Confidence Intervals for an Autoregressive Coeffi cient Near One Based on Optimal Selection of Sequences of Point Optimal Tests

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1 Confidence Intervals for an Autoregressive Coeffi cient Near One Based on Optimal Selection of Sequences of Point Optimal ests Muhammad Saqib, David Harris Department of Economics, he University of Melbourne, Melbourne, Australia Department of Econometrics and Business Statistics Monash University, Clayton, Australia July 2, Abstract In this paper, we reconsider Elliot and Stock s 2001 method to construct confidence intervals for an autoregressive coeffi cient by inverting a sequence of invariant point optimal tests. We show that the power properties of the point optimal tests can vary greatly with the choice of the point under the alternative at which power is maximised. Some choices are shown to lead to inconsistent tests, including, in some cases, the rule of thumb proposed by Elliott, Rothenberg and Stock We propose the optimisation of a weighted power criterion as an alternative method to specify the point optimal tests and demonstrate that this provides tests with desirable asymptotic local power properties. Keywords Point optimal test, GLS detrending, asymptotic power, weighted power maximization, invariant Jeffreys prior. 1 Introduction In the last three decades enormous work has been done by different researchers in the area of unit root testing and construction of confidence intervals for the autoregressive root including local to unity framework proposed by Bobkovski 1983, Cavanagh 1985, Phillips 1987, and Chan and Wei As mentioned in Stock 1991, Cavanagh first constructed the confidence interval for the autoregressive root ρ in an AR1 model without deterministics using local to unity setting with ρ = 1 + c/ based on t-test statistic. However Stock 1991 extended Cavanagh s work by considering deterministic component in ARp model and constructed confidence interval for the largest autoregressive root by inverting the Dickey-Fuller t-test and Modified Sargan-Bhargava test statistics. Elliot and Stock 2001 henceforth ES proposed new asymptotic methods to construct confidence intervals for AR parameter using sequence of tests approach by coupling Stock s method with ideas given in Elliot, Rothenberg, and Stock 1996 henceforth ERS. ERS followed local to unity asymptotic approximations to the point optimal invariant test of King 1

2 1987 and based on Neyman-Pearson Lemma after assuming Gaussian distribution for the errors they described that the best test of ρ = 1 against any given alternative ρ = ρ 1 is the likelihood ratio test. his testing problem with ρ = 1 + c/ is the same as testing H 0 : c = 0 against H 1 : c = c 1 < 0 where the likelihood ratio test is the point optimal test. Contrary to Stock s OLS detrending, ERS suggested GLS transformation of the data to deal with deterministics that utilizes c 1 value under the local alternative. his constant is chosen such that the power curve of point optimal test is tangent to the power envelope at power one-half and hence results in famous choices of c 1 = 7 when x t = 1 and c 1 = 13.5 when x t = 1 t. ERS also provide simulation evidence that the tests based on GLS-demeaning/detrending of the data in the presence of deterministics are more powerful tests with power curves indistinguishable from the Gaussian power envelope compared to the same tests based on OLS transformation with poor power properties. By combining the ideas of Stock and ERS, ES use sequence of tests to obtain confidence set for the largest autoregressive root as a set of those values that are not rejected by the sequence of tests. Under this approach each test in the sequence is point optimal test of a particular null versus a particular alternative. he key inspiration behind their approach is the powerful test of a particular parameter value against different alternatives will produce accurate confidence intervals. he main objective behind this work is to revisit ES work where we generalize their testing problem to H 0 : c = c 0 for any c 0 instead of only c 0 0 for the purpose of constructing confidence interval of the autoregressive root under local to unity set up. his requires modification of the point optimal test and its asymptotic distribution. We do not ignore the possibility of mildly explosive root of a time series and take into account testing positive values of c 0. Our analysis shows that when testing for any positive c 0 the point optimal test can become inconsistent with lower sided alternatives and also has inferior power close to the null using upper tail test based on the alternative values proposed by ERS and ES and indicates that the choice of c 1 is more influential when c 0 > 0 then c 0 0. Moreover in two-sided testing case we reconsider with ES s choice of α L = 0.03 and α U = 0.02 where they justify the assignment of higher probability of type-i error to the left tail by the argument that the power curves are steeper at right tails. his choice is pragmatic and without any formal justification and we have strong conjecture that the choice of α L and α U is sensitive to varying c 0. hese observations also pointed out by ES in their conclusion motivate us to choose model parameters c 1L, c 1U for one-sided tests and c 1L, c 1U, α L, α U for two-sided tests based on some formal method using optimality criteria e.g. maximization of weighted powers. Simulation analysis proves the superiority of power curves of the test based on our proposed parameters compared to those based on ERS and ES and implies accuracy of resulting confidence intervals. he paper is organized as follows. he model and the method of construction of confidence intervals as discussed in ES is summarized in the next section. In section 3, we discuss about the 2

3 impact of different values of c 1 on one-tailed tests, inconsistency of the test caused by some of c 1 values and, alternative criteria for choosing c 1 optimally. We extend the analysis to two-sided tests and provide some formal methods to choose model parameters in section 4. Results are provide in section 5. Section 6 concludes. All proofs are provided in Appendix A. 2 he Model and Hypothesis esting Problem Consider a time series y t with the data generating process given by y t = x tγ + u t, t = 1,...,, 2.1 u t = ρu t 1 + ε t, u 0 = 0, 2.2 ρ = 1 + c/, ε t i.i.d.0, σ 2. where we treat x tγ in the DGP as deterministic component with usual two cases of constant mean x t = 1 and, constant and linear time trend x t = 1 can be extended to higher order time polynomials. t but of course the analysis Similarly the assumption that the initial condition u 0 = 0 is vital and follows ERS and will be maintained in this paper. Since we are interested only in the asymptotic properties of the point optimal test therefore our analysis here is restricted to disturbances ε t being i.i.d., however the approach is flexible to allow for other error structures for practical purposes. Also ε t obeys the Functional Central heorem such that 1/2 [ s] j=1 ε d j σbs, 0 s 1 where [.] represents greatest lesser integer function, Bs is a standard Brownian motion defined on C[0, 1], and d denotes weak convergence in distribution. Sequence test approach Suppose we want to construct 1001 α% confidence set CS α y for the autoregressive parameter ρ by inverting the sequence of tests with y as the test statistic of asymptotic size α that rejects for small values. For the hypothesis H 0 : ρ = ρ 0 against H 1 : ρ ρ 0, the test is performed over a range of values ρ 0 and the confidence set is then derived as the set of those values of ρ 0 that are not rejected by the tests as CS α y = {ρ : y > cv α ρ 0 } where cv is the critical value. his set exhibits the property that it has asymptotic coverage probability of at least 1 α for any true value of ρ, i.e. lim Pr [ρ CS α y] 1 α. We define the rejection probability of the test as π ρ = Pr y < cv α ρ 0. If the test fails to reject the false null H 0 : ρ = ρ provided that ρ 0 is true then the probability of inclusion of false value ρ in CS α y, is given by β = Pr ρ0 ρ CS α y. However accurate confidence intervals require the test to have smaller β implying high power. his paper therefore considers the construction of tests of good power properties. Under local to unity setting the autoregressive root is ρ = 1 + c/ and for any fixed value ρ 0 = 1 + c 0 / we can construct the test of H 0 : ρ = ρ 0 against the one-sided alternatives where 3

