Estimation of Average Local-to-Unity Roots in Heterogenous Panels

Estimation of Average Local-to-Unity Roots K.7 in Heterogenous Panels Erik Hjalmarsson International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 852 December 2005

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 852 December 2005 Estimation of Average Local-to-Unity Roots in Heterogenous Panels Erik Hjalmarsson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

Estimation of Average Local-to-Unity Roots in Heterogenous Panels ∗ Erik Hjalmarsson † Division of International Finance Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA December 2005 Abstract Thispaperconsiderstheestimationofaverageautoregressiveroots-near-unityinpanelswhere the time-series have heterogenous local-to-unity parameters. The pooled estimator is shown to have a potentially severe bias and a robust median based procedure is proposed instead. This median estimatorhasasmallasymptoticbiasthatcanbeeliminatedalmostcompletelybyabias correctionprocedure. Theasymptoticnormalityoftheestimatorisproved. Themethodsproposed in the paper provide a useful way of summarizing the persistence in a panel data set, as well as a complement to more traditional panel unit root tests. JEL classification: C22, C23. Keywords: Local-to-unity; Panel data; Pooled regression; Median estimation; Bias correction. IamgratefultoPeterPhillipsandDonAndrewsforprovidingmuchusefuladvice. Otherhelpfulcommentshavealso ∗ been provided by Lennart Hjalmarsson, Randi Hjalmarsson, Catalin Starica, and Jonathan Wright as well as seminar participants at the European meeting of the Econometric Society in Madrid, 2004, and the econometrics seminar at Göteborg University. Tel.: +1-202-452-2436; fax: +1-202-263-4850; email: erik.hjalmarsson@frb.gov. The views presented in this paper † are solely those oftheauthorand donotrepresentthose ofthe FederalReserve Board orits staff.

1 Introduction Few concepts have had such an impact on recent econometric practice as unit roots. The modern asymptotic theory developed for integrated processes clearly shows that a failure to account for the order of integration of the data can lead to flawed inference. However, many economic time-series exhibit a nearly persistent behavior with the largest auto-regressive root close to one, which often makes it difficult to distinguish between stationary and non-stationary series in practice. This has led to the increasing popularity of so called nearly integrated processes as a modelling device; rather than maintaining a strict dichotomy between integrated and non-integrated time-series, the largest auto-regressive root is treated as being local-to-unity which allows for a smoother transition between the stationary and non-stationary worlds. Originally, nearly intergrated processes were mainly used for theoretical excercises, such as evaluating the local power properties of unit-root tests (e.g. Phillips and Perron, 1988, and Elliot et al., 1996). Lately, however, they have also become increasingly popular in practical inference (e.g. Cavanagh et al., 1995, and Campbell and Yogo, 2003). Although the generalization from a standard unit-root environment to a near integrated environment provides more flexibility, it suffers from the drawback that the key characterstic parameter of such a model, the local-to-unity parameter, cannot be estimated in a time-series setting.1 However, as shown in a series of papers by Moon and Phillips (1999, 2000, and 2004), the local-to-unity parameter can be estimated using a panel of observations, when all of the time-series have identical local-to-unity parameters. In practice, the assumption that all of the time-series in the panel have an identical degree of persistence is obviously very restrictive. In this paper I, therefore analyze the estimation of local-to-unity roots in panels where the degree of persistence varies between the time-series. Thepurposeofthispaperistwofold. First,Iconsiderthepropertiesofthepooledestimatoroflocalto-unityparametersproposedbyMoonandPhillips(2000)inthecasewheretheindividualtime-series possess differing degrees of persistence. Second, I propose a new estimator for the average local-tounity root in a panel, based on applying the median operator to extract the crucial cross-sectional information in the panel. When there is no longer a common local-to-unity parameter in a panel, a desirable property of a panelbasedestimatorwouldbethatitconsistentlyestimatesthemean,oraverage,parametervaluein 1Phillips et al. (1998) do provide a method of estimating local-to-unity roots from a single time-series using a block model. However,their specification of the local-to-unity model is somewhat different from the one typically adopted in theliterature. 1

