Extreme Values and Local Power for KPSS Tests

Transcription

1 Extreme Values and Local Power for KPSS Tests Kenneth N. Hightower and William R. Parke 1 Department of Economics University of North Carolina Chapel Hill, NC November 14, 21 1 Comments are welcome: Kenneth.Hightower@unc.edu or parke@ .unc.edu.

2 Abstract We establish upper bounds on the statistics bη µ and bη τ proposed by Kwiatkowski, Phillips, Schmidt, and Shin (1992) for testing the null hypothesis of stationarity. The bounds are attained by cosine functions that do not exhibit the exploding variance often associated with the alternative hypothesis. We also derive extreme value results for time trends, structural breaks, and unit roots. Using these results for extreme KPSS statistics, we develop analytic results for the power against local alternatives that combine one of these processes with a short memory component. The results are surprising, given the origins of the statistic bη µ, in that the rejection probabilities for the test based on bη µ are very similar for structural breaks and unit roots. We provide a theoretical basis for this last result by showing that bη µ is an algebraic special case of the statistic proposed by Andrews and Ploberger (1994) for testing for structural breaks at an unknown breakpoint.

3 1. Introduction We establish upper bounds for the statistics bη µ and bη τ proposed by Kwiatkowski, Phillips, Schmidt, and Shin (1992). While these bounds are of some interest in their own right because they define an extreme opposite to the null hypothesis, they also provide a basis for analyzing the power of tests based on bη µ and bη τ. In particular, our results for extreme values place an analytic perspective on power comparisons for unit root, time trend, and structural break alternatives. We demonstrate that bη µ is about as effective as a test for structural breaks as it is as a test for unit roots. Our results add to earlier findings that have already revised the interpretation of these statistics. Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) derive bη µ as an LM statistic where the alternative hypothesis is the sum of a unit root process and a stationary component. Their stated null hypothesis is a short-memory, stationary process. On that basis, they describe the value of bη µ as a test of the null hypothesis of stationarity. Lee and Schmidt (1996) and Liu (1998) show that bη µ is consistent for fractionally integrated I(d) processes with d>. This result necessarily revises the interpretation of the KPSS test because an I(d) process with <d<1/2 is stationarity. Lee and Schmidt propose that the KPSS test is more properly viewed as a test of the null hypothesis of short memory. The alternative hypothesis would then be long memory, which includes unit roots and fractionally integrated processes. Unit roots and fractionally integrated processes do not, however, yield the largest values for bη µ and bη τ. For a sample of size T, we establish the bounds bη µ π 2 T and bη τ (2π) 2 T. These bounds are attained for the processes y t = cos(πt/t ) and y t = cos(2πt/t ), respectively. These cosine functions provide benchmarks for other possible alternative hypotheses, and they give a clear picture of the types of realizations that actually trigger large values for bη µ and bη τ. They show that bη µ is most sensitive to a large change in the value of y t between the beginning and theendofthesampleandthatbη τ is most sensitive to data with a complete cycle in the data period. 1