4 the side depends whether ρ 1 < ρ 0 or ρ 1 > ρ 0, i.e. H 1L : ρ = ρ 1 < ρ 0 or H 1U : ρ = ρ 1 > ρ 0. his problem is analogous to testing H 0 : c = c 0 vs. H 1 : c = c 1. For lower sided test we have c 1 < c 0 and upper sided test requires c 1 > c 0. One sided confidence set can be constructed as a set of those values of c 0 that are not rejected by the test, i.e. CS α,i y = {c 0 : i y does not reject}, i = L, U. where for upper sided test we have ĉ U = sup CS α,u y and the resulting confidence interval would be, ĉ U. Similarly ĉ L = inf CS α,l y and provides ĉ L, as the lower sided confidence interval for c. Construction of two-sided confidence interval in this framework for H 0 : c = c 0 requires inverting two one-sided tests corresponding to H 1L : c = c 1L < c 0 and H 1 : c = c 1U > c 0. Given the constraint that the probability of rejecting the true null is some fixed level α, the twosided confidence interval consists of those values of c 0 that are not rejected by both tests. We will explain more about this shortly. Once we obtain the confidence interval for c, confidence interval for ρ can be obtained simply by applying the transformation ˆρ = 1 + ĉ/. For above hypothesis with ρ j = 1 + c j /, j = 0, 1, ES have proposed point optimal invariant test that accounts for values of ρ under both the null and alternative as P c 0, c 1 = 1ˆσ 2 ˆε2 1,t 1 + c 1/ 1 + c 0 / ˆε2 0,t where 2.3 ˆε j,t = û j,t ρ j û j,t 1, j = 0, 1 û j,t = y t x tˆγ j, ˆγ j = arg min y j,t x γ j,tγ y j,t x j,tγ, j and y j,t, x j,t are quasi-differenced series obtained as From above equations we can also write ˆε j,t as { z t, t = 1, z j,t = z t 1 + c j / z t 1, t = 2,...,. ˆε j,t = y j,t x j,tˆγ j, j = 0, 1 Since we are interested in a test with high power at a specific value of the parameter against alternatives in either direction, so the asymptotic rejection probability of the point optimal test is given by πc; c 0, c 1 := lim PrP c 0, c 1 < k α ρ = 1 + c/. where k α = k α c 0, c 1 is the asymptotic critical value. his rejection probability πc; c 0, c 1 represents asymptotic power of the test where the test P c 0, c 1 rejects the null for small values 4

5 and for true c 0 has asymptotic size equal to power given by πc 0 ; c 0, c 1 = α. Since the critical value and the power both depend on c 0 and c 1 the values under the null and local alternative respectively therefore there is no single uniformly most powerful invariant UMPI test however a family of Neyman-Pearson tests indexed by c 1. Each test in the family is most powerful at the point c 1 = c and forms the power envelope defined by πc; c 0, c πc; c 0, c 1. For one-sided confidence intervals for ρ, the test of H 0 : c = 0 with lower tail stationary alternative H 1 : c = c 1 < 0 is usual unit root test and the point optimal test in this case reduces to the one introduced and analyzed in detail by ERS. For the choice of c 1, ERS proposed inverting the envelope power function such that πc 1 ; 0, c 1 = 0.50 and the tests have power functions not too far from the power envelope over a substantial range. his results in c 1 = 7 for x t = 1 and c 1 = 13.5 for x t = 1, t. For two-sided confidence interval two different values of c 1 as c 1L and c 1U in ρ 1 = 1 + c 1 / are required to test the null H 0 : c = c 0 against the alternatives H 1 : c 1 = c 1L < c 0 and H 1 : c 1 = c 1U > c 0. For the sake of confidence interval construction, ES extend the testing problem to different values of c 0 = 0, 5, 10 and recommend using alternatives that are fixed distance from the null as H 1i : c 1 = c 0 + c 1i, i = L, U. hey use c 1L, c 1U = 7, 2 and 13.5, 5 respectively for x t = 1 and x t = 1, t. o make notations simple we define the events R αi c 0, c 1i = {P c 0, c 1i < k αi c 0, c 1i }, i = L, U where for instance R αl c 0, c 1L denotes the event that occurs when the lower tail test rejects the null at α i significance level. ES set the overall size of the test as α L +α U = α to have confidence intervals with desired coverage probability of 1001 α% where α i = Pr R αi c 0, c 1i ρ 0 = 1 + c 0 /, i = L, U is the size of the individual test. We now define the rejection probability of the two-sided test as π α c; c 0, c 1L, c 1U, α L, α U = Pr R αl c 0, c 1L R αu c 0, c 1U ρ = 1 + c/. While size of the joint test is given by π α c 0 ; c 0, c 1L, c 1U, α L, α U = Pr R αl c 0, c 1L R αu c 0, c 1U ρ 0 = 1 + c 0 / = Pr R αl c 0, c 1L ρ = 1 + c 0 / + Pr R αu c 0, c 1U ρ = 1 + c 0 / Pr R αl c 0, c 1L R αu c 0, c 1U ρ 0 = 1 + c 0 / = α L + α U Pr R αl c 0, c 1L R αu c 0, c 1U ρ 0 = 1 + c 0 / α his implies that size of the joint test is bounded by α and if simultaneous rejections occur then choosing α L + α U < α results in a conservative test that yields unnecessary long confidence interval. herefore in our simulation experiments we will also keep eye on the size of the joint tests. Using α U = α α L, the rejection probability of the two-sided test reduces to 5