the panel. As is shown, however, the pooled estimator of Moon and Phillips (2000) can be a severely biased estimator of the average parameter, even for relatively modest deviations from the case of identical local-to-unity roots. Thebasicideaofthepooledestimatoristhataconsistentestimatorcanbeobtainedbytakingthe inconsistentOLStime-seriesestimatorofthelocal-to-unityparameterandsummingupoverthecrosssection in both the numerator and the denominator. Since this method fails when the local-to-unity parameters are no longer identical, I propose a more robust approach by applying the sample median estimator, rather than the sample mean, in both the numerator and denominator of the time-series estimator. Thebiasandconsistencypropertiesoftheresultingestimatorcannotbeanalyticallyevaluated,but results based on numerical integration are straightforward to obtain. After a simple bias correction, theestimatorisshowntobeconsistentinthecasewithidenticallocal-to-unityparametersinthepanel. More importantly, under the additional assumption that the local-to-unity parameters are normally distributed, it is shown that the estimator converges to a quantity that is very close to the average local-to-unityparameter,regardlessofthevarianceinthedistributionofthelocal-to-unityparameters. That is, in the case of identical near unit-roots in the panel, the estimator is consistent and it is very close to consistent in the case of non-identical roots. The bias in the non-identical case is small, and likely to be negligible compared to the variance of the estimates in any finite sample. The asymptotic normality of the estimator is also shown, as well as the estimation of standard errors and confidence intervals. Monte Carlo simulations support these results and also indicate that the estimator works well in cases where the local-to-unity parameters are not normally distributed. The results developed in this paper are useful along several dimensions. First, they highlight the potential hazards of applying estimators of near-unit roots designed for the case of identical local-tounity roots throughout the panel, when there is in fact a possibility that the roots are non-identical. Second, it is shown how to estimate the average near unit-root in a panel data set. This can be useful bothasacharacterizationofthedatainitself,aswellasastartingpointforfurtherempiricalanalysis. Italsoprovidesacomplementtopanelunit-roottests,whichhaverecentlybecomeverypopular.2 The methods in this paper provide a simple diagnostic addition to these tests by estimating the average auto-regressiverootinthepanel. Sinceconfidenceintervalsforthisaveragerootcanalsobecomputed, further conclusions can also beobtained. For instance, a confidence interval that is strictlybelow zero 2See for instance Quah (1994),Maddala and Wu (1999),Choi(2001),Levin et al. (2002),Moon and Perron (2003), and Moon etal. (2003). 2

revealsthattheaveragerootissignificantlylessthanzero;hence,someoftheactualrootsinthepanel must also be negative. The rest of the paper is organized as follows. Section 2 details the setup and main assumptions and Section 3 derives the bias properties of the pooled estimator. The main results of the paper are developedinSection4,wheretheasymptoticpropertiesofthemedianbasedestimatorarederived,and Section 5 concludes. All proofs and details of the numerical calculations are found in the Appendix. A word on notation, denotes weak convergence of the associated probability measures and p ⇒ → denotes convergence in probability. I write (n,T ) when n and T go to infinity simultaneously →∞ and(T,n ) whenT goestoinfinityfirstwhilekeepingnfixed,andthenlettingngotoinfinity. →∞ seq 2 Model and assumptions Let the data generating process for each individual time series, z , satisfy i,t z = β +y , i=1,...,n; t=1,...,T, (1) i,t i,0 i,t c c+η y = a y +(cid:18) , a =1+ i =1+ i, i,t i i,t − 1 i,t i T T where y is a near integrated process with local-to-unity parameter c =c+η . The focus of interest i,t i i in this paper is the estimation of the average, or mean, local-to-unity parameter c. The following assumptions on the error processes and the local to unity parameters, c , will be i useful. Assumption 1 (cid:18) are linear processes satisfying i,t (a) (cid:18) i,t =D i (L)u i,t = ∞j=0 D i,j u i,t − j , ∞j=0 jb | D i,j | < ∞ for some b ≥ 1, | D i (1) |6 =0. (b) u are iid across iPand over t with EP(u )=0, E u2 =1, and finite fourth order moments. i,t i,t i,t ¡ ¢ LetD i =D i (1),Ω i =D i 2,andΛ i = ∞j=1 D i,0 D i,j ,sothatΩ i andΛ i specifythelong-runvariance and the one-sided long-run covariance mPatrix, respectively, of (cid:18) . i,t Assumption 2 The random variables c , i = 1,...,n are normally distributed with mean c and varii ance σ2. c Under Assumption 1, it is well known that as T , y √T D J (r), where J (r)= →∞ i,[Tr] ⇒ i i,ci i,ci 0 r e(r − s)cidW i (s) and W i (r) is a standard Brownian motion (e.g±. Phillips, 1987). R 3

