the predictions of the asymptotic theory are relevant in practice. In Section 1.6 we make some practical recommendations for empirical researchers and Section 1.7 concludes. All proofs are given in the Appendix. 1.2 1.2.1 The Model and Estimation Model and Assumptions Suppose the univariate time series y1t and y2t are given by y1t = 1 + 1t + u1t ; (1) y2t = 2 + 2t + u2t ; (2) where u1t and u2t are mean zero covariance stationary processes. Assume that 3 2 [rT ] X 6 u1t 7 T 1=2 4 5) t=1 u2t W(r) 2 3 6 B1 (r) 7 4 5; B2 (r) where r 2 [0; 1], [rT ] is the integer part of rT and W (r) is a 2 dent standard Wiener processes. (3) 1 vector of indepen- is not necessarily diagonal allowing for correlation between u1t and u2t . In addition to (3), we assume that u1t and u2t are ergodic for the 3 …rst and second moments. Suppose that 2 6= 0 and we are interested in estimating the parameter 1 = 2 which is the ratio of trend slopes. Equation (1) can be rewritten so that y1t depends on through y2t . Rearranging (2) gives t= 1 [y2t 2 u2t ] ; (4) 2 and plugging this expression into Equation (1) and then rearranging, we obtain y1t = ( 1 1 2) + 2 De…ning = 1 1 y2t + (u1t 2 2 and 1 u2t ) = ( t( ) = u1t t( 2) u2t ): t( ): (5) ), it immediately follows from (3) that T 1=2 [rT ] X t=1 t( )) w(r); (6) where w(r) is a univariate standard Wiener process and 2 is the long run variance of + y2t + (u1t u2t gives the regression model y1t = + y2t + Given the de…nition of 1 2 t( = 0 1 ). 4 1 0 1.2.2 Estimation of the Trend Slope Ratio Using regression (5), the natural estimator of is de…ned as 1 where y 1 = T T X T X (y2t e= 2 y2) t=1 y1t and y 2 = T 1 t=1 (y2t y 2 )(y1t y 1 ); (7) t=1 T X y2t . Standard algebra gives the relationship = T X 2 y2) (y2t t=1 Alternatively, one could estimate 2 1 T X t=1 e and ! is ordinary least squares (OLS) which ! 1 T X (y2t y2) t ( ) : (8) t=1 by the analogy principle by simply replacing with estimators. Let b1 and b2 be the OLS estimators of 1 and 2 1 based on regressions (1) and (2): b1 = b2 = where t = T 1 T X (t t)2 t=1 T X (t t=1 t) 2 ! ! 1 T X (t t)(y1t y 1 ); (9) 1 T X (t t)(y2t y 2 ); (10) t=1 t=1 T X t is the sample average of time and de…ne b = b1 = b2 . Simple algebra t=1 shows that b= b1 b2 = T X (t t)(y2t y2) t=1 ! 1 T X (t t)(y1t y1) (11) t=1 which is the instrumental variable (IV) estimator of in (5) where t has been used as an instrument for y2t . Standard algebra gives the relationship b = T X (t t)(y2t y2) t=1 5 ! 1 T X t=1 (t t) t ( ) : (12) 1.2.3 Asymptotic Properties of OLS and IV We now explore the asymptotic properties of the OLS and IV estimators of . The asymptotic behavior of the estimators depends on the magnitude of the trend slope parameters relative to the variation in the random components, u1t and u2t , i.e. the noise. The following theorem summarizes the asymptotic behavior of the estimators 1=2 . for …xed s and for s that are modeled as local to zero at rate T Theorem 1 Suppose that (6) holds and 1; 2 are …xed with respect to T . The follow- ing hold as T ! 1. Case 1 (large trend slopes): For T 3=2 T 3=2 e b 2 2 ) ) Z 1 s 0 Z 2 2 1 2 1 ds 2 1 2 s 0 ds Case 2 (medium trend slopes): For T e T b ) 2 2 Z 1 s 0 1 2 1 2 ds ) 2 0 1 s 1 2 2 ! 2 = 1 Z s 1 s 1 2 2 =T 0 1=2 1; Z 1 2 1 2 12 2 E(u2t t ); 2 ! 1 Z 1 ds 0 2 2 s ! = 2 2, dw(s) dw(s) 1=2 1 2 s 0 1, 1 2 0 1 =T ! Z 1 12 N Z ! 1 0; N N 0; 2 12 2 2 2 12 2 2 ! ! ; : 2, dw(s) + E(u2t t ( )) ; 1 2 dw(s) N 0; 2 12 2 2 ! : Theorem 1 makes some interesting predictions about the sampling properties of OLS and IV. When the trend slopes are …xed, i.e. when the trend slopes are large relative to the noise, OLS and IV converge to the true value of at the rate T 3=2 and are asymptotically normal with equivalent asymptotic variances. The precision of both estimators improves when there is less noise ( 2 the trend slope parameter for y2t increases ( is larger). 2 is smaller) or when the magnitude of When the trend slopes are modeled as local to zero at rate T 1=2 , i.e. when trend slopes are medium sized relative to the noise, asymptotic equivalence of OLS and IV no longer holds. The IV estimator essentially has the same asymptotic behavior as in 6 the …xed slopes case because the implied approximations are the same: Case 1 (large trend slopes): b N Case 2 (medium trend slopes): b ; N 12 2 T3 2 2 ; ! 12 2 T2 2 2 N ! ; N 12 T3 ; 2 ; 2 2 12 T3 2 2 2 : In contrast, the result for OLS is markedly di¤erent in Case 2. While OLS consistently estimates and the asymptotic variance is the same as in Case 1, OLS now has an asymptotic bias that could matter when trend slopes are medium sized. The fact that OLS is asymptotically biased in Case 2 is not that surprising because t( ) is correlated with y2t through the correlation between t( ) and u2t . In Case 2, the trend slopes are small enough so that the covariance between u2t and asymptotically a¤ects the OLS estimator. Because E(u2t t ( )) = E(u1t u2t ) t( ) E(u22t ), the asymptotic bias will be non-zero unless E(u1t u2t ) = E(u22t ) which only happens in very particular special cases. In general, OLS will have an asymptotic bias when trend slopes are medium sized. In the next subsection we propose a bias correction for OLS estimator. 1.2.4 Bias Corrected OLS According to Theorem 1, for the case of medium sized trend shifts OLS is asymptotically biased and this bias is driven by covariance between t( ) and u2t . The approximate bias of OLS suggested by Theorem 1 is given by the quantity bias( e) We can estimate 2R1 2 0 E(u2t t ( )) using T T s PT 1 1 2 2 1 2 ds 2 b2tet t=1 u Z 1 s 0 using T 1 2 2 2 ! 1 ds PT t=1 (y2t E(u2t t ( )): y 2 )2 , and we can estimate where et are the OLS residuals from regression (5) and u b2t are the OLS residuals from regression (2). This leads to the bias corrected 7 OLS estimator of ec = e given by 0 B B 1B T B BT @ 2 T X T 1 u b2tet PT t=1 t=1 (y2t 1 C C C e C= 2 y2) C A 1 T 1 T T X u b2tet t=1 T X : (13) y 2 )2 (y2t t=1 The next theorem gives the asymptotic behavior of the bias corrected OLS estimator for the same cases covered by Theorem 1. Theorem 2 Suppose that (6) holds and 1; are …xed with respect to T . The follow- 2 ing hold as T ! 1. Case 1 (large trend slopes): For T 3=2 ec ) 2 2 Z 1 s 0 1 2 2 ds Case 2 (medium trend slopes): For T ec ) 2 Z 0 1 s 1 2 1 ! 2 =T 2 ds ! Z 1 1 1 = 1 1 2 s 0 1=2 1; Z 0 2 1 s 1, =T 1 2 2 = 2, dw(s) 1=2 N 0; 2 12 2 2 ! : 2, dw(s) N 0; 2 12 2 2 ! : As Theorem 2 shows, the bias corrected OLS estimator is asymptotically equivalent to the IV estimator for both large and medium trend slopes. 1.2.5 Asymptotic Properties of Estimators for Small Trend Slopes As shown by Theorem 1, the magnitudes of the trend slopes relative to the noise can a¤ect the behavior of estimators of the trend slope ratio, . Intuitively, we know that as the trend slopes become very small in magnitude, we approach the case where the trend slopes are zero in which case is not well de…ned. While it is clear that OLS and possibly bias-corrected OLS will have problems when trends slopes are very small, IV is also expected to have problems in this case. If the trend slopes are very small, then the sample correlation between t and y2t also becomes very small and t becomes a weak instrument for y2t . It is well known in the literature that weak instruments have important implications for IV estimation (see Staiger and Stock 1997?) and estimation 8 of is no exception. The next two theorems provide asymptotic results for the estimators of 1 slopes that are local to zero at rates T 3=2 and T for trend with the latter case corresponding to trend slopes that are very small relatively to the noise. Theorem 3 Suppose that (3) and (6) hold and 1; 2 are …xed with respect to T . The following hold as T ! 1. Case 3 (small trend slopes): For T 1=2 (b )) 2 Z e 1 s 0 p ! <; 1 2 2 ! Z ds 1 =T 1; 2 =T 1 2, p ec 1 1 ! 0 and c21 4c2 c0 0. In this case, the roots are real and p( 0 ) opens upwards. The con…dence interval is the values of i.e. 0 0 between the two roots, 2 [r1 ; r2 ]. The inequality c2 > 0 is equivalent to the inequality b2 2 b 22 T X t)2 (t t=1 ! 1 > cv 2 =2 : (23) Notice that the left hand side of (23) is simply the square of the HAC robust t-statistic for testing that the trend slope of y2t is zero. Inequality (23) holds if the trend slope of y2t is statistically di¤erent from zero at the level. This occurs when the trend slope for y2 is large relative to the variation in u2t . Mechanically, c2 will be positive when the t-statistic for testing that 2 = 0 is large in magnitude. Although not obvious, if c2 > 0; it is impossible for c21 4c2 c0 < 0 to hold. In this case p( 0 ) opens upward and has its vertex above zero and the roots are complex; therefore, there are no solutions to (22) and the con…dence interval would be empty. This case is impossible because the con…dence interval cannot be empty because b1 0 = b = b1 = b2 is always contained in the interval given that b = 0 in which case (21) must hold (equivalently (22) must hold). 0 2 Case 2: Suppose that c2 < 0 and c21 4c2 c0 > 0. In this case, p( 0 ) has two real roots and opens downwards and the con…dence interval is the values of 0 not between the two roots. In this case the con…dence interval is the union of two disjoint sets and is given by 0 2 ( 1; r1 ] [ [r2 ; 1). Case 3: Suppose that c2 < 0 and c21 4c2 c0 0. In this case, p( 0 ) opens downward and has a vertex at or below zero. The entire p( 0 ) function lies at or below zero and the con…dence interval is the entire real line: 0 2 ( 1; 1). Although it is a zero probability event, should c2 = 0 then the con…dence interval will 17 take the form of either 2 ( 1; c0 =c1 ] when c1 > 0 or 0 0 2 [ c0 =c1 ; 1) when c1 < 0. While the con…dence intervals constructed using t 0 can be wide when the trend slopes are small and there is no guarantee that these con…dence intervals will contain the OLS or bias-corrected OLS estimators of , t other t-statistics. Recall that we can write t t 0 =v u u t b 11 b1 2 0 b 12 + 0 has a major advantage over the 0 as b 0 2 T X (t 2b 0 22 t)2 t=1 ! 1 : The denominator is a function u b1t and u b2t each of which are exactly invariant to the true values of 1 and 2: write the numerator of t b1 Because b1 When H0 is true, it follows that 0 b = b1 and b2 b 0 2 2 ( = b1 2 0) 1 1 and is exactly invariant to the true values of 0 = 0 and we can 1 0 b2 2 : are only functions of t and u1t ; u2t , the numerator of t also exactly invariant to the true values of of t 2 0 as 0 2 1 1 2. 1 0 is Therefore, the null distribution and 2 including the case where both trend slopes are zero. In contrast, the other t-statistics have null distributions that depend on the magnitudes of and 2 under the null, t magnitudes of 1.4.3 1 and 0 2) 1 and 2. Because of its exact invariance to 1 will deliver much more robust inference (with respect to the than the other t-statistics. Asymptotic Results for t-statistics In this section we provide asymptotic limits of the four t-statistics described in the previous sub-section. We derive asymptotic limits under alternatives that are local to the null given by (14). Suppose that 2 =T 18 2. Then the alternative value of 1 is modeled local to as 0 1 The parameter = 0 +T 3=2+ : (24) measures the magnitude of the departure from the null under the local alternative. In the results presented below, the asymptotic null distributions of the t-statistics are obtained by setting Recall that for the t 0 = 0. statistic, under the null that = 0 it follows that 1( 0) = 0. ; (25) Under the local alternative (24), it follows that 1( 0) = 2( 1 0) = 2T 3=2+ =T 2T 3=2+ =T 3=2 2 regardless of the magnitude of the trend slopes. Therefore, the asymptotic limit of t 0 is invariant to the magnitude of the trend slopes under both the null and local alternative for . We derive the limits of the various HAC estimators using …xed-b theory following Bunzel and Vogelsang (2005). The form of these limits depends on the type of kernel function used to compute the HAC estimator. We follow Bunzel and Vogelsang (2005) and use the following de…nitions. De…nition 1 A kernel is labelled Type 1 if k (x) is twice continuously di¤ erentiable everywhere and as a Type 2 kernel if k (x) is continuous, k (x) = 0 for jxj 1 and k (x) is twice continuously di¤ erentiable everywhere except at jxj = 1: We also consider the Bartlett kernel (which is neither Type 1 or 2) separately. The …xed-b limiting distributions are expressed in terms of the following stochastic functions. De…nition 2 Let Q(r) be a generic stochastic process. De…ne the random variable 19 Pb (Q(r)) as 8 R1R1 > > k 00 (r s) Q(r)Q(s)drds if k (x) is Type 1 > 0 0 > > > > > > > > > > RR > 00 < s) Q (r) Q (s) drds jr sj 1 b > +2k 0 (b) 0 Q (r + b) Q (r) dr if k (x) is Type 2 > > > > > > > > > > > > : 2 R 1 Q (r)2 dr 2 R 1 b Q (r + b) Q (r) dr if k (x) is Bartlett b 0 b 0 where k (x) = k x b 0 and k is the …rst derivative of k from below. The following theorems summarize the asymptotic limits of the t-statistics for testing (14) when the alternative is given by (24). Theorem 4 (Large Trend Slopes) Suppose that (6) holds. Let M = bT where b 2 (0; 1] is …xed. Let 1 = 0 +T 1 3=2 = 1; 2 = 2 where 1; 2 are …xed with respect to T , and let . Then as T ! 1, tOLS ; tBC ; tIV ) p t 0 )p Z Pb (Q(r)) Z Pb (Q(r)) +q 12 +q 12 2 ; 2 P (Q(r)) 1 b 2 ; 2 P (Q(r)) 0 b R1 where Z N (0; 1), Q(r) = w(r) e 12L(r) 0 s 21 dw(s), w(r) e = w(r) Rr L(r) = 0 s 21 ds and Z and Q(r) are independent. rw(1), Theorem 5 (Medium Trend Slopes): Suppose that (6) holds. Let M = bT where b 2 (0; 1] is …xed. Let 1 =T 1=2 1; 2 =T 20 1=2 2 where 1; 2 are …xed with respect to T , and let 1 tOLS = 1 +T 0 . Then as T ! 1, 1 12 E(u2t t ( )) +q 2 +q )p Pb (H1 (r)) 12 21 Pb (H1 (r)) 12 Z Z tBC ; tIV ) p +q Pb (Q(r)) 12 t where H1 (r) = Q(r) 0 )p Z +q Pb (Q(r)) 12 1 12 2 1 2 ; 2 P (H1 (r)) 1 b 2 ; 2 P (Q(r)) 1 b 2 ; 2 P (Q(r)) 0 b L(r) E(u2t t ( )). Theorem 6 (Small Trend Slopes): Suppose that (6) holds. Let M = bT where b 2 (0; 1] is …xed. Let and let 1 = 0 1 1=2 +T 1 =T 1; 2 =T 1 2 where 1; 2 are …xed with respect to T , . Then as T ! 1, 1 d tOLS ; tBC ! r 2 2 Pb (L(r)) 2R1 2 0 (s Z tIV ) p +q Pb (Q(r)) 12 t 0 )p 1 2 2 ) ds 2 ; ; 2 P (Q(r)) 1 b Z +q Pb (Q(r)) 12 +E 1 u22t 2 : 2 P (Q(r)) 0 b Theorem 7 (Very Small Trend Slopes): Suppose that (3) and (6) hold. Let M = bT where b 2 (0; 1] is …xed. Let respect to T , and let 1 = 0 1 =T + 3=2 1; 2 3=2 =T 2 where 1; 2 are …xed with . Then as T ! 