Optimal Jackknife for Discrete Time and Continuous Time Unit Root Models

Transcription

1 Optial Jackknife for Discrete Tie and Continuous Tie Unit Root Models Ye Chen and Jun Yu Singapore Manageent University January 6, Abstract Maxiu likelihood estiation of the persistence paraeter in the discrete tie unit root odel is known for suffering fro a downward bias. The bias is ore pronounced in the continuous tie unit root odel. Recently Chabers and Kyriacou introduced a new jackknife ethod to reove the first order bias in the estiator of the persistence paraeter in a discrete tie unit root odel. This paper proposes an iproved jackknife estiator of the persistence paraeter that works for both the discrete tie unit root odel and the continuous tie unit root odel. The proposed jackknife estiator is optial in the sense that it iniizes the variance. Siulations highlight the perforance of the proposed ethod in both contexts. They show that our optial jackknife reduces the variance of the jackknife ethod of Chabers and Kyriacou by at least % in both cases. Keywords: Bias reduction, Variance reduction, Vasicek odel, Long-span Asyptotics, Autoregression JEL classification: C, C5 Introduction As a subsapling ethod, the jackknife estiator was first proposed by Quenouille 99 for reducing the bias in the estiation of the serial correlation coeffi cient. Tukey 958 used the jackknife ethod to estiate the variance. Miller 97 reviewed the literature of the jackknife, ephasizing its properties in bias reduction and interval estiation. More recently, the jackknife ethod has been found useful Ye Chen, School of Econoics and Si Kee Boon Institute for Financial Econoics, Singapore Manageent University, 9 Staford Road, Singapore, 7893; Eail: chen.ye.9@phdecons.su.edu.sg. Jun Yu, Si Kee Boon Institute for Financial Econoics, School of Econoics and Lee Kong Chian School of Business, Singapore Manageent University, 9 Staford Road, Singapore Eail: yujun@su.edu.sg. We acknowledge coents fro Peter Phillips and Denis Leung.

2 in econoetrics. Hahn and Newey 3 deonstrated the use of the jackknife to reduce the bias arising fro the incidental paraeters proble in the dynaic panel odel with fixed effects. Phillips and Yu 5, PY hereafter introduced a jackknife ethod based on non-overlapping subsaples, and showed that it can be applied to the coeffi cients in continuous tie odels as well as the asset prices directly. Chiquoine and Hjalarsson 9 provided the evidence that the jackknife can be successfully applied to stock return predictability regressions. In the tie series context, the jackknife ethod proposed by PY is very easy to ipleent and also quite effective in reducing the bias. In effect, this jackknife estiator reoves the first order bias in the original estiator by taking a linear cobination of the full saple estiator and a set of subsaple estiators. The validity for reoving the first order bias can be explained by the Nagar approxiation of the oents of the original estiator, in ters of the oents of the estiator s Taylor expansion as a polynoial of the saple oents of the data. In particular, under regular conditions, the first order bias is a reciprocal function of the saple size with a coon coeffi cient. Consequently, the weights used for the subsaples are the sae and are proportional to the weight used for the full saple. For stationary autoregressive odels, Chabers obtained the liit distribution of the jackknife estiator of PY and exained its perforance relative to alternative jackknifing procedures. However, in the context of a discrete tie unit root odel, Chabers and Kyriacou, CK hereafter pointed out that the jackknife estiator of PY cannot copletely reove the first order bias. This is because the first order bias of the least squares LS or axiu likelihood ML estiator of the autoregressive coeffi cient has distinctive functional fors in different subsaples. This distinctiveness is anifest in the fact that the liit distribution of the LS/ML estiator depends on the initial condition. A revised jackknife estiator with new weights was proposed in CK. Unlike the PY estiator, the weight used in CK for the full saple is no longer proportional to those for the subsaples. However, the weights used for the subsaples are required to be identical to each other in CK. CK showed that the odified jackknife estiate perfors better than the PY estiator for bias reduction. While the jackknife ethod of CK reduces the bias of the original estiator, it always increases the variance. This is not surprising because subsaple estiators are used in the construction of the jackknife. As a result, the trade-off between the bias reduction and the increase in variance is iportant. Siulation results in CK suggest that their jackknife ethod soeties gains enough in bias reduction without a significant increase in variance, so that there ay be an overall gain in root ean square error RMSE for the discrete tie unit root odel. In this paper, we propose an iproved jackknife estiator for unit root odels. Our estiator is optial in the sense that it not only reoves the first order bias, but also iniizes the variance, and hence, has better finite saple properties than the CK estiator. Like the estiators of PY and CK,

