Estimation and Inference of Linear Trend Slope Ratios

Transcription

1 Estimation and Inference of Linear rend Slope Ratios imothy J. Vogelsang and Nasreen Nawaz Department of Economics, Michigan State University October 4 Abstract 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 (FCL). 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 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. Keywords: rend Stationary, Bias Correction, HAC Estimator, Fixed-b Asymptotics Correspondence: Department of Economics, Michigan State University, Marshall-Adams Hall, East Lansing, MI Phone: , Fax: , tjv@msu.edu

2 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 (997), Vogelsang (998), Bunzel and Vogelsang (5), Harvey, Leybourne and aylor (9), and Perron and Yabu (9). 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 (5). 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 (994), Evans (997) and omljanovich and Vogelsang (). 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. here 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, aylor, horne, Wehner, Gleckler et al. (5), Klotzbach, Pielke, Christy and McNider (9) and the references cited 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. his 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 (FCL). 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

3 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. herefore, 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. he remainder of the paper is organized as follows: Section describes the model and analyzes the asymptotic properties of the three estimators of the trend slope ratio. Section 3 provides some nite sample evidence on the relative performance of the three estimators. Section 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. he 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 5 indicate that the predictions of the asymptotic theory are relevant in practice. In Section 6 we make some practical recommendations for empirical researchers and Section 7 concludes. All proofs are given in the Appendix. he Model and Estimation. Model and Assumptions Suppose the univariate time series y t and y t are given by y t = + t + u t ; () y t = + t + u t ; () where u t and u t are mean zero covariance stationary processes. Assume that [r ] = ut u t ) W(r) B (r) B (r) ; (3)

4 where r [; ], [r ] is the integer part of r and W (r) is a vector of independent standard Wiener processes. is not necessarily diagonal allowing for correlation between u t and u t. In addition to (3), we assume that u t and u t are ergodic for the rst and second moments. Suppose that 6= and we are interested in estimating the parameter = which is the ratio of trend slopes. Equation () can be rewritten so that y t depends on through y t. Rearranging () gives t = [y t u t ] ; (4) and plugging this expression into Equation () and then rearranging, we obtain y t = ( ) + y t + (u t u t ) = ( ) + y t + (u t u t ): De ning = and t () = u t u t gives the regression model Given the de nition of t (), it immediately follows from (3) that y t = + y t + t () : (5) [r ] = where w(r) is a univariate standard Wiener process and is the long run variance of t (). t () ) w(r); (6) =. Estimation of the rend Slope Ratio Using regression (5), the natural estimator of is ordinary least squares (OLS) which is de ned as e =! (y t y ) (y t y )(y t y ); (7) where y = y t and y = y t. Standard algebra gives the relationship e =! (y t y ) (y t y ) t () : (8) 3

5 Alternatively, one could estimate by the analogy principle by simply replacing and with estimators. Let b and b be the OLS estimators of and based on regressions () and (): b = b =! (t t) (t t)(y t y ); (9)! (t t) (t t)(y t y ); () where t = t is the sample average of time and de ne b = b = b. Simple algebra shows that b = b b =! (t t)(y t y ) (t t)(y t y ) () which is the instrumental variable (IV) estimator of in (5) where t has been used as an instrument for y t. Standard algebra gives the relationship! b = (t t)(y t y ) (t t) t () : ().3 Asymptotic Properties of OLS and IV We now explore the asymptotic properties of the OLS and IV estimators of. he asymptotic behavior of the estimators depends on the magnitude of the trend slope parameters relative to the variation in the random components, u t and u t, i.e. the noise. he following theorem summarizes the asymptotic behavior of the estimators for xed s and for s that are modeled as local to zero at rate =. heorem Suppose that (6) holds and ; are xed with respect to. he following hold as!. Case (large trend slopes): For =, =, Z 3= e ) Z s ds! s 3= b Z ) s ds! Z s! dw) N ; ;! dw) N ; : Case (medium trend slopes): For = = ; = =, Z e ) Z! s ds! s dw) + E(u t t ()) N E(u t t ); ; Z b Z! ) s ds! s dw) N ; : 4