1, 1 T tIV ) R1 0 1=2 (s E(u2 ) E(u2t t ( )) + ; tOLS ; T tBC = q 2t 1 Pb (H2 (r)) E u22t R1 R1 1 1 2 1 2 0 (s 2 )dw(s) + 2 ) ds + 0 (s 2 )dB2 (s) q R1 Pb (H3 (r)) 0 (s 21 )2 ds 1=2 t 0 )p Z Pb (Q(r)) +q 12 21 2 2 P (Q(r)) 0 b ; 1 ; where H2 (r) = w(r) e Z 2 1 (s 0 H3 (r) = w(r) e e2 (r) = B2 (r) and B Z 1 1 2 ) ds + 2 (s 0 1 )dw(s) 2 e 2 L(r) + B2 (r) Z 1 1 s 2 0 12 2 L(r) 2 Z 1 1 )dB2 (s) 2 (s 0 e2 (r) ; +B + 12 Z 0 1 s 1 2 1 1 dB2 (s) dw(s); rB2 (1). Some interesting results and predictions about the …nite sample behavior of the t- statistics are given by the Theorems 4-7. First examine the limiting null distributions that are obtained when = 0. For large trend slopes, all four t-statistics have the same asymptotic null limit and the limiting random variable is the same …xed-b limit obtained by Bunzel and Vogelsang (2005) for inference regarding the trend slope in a simple linear trend model with stationary errors. Therefore, …xed-b critical values are available from Bunzel and Vogelsang (2005). As the trend slopes become smaller relative to the noise, di¤erences among the t-statistics emerge. As anticipated, t 0 has the same limiting null distribution regardless of the magnitudes of the trend slopes. Except for very small trends slopes, tIV has the same limiting null distribution as t 0 . The bias in OLS a¤ects the null limit of tOLS for medium, small and very small trend slopes. The bias correction helps for medium trend slopes in which case tBC has the same null limit as tIV and t 0 . For small and very small trend slopes the bias correction no longer works e¤ectively and tBC has the same limiting behavior as tOLS . Both tests will tend to over-reject under the null when trends slopes are very small given that they diverge with the sample size. In terms of …nite sample null behavior of the t-statistics, the asymptotic theory predicts that tOLS and tBC will only work well when trend slopes are relatively large whereas tIV should work well except when trend slopes are very small. The most reliable test in terms of robustness to magnitudes of trend slopes under the null should 22 be t 0 . When 6= 0; in which case we are under the alternative, the t-statistics have additional terms in their limits which push the distributions away from the null distributions giving the tests power. When trend slopes are large, all four t-statistics have the same limiting distributions with the only minor di¤erence being that t on 2 1 2 0 rather than 2 1 0 depends 2 as for the other t-statistics. In general we cannot rank 0 and as any di¤erence depends on the joint serial correlation structure of u1t and u2t . Unless 2 0 and 2 1 are nontrivially di¤erent, we would expect power of the tests to be similar in the large trend slope cases. As the trends slopes become smaller, power of tOLS and tBC becomes meaningless given that the statistics have poor behavior under the null. In Theorem 6, the limits of tOLS and tBC do not depend on which sug- gests that power will very low when trend slopes are small. In Theorem 7, tOLS and tBC diverge with the sample size which suggests large rejections with very small trend slopes. In constrast power of tIV should be similar to t except when trend slopes are 0 very small. As in the case of null behavior, the asymptotic theory predicts that t 0 should perform the best in terms of power. One thing to keep in mind regarding the limit of t 0 in Theorems 4-7 is that while the limit under the alternative is the same in each case, the relevant values of are farther away from the null in the case of smaller trends slopes compared to the case of larger trend slopes. Therefore, 1 needs to be much farther away from 0 of small trend slopes than for the case of large trend slopes for power of t same in both cases. In other words, for a given value of 1, power of t 0 in the case 0 to be the decreases as the trend slopes become smaller. This relationship between power and magnitudes of the trend slopes can be seen clearly in Theorem 4 where we can see that as limiting distribution under the local alternative for and power decreases. 23 2 ! 0 the approaches the null distribution 1.5 Finite Sample Null Rejection Probabilities and Power Using the same DGP as used in Section 3 we simulated …nite sample null rejection probabilities and power of the four t-statistics. Table 2 reports null rejection probabilities for 5% nominal level tests for testing H0 : alternative H1 : = 0 = 2 against the two-sided 6= 2. Results are reported for the same values of 1; 2 as used in Table 1 for T = 50; 100; 200 and 10; 000 replications are used in all cases. The HAC estimators are implemented using the Daniell kernel. Results for three bandwidth sample size ratios are provided: b = 0:1; 0:5; 1:0. For a given sample size, T , we use the bandwidth M = bT for each of the three values of b. We compute empirical rejections using …xed-b asymptotic critical values using the critical value function cv0:025 (b) = 1:9659 + 4:0603b + 11:6626b2 + 34:8269b3 13:9506b4 + 3:2669b5 , as given by Bunzel and Vogelsang (2005) for the Daniell kernel. The patterns in the empirical null rejections closely match the predictions of the asymptotic results. When the trend slopes are large, 4, 1 2, null rejections are 2 the essentially the same for all t-statistics and are close to 0.05 even when T = 50. This is true for all three bandwidth choices which illustrates the e¤ectiveness of the …xed-b critical values. For medium sized trend slopes, 0:1 1 0:4, 0:05 0:2, tOLS 2 begins to show over-rejection problems that become very severe as the trend slopes decrease in magnitude. The bias-corrected OLS t-statistic, tBC , is less subject to overrejection problems especially when T is not small, although for T = 50, tBC shows nontrivial over-rejection problems. In contrast both tIV and t 0 have null rejections close to 0.05 for medium sized trend slopes. When the trend slopes are small or very small, 1 0:04, 2 0:02, the tOLS and tBC statistics have severe over-rejection problems and can reject 100% of the time. While tIV has less over-rejection problems in this case, the over-rejections are nontrivial and are problematic. In contrast, t null rejections that are close to 0.05 regardless of the magnitudes of the case of 1 = 2 = 0. In fact, the rejections are identical for t 24 0 1; 2 0 has including across values of 1; 2. This is because t 0 is exactly invariant to the values of of null rejection probabilities that t Given that t 0 0 1; 2. It is clear in terms is the preferred test statistic. is the preferred statistic in terms of size, we computed, for each of the parameter con…gurations in Table 2, the proportions of replications that lead to the three possible shapes of con…dence intervals one obtains by inverting t 0 . Recall that the cases are given by Case 1: 3: 0 2 [r1 ; r2 ], Case 2: 0 0 2 ( 1; r1 ] [ [r2 ; 1) and Case 2 ( 1; 1). Table 3 gives these results. For large trend slopes Case 1 occurs 100% of the time. As the trend slopes decrease in magnitude, Case 2 occurs some of the time and as the trend slopes decrease further, Case 3 can occur frequently if the trend slopes are very small. As T increases, the likelihood of Case 1 increases for all trend slope magnitudes. The relative frequencies of the three cases also depends on the bandwidth but the relationship appears complicated. This is not surprising given the complex manner in which the bandwidth a¤ects the null distribution of t 0 . Overall, unless trends slopes are small or very small, Case 1 is the most likely con…dence interval shape. While t 0 is the preferred test in terms of size, how do the t-statistics compare in terms of power? Table 4 reports power results for a subset of the grid of Tables 2,3. For a given value of in the range 2 [2; construction 1 that 2 = max ] 2 2, where 2 as used in we specify a grid of six equally spaced values for = 0 = 2 is the null value and max = 0:01= 2 . By in all cases. Given the way we de…ne the grid for , we ensure is the same for all values of 2. Results are reported for T = 100. Results for other values of T are qualitatively similar and are omitted. The patterns in power given in Table 4 are what we would expect given the local asymptotic limiting distributions. For large trend slopes ( 2 = 10; 2), power of the four tests is essentially the same as predicted by Theorem 4. As the bandwidth increases, power of all the tests decreases. This inverse relationship between power and bandwidth is well known in the …xed-b literature (see Kiefer and Vogelsang 2005). For medium sized trend slopes ( 2 = 0:2; 0:1) di¤erences in power begin to emerge 25 with tOLS having substantially lower power than the other tests. This lower power occurs even though tOLS over-rejects under the null when a small bandwidth is used (b = 0:1). With larger bandwidths, tOLS has no power at all. Both tBC and tIV have good power that is somewhat lower than t 0 , which, according to Theorem 5 would result from di¤erences between 2 0 2 and 1 . For small and very small trends slopes ( 2 = 0:01; 0:001), both tOLS and tBC are distorted under the null with severe over-rejections although tOLS severely under-rejects with a large bandwidth (b = 1:0) when = 0:01. While tIV is less size distorted than 2 tOLS and tBC , it has no power regardless of bandwidth. In contrast to other three statistics, t 0 continues to have excellent size and power. The superior power of t 0 relative to the other statistics is completely in line with the predictions of Theorems 6 and 7. In summary, the patterns in the …nite sample simulations are consistent with the predictions of Theorems 4-7. Clearly t 0 is the recommended statistic given its superior behavior under the null and its higher power under the alternative. 1.6 Practical Recommendations For point estimation, we recommend the IV estimator given its relative robustness to the magnitude of the trend slopes. OLS and bias-corrected OLS are not recommended given that they can become severely biased for small to very small trend slopes. For inference, we strongly recommend the t 0 statistic given its superior behavior under the null and the alternative both theoretically and in our limited …nite sample simulations. Good empirical practice would be to report the IV estimator, b, along with the con…dence interval constructed by inverting t 0 . Because this con…dence interval must contain b, we avoid situations where the recommended point estimator lies outside the recommended con…dence interval. For con…dence interval construction, there is also the practical need to choose a kernel and bandwidth. We do not explore this choice here but encourage empirical researchers to use the …xed-b critical values provided by Bunzel and Vogelsang (2005) 26 once a kernel and bandwidth have been chosen. 1.7 Conclusion and Directions for Future Research In this paper we analyze estimation and inference of the ratio of trend slopes of two time series with linear deterministic trend functions. We consider three estimators of the trend slope ratio: OLS, bias-corrected OLS, and IV. Asymptotic theory indicates that when the magnitude of the trend slopes are large relative to the noise in the series, the three estimators are approximately unbiased and have essentially equivalent sampling distributions. For small trend slopes, the IV estimator tends to remain unbiased whereas OLS and bias-corrected OLS can have substantial bias. For very small trend slopes all three estimators become poor estimators of the trend slopes ratio. We analyze four t-statistics for testing hypotheses about the trend slopes ratio. We consider t-statistics based on each of the three estimators of the trend slopes ratio and we propose a fourth t-statistic based on an alternative testing approach. Asymptotic theory indicates that the alternative test dominates the other three tests in terms of size and power regardless of the magnitude of the trend slopes. Finite sample simulations show that the predictions of the asymptotic theory tend to hold in practice. Based on the asymptotic theory and …nite sample evidence we recommend that the IV estimator be used to estimate the trend slopes ratio and that con…dence intervals be computed using our alternative test statistic. A nice property of our recommendation is that the IV estimator is always contained in the con…dence interval even though the con…dence interval is not constructed using the IV estimator itself. A future research direction related to extension of the results in this paper is as follows: There may be empirical settings with more than two trending time series in which case more than one trend slope ratio can be estimated. It would be interesting to investigate whether panel methods can deliver better estimators of the trend slope ratios than applying the methods in this paper on a pairwise basis. 27 APPENDIX 28 Proofs of Theorems Before giving proofs of the theorems, we prove a series of lemmas for each of the trend slope magnitude cases: large, medium, small and very small. The lemmas establish the limits of the scaled sums that appear in the estimators of and the HAC estimators. Using the results of the lemmas, the theorems are easy to establish using straightforward algebra and the continuous mapping theorem (CMT). We begin with a lemma that has limits of scaled sums that are exactly invariant to the magnitudes of the trend slopes followed by four lemmas for each of the trend slope cases. Throughout the appendix, we use t to denote t ( ). Lemma 1 Suppose that (3) and (6) hold. The following hold as T ! 1 for any values of 1 ; 2 : 3 T T X 2 (t t) ! t=1 2 T [rT ] X (t t) ! t=1 3=2 T T T X t)( (t t=1 T X 1 (u2t Z 1 0 Z r 1 0 )) t u2 )( T X (u2t t=1 T 3=2 T X (t t=1 T 3=2 b2 2 ) t)(u2t Z 0 1 )ds = L(r); 2 Z 1 1 (s )dw(s); 2 0 (s p ) ! E (u2t t ( )) ; t t=1 T 1 2 1 ) ds = ; 2 12 (s 1 (r p u2 )2 ! E u22t ; u2 ) ) Z 1 2 ) dr 2 1 (s 1 )dB2 (s); 2 1Z 1 (s 0 0 1 )dB2 (s): 2 Proof: The results in this lemma are standard given the FCLTs (3) and (6) and ergodicity of u1t and u2t . See ?. Lemma 2 (Large trend slopes) Suppose that (3) and (6) hold and 29 1; 2 are …xed with respect to T . The following hold as T ! 1 for T 3=2 3=2 T T X (y2t y 2 )( t=1 T X )) t y 2 )(u2t (y2t 3 T T 3 T X (y2t t=1 T X p T Z [rT ] X (y2t 2 = 2, (s 1 )dw(s); 2 (s 1 )dB2 (s); 2 1 2 1 2 ) ds = 2 ; 2 12 (s Z 1 1 2 ) ds = 2 ; 2 12 (s 0 y2) ) t=1 1 0 1 2 2 0 Z 0 y2) ! t=1 3=2 2 p t)(y2t (t 2 2 y 2 )2 ! 