3 The bias properties of the pooled estimator Ifirstshowthatthepooledestimatorofcdoesnotworkwellwhenthec arenon-identicalforalli. To i keepthediscussionas transparent aspossible, considerthesimplecasewhere(cid:18) areiid 0,σ2 across i,t (cid:18) i and t; the arguments presented could easily be modified to account for the general erro¡r proc¢esses in Assumption 1. Also, to keep the discussion short, only sequential limit arguments are presented. Noting that a i =1+ c T i =1+ c+ T η i =a+ η T i, I consider estimators of the form cˆ=T (aˆ − 1). The pooled estimator of a is given by, n T z z aˆ= i=1 t=1 i,t − 1 i,t , (2) n T z2 P i=P1 t=1 i,t 1 − and the corresponding pooled estimator of cPis cˆ=PT(aˆ 1). By equation (2), − 1 1 n 1 T − 2 T (aˆ a) = β +y − " n T2 i,0 i,t − 1 # i=1 t=1 X X¡ ¢ n T 1 1 η β +y (1 a)β +(cid:18) + iy × " n T i,0 i,t − 1 − i,0 i,t T i,t − 1 # X i=1 X t=1h¡ ¢³ ´i 1 n 1 − 1 J (r)2dr ⇒ " n i=1Z0 i,ci # X 1 n 1 1 n 1 J (r)dW (r)+ η J (r)2dr , (3) n i,ci i n i i,ci " i=1Z0 i=1 Z0 # X X as T . As shown below, both E 1 J (r)2dr and E η 1 J (r)2dr are finite when the → ∞ 0 i,ci i 0 i,ci c s are normally distributed, and clear h lyRE 1 J (r i )dW (r) h =R0. Thus, by t i he weak law of large i 0 i,ci i h i numbers (WLLN), as n , R →∞ 1 n 1 1 J (r)2dr E J (r)2dr , (4) n i,ci → p i,ci i=1Z0 ∙Z0 ¸ X 1 n 1 J (r)dW (r) 0, (5) n i,ci i → p i=1Z0 X and 1 n 1 1 η J (r)2dr E η J (r)2dr . (6) n i i,ci → p i i,ci i=1 Z0 ∙ Z0 ¸ X 4

Combining these results, as (T,n ) →∞ seq 1 1 1 T (aˆ a) E J (r)2dr − E η J (r)2dr . (7) − → p i,ci i i,ci ∙Z0 ¸ ∙ Z0 ¸ Under the assumption of normally distributed η s, the two expectations in (7) can be calculated i more explicitly. Using the properties of conditional expectations and the moment generating function (mgf) of the normal distribution, M ci (t)=E[etci]=ect+σ2 c t2/2, 1 E J (r)2dr i,ci ∙Z0 ¸ 1 r = E e2(r − s)ci dsdr Z0 1 Z0 h i = 8σ3 2σ c 1 − e2(c+σ2 c ) c h ³ ´ c2 1 c 1 c+2σ2 − √2πe−2σ2 c c+2σ2 c i Φ i √2σ − i Φ i √2σ µ µ c¶ µ ¶¶¸ ¡ ¢ Ψ (c,σ ). (8) 1 c ≡ where Φ(x)= 2 x e t2dt denotes the error function and i=√ 1. Further, √π 0 − − R 1 E η J (r)2dr i i,ci ∙ Z0 ¸ 1 r = E c i e2(r − s)ci cE e2(r − s)ci dsdr − Z0 1 Z0 ³ h i h i´ = 8σ3 − 2σ c c − ce2(c+σ2 c ) +2σ2 c c h ³ ´ c2 1 c 1 c+2σ2 +√2πe−2σ2 c c2 − σ2 c +2cσ2 c i Φ i √2σ − i Φ i √2σ µ µ ¶ µ ¶¶¸ ¡ ¢ Ψ (c,σ ). (9) 2 c ≡ Thus, in sequential limits, as (T,n ) , →∞ seq cˆ c=T(aˆ a) Ψ 1(c,σ )Ψ (c,σ ), (10) − − → p −1 c 2 c and the pooled estimator, cˆ, provides inconsistent estimates of c. Define the function f(c,σ ) as the c limit of cˆ, f(c,σ ) c+Ψ 1(c,σ )Ψ (c,σ ). (11) c ≡ −1 c 2 c 5