6 heorem makes someinteresting 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 3= and are asymptotically normal with equivalent asymptotic variances. he precision of both estimators improves when there is less noise ( is smaller) or when the magnitude of the trend slope parameter for y t increases ( is larger). When the trend slopes are modeled as local to zero at rate =, i.e. when trend slopes are medium sized relative to the noise, asymptotic equivalence of OLS and IV no longer holds. he IV estimator essentially has the same asymptotic behavior as in the xed slopes case because the implied approximations are the same: Case (large trend slopes): b N Case (medium trend slopes): b N ; 3! ; N! N ; 3 ; 3 In contrast, the result for OLS is markedly di erent in Case. While OLS consistently estimates and the asymptotic variance is the same as in Case, OLS now has an asymptotic bias that could matter when trend slopes are medium sized. he fact that OLS is asymptotically biased in Case is not that surprising because t () is correlated with y t through the correlation between t () and u t. In Case, the trend slopes are small enough so that the covariance between u t and t () asymptotically a ects the OLS estimator. Because E(u t t ()) = E(u t u t ) E(u t ), the asymptotic bias will be non-zero unless E(u t u t ) = E(u t ) 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..4 Bias Corrected OLS According to heorem, for the case of medium sized trend shifts OLS is asymptotically biased and this bias is driven by covariance between t () and u t. he approximate bias of OLS suggested by heorem is given by the quantity bias( e ) Z s! ds E(u t t ()): We can estimate R s ds using P (y t y ), and we can estimate E(u t t ()) using P bu te t where e t are the OLS residuals from regression (5) and bu t are the OLS residuals he fact that IV is not asymptotically biased in Case is also not suprising given that t, the instrument for y t, is nonrandom and cannot be correlated with t(). ; : 5

7 from regression (). his leads to the bias corrected OLS estimator of given by e c = e bu t e t bu t e t P (y = t y ) C e : (3) A (y t y ) he next theorem gives the asymptotic behavior of the bias corrected OLS estimator for the same cases covered by heorem. heorem Suppose that (6) holds and ; are xed with respect to. he following hold as!. Case (large trend slopes): For =, =, 3= e c ) Z s ds! Z s dw) N ;! : Case (medium trend slopes): For = = ; = =, e c Z ) s ds! Z s dw) N ;! : As heorem shows, the bias corrected OLS estimator is asymptotically equivalent to the IV estimator for both large and medium trend slopes..5 Asymptotic Properties of Estimators for Small rend Slopes As shown by heorem, 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 y t also becomes very small and t becomes a weak instrument for y t. It is well known in the literature that weak instruments have important implications for IV estimation ee Staiger and Stock 997) and estimation of is no exception. he next two theorems provide asymptotic results for the estimators of for trend slopes that are local to zero at rates and 3= with the latter case corresponding to trend slopes that are very small relatively to the noise. 6

8 heorem 3 Suppose that (3) and (6) hold and ; are xed with respect to. he following hold as!. Case 3 mall trend slopes): For = ; =, = ( b ) ) Z s e p! <; e c ds! Z! p < c ; s dw) N ;! ; where < = Z s! ds + E(u t) E(u t t ()); < c = < E(u t ) E(u t t ()) : Case 4 (very small trend slopes): For = 3= ; = 3=, Z b ) e p! E(u t t ()) E(u t ) ; e c ) ds + Z! p E(u t t ()) E(u t ) ; Z )db ) )dw): heorem 3 shows that for small trend slopes, OLS and bias-corrected OLS are biased and inconsistent. In contrast, IV has the same asymptotic behavior as for large and medium trend slopes. When trend slopes are very small, all three estimators are inconsistent and biased. he IV estimator converges to a ratio of normal random variables that are correlated with each other because B (r) is correlated with w(r) as long as u t is correlated with t (). heorem 3 predicts that none of the estimators of will work well when trend slopes are very small. his is not surprising because it is very di cult to identify the ratio of trend slopes when the noise dominates the information in the sample regarding the trend slopes themselves..6 Implications (Predictions) of Asymptotics for Finite Samples heorems -3 make clear predictions about the nite sample behavior of the three estimators of. For large s the three estimators should have similar bias, variance and sampling properties given that they are asymptotically equivalent. When s are of medium size, IV and bias-corrected OLS are asymptotically equivalent and should have similar behavior compared to the large s case. In contrast, OLS should exhibit some nite sample bias in the medium s case. With small or very small s, OLS and bias-corrected OLS are inconsistent and biased. In contrast IV should continue to perform well even with small s with the main implication of small s being less precision given that the asymptotic variance of IV is inversely related to the magnitude of. For very small s IV is inconsistent and can exhibit substantial variability given that IV is approximately a ratio of two normal random variables. 7