4 The results for time trends and structural breaks confirm this observation. The value bη µ = 1 T is attained by a simple time trend, y 1 t = αt, whereα is a constant. The difference between 1 T for a time trend and 1 π 2 T =.113 T for a cosine function is minimal. A structural break process with a break at the sample midpoint is a crude approximation to a montone trend, and that process attains bη µ = 1 T. 12 Unit roots, which form the original basis for the statistics, generate realizations that may or may not resemble cosine functions. The values of bη µ for unit root realizations are more or less uniformly distributed between the lower bound of and the upper bound of π 2 T. The mean is at the midpoint of 1 T, which is about half of the bound 2 π 2 T =.113 T. Only about 12% of realized unit roots have a value of bη µ aslargeasthevalue 1 T 12 for a structural break known to be at the sample midpoint. The mean of 1 T for a break with an unknown, uniformly distributed breakpoint is also 18 larger than the mean for a unit root. These extreme value results establish a framework for studying the power of the KPSS tests for alternatives that include unit roots and structural breaks. We consider an alternative hypothesis based on the process z t = x t + γ T y t composed of the sum of a short memory process x t and a long memory process y t. The scalar γ T sets the distance between the null and the alternative for asampleofsizet. We show that the power for a small γ T is closely related totheextremevalueresultsfortheprocessy t alone. Section 2 of this paper establish the bounds for bη µ and bη τ. Section 3 presents a result on asymptotic local power for processes that combine short memory and long memory components. Section 4 then presents some calculationsforvariousunitroots,timetrends, and structural breaks. Simulation results in Section 5 give detailed power results for various alternatives. We provide a theoretical basis for our finding that bη µ is about as effective in testing for structural breaks as it is in testing for unit roots by showing that it is an algebraic special case of the test Andrews and Ploberger (1994) propose for structural breaks. This derivation is given in Section 6, and Section 7 presents our summary and conclusions. 2

5 2. Bounds on KPSS Statistics The KPSS statistics bη µ and bη τ are based on two types of residuals e t calculated for a process y t. The statistic bη µ is based on residuals calculated by regressing y t on an intercept. The statistic bη τ is based on residuals calculated by regressing y t on a constant term and a time trend. In both cases, the fundamental quantity is the partial sum of residuals S t = If the residuals are short memory, the long-run variance tx e t. (2.1) t=1 σ 2 = lim T T 1 E(S 2 T ) (2.2) is finite. If not, the residuals e t aresaidtohavethelongmemoryproperty. The KPSS statistics can be written as bη = T P 2 T t=1 S2 t. (2.3) s 2 (`) This statistic is generally denoted bη µ for residuals from a regression on an intercept and bη τ for residuals from a regression on an intercept and a time trend. The denominator is a kernel-based estimate of the long-run variance σ 2 : TX `X TX s 2 (`) =T 1 e 2 t +2T 1 w(s, `) e t e t s, (2.4) t=1 s=1 t=s+1 where w(s, `) = 1 s/(` +1) is the Bartlett window with bandwidth `. For testing under the null hypothesis of short memory, ` is chosen so that ` = o(t 1/2 ) to obtain a consistent estimate of σ 2. For the present study of extreme values for bη µ and bη τ we choose ` = even though σ 2 does not exist and cannot be consistently estimated by any choice of the bandwidth. We are interested here in extreme values for bη µ and bη τ, which occur for process that are not short memory. The sample variance s 2 () does, nonetheless, have a finiteexpectationinafinite sample, and its value provides a convenient reference point that we use in the next section to examine the powers of the statistics. 3

6 We also simplify the analysis by studying continuous time limits. For time trends and structural breaks, the continuous time version is immediate and we will use the notation e t e(t) to denote convergence to that limit. For stochastic process, the notation e t e(t) will denote weak convergence in measure to a demeaned Brownian motion for a unit root or to demeaned fractional Brownian motion for a fractionally integrated process. The fundamental problem we address in this section is thus maximizing the functional R 1 R a bη = e(s) 2 da R 1, (2.5) e(s)2 ds over e(s), s [, 1]. While the limits e(t) for stochastic processes are not smooth, we will assume that the maximum for bη is attained for a smooth, differentiable process. We let bη µ denote the maximum subject to the constraint e(s)ds =. (2.6) We let bη τ denote the maximum subject to (2.6) and the additional constraint s e(s) ds =. (2.7) Proposition 1: For bandwidth ` =, the limiting KPSS statistics defined above satisfy the following bounds: bη µ π 2 T Proof: Appendix A. bη τ (2π) 2 T The bound bη µ = π 2 T occurs for the data y t = y cos(πt/t ), whichisa cosine wave with a period of 2T. This function has a peak at the beginning of the sample, and a trough at the end. (The function y t = y cos(π(1 t/t )) is a mirror image that also attains the maximum.) The largest possible detrendedkpssstatisticbη τ occurs for the data y t = y cos(2πt/t ). This cosine function has a period equal to the sample size with peaks at the beginning and end of the sample. 4

