Testing for a Trend with Persistent Errors

Transcription

1 Testing for a Trend with Persistent Errors Graham Elliott UCSD August 2017 Abstract We develop new tests for the coe cient on a time trend in a regression of a variable on a constant and time trend where there is potentially strong serial correlation. This serial correlation can also include a unit root. We obtain tests under two different assumptions on the initial value for the stochastic component of the variable being examined, either this being zero asymptotically (as in previous examinations of this hypothesis) and also allowing the initial condition to be drawn from its unconditional distribution. We nd that statistics perform better under the second of these assumptions, which is the more natural assumption to make. JEL classi cation: C12; C21; C22 Keywords: composite hypothesis, trend I thank participants at seminars at Monash University, Norges Bank and the IAAE 2017 conference in Sapporo for comments. All errors are mine.

2 1 Introduction We examine the problem of conducting hypothesis tests on the coe cient on a time trend in a regression of a variable that is serially correlated on a constant and time trend. Consider the model x t = a + bt + w t (1) w t = rw t 1 + u t u 1 = : where fx t g T t=1 are observed and fa; b; r; g and the variance of u t are unobserved (for now consider u t to be serially uncorrelated, this will be generalized later in the paper). We are interested in testing the hypothesis H 0 : b = 0 (2) H a : b > 0: The primary complication in deriving tests that control size and have good properties is due to the nuisance parameter r that describes the largest root in a model of the serial correlation. If this parameter were known then x t can be quasi di erenced and (at least asymptotically) a UMP test exists for the one sided test and a UMPU test exists for the two sided hypothesis. Alternatively if this parameter is bounded in absolute value with a bound that is (given the sample size) far enough from one then standard tests discussed below are available and can be employed. These too have asymptotic optimality properties. We are primarily concerned with testing situations where such a bound does not exist, and we would like to allow for r to be close to or equal to one. When the parameter approaches one or could be one, a number of papers have examined the regression coe cient under study (Canjels and Watson (1997) for estimation, and Bunzel and Volgelsang (2005), Harvey et. al. (2007) and Perron and Yabu (2009) have each provided tests). The following gure (Figure 1) shows the problem of size control for models where r becomes close to or equal to one and also provides a guide to some of the previous solutions to providing tests that can potentially control size uniformly over values for r. For the model in (1) with u t N(0; 1) we examine the size of tests of (2) undertaken in a few di erent ways. The rst consideration is to simply construct the usual one sided test on the OLS estimate for b from a regression of x t on a constant, time trend and lag of x t using the data from t = 2; :::T: The second consideration is to use a t test based on the Feasible GLS estimator for b where rather than knowing r we replace it with an estimator based on 1

3 Figure 1: Solid lines are size for FGLS (lower line is for = 0) and the dotted line is for OLS. The dash-dot line in the second panel is GLS with a unit root imposed. a regression of the residuals from a regression of x t on a constant and time trend on the residuals lagged one period (again dropping the rst observation because of the lag). The rst panel of Figure 1 shows the size from a Monte Carlo for various values for r (the x axis reports this as = T (1 r); anticipating the asymptotic problem that is examined below) where T = 500 and results reported are averages over replications. Two variations on are considered, either setting this value to zero or instead allowing to be zero for r = 1 and drawn from a normal distribution with mean zero and variance 1=(1 r 2 ): Whilst the numbers are not identical the e ect of the di erent initial conditions on OLS is not noticeable in the gure. It is clear that when is closer to zero (r closer to one) that standard tests fail to control size. As becomes larger, standard tests do indeed control size. Hence the di cult part of the testing problem to be solved is in the region of r = 1, since any test that has good properties in that region could be mixed or switched with the standard one when r is small enough. The second panel of Figure 1 reports the same values as in the rst panel focussing on values for r closer to one, where 0 20: Added to the gure is the size of tests based on the MLE for b when we assume r = 1. The test imposing r = 1 is undersized for r < 1, quickly converging to zero. It is here that one strategy that could be employed becomes clear constructing a test that mixes the test at r = 1; which controls size at the boundary point and is otherwise undersized, with the test that works for r su ciently 2