1, Z u2 ) ) t=1 = 1 2 L(r): y2 = Proof: The results of the lemma are easy to establish once we substitute y2t t + (u2t u2 ) into each expression: 2 t T 3=2 T X (y2t y 2 )( )= t 3=2 2T T X (t t)( )+T t 3=2 t=1 t=1 3=2 2T = T X (t T X (u2t u2 )( t ) t=1 t)( ) + op (1); t t=1 T 3=2 T X (y2t y 2 )(u2t 2T u2 ) = 3=2 t=1 T X (t t)(u2t u2 ) + T 3=2 t=1 2T = 3=2 T X (t T X (u2t u2 )2 t=1 t)(u2t u2 ) + op (1); t=1 3 T T 3 T X (y2t t=1 T X (t y 2 )2 = 2 2T 3 T X t=1 t)(y2t y2) = 2T t=1 3=2 X (y2t 3 T X (t t)2 + op (1); t=1 [rT ] T t)2 + op (1); (t [rT ] y2) = 2T t=1 2 X (t t) + op (1): t=1 The limits follow from Lemma 1. Lemma 3 (Medium trend slopes) Suppose that (3) and (6) hold and 30 1; 2 are …xed with respect to T . The following hold as T ! 1 for 1 T T X (y2t y 2 )( t=1 T X 1 T )) t y 2 )(u2t (y2t T 5=2 2 T X (y2t t=1 T X 1 Z y2) ! [rT ] X (y2t 2 =T (s 1 2 ) ds = 2 ; 2 12 1=2 2, 2 0 Z 1 1 2 ) ds = 2 ; 2 12 (s 0 y2) ) t=1 1, 1 )dB2 (s) + E u22t ; 2 1 2 1=2 (s 0 p t=1 T 2 2 Z =T 1 )dw(s) + E (u2t t ( )) ; 2 (s 2 p 3=2 1 0 y 2 )2 ! t)(y2t (t 2 u2 ) ) t=1 T Z 1 2 L(r): Proof: The results of the lemma are easy to establish once we substitute y2t t + (u2t u2 ) into each expression: 2 t T 1 T X y 2 )( (y2t t )= 2T 3=2 t=1 T 1 T X y 2 )(u2t (y2t u2 ) = 2T 3=2 2 T T X (y2t y 2 )2 = 2 2T 3 t=1 5=2 t=1 T X (t t)( t )+T 1 T X T X (t t)(y2t y2) = 2T 3 t=1 T X u2 )( t ); t)(u2t u2 ) + T 1 T X (u2t u2 )2 ; t=1 (t t)2 + op (1); (t t)2 + op (1); (t t) + op (1): t=1 t=1 T T X (u2t t=1 t=1 t=1 T T X (t y2 = [rT ] 3=2 X (y2t [rT ] y2) = 2T t=1 2 X t=1 The limits follow from Lemma 1. Lemma 4 (Small trend slopes) Suppose that (3) and (6) hold and 31 1; 2 are …xed with respect to T . The following hold as T ! 1 for 1 T T X t=1 T X 1 T (y2t T 1 p t=1 T 2 2 y 2 )2 ! (y2t 2 T X 1 0 t=1 1 T 2 =T 1 2, p 2 1 2 ) ds + E u22t = 2 + E u22t ; 2 12 Z 1 1 2 p y2) ! 2 (s ) ds = 2 ; 2 12 0 (s t)(y2t (t 1, u2 ) ! E u22t ; y 2 )(u2t Z 1 ) ! E (u2t t ( )) ; t t=1 T X =T p y 2 )( (y2t 1 [rT ] X (y2t y2) ) t=1 2 L(r): y2 = Proof: The results of the lemma are easy to establish once we substitute y2t t + (u2t u2 ) into each expression: 2 t T 1 T X (y2t y 2 )( )= t 2T 2 T X (t t)( )+T t 1 t=1 t=1 =T 1 T X (u2t T X (u2t u2 )( t ) t=1 u2 )( ) + op (1); t t=1 T 1 T X (y2t y 2 )(u2t 2T u2 ) = 2 t=1 T X (t t)(u2t u2 ) + T t=1 =T T X T 1 (y2t y 2 )2 = 1 2 2T t=1 T T X 2 (t t)(y2t 1 u2 )2 (u2t u2 )2 + op (1); (u2t t=1 T X 3 t)2 + T (t 1 T X (u2t u2 )2 + op (1); t=1 y2) = 2T t=1 T T X t=1 t=1 T X 1 3 T X (t t)2 + op (1); t=1 [rT ] X [rT ] (y2t y2) = 2T t=1 2 X (t t) + op (1): t=1 The limits follow from Lemma 1. Lemma 5 (Very small trend slopes) Suppose that (3) and (6) hold and 32 1; 2 are …xed with respect to T . The following hold as T ! 1 for 1 T T X t=1 T X 1 T (y2t 1 T 3=2 (t p t)(y2t y2) ! t=1 2 1=2 T Z 1 (s 0 [rT ] X (y2t t=1 3=2 =T p p 1 2 ) ds + 2 Z y2) ) 2 L(r) 1 1 )dB2 (s) = 2 + 2 12 (s 0 Z 1 T T X y 2 )( (y2t )= t 5=2 2T T X (t 0 e2 (r): +B t)( 1 )+T t t=1 t=1 =T 1 T X (u2t T X (u2t 1 )dB2 (s); 2 (s Proof: The results of the lemma are easy to establish once we substitute y2t t + (u2t u2 ) into each expression: 2 t 1 2, y 2 )2 ! E u22t ; (y2t t=1 T X 2 u2 ) ! E u22t ; y 2 )(u2t T X 1, ) ! E (u2t t ( )) ; t t=1 T 3=2 =T p y 2 )( (y2t 1 u2 )( y2 = ) t t=1 u2 )( t ) + op (1); t=1 T 1 T X (y2t y 2 )(u2t 5=2 2T u2 ) = t=1 T X (t t)(u2t u2 ) + T t=1 =T T X T 1 (y2t y 2 )2 = 1 2 2T t=1 =T 1 T X T X u2 )2 (u2t t=1 u2 )2 + op (1); (u2t t=1 T X 4 t)2 + T (t t=1 T X 1 1 T X (u2t u2 )2 + op (1) t=1 u2 )2 + op (1); (u2t t=1 T 3=2 T X (t t)(y2t y2) = 2T t=1 T 3 T X (t t)2 + T 3=2 t=1 1=2 [rT ] X (y2t y2) = t=1 2T 2 [rT ] X (t t=1 T X (t t)(u2t u2 ); t=1 t) + T 1=2 [rT ] X (u2t u2 ): t=1 The limits follow from Lemma 1. Proof of Theorem 1. The proof follows directly from Lemmas 1, 2 and the CMT. 33 For the case of large trend slopes it follows that T T 3=2 3=2 e = 3 T T X 2 (y2t y2) t=1 2 2 ) b = Z (s T X 1 (t 2 ) 2 (s 0 1 y 2 )( ) t 1 )dw(s); 2 (s 3=2 T T X (t t)( t ) t=1 Z 1 (y2t t=1 1 y2) 1 2 ) ds 2 T X 0 ! t)(y2t 1 Z 1 t=1 Z 3=2 T 1 2 ) ds 2 0 3 T 1 ! 1 1 )dw(s): 2 (s 0 For the case of medium trend slopes it follows that T e T b = 2 T T X y2) (y2t 2 t=1 2 2 ) = 1 0 2 Z T X (t 1 T (s 0 T X (y2t Z 2 t ) Z 1 )dw(s) + E (u2t t ( )) ; 2 (s 1 y2) 1 1 0 ! t)(y2t 1 2 ) ds 2 y 2 )( t=1 t=1 1 1 1 1 2 ) ds 2 (s 5=2 T ) Z ! 5=2 T T X (t t)( ) t t=1 1 1 )dw(s): 2 (s 0 P b2tet for the large Proof of Theorem 2. We …rst need to derive the limit of T 1 Tt=1 u and medium trend slope cases. For large trend slopes we have, using Lemmas 1, 2 and Theorem 1: T 1 T X t=1 u b2tet = T T +T 1 T X (u2t u2 )( ) t 1 T T e 3=2 t=1 1 1 T 3=2 b2 T 3=2 e =T 1 T X t=1 (u2t 2 T 3=2 T X (t t)( T 3=2 T X (y2t y 2 )(u2t u2 ) t=1 ) t t=1 T 3=2 b2 u2 )( 2 T 3 T X (t t)(y2t y2) t=1 p t ) + op (1) ! E (u2t t ( )) : For medium trend slopes we have, using Lemmas 1,3 and Theorem 2: 34 (26) T 1 T X t=1 1 u b2tet = T T X (u2t u2 )( ) t T t=1 1 T 1 +T T 3=2 b2 T e 1 =T T X (t ec The results for T ec = T b2 3=2 (u2t 3 T X (y2t y2) 2 T X (y2t y2) 2 t=1 Z 2 2 ) T X (y2t y 2 )(u2t u2 ) t=1 ) t 5=2 T 2 T X (t t)(y2t y2) t=1 ) + op (1) ! E (u2t t ( )) : t (27) are as follows. For large trend slopes 3 T t)( T 1 p u2 )( t=1 = T e t=1 T T X 3=2 T 2 t=1 3=2 1 1 (s 0 ! 1 3=2 T (y2t y 2 )( t ) T 3=2 t=1 ! 1 3=2 T Z 1 1 2 ) ds 2 T X 2 T X t=1 (y2t y 2 )( t ) + op (1) t=1 1 (s 0 T X ! u b2tet ! 1 )dw(s); 2 using Lemma 1, (26) and the CMT. For medium trend slopes T ec = T X 2 T (y2t y 2 )2 t=1 ) = 2 2 2 2 Z Z 1 (s 0 1 0 (s 1 2 ) ds 2 1 2 ) ds 2 ! 1 T X 1 T (y2t y 2 )( t t=1 1 2 1 2 Z Z 1 0 1 0 (s (s ) T 1 T X t=1 u b2tet ( ) 1 )dw(s) + E (u2t t ( )) 2 ! E (u2t t ( )) 1 )dw(s); 2 using Lemma 2, (27) and the CMT. Proof of Theorem 3. We …rst give the proof for the case of small trend slopes. Using 35 Lemmas 1, 4 and the CMT it follows that e T = T p Z 2 2 ! b 1=2 1 T X (y2t t=1 1 ) 2 T 2 ! 1 1 T Z T X (t t)(y2t ! (s 0 Z 1 1 2 ) ds 2 (y2t y 2 )( 1 3=2 T T X (t t)( 1 T X t=1 u b2tet ( ) = T T +T 1 T X (u2t u2 )( 1 1 )dw(s): 2 (s 0 e ) t t=1 1 T 3=2 1=2 e b2 T 2 3=2 T X ) t t=1 To establish the result for ec we …rst need to derive the limit of T Lemmas 1, 4 and the results for OLS T ) t E (u2t t ( )) = <; y2) t=1 1 T X t=1 1 1 2 ) ds + E(u22t ) 2 (s 0 = y2) 2 T 1 T X 1 (y2t PT b2tet . t=1 u y 2 )(u2t Using u2 ) t=1 t)( (t ) t t=1 T 3=2 b2 T X 1 =T (u2t u2 )( 2 e ) t > k 00 (r s) Q(r)Q(s)drds if k (x) is Type 1 > 0 0 > > > > > > > > > > RR > 00 < s) Q (r) Q (s) drds jr sj 1 b > if k (x) is Type 2 +2k 0 (b) 0 Q (r + b) Q (r) dr > > > > > > > > > > > > : 2 R 1 Q (r)2 dr 2 R 1 b Q (r + b) Q (r) dr if k (x) is Bartlett b 0 b 0 where k (x) = k x b and k 0 is the …rst derivative of k from below. The following theorem summarizes the asymptotic limits of the t-statistics for testing (37) when the alternative is given by (39). Theorem 10 (Large Trend Slopes) Let M = bT where b 2 (0; 1] is …xed. Let 2 = and 2 1 where = 0 +T 1; 2 are …xed with respect to T , and let 3=2 1 = 0 for case 2. Then as T ! 1, (Case 1: +T t ; u1t I(1) processes.) tIV ) t 0 ) p p 12 12 R1 (r 12 )w(r)dr p0 +q Pb (Q1 (s)) 12 R1 p0 (r 1 2 )w(r)dr Pb (Q1 (s)) +q 12 Case 2: u1t and u2t are I(1) processes, whereas t Z tIV ) p +q Pb (Q(r)) 12 59 2 2 ; ; P (Q1 (s)) 1 b 2 2 ; : P (Q1 (s)) 0 b is an I(0) process. 2 2 P (Q(r)) 1 b ; 1=2 1 = 1; for case 1 and u2t are all t 0 )p where Z N (0; 1), Q1 (s) = 1 2 )w(r)dr; Q(r) = w(r) e Rr 0 1 2 s Z Pb (Q(r)) Rs +q 12 2 ; 2 P (Q(r)) 0 b 1 R1 R1 R1 1 2 w(r)dr s 2 ) dr 0 w(r)dr L(s) 0 (r 0 (r R1 e = w(r) rw(1), L(r) = 12L(r) 0 s 12 dw(s), w(r) 0 ds and Z and Q(r) are independent. Theorem 11 (Medium Trend Slopes): Let M = bT where b 2 (0; 1] is …xed. Let T 1=2 1; 2 =T for case 1 and 1 1=2 2 = 0 where 1; 1 +T 2 are …xed with respect to T , and let for case 2. Then as T ! 1, (Case 1: 1 = t ; u1t 0 1 = + and u2t are all I(1) processes.) tIV ) +r 2 ; 1 Q(P (s)) t 0 ) R1 2 0 p 12 12 1 2 2 ) dr (r R1 p0 p (r R1 (r 21 )w(r)dr p0 Pb (Q1 (s)) + R1 2 0 1 2 )w(r)dr Pb (Q1 (s)) (r +q 12 Case 2: u1t and u2t are I(1) processes, whereas +r 1 2 )w2 (r)dr t 2 R1 0 2 2 ; ; (r 1 2 2 ) dr (r 1 2 2 ) dr : P (Q1 (s)) 0 b is an I(0) process. Z tIV ) p Pb (Q0 (s)) 2 1 Pb (Q0 (s)) R1 2 0 t 0 (r )p 1 2 2 ) dr + R1 2 0 (r Z +q Pb (Q(r)) 12 2 1 2 )w2 (r)dr 2 2 R1 0 ; ; P (Q(r)) 0 b 1 R1 Rs R1 R1 where Q1 (s) = 0 w(r)dr s 0 w(r)dr 2 L(s) 2 0 (r 12 )2 dr + 2 0 (r 21 )w2 (r)dr 1R1 R1 R1 R1 1 1 2 1 0 (s) = w(s) (r )w(r)dr; and Q e L(s) (r ) dr + (r )w (r)dr 2 2 2 2 2 2 2 0 0 0 0 (r 1 2 )dw(r). Theorem 12 (Small Trend Slopes): Let M = bT where b 2 (0; 1] is …xed. Let 60 1 = T 1 1; 2 case 1 and 1 =T 1 = 2 0 where +T 1; 1 2 are …xed with respect to T , and let 1 for case 2. Then as T ! 1, (Case 1: = t ; u1t 0 + for and u2t are all I(1) processes.) tIV ) p R1 12 0 (r 12 )w(r)dr p +r Pb (Q1 (s)) t 0 ) p 12 R1 p0 (r 2 ; 1 R1 Pb (Q1 (s)) 1 2 )w(r)dr Pb (Q1 (s)) 2 0 Z tIV ) r +r e Pb K(s) t where Q1 (s) = Rs 0 e and K(s) = w(r) e 0 2 1 )p w(r)dr s 2 J(s) Pb Z Pb (Q(r)) R1 t R1 e K(s) 2 +q 12 Case 2: u1t and u2t are I(1) processes, whereas 2 0 w(r)dr 2 L(s) R1 1 2 0 (r 2 )w2 (r)dr 0 2 ; R1 2 0 ; (r 1 2 2 ) dr : P (Q1 (s)) 0 b is an I(0) process. 1 2 )w2 (r)dr (r +q 12 (r 1 2 )w2 (r)dr 2 2 R1 0 ; 1 2 2 ) dr (r ; 2 P (Q(r)) 0 b R1 2 0 (r 1R1 0 (r 1 2 )w2 (r)dr 1 2 )dw(r): 1R1 0 (r 1 2 )w(r)dr; Some interesting results and predictions about the …nite sample behavior of the t-statistics are given by Theorems 10-12. First examine the limiting null distributions that are obtained when = 0. The bias in the IV estimator for medium and small trend slopes a¤ect the asymptotic distribution of the t-statistic when t( ) is an I(1) process. We get di¤erernt limits for large, medium and small trend slopes both for the cases when t( ) is I(1) as well as I(0). The unit roots in u1t and u2t do not change the critical values if t is an I(0) process, and 1; 2 are large as is evident from the limit of the t-statistics for large trend slopes which is exactly the same as in Chapter one. When 6= 0; in which case we are under the alternative, the t-statistics have additional terms in their limits which push the distributions away from the null distributions giving the tests power. As the limiting distributions for t 0 are di¤erent for I(0) and I(1) regression errors, 61 it may be useful to have a test robust to both I(0) and I(1) errors. A robust test has been proposed in Bunzel and Vogelsang (2005), and the t-statistic is de…ned as follows: t 0 =v u u tb2 0 b1 ( 0 ) T X (t t)2 t=1 ! 1 exp( cU R); exp( cU R) is a scaling factor, U R denotes either the J or BG unit root statistics and c is a constant. J is de…ned as follows: Consider the regression zt ( 0 ) = 0( 0) + 1 ( 0 )t + 9 X i it + vt ( 0 ) ; i=2 then the J statistic is de…ned as J= SSR(1) SSR(2) ; SSR(2) where SSR(2) is the sum of squared residuals obtained from the estimation of above equation by OLS, and SSR(1) be the sum of squared residuals from the OLS estimation of (18). The value of scalar c can be found as follows: c(b) = where the values of 0 + i ’s 1b + 2 2b + 3 3b + 4 4b + 5 5b + 6 6b + 7 7b ; and the methodology for the data dependent bandwidth rule can be found in Bunzel and Vogelsang (2005). For con…dence interval construction, however, a lot more work needs to be done as the scaling factor depends on the true value of which is unknown in practice. This is left as a future research task. 2.5 Finite Sample Null Rejection Probabilities and Power Using the same DGP’s as used in Section 2.3 we simulated …nite sample null rejection probabilities and power of the IV t-statistics both for 62 t( ) as I(0) and I(1) cases and the t-statistics for linear hypothsis (t 0 ) based on I(1) errors as well as robust to the nature of serial correlation in the error term. Tables 2.2a and 2.2b report null rejection probabilities for 5% nominal level tests for testing H0 : sided alternative H1 : = 0 = 2 against the two- 6= 2 for I(1) and I(0) regression errors respectively. Results are reported for the same values of 1; 2 as used in Tables 2.1a and b, for T = 50; 100; 200 and 10; 000 replications are used in all cases. The HAC estimators are implemented using the Daniell kernel. Results for three bandwidth sample size ratios are provided: b = 0:1; 0:5; 1:0. For a given sample size, T , we use the bandwidth M = bT for each of the three values of b. We compute empirical rejections using …xed-b asymptotic critical values as given by Bunzel and Vogelsang (2005) for the Daniell kernel. For IV, and the linear hypothesis for I(0) errors, the critical values speci…c to I(0) errors have been used, whereas for IV, and the linear hypothesis based on I(1) errors, the critical values speci…c to the I(1) errors have been used. For the robust linear test, I(0) critical values have been used. The patterns in the empirical null rejections closely match the predictions of the asymptotic results as in Theorems 10-12. When the trend slopes are large, 4, 1 2, null rejections are essentially the same for all t-statistics and are close to 2 0.05 even when T = 50. This is true for all three bandwidth choices which illustrates the e¤ectiveness of the …xed-b critical values. For medium and small trend slopes, 0:2 1 2, 0:1 2 1, IV begins to show over-rejection problem for the I(1) case, that becomes very severe as the trend slopes decrease in magnitude. In contrast, t null rejections that are close to 0.05 regardless of the magnitudes of the case of 1 = 2 1; 2 0 has including = 0. It is clear in terms of null rejection probabilities that t 0 is the preferred test statistic. For I(0) case, the IV shows under-rejections for small trend slopes, whereas t 0 has null rejections that are close to 0.05 regardless of the magnitudes of trend slopes. Tables 2.3a and b report power results for a subset of the grid of 2.1a and b. For a given value of in the range 2 [2; max ] 2, where 2 as used in Tables we specify a grid of six equally spaced values for = 0 63 = 2 is the null value. By construction 1 = 2 in all cases. Results are reported for T = 100. Results for other values of T are qualitatively similar and are omitted. The power of t 0 large trend slopes ( for I(1) errors is the highest among all three test statistics. For 2 = 10; 2), power of the IV test is lower than that of t 0 for I(1) errors, however, it is higher than the robust t 0 . As the trend slopes decrease in magnitude, the power of IV test decreases substantially. The power of the robust t 0 test is the lowest among all three. The power of the test for I(0) errors decreases as the trend slopes decrease in magnitude. This is evident from the results reported in table 2.3b. These results suggest that if empirical researchers have a reason to believe that the noise term is an I(1) process, it is better to use the test based on I(1) errors as compared to the one which is robust to the nature of serial correlation in the noise term. As the bandwidth increases, power of all the tests decreases. This inverse relationship between power and bandwidth is well known in the …xed-b literature (see Kiefer and Vogelsang 2005). In summary, the patterns in the …nite sample simulations are consistent with the predictions of Theorems 10-12. Clearly t 0 is the recommended statistic given its superior behavior under the null and its higher power under the alternative. 