6 π α c; c 0, c 1L, c 1U, α L = Pr R αl c 0, c 1L R α αl c 0, c 1U ρ = 1 + c/ he confidence set is a set of those values of ρ 0 that are not rejected by either of the two tests or both. his implies that neither of the two events R αl c 0, c 1L and R αu c 0, c 1U should occur. However due to the possibility of simultaneous rejection of lower and upper tail tests the overall size of the tests follows the constraint α L + α U α whereas α works as upper bound for the size of the test. Confidence set and the resulting confidence interval given α L and α U is CS = {c 0 : R αl c 0, c 1L c R αu c 0, c 1U c }, CI = ĉ L, ĉ U where ĉ L = inf c0 CS and ĉ U = sup c0 CS. he confidence interval for ρ is then ˆρ L, ˆρ U where ˆρ L = 1 + ĉ L / and ˆρ U = 1 + ĉ U /. 3 Modified Point Optimal est Ng and Perron 2001 have considered the modified point optimal tests and provide their limiting distributions that coincide with the feasible point optimal test of ERS when testing c 0 = 0. Since our objective is to extend the analysis to the general testing problem c = c 0 in ρ = 1 + c/, so we modify the point optimal test of Ng and Perron 2001 and denote it by MP. he test statistic and its limiting distribution for testing H 0 : c = c 0 against H 1 : c = c 1 are given for 1. Constant Case c MP c 0, c 1 = ˆσ c d c 2 1 c û 2 1,t 1 c 1 c 0 1 û 2 1, B c s 2 ds c 1 c 0 B c Constant and Linear rend Case MP c 0, c 1 = ˆσ 2 d c c û 2 1,t c 1 1 û 2 1, c V c,c1 s 2 ds + 1 c 1 V c,c1 1 2 c û 2 0,t 1 1 c 0 1 û 2 0, V c,c0 s 2 ds 1 c 0 V c,c

7 [ where V c,ci s = B c s s λb c λ i ] 1 0 sb csds, λ i = B c s is the Ornstein-Uhlenbeck process given by 1 c i 1 c i + 1, i = 0, 1, and 3 c2 i B c s = with B r as standard Brownian motion. 4 One-Sided esting s 0 exp c s r db r In this section we investigate the influence of different choices of c 1 under the alternative on asymptotic local power properties of the point optimal test for different c 0. We consider c 0 = 4 as a representative case for negative choices of c 0, c 0 = 4 for positive values and, the famous unit root case when c 0 = 0. All simulations in this paper are performed in GAUSS using = 1000 and 50, 000 replications for cases x t = 1 and x t = 1, t. 4.1 he effect of c 1 on lower tail tests he first two panels in figures 4.1 and 4.2 respectively correspond to c 0 = 0 and 4 where we observe that ERS and ES choices of c 1 result in power curves that are quite indistinguishable from the power envelope across all c. his property is also fairly robust to variations in c 1 as evident from the power curves for some arbitrary choices of c 1 that correspond to power higher than 50% at the power envelope. he most interesting behavior emerges for the final case as can be seen from third panel of above figures, where we have quite different findings for c 0 = 4. We observe that the choice of c 1 values can matter a great deal to the power properties of the point optimal test. Based on 50% rule of ERS, c 1 = 3.75 for demeaned case and c 1 = 2.25 for constant and linear trend case produce tests with non-monotonic power curves that are close to the envelope near the null but then decline towards zero as c moves away from the null. his behavior is explored further below. However c 1 equal to 3 for x t = 1 and 9.5 for x t = 1, t based on ES fixed distance from the null rule behaves reasonably well but unlike the examples of c 0 = 4, 0, there does not exist any single choice of c 1 that produces a test with power uniformly close to the power envelope. herefore the choice of c 1 when we test some positive value of c 0 is important and will be considered below in more detail Why is the test inconsistent for some c 1 when c 0 > 0? From above discussion we know that the point optimal test when applied to test c 0 = 4 against the lower tail local alternative based on c 1 of ERS and arbitrary value of 2.5 has non-monotonic 7

8 Figure 4.1: Power Curves: Constant Mean 8

9 Figure 4.2: Power Curves: Constant and linear trend 9

10 asymptotic power curves that approach to zero as c diverges from the null. his observation suggests that the test may be inconsistent. A possible explanation for this inconsistency of the test may be as follows. he point optimal test without deterministics as shown in appendix A is approximately P c 0, c 1 = 1ˆσ c c y2 t 1 + c 0 c 1 1 y and we reject the null for small values i.e. the test rejects when P c 0, c 1 < k α. Since the asymptotic distribution of point optimal test is independent of ˆσ 2, so we can replace it with known σ 2 to keep things simple. First consider the case of testing unit root against the stationary local alternatives that requires c 0 = 0 and c 1 < 0. he test given above becomes P 0, c 1 = 1 σ 2 c y2 t 1 c 1 1 y 2, and consists of entirely positive terms and leads to k α > 0. hus the test is consistent against fixed alternatives and its consistency can be shown as below. With ρ < 1, we have and 1 yt 1 2 p E yt 1 2 σ 2 = 1 ρ 2 y 2 d y 2, where y 2 represents a random variable with stationary distribution of y t, i.e. y σ 2 0, 1 ρ 2. hus P 0, c 1 d c2 1 1 ρ 2 c y 2 1. σ his convergence in distribution shows that under fixed alternatives P 0, c 1 = O p 1 or P 0, c 1 = O p 1. his implies that P 0, c 1 p 0 and hence we observe Pr P 0, c 1 < k α ρ < 1 1, i.e. the test statistic will be less than any positive critical value with probability approaching 1. herefore the test is consistent. Now consider the case such that c 0 > 0 and we test c 0 against the lower tail alternative c 1 < c 0. he signs of terms c 2 1 c2 0 2 y2 t 1 + c 0 c 1 1 y 2 of point optimal test in 4.1 depend on magnitudes of c 0 and c 1 contrary to the unit root test case with c 0 = 0 and c 1 < 0. Under this new situation with c 0 > 0 and c 1 < c 0, we have c 0 c 1 > 0 and so the second term c 0 c 1 1 y 2 > 0. Now we have two cases to conjecture about the sign of the first term as: 1. If c 1 > c 0, then c 2 1 c2 0 > 0 leads to c 2 1 c2 0 2 y2 t 1 > 0 and we are back to the situation with consistent test. 2. For c 1 < c 0, we have c 2 1 c2 0 < 0 implying c 2 1 c2 0 2 of c 1 we may have c 2 1 c2 0 2 y2 t 1 c 0 c 1 1 y 2 test P c 0, c 1 is not guaranteed to be positive or negative. 10 y2 t 1 < 0. For different values and so the point optimal