4 less than one could, depending on how the tests are mixed control size. A test could control size by putting enough weight on the tests at r = 1 when r < 1 but in the region where the regular tests are oversized could control size in this region as it is mixed with a severely undersized test, and then for values for r further from one place more weight on the usual test. This is precisely the insight of Harvey et. al. (2007), who suggest a test statistic of the form z i = z 0(1 (x))+(x)r i (x)z 1 where z 0 is the test when r is far less than one and z 1 is the test based on imposing r = 1: The mixing statistic (x) is a random variable depending on the data that has the properties that (pointwise) is such that (x)! p 1 if r = 1 and (x)! p 0 for r < 1: The pointwise properties are not su cient in themselves to ensure uniform size control, however they show in Monte Carlo results that size does appear to be controlled for values of r near one for many sample sizes. The role of R i (x) is because when they set R i (x) = 1 (as for the z statistic) the test is severely undersized asymptotically for near zero, so alternative choices were made to lessen this issue. Perron and Yabu (2009) criticize this formulation of the mixture on a number of grounds. From a practical point of view the strongest criticism is that the mixing coe cient (x), although converging to one asymptotically when r = 1, has in nite samples a distribution that places positive weight on smaller values for r which can impact the performance of the test. Indeed, to construct (x) Harvey et. al. (2007) make a number of ad hoc choices motivated by the need to control size and choose power curves, resulting in three variations of the test (i.e. choices of R i (x)). The e ect of this is that in any sample the test involves both z 0 and z 1 : Instead, Perron and Yabu (2009) alter the strategy in two ways. First, rather than have (x) take values between zero and one, they essentially suggest a statistic that chooses either the values of zero and one, essentially choosing the statistic z 1 whenever r is close enough to one (based on the data) and choosing a version of the standard test otherwise. This has the virtue of essentially always choosing z 1 when r is close to one. The second innovation is to improve the statistic chosen when r < 1: This essentially involves using a di erent estimator for r in the feasible GLS regression (Harvey et. al. (2007) use OLS) 1 in order to have better properties for r large but not equal to one. For problems where r 2 [0; 1], no tests will be optimal uniformly in r: From a hypothesis testing problem we have a nuisance parameter r that is not simple under either the null or alternative hypotheses. As is seen in Figure 1, use of a (pointwise) consistent estimator such as OLS for r will not result in size control. Hence previous approaches, although ad hoc, make sense since constructing tests that control size uniformly for these problems is di cult. 1 This alternative estimator is su ciently complicated that rather than repeating the method here, we refer readers to the original paper. 3

5 For any classical test, choosing a test to use amongst tests that control size is essentially the choice over power functions of the tests. Here, because of the nuisance parameter, we have a surface of power functions from which to choose (or alternatively, asymptotically a power function for each choice of ): Indeed, examining the power functions presented in either Harvey et. al. (2007) (which were compared with the method of Bunzel and Vogelsang (2005)) or Perron and Yabu (2009) (which compared with both the previous approaches) there is no uniformly dominant method. Each paper argued that despite this their approach was preferred over their predecessor based on having better power properties over the majority of parameterizations considered. We take the same approach here, constructing new tests that pay close attention to the properties of the test when r is large or potentially one, and switching to standard tests for values of r further from one. At issue statistically here is that there is a nuisance parameter, the local size of the largest root of w t, that complicates the testing. By utilizing the methods 2 of Elliott, Mueller and Watson (2015), denoted below as EMW, we are able to construct tests that (asymptotically) control size directly for a large region for including zero rather than rely on a test that has very small size (z 1 ) in this region. Because we can control size for a large region of r near one we switch to standard methods for values of r much further from 1 than previous tests. Our approach, by virtue of the explicit dependence on how power is directed towards the alternative hypothesis, results not in a single test but a family of tests dependent on these choices. In addition to providing results when is asymptotically irrelevant, the situation assumed in previous work, we also provide tests that are constructed for the case where is drawn from the unconditional distribution for w t when r < 1: This is a more natural assumption to make in practice and we will show that tests derived under this assumption have better properties. We develop these tests in the next section, and in section 3 examine this test along with those discussed above. The actual construction of the tests allowing for serial correlation is presented in a self contained section in the appendix. 2 Constructing the Test The form of the test we propose is a test that has the same switching approach as discussed above, namely T R = ' 0 (x)(1 (x)) + (x)' 1 (x) 2 Other methods are possible, for example King (1996) 4