Panel A in Table 1 gives the numerical values of the functionf(c,σ ) for various combinations c of c and σ . It is readily apparent that the asymptotic bias of the pooled estimator for c is already c large for fairly small values of σ and grows very large as σ increases. Panel B in Table 1 shows c c the mean values of cˆ from a Monte Carlo simulation with n = 100 and T = 1,000 using 10,000 repetitions. Thedistributionofthec sisnormalandtheinnovationprocesses,(cid:18) ,arealsoiidnormal i i,t with σ2 = 1. Figure 1 shows the density estimates of cˆ, from the same simulation exercise, for the (cid:18) cases where c = 10,0,5 and σ = 0,5,10. The graphs clearly illustrate that the pooled estimator c − performs excellently for σ = 0, but that its density starts drifting to the right as σ increases. This c c is by any measure a large panel, and the c s are drawn from a normal distribution, as was assumed i when deriving the asymptotic limit function f(c,σ ). c BycomparingthevaluesinPanelAandPanelBinTable1,itisobviousthattheasymptoticlimit of cˆ, givenbyf(c,σ ), provides a verypoorapproximation infinite samples assoonas thevarianceof c the local-to-unity parameter starts to increase; the size of the sample in the Monte Carlo simulation was chosen to illustrate that this remains true also in very large samples. Two conclusions are thus immediate. First, the asymptotic limit of cˆ given by f(c,σ ) cannot be used as a basis for a bias c correcting procedure of cˆsince it does not provide a good approximation in finite samples. Of course, even if f(c,σ ) did provide a good approximation, any bias correction scheme based on it would be c complicated by the fact that σ is unknown. Second, the pooled estimator works very poorly as soon c as there is any variance, or heterogeneity, in the c s. Thus, applying the pooled estimator for c to i a panel, without any strong prior evidence or theory that the c s are nearly identical, could lead to i seriously biased inference. Howdoesoneexplainthepoorfinitesampleperformanceoftheasymptoticbiasfunction? Observe thattheactualfinitesamplebiasistypicallymuchsmallerthantheasymptoticbias,asσ growslarge. c However,thegapbetweentheasymptoticresultsandthefinitesampleresultsisnotmerelyafunction of the standard deviation, σ . For smaller values of c, a larger standard deviation is needed before the c asymptotic valuedeviates substantially fromthe finite sample result. In fact, for largenegative values of c, there is a very sharp increase in the asymptotic bias after σ exceeds some value. For example, c for c= 50, the asymptotic limit for σ =7 is equal to 48.8, and for σ =8 the limit is 76.3. Before c c − − this breakpoint, the finite sample results are similar to the asymptotic ones, but afterwards, they are vastly different. As c becomes less negative, this effect becomes less distinct, and the growth of both the asymptotic bias and its deviation from the finite sample bias become smoother. 6

Given these observations, a tentative explanation for the large difference between the asymptotic and finite sample results is the following. When c > 0, the corresponding process is non-stationary i and explosive. For positive c , the quantities η 1 J (r)2dr and 1 J (r)2dr will therefore grow i i 0 i,ci 0 i,ci veryquicklyinc . Thus, theirmeanvalueswillbRehighlyinfluencedRbythetailbehavior,ormaximum i value, of c . This causes no problems when calculating their analytical means, of course, but leads i to problems when one tries to simulate them, which is essentially what is done in the Monte Carlo simulation. If the mean value depends on the tail behavior, it might be the case that extremely large samplesizesareneededbeforethesimulatedmeansapproachtheanalyticalones.3 Sincethefunctions η 1 J (r)2dr and 1 J (r)2dr do not grow fast in c for c non-positive, the above-mentioned i 0 i,ci 0 i,ci i i proRblems only manifesRt themselves when there is a large enough probability for c to be positive that i it will significantly affect the mean. Otherwise, the tail behavior will have less of an impact on the mean. This would explain why a larger variance is needed for small c before the gap between the i finite sample value and the asymptotic value grows large. This also provides some intuition for the extremely large asymptotic biases from which the pooled estimator suffers. The above reasoning suggests that the asymptotic bias approximation might perform better in cases where the support of the distribution of the local-to-unity parameters is bounded from above. To analyze this possibility, consider the case where the c s are uniformly distributed on an interval i (c ,c ). Inthiscase, c=E[c ]=(c +c)/2andtheasymptoticbiasfunctionofthepooledestimator l u i u l can be written as c +c fu(c l ,c u )= u 2 l +Ψu 1 (c l ,c u )− 1Ψu 2 (c l ,c u ), (12) where Ψu(c,c ) E η 1 J (r)2dr and Ψu(c ,c ) E η 1 J (r)2dr . 1 l u ≡ i 0 i,ci 2 l u ≡ i 0 i,ci The numerical valu h es fRor the functio i n fu(c ,c ), obtained h byRusing the mgf i of the uniform distribl u ution, are given in Panel A of Table 2. Panel B presents the corresponding mean pooled estimates of c from a Monte Carlo simulation identical to the one described above, except that the local-to-unity parameters are now uniformly distributed. If the asymptotic limit function fu(c ,c ) provides a good l u approximation in finite samples, the corresponding values in Panel A and Panel B should be close. They are indeed much closer than in the normal case, and the asymptotic results do provide a good approximation to the finite sample values, lending some credibility to the explanation offered above. However, though the asymptotic results correspond better to the finite sample values in the uniform 3Steele (2001) gives an illustrative example of the problems of simulating tail probabilities. He argues that if one attemptstosimulatethevalueofE[1(x 30)],wherexisstandardnormal,bynaivemethods,thenumberofsimulations needsto be ofan ordergreaterthan 101≥00. 7