2.6 Unit Root Tests for t( ) The inference section suggests that it is important for the empirical researchers to conduct a unit root test on t( ) to determine which critical values they should use for inference on : In this section, we show how to carry out a unit root test on t( ), and how to improve the power of the unit root test through the ADF-GLS transformation. We focus on the case where t( ) is an AR(1) process. The additional serial correlation can be handled in the usual way by including lagged …rst di¤erences. In order to test the null hypothesis of a unit root in regression equation: t, we need to regress bt on bt bt = bt 64 1 + t: 1 in the following OLS gives T bb t=2 t t 1 : T b2 t=2 t 1 b= Centering the estimator b around one and scaling it by T , we obtain T (b 1) = T b bt 1 ) t=2 t 1 (bt 2 T 2 T t=2 bt 1 1 T = 1 T T T b t=2 t 1 bt : 2 T b2 t=2 t 1 The t-statistic for testing the null hypothesis of a unit root in T (b t b=1 = q s2 T 2 where s2 = 1 T 1) T t=2 (bt 2 1 T b2 t=2 t 1 bbt t( ) is given as follows: ; 2 1) : The next theorem provides the limit of t b=1 under the null that Theorem 13 Suppose that t( 1; ) is an I(1) process and 2 R1 Case 2 (medium trend slopes): For t b=1 ) 1 2 h 2 ; Case 3 (small trend slopes): For 1 h 2 ; t b=1 ) 1 2 1 =T 1=2 1; 2 =T 2 w 2 w b (1)2 ; b (0) q R 1 L 0 w b (r)2 dr =T 1 1; 2 =T 1 2, : 2, 2 w 2 w b (1)2 ; b (0) q R 1 L0 0 w b (r)2 dr 65 1=2 i L L0 ) is I(1). are …xed with respect to T . The following hold as T ! 1. Case 1 (large trend slopes): For b 0 w(r)dw(r) t b=1 ) q : R1 2 dr w(r) b 0 t( i : 1 = 1, 2 = 2, Case 4 (zero trend slopes): For 1 1 2 t b=1 ) = 0; 2b 1w where Z w(r) b = w(s) 2w (1)2 b (0)2 q R 1 00 1 w (r)2 dr 1 L 0 b 1 w(r)dr w b (r) = w(s) R1 2 2 0 Z 1 ) 2 (s 0 R1 = 0, 2 w(r)dr L00 1 1 2 (r ) dr 2 0 h 1 ) + 2 w(s) 2 2 (s ; 1Z 1 (r 0 R1 0 w(r)dr 1 )w(r)dr; 2 i ; 1R1 1 1 (r )w (r)dr (r )w(r)dr 2 0 2 2 2 2 0 i h R1 R1 w(s) 2 w(s) 0 w(r)dr 0 w(r)dr w b (r) = 1R1 R1 1 1 2 0 (r 2 )w2 (r)dr 2 )w(r)dr 0 (r i i h h R1 R1 w (r)dr w (r)dr w (s) w (s) 2 1 2 1 1 0 0 ; w b (s) = 1 R1 R 1 1 1 (r )w (r)dr )w (r)dr (r 2 1 1 2 2 0 0 L= 2 2 ; 6 41 + ; L0 = 2 0 (r 2 2 2 ; ; 00 R1 26 24 " 2 Z (r " 2 1 2 2 2 " Z Z 1 (r 0 1 R1 (r Z 1 2 1 (r R1 2 0 1 1 2 )w2 (r)dr (r (r 0 1 )w2 (r)dr 2 1Z 1 1 )w2 (r)dr 2 1 )w2 (r)dr 2 1Z 1 0 0 1 (r 0 # 1 )w(r)dr ; 2 1 )w(r)dr 2 # #2 3 1 )w(r)dr : 2 (r 1 (r 1Z 1 32 3 7 7 5 5+ 1 2 )w(r)dr 1 )w2 (r)dr 2 (r 0 + 0 2 " Z R1 1 2 2 ) dr (r 0 + + 1 2 ) dr + 2 0 41 + L = 2 0 1 2 2 2 1 2 2 ) dr Z 0 1 (r 1 )w1 (r)dr 2 5+ #2 When the trend slopes are …xed, i.e. when the trend slopes are large relative to the noise, the limit of the unit root test statistic is the same as ADF limit when an 66 intercept and time trend are included in the ADF regression equation. When the trend slopes are not large in magnitude, the asymptotic limits are no longer the same. We can boost the power of the unit root test through the ADF-GLS transformation. Straight forward algebra allows us to write: t ) =( t ) b =( t ) 2 bt = ( b (y2t y2) ; 2 (t b (40) t) + (u2t t) (t u2 ) ; b (u2t u2 ): Now let bGLS = bt t %b1 %b2 t: The estimators %b1 ; %b2 are GLS estimators obtained from regression of where bt = bt b1 = b1 ; bt 1; bt on dt ; on dt : t = 2; :::; T dt = (1; t)0 ; %0 = (%1 ; %2 ) ; dt = dt dt 1; t = 2; :::; T d1 = d1 : De…ne GLS detrended residuals as GLS t = t b1 b2 t: The estimators b1 ; b2 are GLS estimators obtained from regression of 67 t Similarly, b1 ; b2 are GLS estimators obtained from regression of u2t on dt leading to the GLS residuals b1 uGLS = u2t 2t b2 t: Applying GLS detrending to both sides of eq. (40) gives bGLS = t b GLS t uGLS 2t , noting that GLS detrending eliminates anything that is constant (like a sample average) or proportional to t and replaces OLS demeaned (or detrended) quantities with GLS detrended quantities. To test the null hypothesis of unit root in regress t, we bGLS on bGLS t t 1 as follows: OLS gives b= Tb = T bGLS bGLS t t=2 t 1 ; T b2GLS t=2 t 1 T 1 Tt=2bGLS bGLS t t 1 : T 2GLS 2 T t=2 bt 1 The t-statistic for testing the null hypothesis H0 : where tGLS b=0 = q s2 T s2 = 1 T 2 2 T t=2 = 0 is as follows: Tb T b2GLS t=2 t 1 bGLS t 1 bbGLS t 1 The asymptotic limits of ADF-GLS unit root tests for ; 2 t( ) for various magnitudes of beta’s are presented in the following theorem. Theorem 14 Suppose that t( ) is an I(1) process and 68 1; 2 are …xed with respect to T . The following hold as T ! 1. Case 1 (large trend slopes): For 1 2 tGLS b=0 ) Kc (1; c)2 Kc (0; c)2 qR 1 2 0 Kc (r; c) dr Case 2 (medium trend slopes): For tGLS b=0 ) 1 2 2L Case 3 (zero trend slopes): For 1 2 tGLS b=0 ) h 2 ; 1=2 =T 1; 2 =T 1=2 1; ; 2; 2 2 = 2, 2, 2 2 L (0; c; 2 2) c 2) q R 1 z 0 Lc (r; c; 2 )2 dr = 0; 1, : c (1; c; 1 Lc (1; c; 1 1 = 1 z : = 0, )2 2 ; q R 1 z 0 Lc (s; c; Lc (0; c; 1; 2; 1; 2; )2 z )2 dr i : where Kc (c) = 3$wc (1) + 3(1 $) Z 1 rwc (r)dr; 0 $ = (1 1 c) =(1 $ = c2 =(1 Kc (s; c) = [wc (s) 2 6 6 Lc (s; c; 2 ) = 6 6 4 z= +2 2 ; c + c2 ): sKc (c)] 2 wc (s) s c + c2 ); 2 Kc (c) " R1 (r R1 2 0 (r 2 0 Z 1 wc (s) sKc (c) R1 1 2 1 ) dr + 2 2 2 )w2 (r)dr 0 (r R 1 1 1 2 2 ) dr + 2 0 (r 2 )w2 (r)dr 1 2 ) dr + 2 + (r 2 2 0 Z 1 Z 1 1 2 2 (r ) dr + (r 2 2 2 2 0 0 2 2 69 Z 0 1 (r 1R1 0 (r 1R1 0 1 0 Z 0 1 (r 1 2 )w(r)dr (r Z 1 1 )w2 (r)dr 2 1 )w2 (r)dr 2 1 2 )w(r)dr+ 1 (r 3 7 7 7 7 5 1 )w(r)dr 2 1 )w(r)dr 2 #2 z = 2 ; +2 2 2 + 2 2 " " 2 Z 1 (r 0 2 Z 0 1 (r 1 1 )w2 (r)dr 2 1 1 (r 0 1 1 )w2 (r)dr 2 Z 1 Z 0 1 (r 1 )w1 (r)dr 2 1 )w1 (r)dr 2 # #2 : When trend slopes are large in magnitude, the asymptotic limit of tGLS b=0 is the same as the usual ADF-GLS limit obtained by Elliott, Rothenberg and Stock (1996). When the trend slopes are not large in magnitude, the asymptotic limits are no longer the same. 2.7 Finite Sample Null Rejection Probabilities and Power of Unit Root Tests Using the same DGP as used in section 2.3 we simulated …nite sample null rejection probabilities of the unit root tests. For power of the ADF and ADF-GLS tests for t( ) for the cointegration case, the DGP is the same as used in section 2.3 for theorem 9, i.e. u2t = u2t 1 + "2t ; u1t = u2t + "1t "1t = "1t "2t ; t 1 + t; i:i:d: N (0; I2 ); u10 = u20 = 0; whereas for the case where both u1t and u2t are I(0), the DGP is as follows: 70 u1t = 1 u1t 1 + "1t ; u2t = 2 u2t 1 + "2t ; ["1t ; "2t ]0 i:i:d: N (0; I2 ); u10 = u20 = 0: Table 2.4 reports null rejection probabilities for 5% nominal level for testing the null hypothesis of a unit root in value = t( ) (ADF critical value = 3:03). Results are reported for the same values of 3:451, ADF-GLS critical 1; 2 as used in Tables 2.1a and b, for T = 50; 100; 200 and 10; 000 replications are used in all cases. The null rejection probabilities are very similar for both ADF and ADF-GLS and are close to 0.05 in all cases. Table 2.5 reports power results for six equally spaced values of 1 = 2 = , which clearly shows that ADF-GLS test performs better than ADF in terms of power. 2.8 Conclusion In this chapter we analyze estimation and inference of the ratio of trend slopes of two time series with linear deterministic trend functions under the assumption that the stochastic parts of both series are I(1) processes. We consider the IV estimator of the trend slopes ratio both for the cases where the noise term in the IV regression equation follows an I(0) and I(1) process. Asymptotic theory indicates that when the magnitude of the trend slopes are large relative to the noise, the IV estimator tends to remain unbiased, whereas for medium to smaller trend slopes, the IV estimator becomes a poor estimator of the trend slopes ratio when the noise term is an I(1) process. In contrast (see Chapter 1), the IV estimator is consistent for all three cases, i.e. large, medium and small trend slopes when the noise term is an I(0) process. We analyze t-statistics based on the IV estimator for testing hypotheses about the trend slopes ratio both for the I(0) and I(1) regression errors. We also consider the 71 t-statistic based on the alternative linear in slopes testing approach. Simulations show that the alternative test dominates the test based on the IV estimator in terms of size and power regardless of the magnitude of the trend slopes. We propose an ADF-GLS test for a unit root in the noise term of the IV regression equation. The power of the ADF-GLS test is higher than that of ADF test as shown by the …nite sample simulations. The test may help empirical researchers choose a test for the trend slopes ratio with higher power as compared to a test which is robust to the nature of serial correlation in the noise term. Finite sample simulations show that the predictions of the asymptotic theory tend to hold in practice. 72 APPENDIX 73 Proofs of Theorems Before giving proofs of the theorems, we prove a series of lemmas for each of the trend slope magnitude cases: large, medium, small and very small. The lemmas establish the limits of the scaled sums that appear in the estimators of and the HAC estimators. Using the results of the lemmas, the theorems are easy to establish using straightforward algebra and the continuous mapping theorem (CMT). We begin with a lemma that has limits of scaled sums that are exactly invariant to the magnitudes of the trend slopes followed by four lemmas for each of the trend slope cases. Throughout the appendix, we use t to denote t ( ). Lemma 6 Suppose that (34) holds. The following holds as T ! 1 for any values of 1; 2 : Z 1 T X 1 5=2 T (t t)( t )) ; (r )w(r)dr: 2 0 t=1 Proof: The result in this lemma is standard given the FCLT (34). See Hamilton (1994). Lemma 7 (Large trend slopes when t ( ) is an I(1) process) Suppose that (34) holds and 1 ; 2 are …xed with respect to T . The following holds as T ! 1 for 1 = 1 , 2 = 2, Z 1 T X 1 2 3 T (t t)(y2t y 2 ) ! 2 ) ds: (s 2 0 t=1 Proof: The result of the lemma is easy to establish once we substitute y2t t + (u2t u2 ) into the above expression: 2 t T 3 T X (t t)(y2t y2) = 3 2T t=1 T X (t y2 = t)2 + op (1): t=1 The limit follows from Lemma 1 in chapter one. Lemma 8 (Medium trend slopes when t ( ) is an I(1) process) Suppose that (34) holds and 1 ; 2 are …xed with respect to T . The following holds as T ! 1 for 1 = T 1=2 1 , 1=2 2 =T 2, T 5=2 T X (t t)(y2t t=1 y2) ) 2 Z 1 (r 0 1 2 ) dr + 2 2 Z 1 (r 0 1 )w2 (r)dr: 2 Proof: The result of the lemma is easy to establish once we substitute y2t T 1=2 2 t t + (u2t u2 ) into the above expression: T 5=2 T X (t t=1 t)(y2t y2) = 2T 3 T X (t t=1 The limits follow from Lemma 1 in chapter one. 74 t)2 + T 5=2 T X t=1 (t t)(u2t u2 ): y2 = Lemma 9 (Small trend slopes when t ( ) is an I(1) process) Suppose that (34) holds and 1 ; 2 are …xed with respect to T . The following holds as T ! 1 for 1 = T 1 1 , 1 2 =T 2, T 5=2 T X t)(y2t (t y2) ) t=1 2 Z 1 1 )w2 (r)dr: 2 (r 0 Proof: The result of the lemma is easy to establish once we substitute y2t T 1 2 (t t) + (u2t u2 ) into the above expression: T 5=2 T X (t t)(y2t y2) = 7=2 2T t=1 T X (t t)2 + T 5=2 t=1 T X (t t)(u2t y2 = u2 ): t=1 The limits follow from Lemma 1 in chapter one and two. Lemma 10 (Medium trend slopes when t ( ) is an I(0) process) Suppose that are …xed with respect to T . The following holds as T ! 1 for 1 = T 1=2 1 , T 1=2 2 , 3=2 T [sT ] X (y2t t=1 T [sT ] X (y2t y2) = 2 2T t=1 [sT ] X (t t) + T 3=2 t=1 [sT ] X (u2t T y2) ) t=1 T 5=2 T X (t 2 Z s w2 (r)dr s 0 t)(y2t t=1 u2 ): Z 3=2 [sT ] X (y2t t=1 T X T 5=2 (t t=1 t)(y2t y2) = 2T 2 1 are 2, 1 w2 (r)dr 2 J(s); 0 y2) ) 2 Z 1 (r 0 1 )w2 (r)dr: 2 Proof: The result of the lemma is easy to establish once we substitute y2t T 1 2 t t + (u2t u2 ) into the above expression: T y2 = t=1 Lemma 11 (Small trend slopes when t ( ) is an I(0) process) Suppose that 1 ; …xed with respect to T . The following holds as T ! 1 for 1 = T 1 1 , 2 = T [sT ] X (y2t 2 p The limits follow from Lemma 1 in chapter one. 3=2 2 = y 2 ) ! L(s): Proof: The result of the lemma is easy to establish once we substitute y2t T 1=2 2 t t + (u2t u2 ) into the above expression: 3=2 1; 5=2 [sT ] X (t t) + T t=1 T X y 2 ) = 2 T 7=2 (t t=1 75 t)2 + T 3=2 [sT ] X (u2t t=1 T X 5=2 t=1 (t u2 ) t)(u2t u2 ) y2 = The limits follow from Lemma 1 in chapter one and two. Proof of Theorem 8. The proof follows directly from Lemmas 1, 2, 3, 4 and the CMT. For the case of large trend slopes it follows that T b 1=2 = 3 T T X (t t)(y2t y2) t=1 ) Z 2 1 1 1 2 ) dr 2 (r 0 ! ; 1 5=2 T T X (t t)( t ); t=1 Z 1 1 )w(r)dr: 2 (r 0 For the case of medium trend slopes it follows that b = 5=2 T ) 2 Z T X t)(y2t (t y2) t=1 1 1 2 ) dr + 2 (r 0 2 Z ! 1 5=2 T T X (t t)( ); t t=1 1 (r 0 Z 1 1 )w2 (r)dr 2 ; 1 1 )w(r)dr: 2 (r 0 For the case of small trend slopes it follows that b = 5=2 T ) 2 Z T X (t t)(y2t y2) t=1 1 (r 0 ! 1 Z 1 1 )w2 (r)dr 2 5=2 T ; T X t)( (t ); t t=1 1 1 )w(r)dr: 2 (r 0 Proof of Theorem 9. The proof follows directly from Lemmas 1, 2, 3, 4, 5, 6 and the CMT. For the case of large trend slopes it follows that T b 3=2 = 3 T T X (t t)(y2t ! y2) t=1 ) 2 Z 1 (r 0 1 2 ) dr 2 1 1 T Z 3=2 T X (t t)( t ) t=1 1 (r 0 1 )dw(r): 2 For the case of medium trend slopes it follows that T b = ) 5=2 T 2 Z 0 T X (t t)(y2t y2) t=1 1 (r 1 2 ) dr + 2 ! 2 Z 1 T 1 (r 0 For the case of small trend slopes it follows that 76 3=2 T X (t t)( t=1 1 )w2 (r)dr 2 1 ) t Z 0 1 (r 1 )dw(r): 2 T b = 5=2 T ) 2 T X (t t)(y2t y2) t=1 Z 1 0 1 3=2 T T X (t t)( ) t t=1 Z 1 1 )w2 (r)dr 2 (r ! 