7 3. Local Power The bounds on bη µ and bη τ have implications for the more practical problem of detecting a long memory component in a process that also contains a substantial short memory component. Consider a process z t = x t + γ T y t (3.1) composed of the sum of a short memory process x t and a long memory process y t. The scalar γ T sets the extent to which the alternative hypothesis differs from the null. We assume that γ 2 T = O(T 1 ). Let bη x and bη z be the KPSS statistics for x t and z t calculated using bandwidth `. (For notational clarity, we will generally drop the subscripts µ and τ. We note that the algebra for the two cases is the same.) The critical values for the test are based on the distribution of bη x. If we let g(η) denote the density for the KPSS statistic, then Pr(bη x >c)= Z c g(η)dη We let c denote a particular critical value with Pr(bη x >c )=α. We will show that the local power of the test depends on y t only through the expectation of TX Ψ = T 2 Sy,t. 2 While E(Ψ) generally grows without bound, it does exist for a finite T. We define v T = E(Ψ) T and assume that the parameters of the process y t are chosen so that v T V y for a constant V y. In the cases we consider, s 2 y() will be constant when v T V y. Proposition 2: If γ 2 T = T 1, then the rejection probabilities under the alternative satisfy t=1 Pr(bη z >c ) α g(c ) V y σ 2 x as T. (3.4) 5

8 Proof: Appendix A. The asymptotic local power of the KPSS test depends on the alternative only through V y /σ 2 x and will thus be the same for all alternatives with a given value for V y. Proposition 2 establishes a square root rule; quadrupling T and halving γ T yields constant rejection probabilities. Proposition 1 can be used to establish a lower bound on the sample variance E(s 2 y()) necessary to achieve a given value for V y. In the case of bη µ, for example, the bound in Proposition 1 implies that s 2 y() π 2 T 3 TX t=1 S 2 y,t. Taking expectations of both sides yields E(s 2 y()) π 2 V y. (3.6) The cosine function that establishes the bound for bη µ also has the minimum possible variance necessary to achieve a given value for V y. In the next section, we address the practical question of how large E(s 2 y()) must be to achieve a given V y for processes that do not attain the bound in Proposition Alternative Hypotheses Cosine functions are certainly not the most frequently mentioned alternative hypotheses for the KPSS tests. That distinction accrues to the unit root process, with fractional integration a distant second. We augment this range with a variety of structural break processes and time trends. Table 1 specifies a variety of process that have large values for bη µ. The parameters of each process are chosen to achieve V y = 1 so that all the processes have the same asymptotic local power as given in Proposition 2. Table 1 presents E(s 2 y()) and E(bη µ ) for these processes. These results are derived in Appendix B. The unit root process that motivated the original derivation of bη µ does not dominate the set of extreme values for the KPSS statistics. The expected value of bη µ for a pure unit root (process UR) is very close to 1 T,whichis 2 6