11 Note that for this second case, the test is still the likelihood ratio test but may cause inconsistency in some cases. Under the fixed alternative case ρ < 1 we have P c 0, c 1 d c 2 1 c ρ 2 + c y 2 0 c 1. σ his implies that P c 0, c 1 = O p 1 or P c 0, c 1 p 0 for fixed alternative case as before. But the inconsistency arises given c 2 1 c2 0 < 0 and the critical value may be negative i.e. k α < 0. he reason why this negative critical value causes inconsistency is that if P c 0, c 1 p 0 and k α < 0 then Pr P c 0, c 1 < k α converges to zero instead of one. For instance, in demeaned case for c 0 = 4 if we choose c 1 such that < c 1 < 4, the resulting test is inconsistent. his is the underlying reason for non-monotonic power curves when testing some positive value of c 0. As c, the local to unity model behaves like a stationary model and the point optimal test does not have power against stationary models. his logic is not applicable to local to unity model itself, i.e. the test with c 1 < 4 e.g. c 1 = 2.5 as evident from figure 4.1 may have good local power properties near the null but has poor properties further from the null suggesting not to choose that test. hus based on our simulation experiments we find that; for a test with constant we need c , and with constant and linear trend we need c 1 < in order for the P test to be consistent against stationary alternatives. 4.2 he effect of c 1 on upper tail tests o see if c 1 matters in case of upper tail testing, power curves are presented in figures 4.3 and 4.4 respectively for demeaned and detrended cases. First two panels in these figures correspond to c 0 = 0 and c 0 = 4 respectively. In both specifications of x t we see that the power curves produced by using different values of c 1 are fairly close to the power envelope and we are indifferent in using either of these values with an exception of the subjective choice of c 1 = 4 in case of c 0 = 0. he test when used with c 1 = 4 a value that is a bit far from the null has low power for all c that are close to the null but as c moves away from the null the power gets closer to the power bound. his power loss for values near the null make c 1 = 4 an inferior choice compared to ERS and ES. As evident from final panels of above figures, some different behavior emerges for c 0 = 4. We have c ERS 1 = and c ES 1 = 6 for constant mean case and c ERS 1 = 4.5 and c ES 1 = 9 for constant and linear trend case with some arbitrary choices of c 1 as a value between c ERS 1 and c ES 1 or a value greater than c ES 1. We observe that in both cases power curves of the test based on c ERS 1 are reasonably close to the power envelope until their tangency to the envelope at c 1 itself but afterwards remain below and, contrary to other two cases of c 0 = 4, 0, never get quite close to the envelope. On the other hand, power of the test based on c ES 1 remains substantially low for c that are near to the null as manifested by nearly horizontal segment of the power curve. However as c increases further away from the null the power curve starts rising quickly towards the power 11

12 Figure 4.3: Power Curves: Constant mean 12

13 Figure 4.4: Power Curves: Constant and linear trend 13

14 bound in demeaned case but behaves rather poorly in linear trend case. In summary, the choice of c 1 has a substantial effect on the power properties of the tests for c 0 = 4. It will therefore be necessary to consider ways of choosing c Is the test inconsistent for upper tail testing? o answer this question of inconsistency for the upper tail test we again focus on the two terms given in the point optimal test. Since c 1 > c 0, so the second term c 0 c 1 1 y 2 in point optimal test is always negative. he first term c 2 1 c2 0 2 y2 t 1 might be positive or negative depending on c 2 1 c2 0. However, for upper tail test none of this matters in the same way that it does for the consistency of lower tail tests against stationary alternatives. In the latter case the statistic converges to zero, so the sign of the critical value is important for the consistency or otherwise of the test. In considering the consistency of a test against explosive alternatives, we have both 2 + and 1 y 2 p +. More specifically, for the DGP y t = ρy t 1 + ε t with ρ > 1, we find from heorem 2 of Lai and Wei 1983 see equations 2.1 and 2.3 that ρ 2 y 2 = O p 1 and ρ 2 y2 t 1 = O p 1, so the two terms in the test statistic are actually the same order when ρ > 1. But notice that y2 t 1 is divided by 2 while y 2 is only divided by in the statistic, so the asymptotic behavior of the statistic when ρ > 1 is determined entirely by c 0 c 1 1 y 2. For any upper tailed test i.e. any c 0, c 1 such that c 0 < c 1 we have c 0 c 1 1 y 2 p, and the test rejects for small values of P, so it will reject with probability converging to one against explosive alternatives. hus inconsistency is not an issue here. y2 t 1 p 5 Alternative criteria to choose c 1 Based on previous discussion we find that the alternatives proposed in the literature work fairly well when applied to testing c 0 taking zero or negative values. But for testing positive values of c 0, the test becomes inconsistent for lower sided testing or has low power near the null for upper tail testing. his issue motivates us to find some alternative choices of the model parameters. As discussed in Patterson 2011, Cox and Hinkley 1974 provide the following three possible choices when there does not exist any uniformly most powerful test. i. Use the most powerful test for a representative ρ 1 value under the alternative. ii. Maximize power for an alternative very local to the null. iii. Maximize the weighted power for a range of local alternatives. o achieve our objective we will use i and iii and some other criteria as well. o this end, let π α c; c 0, c 1i and π α c; c 0, c respectively denote the asymptotic local power function and the asymptotic power envelope of test of size α and 14