6 where ' 0 is a test function (so takes a value zero for failure to reject and one for rejection) for the hypothesis in (2) when r is far less than one (this is the simple test in the jargon of EMW), (x) is a switching rule that, as in Perron and Yabu (2009), takes values of zero or one depending on whether the data indicates we are in the simple region or not, and nally ' 1 (x) is a new test function derived in this paper for testing (2) when r is in the neighborhood of one. So for any sample fx t g T t=1 the test function T R as a function of the sample takes on a value of one if the hypothesis is rejected and zero otherwise. A primary di erence here as opposed to previous tests is that our test switches to the standard test for alternatives much further from one than earlier tests. We are able to do this and still control size as a result of the test when r is near one being constructed to control size not just when r = 1 but also for local alternatives as well. Hence rather than relying on the test assuming a unit root in the neighborhood of a unit root being conservative to keep the mixture test size controlled, we rely on the construction of the test function ' 1 (x) having good size properties over a wide range of models. 2.1 The test when r is near one We turn to the construction of ' 1 (x), which provides a test of the hypothesis when r is near one. The test statistics we propose have the form W LR = P m i=1 P J j=1 w a i L(x; i ; ij ) P m i=1 w 0 i L(x; i ) where i = T (1 r i ); ij are chosen points under the alternative hypothesis local to zero; w ai are a set of weights that sum to one, w 0i are a set of weights (that do not sum to one generally for reasons discussed below), L(x; i ; i ) is an approximation to the likelihood local both to r = 1 and b = 0 and L(x; i ) = L(x; i ; 0), the approximation to the likelihood when b = 0 (i.e. under the null hypothesis). The statistic rejects when log(w LR) is greater than zero, i.e. when the weighted average likelihood under the alternative is large relative to the weighted average likelihood under the alternative. The critical value here is subsumed into the weights under the null, which explains why these weights do not sum to one. To see this, de ne ~w 0i as a set of weights that sum to one, and notice that we could write the test function as rejecting for x when P m i=1 w a i L(x; i ; i ) P m i=1 ~w > cv 0 i L(x; i ) 5

7 but this is equivalent to rejecting for x when P m i=1 w a i L(x; i ; i ) P m i=1 (cv ~w 0 i )L(x; i ) > 1 so we directly construct the test with w 0i = cv ~w 0i and so the sum of the weights is the critical value in the standard set up of a likelihood ratio testing problem. The form of the suggested statistic is a likelihood ratio test for (2) where we deal with the nuisance parameter describing the serial correlation by taking weighted averages over this nuisance parameter. Under the alternative hypothesis (the numerator of the statistic) the statistic then weights di erent models, with the choice of weights directing power to the alternatives. This is a standard approach to constructing tests that have optimality properties against the chosen weighted alternative, see Andrews and Ploberger (1994). Under the null hypothesis the weights ~w 0i are an estimate of the least favorable distribution for the testing problem given the choice of the alternative. Tests constructed in this way have, by virtue of their form as a Neyman Pearson statistic, optimality properties. See EMW for an extensive discussion with relation to the approach used here. To construct the test then we need to de ne the form of the approximate likelihood L(x; i ; i ); the weights w ai, and then estimate the critical value adjusted weights w 0i using the methods in EMW: We rst consider L(x; i ; i ): For the testing problem addressed here, the form of the likelihood even asymptotically depends on the assumption on the initial condition, in much the same way that it does for testing in the unit root literature. We will consider two assumptions, the rst allowing to be xed so that it is asymptotically irrelevant and can be set to zero in the approximation, and second drawing the initial value w 1 from the unconditional distribution for w t when r < 1: Note that because under both assumptions we remove the nuisance parameter a from the likelihood through invariance to translations of the form fx t g T t=1 7! fx t + ~ag T t=1, ~a 2 R, then this also removes the initial value at r = 1: The Likelihood when = 0: Consider the model in (1) with = 0 and additionally assume that u t are serially uncorrelated and have N(0; 2 ) marginal distributions. In this case the likelihood for the statistic invariant to the translation discussed in the previous paragraph is proportional to 2 (x x 1 bz) 0 (A(r) 0 A(r) A(r) 0 A(r)( 0 A(r) 0 A(r)) 1 0 A(r) 0 A(r))(x x 1 bz) where x = (x 1 ; x 2 ; :::; x T ) 0, z = (1; 2; :::; T ) 0, is a T x1 vector of ones and A(r) is a T xt di erencing matrix with ones on the diagonal, r on the immediately diagonal below the 6