case, any inference method relying on these results would face the problem that the limit function fu(c,c ) is not monotone in both c and c for all values of c and c ; the limit of cˆfor c = 10 is l u l u l u u constant for all values of c that are considered. Though this does appear to be a problem for large, l positive c , it may not be relevant in practical applications, where c is likely to be less than or equal u u to zero. 4 A median based estimator 4.1 Bias properties Given the poor performance of the pooled estimator in the previous section, an alternative estimator is proposed in this section. Rather than summing up over the cross-section, consider applying the sample median instead. The intuition behind this approach is simple. The median is generallya more robust estimator than the mean and can perform better in cases where the mean performs poorly. Let Assumption 1 hold, and let Λˆ and Ωˆ be consistent estimators, as T , of Λ and Ω , i i i i → ∞ respectively (see Moon and Phillips, 2000, for details). Begin with the inconsistent estimator of c , i 1 1 T z z TΛˆ 1 T z2 c˜ =T (a˜ 1)= Ωˆ i T t=1 i,t − 1 i,t − i − T t=1 i,t − 1 m 1,i,T, (13) i i − h ³ P 1 1 T z2´ P i ≡ m 2,i,T Ωˆ i T2 t=1 i,t − 1 P whereTΛˆ istheserialcorrelationbiascorrectiontermandm andm aredefinedintheobvious i 1,i,T 2,i,T manner. Define the median based estimator cˇas follows, med(m ) cˇ= 1,i,T , (14) med(m ) 2,i,T where med() denotes the sample median. As T , for fixed i, · →∞ 1 1 m c J (r)2dr+ J (r)dW (r)=m , (15) 1,i,T ⇒ i i,ci i,ci i 1,i Z0 Z0 and 1 m J (r)2dr =m . (16) 2,i,T ⇒ i,ci 2,i Z0 The division by Ωˆ , in both the numerator and denominator in equation (13) enables the derivation i of standardized results that are independent of the Ω s; it is not necessary when the Ω s are identical i i 8

for all i. Define ˆθ , k =1,2, as k n 1 1 1 ˆθ =arg min H (θ )+O =arg min m θ +O , (17) k θk∈ Θk k,n,T k p µ n ¶ θk∈ Θk n i=1 | k,i,T − k | p µ n ¶ X and thus, cˇ= ˆθ ˆθ . (18) 1 2 . Let θ0 and θ0 denote the medians of m and m , so that 1 2 1,i 2,i 1 1 =Pr m θ0 and =Pr m θ0 . (19) 2 1,i ≤ 1 2 2,i ≤ 2 ¡ ¢ ¡ ¢ Assumption 3 θ0 1 ,θ0 2 ∈ Θ,and Θ is a compact subset of R. Theorem 1 Under Assumptions 1 and 3, as (n,T ), →∞ ˆθ θ0, ˆθ θ0, and cˇ=ˆθ /ˆθ θ0/θ0. (20) 1 → p 1 2 → p 2 1 2 → p 1 2 So far, it has not been necessary to invoke Assumption 2; the above results hold for general distributions of the c s. However, in order to calculate θ0 and θ0, additional structure needs to be i 1 2 added to the problem. Analytical expressions for the medians of m and m are most likely not 1,i 2,i attainable, except for very special cases, but numerical results, given a distributional assumption on the c s, can be obtained. Therefore, I now make use of Assumption 2, and calculate numerical values i for θ0 and θ0, for different combinations of c and σ . The numerical methods used are described in 1 2 c the Appendix. Panel A in Table 3 presents the numerical values of θ0 θ0, under Assumption 2, for various 1 2 combinationsofcandσ . Ifcˇwereaconsistentestimatorofc,±regardlessofσ , allthesevaluesshould c c equal their corresponding value for c. As is seen, this is not quite the case, but cˇ still turns out to have several desirable properties. First, for all combinations of c and σ recorded in Panel A of Table c 3, which arguably covers most empirically interesting cases, the bias is seen to be small and below 1.3 in absolute value. Indeed, for positive values of c, the bias is almost zero. Second, and just as importantly, for a fixed c, the bias varies only slightly with the variance parameter σ . The maximum c difference observed, between σ =0 and σ =10, is no larger than 0.3 in absolute value, and is likely c c to be insignificant next to the variance of the estimates in any finite sample. This suggests that the 9