9 half the bound set by the cosine function. Table 3 shows the distribution, which is surprisingly uniform between and π 2 T. This distribution reflects anattributeofunitroots; someunitrootrealizationsareclearlynonstationary, but others are by chance not all that distinct in their appearance from stationary data. Indeed, bη µ is more sensitive to a variety of processes than it is to process UR. The value bη µ = 1 T for a simple time trend (process TT) is a close 1 second to the value for a cosine function. The difference between 1 and 1 π 2 =.1132 isjustalittleover1%. Thevaluebη µ = 1 T for a simple 12 break at the sample midpoint (process D 1/2 ) is greater than bη µ for 88% of realized unit roots. If the break is at the point [θt ], where[ ] denotes the integer closest to a real number, then bη µ = 1 θ(1 θ)t. The average for a 3 break uniformly distributed in [,1] is 1 T. 18 These results have parallel implications for asymptotic local power. The sample variances needed to achieve V y =1are 9.87, 1, and 12 for COS, TT, and D 1/2, but 15 for UR. The latter figure is slightly smaller that the sample variance of 16 needed to attain V y =1for the unknown breakpoint process (process D θ ). Table 2 also shows corresponding results for the bη µ test allowing for a time trend. Again, the cosine sets the bound on bη τ and has the smallest sample variance that achieves V y =1. A close second place goes to the broken trend that goes up linearly over the firsthalfofthesampleanddown linearly over the second half. The statistic bη τ equals 1 T. This value of 4.25T is only slightly less than (2π) 2 T =.2533T. The double break process that takes on one value between 1/2 and 3/4 and another value on the other two regions has bη τ = 1 T or.28t. 48 Table 3 shows the distribution of bη τ for a pure unit root. The distribution covers the range between and (2π) 2 T, but it is not as uniform as the distribution of bη µ. The mean is about.19t. 5. Power Comparisons The powers of bη µ and bη τ for the various alternatives described above are partially dictated by Propositions 1 and Proposition 2. Proposition 1 suggests that the power will be bounded from above by the powers for the cosine functions. Proposition 2 gives an asymptotic local power approximation, showing the local powers are equal for processes with equal values for 7

10 V y. The remaining issue is the very practical question of power for larger values of γ for processes that do not attain the bound in Proposition 1. We approach this issue using simulations. Tables 4 and 5 presents rejection probabilities for a variety of alternative hypotheses for bη µ and bη τ, respectively. In all cases, the short memory component is x t N(, 1). The alternatives y t are described in Tables 1 and 2. The alternatives are parameterized so that V y =1,equatingasymptotic local powers, and that characteristic is evident in the results for small γ t. The results also confirm that the cosine functions bound the powers. The results (after scaling) are very nearly identical across sample sizes of 5, 2, and 8. The results for the three nonstochastic processes (COS, TT, and D 1/2 in Table 4 and COS2, TT2, and D2 in Table 5) are very nearly identical for all values of γ T. The precise shape of the y t process is evidently less important than the fact that the shape is nonstochastic. The power for a unit roots with the same V y is similar for small values of γ T but lower for larger values of γ T. To put the power for unit roots into perspective, we compare those figures with powers for four types of structural break processes in Table 4. 1 There are two sources of randomness involved: the location of the break and its size. Appendix B shows that V y = 1 3 θ2 (1 θ) 2 δ 2 (5.1) for a break of magnitude δ at point θ. If a randomly located break is too near a sample endpoint, it is difficult to detect. In terms of (5.1), Ψ is small if θ is near or 1. This causes the rejection probabilities to be smaller for D θ with a uniformly distributed breakpoint than for D 1/2. If a random-sized breakissmall,itisalsodifficult to detect because Ψ is small if δ 2 is small. The simulations use a normally distributed break size, which puts the highest probability density on the area around a break of size zero, emphasizing this effect. As a consequence, the rejection probabilities are larger for D 1/2 and D θ than for D 1/2,N and D θ,n. The unit root process, which motivated the original derivation of bη µ,has rejection probabilities that fall between those for D 1/2 and D θ and those for D 1/2,N and D θ,n. For large γ T, a break is easier to detect than a unit root 1 For large γ, the powers of these tests depend on a complex interaction between the distribution ofs Ψ and s 2 z(`), which are not independent, and the density g(bη x ). 8