9 For a given sample size and variability of the noise, as the trend slopes decrease from being large to becoming very small, we should see the performance of all three estimators deteriorating with OLS deteriorating quickest followed by bias-corrected OLS followed by IV. 3 Finite Sample Means and Standard Deviations of Estimators In this section we illustrate the nite sample performance of the estimators via a Monte Carlo simulation study. For the data generating process (DGP) that we consider, the nite sample behavior of the three estimators closely follows the predictions suggested by heorems -3. he following DGP was used. he y t and y t variables where generated by models () and () where the noise is given by u t = :4u t + :3u t + " t ; u t = :5u t + " t ; [" t ; " t ] i:i:d: N(; I ); u = u = : Given that all three estimators are exactly invariant to the values of and, we set = ; = without loss of generality. We report results for various magnitudes of and where it is almost always the case that = = =. he exception is when =, = in which case is not de ned. We report results for = 5; ; with ; replications used in all cases. Given that the bias-corrected OLS estimator uses a bias correction based on the OLS residuals from (5), we experimented with an iterative procedure for the bias-corrected OLS estimator that improved its nite sample performance. We rst compute ec as given by (3). hen we updated the OLS residuals using ec in place of e and recalculated ec. We iterated between updated residuals and bias-correction times. able reports estimated means and standard deviations of the three estimators across the ; replications. Focusing on the = 5 case we see that when the trend slopes are large ( = ; 5), the means and standard deviations of the three estimators are the same and none of the estimators shows any bias. For medium sized trend slopes ( = ; :; :5; :), bias-corrected OLS and IV have the same means and standard deviations with little bias being present. In contrast, OLS shows bias that increases substantially as decreases. For all three estimators we see that the standard deviations increase as decreases as expected. For small and very small trend slopes ( = :5; :; :), OLS and bias-corrected OLS show substantial bias. It is di cult to determine whether IV is biased given the very large standard deviation of IV in this case. Overall, IV has the least bias but IV becomes very imprecise as the trend slopes approach zero. Results for the cases of = ; are similar to the = 5 case. he only di erence is 8

10 that the bias of OLS and bias-corrected OLS kicks in more slowly as decreases. With =, bias-corrected OLS and IV have the same means and standard deviations for as small as :. he results for =, = at rst may look surprising but make sense upon deeper inspection. he OLS estimator is no longer estimating which is not de ned. Instead, OLS is estimating the population quantity E(u t t ())=E(u t ) which is very close to :469 in our DGP. Bias-corrected OLS is attempting to correct the wrong bias and the IV estimator is based on an instrument that has zero correlation with y t. In fact, the estimators are behaving as expected when the trend slopes are zero. Overall, the nite sample means and variances exhibit patterns as predicted by the asymptotic theory. IV and bias-corrected OLS work equally well for large, medium and somewhat small trend slopes. As trend slopes become very small, none of the estimators are very good and this is to be expected given that the data has relatively little information about the trend slope ratio when trend slopes are small relative to the noise and/or the sample size is small. 4 Inference In this section we analyze test statistics for testing simple hypotheses about. Suppose we are interested in testing the null hypothesis H : = ; (4) against the alternative hypothesis H : = 6= : It is straightforward to construct HAC robust statistics using the three estimators of as t OLS = t BC = ( e ) v " u # ; (5) te (y t y ) ( ec ) v " u # ; (6) te c (y t y ) t IV = ( b ) v " u # ; (7) tb (t t)(y t y ) (t t) 9

11 where the estimated long run variance estimators are given by e = e + k e c = ec + k j= j= b = b + k j e j, e j = e t e t j ; M t=j+ j e c M j, e j c = e c t e e c c t j e c ; t=j+ j= j b j, b j = b t b t j ; M t=j+ with e t = y t e e yt being the OLS residuals from (5), e c t = y t e e c y t being the bias-corrected OLS residuals, and b t = y t b b yt being the IV residuals from (5). Because e c t = y t e e yt e c e y t = e t e c e y t ; the bias-corrected OLS residuals do not sum to zero. herefore, we construct e c using ec t e c where e c = e c t, i.e. we demean e c t before computing e c. he long run variance estimators are constructed using the kernel weighting function k(x) and M is the bandwidth tuning parameter. 4. Linear in Slopes Approach Because all three estimators of deteriorate as the trend slopes approach zero, we consider a fourth test statistic that is exactly invariant to the true values of the slope parameters under H. Given the null value of, H and H can be written in terms of the trend slopes as H : = ; H : = which is a nonlinear restriction on the trends slopes. Obviously, the restrictions implied by these hypotheses can be written as linear functions of and as Given, de ne the univariate time series where it follows from () and () that H : = ; H : = = ( ) 6= : z t ( ) = y t y t ; z t ( ) = ( ) + ( )t + v t ( ) ; (8)

12 where ( ) =, ( ) = and v t ( ) = u t u t. Notice that v t ( ) = t ( ) and therefore the long run variance of v t ( ) is = : Under H it follows that ( ) = whereas under H it follows that ( ) = ( ) 6=. We can test the original null hypothesis given by (4) by testing H : ( ) = in (8) against the alternative H : ( ) 6= using the following t-statistic: t = v u tb b ( )! ; (9) (t t) where b ( ) =! (t t) (t t) (z t ( ) z ( )) ; b = b + k j= j M b j, b j = bv t ( ) bv t j ( ) ; t=j+ bv t ( ) = z t ( ) z ( ) b ( ) t t : Note that b ( ) is simply the OLS estimator of ( ) from (8) and bv t ( ) are the corresponding OLS residuals. 4. Con dence Intervals Using t Con dence intervals for can be constructed by nding the values of such that jt j cv = () where cv = is the two-tail critical value for signi cance level. Because both b and b are functions of, nding the values of that result in a non-rejection, i.e. satisfy (), is equivalent to nding the roots of a particular second-order polynomial. Depending on whether the roots are real or complex, the con dence interval for can be a closed interval on the real line, the complement of an open interval on the real line, or the entire real itself. he form of the con dence interval depends on the magnitudes of the trend slopes relative to the noise as we now explain. Let b = b + k j= j ( M b j + b j )