8 main diagonal and zeros everywhere else (so A(r)y for any vector y is (y 1 ; y 2 ry 1 ; :::; y T ry T 1 ) 0 ): See Mueller and Elliott (2003) for a derivation of this likelihood. Let ij = b ij = p T then we can rewrite this for choices ( i ; ij ) as h(x) + 2 i 2 T 2 ~x 0 1~x i 2 T 1 ~x 0 1(~x ~x 1 ) (3) i 1=2 T 1=2 ~x T + 2 i 1 T P 2 3=2 T t=2 ~x t ij 2 ij T (1 + (T 1)(1 r) 2 ) " T 1 T i s 2 + r 2 T r i i ( T 1 + 2r i i s ) + 2 # T 2 i s 1 1 T (1 + (T 1)(1 r i ) 2 ) 2 3 (r i + i ) 1 T 1=2 ~x T + 2 i 1 T 5=2 z 0 ~x 1 2 i 1 T 5=2 ~x T 1 (1+r i i ( T 1 T )+2 i s 1) i 1=2 T 1=2 ~x T + 2 i 1 T P 5 3=2 T : 1+(T 1)(1 r i ) 2 t=2 ~x t 1 where ~x = x x 1 and ~x 1 = (0; ~x 1 ; :::; ~x T 1 ) 0, s 1 = T P 2 T t=2 t and s 2 = T P 3 T t=2 t2 : Since h(x) is not a function of the nuisance parameters (apart from the scale parameter which is consistently estimable) we can ignore this as it cancels in the numerator and denominator for the test statistic W LR: The remaining terms are either O p (1) or disappear asymptotically. After subtracting h(x) then the exponent of the remainder of the terms in (3) multiplied by minus one half is de ned as L 0 (x; ; ) where the zero subscript denotes computation under the assumption that = 0: Then using standard FCLT and CMT results in Z Z 2 ln L 0 (x; i ; ij ) ) 2 i W ; (s) 2 ds + 2 i W ; (s)dw ; (s) + 2 ij i 3 + i Z 2 ij (1 + i )W (1) + 2 i sw (s)ds = H( i ; ij ) where W ; (s) = R s e (s ) dw () + and W () is a standard Brownian Motion. The 0 notation suppresses dependence on the true values for (; ) that arise through W ; (s): The limit distribution of the test statistics are then equal to W LR ) P m i=1 P J i=j w a i exp( 0:5H( i ; ij )) P m i=1 w 0 i exp( 0:5H( i )) for various choices over w ai, m; and b ij : Each of these choice variables indicate the choice of the alternative in the weighted average under the alternative. Our actual statistic, i.e. the construction of L(x; i ; i ); is based on the term (3) after removing h(x) but has the same limiting distribution as stated immediately above. Exact 7

9 calculation details are given in the appendix. For the null likelihood we set = 0 so the second two terms do not appear. Because we make a number of choices for the values under the alternative hypothesis (w ai ; ij ) we essentially derive a family of tests here indexed by these choices. We make three di erent choices for the case where = 0, the statistics will be denoted W LR 1, W LR 2 and W LR 3 with the particular choices made clear in the appendix deriving the tests The Likelihood when comes from the unconditional distribution. Consider now the same model from the previous subsection, however now N(0; 2 =(1 r 2 )) and independent of fu t g T t=2 for r < 1: Note that the statistic invariant to translations of the form fx t g T t=1 7! fx t + ~ag T t=1, ~a 2 R is invariant to the value of at r = 1: For this case the likelihood is proportional to 2 (x x 1 bz) 0 (A(r) 0 V 1 A(r) A(r) 0 V 1 A(r)( 0 A(r) 0 V 1 A(r)) 1 0 A(r) 0 V 1 A(r))(x x 1 bz) where V 1 is the identity matrix except that it has (1 r 2 ) in the (1,1) element if r < 1, and is the identity matrix otherwise. Expanding as above for choices ( i ; ij ) we obtain that this can be rewritten for r i < 1 as 3 ~h(x) + 2 i 2 T 2 ~x 0 1~x i 2 T 1 ~x 0 1(~x ~x 1 ) (4) i 1=2 T 1=2 ~x T + 2 i 1 T P 2 3=2 T t=2 ~x t ij 2 ij T (1 ri 2 + (T 1)(1 r)2 ) " T 1 (1 r i )T i s 2 + r 2 i T 1 + r i i # i s 1 + 2r i i s 1 T T (1 ri 2 + (T 1)(1 r i) 2 ) 2 3 (r i + i ) 1 T 1=2 ~x T + 2 i 1 T 5=2 z 0 ~x 1 2 i 1 T 5=2 ~x ( i T 1 +r i i + 2 i s 1) T (1 ri 2+(T 1)(1 r i) 2 ) i 1=2 T 1=2 ~x T + 2 i 1 T P 5 3=2 T t=2 ~x : t 1 Again ~ h(x) does not depend on nuisance parameters and so it will cancel from the numerator and denominator in W LR: Ignoring this term, multiplying by minus one half and taking limits using standard FCLT and CMT results yields a limit distribution of 2 ln L 1 (x; i ; ij ) ) 2 i Z M ; (s) 2 ds + 2 i Z M ; (s)dm ; (s) 3 The expression for r i = 1 is as for the zero initial condition case. R i M ; (1) + 2 i M; (s)ds 2 2 i + 2 i 8