1 1 )dw(r): 2 (r 0 Proof of Theorem 10. With large trend slopes, the scaling and limits of the partial sum processes are given as follows where the limits follow from Lemmas 1, 2 and Theorems 1, 2. Case 1: T ) = 3=2 b S[sT ] ; Z [rT ] X 3=2 =T ( t=1 s w(r)dr s ; 0 ; t Z T 1=2 b ) 1 w(r)dr Z L(s) 0 Q1 (s): 2 T [rT ] X 1=2 b S[rT ] ) y2) t=1 1 1 1 2 ) dr 2 (r 0 Case 2: T (y2t ; Z 1 (r 0 1 )w(r)dr 2 Q(r). Using …xed-b algebra and arguments from Kiefer and Vogelsang (2005), it follows that Case 1: T 2 b2 ) 2; Pb (Q1 (s)); Case 2: b2 ) 2 Pb (Q(r)) : The limits of tIV for case 1 and 2 are as follows: Case 1: tIV = v u u tT =v u u tT R1 (r r " p 12 3 T 3 T 2 ; T X (t 0) # t)(y2t T 1=2 ( b T X (t 1) t)(y2t y2) 1 ; 1 R1 0 Pb (Q1 (s)) (r 12 )w(r)dr p0 +q Pb (Q1 (s)) 12 77 2 ; # t)2 (t t=1 ; 2 T 3 T X 1 2 2 ) dr t)2 (t t=1 1 2 )w(r)dr (r 2R1 2 0 (r R1 3 T T X + t=1 1 ; 2 y2) t=1 1 2 2 ) dr 2 0 ) = 2 b2 2 b2 " T 1=2 ( b + ; 1 2 P (Q1 (s)) 1 b : Case 2: tIV = v " u u tb2 T =v " u u tb2 T R1 = p t)(y2t T 3=2 ( b 1) t)(y2t 1 1 R1 0 R1 (r 12 )dw(r) p0 +q Pb (Q(s)) 12 Z +q =p Pb (Q(s)) 12 ; 2 3 T T X t)2 (t t=1 ; T 3 T X (t t)2 t=1 1 2 )dw(r) (r 2R1 2 0 (r Pb (Q(s)) # 2 y2) 1 1 2 2 ) dr 2 y2) + # t=1 (r r 12 (t 0) t=1 T X 3 (t 2 0 ) 3 T X T 3=2 ( b + ; 1 1 2 2 ) dr 2 2 1 Pb (Q(s)) 2 ; 2 P (Q(s)) 1 b R1 1 . using the fact that 0 (r 21 )dw(r) N 0; 12 Proof of Theorem 11. With medium trend slopes, the scaling and limits of the partial sum processes are given as follows where the limits follow from Lemmas 1, 2, 3, 4 and Theorems 1, 2. Case 1: T ) 3=2 b S[sT ] ; Z =T w(r)dr 0 ; Case 2: T ) w(s) e 2 Q1 (s): Z s ; ( ) t Z b 1 1 1 2 ) dr + 2 (r 0 2 Z 0 =T 1=2 [rT ] X 2 Z (r [rT ] X (y2t y2) 1 Z t=1 1 1 )w2 (r)dr 2 (r 0 ( ) t t=1 1 T 3=2 w(r)dr 0 1=2 b S[rT ] 2 L(s) [rT ] X t=1 s 2 L(s) = 3=2 T b Z 1 1 2 ) dr + 2 (r 2 0 = Q0 (s): T 1 )w2 (r)dr 2 ; 1 (r 0 3=2 [rT ] X t=1 1 (y2t Z 0 1 )w(r)dr 2 y2) 1 (r 1 )dw(r) 2 Using …xed-b algebra and arguments from Kiefer and Vogelsang (2005), it follows that 78 Case 1: T 2 b2 ) 2; Pb (Q1 (s)); Case 2: b2 ) 2 Pb Q0 (s) : The limits of tIV are as follows: Case 1: tIV = v u u tT =v u u tT )r = p 2 b2 " R1 2 0 2 ; 12 +r 2 b2 T (r " T 5=2 5=2 T X (b (t 0) t)(y2t T X (t 1) 1 2 2 ) dr Pb (Q1 (s)) + R1 2 0 2 0 (r y2) Pb (Q1 (s)) R1 2 0 (r # 1 2 2 ) dr + 1 2 2 ) dr + 79 t)2 (t t=1 ; 2 3 T T X 1 2 )w2 (r)dr (r (r 12 )w(r)dr p0 Pb (Q1 (s)) 2 ; 3 T T X + t)(y2t R1 ; 2 y2) t=1 (b t=1 R1 # R1 2 0 R1 2 0 (r (r t)2 (t t=1 1 ; R1 0 (r 1 2 )w2 (r)dr 1 2 )w2 (r)dr 1 2 )w(r)dr 2 R1 0 2 R1 0 (r + ; 1 2 2 ) dr : (r 1 2 2 ) dr Case 2: tIV = v " u u tb2 T =v " u u tb2 T )r = p 5=2 1 2 +r 2 1 T X (t Pb 1) + R1 2 0 y2) y2) R1 (Q0 (s)) 2 0 R1 (Q0 (s)) 2 0 R1 2 0 (r ; 2 3 T T X t)2 (t t=1 + t)(y2t Z =p Pb (Q0 (s)) 1 t)(y2t T (b 1 2 2 ) dr (r Pb (Q0 (s)) Pb # t=1 R1 12 0 (r 12 )dw(r) p Pb (Q0 (s)) +r (t 0) t=1 R1 2 0 2 5=2 T X T (b # ; 2 3 T T X t)2 (t t=1 1 1 2 )w2 (r)dr (r 1 2 2 ) dr (r 1 2 2 ) dr (r 1 2 2 ) dr + + + R1 2 0 R1 2 0 R1 2 0 1 R1 0 1 2 )w2 (r)dr (r 1 2 )dw(r) (r (r 1 2 )w2 (r)dr (r 1 2 )w2 (r)dr 2 R1 0 2 R1 0 2 R1 0 + (r ; 1 2 2 ) dr ; (r 1 2 2 ) dr (r 1 2 2 ) dr : Proof of Theorem 12. With small trend slopes, the scaling and limits of the partial sum processes are given as follows where the limits follow from Lemmas 1, 2, 3, 4, 5, 6 and Theorems 1, 2. Case 1: T ) 3=2 b S[sT ] ; Z = =T ; [rT ] X t=1 s w(r)dr 0 2 L(s) 3=2 2 Q1 (s): Z 0 s ( ) t Z 1 w(r)dr b T (r 1 )w2 (r)dr 2 80 [rT ] X (y2t t=1 0 1 3=2 1 ; Z 0 1 (r 1 )w(r)dr 2 y2) 1=2 b S[rT ] Case 2: T =T 1=2 [rT ] X ( =T [rT ] X ( T b ) t t=1 ) t 1=2 2 J(s) 2 Z 1 0 t=1 T Z 1 (y2t 1 (r 0 e K(s) ) [rT ] X y2) t=1 1 )w2 (r)dr 2 (r 3=2 1 )dw(r) 2 Using …xed-b algebra and arguments from Kiefer and Vogelsang (2005), it follows that 2 b2 Case 1: T Case 2: b2 ) 2 The limits of tIV are as follows: Case 1: tIV = v u u tT =v u u tT )r = p 2 b2 2 b2 " R1 2 0 2 ; T (r " T 5=2 5=2 T X (b (t 2 ; ) Pb (Q1 (s)); e Pb K(s) : 0) # t)(y2t y2) t=1 T X (b (t 1) y2) t=1 Pb (Q1 (s)) R1 2 0 (r R1 12 0 (r 12 )w(r)dr p +r Pb (Q1 (s)) 3 T T X 1 ; # R1 0 t=1 ; 2 T (r 1 2 )w2 (r)dr 2 ; Pb (Q1 (s)) 81 t)2 (t + t)(y2t 1 2 )w2 (r)dr ; 2 3 T X (t t)2 t=1 1 2 )w(r)dr 2 R1 0 (r R1 2 0 (r + ; 1 2 2 ) dr 1 2 )w2 (r)dr 2 R1 0 : (r 1 2 2 ) dr Case 2: tIV = v " u u tb2 T =v " u u tb2 T )r p = 5=2 1 (t 0) # t)(y2t T X (t T (b 1) t)(y2t R1 e Pb K(s) 2 0 y2) Z =r +r e Pb K(s) 2 1 # ; 2 3 T T X 0 1 2 1 t)2 (t t=1 R1 1 (r Pb t)2 (t t=1 1 2 )dw(r) (r 0 R1 e Pb K(s) 2 0 R1 e K(s) R1 2 1 2 )w2 (r)dr R1 12 0 (r 12 )dw(r) r +r e Pb K(s) 3 T T X + 1 2 )w2 (r)dr (r ; 2 y2) t=1 t=1 R1 2 0 2 5=2 T X T (b 2 0 + ; (r 1 2 2 ) dr (r 1 2 )w2 (r)dr (r R1 0 R1 2 1 2 )w2 (r)dr 2 0 (r ; (r 1 2 2 ) dr : 1 2 2 ) dr Proof of Theorem 13. The asymptotic limit of the unit root test statistic for t ( ) for all the three cases is derived as follows: Case 1, 1 = 1 ; 2 = 2 : T 5=2 T t=1 ( t T t=1 ( t as T t=1 ( t 5=2 T t=1 (t Z 1 5=2 y2) = T ) (y2t = 2T ) 2 ) t) ( t) + (u2t u2 ) ) + op (1) t 1 )w(r)dr; 2 (r ; 2 (t 0 u2 ) = Op (T 2 ): ) (u2t Also, the following holds: T ) ; 1=2 Z w(s) 1=2 =T [sT ] [sT ] T T w(r)dr T t=2 ( t 1 1 3=2 T t=2 [sT ] 1 ; N (s); 0 2 T y2[sT ] 2 ) =T y2 = T ! 1 1 2 (s T t=1 h T 2 ([sT ] 1=2 ( t 1 ) i2 t) + (u2[sT ] ) u2 ) 2 ; Z 1 N (r)2 dr 0 1 ): 2 For the limit of the unit root test statistic, we need to compute the limit of the following 82 expression: T (b T 1) = T b t=2 t 1 bt : 2 T b2 t=2 t 1 1 T Let us …rst derive the asymptotic limit of the denominator as follows: bt = ( ) t After scaling the partial sums of bt by T T 1=2 1=2 b[sT ] = T ) " ; ; where T 1=2 b [sT ] Z w(s) 1 w(r)dr b (y2t 1=2 we get 1 T (s y2[sT ] Z 1 ) 2 0 y 2 ): 1 (r 0 y2 1Z 1 1 2 ) dr 2 1 )w(r)dr 2 (r 0 # w(s); b Z w(s) b = w(s) 1 w(r)dr (s 0 1 ) 2 Z T 1=2 1 1 2 ) dr 2 (r 0 1Z 1 (r 0 1 )w(r)dr: 2 It follows that T T 2 t=2 bt 1 2 1 =T T t=2 bt 2 ) 1 2 ; Z 0 1 w(r) b 2 dr: The asymptotic limit of the numerator is obtained as follows. Straightforward calculations give 2 + 2t = 2 + 2 (t y2t = 2 + y2t y2t = y2t 1 In order to compute the limit of T bt = ( t T t=2 1 ) First di¤erence both sides to obtain bt bt 1 = bt = Using the formula for t bt gives + u2t ; 1) + u2t 1 1; + w2t : b2t ; we proceed as follows: b b (y2t y2t = 83 t y2) : b (w2t + 2 ): T T t=2 1 b2t = T =T 1 T 2 t=2 t +T 1 T 2 t=2 t +T b 2T 1=2 b T T 2 T b2 ; t=2 t 1 + 2) 2 2T 1 2 2) + T t=2 t b (w2t + : and T 1 T t=2 b2t ; we obtain the limit of T = T 1 Tt=2 (bt 1 + bt )2 = T 1 Tt=2b2t 1 + T 1 Tt=2 b2t + 2T 1 bt = T 1 Tt=2b2t T 1 Tt=2b2t 1 T 1 Tt=2 b2t 2 1 = T 1 (b2T b21 ) T 1 Tt=2 b2t 2 2 1 2 2 2 ) w(1) b 2 w(0) b 2 = w(1) b 2 w(0) b 2 1 : 2 2 T b t=2 t 1 1 1 T t=2 bt 1 Now the asymptotic distribution of t-statistic is as follows: T (b t b=1 = q s2 T 2 2 ; 2 b 2 [w(1) R 2 )r ; 2 ; 2 ; 1) 1 T b2 t=2 t 1 1 0 w(0) b 2 1] w(r) b 2 dr R1 0 1 w(r) b 2 dr = R1 b w(1) b 2 w(0) b 2 1 0 w(r)dw(r) qR = q ; R 1 1 2 dr 2 dr w(r) b w(r) b 0 0 1 2 which is the usual DF limit for t b=1 when an intercept and time trend are in DF regression. It is easy to show that plims2 = 2; as follows: s2 = = = = 1 T 2 1 T 2 1 T 2 1 T Case 2, 2 1 T t=2 (bt T t=2 ( T t=2 T t=2 =T bt 2 1) bbt (b T 2 T 1; 2 2 1) 1 p 2 ; =T 1=2 b2t + op (1) ! 1=2 1 T 1)bt 2 b2t = T t=2 (bt 2 T t=2 bt 1 : 2) 2) T 2 t=2 bt 1 T t=2 bt 1 1 p (w2t + T t=2 (w2t 2 T T t=2 t (w2t + op (1) ! 2 Using the limits of T as follows: 2 b 2 3=2 T T 2 t=2 t 1 =T T t=2 1 bt bt T (b 2 84 1 (b 1) + 1)bt 1 T 2 T 1) 2 2 T 2 2 t=2 bt 1 T (b 1)2 bt bt 1=2 T y2[sT ] 1=2 b[sT ] = T ) ; ; This implies that R1 w(s) 4 R1 2 T 2 t=2 bt 1 2 1=2 (u2[sT ] u2 ) Z 1 w(r)dr : w(s) 0 0 b 1=2 ; b =T 1=2 T w(r)dr 0 (r 1 2 2 ) dr + T t=2 T 1 (y2t y2) we get y2[sT ] y2 h 1 ) + w(s) 2 2 2 (s 2 w b (s): 2 T [sT ] 2 1 )+ 2 ) t Scaling the partial sums of bt by T 1=2 t) + T 1); the asymptotic limit of the denominator is as follows: bt = ( T 2 (t 2 (s ) In the expression for T (b 1 y2 = T R1 1=2 bt 2 1 ) 2 ; Z 1 0 The asymptotic limit of the numerator is obtained as follows: T 1 =T T t=2 t w2t = T 1 T t=2 w1t w2t In order to compute the limit of T bt bt 1 = bt = Using the formula for t b 1 1 T t=2 1R1 1 2 )w2 (r)dr 0 (r R1 0 w(r)dr i 1 2 )w(r)dr 0 (r w b (r)2 dr: T w2t )w2t t=2 (w1t 1 T 2 2 T t=2 w2t ! 2: b2t ; we proceed as follows: y2t = bt ; we obtain 85 t b (w2t + T 1=2 2 ): 3 5 T 1 T t=2 T 2 t=2 t b2t = T 1 =T 1 T 2 t=2 t 2 b T 2 2 ; ) 2 ; 2 + b T t=2 t (w2t 1 R1 24 2 " 2 Z 2 0 1 +T 1 2 ) dr + 2 (r 0 1=2 2 Z 2 2) +T 1=2 2 2) 2) R1 1 2 2 ) dr + R1 0 (r (r 1=2 2) T t=2 (w2t 1 T (w2t + T 1=2 (w2t + T 2 41 + 2 2 b T t=2 t 1 2T 2 b T t=2 1 +T 1 2 )w2 (r)dr 2 0 (r 1 2 )w(r)dr 1 (r 0 1Z 1 1 )w2 (r)dr 2 1 32 3 5 5+ 1 )w(r)dr 2 (r 0 # L: 1 T T b2 ; t=2 t 1 2 Using the limits of T as follows: T t=2 bt 1 and T 1 T t=2 1 b2t ; we obtain the limit of T 1 T 1 (b2T b21 ) 2 1 2 ) b (1)2 ; w 2 bt = T t=2 1 T w b (0)2 2 ; T b t=2 t 1 b2t L Now the asymptotic distribution of t-statistic is as follows: T (b t b=1 = q s2 T 2 1 2 2 [ ; )r L R1 2 ; 0 where s2 = = = = 1 T 2 1 T 2 1 T 2 1 T Case 3, 2 1 T t=2 (bt T t=2 ( T t=2 b2t T t=2 =T bt 1 bbt 2 1) (b w b = (r)2 dr 1 T 1)bt 2 T 2 T 2 1) 1 p b2t + op (1) ! L: 1; 2 1 T b2 t=2 t 1 w b (1)2 L] R1 b (r)2 dr 0 w ; 2 1) =T 1 1 = T t=2 (bt 2 T t=2 bt 1 1 2 h bt bt T (b 2 86 2 2 w w b (1)2 ; b (0) q R 1 b (r)2 dr L 0 w ; 2 ; 1 (b 1) + 1)bt 1 T 2 T 1) i L ; 2 2 T 2 2 t=2 bt 1 T (b 1)2 bt 1=2 T y2[sT ] ) In the expression for T (b follows: We know that 1=2 b[sT ] = T ) ; T 2 t=2 bt 1 b ) 1=2 ; (y2t y2) : we get b [sT ] 1=2 T R1 h w b (s): T t=2 1 =T 1=2 T bt 2 1 Z 2 ; ) 1 0 The asymptotic limit of the numerator is obtained as follows: bt bt T t=2 b2t = T 1 =T 1 T 2 t=2 t 2 b T 1 bt = = Using the formula for T 1 ) 2 2 ; ; 41 + Using the limits of T as follows: 2 2 + b " Z 2 Z +T 1 (r (r 0 T b2 ; t=2 t 1 1 )w2 (r)dr 2 and T 1 87 T t=2 1 2) 1 2) 1 2 ): 2 2) +T 2 2) 1Z 1 1 )w2 (r)dr 2 0 1 1 w b (r)2 dr: (w2t + T (w2t + T T t=2 (w2t 1 T T t=2 t (w2t 1 1 (w2t + T b t 2 b T t=2 b 2 2 2 1 +T T t=2 t " y2t = bt ; we obtain 2 2 2 b t T 2 t=2 t 1 2T u2 ) y2[sT ] y 2 i 3 R1 w(r)dr w(r)dr w(s) w(s) 2 0 0 4 5 1R1 R1 1 1 )w (r)dr (r )w(r)dr 2 0 (r 2 2 2 0 It follows that T t) + T 1=2 (u2[sT ] Z 1 w(r)dr : 0 2 ; 2 w(s) 2 t Scaling the partial sums of bt by T 1=2 2 (t 1); the asymptotic limit of the denominator is obtained as bt = ( T 3=2 y2 = T 1Z 1 0 0 (r (r 1 )w(r)dr 2 # 1 )w(r)dr 2 #2 3 5+ L0 : b2t ; we obtain the limit of T 1 T b t=2 t 1 bt 1 T 1 (b2T b21 ) 2 1 2 ) b (1)2 ; w 2 T t=2 bt 1 1 T bt = T t=2 T 1 2 ; w b (0)2 b2t L0 : Now the asymptotic distribution of t-statistic is as follows: T (b tGLS b=1 = q s2 T 2 1 2 [ 2 ; )r w b ; where s2 = = = = 1 T 2 1 T 2 1 T 2 1 T Case 4, follows: Now 2 1 T t=2 (bt T t=2 ( T t=2 T t=2 = 0; bt b2t 1 T b2 t=2 t 1 (1)2 R1 2 2 ; L0 1) 0 2 bbt 0 (r)2 dr w b R1 = 1 T 1)bt 2 T 2 2 1) 1 T p b2t + op (1) ! L0 2 1 w b (r)2 dr 2 1) (b (0)2 L0 ] w b ; 1 2 = T t=2 (bt 2 ; bt 2 T t=2 bt 1 h 2 w 2 w b (1)2 ; b (0) q R 1 L0 0 w b (r)2 dr ; (b 1 bt T (b 1) + 1)bt 1 T 2 T 1) T 1=2 5=2 5=2 T (t t=2 T (t t=2 (u1[sT ] t)u1t ) t)u2t u1 ) ) 1 R1 1 0 (r R1 2 0 (r w1 (s) i ; 2 2 T 2 2 t=2 bt 1 T (b = 0: The asymptotic limit of the IV estimator of b= T T L0 1)2 is derived as 1 2 )w1 (r)dr : 1 )w (r)dr 2 2 Z 1 w1 (r)dr : 0 and T 1=2 (u2[sT ] In the expression for T (b as follows: u2 ) ) 2 w2 (s) Z 1 w2 (r)dr : 0 1); the asymptotic limit of the denominator is obtained bt = (y1t = (u1t y1) u1 ) 88 b (y2t b (u2t y2) ; u2 ) 1=2 ; Scaling the partial sums of bt by T 1=2 T bT 1=2 u2[sT ] u1[sT ] u1 h i h R1 w (s) w (r)dr w2 (s) 1 1 1 0 1 R1 R1 1 1 0 (r 2 )w2 (r)dr 0 (r 1=2 b[sT ] = T ) b 1w It follows that 2 T we get T 2 t=2 bt 1 u2 R1 0 w2 (r)dr 1 2 )w1 (r)dr i (s): =T T t=2 1 T 1=2 bt 2 1 ) 2 1 Z 1 0 (r)2 dr: w b The asymptotic limit of the numerator is obtained as follows: T 1 bt bt T t=2 b u2t = w1t bw2t 2 2 b2t = T 1 Tt=2 w1t + b2 T 1 Tt=2 w2t 2 bT 1 Tt=2 w1t w2t #2 " Z Z 1 1 1 1 1 2 (r ) 1+ )w2 (r)dr (r )w1 (r)dr 1 2 2 0 0 Z 1 Z 1 1 1 1 2& 2 )w2 (r)dr (r )w1 (r)dr (r 1 2 2 0 0 L00 ; 1 = bt = u1t assuming w1t = &w2t + v1t ; where v1t T 1 Tt=2 b2t ; we obtain the limit of T T 1 2 v ): Using the limits T b t=2 t 1 bt as follows: (0; 1 1 T 1 (b2T b21 ) 2 1 2 ) w b (1)2 2 1 T t=2 bt 1 bt = T 2 b 1w 1 T t=2 (0)2 of T 2 T b2 ; t=2 t 1 b2t L00 : Now the asymptotic distribution of t-statistic is as follows: T (b t b=1 = q s2 T 2 1 2 [ )r 2b 1w L00 where 1) T b2 t=2 t 1 (1)2 R 2 1 b 1 0 w 2b 1w R 2 1 b 1 0 w 1 (0)2 L00 ] (r)2 dr (r)2 dr 1 = 89 1 2 2b 1w 2w (1)2 b (0)2 q R 1 00 1 w (r)2 dr 1 L 0 b L00 ; and s2 = = = = 1 T 2 1 T 2 1 T 2 1 T 2 T t=2 (bt T t=2 ( T t=2 bt bbt b2t T t=2 2 1) (b 1 T 1)bt 2 T = 2 1) p bt 2 T t=2 bt 1 1 T T t=2 (bt 2 bt T (b b2t + op (1) ! L00 : (b 1 1) + 1)bt 1 T 2 1) 2 2 T T 2 2 t=2 bt 1 T (b 1)2 Proof of Theorem 14. The asymptotic limit of the ADF-GLS test statistic for t ( ) for large and medium trend slopes is derived as follows: Case 1, 1 = 1 ; 2 = 2 : We already know that t ) =( t ) b =( t ) 2 bt = ( b Let the regression equation of t t 0 y2) ; (y2t 2 (t b t) + (u2t t) (t on dt be as follows: 0 = dt + = ( 1; t = 1 + u2 ) ; b 2t u2 ): (u2t + t; 2) ; 0 dt = (1; t) ; =' t c '= ; T t 1 + t; then we can write t 0 = In the true data generating process, ( 1 ; to (41), we obtain Now b= + dt = T X dt + 2) dt ( = (0; 0) ; and dt ) t=1 dt + (1 t: ) dt 0 ! 1 = 1 T X (41) t = t: cdt 1 =T; t=1 dt d1 = (1; 1) ; c(t 90 Through OLS applied dt 0 dt = [ c=T; 1 t: 1)=T ] for t 2: This implies that T X T X dt ) = (1; 1) (1; 1)+ 0 dt ( t=1 1 0 0 T 1=2 1 T X 0 dt ( (c=T )2 c(t 1)=T ) =T c (1 t=2 Now let DT = DT 0 c (1 (1 c(t c(t 1)=T ) =T 1)=T )2 ; then 1 p 1 dt ) DT = 1= T t=1 + T X (c=T )2 c(t 1)=T ) =T 3=2 c (1 t=2 1 0 0 0 = p 1= T 1=T + T X 0 0 (1 t=2 c (1 c(t 1)=T ) =T 3=2 (1 c(t 1)=T )2 =T 0 c(t 1)=T )2 =T 1 + p Op (1): T Therefore DT 1 T X dt ( dt ) DT ! t=1 1 0 = DT 0 c + c2 =3 1 1 T X 0 dt ( DT 1 T X dt t + dt ) DT = DT 1 (1; 1)0 = DT 1 (1; 1)0 t = t c t 1 =T for t T 1 X p T t=1 t 1 0 0 0 (1 cr)2 0 1 ! dr 1 1 2; 1 =p T 1 =p T 1 1 0 p ! + DT 1 t=1 As Z ; and t=1 Also 1 0 0 0 1 p 0 1 +p T 1 = 3=(1 T X 1; (1 t=2 and T t=2 T 1 X 1+ p T t=2 T T c X T 3=2 t=2 91 t t 1: : (c=T ) t c(t 1)=T ) p (c t )= T c(t 1)=T ) (1 t=2 T X 0 c + c2 ) t = T c X T 3=2 t=2 T t 1 t : t 1; therefore : Let = 1 + c=T , then T 1 p T 1 X =p T j=1 t t j j; [sT ] 1 X [sT ] j wc (s); j ) [sT ] = p T j=1 Z s exp(c (s u))dw0 (u): wc (s) = 1 p T 0 This implies that T 1 X p T t=1 As T (t t=2 1) T 1 X p 1 T t=2 t t c(t = 1 =p T 1 T c X T + (T T 3=2 t=2 1) T 1 t=2 t ; T T 1) t T ) t 1 1 X =p 1 T t=2 (1 c) = p T T 1) T 3=2 t=2 1) T X1 T 3=2 ) T (1 ; 1) T c X t T X t t=2 + op (1) (1 c) = p T (t t 1 T 3=2 t=2 T c2 X + 5=2 t T t=2 t=2 t 1 2 c) wc (1) + c t 1 Z ! + T c2 X (t T 5=2 t=2 + op (1) 1 rwc (r)dr : 0 It follows that DT 1 T X T dt t = DT t=1 = ) " " 1 1 X (1; 1) 1 + p T t=2 0 1 op (1) + (1 ; h + c(t t= R1 0 p (c c(t p 1)=T ) c) wc (1) + c2 92 (1 p1 Op (1) T 1 (1 t 1 t 1 c + wc (r)dr : c T t T c X T c2 X (t T 5=2 t=2 + 1 therefore T t c 0 c(t T 1 X =p T t=2 wc (1) ; Z t )= T 1)=T ) # rwc (r)dr i # T t 1) t 1 This implies that DT (b )= 1 DT T X 0 dt ( dt ) DT t=1 1 0 ) " 0 3=(1 c + c2 ) 1 = ; 1 ! 