3 our optial jackknife estiator is a linear cobination of the full saple estiator and the subsaple estiators. However, we do not require the weight for the full saple to be proportional to those for the subsaple, nor do we require the weights for the subsaple be the sae. Indeed, all the weights are obtained by iniizing the variance under the condition that the first order bias is reoved. The calculation of the weights involves obtaining the covariances between the full saple estiator and the subsaple estiators, and the covariances between the subsaple estiators. To do so, we eploy the joint oent generating function MGF of the liit distributions of these estiators. The idea is applied to both the discrete tie and the continuous tie unit root odels. Optial weights are derived and the finite saple perforance of the new estiator is exained for both odels. It is found that the optial jackknife estiator offers over % reduction in variance over the CK estiator without coproising bias reduction in both cases. The paper is organized as follows. Section derives the general for of the optial jackknife estiator in the discrete tie unit root odel. Section 3 extends the results to the continuous tie unit root odel where the bias of the persistent paraeter is known to be ore serious. Section presents the Monte Carlo siulation results. Section 5 concludes. Appendix collects all the proofs. Throughout the paper, we adopt the following notations. The full saple size is denoted by n, the tie span of data by T, the paraeter of interest by θ or β or κ, and the true value of it by θ or β or κ. The signal is used to indicate the convergence of the associated probability easures as n or T goes to infinity while denotes the equality in distribution, and the nuber of subsaples. θ j eans the LS/ML estiator of θ fro the jth subsaple of saple length l i.e., l = n, θ P Y is the jackknife estiator of θ proposed by PY, θ CK is the jackknife estiator proposed CY by CK, and θ is the jackknife estiator proposed in the present paper. Following CK, we define Z = W dw/ W, Z j = j/ j / W dw/ j/ j / W, µ = EZ and µ j = EZ j, where W is a standard Brownian otion. Optial Jackknife for the Discrete Tie Unit Root Model In this section, we first briefly review the literature, focusing especially on the approach of CK. We then introduce our optial jackknife estiator in the context of the discrete tie unit root odel.. A Literature Review Considering a siple unit root odel with initial value y = O p : y t = y t + ε t, ε t iid, σ ε, t =,,..., n.. 3

4 The LS estiator of the autoregressive AR paraeter of, say, β, is β = n t= y t y t / n t= y t. When ε t is norally distributed, β is also the ML estiator of β, conditional on y. The liit distribution of β was obtained by Phillips 987a using the functional central liit theory and the continuous apping theore: n β W dw W.. The liit distribution is skewed for two reasons. First, W dw is not centered around zero. Second, W dw and W are dependent. This skewness gives rise to the first order bias in the LS/ML estiator, even asyptotically. Under the assuption that the initial value y is, Phillips 987a obtained the asyptotic expansion of the liit distribution of n β : n β = W dw W / nξ W + O pn,.3 where ξ N,, and is independent of W. Taking the expectation of.3 and noting that µ = EZ =.78, we have E β =.78 n + On.. This result is in contrast to the Kendall bias forula for the stationary AR odel i.e. β < : when E β β = β n + On..5 Following the original work of Quenouille, PY 5 utilized the subsaple estiators of β to achieve bias reduction with the following linear cobination: β P Y = β j= β j,.6 where β is the LS/ML estiator of β based on the full saple, i.e., y,..., y n ; βj is the LS/ML estiator of β based on the jth subsaple, i.e., y j l+,..., y jl. There are two iportant features in this estiator. First, the weight given to the full saple estiator is proportional to i.e. ties as large as that assigned to the average of the subsaple estiators. Second, the weights are the sae for all the subsaple estiators. The validity of this jackknife ethod can be explained by a general result based on the Nagar expansion: E β = β + b n + O n,.7

5 which is assued to hold for soe constant b. Two observations can be ade fro this general result. First, to the first order, the bias is a reciprocal function of the saple size. This explains why in the PY ethod, the weight assigned to the full saple estiator is ties as big as that assigned to the subsaple estiators. Second, to the first order, the bias is the sae across different subsaples. This explains why in the PY ethod, the weights are the sae for all the subsaple estiators. It is Y easy to show that the jackknife estiator reoves the first order bias, i.e., E βp = β + O n, when.7 holds true. Particularly effective bias reduction can be achieved by choosing =. In this case, this estiator becoes: β P Y = β β + β..8 Both PY and Chabers have reported evidence to support this ethod for the purpose of bias reduction in different contexts. It should be pointed out that any jackknife procedures have been proposed, including the delete- jackknife of Tukey 958, the delete-d jackknife of Wu 986, and the oving-block jackknife of Chabers. The Nagar approxiation is a general result and ay be verified by Sargan s 976 theore. Given the ild conditions under which Sargan s 976 theore hold, it is perhaps rather surprising that the jackknife ethod fails to reove the first order bias in a unit root odel. This failure was first docuented in CK. The basic arguent of CK is that in.7, b is not a constant any ore in the unit root odel. Instead, it depends on the initial condition. As the initial condition varies across different subsaples, the jackknife cannot eliinate the first order bias ter. To be ore specific, the liit distribution of the noralized subsaples estiator, l β j, is j/ j / W dw/ j/. j / W j/ CK showed that µ j = E j / W dw/ j/ j / W depends on j. To eliinate the first order asyptotic bias, CK proposed the following odified jackknife estiator: where β CK = b CK β j= δ CK β j,.9 b CK j= = µ j j= µ j µ, µ δck =.. j= µ j µ When µ = = µ = µ, b CK = / = b P Y, and δ CK = /, the CK estiator is the sae as the PY estiator. Under odel., CK showed that µ = µ =.783, µ =.389, µ 3 =.9399, µ =.833, etc. That is, the bias becoes saller and saller as we go deeper and deeper into subsapling. Substituting these expected values into the forulae., we can calculate the weights. Table reports the weights when =. We also report the weights of PY for coparison. As µ is closer to zero than µ and µ, a larger weight is assigned to the full saple 5