15 where π α c 0 ; c 0, c 1i = α. Criterion 1 Power curves tangency π α c; c 0, c 1i := lim Pr P c 0, c 1i < k α ρ = 1 + c/ Based on i above, we can apply power curves tangency rule to obtain c 1i as a value such that the asymptotic power curve is tangent to the power envelope at some pre-decided power π, i.e. π α c 1 ; c 0, c 1i = π where π = 50% or 80%. Note that ERS s choice of c 1i is based on this criterion using π = 0.50 with emphasis on the treatment of deterministics under the alternatives local to the null. Criterion 2 Minimax criterion In addition to the criteria above, we could choose the optimal value of c 1 as the value that minimizes the largest power loss relative to the power envelope, i.e. c 1i = arg min c 1i Criterion 3 Optimal weighted average power max π α c; c 0, c π α c; c 0, c 1i. c Based on rule iii given above following Cox and Hinkley 1974, we can define the optimal weighted average power function as c 1i = arg max c 1i C j π α c; c 0, c 1i wcdc, j = l, u where π α c 0 ; c 0, c 1i = α and wc 0 represents weights as a function of c, Cl =, c 0 ] for lower tailed test and C u = [c 0, when the test is upper tailed. C j One problem with the set C j is that the integral Cj π α c; c 0, c 1i wcdc need not exist if is unbounded. We can either choose w c such that Cj w c dc < or restrict wc to interval/truncated domain C j C j so that the integral π α c; c 0, c 1i wcdc exists when defined on C j with C l = [c l, c 0 ] and C u = [c 0, c u ]. For instance, for lower tailed test the integral can become c0 b π α c; c 0, c 1L wcdc. Now the question arises of how to choose b? As a practical matter we can choose b such that for any small ε > 0 we observe power envelope within ε of maximum attainable power of 1, i.e. 1 π α b; c 0, b ε. For two-sided tests we will stick to the choice of weighted average power maximization. As given above the rejection probability of the two-sided test using α U = α α L is π α c; c 0, c 1L, c 1U, α L = Pr R αl c 0, c 1L R α αl c 0, c 1U ρ = 1 + c/ 15

16 So in general we can set about choosing c 1L, c 1U, α L to maximize weighted power, i.e. {c 1L, c 1U, α L} = arg max π α c; c 0, c 1L, c 1U, α L w c dc. {c 1L,c 1U,α L } C Using the optimal value α L, we can find optimal value of α U from α U = α α L. However for fixed α L and hence α U like the case of ES, we can obtain separately the optimal values of c 1L and c 1U as c 1i = max c 1i π α c; c 0, c 1L, c 1U w c dc, C i i = L, U. But rather than setting an arbitrary value for α L, we will investigate to choose it optimally with other parameters. Some possible choices of weight function wc are as follows. { 1 if c Cj, i. Uniform weights: wc = 0 if c / C j. his choice of weight function results in simple average of power, so the optimal value of c 1 for the local alternatives is chosen as a value that maximizes simple average of powers. For one-sided tests c 1i = arg max π α c; c 0, c 1 dc, j = l, u c 1 C j and for two-sided tests {c 1L, c 1U, α L} = arg max {c 1L,c 1U,α L } C π α c; c 0, c 1L, c 1U, α L dc ii. wc = I c 1/2 Jeffreys Prior ii.a. Jeff reys prior based on full likelihood Phillips 1991 suggests using Jeffreys prior instead of flat priors in response to the criticism by Sims and Uhlig 1991 on frequentists approach to unit roots. Phillips considers fixed ρ in the autoregressive model with and without deterministic trend and both conditional and unconditional cases. However for our purpose we assume the conditional case u 0 = 0, known σ 2 and with no deterministics to translate his results using ρ = 1 + c/ to take asymptotic approximations. he corresponding Fisher s information matrix see appendix for the derivation is given by { 1 e 2c 1 2c 2c 1 if c 0, I c = 1 2 if c = 0. he prior is plotted in figure 5.1 below. From this diagram we observe that the prior increases slowly to the value 1/ 2 at c = 0 as the information content increases with, but 16

17 wc 3 JP IJP c Figure 5.1: Jeffreys prior and Invariant Jeffreys prior then it starts growing exponentially for all c > 0. his higher density for c > 0 is due to our prior knowledge about the parameter that when true value of c 0 > 0 in ρ = 1 + c/, the data will carry more information about c 0. ii.b. Jeff reys prior based on invariant likelihood Since in this analysis we are applying point optimal test of King 1987 that is invariant to transformations of the form y y + Xb, where y = y 1,..., y and X = x 1,..., x, following King 1980 and King and Hillier 1985, we can derive Jeffreys prior based on invariant likelihood. Although King and Hillier discuss both invariance to the regressors and to scaling but we follow ERS and use invariance only to the regressors and not to scaling which is handled by dividing the point optimal statistic by an estimate of the variance. By assuming Gaussian errors the information matrix using invariant maximum likelihood when x t = 1 is the same as the one we have obtained from full maximum likelihood. However for x t = 1 t, this information matrix is given by I c = 1 2c e 2c 1 2c 1 1 c 2 3c 2e 2c + ce 2c + 2c c 3 1 c + c c + c c 6 1 c + c2 3 2, c 0 Note that I c 0 as c 0 and is evident from the plot of this prior given in figure 5.1. his observation is also pointed out by Marsh