1 DT 1 where Kc (c) = 3$wc (1) + 3(1 $) Z dt t; t=1 h ; ; Kc (c) T X 1 c) wc (1) + c2 (1 R1 0 rwc (r)dr dt t i # ; 1 rwc (r)dr; 0 $ = (1 $ = c2 =(1 1 c + c2 ); c) =(1 c + c2 ): Similarly, let the regression equation of u2t on dt be as follows: 0 u2t = 0 =( dt + 1; t = 1 + 2t + t; 2) ; 0 dt = (1; t) ; ={ t c {= : T t 1 + w2t ; This implies that b DT = DT 1 T X 0 dt ( dt ) DT t=1 ) 1 1 ! 1 DT 1 T X t=1 : 2 Kc (c) Let us …rst derive the asymptotic limit of the denominator in the expression of T b as follows: bGLS = t Substituting the expressions for bGLS = t = t t b1 (b1 b2 t 1) b (b2 b GLS t GLS ; t and uGLS 2t ; we get h u2t b1 2 )t Scaling the partial sum of bGLS by T t uGLS 2t ; 1=2 ; b i b2 t h we get 93 t ( b1 1) ( b2 i 2 )t : T T 1=2 GLS bt 1=2 GLS b[sT ] =T 1=2 +T 1 =T 1=2 =T 1=2 ) ; 1=2 T t T 1=2 b 1=2 T t (b1 T 1=2 (b2 1) ( b1 1) T 1=2 (b2 [wc (s) sKc (c)] T 1=2 T 1=2 (b2 Op (1) [sT ] 1=2 +T t T b 2) 2) T 1 tT b 1=2 T 1=2 ( b2 2) t T t + op (1); T [sT ] + op (1) T ; Kc (s; c): 2) This implies that T 2 T 2GLS t=2 bt 1 =T T t=2 1 2 1=2 GLS bt 1 T Z 2 ; ) 1 Kc (r; c)2 dr: 0 The asymptotic limit of the numerator in the expression of T b is obtained as follows: bGLS = t bGLS t 1 = T bGLS t 1 T t=2 (b1 t (b1 h t 1 b bGLS t 1 = b2GLS =T t t T t=2 2 t 3 T t=2 T b 3=2 + 2T 2 + 2T 2 p t 1 1 2T 5=2 2T 5=2 2 ; Using the limits of T 2 is obtained as follows: (b2 2) T t=2 T (b2 2 2 T ( b2 t T 1=2 ( b2 1=2 T 1=2 (b2 t=2 T 2 T b2GLS ; t=2 t 1 and T b 2 )T 1=2 T 1=2 ( b2 1 T t=2 94 h 1) ( b1 t 2 )(t b 2) T t=2 2 )T b ( b2 1) T b : 2 )(t ( b1 T 1=2 b T 1=2 (b2 2 )t (b2 +T T 1=2 b 2T (b2 1) bGLS = t +T ! 1) t 2) 2 2 +T 2T i 1) ; + b 3=2 1 T b ( b2 2 T 1=2 (b2 i ( b2 1) T 2 )t ; 2) 1 T t=2 2 t T 2 ) t=2 t T bGLS t=2 t 1 bGLS t t T 2 ) t=2 t T t=2 t b T 2 ) t=2 T 1=2 ( b2 2) t b2GLS ; the limit of T t 1 T T 1 1 T 2GLS t=2 bt T GLS t=2 bt 1 bGLS t 1 =T T GLS t=2 bt 1 1 T GLS t=2 bt 1 + bGLS t 2 =T 1 T 2GLS t=2 bt 1 +T 1 T t=2 + 2T bGLS ; t 1 T 1 Tt=2b2GLS T 1 Tt=2b2GLS T 1 Tt=2 b2GLS = t t 1 t 2 1 T 1 b2GLS b2GLS T 1 Tt=2 b2GLS = T 1 t 2 1 2 2 2 2 2 ) ; Kc (1; c) ; Kc (0; c) ; 2 2 ; = Kc (1; c)2 2 Kc (0; c)2 b2GLS t 1 : Now the asymptotic distribution of t-statistic is as follows: t b=1 = q s2 T Tb 2 ; 2 2 [Kc (1;c) R 2 ; )r 2 ; 2 ; 1 0 Kc (0;c)2 1] Kc (r;c)2 dr R1 0 s2 = = = 1 T 1 T 2 1 T 2 2 T t=2 T t=2 T t=2 2 ; T 2 T bbGLS t 1 2 1 p b2GLS + op (1) ! t Case 2, 1 = T 1=2 1 ; 2 = T We already know that bt = ( T t=2 2 T t ) =( t ) =( t ) 1=2 2 ; T 1=2 1 2 Kc (1; c)2 Kc (0; c)2 qR 1 2 0 Kc (r; c) dr bGLS t bbGLS t 1 : 2 T GLS t=2 bt 1 : (y2t y 2 ) ; h T 1=2 2 (t 2 1 2 bGLS Tb + t 1 T 2 T 2 T 2GLS 2 2 t=2 bt 1 T b 2: b b = as follows: 1 bGLS t b2GLS t 1 Kc (r; c)2 dr It is easy to show that plims2 = s2 = s2 = 1 T e2 t=2 t 1 2 b (t 95 t) + (u2t t) b i u2 ) ; (u2t u2 ): T t=2 t w2t = T 1 T t=2 w1t w2t 1 T =T T w2t )w2t t=2 (w1t 1 T 2 2 T t=2 w2t ! 2: 1 c c p 2 t 1 + t )( u2t 1 + w2t ) ! 2: T T The asymptotic limit of the denominator in the expression of T b can be derived as follows: T T t=2 1 bGLS = t b1 t = =T 1) b b2 t (b1 t t t T t=2 ( 1 h u2t (b2 T 1=2 GLS bt =T 1=2 +T 1=2 =T 1=2 1=2 T t b 1=2 ; (b1 ( b1 we get + b 2) t T 2) T 1=2 [sT ] T T t 1=2 GLS b[sT ] 1=2 =T + b ) 2 ; ; 6 6 6 6 6 6 6 4 T 1=2 (b2 [sT ] T 1=2 ( b2 R1 2 0 Lc (s; c; s 2) 2) 2 Kc (c) + R1 2 0 2 ): R1 2 0 (r t t T + b i )t : 2 b T 1=2 ( b2 2 T 2GLS t=2 bt 1 =T 1 T t=2 T 1 2 )w2 (r)dr R1 1 2 ) dr + 2 0 (r R 12 1 2 )w(r)dr 0 (r 1=2 GLS bt 1 2 ) 2 ; Z 7 7 7 1 7 2 )w(r)dr+ 7 0 (r 7 1 1 7 )w (r)dr 2 5 2 1R1 1 0 The asymptotic limit of the numerator is obtained as follows: 96 t T 3 It follows that T 2) b [sT ] sKc (c) 2 wc (s) (r 1=2 2) [sT ] + op (1) T wc (s) 1 2 2 ) dr (r T b 1=2 ( b2 1) t T T 1=2 ( b2 + op (1); T ( b1 t T 1=2 (b2 1) 1) T 1=2 (b2 t b 2 )t by T Scaling the partial sum of bGLS t i b2 t h b1 Lc (r; c; 2 2 ) dr: bGLS = t bGLS t 1 = T bGLS t 1 b b2GLS =T t 1 T t=2 2 T t=2 2T + 2T 1 3=2 2 2T 3=2 2 ; b T ( b2 2 24 2 + 2 z: Z 0 (r T 1 1 T 2GLS t=2 bt T GLS t=2 bt 1 bGLS t =T 1 2) T GLS t=2 bt 1 1 T GLS t=2 bt 1 + 1 2 + 2T 3=2 3=2 t T 2 ) t=2 1 2 2 ) dr T t=2 bGLS t + b b 2 =T 1 ; 2 ); T t=2 2 t T 1=2 ( b2 t T 2 ) t=2 t 2) t R1 0 (r 1 2 )w(r)dr 1 1 )w2 (r)dr 2 1 T 2GLS t=2 bt 1 32 1 1 2 )w2 (r)dr ; Z Now the asymptotic distribution of t-statistic is as follows: 5 1 (r 0 1 +T T bGLS t=2 t 1 1 T t=2 + 2T bGLS ; t 1 T 1 Tt=2b2GLS T 1 Tt=2b2GLS T 1 Tt=2 b2GLS = t t 1 t 2 1 = T 1 b2GLS b2GLS T 1 Tt=2 b2GLS T 1 t 2 1 2 2 2 2 z : ) ; Lc (0; c; 2 ) ; Lc (1; c; 2 ) 2 97 2 )t T 2 ) t=2 T 1=2 (b2 ; the limit of T b2GLS t 2 ( b2 T i ( b2 1) i 1) ; + b T 1=2 ( b2 + 2 R1 ; 0 (r Z 1 1 2 ) dr + 2 (r 2 0 Using the limits of T 2 Tt=2b2GLS t 1 ; and T can be found as follows: T t 2T b ( b1 t 2 )(t T t=2 T 1=2 ( b2 2 0 (r b 2) R1 1 t 1) 2 2) t 2) 2 b b T t=2 T (b2 T t=2 T 1=2 (b2 2 2 +2 2 b 2) 2 +T h ( b2 1) (b2 t 2 t 2 )(t ( b1 t 1 b 2 )t (b2 T 1=2 (b2 t=2 T 2T p (b2 1) bGLS = t +T ! 1) (b1 h t 1 bGLS t 1 = T t=2 (b1 t 1 )w(r)dr 2 bGLS t b2GLS t t b=1 = q 1 2 s2 T 2 [ ; Tb 2 Lc (1;c; 2 ; ) r 2 ; z 1 T e2 t=2 t 1 2 2 L (0;c; c 2) ; R1 L (r;c; )2 dr c 2 0 R1 0 Lc (r; c; 2) 2 z] 1 2 2 ) dr = 1 2 h 2 ; 2 L (0; c; 2 Lc (1; c; 2 )2 c 2) ; q R 1 z 0 Lc (r; c; 2 )2 dr ; i z : It is easy to show that plims2 = z as follows: s2 = s2 = = = 1 T 1 T 2 1 T 2 2 T t=2 T t=2 T t=2 b2GLS t bGLS t 2 T bbGLS t 1 2 T p 1 b2GLS + op (1) ! z: t 2 T GLS t=2 bt 1 Case 3, 1 = 0; 2 = 0: The asymptotic limit of the IV estimator of b) R1 1 0 (r R1 2 0 (r bGLS Tb + t 1 T 2 T 2 T 2GLS 2 2 t=2 bt 1 T b is as follows: 1 2 )w1 (r)dr : 1 2 )w2 (r)dr The asymptotic limit of the denominator in the expression of T b can be obtained as follows: 98 bGLS = t = T 1=2 GLS bt b1 t (b1 t =T 1=2 +T 1=2 =T 1=2 1) b t (b2 1=2 T t h u2t b b2 t b1 b 2 )t (b1 1) ( b1 1) T 1=2 (b2 T 1=2 (b2 + b 2) t T b2 t h 1=2 GLS b[sT ] 1=2 =T + b ) ; ; 2 6 6 6 6 4 T 1=2 (b2 [sT ] T 1=2 ( b2 1 ; s 1 Lc (s; c; 2) wc (s) 2 Kc (c) ; 1; 2) 2; ; t t T 2) ( b1 T 1=2 [sT ] T T t T 1=2 [sT ] ): 1 2 )w2 (r)dr ( b2 2 )t b i T 1=2 ( b2 1 1 2 )w1 (r)dr R1 1 0 (r 1 2 )w1 (r)dr It follows that T 2 T 2GLS t=2 bt 1 =T 1 T t=2 T 1=2 GLS bt 1 2 ) 2 ; Z 0 1 Lc (s; c; The asymptotic limit of the numerator is obtained as follows: 99 2) t T b [sT ] + op (1) T wc (s) sKc (c) 1 R1 R1 1 (r )w (r)dr 2 1 0 (r 2 0 R1 2 0 (r t 2) b 1=2 1) t T + b T 1=2 ( b2 + op (1) T i 1; 2; ; )2 dr: 3 7 + 7 7 7 5 bGLS = t bGLS t 1 = T bGLS t 1 b b2GLS =T t 1 T t=2 2 T t=2 1 2T 3=2 + 2T 2 2T 3=2 2 ; +2 b T ( b2 " " 2 2 b T 1=2 ( b2 Z 1 1 (r 0 b t 2) 2 2T + 2T 3=2 b i 1) ; + b 3=2 b t T 2 ) t=2 ( b2 1 T 2 )t 2) T t=2 T 2 ) t=2 T 1=2 (b2 T 1=2 ( b2 T 1=2 ( b2 2 t t T 2 ) t=2 t 2) t 1 1 Z 1 1 Z 1 #2 1 )w1 (r)dr 2 # (r 0 1 1 )w2 (r)dr 2 2 i ( b2 1) + b 1 )w2 (r)dr 2 (r ( b1 t 2 )(t T t=2 2) 0 Z t 1) 2 2) t 2) 2 b 2 2 T t=2 T (b2 T t=2 T 1=2 (b2 + 2 2 2 b 2) 2 +T h ( b2 1) (b2 t 2 t 2 )(t ( b1 t 1 b 2 )t (b2 T 1=2 (b2 t=2 T 2T ! (b2 1) bGLS = t +T p 1) (b1 h t 1 bGLS t 1 = T t=2 (b1 t 1 )w1 (r)dr 2 (r 0 z : Using the limit of T 2 Tt=2b2GLS t 1 ; and T can be obtained as follows: T T 1 1 T 2GLS t=2 bt T GLS t=2 bt 1 bGLS t =T 1 T GLS t=2 bt 1 1 T GLS t=2 bt 1 1 + T t=2 b2GLS ; the limit of T t bGLS t + 2T bGLS ; t 1 = T 1 Tt=2b2GLS T t 2 1 = T 1 b2GLS b2GLS T 1 2 1 2 ) ; Lc (1; c; 1 ; 2 ; 2 1 2 =T T 2GLS t=2 bt 1 T ; 1 )2 1 T 2GLS t=2 bt 1 T t=2 2 ; +T T t=2 1 b2GLS t Lc (0; c; Now the asymptotic distribution of t-statistic is as follows: 100 T 1; T bGLS t=2 t 1 1 2; 1 T t=2 b2GLS t ; )2 bGLS t b2GLS t z : ; t b=1 = q s2 T 1 2 [ 2 Tb 2 Lc (1;c; ; T e2 t=2 t 1 1; 2; 2 ) = ; r z 1 2 h 2 ; 2 ; R1 0 R1 2 ; Lc (s;c; 1; ; )2 2; Lc (0;c; 1; 2; Lc (s; c; 0 Lc (1; c; ; 1 ; 1; ; 1; 2; )2 z ] )2 dr 2; )2 ; ; 2 ; q R 1 z 0 Lc (s; c; )2 dr 1 Lc (0; c; 1; 2; 1; ; 2; ; )2 )2 dr z i : It is easy to show that plims2 = z as follows: s2 = s2 = = = 1 T 1 T 2 1 T 2 2 T t=2 T t=2 T t=2 b2GLS t bGLS t 2 T bbGLS t 1 2 T p 1 2 T GLS t=2 bt 1 b2GLS + op (1) ! z : t 101 bGLS Tb + t 1 T 2 T 2 T 2GLS 2 2 t=2 bt 1 T b Table 2.1a: Finite Sample Mean and Standard Deviation, t u1t ; u2t I(1); 10,000 Replications, = 2. Mean Standard Deviation T IV IV 1 2 50 20 10 2.000886 .0346252 14 7 1.999809 .0484356 10 5 1.998806 .0668215 8 4 2.004179 .0856235 6 3 2.004114 .1190519 4 2 2.010776 .1805444 2 1 2.038802 .355008 .4 .2 1.854142 27.48011 .2 .1 .3588796 53.2373 0 0 3.625984 84.86976 100 200 20 14 10 8 6 4 2 .4 .2 0 20 14 10 8 6 4 2 .4 .2 0 Note: IV 10 7 5 4 3 2 1 .2 .1 0 10 7 5 4 3 2 1 .2 .1 0 denotes 1.999302 1.999138 2.001098 2.001684 2.000639 2.002888 2.029194 2.003915 1.833327 .088921 .0236247 .0337489 .0503667 .0630423 .0810775 .1222339 .2628126 18.49181 42.03659 12.9427 2.000063 .0171978 1.999372 .0245572 2.000412 .0354813 2.004122 .0421344 2.001642 .0594053 2.001649 .0845398 2.006317 .1758927 2.540382 2.56661 2.845896 22.6069 2.227734 102.2849 the estimator given by (11) 102 I(1) Table 2.1b: Finite Sample Mean and Standard Deviation, u1t ; u2t I(1); 10,000 Replications, = 2. Mean Standard Deviation T IV IV 1 2 50 20 10 2.000047 .0011734 14 7 2.000029 .0016992 10 5 1.999854 .0024306 8 4 2.000055 .003043 6 3 1.999914 .0039232 4 2 1.999913 .0060052 2 1 1.999915 .0122615 .4 .2 2.063545 2.008758 .2 .1 2.043028 .8976636 0 0 2.089652 3.950261 100 200 20 14 10 8 6 4 2 .4 .2 0 20 14 10 8 6 4 2 .4 .2 0 Note: IV 10 7 5 4 3 2 1 .2 .1 0 10 7 5 4 3 2 1 .2 .1 0 denotes 1.999994 1.999968 1.999987 1.999995 1.999947 1.999985 1.999895 2.001467 2.083594 2.000129 .0004378 .0006101 .0008715 .0010907 .0014223 .0022623 .0042304 .0905835 2.250663 2.027314 2.000008 .0001443 2.000004 .0002254 2.000009 .0003098 1.999989 .0003664 2.000035 .0005234 1.999999 .000771 2.000043 .0015436 2.001051 .0344767 2.004925 .4323694 1.955252 .9218907 the estimator given by (11) 103 t I(0) Table 2.2a: Empirical Null Rejection Probabilities, 5% Nominal Level, t I(1): 10,000 Replications, b for t 0 (R Robust)is data dependent; H0 : = 0 = 2, H1 : 6= 2: b = 0:1 b = 0:5 b = 1:0 T tIV t0 t 0 (R) tIV t0 t 0 (R) tIV t0 t 0 (R) 1 2 50 20 10 .053 .054 .053 .051 .052 .049 .047 .047 .048 14 7 .048 .049 .039 .053 .049 .048 .055 .056 .053 10 5 .040 .037 .045 .050 .050 .049 .046 .045 .045 8 4 .050 .050 .049 .055 .053 .054 .050 .051 .047 6 3 .059 .055 .051 .049 .052 .051 .045 .046 .040 4 2 .054 .051 .055 .055 .049 .054 .050 .053 .052 2 1 .044 .039 .044 .059 .052 .049 .056 .062 .053 .4 .2 .093 .048 .049 .065 .035 .051 .044 .043 .033 .2 .1 .144 .042 .037 .111 .058 .044 .085 .045 .047 0 0 .218 .049 .048 .141 .058 .056 .124 .039 .040 100 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .045 .046 .046 .047 .043 .037 .035 .218 .190 .207 .051 .052 .052 .046 .047 .049 .052 .051 .049 .052 .052 .052 .052 .052 .046 .054 .050 .049 .048 .052 .051 .053 .046 .043 .064 .046 .052 .053 .096 .135 .051 .051 .046 .043 .068 .049 .049 .051 .039 .050 .052 .047 .046 .050 .066 .053 .050 .049 .038 .054 .049 .047 .052 .048 .050 .046 .051 .051 .068 .122 .050 .052 .052 .049 .049 .047 .053 .052 .049 .046 .051 .050 .053 .050 .048 .046 .052 .049 .048 .049 200 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .044 .046 .054 .054 .049 .051 .040 .077 .099 .225 .041 .045 .053 .053 .050 .053 .045 .053 .053 .050 .042 .046 .057 .060 .045 .043 .046 .057 .060 .045 .053 .050 .043 .052 .046 .043 .053 .061 .074 .147 .050 .049 .052 .051 .050 .045 .049 .054 .049 .043 .050 .048 .051 .049 .050 .040 .048 .053 .045 .043 .052 .059 .047 .053 .051 .049 .053 .046 .075 .123 .049 .056 .046 .050 .052 .048 .051 .047 .061 .049 .048 .049 .043 .051 .049 .049 .049 .046 .057 .060 104 Table 2.2b: Empirical Null Rejection Probabilities, 5% Nominal Level, t I(0): 10,000 Replications, b for t 0 (Robust)is data dependent; H0 : = 0 = 2, H1 : 6= 2: b = 0:1 b = 0:5 b = 1:0 T tIV t0 t 0 (R) tIV t0 t 0 (R) tIV t0 t 0 (R) 1 2 50 20 10 .047 .053 .033 .047 .052 .036 .052 .058 .036 14 7 .047 .051 .039 .051 .060 .035 .052 .059 .038 10 5 .046 .049 .038 .044 .048 .033 .051 .054 .039 8 4 .055 .058 .032 .047 .048 .036 .051 .060 .031 6 3 .051 .049 .035 .047 .048 .031 .046 .054 .030 4 2 .051 .056 .039 .050 .053 .034 .047 .051 .039 2 1 .047 .059 .032 .049 .065 .039 .048 .055 .031 .4 .2 .034 .067 .030 .032 .055 .038 .031 .060 .031 .2 .1 .033 .059 .032 .026 .058 .030 .023 .055 .039 0 0 .028 .053 .035 .027 .065 .035 .025 .061 .038 100 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .047 .051 .050 .051 .046 .046 .045 .038 .032 .021 .048 .054 .056 .055 .053 .056 .055 .051 .040 .048 .035 .033 .039 .030 .037 .038 .028 .039 .039 .023 .048 .051 .055 .049 .051 .048 .040 .041 .030 .024 .051 .054 .057 .051 .054 .055 .047 .045 .063 .060 .038 .032 .032 .038 .030 .036 .039 .038 .035 .028 .047 .046 .051 .053 .056 .052 .044 .032 .023 .025 .052 .048 .049 .057 .058 .058 .054 .054 .061 .065 .035 .039 .038 .039 .032 .035 .039 .035 .036 .030 200 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .048 .046 .056 .049 .050 .047 .045 .038 .019 .026 .054 .056 .057 .052 .055 .055 .056 .055 .054 .050 .031 .038 .035 .031 .033 .031 .037 .035 .039 .034 .051 .049 .051 .049 .049 .052 .048 .046 .045 .035 .053 .049 .053 .049 .051 .056 .055 .053 .061 .060 .032 .039 .034 .031 .032 .039 .031 .039 .033 .030 .045 .048 .052 .056 .055 .046 .047 .035 .025 .023 .045 .050 .055 .053 .056 .052 .057 .060 .048 .049 .035 .030 .033 .030 .031 .033 .039 .037 .039 .029 105 Table 2.3a: Finite Sample Power, 5% Nominal Level, t I(1): T = 100. Two-sided Tests, 10,000 Replications, H0 : = 0 = 2; H1 : = 1 , 1 = 1 2 . b(IV; t 0 ) = 0:1 b(IV; t 0 ) = 0:5 b(IV; t 0 ) = 1:0 tIV t0 t 0 (R) tIV t0 t 0 (R) tIV t0 t 0 (R) 2 1 10 2.000 .045 .052 .052 .051 .051 .052 .051 .052 .050 2.030 .200 .445 .106 .115 .123 .101 .110 .116 .106 2.060 .528 .750 .192 .295 .308 .192 .233 .243 .193 2.090 .876 .971 .293 .484 .506 .262 .377 .388 .263 2.120 .962 .999 .308 .588 .619 .308 .474 .498 .325 2.150 .997 1.00 .347 .730 .770 .378 .578 .613 .348 2 2.000 2.150 2.300 2.450 2.600 2.750 .046 .137 .432 .751 .902 .984 .047 .393 .784 .947 .996 1.00 .046 .099 .192 .267 .323 .362 .052 .119 .229 .358 .524 .639 .049 .140 .275 .441 .635 .777 .050 .112 .187 .245 .310 .403 .046 .076 .193 .327 .401 .468 .047 .092 .249 .383 .496 .605 .046 .090 .183 .273 .315 .341 1 2.000 2.300 2.600 2.900 3.200 3.500 .059 .092 .331 .653 .804 .920 .055 .375 .766 .960 .994 1.00 .056 .098 .185 .285 .318 .347 .052 .086 .193 .313 .403 .494 .049 .118 .281 .483 .637 .747 .050 .091 .167 .273 .308 .381 .051 .104 .172 .263 .328 .386 .053 .146 .239 .375 .494 .604 .052 .115 .186 .248 .324 .356 .4 2.0 2.75 3.5 4.25 5.00 5.75 .053 .027 .102 .264 .373 .486 .052 .406 .750 .834 .963 .995 .048 .088 .192 .263 .325 .348 .052 .053 .113 .176 .207 .221 .051 .135 .288 .481 .622 .747 .053 .106 .193 .263 .325 .348 .051 .057 .106 .147 .208 .231 .049 .123 .272 .371 .518 .605 .047 .098 .202 .255 .330 .356 .2 2 3.5 5.0 6.5 8.0 9.5 .067 .050 .080 .048 .108 .134 .043 .190 .524 .851 .956 .998 .042 .088 .180 .241 .323 .362 .053 .016 .055 .078 .097 .129 .051 .104 .299 .465 .650 .763 .049 .090 .183 .264 .323 .362 .051 .023 .051 .083 .089 .117 .052 .097 .254 .378 .479 .622 .049 .106 .202 .263 .300 .346 .1 2 5 8 11 14 17 .130 .012 .009 .026 .022 .043 .045 .188 .551 .822 .964 .991 .046 .099 .210 .261 .301 .361 .096 .023 .035 .054 .043 .044 .039 .118 .290 .505 .640 .767 .038 .098 .185 .285 .318 .347 .068 .021 .020 .040 .065 .047 .049 .146 .239 .375 .494 .604 .048 .115 .186 .248 .324 .356 106 Table 2.3b: Finite Sample Power, 5% Nominal Level, t I(0): T = 100. Two-sided Tests, 10,000 Replications, H0 : = 0 = 2; H1 : = 1 , 1 = 1 2 . b(IV; t 0 ) = 0:1 b(IV; t 0 ) = 0:5 b(IV; t 0 ) = 1:0 tIV t0 t 0 (R) tIV t0 t 0 (R) tIV t0 t 0 (R) 2 1 10 2.0000 .046 .056 .038 .045 .040 .030 .050 .054 .036 2.0005 .153 .156 .143 .131 .133 .120 .171 .179 .154 2.0010 .583 .579 .491 .283 .297 .470 .263 .259 .504 2.0015 .856 .859 .860 .532 .547 .850 .453 .456 .843 2.0020 .994 .995 .991 .702 .709 .991 .553 .539 .998 2.0025 1.00 1.00 1.00 ,732 .749 1.00 .624 .629 1.00 2 2.0000 2.0025 2.0050 2.0075 2.0100 2.0125 .045 .172 .495 .895 .971 1.00 .064 .175 .509 .899 .969 1.00 .036 .153 .375 .834 .980 1.00 .058 .092 .283 .412 .662 .802 .059 .126 .299 .418 .678 .822 .038 .172 .496 .812 1.00 1.00 .064 .069 .182 .494 .492 .682 .068 .070 .205 .448 .472 .694 .029 .152 .446 .886 1.00 1.00 1 2.0000 2.0050 2.0100 2.0150 2.0200 2.0250 .046 .187 .511 .901 .980 .991 .053 .189 .534 .916 .985 .995 .030 .123 .452 .882 .971 .981 .046 .124 .223 .342 .621 .824 .050 .137 .241 .363 .689 .828 .032 .179 .513 .774 1.00 1.00 .056 .108 .192 .405 .546 .634 .050 .089 .215 .457 .579 .628 .029 .120 .459 .887 .981 1.00 .4 2.0000 2.0125 2.0250 2.0375 2.0500 2.0625 .053 .112 .472 .851 .981 1.00 .056 .097 .485 .863 .986 1.00 .029 .091 .430 .850 .990 1.00 .048 .102 .261 .456 .532 .569 .049 .099 .287 .524 .694 .742 .031 .140 .489 .903 .983 .995 .050 .102 .201 .327 .456 .487 .046 .116 .217 .419 .558 .564 .029 .125 .459 .889 .978 .982 .2 2.0000 2.0250 2.0500 2.0750 2.1000 2.1250 .029 .165 .351 .760 .890 .885 .040 .262 .423 .839 1.00 1.00 .021 .161 .401 .831 .981 1.00 .040 .059 .151 .293 .309 .445 .041 .143 .274 .408 .681 .768 .027 .197 .498 .873 .978 1.00 .040 .054 .136 .256 .309 .318 .064 .102 .194 .392 .517 .551 .028 .098 .439 .857 .992 1.00 .1 2.0000 2.0500 2.1000 2.1500 2.2000 2.2500 .040 .142 .362 .611 .683 .725 .042 .167 .558 .902 .994 1.00 .030 .170 .534 .885 .981 1.00 .035 .065 .112 .191 .264 .291 .065 .154 .183 .423 .618 .867 .030 .146 .485 .801 1.00 1.00 .040 .039 .115 .246 .258 .249 .050 .132 .205 .461 .497 .671 .025 .162 .408 .839 .990 1.00 107 Table 2.4: Finite Sample Performance of ADF and ADF-GLS Unit Root Test for bt : Null Rejections T DF DF-GLS 1 2 50 20 10 .054 .042 14 7 .058 .041 10 5 .053 .061 8 4 .068 .059 6 3 .054 .057 4 2 .044 .045 2 1 .052 .058 .4 .2 .051 .052 .2 .1 .033 .051 0 0 .055 .060 100 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .049 .053 .052 .055 .059 .055 .052 .043 .050 .056 .053 .050 .049 .068 .053 .053 .056 .053 .067 .059 500 20 14 10 8 6 4 2 .4 .2 0 10 7 5 4 3 2 1 .2 .1 0 .057 .060 .053 .056 .039 .046 .040 .050 .047 .049 .059 .059 .058 .062 .041 .064 .047 .054 .051 .057 108 Table 2.5: Finite Sample Power of ADF & ADF-GLS Unit Root Test for bt . 5% Nominal Level, T = 100: No Cointegration Cointegration DF DF-GLS DF DF-GLS 1 2 20 10 1.00 .049 .053 0.98 .059 .058 .071 .062 0.96 .074 .098 .059 .105 0.94 .107 .139 .085 .126 0.92 .142 .201 .089 .139 0.90 .194 .260 .179 .286 14 10 8 6 4 7 5 4 3 2 1.00 0.98 0.96 0.94 0.92 0.90 .053 .054 .079 .098 .134 .202 .050 .047 .101 .131 .189 .280 .063 .081 .083 .130 .187 .073 .065 .154 .219 .285 1.00 0.98 0.96 0.94 0.92 0.90 .052 .042 .065 .105 .132 .201 .049 .060 .089 .143 .205 .283 .057 .143 .139 .187 .210 .063 .139 .190 .244 .218 1.00 0.98 0.96 0.94 0.92 0.90 .055 .064 .074 .117 .143 .224 .068 .078 .102 .147 .194 .296 .049 .107 .089 .152 .234 .058 .143 .171 .159 .269 1.00 0.98 0.96 0.94 0.92 0.90 .059 .061 .086 .099 .143 .178 .053 .066 .109 .151 .203 .257 .038 .054 .131 .140 .192 .045 .063 .185 .216 .286 1.00 0.98 0.96 0.94 0.92 0.90 .055 .061 .074 .099 .150 .203 .053 .066 .096 .153 .210 .278 .049 .059 .103 .152 .189 .055 .086 .155 .220 .268 109 3 3.1 SPEED OF ECONOMIC CONVERGENCE OF U.S. REGIONS Introduction There is a large literature in economics regarding the economic convergence of countries, regions, etc. in terms of per-capita income. -convergence occurs when poor economies grow faster than rich ones. The empirical …ndings on convergence are mixed, e.g. Baumol 1986 …nds some evidence in the developed economies. Barro and Sala-i Martin (1990) …nd evidence of -convergence in USA, whereas Brown, Coulson and Engle (1990) …nd no convergence in US states. Quah (1993) does not …nd any evidence of convergence. Carvalho and Harvey (2005) show that all but the two richest US regions are converging to the average. DeJuan and Tomljanovich (2005) …nd support for both stochastic convergence (convergence in growth rates) and -convergence (convergence in levels) on the basis of personal income data for the majority of Canadian provinces, after allowing for a structural break in the data. Ko cenda, Kutan and Yigit (2006) show slow but steady per-capita real income convergence of 10 European Union (EU) members toward EU standards. Cuñado and de Gracia (2006) examine the real convergence hypothesis in 43 African countries (both toward an African average and the U.S. economy). They …nd convergence both toward the African average and the US economy. Rodríguez (2006) provides some evidence of -convergence in Canada. Cuñado and de Gracia (2006) …nd evidence of convergence during the nineties-2003 period for Poland, Czech Republic and Hungary toward Germany and only for Poland toward the US economy. Kutan and Yigit (2007) also provide some evidence in the support of convergence in Eurpoean Union. Galvao Jr and Reis Gomes (2007) investigate the occurrence of per capita income convergence in 19 Latin American countries. Their results indicate that there is substantial evidence in favor of conditional convergence in Latin America. Dawson and Sen (2007) provide evidence that the relative income series of 21 countries are consistent with stochastic convergence, and that -convergence has occurred in at least 16 countries at some point during the twentieth century. Heckelman (2013) performs convergence tests on the U.S. states for per capita income from 1930 110 to 2009, and …nd that about half of the states exhibit stochastic and -convergence. Ayala, Cunado and Gil-Alana (2013) investigate the real convergence of 17 Latin American countries to the US economy for the period 1950 to 2011. They …nd real convergence (productivity catch-up) to the US for three Latin American countries: Chile, Costa Rica and Trinidad and Tobago, with these countries also presenting evidence of stochastic and -convergence. Where -convergence is found, it is interesting to obtain a measure of the speed of convergence and estimate it. This chapter develops a simple measure of the speed of convergence that can be expressed as a ratio of two trend slopes. Using the methodology developed in the …rst two chapters, we estimate the speed of convergence in practice. We apply our approach to U.S. regions and document the speed of convergence for regions that exhibit -convergence. The remainder of this chapter is organized as follows: Section 3.2 describes the model. Section 3.3 presents the estimation results and section 3.4 concludes. 3.2 Model Suppose there are two countries; country 1 is poor and country 2 is rich. The initial incomes of the two countries are Y10 and Y20 respectively, and Y10 < Y20 . Assume income grows as follows: Y1t = (1 + t 1 ) Y10 ; Y2t = (1 + t 2 ) Y20 ; 2 < 1: The above inequalities suggest that the richer country with higher initial income must have a growth rate lower than that of the poorer country for -convergence to occur. We now develop a measure of the speed of -convergence. Suppose incomes of the two 111 countries equalize at date t = t ; then at t = t ; Y1t = Y2t ; giving (1 + t 1) Y10 = (1 + t 2) Y20 : Solving for t gives t = log (Y20 =Y10 ) log ((1 + 1 )=(1 + 2 )) : Now the speed of convergence, SC; of income of country 1 to income of country 2 is given by SC = (Y20 Y10 ) t = log 1+ 1+ 1 2 (Y20 Y10 ) : log (Y20 =Y10 ) For given initial income levels, the speed of convergence obviously depends on the magnitude of = 1+ 1+ 1 ; 2 which can be expressed as a ratio of linear trend slopes as follows. Taking the natural log of the expressions for Y1t , Y2t , and denoting y1t = log Y1t and y2t = log Y2t ; we obtain y1t = t log(1 + 1) + y10 y10 + 1 t; y2t = t log(1 + 2) + y20 y20 + 2 t: The estimation equations for y1t and y2t can be written as 112 y1t = y10 + 1t + u1t 1 + 1t + u1t ; y2t = y20 + 2t + u2t 2 + 2t + u2t : Adding t to both sides of each equation gives y1t = y1t + t = 1 + (1 + 1 )t + u1t ; y2t = y2t + t = 2 + (1 + 2 )t + u2t : Given the analysis in Chapter 1 and 2, the regression of y1t on y2t estimates : The regression y1t = + y2t + t ; estimates the parameter given by = 1+ 1+ 1 : 2 The speed of convergence is directly proportional to the parameter : For given initial income levels, i.e. Y10 and Y20 ; the speed of convergence is greater for higher values of 3.3 and vice versa. Estimation Results We collected annual per capita income series from 1929 2013 for eight US regions: Southeast, Southwest, Rocky Mountains, Plains, Great Lakes, Far West, Mideast and New England. According to Carvalho and Harvey (2005), all regions but the Mideast and New England are converging ( -convergence). Figure 1 shows plots of natural log of per capita income of all regions against time. Table 3.1 reports OLS estimated 113 parameters from the regression of each log series (in order from poorest to richest) on an intercept and time trend. Each series was tested for a unit root (around the linear time trend) using the ADF-GLS test. Those results are also given in table 3.1. For the trend slope estimators in Table 3.1, the standard errors have been calculated using the following formula: where se( b) = t=T 1 s T X c2 T (t t=1 t; t=1 T X1 c2 = b0 + 2 t)2 k j=1 j M bj ; and bj = T 1 T t=j+1 bt bt j : bt are the OLS residuals from the regression of the regions’series on time variable. The …xed-b critical values from Bunzel and Vogelsang (2005) have been used to assess statistical signi…cance. Because of great depression of 1946 and 1973-75 recession, subsamples of data, i.e. post 1946 and post 1973 have been considered separately for analysis. The trend slopes are signi…cant at 95% con…dence level. The poorer regions tend to have b0 s that are bigger than the more wealthy regions which is consistent with -convergence. The con…dence intervals in table 3.1 are reported for the trend slopes using I(1) critical values except for Plains (post 1946) for which I(0) critical value has been used, as we can reject the null of a unit root for the series of Plains (post 1946). In Table 3.2a, the IV point estimates of shown for the following regression equation: 114 and the 95% con…dence intervals are y1t = + y 2t + t ; where y1t = y1t + t, for a given region and y 2t = y 2t + t, where y 2t is the average across all regions. ADF-GLS statistic is also reported for the residuals. The purpose of carrying out a unit root test for the residuals is to use the right critical value which is di¤erent for I(0) and I(1) errors. Although the ADF-GLS statistic values suggest that the residuals (for all but Great Lakes) are integrated of order one, however, the con…dence intervals based on both I(0) and I(1) errors have been reported. The null hypothesis of = 1; is rejected for Southeast, Great Lakes and Far West, which is a statistical evidence of convergence of these regions to the average income level. In Table 3.2b, the IV point estimates of and the 95% con…dence intervals similar to those in Table 3.2a are reported for the subsample of post 1946 period. As in Table 3.2a, for IV residuals, the null of unit root is rejected only for Great Lakes. The null of = 1; is rejected for Great Lakes and Far West. In Table 3.2c, the IV point estimates of and the 95% con…dence intervals similar to those in Tables 3.2a and b are reported for the subsample of post 1973 period. As in Tables 3.2a and b, for IV residuals, the null of unit root is rejected only for Great Lakes. The null of = 1; is rejected for Great Lakes and Far West. In Table 3.3a, pairwise IV point estimates and the 95% con…dence intervals for all the series are reported. The regions in the descending order of initial income are as follows: Mideast, Far West, New England, Great Lakes, Rocky Mountain, Plains, 115 Southwest, Southeast. For Southeast the value of b increases from Southwest to Great Lakes, then it falls slightly for New England; it is higher for Far West and decreases for Mideast again. For Southwest again, the estimates increase across the row up to Great Lakes, then the estimate decreases for New England, increases for Far West and decreases for Mideast. This pattern remains the same for each and every row in Table 3.3a, i.e. the estimates increase up to Great Lakes, they all decrease for New England, then increase for Far West and decrease for Mideast. Therefore except for New England and Mideast, as we move across a row, we see increasing b0 s, which is an evidence that convergence is occurring faster, the larger the initial income gap. The estimates are statistically di¤erent from one for majority of the estimates in Table 3.3a for I(0) critical values, which is a statistical evidence of -convergence of regions. For I(1) critical values, the estimated ratio of trend slopes is statistically di¤erent from one only for Southeast when regressed against Great Lakes and Far West. There is no evidence of convergence of other regions based on I(1) critical values. In Table 3.3b, i.e. for post 1946 subsample, the pattern of convergence for I(0) critical values is pretty similar to that in Table 3.3a, however, for I(1) critical values, there is evidence of convergence of only Plains and Far West. In Table 3.3c, where post 1973 subsample has been used for estimation of , the pattern of increasing b0 s as we move across a row remains the same as that in Table 3.3a except for Southeast when regressed against Southwest and Plains; the estimates decrease from Southwest to Plains instead of increasing. There is no evidence of convergence for any of the regions based on I(1) critical values, whereas based on I(0) critical values, there is evidence of convergence of Southeast to Great Lakes, Plains to Far West, New England to Far West and Mideast, and Far West to Mideast. 