13 where Partition b as It is easy to show that and b j = t=j+ bu t bu t j; bu t = [u t ; u t ] : " b b b b b # b ( ) = b b ; b = b b + b ; : allowing us to write () as b b v u t b b + b! (t t) or equivalently b b cv = ; b b + b he inequality given by () can be rewritten as where! cv= : () (t t) c + c + c ; () c = b b ; c = b b b, c = b b ; = cv =! (t t) : he values of that solve () depend on the roots of the polynomial p( ) = c + c + c. Let r and r denote the roots of p( ) and when r and r are real, order the roots so that r r. here are three potential shapes for the con dence interval for : Case : Suppose that c > and c 4c c. In this case, the roots are real and p( ) opens upwards. he con dence interval is the values of between the two roots, i.e. [r ; r ]. he inequality c > is equivalent to the inequality b b! > cv= : (3) (t t)

14 Notice that the left hand side of (3) is simply the square of the HAC robust t-statistic for testing that the trend slope of y t is zero. Inequality (3) holds if the trend slope of y t is statistically di erent from zero at the level. his occurs when the trend slope for y is large relative to the variation in u t. Mechanically, c will be positive when the t-statistic for testing that = is large in magnitude. Although not obvious, if c > ; it is impossible for c 4c c < to hold. In this case p( ) opens upward and has its vertex above zero and the roots are complex; therefore, there are no solutions to () and the con dence interval would be empty. his case is impossible because the con dence interval cannot be empty because = b = b = b is always contained in the interval given that b () must hold (equivalently () must hold). b = in which case Case : Suppose that c < and c 4c c >. In this case, p( ) has two real roots and opens downwards and the con dence interval is the values of not between the two roots. In this case the con dence interval is the union of two disjoint sets and is given by ( ; r ] [ [r ; ). Case 3: Suppose that c < and c 4c c. In this case, p( ) opens downward and has a vertex at or below zero. he entire p( ) function lies at or below zero and the con dence interval is the entire real line: ( ; ). Although it is a zero probability event, should c = then the con dence interval will take the form of either ( ; c =c ] when c > or [ c =c ; ) when c <. While the con dence intervals constructed using t 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 has a major advantage over the other t-statistics. Recall that we can write t as b b t = v u t b b + b! : (t t) he denominator is a function bu t and bu t each of which are exactly invariant to the true values of and : When H is true, it follows that = and we can write the numerator of t as b b = b b ( ) = b b : Because b and b are only functions of t and u t ; u t, the numerator of t is also exactly invariant to the true values of and. herefore, the null distribution of t is exactly invariant to the true values of and 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. Because of 3

15 its exact invariance to and under the null, t will deliver much more robust inference (with respect to the magnitudes of and ) than the other t-statistics. 4.3 Asymptotic Results for t-statistics In this section we provide asymptotic limits of the four t-statistics described in the previous subsection. We derive asymptotic limits under alternatives that are local to the null given by (4). Suppose that =. hen the alternative value of is modeled local to as = + 3=+ : (4) he parameter 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 statistic, under the null that = it follows that ( ) =. Under the local alternative (4), it follows that ( ) = ( ) = 3=+ = 3=+ = 3= ; (5) regardless of the magnitude of the trend slopes. herefore, the asymptotic limit of t 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 (5). he form of these limits depends on the type of kernel function used to compute the HAC estimator. We follow Bunzel and Vogelsang (5) and use the following de nitions. De nition A kernel is labelled ype if k (x) is twice continuously di erentiable everywhere and as a ype kernel if k (x) is continuous, k (x) = for jxj and k (x) is twice continuously di erentiable everywhere except at jxj = : We also consider the Bartlett kernel (which is neither ype or ) separately. limiting distributions are expressed in terms of the following stochastic functions. he xed-b De nition Let Q(r) be a generic stochastic process. De ne the random variable P b (Q(r)) as 8 R R k (r s) Q(r)Q)drds if k (x) is ype >< R R jr sj<b k (r s) Q (r) Q ) drds P b (Q(r)) = +k (b) R b Q (r + b) Q (r) dr if k (x) is ype >: R b Q (r) dr b R b Q (r + b) Q (r) dr if k (x) is Bartlett where k (x) = k x b and k is the rst derivative of k from below. 4