18 Since Jeffreys prior is the square root of the information matrix, so we can use the weight function wc = I c 1/2 and the optimal choice of c 1 is given by c 1i = arg max π α c; c 0, c 1 I c 1/2 dc, j = l, u c 1i C j {c 1L, c 1U, α L} = arg max π α c; c 0, c 1L, c 1U, α L I c 1/2 dc. {c 1L,c 1U,α L } C iii. wc based on the Symmetrized Asymptotic Reference Prior SARP Berger and Yang 1994 provide the symmetrized asymptotic reference prior that maximizes the Kullback-Leibler divergence between prior and posterior that results in the following prior. For large, this is approximately π ρ exp π ρ E ρ log π ρ y 1 2 E ρ log but has different orders of for different ρ. Berger and Yang 1994 suggest using the normalized π ρ for ρ < 1, then use transformation ρ 1/ρ for ρ > 1.his leads to the following asymptotic symmetric reference prior. y 2 t 1 { 1/ 2π 1 ρ 2, ρ < 1 π SR = 1/ 2π ρ ρ 2 1, ρ > 1 Berger and Yang discuss some of the important features of this prior as; 1. it a proper prior since it integrates to one, 2. it assigns equal probability of one-half to ρ < 1 and ρ > 1, 3. contrary to Jeffreys prior as a weight function that assigns finite weight to the stationary case where ρ < 1 but unreasonable infinite weight to the explosive case ρ > 1, this prior assigns more weight to the values that are close to c = 0 but as c moves away from zero the weights start shrinking; in particular, for c > 0 the weights converge to zero, 4. usually we can not use improper priors for testing but above mentioned properties of SR prior make it suitable for use in testing. his asymptotic reference prior makes sense under local to unity setting ρ = 1 + c/ with and becomes 1 π R c exp 2 E c log B c s 2 ds

19 where B c s is the Ornstein-Uhlen process. his prior is similar to Jeffreys prior but is flexible to imposing symmetry about c = 0. For computational purpose we approximate E c log 1 0 B c s 2 ds by simulation, as E c log 1 0 B c s 2 ds 1 R R 1 log n 2 r=1 n z 2 c,r,t where z c,r,t is generated from z c,r,t = 1 + c z c,r,t 1 + η r,t, with z c,r,0 = 0 and { } n η r,t being pseudo-random drawings from the i.i.d. standard normal distribution. he sample size n and number of repeated samples R are chosen to be large in the usual way to reduce simulation error. he resultant reference prior is 1 1 R 1 n π R c = exp log 2 R n 2 zc,r,t 2. r=1 Following diagram presents asymmetric and symmetric versions of the reference prior along with Jeffreys. Figure 5.2: Asymmetric and Symmetric Reference Prior Now we assume the weight function wc = π R c and obtain optimal values of c 1 for both one and two-sided tests as c 1i = arg max π α c; c 0, c 1 π R c dc, i = l, u c 1 C i {c 1L, c 1U, α L} = arg max π α c; c 0, c 1L, c 1U, α L π R c dc. {c 1L,c 1U,α L } C 19

20 6 Results 6.1 One-sided tests Constant mean case In able 6.1 below, we have reported optimal values of c 1L and c 1U obtained after using different criteria discussed above for individual one-sided point optimal test when x t = 1. Power curves corresponding to these optimal values for c 0 = 0, 4 are presented in figure 6.1. We observe that these power curves for lower tailed alternatives given in panel 1 and 2 are indistinguishable from the power envelope. Similarly for upper tailed alternatives we find that some of optimal c 1L values produce power curves slightly below the envelope near the null only. But these differences are negligible, except for invariant Jeffreys prior case when c 0 = 0, and hence we can happily use any of these values for confidence interval construction. However looking at figure 6.2 for c 0 = 4 we observe that some of the power curves given in the first panel for lower tailed test indicate the inconsistency of the test for some c 1L values. In particular, we find c 1L values obtained by using ERS, minimax, π α c 1 ; c 0, c 1 = 0.85 rules and, the Jeffreys prior as weight function are all greater then the threshold value of c 1 = for constant case. As a consequence of using these values we get inconsistent tests with powers converging to zero instead of one as c moves away from the null. Further we also find that none of these criteria produce power uniformly close to the power envelope. Specifically power curves based on ES behave poorly for alternatives not far from the null compared to other three choices. able 6.1: Optimal values of model parameters for one- sided tests: Constant only H 0 : c 0 = 4 c 0 = 0 c 0 = 4 Criterion c 1L c 1U c 1L c 1U c 1L c 1U ERS/ π α c 1 ; c 0, c 1 = π α c 1 ; c 0, c 1 = Elliot and Stock Mini-Max Simple Average Jeffreys Prior Invariant JP Symmetric Reference Prior Power curves for upper tail test of c 0 = 4 using the same criteria are presented in second panel of figure 6.2 where inconsistency is not an issue. he best choice of c 1U depends on the position of the power curve relative to the power envelope, that is, the value of c 1U that produces power curve closer to the power envelope is preferred. But we observe none of the curve is uniformly close to 20

21 Figure 6.1: Lower-sided power curves with different criteria x t = 1 21

22 Figure 6.2: Lower-sided power curves with different criteria x t = 1 22

23 the power envelope and hence we can not select such value. However we find that value of c 1U proposed by ES has very low power for all the alternatives that are close to the null 4 < c < 4.85 but as c moves away from the null there is a sharp rise in the power and the curve gets close to the power bound. All other criteria provide c 1U values with power curves that are fairly close to the envelope so we can use any of these values Constant and linear trend case able 6.2 reports the optimal values of parameters c 1L and c 1U for constant and linear trend case and corresponding power curves are presented in figures 6.3 and 6.4 below. Like demeaned case above, we find that when testing c 0 = 4 and c 0 = 0, c 1L and c 1U for lower and upper sided tests respectively produce power curves close to the power envelope and hence make these c 1 values from certain criteria feasible. able 6.2: Optimal values of model parameters for one- sided tests: Drift and Linear rend H 0 : c 0 = 4 c 0 = 0 c 0 = 4 Criterion c 1L c 1U c 1L c 1U c 1L c 1U ERS/π α c 1 ; c 0, c 1 = π α c 1 ; c 0, c 1 = Elliot and Stock Mini-Max Simple Average Jeffreys Prior Invariant Jeffreys Prior Symmetric Reference Prior From figure 6.4 for c 0 = 4 case we observe that only ERS rule does not provide monotonic power curve and hence causes an inconsistent test for lower sided alternatives. Simulation evidence shows that all c 1L > will produce negative critical values and hence will make the test inconsistent. We also observe that the value of c 1L based on ES rule produces a test that has lower power for alternatives near the null. We also examine that c 1L based on minimax criterion produces the power curve that is not close to the envelope for values far from the null and also making it an inferior choice compared to other c 1L values. Similarly for upper tailed testing we observe that c 1U based on ES rule works poorly since the corresponding power curve has a flatter portion for 4 < c < 6.5 with very low power, then a big jump in power towards power envelope near c = 6.5. Also power curve based on c 1U from reference prior produces slightly low power for alternatives close to the null but then has power very close to the envelope therefore we should not rule out this value from the list of optimal values of model parameters. 23