10 + 2 ij i 12 + i 2 2 ij (1 = H 1 ( i ; ij ) i 2 )M ;(1) + 2 i Z sm ; (s)ds 2 i 2 Z M ; (s)ds (5) where M ; (s) is de ned as ( M ; (s) = W (s) + for = 0 (e s )(2) 1=2 + R s 0 e (s ) dw () + else : (6) As in the previous case this establishes the limit distributions for the test statistics. As with the previous case, we really de ne a family of tests based on the choices made under the alternative hypothesis. Here we make two choices, denoted W LR 4 and W LR 5 : This is discussed in the next subsection and more concretely de ned in the Appendix Weights for the Likelihood under H a The choice of the weights for the likelihood under the alternative in each case involves choosing ij and w ai. The choices impact the statistics in a number of ways. In terms of the optimality properties of the statistics the statistics have such properties against the alternatives chosen. More practically we can consider the choice as one that generates the power properties of the tests. Finally, for the methods of EMW to work well the choices can result in better or poorer estimation of the least favorable weights. In practice some searching over choices was undertaken to product tests with good asymptotic properties. Since this (and the nite sample performance of the tests) is what we are ultimately interested in then it seems reasonable to make a number of choices that have explicit trade-o s. This will be evident in the results below. The actual choices are available in the description of the test statistic construction below, but generally the weights are chosen so that w ai are even across points in the range for from zero to 81 or by choosing weights that upweight values for from zero to 10 relative to larger values of : For ij, q we choose b ij = c j = 1 2 i =d which is in the spirit of point optimal test choices (see King (1987) and King and Sriananthakumar (2017). This achieves a number of objectives. First, ij becomes smaller as i increases because the power of tests for a trend increase quickly as the data becomes more stationary. This is the primary reason for focussing on power when is small, because this is the region where power is low for the tests (this is evident in the large sample gures below, where as becomes larger the scale of the alternatives for 9

11 which power is below one becomes smaller). The rate of decline is increased by choosing a larger value for d: Second, the entire function is shifted up through choices of c j : For the best results it was found that choosing two values for c j improved the shape of the asymptotic power functions. For each di erent choice in the weights and ij di erent weights under the null are estimated Switching For the tests we need to de ne (x), which will equal one when we employ the W LR statistic and zero when the standard test controls size. We do so e ectively through a pre-test for r = 1 but one which only rejects for values for very far from zero (so a test with e ectively zero size for a wide range of ): The idea is to only switch when the standard test is in a range for that controls size well. From Figure 1 we see that this is in the range for beyond 40 or so. Consider the regression x t = a + bt + p 0 x t 1 + LX p l x t 1 + v t (7) l=1 where if L = 0 no additional lagged changes are included and L is chosen by the BIC criterion (one could also choose the MAIC criterion of Perron and Ng (2006)). If L=0 is imposed (because we know there is no serial correlation in u t ) then we construct the statistic (x) = 1( T ^p 0 < 50): For situations where there is some potential serial correlation the statistic needs to be adjusted. So for L > 0 we construct (x) = 1( T ^p 0 q^! 2 u=^ 2 v < 50) where ^ 2 v = (T L 3) 1 P T L+1 ^v2 t where ^v t are the OLS residuals from (7) and ^! 2 u is an estimate of the spectral density of ^u t where ^u t are the OLS residuals of (7) with L set to zero. This approach to switching is employed for both assumptions on the initial condition The Test when r is far below one There are two issues in choosing the test function ' 0 (x) for use when r is far below one. The rst is size control, the second is the properties under more general serial correlation. Whilst use of the feasible GLS estimator may allow some power advantages over using least squares when r is far from one, our (unpresented) Monte Carlo evaluation found that using the t test from the regression (7) with L chosen using the BIC criterion had the advantages that it accounts fairly well for serial correlation of various types and it simpli es construction of the test statistic since we require estimation of the same regression as for constructing the switching statistic. 10