11 even if the break is randomly located as long as the break size is clearly not zero. The unit root is a little easier to detect than the break process with normally distributed break sizes. This would likely not be the case if the break size were stochastic, but bounded away from zero. Overall, however, the differences in rejection probabilities are not huge. While the precise ordering depends on the specifics of the structural break process, the KPSS tests are basically as effective in detecting structural breaks as they are in detecting in unit roots. 6. The KPSS Test for Structural Breaks? It is not entirely surprising that bη µ is sensitive to structural breaks. In fact, bη µ is an algebraic special case of the statistic Andrews and Ploberger (1994) propose to test for structural breaks. The general form of the test they propose is a weighted average of LM statistics Z J(θ)LM(θ)dθ, (6.1) where θ (θ 1, θ 2 ) [, 1] parameterizes the location of the break in the sample and J(θ) is a weighting function. For a structural break with a uniformly distributed breakpoint, Andrews and Ploberger recommend uniform weighting J(θ) =1, producing the test statistic Z LM(θ)dθ. (6.2) Tocomparethisstatisticwithbη µ, it is necessary to examine the details of the LM statistics. Consider a model y t = α + βd i,t + ε t where d i,t is a dummy variable equal to 1 for t i and for t>i. likelihood function The log SSR = 1 2σ 2 TX (y t α βd i,t ) 2, t=1 9

12 where σ 2 is the variance of ε t, has a gradient with respect to β given by SSR β = 1 σ 2 TX t=1 d i,t e t = S i σ 2. This is the same S i that enters the KPSS calculations (2.1). Using V (S i )= θ(1 θ)σ 2,wherei =[θt ], and substituting the estimate s 2 (`) for σ 2 yields St 2 LM(θ) = T θ(1 θ)s 2 (`) WecanthuswritetheKPSSstatisticas Z bη µ = θ(1 θ)lm(θ)dθ, (6.3) which is (6.1) with the weighting J(π) =θ(1 θ). The differences between (6.2) and (6.3) are minimal because θ(1 θ) is nearly constant except at the extremes of the [, 1] range. In fact, Andrews and Ploberger state that the calculations must omit values of θ near or 1 to avoid technical problems. The summation for bη µ is not subject to this limitation because the factor θ(1 θ) is effectively a continuous counterpart tothesampletrimmingandrews and Ploberger recommend. Indeed, this similarity has been mentioned in passing by Andrews, Lee and Ploberger (1996) in the context of Nyblom s (1989) test for parameter changes under martingale alternatives. This general alternative specification encompasses several interesting departures from constancy including a single jump at an unknown point in time (the change-point problem) and slow random variation (a random walk). KPSS reference Nyblom s (1986) paper. 7. Summary and Conclusions One way to describe a test is in terms of the assumptions accompanying its derivation. The statistics bη µ and bη τ were originally developed to test for the existence of a unit root component. Lee and Schmidt refined this description to portray bη µ and bη τ as tests for the existence of a long memory component. Andrews and Ploberger (1994) derive bη µ as a test for structural breaks. On this basis, one might describe bη µ as a test of the null hypothesis 1

13 of short memory against an alternative that includes unit roots, fractionally integrated processes, and structural breaks. One can also characterize a test in terms of what it actually detects. We show that bη µ is most sensitive to a cosine function with half a cycle during the sample period and that bη τ is most sensitive to a cosine function with a full cycle during the sample period. On this basis, one might describe bη µ and bη τ as tests for cosine functions or, more generally, as tests for realized processes that happen to resemble cosine functions. In the case of bη µ,thevaluefora time trend very nearly matches the bound set by the cosine function and the value for a structural break at the sample midpoint is only a little smaller. We also consider the more practical comparison between the unit root alternative and structural breaks at an unknown breakpoint. While bη µ is greater for a pure structural break process with a break in the middle of the sample than for 88% of realized unit roots, an unknown breakpoint uniformly distributed in the sample has an average bη µ only somewhat higher than the mean for realized unit roots. The simulation results here show that bη µ is equally effective in detecting unit roots and in detecting breaks at an unknown breakpoint. An important caution is implied by the fact that bη µ is described as a test for unit roots by KPSS and as a test for structural breaks by Andrews and Ploberger. If bη µ is large enough to reject the null hypothesis of short memory, it would be entirely misleading to cite KPSS and claim discovery ofaunitrootjustasitwouldbemisleadingtociteandrewsandploberger and claim evidence of a structural break. The only appropriate conclusion is a rejection of the null hypothesis of short memory. 11