24 Figure 6.3: One-sided power curves with different criteria constant and linear trend 24

25 Figure 6.4: One-sided power curves with different criteria constant and linear trend 25

26 6.2 wo Sided est Following the discussion in section 3 about two-sided testing and using only the criterion of optimal weighted average power we have computed optimal values of parameters α L, α U, c 1L and c 1U for both demeaned and detrended cases for confidence interval construction. he results are reported in table 6.4 and the corresponding power curves for different c 0 in 6.2 below. able 6.3 reports values based on the Elliot and Stock s rule discussed above with fixed α L = 0.03 and α U = able 6.3: Model Parameters based on ES Rule Demeaned Case Detrended Case c 0 c 1,L c 1,U c 1,L c 1,U From the following table we have quite different optimal values of model parameters from those proposed by ES. We find varying values of α L, α U across different choices of c 0 instead of fixed values proposed by ES. We observe that for positive c 0, all the different choices of weight functions assign higher probability of type-i error to the left tail i.e. higher α L that is even higher than the ES value of α L = 0.03 for a test of overall size α = able 6.4: Optimal values of model parameters for two sided tests Weight Function Demeaned Case Detrended Case H 0 : c 0 = 4 c 1L c 1U α L c 1L c 1U α L Simple Average Jeffreys Prior Symmetric Reference Prior Invariant Jeffreys Prior Same as Jeffreys prior H 0 : c 0 = 0 Simple Average Jeffreys Prior Symmetric Reference Prior Invariant Jeffreys Prior Same as Jeffreys prior H 0 : c 0 = 4 Simple Average Jeffreys Prior Symmetric Reference Prior Invariant Jeffreys Prior Same as Jeffreys prior

27 In diagram 6.5 below, left and right panels correspond respectively to the power curves for demeaned case and the detrended case. We find well behaved power curves using the optimal values of c 1L, c 1U, α L, α U obtained from multiple choices of weights. On comparison we observe varying slope of the curves particularly at right tails for different c 0. We also find that our proposed values produce power curves that outperform the power curves based on ES in almost all cases especially at the lower tails. We observe significant power gains using our proposed values over those of ES particularly when we test some positive value of c 0. hese vital differences support our hypothesis of varying slope of the power curves for different c 0 values and hence different values for the model parameters. Figure 6.5:Power Curves for two-tailed test with different criteria On comparison we observe that power curves corresponding to the parameter values from the weighted powers using the uniform and symmetric reference prior weights are superior to those from Jeffreys prior weights at the left tails for each c 0. While the test achieves higher power in detrended case with Jeffreys invariant prior weights compared to Jeffreys prior weights when c 0 is either 4 or 0. By looking at the right tails of the curves we find that for all alternative c values the test has almost identical powers using optimal values. he most interesting case is that of testing c 0 = 4 where we find ES as a poor choice since it has low power close to the null. Also 27

28 both types of Jeffreys prior weights have identical power at right tail but have slight differences in powers at left tail. However for detrended case, when we compare the powers at left tails we find significantly lower powers from Jeffreys prior weights compared to powers from uniform and symmetric reference prior weights. Given this simulation analysis we observe that there does not exist any unique set of parameters with uniformly better power across the grid of c values. Despite this outcome we find that our proposed set of values produce powers that are relatively higher than powers proposed by ES. In particular we observe that the point optimal test has good power properties when computed from the parameters generated from uniform and symmetric reference prior weight functions for all c 0 and for both demeaned and detrended cases. 7 Conclusions In this paper, we have analyzed and extended Elliot and Stock s method to construct confidence intervals, by inverting a sequence of invariant point optimal tests, for the autoregressive root local to one. Our analysis of one-sided testing problems shows that in the presence of deterministics the choices of model parameters based on ERS and ES work fairly well when testing zero or negative values of parameter c 0. However for some positive c 0, the test based on the ERS rule becomes inconsistent for lower sided stationary alternatives. Also the test based on the ES rule has low power for alternative values close to the null. For upper tail test, this behavior remains the same whether the deterministic component involves constant mean or constant and linear trend. We provide the reason of the inconsistency of the test for lower sided alternatives for some choices of parameter c 1L. Similarly for two-sided test, instead of using fixed values of α L = 0.03 and α U = 0.02 and c 1L, c 1U that are fixed distance from the null, we search for optimal values of these parameters that are jointly determined by the maximization of weighted powers using different weight functions. Our results indicate that optimal values of these parameters are very sensitive to changes in c 0. In particular we find that all types of weight functions considered here, except Jeffreys prior, provide optimal α L > 0.03 for different c 0. However, weights based on symmetric reference prior result in the largest α L close to the nominal level of We acknowledge that none of these weight functions produce uniformly higher power so that we prefer one weight function over the other. But simulation results prove that our proposed values of the parameters particularly from uniform and symmetric reference prior weights result in more powerful test compared to those provided by ES. Based on these findings we can say that out proposed values may yield more accurate confidence intervals. Possible extensions to this work include; the derivation of an invariant symmetric reference prior following the same approach we used for the invariant Jeffreys prior, allowing for serially correlated errors in the usual way, and consideration of the effect of the initial condition on these 28