12 3 Evaluation of the Methods For the model when = 0, we construct three statistics, each with di erent properties with none dominating the other. The test statistics are denoted T R 1 ; T R 2 and T R 3 and were designed to give di erent options as to the set of power functions. The precise construction and weighting choices under the alternative are detailed in the appendix. We compare them to the z m1 statistic of Harvey et. al. (2009) and the M U statistic in Perron and Yabu (2009). A more detailed examination of other statistics from these two papers is available in a supplemental appendix. We now turn to large sample power properties of the statistics suggested in this paper and earlier work. Figure 2 reports results from a Monte Carlo exercise with the sample size set to 500 and Monte Carlo draws. Each panel shows power as a function of = b= p T where the data is drawn from (1) with = 0: Di erent panels accord with di erent choices of the nuisance parameter, which takes the values 0; 5; 10; 15; 20; and 30: At = 0, we note rst that the only statistic that is (essentially) equivalent to the infeasible MLE power bound is the Perron and Yabu (2009) MU statistic, which comes about by virtue of that statistic forcing itself to be equal to the MLE with a unit root for roots near one (Harvey et. al. (2007) also have a statistic (they denote as z ) that achieves this bound however the one reported here has much better power than this statistic for other alternatives). This comes at a cost in power as gets larger, for example for = 10 through 20 the power of the MU statistic (the dotted line with the at part) is far below all other statistics for most of the local alternatives. The z m1 statistic (the dotted line with monotonically increasing power, but size near zero for near 0) has lowest power at = 0 and, because of poor asymptotic size properties for > 0 low power for near alternatives. Despite this, at = 15 this statistic has a power function that is not dominated by other methods, however at other values for (larger and smaller) other statistics have better local power. Similar to the comparison between the MU and z m1 statistics, the T R family of tests have power functions that do not dominate each other over all : The di erent choices for the weighted averaging allow construction of tests that, like z m1, have poor size properties for greater than but near zero but steep power curves, T R 2 (dashed line) provides an example of this set of power curves. For each power does not atten out for any range of, but the size properties mean that power for small is lower than other tests. However it is clear from the gure that although T R 2 and z m1 share similar qualitative properties and the two sets of power curves are quite similar for = 5 and 10 the T R 2 statistic is dominant for wide ranges of for other values for : For > 10 this occurs because the size becomes closer to 11

13 Figure 2: Upper solid line is the infeasible MLE, dotted lines are the MU and z m1 statistics distinguishable as follows: For = 0 the visible dotted line is z m1 and MU coincides with the infeasible MLE, all others MU is the curve with a kink and power slow to rise for large : Lower solid line is T R 1, dashed line is T R 2 and dash dot is T R 3 : 12

14 nominal size faster as gets larger. It also clearly dominates at = 0: The choice of weighting that results in the statistic T R 1 has similar qualitative properties as the MU statistic, in that this statistic has less of a tendency to be undersized for near zero but this comes at a cost of wiggles in the power function for moderate values of : It is dominated by MU at = 0, however at values larger than zero T R 1 strongly dominates MU for smaller values of and is generally only slightly below that of MU for larger values of : The point at which the two power curves cross occurs at higher and higher power as increases and for = 20 there is virtually no di erence when they are close and at = 30 T R 1 dominates. Indeed, over the majority of (; ) pairs T R 1 dominates all of the other tests, sometimes by a large margin, and is virtually equal to the infeasible MLE at = 30: The T R 3 statistic has properties that are in between those of T R 1 and T R 2 : The power functions have a smaller wiggle than T R 1 but retain some of that statistics good power properties for small when is small and moves quickly towards the infeasible power bound as gets larger. At = 20 for example it has power well above all other tests apart from T R 1 for power less than 80%: When for r < 1 our model assumes N(0; 2 =(1 r 2 )); as shown in Section 2 the limit distributions under both the null and alternative hypothesis di er from those when we assume = 0. And, as we noted above, the optimal statistics generated in this paper will have di erent weights for the same chosen alternative distributions. So now we turn to large sample results under this assumption, and rst consider the statistics optimized for this distribution. We present results for two statistics, T R 4 and T R 5. As in the previous case, we evaluate power for various values for. As before, the infeasible MLE (under the new assumption on the initial condition) is provided as a benchmark even though for most the power of this statistic cannot be matched as it knows r: Figure 3 shows the results for this statistic and T R 4 and T R 5 : The two sets of weights under the alternative were chosen not only to maximize power for these cases but also to capture that for some weighting functions we still have wiggles in the power function for moderate values for : However unlike the = 0 case the statistic with the smoother power functions (T R 4 ) does not su er as much from being undersized for small but non zero : This means that despite neither of the power functions dominating each other for all and the T R 4 statistic does seem to be the better choice. The power of T R 5 does dominate at = 0 however there are very similar at = 5 and 10 after which the T R 4 test dominates. However both have power functions that are close to the infeasible bound at =0 and at = 30. We now turn to small sample evaluation of the methods. We extend the model in (1) to 13