14 Appendix A Proofs where and Proof of Proposition 1. We seek to maximize S t = s 2 = Z t bη µ = R 1 S2 t dt s 2 y(s)ds t µ y(t) y(r)dr y(r)dr 2 dt. (A.1) (A.2) Our strategy is as follows. Differentiating (A.1) with respect to y(a) yields a condition that y(a) must satisfy. Differentiating this condition with respect to a then yields a differential equation that y(a) must satisfy for all a. Differentiating a second time yields a second-order differential equation that does not involve the integrals in (A.2). We differentiate (A.1) with respect to y(a) using the Frechet derivatives ½ ¾ S t 1 t, t a y(a) = t, t < a and s 2 y(a) =2(y(a) we obtain the derivative y(r)dr), R a 1 bη 2 y(a) = ( t)s R 1 tdt + (1 t)s R 1 a tdt S2 t dt s 2 s 2 (s 2 ) (y(a) 2 Equating this to zero yields 12 y(r)dr).

15 (y(a) R 1 y(r)dr) R a = ( t)s tdt R s S2 t dt Differentiating both sides with respect to a yields y(a)/ a = as a R s 2 1 S2 t dt (1 a)s a R 1 S2 t dt, which can be simplified to S a R 1 S2 t dt = y(a)/ a. s 2 R 1 (1 t)s a tdt R 1. S2 t dt We thus have that the maximum KPSS statistic is attained for data following the recursion y(a) a = bη 1 µ S a. Substitute the definition for S a µz y(a) a a = bη 1 µ y(i)di a We then need to find solutions to The residuals have the property that 2 y(a) a 2 We can thus rewrite (A.3) as = bη 1 µ µ y(a) e(a) =y(a) 2 e(a) a 2 2 e(a) a 2 y(r)dr = 2 y(a) a 2. y(r)dr. y(r)dr. (A.3) = bη 1 µ e(a). (A.4) The solutions to this differential equation are of the form y(t) =y() cos(kt + c), 13

16 where k 2 = bη 1 µ. (A.5) We are, therefore, looking for solutions to (A.4) with Using Z a the numerator is bη = R 1 R a R 1 cos2 (kt + c)dt cos(kt + c)dt 2 da h R 1 cos(kt + c)dt i 2. (A.6) cos(kt + c)dt = k 1 sin(ka + c) k 1 sin(c), µ 2 cos(kt + c)dt = k 2 sin 2 (kt+c)dt 2 k 2 sin(kt+c)sin(c)dt+ = k 2 2 k 3 2 sin(k)cos(k)+2k 2 sin(c)(cos(k + c) cos(c)) + k 2 sin 2 (c). The denominator is the sum of k 2 sin 2 (c)dt cos 2 (kt + c)dt = k 1 sin(k + c)cos(k + c) 1 2 k 1 sin(c)cos(c) and cos(kt + c)dt Putting all this together, we have 2 = k 2 sin 2 (k + c). 1 bη µ = k k 1 sin(k + c)cos(k + c)+2sin(c)(cos(k + c) cos(c)) + sin 2 (c) k 1 sin(k + c)cos(k + c) 1. 2 k 1 sin(c)cos(c) k 2 sin 2 (k + c) We need to find the values of k and c for which the fraction on the right equals 1. This happens if sin(c) = and sin(k + c) = implying that the maximum occurs for c =and k in the set of values π, 2π, 3π,... Checking the values of (A.6) numerically shows that the k = π determines the largest possible KPSS statistic π 2 T. 14