16 he following theorems summarize the asymptotic limits of the t-statistics for testing (4) when the alternative is given by (4). heorem 4 (Large rend Slopes) Suppose that (6) holds. Let M = b where b (; ] is xed. Let = ; = where ; are xed with respect to, and let = + 3=. hen as!, Z t OLS ; t BC ; t IV ) p Pb (Q(r)) + q ; P b (Q(r)) Z t ) p Pb (Q(r)) + q ; P b (Q(r)) where Z N(; ), Q(r) = ew(r) L(r) R s dw), ew(r) = w(r) and Z and Q(r) are independent. rw(), L(r) = R r s ds heorem 5 (Medium rend Slopes): Suppose that (6) holds. Let M = b where b (; ] is xed. Let = = ; = = where ; are xed with respect to, and let = +. hen as!, Z t OLS ) p Pb (H (r)) + E(u t t ()) q + q ; P b (H (r)) P b (H (r)) Z t BC ; t IV ) p Pb (Q(r)) + q ; P b (Q(r)) Z t ) p Pb (Q(r)) + q ; P b (Q(r)) where H (r) = Q(r) L(r) E(ut t ()). heorem 6 (Small rend Slopes): Suppose that (6) holds. Let M = b where b (; ] is xed. Let = ; = where ; are xed with respect to, and let = + =. hen as!, d t OLS ; t BC! r P b (L(r)) R ; ) ds + E u t Z t IV ) p Pb (Q(r)) + q ; P b (Q(r)) Z t ) p Pb (Q(r)) + q : P b (Q(r)) 5

17 heorem 7 (Very Small rend Slopes): Suppose that (3) and (6) hold. Let M = b where b (; ] is xed. Let = 3= ; = 3= where ; are xed with respect to, and let = +. hen as!, t IV ) = t OLS ; = t BC = R )dw) + R E(u t ) E(ut t ()) + q P b (H (r)) ; E u t ) ds + R )db ) q P b (H 3 (r)) R ) ds Z t ) p Pb (Q(r)) + q ; P b (Q(r)) ; where H (r) = ew(r) H 3 (r) = ew(r) Z and e B (r) = B (r) rb (). ) ds + Z L(r) + e B (r) + Z )db ) Z s db ) )dw) Z L(r) + B e (r) s dw); Some interesting results and predictions about the nite sample behavior of the t-statistics are given by the heorems 4-7. First examine the limiting null distributions that are obtained when =. 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 (5) for inference regarding the trend slope in a simple linear trend model with stationary errors. herefore, xed-b critical values are available from Bunzel and Vogelsang (5). As the trend slopes become smaller relative to the noise, di erences among the t-statistics emerge. As anticipated, t ; has the same limiting null distribution regardless of the magnitudes of the trend slopes. Except for very small trends slopes, t IV has the same limiting null distribution as t. he bias in OLS a ects the null limit of t OLS for medium, small and very small trend slopes. he bias correction helps for medium trend slopes in which case t BC has the same null limit as t IV and t. For small and very small trend slopes the bias correction no longer works e ectively and t BC has the same limiting behavior as t OLS. 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 t OLS and t BC will only work well when trend slopes are relatively large whereas t IV should work well except when trend slopes are very small. he most reliable test in terms of robustness to magnitudes of trend slopes under the null should be t. 6

18 When 6= 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 depends on rather than as for the other t-statistics. In general we cannot rank and as any di erence depends on the joint serial correlation structure of u t and u t. Unless and 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 t OLS and t BC becomes meaningless given that the statistics have poor behavior under the null. In heorem 6, the limits of t OLS and t BC do not depend on which suggests that power will very low when trend slopes are small. In heorem 7, t OLS and t BC diverge with the sample size which suggests large rejections with very small trend slopes. In constrast power of t IV should be similar to t except when trend slopes are very small. As in the case of null behavior, the asymptotic theory predicts that t should perform the best in terms of power. One thing to keep in mind regarding the limit of t in heorems 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. herefore, needs to be much farther away from in the case of small trend slopes than for the case of large trend slopes for power of t to be the same in both cases. In other words, for a given value of, power of t decreases as the trend slopes become smaller. his relationship between power and magnitudes of the trend slopes can be seen clearly in heorem 4 where we can see that as! the limiting distribution under the local alternative for approaches the null distribution and power decreases. 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. able reports null rejection probabilities for 5% nominal level tests for testing H : = = against the two-sided alternative H : 6=. Results are reported for the same values of ; as used in able for = 5; ; and ; replications are used in all cases. he HAC estimators are implemented using the Daniell kernel. Results for three bandwidth sample size ratios are provided: b = :; :5; :. For a given sample size,, we use the bandwidth M = b for each of the three values of b. We compute empirical rejections using xed-b asymptotic critical values using the critical value function cv :5 (b) = : :63b + :666b + 34:869b 3 3:956b 4 + 3:669b 5, as given by Bunzel and Vogelsang (5) for the Daniell kernel. 7