15 Figure 3: Solid line is infeasible MLE, Dash dot line is T R 4 and the dashed line is for T R 5 : 14

16 allow for serial correlation in u t, we consider data generating processes of the form (1 1 L)u t = (1 + 2 L)" t where " t are serially independent N(0; 1) random variables and we consider various values for 1 and 2 : For the test we require that the test be adjusted to allow for serial correlation this is accomplished through using an estimator of the spectral density at frequency zero of u t in place of the variance as well as (for one of the terms) adopting the correction in Phillips (1987) for one of the terms. The details are in the appendix where the construction of the statistics are presented in their full generality. We report results for T R 1 ; T R 2 and T R 3 (Tables 1 and 2) when the initial values for u t are set to zero and for T R 4 and T R 5 (Tables 3 and 4) where we approximate the unconditional case by allowing for 500 burn in observations prior to the sample that is employed for the tests. In each case we allow up to four lags in estimating robust statistics. Results reported are averages across 10; 000 Monte Carlo simulations. First, consider the model where the initial values are set to zero, which can be compared to previously suggested tests. Results for these cases are in Tables 1 and 2 for T = 100 and T = 250: When w t has a unit root ( = 0) size is fairly well controlled except when there is strong negative serial correlation, either 1 or 2 at 0:5: For the rst of these models the size distortion is small as adding lags (which is how we estimate the spectral density) works reasonably well however for the latter case it is large. This is a well known issue in scaling partial sums, see Ng and Perron (2001) for details. In unreported results we examined use of their MAIC criterion in choosing the lag length (here we use BIC) however found little improvement. As gets larger but remains in the near zero region for most models the tests become undersized. This is to be expected from the large sample results, where the same phenomenon appeared 4. As noted earlier, the di erent tests were chosen to have di erent properties here T R 1 tends to have better size control for moderate serial correlation across these values for whereas T R 2 is the most severely undersized test and T R 3 is between the two tests in terms of properties. As gets larger however this undersized nature of the tests begins to disappear, with nominal size close to desired size for no serial correlation and the typical small oversized properties of tests based on scaled partial sums when the serial correlation is accounted for. In the unconditional initial condition case, similar results hold although both T R 4 and T R 5 control size well when there is mild serial correlation. The extent of the undersizedness 4 This is true of prevoiusly suggested tests, see Table 2 of Harvey et. al. (2007) for example. 15

17 when is nonzero but small is much less evident (though still there). Overall the tests, even with 100 observations, are capable for many of the models to control size reasonably well. When serial correlation is more extreme the size control is again better than the zero initial condition case. 4 Conclusion We considered here the problem of testing for the value of a coe cient on a time trend in a linear model of a variable with a mean and time trend. Testing for this coe cient being zero then tests for the presence of the time trend. To test other values simply subtract 0 t from the variable to be tested and one can use the tests described in this paper. Multiplying the variable to be tested by minus one yields a test that the coe cient is less than zero instead of greater than zero as in the paper. This test is non trivial because the nuisance parameter describing the largest root in a linear model of the serial correlation is a nuisance parameter under both the null and alternative hypotheses and hence any test must control size over all values of this nuisance parameter. With su ciently mean reverting serial correlation, this is not much of an issue however when this root is near the unit circle then the value of this coe cient impacts tests greatly. Standard methods do not control size in these cases. For this problem, power is much higher the more mean reverting the data, and so it makes sense in developing new tests to concentrate on the power properties of the tests for models where the serial correlation has a unit root or near unit root. This is the focus of this paper. In previous work, authors have constructed tests that appear to control size but at unknown costs to power. In this paper we use the methods of Elliott, Mueller and Watson (2017) to construct tests paying attention to the size for these di cult values of the nuisance parameter. In the case where the initial condition of the stochastic component of the variable being examined is assumed asymptotically irrelevant - which is the assumption of the previous literature - we are able to construct a family of tests for the hypothesis. Further, some members of this family of tests mimic the properties of previously suggested methods however compare well in terms of asymptotic power. In addition we construct another test, T R 3, that we would recommend that provides a good power function trade-o across di erent values for the largest root in the serial correlation. The assumption of a zero initial condition is perhaps less natural than choosing the initial value so that for mean reverting models the serial correlation is stationary. We derive the 16

18 1 2 T = 100 T = 250 T R 1 T R 2 T R 3 T R 1 T R 2 T R 3 = = = Table 1: Size, initial value set to zero. 17

19 1 2 T = 100 T = 250 T R 1 T R 2 T R 3 T R 1 T R 2 T R 3 = = = Table 2: Size, initial value set to zero (continued). 18

20 1 2 T = 100 T = 250 T R 4 T R 5 T R 4 T R 5 = = = Table 3: Size, initial observation drawn from the unconditional distribution. 19

21 1 2 T = 100 T = 250 T R 4 T R 5 T R 4 T R 5 = = = Table 4: Size, initial observation drawn from the unconditional distribution (Continued) 20