17 The bound for bη τ follows directly from this result. The statistic bη τ can be calculated for a variable w(t) by detrending that variable to produce residuals denoted y(t) and then calculating bη µ for y(t). The bound on this bη µ is determined by the above analysis with the further condition that y(t) must be detrended. The choice k =2π produces the largest bη µ with this property because y(t) is trend-free for k =2π, 4π,... End of proof. or as Proof of Proposition 2 We can write the KPSS statistic as bη z = T 2 X T bη z = T X 2 T (S x,t + γ T S y,t ) 2 t=1 s 2 z(`) t=1 S2 x,t +2γ T T X 2 T S x,ts y,t + γ 2 t=1 T T X 2 T s 2 x(`)+s 2 z(`) s 2 x(`) It will prove convenient to reoganize these terms as bη z = bη x + A 1+B. where T X 2 T A =2γ S x,ts y,t T X 2 T t=1 T + γ 2 t=1 S2 y,t s 2 T x(`) s 2 x(`) and B = s2 z(`) s 2 x(`). s 2 x(`) This allows to express the rejection probability as Pr(bη z >c )=Pr(bη x >c (1 + B) A). Integrating over the densities for x and y yields. ZZ Pr(bη z >c )= G (c (1 + B) A) f(y)f(x) y x t=1 S2 y,t 15

18 where G(c) =Pr(bη x >c). Using Pr(bη x >c )=α and G(c)/ c = g(c), we can expand G (c (1 + B) A) G (c ) to obtain ZZ Pr(bη z >c ) α = g(c ) (A c B) f(y)f(x) y x. (A.7) We can analyze A and B as follows. The fact that x t and y t are independent implies that Z Z TX γ 2 T T 3 S x,t S y,t f(y)f(x)dydx =. The definition of V y then yields Z T 3 t=1 TX Sy,tf(y)dy 2 = V y. Together, these give us RR Af(y)f(x) y x γ 2 T T = V y γ 2 T T. σ2 x The fact that x t and y t are independent also implies that t=1 s 2 z(`) =s 2 x(`)+γ 2 T s 2 y(`). From this, we obtain RR Bf(y)f(x) y x γ 2 T T = s2 y(`) Ts 2 x(`). We can now substitute these results into (A.7) to obtain End of proof. V y Pr(bη z >c ) α g(c ) γ 2 T T. σ2 x 16

19 Appendix B NotesonCalculationsforTables1and2 Trends TT: y t = t. S t = s 2 = Z t (t 1 2 )2 dt = 1 12 (t 1 2 )dt = 1 2 (t 1 2 )2 1 8 S 2 t dt = 1 (t )4 1(t ) dt = 1 (t )5 1 (t )3 + 1 t 16 1 = = = TT2: y t = t for t 1/2; y t =1/2 t for 1/2 t 1. From to 1/2, s 2 =2 /2 (t 1 4 )2 dt = 1 48 S 2 t dt =2 /2 S t = 1(t ) (t )4 1 (t ) dt = Breaks We derive the result for an arbitrary break located at θ. y t = δ for t [θt ]; y t =for t>[θt ]. s 2 = θ[(1 θ)δ] 2 +(1 θ)[θδ] 2 17

20 S 2 t dt = Z θ [(1 θ)δt] 2 dt + s 2 = θ(1 θ)δ 2 D 1/2 : Do the obvious substitution. D θ : Break with θ uniformly distributed on [,1]. θ θ(1 θ)dθ = 1 6 θ 2 (1 θ) 2 dθ = 1 3 [θδ(1 t)] 2 dt = 1 3 (1 θ)2 θ 2 δ 2 D 1/2,N and D θ,n : Use the expectation of the square. D2: y t =1for (, 1/4) and (3/4, 1); y t = 1 for (1/2, 3/4). Unit Roots S 2 t dt =4 s 2 = 1 4 /4 ( 1 2 t)2 dt = UR: y t = y t 1 + ε t, ε t i.i.d., E(ε t )=, V (ε t )=1. Wewillusethetwo lemmas: µz a Z a E yt 2 dt = tdt ((1)) Ã µz a! 2 Z a E y t dt = t 2 dt ((2)) E(s 2 )= = tdt µz t µ E y i di µ yt 2 dt t 2 y t dt t 2 dt = = 1 6 y i di =(1 t) 18 Z t idi ((3))