samebiascorrectionschemecanbeusedforaspecificc,regardlessofthevalueofσ . Thisisextremely c convenient, since no estimate of σ is then needed. Also, the bias correction is most naturally based c on the case of σ = 0, unless some specific prior information is available, for which the calculation of c the bias is greatly simplified as compared to the case σ >0. Finally, for σ =0, the ratio θ0 θ0 is a c c 1 2 monotone function of c, making bias correction feasible. ± Given the experience with the pooled estimator, one would naturally wish to evaluate the correspondence between the asymptotic results presented in Panel A in Table 3 and the finite sample properties of the estimator. Panel B in Table 3 shows the results from a Monte Carlo study with a relatively large panel. The setup is the same as that used in the pooled case. Each simulated panel consists of n = 100 time series, with T = 1,000 observations in each. The innovation processes are normaliid(0,1)andthec sarenormallydistributed. 10,000repetitionswereperformedandthemean i values of the estimates are reported in Panel B of Table 3, for each combination of c and σ . The c estimates have not been bias corrected in any way, and the serial correlation correction term of the estimator is not included. Since the error terms all have the same variance, the division by Ωˆ in the i numerator and denominator is not performed either. If the asymptotic results are valid finite sample approximations, the values in Panel A and Panel B of Table 3 should be close for corresponding values of c and σ . This also turns out to be the case, c and the median estimator does appear to be robust with regard to the variance of the local-to-unity parameter. Sincetheasymptoticbiasseemslikea reasonableapproximationof the finitesamplebias, a simple bias-correction scheme, based on the asymptotic results, can be implemented. Denote g(c) = θ0 θ0 1 2 for σ =0. Table 4 tabulates the values of g(c) for c [ 50,10]. As is seen, g(c) is strictly increas±ing c ∈ − in c. A bias corrected version of cˇ, which we will denote cˇ+, is now obtained by setting cˇ+ =g 1(cˇ). − The estimator cˇ+ is now a nearly consistent estimator of c, in the general case of σ >0, and exactly c consistentforthespecialcaseofσ =0. Thebiascorrectionschemeisparticularlysimpleforthecases c of θ0 θ0 8.78 and for θ0 θ0 4.90. According to the results of Table 4, θ0 θ0 = c 1.28 for 1 2 ≤ − 1 2 ≥ 1 2 − θ0 θ±0 8.78 and θ0 θ0 =±c for θ0 θ0 4.90. ± 1 2 ≤− 1 2 1 2 ≥ ±Performing an iden±tical Monte C±arlo simulation as the one described above, the bias corrected estimatescˇ+ arecalculatedandtheestimateddensitiesoftheseestimatesareplottedinFigure2. The densities of the estimates are centered very close to the true value of c, even for large values of σ . c Panels encountered in empirical practice are seldom as large as the ones used in the Monte Carlo 10