19 he patterns in the empirical null rejections closely match the predictions of the asymptotic results. When the trend slopes are large, 4,, null rejections are the essentially the same for all t-statistics and are close to.5 even when = 5. his is true for all three bandwidth choices which illustrates the e ectiveness of the xed-b critical values. For medium sized trend slopes, : :4, :5 :, t OLS begins to show over-rejection problems that become very severe as the trend slopes decrease in magnitude. he bias-corrected OLS t-statistic, t BC, is less subject to over-rejection problems especially when is not small, although for = 5, t BC shows nontrivial over-rejection problems. In contrast both t IV and t have null rejections close to.5 for medium sized trend slopes. When the trend slopes are small or very small, :4, :, the t OLS and t BC statistics have severe over-rejection problems and can reject % of the time. While t IV has less over-rejection problems in this case, the over-rejections are nontrivial and are problematic. In contrast, t has null rejections that are close to.5 regardless of the magnitudes of ; including the case of = =. In fact, the rejections are identical for t across values of ;. his is because t is exactly invariant to the values of ;. It is clear in terms of null rejection probabilities that t is the preferred test statistic. Given that t is the preferred statistic in terms of size, we computed, for each of the parameter con gurations in able, the proportions of replications that lead to the three possible shapes of con dence intervals one obtains by inverting t. Recall that the cases are given by Case : [r ; r ], Case : ( ; r ][[r ; ) and Case 3: ( ; ). able 3 gives these results. For large trend slopes Case occurs % of the time. As the trend slopes decrease in magnitude, Case 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 increases, the likelihood of Case increases for all trend slope magnitudes. he relative frequencies of the three cases also depends on the bandwidth but the relationship appears complicated. his is not surprising given the complex manner in which the bandwidth a ects the null distribution of t. Overall, unless trends slopes are small or very small, Case is the most likely con dence interval shape. While t is the preferred test in terms of size, how do the t-statistics compare in terms of power? able 4 reports power results for a subset of the grid of as used in ables,3. For a given value of, we specify a grid of six equally spaced values for in the range [; max ] where = = is the null value and max = :=. By construction = in all cases. Given the way we de ne the grid for, we ensure that is the same for all values of. Results are reported for =. Results for other values of are qualitatively similar and are omitted. he patterns in power given in able 4 are what we would expect given the local asymptotic limiting distributions. For large trend slopes ( = ; ), power of the four tests is essentially the same as predicted by heorem 4. As the bandwidth increases, power of all the tests decreases. his inverse relationship 8

20 between power and bandwidth is well known in the xed-b literature ee Kiefer and Vogelsang 5). For medium sized trend slopes ( = :; :) di erences in power begin to emerge with t OLS having substantially lower power than the other tests. his lower power occurs even though t OLS over-rejects under the null when a small bandwidth is used (b = :). With larger bandwidths, t OLS has no power at all. Both t BC and t IV have good power that is somewhat lower than t, which, according to heorem 5 would result from di erences between and. For small and very small trends slopes ( = :; :), both t OLS and t BC are distorted under the null with severe over-rejections although t OLS severely under-rejects with a large bandwidth (b = :) when = :. While t IV is less size distorted than t OLS and t BC, it has no power regardless of bandwidth. In contrast to other three statistics, t continues to have excellent size and power. he superior power of t relative to the other statistics is completely in line with the predictions of heorems 6 and 7. In summary, the patterns in the nite sample simulations are consistent with the predictions of heorems 4-7. Clearly t is the recommended statistic given its superior behavior under the null and its higher power under the alternative. 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 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. 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 (5) once a kernel and bandwidth have been chosen. 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, Some readers may notice that for a given bandwidth, power of t is the same regardless of the value of. his occurs because we have con gured the grids for so that max = : is the same in all cases. 9

21 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. here are at least two directions for future research related to extensions of the results in this paper. First, it would be interesting to see how the properties of the estimators and test statistics change if the stochastic part of the time series have unit roots. It should still be possible to consistently estimate the trend slopes ratio in this case although trend slopes may need to be relatively large for estimation to be accurate. Inference will almost certainly be a ected by unit roots and inference procedures that are valid whether or not it is known that the series have unit roots would be highly desirable in practice. Second, 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. 8 Appendix: Proofs of heorems 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. he 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 (CM). 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. hroughout the appendix, we use t to denote t ().