21 µ 2 E y i di = t t i 2 di + t(1 t) 2 ((4)) Rearrange as S t = Z t y i di t y i di S 2 t =(1 t) 2 µz t Z t S t =(1 t) y i di t t y i di 2 µz t µ y i di 2t(1 t) y i di Use independence (3) on the middle term. t µ y i di + t 2 t 2 y i di Z t Z t t E(St 2 )=(1 t) 2 i 2 di 2t(1 t) 2 idi + t 2 i 2 di + t 3 (1 t) 2 E(S 2 t )=(1 t) 2 t 3 /3 2t(1 t) 2 t 2 /2+t 2 (1 t) 3 /3+t 3 (1 t) 2 E(S 2 t )=(1 t) 2 t 2 [t/3 t +(1 t)/3+t] E(St 2 )=(1 t) 2 t 2 [1/3] µ E St 2 = 1 (1 t) 2 t 2 dt = (1 t) 2 t 2 dt = = 1 3 UR2: These results are derived from simulations. 19

22 Table 1 Expected Values for bη µ and s 2 y() for V y =1 Process bη µ s 2 y() COS y t =2 1/2 πcos(πt/t ) π 2 T π 2 = TT y t = βt, β = 12 1/2 T T 1 D 1/2 y t = δd 1/2,t, δ =48 1/ T 12 Process E(bη µ ) E s 2 y() D 1/2,N y t = δd 1/2,t, δ N(, 48) 1 12 T 12 D θ y t = δd θ,t, δ =9 1/ T 16 D θ,n y t = δd θ,t, δ N(, 9) 1 18 T 16 UR y t = y t 1 + ε t, σ 2 ε =9/T 1 2 T 15 d θ,t = {1 for t<[θt ], otherwise}. ε t N(, σ 2 ε). 2

23 Table 2 Expected Values for bη τ and s 2 y() for V y =1 Process bη τ s 2 y() COS2 y t =2 3/2 πcos(2πt/t ) (2π) 2 T (2π) 2 TT2 y t = {(48)1/2 t for t 1 2, (48) 1/2 ( 1 2 t) for t> 1 2 } 1 4 T 4 D2 y t = { 481/2 for t [ 1 4 T, 3 4 T ], +48 1/2 otherwise} 1 T Process E(bη τ ) E s 2 y() UR2 y t = y t 1 + ε t, σ 2 =12/T.19T 8 21

24 Table 3 Distributions of bη µ and bη τ for a Pure Unit Root α Pr(bη µ < απ 2 T ) Pr(bη τ < α(2π) 2 T ) Calculated for T=8 using 1, simulations. 22

25 23

26 24

27 25

28 Bibliography Andrews, D.W.K., I. Lee, and W. Ploberger (1996), Optimal Changepoint Tests for Normal Linear Regression, Journal of Econometrics 7, Andrews, Donald W.K. and Werner Ploberger (1994): Optimal Tests When a Nuisance Parameter is Present Only under the Alternative, Econometrica 62, Kwiatkowski, Denis, Peter C.B. Phillips, Peter Schmidt, and Yongcheol Shin (1992): Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root, Journal of Econometrics 54, Lee, Dongin and Peter Schmidt (1996): On the Power of the KPSS Test of Stationary Against Fractionally-Integrated Alternatives, Journal of Econometrics 73, Liu, Ming (1998): Asymptotics of Nonstationary Fractional Integrated Series, Econometric Theory 14, Nyblom, J. (1986), Testing for Deterministic Time Trend in Time Series, Journal of the American Statistical Association Nyblom, J. (1989), Testing for the Constancy of Parameters over Time, Journal of the American Statistical Association 84,