simulation above. In Table 5 and Figure 3, I show the results from a Monte Carlo simulation with n=20 and T =100. The local-to-unity parameters are once again drawn from normal distributions, andtheinnovationprocessesarealsoiidnormal,withunitvariance. Themeanvaluesoftheestimates presented in Table 5 generally look good. Considering the estimated densities, shown in Figure 3, the dispersion of the estimates for large values of σ is, of course, fairly large, given the small sample size. c But, for reasonable values, likeσ =5 and c= 10, theestimator still appears to performacceptably, c − given the sample size. Simulation results not reported in this paper also illustrate that the estimator cˇ+ works well for estimatingaveragelocal-to-unityparameterswhenthedistributionofthec sisnotnormal. Inthetwo i caseswherethec sweredrawnfromeitheruniformdistributionsorCauchydistributions,theestimator i cˇ+ was shown to deliver nearly unbiased estimates in finite samples. These results are available from the author upon request. 4.2 Asymptotic normality and standard error estimation Havingestablishedconvergenceofcˇto θ0 θ0 as(n,T ),Inowderivetheasymptoticdistribution 1 2 →∞ of the estimator. Since the bias correcte±d estimator cˇ+ is merely a shifted version of cˇ, it will have the same asymptotic variance, but its distribution will be centered on g 1 θ0 θ0 , rather than on − 1 2 θ0 θ0.4 ¡ ± ¢ 1 2 ± Theorem 2 Under Assumptions 1-3, as (T,n ) , →∞ seq √n cˇ θ0 θ0 − 1 2 N ¡ 0, ± 1¢ + θ0 1 2 v 12 θ0 1 ,θ0 2 . (21) ⇒ Ã 4 θ0 1 2 f 1 θ0 1 2 4 θ0 2¡ 4 f¢2 θ0 2 2 − 2 θ0 2 2 f 1¡ θ0 1 f¢2 θ0 2 ! ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ where f (θ )= d F (θ ), F () is the cumulative distribution function for m for k =1,2, and k k dθk k k k · k,i v (θ ,θ )=E[sign(m θ )sign(m θ )]. (22) 12 1 2 1,i 1 2,i 2 − − In order to perform inference on cˇ and cˇ+, an estimate of the limit variance, given in equation (21), is needed. If one is willing to work with a specific parametric distribution for the c s, such as i 4The asymptotic normality of the estimator is only shown for sequential limits. Subject to some additional rate restrictionsonnandT theresultalsolikelyholdsinjointlimitsas(n,T ). However,duetothenon-linearnature →∞ of the median operator, the proof for joint limits becomes very technical and is not crucial to the relatively applied discussion ofthis paper. 11

the normal distribution, then the densities f () and f () can be calculated numerically for given θ 1 2 1 · · andθ ,andestimatesoff θ0 andf θ0 aregivenbynumericalcalculationoff ˆθ andf ˆθ . 2 1 1 2 2 1 1 2 2 Similarly, the expectation v¡ ¢θ0,θ0 c¡oul¢d be numerically calculated. These num ³ eric ´ al calcula ³ tion ´ s 12 1 2 are straightforward extensions¡of the¢methods used for finding the medians of m and m , and will 1 2 not be detailed here. However, by using non-parametric methods, estimates of the desired quantities can be obtained without making any distributional assumptions. As argued above, F (θ )=Pr(m θ ) Pr(m θ )=F (θ ), (23) k,T k k,i,T k k,i k k k ≤ → ≤ as T , for k = 1,2 and i = 1,...,n. Since F () is continuously differentiable with a continuous k → ∞ · derivative f (), it follows that k · d d f (θ )= F (θ ) F (θ )=f (θ ) (24) k,T k dθ k,T k → dθ k k k k k k as T . A consistent estimator for f (θ ) is given by the kernel density estimator, k k →∞ n 1 m θ fˆ (θ )= K k,i,T − k , (25) k k nh h i=1 µ ¶ X where K() is a kernel function and h is the bandwidth parameter. Consistent estimates of f θ0 · 1 1 and f θ0 are now given by fˆ ˆθ and fˆ ˆθ . Since the m s are iid, standard results¡fo¢r 2 2 1 1 2 2 k,i,T consiste¡ncy¢of fˆ (θ ) apply (e.g. P ³ ag ´ an and Ul ³ lah ´ , 1999). k k Finally, a consistent estimator of v (θ ,θ ) is given by 12 1 2 n 1 vˆ (θ ,θ )= sign(m θ )sign(m θ ), (26) 12 1 2 n 1,i,T − 1 2,i,T − 2 i=1 X and a consistent estimate of v θ0,θ0 is provided by vˆ ˆθ ,ˆθ . 12 1 2 12 1 2 Thenon-parametricapproach¡isobv¢iouslymorerobustt ³ hanth ´ eparametriconefirstdescribedand is recommended in general. It is also the analogue of estimation procedures of the limiting covariance matrix in standard Least Absolute Deviations (LAD) regressions. 12