22 rst tests for this more natural model, and nd that the properties under this assumption provide tests that, whilst still undersized when roots are near zero, have power functions that appear to provide a reasonable trade-o in power across models of the serial correlation. 5 Appendix We detail construction of the tests 5.1 Construction of the Tests For all versions of the T R statistics, we employ the following steps; (1) Choose the lag length for constructing robust tests by using the BIC criterion with a maximum of p max lags of x(t) in a regression of x(t) that includes a constant and time trend. Call the chosen lag length p: (2) From regressions in (7) both including and excluding p lags of x t construct an estimate ^ as ^ = T ^p0 s ^! 2 where ^p 0 is an estimate from the regression including the lags, ^! 2 is an estimate of the spectral density at frequency zero using the residuals when the lagged changes of x t are excluded and ^ 2 v is an estimate of the variance of the residuals from the regression with the lagged changes of x t included. (3) If ^ > 50; the test for b = 0 is the usual t-test of the hypothesis in (2) in the regression in step 2 where the lagged changes of x t have been included in the regression. (4) If ^ 50; we construct the WLR test. For this test we construct ~x t = x t x 1 : We construct from the data the following statistics: y1 = T 2^! 2 TX t=2 ~x 2 t 1 ^ 2 v y2 = T 2^! TX 2 ~x t 1 ~x t t=2 y3 = T 1=2^! 1 ~x T y4 = T 5=2^! TX 1 t~x t 1 t=2 ^ 2 w ^!

23 y5 = T 3=2^! 1 TX t=2 ~x t 1 where ^ 2 w is an estimator of the variance of the rst di erence of residuals from a regression of x t on a constant and time trend and ^! 2 is as in step 2. For the case where = 0 is assumed we have ^H( i ; ij ) = 2 T 1 i y i y 5 i y i y 2 (1 + (T 1)(1 r i ) 2 ) " T ij T i s 2 + r 2 T r i i ( T 1 + 2r i i s ) + 2 # T 2 i s 1 1 T (1 + (T 1)(1 r i ) 2 ) " 2 ij (r i + i )y i y 4 2 i T r i i ( T 1 y ) + T 2 i s 1 ) # 5 T (1 + (T 1)(1 r i ) 2 ) iy i y 5 : where s 1 = T P 2 T t=2 t and s 2 = T P 3 T t=2 t2 : For the case where the initial observation is drawn from its unconditional distribution we have ^H 1 ( i ; ij ) = 2 i y i y 5 i y i y 2 T (1 ri 21( i > 0) + (T 1)(1 r i ) 2 ) + 2 ij " 2 ij " (1 r i 1( i > 0))T i s 2 + r 2 T 1 T (8) 1 ri 2 1( i > 0) + r i i ( T 1 + 2r i i s ) + 2 T 2 i s 1 1 T (1 ri 21( i > 0) + (T 1)(1 r i ) 2 ) (r i + i )y i y 4 2 i T 1 1 ri 2 1( i > 0) + r i i ( T 1 y ) + T 2 i s 1 5 T (1 ri 21( i > 0) + (T 1)(1 r i ) 2 ) iy i y 5 These forms allow vectorization of the programming of the test statistic which increases speed of computation and allows a simpler generalization to serial correlation corrections when there is serial correlation in u t : The W LR statistic P m P J i=1 i=j W LR = w a i ^H(x; i ; ij ) P m i=1 w 0 i ^H(x; i ) is then constructed using the weights w ai and w 0i from Tables 5,6 and 7 for W LR 1 :W LR 2 and W LR 3 (using the zero initial version of ^H) respectively and Tables 8 and 9 for W LR 4 and W LR 5 respectively using ^H 1 ( i ; ij ), i = f0; 0:25 2 ; 0:5 2 ; ::::; 9 2 g so m = 37, J = 2 with q ij = c j = 1 2 i =d with values for (c j ; d) given below. The version of H(:) used depends on the initial condition assumption. The test rejects if W LR > 1: Di erent versions of the tests given an assumption on the initial condition di er by di erent choices of w ai, c j and d which in turn lead to di erent estimates of w 0i : Values for c j and d under the alternative are as follows: 22 # : (9) #

24 c 1 c 2 d W LR W LR W LR W LR W LR

25 i i1 i2 w ai w 0i Table 5: Values for W LR 1

26 i i1 i2 w ai w 0i Table 6: Values for W LR 2

27 i i1 i2 w ai w 0i Table 7: Values for W LR 3

28 i i1 i2 w ai w 0i Table 8: Values for W LR 4