29 tests. 8 References Berger, J. O., and R. Y. Yang 1994 "Noninformative Priors and the Bayesian esting for the AR1 Model," Econometric heory 10, Bobkovski, M.J "Hypothesis esting in Nonstationary ime Series," Unpublished Ph.D. hesis, Department of Statistics, University of Wisconsin. Cavanagh, C "Roots Local to Unity," Unpublished Manuscript, Department of Economics, Harvard University. Chan, N. H., and C. Z. Wei 1987 "Asymptotic Inference for Nearly Nonstationary AR1 Processes," Annals of Statistics 15, Cox, D. R., and D. V. Hinkley 1974 heoretical Statistics. Chapman & Hall, London. Elliot, G.,. J. Rothenberg, and J. H. Stock 1996 "Effi cient ests for an Autoregressive Unit Root," Econometrica 64, Elliot, G., and J.H. Stock 2001 "Confidence Intervals for Autoregressive Coeffi cients Near One," Journal of Econometrics 103, Lai,. L., and C. Z. Wei 1983 "Asymptotic Properties of General Autoregressive Models and Strong Consistency of Least-Squares Estimates of their Parameters," Journal of Multivariate Analysis 13, King, M. L "Robust ests for Spherical Symmetry and their Applications to least Squares Regression," he Annals of Statistics 8, King, M. L., and G. H. Hillier 1985 "Best Invariant ests of the Error Covariance Matrix of the Linear Regression Model," Journal of Royal Statistical Society. Series B 47, King, M. L "owards a heory of Point Optimal esting," Econometric Reviews 6, Marsh, P "he Available Information for Invariant ests of a Unit Root," Econometric heory 23, Ng, S., and P. Perron 2001 "Lag Length Selection and Construction of Unit Root ests with Good Size and Power," Econometrica 69, Phillips, P. C. B "owards a Unified Asymptotic heory for Autoregression," Biometrika 48, Phillips, P. C. B "o Criticize the Critics: An Objective Bayesian Analysis of Stochastic rends," Journal of Applied Econometrics, vol. 6, Patterson, K Unit Root tests in ime Series Volume 1. Palgrave Macmillan, UK. Sims, C. A., and Uhlig, H "Understanding Unit Rooters: A Helicopter our," Econometrica 59,

30 Stock, J.H "Confidence Intervals for the Largest Autoregressive Root in U.S. Macroeconomic ime Series," Journal of Monetary Economics 28, Appendix A 1 Derivation of Jeffreys Prior We derive here the expression for Jeffreys prior in the autoregressive model under local to unity setting, whereas this prior is the positive square root of Fisher s information matrix. he loglikelihood function based on ε t being i.i.d. in equation 2.2 is given by log Lc σ 2 = 2 log2π 2 log σ2 1 2σ 2 u t 1 + c 2 u t 1, the score function with respect to c is given by log Lc σ 2 c and the derivative of the score function is = 1 σ 2 u t u t c u 2 t 1, 2 log Lc σ 2 c 2 = 2 σ 2 u2 t 1 = 2 σ 2 1 t=0 u2 t. By recursively iterating u t = 1 + c/ u t 1 + ε t, we obtain hus we have u t = t 1 j=0 1 + c j εt j Eu 2 t = σ 2 t 1 j=0 E 1 t=0 u2 t = σ c c 1 + c 2j geometric series 2t 2 2t = σ c 1 c2 2c +. = σ 2 1 t= c c2 2c + 1 = σ c2 2c + 2t 1 1 t=0 1 + c 2t he information matrix is then obtained as 30

31 2 log Lc σ 2 I = E c 2 = 2 σ 2 E 1 t=0 u2 t = 2 1 σ 2 σ2 1 c2 2c + = 1 2c + c2 = 1 2c + c2 = 1 2c + c2 2 c 0 1 4c 2 e 2c 1 2c = 1 2c 1 1 t= c c c 2 1 2c + c2 1 2c + c2 + 1 e 2c 1 2c c 2t 1 + c 2c c If c 0, then by L Hopital s rule we get I =1/2. hus in the limit we have { 1 e 2c 1 2c 2c 1 if c 0, I c = 1 2 if c = 0. 2 Point Optimal and Modified Point Optimal ests and heir Asymptotic Distributions 2.1 Point optimal test in the absence of determinsitcs First of all we derive the point optimal test without deterministics as this will help explaining why the test statistic becomes inconsistent for some choices of c 0 and secondly we can also use this expression to derive modified point optimal test. he point optimal test is given by P ρ 0, ρ 1 = ˆσ 2 ˆε 2 1,t ρ 1 ρ 0 ˆε j,t = û j,t ρ j û j,t 1, û j,t = y t x tˆγ j ˆε 2 0,t, Suppose we have no deterministics i.e. x tγ = 0, so that the original model given in 2.1 and 2.2 becomes y t = u t, and û j,t = y t u t = ρu t 1 + ε t, u 1 = ε 1. 31

32 and he point optimal test of H 0 : ρ = ρ 0 against H 1 : ρ = ρ 1 is then P ρ 0, ρ 1 = ˆσ 2 = ˆσ 2 Consider the term in the parenthesis = y t ρ 1 y t 1 2 ρ 1 ρ 0 1 ρ 1 ρ 0 = 1 ρ 0 ρ 0 ρ 1 y 2 + ε 2 1,t ρ 1 ρ 0 ε 2 0,t y t ρ 1 y t 1 2 ρ 1 ρ 0 y t ρ 0 y t 1 2 = y 2 t 1 y ρ 0 ρ 1 + ρ 2 1 ρ 1 ρ 0 y 2 t 1. y t ρ 0 y t ρ 1 ρ 0 y 2 t 1 his statistic is applied with ρ 0 = 1 + c 0 / and ρ 1 = 1 + c 1 /, so 1 ρ 0 ρ 1 = c 0 ρ 0 ρ 1 = c 0 c c 1 so for large which is what concerns us for choice of c 1 P c 0, c 1 c0 c 1 = c 0 c 1 P c 0, c 1 = c 2 1 c y 2 c 0 + c 1 = c 0 + c 1 y 2 t 1 1 y 2 c 0 + c 1 2 y 2 t + ρ 2 1 ρ 1 ρ 0 c 0c 1 2, yt 1 2 y 2 t 1 yt c 0 c 1 1 y and we reject for small values. it s worth not omitting the c 0 c 1 at the front of second last term above so as not to mess with the sign of the statistic, and hence the rejection tail. 2.2 Asymptotic distributions in the presence of deterministics but no autocorrelation in error term ε t Now we incorporate the deterministic component and assuming no serial correlation in error term ε t derive the asymptotic distribution of the test statistic under the local to unity framework. he model is given by 32