22 Lemma Suppose that (3) and (6) hold. he following hold as! for any values of ; : 3 (t t)! [r ] (t t)! Z Z r Z 3= (t t)( t ) ) ) ds = ; )ds = L(r); )dw); (u t u )( t )! p E (u t t ()) ; (u t u )! p E u t ; 3= (t t)(u t u ) ) Z 3= b ) (r Z Z ) dr )db ); )db ): Proof: he results in this lemma are standard given the FCLs (3) and (6) and ergodicity of u t and u t. See Hamilton (994). Lemma (Large trend slopes) Suppose that (3) and (6) hold and ; are xed with respect to. he following hold as! for =, =, Z 3= (y t y )( t ) ) Z 3= (y t y )(u t u ) ) 3 (y t y )! p Z Z 3 (t t)(y t y )! p [r ] 3= (y t y ) ) L(r): )dw); )db ); ) ds = ; ) ds = ;

23 Proof: he results of the lemma are easy to establish once we substitute y t y = t t + (u t u ) into each expression: 3= (y t y )( t ) = 3= (t t)( t ) + 3= (u t u )( t ) = 3= (t t)( t ) + o p (); 3= (y t y )(u t u ) = 3= (t t)(u t u ) + 3= (u t u ) = 3= (t t)(u t u ) + o p (); 3 (y t y ) = 3 (t t) + o p (); 3 (t t)(y t y ) = 3 (t t) + o p (); [r ] 3= he limits follow from Lemma. [r ] (y t y ) = (t t) + o p (): Lemma 3 (Medium trend slopes) Suppose that (3) and (6) hold and ; are xed with respect to. he following hold as! for = =, = =, Z (y t y )( t ) ) Z (y t y )(u t u ) ) (y t y )! p Z Z 5= (t t)(y t y )! p [r ] 3= (y t y ) ) L(r): )dw) + E (u t t ()) ; )db ) + E u t ; ) ds = ; ) ds = ;

24 Proof: he results of the lemma are easy to establish once we substitute y t y = t t + (u t u ) into each expression: (y t y )( t ) = 3= (t t)( t ) + (u t u )( t ); (y t y )(u t u ) = 3= (t t)(u t u ) + (u t u ) ; (y t y ) = 3 (t t) + o p (); 5= (t t)(y t y ) = 3 (t t) + o p (); [r ] 3= he limits follow from Lemma. [r ] (y t y ) = (t t) + o p (): Lemma 4 (Small trend slopes) Suppose that (3) and (6) hold and ; are xed with respect to. he following hold as! for =, =, (y t y )( t )! p E (u t t ()) ; (y t y )(u t u )! p E u t ; (y t y )! p Z Z (t t)(y t y )! p [r ] ) ds + E u t = + E u t ; (y t y ) ) L(r): ) ds = ; 3

25 Proof: he results of the lemma are easy to establish once we substitute y t y = t t + (u t u ) into each expression: (y t y )( t ) = (t t)( t ) + (u t u )( t ) = (u t u )( t ) + o p (); (y t y )(u t u ) = (t t)(u t u ) + (u t u ) = (u t u ) + o p (); (y t y ) = 3 (t t) + (u t u ) + o p (); (t t)(y t y ) = 3 (t t) + o p (); [r ] he limits follow from Lemma. [r ] (y t y ) = (t t) + o p (): Lemma 5 (Very small trend slopes) Suppose that (3) and (6) hold and ; are xed with respect to. he following hold as! for = 3=, = 3=, (y t y )( t )! p E (u t t ()) ; (y t y )(u t u )! p E u t ; Z 3= (t t)(y t y )! p (y t y )! p E u t ; [r ] = ) ds + Z )db ) = + (y t y ) ) L(r) + e B (r): Z )db ); 4

26 Proof: he results of the lemma are easy to establish once we substitute y t y = t t + (u t u ) into each expression: (y t y )( t ) = 5= (t t)( t ) + (u t u )( t ) = (u t u )( t ) + o p (); (y t y )(u t u ) = 5= (t t)(u t u ) + (u t u ) = (u t u ) + o p (); (y t y ) = 4 (t t) + (u t u ) + o p () = (u t u ) + o p (); 3= (t t)(y t y ) = 3 (t t) + 3= (t t)(u t u ); [r ] = he limits follow from Lemma. [r ] [r ] (y t y ) = (t t) + = (u t u ): Proof of heorem. he proof follows directly from Lemmas, and the CM. For the case of large trend slopes it follows that 3= e 3= b! = 3 (y t y ) 3= (y t y )( t ) Z Z ) ) ds )dw);! = 3 (t t)(y t y ) 3= (t t)( t ) Z ) Z ) ds )dw): 5