6 Table : Weights assigned to the full saple and subsaple estiators for alternative jackknife ethods b P Y b CK b CY a CY, a CY, δ P Y δ CK estiator, copared to the PY estiator. The dependence of the bias on the initial condition is a very interesting finding which is explained by a recent result obtained in Phillips and Magdalinos 9, PM hereafter. In PM, the initial condition, y, is assued to be: with y n = k n j= ε j,. k n /n τ, + ].. When τ =, y n is said to be a recent past initialization, which obviously includes y = as a special case. When τ =, +, y n is said to be a distant past initialization. When τ =, y n is said to be an infinite past initialization. PM showed that when τ, +, the liit distribution of the LS/ML estiator of β under the unit root odel is given by: n β W τ + dw W +, τ where W τ + t = W t + τw t, and W is another standard Brownian otion but independent of W. If τ =, the liit distribution is the sae as in.. When τ = +, the liit distribution of the LS/ML estiator of β under the unit root odel is given by: kn n β C, where C stands for a Cauchy variate. The dependence of the liit distribution and hence its first oent on the initial condition are clearly seen fro the above results. The larger the value of τ, the greater is the iportance of the initial condition, and the bigger is the share of τw t in W τ + t. For Model., τ = applied for the first saple and τ = jl/n, for the j + th subsaple. That is, for j =,,...,, we have τ =, /,..., /. As we go deeper and deeper into subsapling, the bigger and bigger is the influence of the initial condition on the liit distribution. Not surprisingly, all the oents, including the ean and the variance, are different for different subsaples. 6

7 . Optial Jackknife It is known that the jackknife estiator of PY increases the variance, when copared to the LS/ML estiator. The sae feature is seen in the estiator of CK. In this paper, we introduce a new jackknife estiator, which can reove the first order bias as well as iniize the variance for a given. The reason why it is possible to reduce the variance of the estiator of CK is that it is not necessary to assign an identical weight to different subsaples, given that the biases of these estiators are different. To fix the idea, the new jackknife estiator is defined as: β CY = b CY β j= a CY j, β j, where b CY and {a CY j, } j= are the weights assigned to the full saple estiator and the subsaple estiators, respectively. Unlike the jackknife estiators of PY and CK, we do not require the weights for different subsaples to be the sae. iniize the variance of the new estiator, i.e.: subject to two constraints: b CY in,{acy j, } j= Our objective is to select the weights b CY, {a CY j, } j=, to V ar b CY = b CY j= b CY µ = β a CY j, β j j=,.3 a CY j, +,. j= a CY j,µ j,.5 where µ = µ. These two constraints are used to ensure the first order bias of the LS/ML estiator is fully reoved. Fro the two constraints, we can express b CY and a CY, as functions of a CY,,..., a CY,: Substituting b CY respect to a CY j,, we get: b CY = a CY µ µ, µ + + µ µ acy, µ +, a CY, = a CY µ µ, µ + + µ µ acy, µ +. and a CY, into the objective function.3, and then taking the derivative with 7

8 = b CY µ µ j +a CY, + for j =,,. i= ] µ Cov β, β Cov β, β j µ V ar β µ µ ] j µ Cov β, β + Cov β, β j + µ µ i µ Cov β, β i + µ µ ] i µ Cov β, β i + Cov β i, β j, µ V ar β µ µ j µ µ j a CY i, a CY,,..., a CY, and hence b CY These first order conditions can be used to obtain analytical expressions for and a CY,, all as functions of µ, µ,..., µ, the variances and the covariances of the full saple and subsaple estiators. To eliinate the first order bias, one ust first obtain µ, µ,..., µ, as CK did. To iniize the variance of the new estiator, one ust also calculate the exact variances and covariances of the finite saple distributions. However, it is known in the literature that the exact oents are analytically very diffi cult to obtain in dynaic odels. To siplify the derivations, we approxiate the oents of the finite saple distributions by those of the liit distributions. We shall check the quality of these approxiations in the siulation studies. While the techniques proposed in White 96 and in CK can be cobined to copute the variances, additional effort is needed to copute the covariances. A technical contribution of the present paper is to show how to copute these covariances. We now illustrate the optial jackknife ethod in a special case where =, that is, we split the full saple into two non-overlapping subsaples. The first subsaple is ade with the observations fro y to y l, with l = n/ and the initial condition, and the reainder belongs to the second subsaple with the initial condition y l. Under the two non-overlapping subsaple schee, we have: β CY = b CY β a CY β + a CY β. The objective function and the constraints are: in b CY,a CY,a CY s.t. b CY V ar β + j= b CY = a CY + a CY +, a CY j b CY µ = a CY µ + a CY µ. V ar βj b CY j= a CY j Cov β, β j + a CY a CY Cov β, β, Fro the two constraints, we express b CY and a CY in ters of a CY, that is: 8

9 b CY = acy µ µ +, µ.6 a CY = acy µ µ +. µ.7 Fro CK, we know that µ = µ =.783, and µ =.389, and hence, we have b CY =.7a CY + and a CY =.779a CY +. The first order condition with respect to a CY gives:.888v ar β.5558v ar β =.336Cov β, β Cov β, β + Cov β, β.9v ar β +.55V ar β + V ar β +.86Cov β, β.888cov β, β.6cov β, β. a CY.8 It is straightforward to check that this is the global iniizer as the objective function is quadratic and convex. To calculate the weights, it is iperative to obtain V ar β, V ar β, V ar β, Cov β, β, Cov β, β, and Cov β, β. Instead of obtaining the variances and the covariances fro the finite saple distributions, we will calculate the fro the corresponding asyptotic distributions. First, the variances can be coputed by cobining the techniques of White 96 and CK. Note that: n V ar β = n E β β ] E = n E W dw/ W + + n = E n + E W dw/ W W dw W µ + o, n E W dw/ W + n + on E W dw/ ] W n + on l V ar β j = E j/ j / W dw j/ j / W µ j + o, j =,. We need to copute the first ter on the right hand side of these two equations. Let Na, b = b a W dw, Da, b = b a W a < b, and M a,b θ, θ denote the joint oent generating function MGF of Na, b and Da, b. White 96 gave the forula for calculating the 9