3.4 Conclusion The speed of convergence of two di¤erent regions’per capita income has been shown to be proportional to the ratio of trend slopes. This ratio has been estimated for all US regions using the methodology developed in the …rst two chapters. For all regions, the 116 IV point estimates and the 95% con…dence intervals have been computed. Unit root tests have been applied to the IV residuals, and the results suggest that the residuals (for all but Great Lakes) are integrated of order one. The con…dence intervals of based on both I(0) and I(1) errors have been reported to compare their performance based on the type of noise. The model suggests that the speed of convergence is higher for the cases where the estimated ratio of trend slopes is larger in magnitude, all else the same. The estimates suggest that for U.S. regions, the convergence is occurring faster, the larger the initial income gap. 117 APPENDIX 118 12 10 8 6 4 1920 1940 1960 1980 2000 year Southwest Southeast Rocky Mountain Plains Great Lakes Mideast Far West New England Figure 1: Natural Log of Per Capita Income of US Regions Data Series: 1929-2013 119 2020 Table 3.1: Regression Results for Regions against Time, and ADF-GLS for Regions, b = 0:1; Critical value (10:97500) for I(1) errors, except for Plains (Post 1946) with b = 0:25; Critical value (4:2027536). The regression is yt = + t + ut : b DF-GLS b (P.1946) DF-GLS b (P.1973) DF-GLS S:east 0:0664 -1.61 0:0636 -1.64 0:0529 -0.64 (0:0023) (0:0026) (0:0065) [:041; :091] [:035; :092] [ :019; :124] S:west 0:0631 -2.42 0:0601 -1.66 0:0507 -0.93 (0:0023) (0:0026) (0:0061) [:038; :088] [:031; :089] [ :016; :118] P lains 0:0621 -2.13 0:0636 -3.91# 0:0516 -0.74 (0:0021) (0:0027) (0:0053) [:039; :085] [:055; :072] [ :006; :109] R:M nts 0:0603 -2.05 0:0584 -2.09 0:0511 -0.57 (0:0019) (0:0021) (0:0047) [:039; :082] [:035; :082] [ :001; :103] G:Lakes 0:0579 -1.52 0:0568 -2.57 0:0494 -1.18 (0:0019) (0:0019) (0:0045) [:037; :079] [:035; :079] [ :000; :099] N:Eng 0:0596 -1.26 0:0614 -2.09 0:0559 -0.77 (0:0018) (0:0019) (0:0028) [:039; :079] [:040; :082] [:025; :087] F:W est 0:0567 -1.48 0:0561 -2.36 0:0486 -0.77 (0:0018) (0:0018) (0:0043) [:037; :076] [:036; :076] [:002; :096] M ideast 0:0581 -1.13 0:0594 -2.35 0:0535 -0.37 (0:0018) (0:0018) (0:0031) [:039; :078] [:039; :079] [:019; :087] * signi…cant at 5% level, # Unit root rejected at 5% level. 120 Table 3.2a: IV Point Estimates and the 95% Con…dence Intervals. t = T ime yit = yit + t; y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: I(0) Errors (b = 0:25) I(1) Errors (b = 0:1) ADF-GLS for Residuals Average Average y1t 1:006011 1:006011 -0.793 [1:0016623; 1:0103534] [:99753952; 1:0143694] y2t 1:002954 1:002954 -1.563 [0:99855986; 1:0073596] [:99361615; 1:0121836] y3t 1:001926 1:001926 -1.775 [0:99925281; 1:0046108] [:99625521; 1:0075406] y4t 1:000278 1:000278 -1.412 [0:99819446; 1:0023757] [:99539561; 1:0051402] y5t 0:9980236 0:9980236 -4.012 [0:99728663; :99876237] [:99591106; 1:0001218] y6t 0:9995676 0:9995676 -1.179 [0:99481678; 1:0043051] [:98979901; 1:0094321] y7t 0:996802 0:996802 -2.834 [0:99657843; :99702737] [:99510981; :9985193] y8t 0:9981557 0:9981557 -1.093 [0:99453362; 1:0017706] [:99055351; 1:0058457] * H0 : = 1 is rejected at 5% level. Table 3.2b: IV Point Estimates and the 95% Con…dence Intervals (Post 1946). yit = yit + t; y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: I(0) Errors (b = 0:25) I(1) Errors (b = 0:1) ADF-GLS for Residuals Average Average 1:00403 -0.804 y1t 1:00403 [0:99793392; 1:0101014] [:99236877; 1:0155019] y2t 1:000696 1:000696 -1.576 [0:99434166; 1:0070414] [:987664; 1:0135328] y3t 1:000447 1:000447 -1.789 [0:99632049; 1:0045694] [:99208869; 1:0086889] y4t 0:9991468 0:9991468 -1.420 [0:99614358; 1:0021536] [:99254768; 1:0056822] y5t 0:9976508 0:9976508 -4.009 [0:99658115; :99871865] [:99504994; 1:0002227] y6t 1:001998 1:001998 -1.179 [0:99508796; 1:0089175] [:98821934; 1:0159674] y7t 0:9969806 0:9969806 -2.833 [0:99643026; 0:99753492] [:99509792; :99889623] y8t 1:000052 1:000052 -1.094 [0:99477325; 1:0053404] [:98940668; 1:0108576] * H0 : = 1 is rejected at 5% level. 121 Table 3.2c: IV Point Estimates and the 95% Con…dence Intervals (Post 1973). yit = yit + t; y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: I(0) Errors (b = 0:25) I(1) Errors (b = 0:1) ADF-GLS for Rersiduals Average Average y1t 1:001059 1:001059 -0.821 [0:9873747; 1:014141] [:97619774; 1:0239544] y2t 0:9989959 0:9989959 -1.586 [0:98827432; 1:0092778] [:97843847; 1:0179989] y3t 0:9998583 0:9998583 -1.794 [0:99418384; 1:0053132] [:98877025; 1:0101407] y4t 0:9993285 0:9993285 -1.419 [0:99657007; 1:0020165] [:99302216; 1:0053122] y5t 0:9977157 0:9977157 -4.009 [0:99674732; :99865932] [:99521983; 1:0001008] y6t 1:003979 1:003979 -1.180 [0:99245282; 1:0160021] [:98289965; 1:0267798] y7t 0:9969747 0:9969747 -2.833 [0:99642172; :99755378] [:99509723; :99894963] y8t 1:001634 1:001634 -1.095 [0:99264083; 1:0110192] [:98514176; 1:0194875] * H0 : = 1 is rejected at 5% level. 122 Table 3.3a: Pairwise IV Point Estimates and the 95% Con…dence Intervals. Top Row: I(1) Errors (b = 0:1); Bottom Row: I(0) Errors (b = 0:25): y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains; y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: y2t y3t y4t y5t y6t y7t # # # # y1t 1:003 1:004 1:006 1:008 1:006 1:009 # :9987; :9997; :9992; 1:001; :989; 1:000; 1:014 1:008 1:01 1:015 1:024 1:018 1:002; 1:002; 1:002; 1:004; :997; 1:005; 1:004 1:006 1:01 1:012 1:015 1:014 # y2t 1:001 1:0027 1:005 1:003 1:006# :9967; :9969; :9965; :984; :9965; 1:005 1:01 1:013 1:022 1:0157 :9993; :9999; 1:001; :994; 1:002; 1:003 1:01 1:009 1:013 1:011 y3t 1:002# 1:004# 1:002 1:005# :9988; :987; :9992; :9985; 1:011 1:018 1:008 1:005 1:002; :995; 1:002; 1:0004; 1:008 1:009 1:006 1:003 y4t 1:002# 1:001 1:003# :9983; :986; :9984; 1:009 1:015 1:006 1:001; :994; 1:001; 1:006 1:007 1:004 y5t :9984 1:001# :9979; :987; 1:004 1:009 1:000; :993; 1:002 1:004 y6t 1:003 :9929; 1:013 :9981; 1:007 y7t * H0 : = 1 is rejected at 5% level for I(1), 123 # H0 : y8t 1:008 :992; 1:0237 :999; 1:016 1:005 :988; 1:022 :997; 1:013 1:004 :991; 1:017 :997; 1:0101 1:002 :989; 1:014 :997; 1:008 :9999 :991; 1:0091 :996; 1:0041 1:001# :9988; 1:004 1:0002; 1:003 :9986 :9909; 1:006 :9951; 1:002 = 1 is rejected at 5% level for I(0). Table 3.3b: Pairwise IV Point Estimates and the 95% Con…dence Intervals (Post Top Row: I(1) Errors (b = 0:1); Bottom Row: I(0) Errors (b = 0:25): y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains; y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: y2t y3t y4t y5t y6t y7t # # # # y1t 1:003 1:003 1:005 1:006 1:002 1:007 # :9970; :9977; :9966; :9999; :992; 1:000; 1:01 1:009 1:013 1:013 1:012 1:014 1:001; 1:001; 1:001; 1:003; :997; 1:004; 1:01 1:006 1:009 1:010 1:007 1:010 # y2t 1:0002 1:001 1:003 :9987 1:004# :9958; :9963; :9969; :989; :9992; 1:005 1:007 1:009 1:008 1:008 :9989; :9989; 1:000; :994; 1:002; 1:001 1:004 1:006 1:003 1:006 y3t 1:001# 1:003# :9984 1:003 # 1:001; :992; :9995; :9978; 1:006 1:005 1:006 1:005 1:003; :995; 1:001; 1:000; 1:004 1:002 1:004 1:002 y4t 1:001# :9971 1:002# :9986; :989; :9976; 1:006 1:004 1:005 1:001; :995; 1:000; 1:003 :9994 1:003 y5t :9957 1:001 :9971; :991; 1:004 1:001 :9995; :994; 1:002 :9975 y6t 1:005# :9993; 1:011 1:002; 1:008 y7t * H0 : = 1 is rejected at 5% level for I(1), 124 # H0 : 1946). y8t 1:004 :995; 1:0127 :999; 1:009 1:001 :992; 1:009 :997; 1:004 1:000 :995; 1:005 :998; 1:003 :9991 :993; 1:005 :998; 1:000 :9976 :993; 1:002 :996; :9988 1:002# :9993; 1:004 1:0009; 1:003 :9969# :9925; 1:001 :9951; :9987 = 1 is rejected at 5% level for I(0). Table 3.3c: Pairwise IV Point Estimates and the 95% Con…dence Intervals (Post 1973). Top Row: I(1) Errors (b = 0:1); Bottom Row: I(0) Errors (b = 0:25): y1t = Southeast; y2t = Southwest; y3t = P lains; y4t = RockyM ountains; y5t = GreatLakes; y6t = N ewEngland; y7t = F arW est; y8t = M ideast: y2t y3t y4t y5t y6t y7t y8t # y1t 1:002 1:001 1:002 1:003 :9971 1:004 :9994 :9905; :994; :9918; :9989; :992; :9968; :995; 1:01 1:008 1:011 1:008 1:003 1:011 1:004 :9959; :997; :9962; 1:002; :995; :9999; :998; 1:01 1:005 1:007 1:005 :9990 1:008 1:001 y2t :9991 :9997 1:001 :9950 1:002 :9974 :992; :9923; :9899; :982; :9956; :986; 1:006 1:007 1:013 1:009 1:008 1:009 :996; :9957; :9953; :988; :9999; :992; 1:002 1:003 1:007 1:002 1:004 1:003 y3t 1:0005 1:002 :9959 1:003# :9982 :992; :9981; :986; :9962; :9945; 1:005 1:008 1:006 1:008 1:006 :995; 1:001; :991; :9989; :9976; 1:001 1:004 1:001 1:005 1:003 y4t 1:002 :9954 1:002 :9977 :987; :9958; :982; :9935; 1:009 1:009 1:009 1:010 :992; :9992; :988; :9972; 1:004 1:005 1:002 1:006 y5t :9938 1:001 :9961 :986; :9933; :989; 1:002 1:008 1:003 :993; :9967; :990; :9991 1:005 :9974 1:007# 1:002# y6t :9972; :9976; 1:016 1:007 1:002; 1:0005; 1:012 1:004 y7t :9953# :9882; 1:003 :9916; :9991 * H0 : = 1 is rejected at 5% level for I(1), # H0 : = 1 is rejected at 5% level for I(0). 125 REFERENCES 126 REFERENCES Ayala, A., Cunado, J. and Gil-Alana, L. A.: (2013), Real convergence: empirical evidence for latin america, Applied Economics 45(22), 3220–3229. Barro, R. J. and Sala-i Martin, X.: (1990), Economic growth and convergence across the united states, Technical report, National Bureau of Economic Research. Baumol, W. J.: 1986, Productivity growth, convergence, and welfare: what the long-run data show, The American Economic Review pp. 1072–1085. Brown, S. J., Coulson, N. E. and Engle, R. F.: (1990), Non-cointegration and econometric evaluation of models of regional shift and share, Technical report, National Bureau of Economic Research. Bunzel, H. and Vogelsang, T. J.: (2005), Powerful trend function tests that are robust to strong serial correlation with an application to the prebisch-singer hypothesis, Journal of Business and Economic Statistics 23, 381–394. Canjels, E. and Watson, M. W.: (1997), Estimating deterministic trends in the presence of serially correlated errors, Review of Economics and Statistics May, 184– 200. Carvalho, V. M. and Harvey, A. C.: (2005), Growth, cycles and convergence in us regional time series, International Journal of Forecasting 21(4), 667–686. Cuñado, J. and de Gracia, F. P.: (2006), Real convergence in africa in the secondhalf of the 20th century, Journal of Economics and Business 58(2), 153–167. Dawson, J. W. and Sen, A.: (2007), New evidence on the convergence of international income from a group of 29 countries, Empirical Economics 33(2), 199–230. DeJuan, J. and Tomljanovich, M.: (2005), Income convergence across canadian provinces in the 20th century: Almost but not quite there, The Annals of Regional Science 39(3), 567–592. Elliott, G., Rothenberg, T. J. and Stock, J. H.: (1996), E¢ cient tests for an autoregressive unit root. Evans, P.: (1997), How fast do economies converge?, Review of Economics and Statistics 79(2), 219–225. Fagerberg, J.: (1994), Technology and international di¤erences in growth rates, Journal of Economic Literature 32, 1147–75. Galvao Jr, A. and Reis Gomes, F.: (2007), Convergence or divergence in latin america? a time series analysis, Applied Economics 39(11), 1353–1360. 127 Hamilton, J. D.: (1994), Time Series Analysis, Princeton University Press, Princeton, N.J. Harvey, D., Leybourne, S. J. and Taylor, A. M. R.: (2009), Simple, robust and powerful tests of the breaking trend hypothesis, Econometric Theory 25, 995–1029. Heckelman, J. C.: (2013), Income convergence among us states: cross-sectional and time series evidence, Canadian Journal of Economics/Revue canadienne d’économique 46(3), 1085–1109. Kiefer, N. M. and Vogelsang, T. J.: (2005), A new asymptotic theory for heteroskedasticity autocorrelation robust tests, Econometric Theory 21, 1130–1164. Klotzbach, P. J., Pielke, R. A., Christy, J. R. and McNider, R. T.: (2009), An alternative explanation for di¤erential temperature trends at the surface and in the lower troposphere, Journal of Geophysical Research: Atmospheres (1984–2012) 114(D21). Ko cenda, E., Kutan, A. M. and Yigit, T. M.: (2006), Pilgrims to the eurozone: How far, how fast?, Economic Systems 30(4), 311–327. Kutan, A. M. and Yigit, T. M.: (2007), European integration, productivity growth and real convergence, European Economic Review 51(6), 1370–1395. Perron, P. and Yabu, T.: (2009), Estimating deterministic trends with an integrated or stationary noise component, Journal of Econometrics 151(1), 56–69. Phillips, P. C.: (1986), Understanding spurious regressions in econometrics, Journal of econometrics 33(3), 311–340. Quah, D.: (1993), Galton’s fallacy and tests of the convergence hypothesis, The Scandinavian Journal of Economics pp. 427–443. Rodríguez, G.: (2006), The role of the interprovincial transfers in the -convergence process: further empirical evidence for canada, Journal of Economic Studies 33(1), 12– 29. Santer, B., Wigley, T., Mears, C., Wentz, F., Klein, S., Seidel, D., Taylor, K., Thorne, P., Wehner, M., Gleckler, P. et al.: (2005), Ampli…cation of surface temperature trends and variability in the tropical atmosphere, Science 309(5740), 1551–1556. Staiger, D. and Stock, J. H.: (1997), Instrumental variables regression with weak instruments, Econometrica 65, 557–586. Tomljanovich, M. and Vogelsang, T. J.: (2002), Are u.s. regions converging? using new econometric methods to examine old issues, Empirical Economics 27, 49–62. 128 Vogelsang, T. J.: (1998), Trend function hypothesis testing in the presence of serial correlation correlation parameters, Econometrica 66, 123–148. Vogelsang, T. J. and Franses, P. H.: (2005), Testing for common deterministic trend slopes, Journal of Econometrics 126, 1–24. 129