27 For the case of medium trend slopes it follows that! e = (y t y ) (y t y )( t ) Z Z ) ) ds! b = 5= (t t)(y t y ) 5= (t t)( t ) Z ) Z ) ds )dw) + E (u t t ()) ; )dw): Proof of heorem. We rst need to derive the limit of P bu te t for the large and medium trend slope cases. For large trend slopes we have, using Lemmas, and heorem : 3= b bu t e t = (u t u )( t ) 3= e 3= (y t y )(u t u ) 3= (t t)( t ) + 3= e 3= b 3 (t t)(y t y ) = (u t u )( t ) + o p ()! p E (u t t ()) : (6) For medium trend slopes we have, using Lemmas,3 and heorem : 3= b bu t e t = (u t u )( t ) e (y t y )(u t u ) 3= (t t)( t ) + e 3= b 5= (t t)(y t y ) = (u t u )( t ) + o p ()! p E (u t t ()) : (7) he results for e c are as follows. For large trend slopes 3= e c! = 3 (y t y )! = 3 (y t y ) ) Z! 3= (y t y )( t ) 3= bu t e t! 3= (y t y )( t ) + o p () Z ) ds )dw); 6

28 using Lemma, (6) and the CM. For medium trend slopes e c! = (y t y ) ) = Z Z using Lemma, (7) and the CM.! (y t y )( t ) bu t e t () Z ) ds Z ) ds )dw) + E (u t t ()) E (u t t ()) )dw); Proof of heorem 3. We rst give the proof for the case of small trend slopes. Using Lemmas,4 and the CM it follows that = b e! = (y t y ) (y t y )( t ) p! Z = ) ds + E(u t) E (u t t ()) = <;! (t t)(y t y ) 3= (t t)( t ) Z ) Z ) ds )dw): o establish the result for e c we rst need to derive the limit of P bu te t. Using Lemmas,4 and the results for OLS 3= b bu t e t () = (u t u )( t ) 3= (t t)( t ) + = e 3= b = (u t u )( t ) e (y t y )(u t u ) (t t)(y t y ) e (y t y )(u t u ) + o p () Z p! E (u t t ()) <E(u t) = ) ds<: (8) 7

29 Using Lemma 4, (8) and the CM, it follows that e!! c = (y t y ) (y t y )( t ) bu t e t () p! Z Z ) ds + E(u t) E (u t t ()) = < E(u t ) E (u t t ()) = < c: ) ds< he proof for very small slopes is similar. Using Lemmas,5 and the CM it follows that e = (y t y )( t ) (y t y ) p! E (u t t ()) E(u t ) ; b! = 3= (t t)(y t y ) 3= (t t)( t ) Z ) ) ds + Z Z )db ) )dw): o establish the result for e c we rst need to derive the limit of P bu te t. Using Lemmas,5 and the results for OLS 3= b bu t e t () = (u t u )( t ) 3= (t t)( t ) + e 3= b = (u t u )( t ) p! E (u t t ()) e (y t y )(u t u ) 3= (t t)(y t y ) e (y t y )(u t u ) + o p () E (u t t ()) E(u t ) E(u t) = : (9) Using Lemma 4, (9) and the CM, it follows that e!! c = (y t y ) (y t y )( t ) bu t e t () = e + o p () p! E (u t t ()) E(u t ) : We now provide proofs of heorems 4-7. he proofs involve two steps. For a given t-statistic, we rst establish the xed-b limit of the relevant HAC estimator and then we establish the limit of 8

30 the t-statistic using heorems -3 and the CM. o establish the xed-b limit of a HAC estimator we rewrite the HAC estimator in terms of scaled partial sum processes using the xed-b algebra of Kiefer and Vogelsang (5). We do not provide details of the xed-b algebra here and we simply establish the scaling and limits of the relevant partial sum processes which are given by es [r ] = P [r ] e t, S e [r c ] = P [r ] e c t e c and b S [r ] = P [r ] b t. Note that no proofs are needed for the limit of t because t is computed using regression model (8) under the local alternative for ( ) given by (5). his is a special case of heorem in Bunzel and Vogelsang (5). Proof of heorem 4. With large trend slopes, the scaling and limits of the partial sum processes are given as follows where the limits follow from the Lemmas, and heorems,. For the OLS estimator we have [r ] = S[r e ] = = ( t ) 3= e ) ew(r) Z = ew(r) L(r) Z [r ] (y t y ) Z ) ds )dw) L(r) )dw) = Q(r): he forms of e S c [r ] and b S [r ] are identical with e replaced by e c or b. Because the three estimators have the same asymptotic distributions, it follows that = e S c [r ] ; = b S[r ] ) Q(r). Using xed-b algebra and arguments from Kiefer and Vogelsang (5), it follows that e ; e c ; b ) P b (Q(r)) : he limit of t OLS is as follows. he arguments for t BC and t IV are similar and are omitted. Simple algebra gives 3= e 3= e + t OLS = v " u te # = v " u 3 (y t y ) te # 3 (y t y ) ) R r p R = p )dw) + Pb (Q(r)) using the fact that R R ) ds P b (Q(r)) q P b (Q(r)) )dw) N ;. )dw) + R ) ds 9 Z = p Pb (Q(r)) + q ; P b (Q(r))