10 second oent of Na, b/da, b as: E Na, b = Da, b w M a,b θ, θ θ= dθ dw. Following Phillips 987b, CK obtained the expression for M a,b θ, θ : θ M a,b θ, θ = exp θ ] { b a λb a] λ where λ = θ. The following proposition calculates the variances. θ + aθ λ ] } / λb a], Proposition. The second derivative of M a,b with respect to θ, evaluated at θ =, is given by.9 b a H / + H 3/ b ah H ] + 3 H 5/ H := va, b,. where H, H, H, H are given in the Appendix. When a, b] =, ], v, = / λ λ 3/ λ λ + 3 λ 5/ λ λ. This leads to the approxiate variance for the full saple estiator and the first subsaple estiator in the discrete tie unit root odel: n V ar β = l V ar β =.3 + On.. When a, b] = /, ], v/, = λ λ + λ ] / λ λ λ + λ λ ] 5/ λ λ. + λ λ ] 3/ λ This leads to the approxiate variance for the second subsaple estiator in the discrete tie unit root odel: l V ar β = On.. Reark. The variance of the full saple estiator, β, noralized by n, is.3. Interestingly, it is the sae as that of the first subsaple estiator, noralized by l. This equality arises because the full saple has the sae initial value as the first subsaple. Reark.3 The variance of the first subsaple estiator is about twice as large as that of the second subsaple estiator. This diff erence is due to the distinctive initial conditions and ay be ade clearer

11 Table : Variances of subsaple estiators j Variance noralized by l by exaining the liit distribution of l β j, l β j j/ j / W dw j/ j / W..3 In.3, the nuerator plays an insignificant role in the variance calculation since it can be rewritten as a difference between two chi-square variates. The iportant part is in the denoinator, which is a partial su of chi-square variates. For the first subsaple estiator, the denoinator starts with the square of zero. However, the denoinator of the second subsaple begins with the square of a rando variable whose agnitude is O p n. This rando initialization of the second subsaple increases the agnitude of the denoinator and, hence, tends to reduce the absolute value of the ratio. To confir our explanation, we carried out a siple Monte Carlo study, in which data were siulated fro the Brownian otion with the saple length being set at 5, and the nuber of replications set at,. The ean of / W is.3, whereas the ean of / W is.369. Not only is the ean affected, but also the variance and the entire distribution. In this case, the two variances are. and.69, respectively. Reark. Table lists the variances of all the subsaple estiators when =. It can be seen that the variance of the subsaple estiator decreases as j increases. The largest diff erence occurs between j = and j =. If is allowed to go to infinity, the liit distribution of the jackknife estiator will be the sae as that of the LS estiator, as pointed out by CK.

12 Second, to calculate the covariances, we note that: n Cov β, β j = E W dw W j/ j / W dw j/ j / W µµ j + On, j =,, / n Cov β, β W dw = E / W dw / W / W µ µ + On. It is required to obtain the expected value of the cross product of rando variables. As for the variances, we also calculate these expected values fro the joint MGF given in the following lea. Lea.5 Let M a,b,c,d θ, θ, ϕ, ϕ denote the MGF of Na, b, Nc, d, Da, b and Dc, d with a < b and c < d. Then the expectation of Na,b Nc,d Da,b Dc,d E Na, b Nc, d = Da, b Dc, d is given by: M a,b,c,d θ, θ, ϕ, ϕ θ ϕ θ=,ϕ = dθ dϕ.. The following proposition obtains the expression for the MGF of Na, b, Nc, d, Da, b and Dc, d, and the covariances when =. Proposition.6 The MGF M,,,/ θ, θ, ϕ, ϕ is given by { λ,,,/ exp θ ϕ ; θ λ,,,/ λ,,,/ the MGF M,,/, θ, θ, ϕ, ϕ is given by: { λ,,/, exp θ ϕ ; θ + ϕ λ,,/, λ,,/, and the MGF M,/,/, θ, θ, ϕ, ϕ is given by: { λ,/,/, exp θ ϕ, ϕ λ,/,/, λ,/,/, ] ] ] η,,,/ η,,/, π,,,/ η,,,/ π,,/, η,,/, η,/,/, η,,,/ η,,/, π,/,/, η,/,/, ]} / ]} / η,/,/, where λ a,b,cd, η a,b,c,d and π a,b,c,d are given in the Appendix. These MGFs, together with forula., ]} /

13 Table 3: Approxiate values for the variances and covariances for the full saple and subsaples when = Variance n V ar β l V ar β l V ar β Covariance n Cov β, β n Cov β, β n Covβ, β lead to the approxiate covariances between the full saple estiator and the two subsaple estiators: n Cov β, β = On ;.5 n Cov β, β = On ;.6 n Cov β, β =. + On..7 Reark.7 There are several interesting findings in Proposition.6. First, the covariances between the full saple estiator and the second subsaple estiator are siilar to, but slightly larger than that between the full saple estiator and the first subsaple estiator, although the variance of the second subsaple estiator is saller. This is because the correlation between the full saple estiator and the second subsaple estiator is larger due to the increased order of agnitude of the initial condition. Second, these two covariances are uch larger than the covariance between the two subsaple estiators. This is not surprising as the data used in the two subsaples estiators do not overlap. Reark.8 Table 3 suarizes the approxiate values of the variances and covariances when =. Theore.9 The optial weights for the jackknife estiator of β in the discrete tie unit root odel when = are b CY =.839, a CY =.677, and a CY =.69. Hence, the optial jackknife estiator is β CY JK =.839 β.677 β +.69 β..8 Reark. The optial jackknife estiator copares interestingly to the estiator of CK: β CK JK =.565 β.785 β β..9 3

14 Relative to the CK estiator, our estiator gives a larger weight to the full saple estiator, and ore so, to the second subsaple estiator. The weight for the second subsaple is nearly twice as uch as that for the first subsaple. Since the variance of the second subsaple estiator is nearly half as that of the first saple estiator, not surprisingly, the variance of our estiator is saller than that of CK s. 3 Optial Jackknife for the Continuous Tie Unit Root Model In the section, we extend the result to a continuous tie unit root odel. In the continuous tie stationary odels, it is well known that the estiation bias of the ean reversion paraeter depends on the span of the data, but not the nuber of observations. In epirically realistic situations, the tie span easured in nuber of years is often sall. Consequently, the bias can be uch ore substantial than that in the discrete tie odels; see for exaple, PY 5 and Yu. Considering the following Vasicek odel with y = : dy t = κydt + σdw t. 3. The paraeter of interest here is κ that captures the persistence of the process. When κ >, the process is stationary and κ deterines the speed of ean reversion. The observed data are assued to be recorded discretely at h, h,, nh= T in the tie interval, T ]. So h is the saple interval, n is the total nuber of observations and T is the tie span. In this case, Yu showed that the bias of the ML estiator of κ is: E κ κ = 3 + e κh + ot. 3. T It is clear fro 3. that the bias depends on T, κ, and h. When κ is close to, or h is close to, which is epirically realistic, the bias is about /T. When T is not too big, this bias is very big, relative to the true value of κ. The result can be extended to the Vasicek odel with an unknown ean and to the square root odel; see Tang and Chen 9. The large bias otivated PY 5 to use the jackknife ethod. When κ =, the odel has a unit root, and the exact discrete tie representation is a rando walk. In this case, it can be shown that the bias of the ML estiator of κ is:.78 T + ot. 3.3a The bias forula for κ is siilar to that for β in.. However, the direction of the bias is opposite and the bias depends on T, not n. When T is not big, this bias is very big relative to the true value

15 of κ. If T, the so-called long span liit distribution of κ is given by: T κ W dw W. See, for exaple, Phillips 987b and Zhou and Yu. Siilarly, the long span liit distributions of the sub-saple estiators are: T κ j j/ j / W dw T, j =,...,. j/ W j / Obviously, the only difference between these liit distributions and those in the discrete tie unit root odel is the inus sign. Hence, the variances and covariances of the two sets of liit distribution are the sae but the expected values of the change the sign. Of course, the rate of convergence changes fro n to T. Consequently, we have the following theore. Theore 3. When =, the approxiate variance for the full saple estiator and the first subsaple estiator in the continuous tie unit root odel is: T V ar κ = T V ar κ =.3 + OT. 3. Siilarly, the approxiate variance for the second subsaple estiator in the continuous tie unit root odel is: T V ar κ = OT. 3.5 The approxiate covariances between the full saple estiator and the two subsaple estiators are: T Cov κ, κ = OT ; 3.6 T Cov κ, κ = OT ; 3.7 T Cov κ, κ =. + OT. 3.8 The optial jackknife estiator of κ in the continuous tie unit root odel when = is: κ CY =.839 κ.677 κ +.69 κ. 3.9 Reark 3. The optial jackknife estiator of κ in the continuous tie unit root odel has the sae weights and the expression as that of β in the discrete tie unit root odel. This is not surprising because there are only two differences in the liit theory, the sign of the bias and the rate of convergence. These differences do not have any ipact on the weights as they are canceled out. 5

16 Reark 3.3 The effect of the initial condition on the bias in the continuous tie odel was recently pointed out in Yu. When the initial value is in Model 3. with κ >, Yu showed that the bias of the ML estiator of κ is: E κ κ = 3 + e κh + ot. 3. T However, when the initial value is N, σ /κ in Model 3. with κ, the bias of the ML estiator of κ is: Monte Carlo Studies E κ κ = 3 + e κh e nκh T T n e κh + ot. 3. In this section, we check the finite saple properties of the proposed jackknife against the CK jackknife using siulated data. This is iportant for two reasons. First, it is iportant to easure the effi - ciency gain of the proposed ethod. Second, the weights are obtained based on the variances and the covariances of the liit distributions but not based on the finite saple distributions. These approxiations ay or ay not have ipacts on the optial weights. Hence, it is inforative to exaine the iportance of the approxiation error. Although it is diffi cult to obtain the analytical expressions for the variances and the covariances of the finite saple distribution, we can copute the fro siulated data in a Monte Carlo study, provided the nuber of replications is large enough. In this paper, we always set the nuber of replications at 5,.. Discrete Tie Unit Root Model First, we siulate data fro the following discrete tie unit root odel with initial value y = : y t = y t + ε t, ε t iid N,, t =,,..., n. We evaluate the perforance of alternative jackknife ethods by applying the to five different saple sizes, i.e. n =,, 8, 96, 8. In particular, we copare three ethods when =, the CK jackknife ethod based on.9, the CY jackknife ethod based on.8 where the weights are derived fro the approxiate variances and covariances, the CY jackknife ethod where the weights are calculated fro the exact variances and covariances obtained fro the finite saple distributions. The results are reported in Table, where for each case we calculate the ean, the variance and the RMSE for the estiates of β, all across 5, saple paths. In addition, we calculate the ratio of the variances and the RMSEs for the CK estiates and the proposed CY estiates to easure the effi ciency loss in using the CK estiate. 6

17 Table : Finite saple perforance of alternative jackknife estiators for the discrete tie unit root odel when = n Statistics CK CY Effi ciency of Exact CY CK relative to CY Bias Var* RMSE* Bias Var* RMSE* Bias Var* RMSE* Bias Var* RMSE* Bias Var* RMSE* Several interesting results eerge fro the table. First, the bias in the estiate of β for the CK ethod and the CY ethod is very siilar in every case. Second, the variance in the estiate of β for the CY ethod is significantly saller than that for the CK ethod in each case. The reduction in the variance is over % in all cases. Consequently, the RMSE is saller for the CY ethod. Third, although the exact CY ethod provides a saller variance, the difference between the two CY ethods is so sall, suggesting that the proposed CY works well, and it is not necessary to bear the additional coputational cost associated with calculating the variances and covariances fro the finite saple distributions. It is worth pointing out that even when n =, a very sall saple size that could have bigger iplications for the approxiation error, the difference between the two CY ethods is still negligible in ters of the bias, variance and RMSE. To understand why the two CY estiates are so siilar, Table 5 reports the weights obtained fro the finite saple distributions. For the purpose of coparison, we also report the weights obtained fro the liit distribution. It can be clearly seen that when n, all the weights converge. When n is as sall as, the weights are not very different fro those obtained when n is infinite.. Continuous Tie Unit Root Model Second, we siulate data fro the following continuous tie unit root odel with initial value y = : dy t = κydt + dw t, 7

18 Table 5: Jackknife weights based on the finite saple distributions and the liit distribution for the discrete tie unit root odel. n b CY a CY a CY Table 6: Finite saple perforance of alternative jackknife estiators for the continuous tie unit root odel when =. n h T Statistics CK CY Effi ciency of CK Exact CY relative to CY /5 Bias Var RMSE /5 5 Bias Var RMSE /5 Bias Var RMSE /5 5 Bias Var RMSE with κ =. Here, we generate data based on its exact discrete for as y t+h = y t +ε t, with ε t N, h. As for the discrete tie odel, we eploy the three jackknife ethod to estiate κ. Table 6 shows the results based on two sapling intervals h = /5, /5, corresponding to the weekly and daily frequency. The tie span, T, is set at years and 5 years. These settings are epirically realistic for odeling interest rates and volatility. Several interesting results also eerge fro the table. First. the bias in the estiate of κ for the CK ethod and the CY ethod is very siilar in each case. Second, the variance in the estiate of κ for the CY ethod is always significantly saller than those for the CK ethod. The reduction in the variance is at least %. Consequently, the RMSE is saller for the CY ethod. Third, the difference between the two CY ethods is very sall, suggesting that the proposed CY works well and it is not necessary to bear the additional coputational cost associated with calculating the variances and covariances fro the finite saple distributions. 8

19 Table 7: Weights based on the finite saple distributions and the liit distribution n h T b CY a CY a CY / / / / fixed To understand why the two CY estiates are so siilar in the continuous tie odel, Table 7 reports the weights obtained fro the finite saple distributions. For the purpose of coparison, we also report the weights obtained fro the liit distribution. It can be clearly seen, as T but not n increases, all the weights get close to those obtained fro the liit distribution. When T is as sall as, the weights are not different fro those obtained when T is infinite. 5 Conclusion This paper has introduced a new jackknife procedure for unit root odels that offers iproveent over the jackknife ethodology of CK. The proposed estiator is optial in the sense that it iniizes the variance of the jackknife estiator while aintaining the desirable property of being able to reove the first order bias. The new ethod is applicable to both discrete tie and continuous tie unit root odels. Siulation studies have shown that this new ethod reduces the variance by ore than % relative to the estiator of CK without coproising the bias. Although it is not pursued in the present paper, it ay be useful to further iprove our ethod by using alternative values for, and hence, further reduce the variance and RMSE. Furtherore, there are other odels where the asyptotic theory is critically dependent on the initial condition. Exaples would include near unit root odels and explosive processes. It ay be interesting to extend the results in the present paper to these odels. We plan to report these results in our later work. A Proof of Proposition. APPENDIX The notations in the proposition are defined as follows. H = Hθ = λb a] λ θ + aθ λ ] λb a]; 9

20 H = H, i.e.: H = λb a] + aλ λb a]; H denotes the first derivative of Hθ with respect to θ, evaluated at θ =, i.e.: H = H θ== λb a]. θ λ Siilarly, H denotes the second derivative of Hθ with respect to θ, evaluated at θ =, i.e.: H = H θ θ== a λ λb a]. The first derivative of M a,b with respect to θ can be written as: M a,b = b a exp θ ] b a H / θ exp θ ] b a 3/ H H. θ The second derivative of M a,b with respect to θ is: M a,b θ = b a exp θ b a + 3 exp θ b a ] H / + exp θ b a ] H 5/ H θ ] H 3/ b a H ] H θ θ Setting θ = and by standard nuerical integration techniques, we can obtain the results in Proposition.. B Proof of Lea.5 Taking the derivative of MGF with respect to θ, we get: M a,b,c,d θ, θ, ϕ, ϕ θ = E Na, b expθ Na, b θ Da, b + ϕ Nc, d ϕ Dc, d]. Setting θ =, taking the derivative with respect to ϕ, and then evaluating it at ϕ =, we have, { ] } Ma,b,c,d θ, θ, ϕ, ϕ θ= / ϕ θ ϕ == E {Na, bnc, d exp θ Da, b ϕ Dc, d]}.

21 Consequently, = = E = E { ] } Ma,b,c,d θ, θ, ϕ, ϕ θ= / ϕ θ ϕ = dθ dϕ E {Na, bnc, d exp θ Da, b ϕ Dc, d]} dθ dϕ Nc, d exp ϕ Dc, d]dϕ Na, b exp θ Da, b]dθ ] Na, b Nc, d. Da, b Dc, d C Proof of Proposition.6 To prove this proposition, we follow CK. STEP : Deriving the MGF for E W dw / W dw W / W Let Xt and Y t t, ] be the OU process defined by. dxt = γxtdt + dw t, X =, dy t = λy tdt + dw t, Y =. The easures induced by X and Y are denoted as µ X and µ Y, respectively. By Girsanov s Theore, dµ X s = exp γ λ dµ Y stdst γ λ ] st dt, where the left side is the Radon-Nikody derivative evaluated at st. Let the MGF of E W dw / W dw be M W,,,/ θ, θ, ϕ, ϕ so that M,,,/ θ, θ, ϕ, ϕ = E / W exp / ] / θ W dw + θ W + ϕ W dw + ϕ W. With the change of easure, we have the following forula by setting γ = : M,,,/ θ, θ, ϕ, ϕ / / = E exp θ Y dy + θ Y + ϕ Y dy + ϕ Y λ Y dy + λ Y ].

22 By Ito s calculus, we get: M,,,/ θ, θ, ϕ, ϕ = exp θ ϕ + λ { θ λ E exp = exp θ ϕ + λ Y + ϕ { E exp / Y + ϕ + θ + λ Y + θ λ Y + ϕ ]} / Y + ϕ Y ]} θ + λ Y / where λ = θ. To calculate the expectation, we first take the expectation of M,,,/ θ, θ, ϕ, ϕ conditional on Ϝ /, which is the siga field generated by W on, ], and then change the easure by Girsanov s Theore with introducing another OU process. Note that, M,,,/ θ, θ, ϕ, ϕ, Ϝ / = E = exp θ ϕ + λ ϕ exp / Y + ϕ M,,,/ θ, θ, ϕ, ϕ Ϝ / ] { Y E exp ] θ λ Y Y ]}. The MGF of the noncentral χ distribution, θ λ Y, is θ λϖ ] / exp θ λ ζy ] with ϖ = expλ λ and ζ = θ λϖ ] expλ. Therefore, M,,,/ θ, θ, ϕ, ϕ, Ϝ / = θ λϖ ] / exp θ ϕ + λ { θ λ E exp ζ + ϕ ] Y } / + ϕ Y. Now we introduce another process Zt on, ], given by dzt = ηztdt + dw t, Z =. By changing the easure and setting ϕ = λ η / to cancel out / Z, we have { θ λ E exp ζ + ϕ ] } / Y + ϕ Y { θ λ = E exp ζ + ϕ ] } / Z + λ η ZdZ ] η λ = exp E exp τz,

23 with τ = θ λ ζ + ϕ + λ η. Considering Z N, ϖ z where ϖ z = expη η, we have E exp { θ λ ζ + ϕ ] Y Finally by cobining the equations above, we obtain M,,,/ θ, θ, ϕ, ϕ = Note that, λ θ λϖ ] exp and that η τϖ z] exp = exp λ = exp η where π = θ λζ + ϕ + λ. Thus, } / + ϕ Y = τϖ z] / exp η λ { } / η + λ θ λϖ ] τϖ z] exp exp θ ϕ. + exp λ θ λ + exp η π η exp λ exp η exp λ =. λ θ λ exp η η = π η η, M,,,/ θ, θ, ϕ, ϕ = H,,,/ θ, θ, ϕ, ϕ / exp θ ϕ, λ, with H,,,/ θ, θ, ϕ, ϕ = λ θ λ ] λ η π η ] η. ] Ma,b,c,d θ STEP : Copute, θ,ϕ, ϕ θ θ= / ϕ ϕ =. Denote H,,,/ θ, θ, ϕ, ϕ and M,,,/ θ, θ, ϕ, ϕ by H and M. Note that M = H / exp θ ϕ where λ = θ and η = θ + ϕ. Taking the partial derivative with respect to θ and evaluating it at θ =, we have, M θ== θ exp ϕ H / exp ϕ H 3/ H θ=. θ Then taking the second derivative with respect to ϕ and evaluating it at ϕ =, we obtain ] M θ θ= ϕ = ϕ = 8 H / + H 3/ H ϕ = + H θ=,ϕ ϕ 8 θ = + 3 H H H 5/ ϕ = θ=,ϕ ϕ θ =, H θ θ= ϕ = ϕ 3

24 fro which we require the following four expressions H = H θ=,ϕ == H ϕ H θ θ=,ϕ == λ λ η + λ η λ η ; ϕ == η λ λ η ; η η η H θ θ= ϕ == λ η ϕ λη. λ ; Finally, we have the following expression that is aenable for nuerical integrations. ] M θ θ= ϕ = ϕ = 8 H / + H 3/ + 3 H 5/ { η η λ 8λ λ η ] λ λ 3 η 8η λ + η η η λ λη ] λ. η } By the sae arguent, we have the MGFs of M,,/, θ, θ, ϕ, ϕ and M,/,/, θ, θ, ϕ, ϕ. To distinguish λ, η and π in the three cases, we use subscripts. To su up: The MGF M,,,/ θ, θ, ϕ, ϕ is given by { λ,,,/ exp θ ϕ, θ λ,,,/ λ,,,/ ] η,,,/ π,,,/ η ]} / η,,,/ where λ,,,/ = θ, η,,,/ = θ ϕ, and π,,,/ = θ λ,,,/ ζ,,,/ + ϕ + λ,,,/ with ζ,,,/ = exp λ,,,/ θ λ,,,/ ϖ ], ϖ = exp,,,/,,,/ τ,,,/ = θ λ,,,/ ζ,,,/ + ϕ + λ,,,/ η,,,/, and ϖ z,,,,/ = exp The MGF M,,/, θ, θ, ϕ, ϕ is given by { λ,,/, exp θ ϕ, θ + ϕ λ,,/, λ,,/, ] η,,/, η,,,/ η.,,,/ π,,/, η,,/, λ,,,/ λ,,,,/ η,,/, ]} /

25 where λ,,/, = θ ϕ, η = θ and π,,/,,,/, = θ + ϕ λ,,/, ζ ] exp λ,,/, λ,,/, with ζ =,,/, θ + ϕ λ,,/, ϖ,,/, τ,,/, = θ+ϕ λ,,/, ζ ϕ,,/, + λ η,,/,,,/,, and ϖ z, = exp,,/, The MGF M,/,/, θ, θ, ϕ, ϕ is given by { λ,/,/, exp θ ϕ where ϖ, λ = exp,/,/, ϕ λ,/,/, λ,/,/, λ,/,/, ζ,/,/, +θ ϕ +λ,/,/, with ζ,/,/, = ] η,/,/,, ϖ,,/,,,/, = exp η,,/, η.,,/, π,/,/, η,/,/, ϕ + λ,,/, λ,,,/, η,/,/, λ, λ,/,/, = ϕ, η,/,/, = θ, and π,/,/, = ϕ,/,/, θ λ,/,/, ϖ,/,/,], expλ,/,/, τ,/,/, = ϕ λ,/,/, ζ,/,/, + θ ϕ + λ,/,/, η,/,/,, and ϖ z = expη,/,/, η,/,/,. The coputation of the second derivative of the MGF siply follows step. For exaple, the second derivative of MGF M,,,/ θ, θ, ϕ, ϕ is M,,,/θ, θ,ϕ, ϕ θ θ= ϕ = ϕ = 8 H /,,,,/ + H 3/,,,,/ { 3 8η,,,/ + 3 H 5/,,,,/ λ,,,/ η,,,/ η,,,/ λ,,,/ 8λ,,,/ λ,,,/ η,,,/ λ,,,/ and H,,,,/ = θ and η = θ + ϕ,,,/. η,,,/ η,,,/ λ,,,/ η,,,/ λ,,,/ η,,,/ ] λ,,,/ η,,,/ + λ,,,/ η,,,/ η,,,/ η,,,/ λ,,,/ λ,,,/ λ,,,/ η,,,/ ]. } with λ,,,/ = ]} / 5

26 The second derivative of MGF M,,/, θ, θ, ϕ, ϕ is M,,/,θ, θ,ϕ, ϕ θ θ= ϕ = ϕ = 8 H /,,,/, + H 3/,,,/, { 3 8η,,/, + 3 H 5/,,,/, λ,,/, η,,/, λ,,/, λ,,/, λ,,,/ 8λ,,/, λ,,/, λ,,/, and H,,,,/ = θ + ϕ and η =,,,/ θ. λ,,/, + η,,/, η,,,/ η,,/, λ,,/, η,,/, ] η,,/, η,,/, + λ,,,/ η,,,/ λ,,/, η,,/, η ] λ,,/,,,/,, η,,,/ The second derivative of MGF M,/,/, θ, θ, ϕ, ϕ is ] M,/,/, θ, θ, ϕ, ϕ ϕ = ϕ = 6 H /,,/,/, + H 3/,,/,/, { 8η,/,/, + 3 H 5/,,/,/, η,/,/, and H,,/,/, = η,/,/, λ,/,/, η,/,/, λ,/,/, 8λ,/,/, λ,/,/, λ,/,/, with λ,/,/, = ϕ and η,/,/, = θ. λ,/,/, λ,/,/, η,/,/, ], λ,/,/, η,/,/, ] η,/,/, + λ,/,/, η,/,/, η,/,/, λ,,,/ η,/,/, λ,/,/, } with λ,,,/ = λ,/,/, η,/,/, } D Proof of Theore.9 Using the results fro Propositions. and.6, and substituting the values in Equation.8, we have the stated result. References 6

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