5 Conclusion In this paper, I analyze the problem of estimating the average local-to-unity parameter from a panel data set, where the local-to-unity parameters are treated as random variables. It is shown that the generalization from the setup with identical local-to-unity parameters raises some real issues in terms of consistency. Thepooledestimatorforthe average local-to-unityparameterisseverelybiasedforevenmoderate variationsinthelocal-to-unityparametersandcouldprovideverymisleadingresultsifusedindiscriminately. An alternative median based estimator is proposed instead. The idea behind this estimator is simple. To obtain more robust estimates than those provided by the pooled estimator, the sample median rather than the sample mean is used to extract the crucial cross-sectional information needed toestimatethelocal-to-unityparameter. Themedian based estimatoris analyzedforthespecific case of normally distributed local-to-unity parameters and is shown to exhibit a small asymptotic bias. Thebias,however,isalmostindependentofthevarianceofthelocal-to-unityparametersandasimple bias-correction procedure is used to obtain nearly consistent estimates. The estimator is shown to work well in finite samples and appears robust against deviations from the normality assumption. One issue not considered in this paper is that of heterogenous deterministic trends. Moon and Phillips (1999, 2000, and 2004) show that in the case of identical local-to-unity parameters, heterogenous trends cause the standard pooled estimator to become inconsistent. The effect of deterministic trendsonthepropertiesofthemedianbasedestimatorproposedinthispaperisleftforfutureresearch. A Appendix A.1 Numerical calculation of the medians of m and m 1,i 2,i First, note that 1 1 J (1)2 =1+2c J (r)2dr+2 J (r)dW (r), i,ci i i,ci i,ci i Z0 Z0 and, thus, 1 1 1 c J (r)2dr+ J (r)dW (r)= J (1)2 1 . i i,ci i,ci i 2 i,ci − Z0 Z0 ³ ´ Further, from Phillips (1987), 1 1 J i,ci (r) | c i = N 0, 2c e2rci − 1 c i = 2c (e2rci − 1)N(0,1) c i . µ i ¶¯ r i ¯ ¡ ¢ ¯ ¯ ¯ ¯ ¯ ¯ 13

Thus, Pr J (r)2 x2 i,ci ≤ ³ ´ ∞ = [Pr( x J (r) x c )]f (c )dc − ≤ i,ci ≤ | i c i i Z−∞ x ∞ = 2Φ 1 f (c )dc . (27) c i i Z−∞ ⎡ ⎛ 2 1 ci (e2rci − 1) ⎞− ⎤ ⎣ ⎝q ⎠ ⎦ The median of m , which we denote θ , is the solution to 1,i 1 1 1 =Pr(m θ )=Pr J (1)2 1 θ =Pr 2θ +1 J (1)2 2θ +1 . 2 1,i ≤ 1 2 i,ci − ≤ 1 − 1 ≤ i,ci ≤ 1 µ ³ ´ ¶ ³ p p ´ θ is obtained by numerically evaluating the integral (27) and finding the value x which sets this 1 integral equal to one half. The median is then given by θ = x2 1 /2. 1 − In order to derive the median of m , I use the character¡istic fu¢nction approach. By a result in 2,i Tanaka (1996, chapter 4), the characteristic function of m , for a fixed c , is given by 2,i i φ (t)=E eitm2,i c =E exp it 1 J (r)2dr c = e − ci/2 c , m2| ci £ ¯ i ¤ ∙ ½ Z0 i,ci ¾¯ ¯ i ¸ cosλ(c i ,t) − c i sin λ( λ c ( i c ,t i ) ,t)¯ ¯ ¯ i ¯ ¯ q ¯ ¯ ¯ ¯ where λ(c ,t)= 2it c2. By Lévy’s inversion theorem for nonnegative random variables, i − i p 1 1 e itx Pr(m x c )= ∞ Re − − φ (t) dt. 2,i ≤ | i π Z0 ∙ it m2| ci ¸ Thus, under the assumption of a random c , i 1 1 e itx Pr(m x)= ∞ Pr(m x c )f (c )dc = ∞ ∞ Re − − φ (t) dtf (c )dc . 2,i ≤ Z−∞ 2,i ≤ | i c i i Z−∞ π Z0 ∙ it m2| ci ¸ c i i This integral is evaluated numerically and θ is given by the solution to 1 =Pr(m x). 2 2 2,i ≤ A.2 Proofs of Theorems Proof of Theorem 1. The first order condition for H (θ ), is k,n,T k n 1 1 G (θ )= [1 m >θ 1 m θ ]=O . k,n,T 1 n { k,i,T k }− { k,i,T ≤ k } p n i=1 µ ¶ X 14

For fixed n, as T , by the continuous mapping theorem (CMT), →∞ n 1 G (θ ) [1 m >θ 1 m θ ]=G (θ ). k,n,T 1 ⇒ n { k,i k }− { k,i ≤ k } k,n k i=1 X The population moment condition for G (θ ) is k,n k 0=G (θ )=E[1 m >θ ] E[1 m θ ]=1 2F (θ ), k k k,i k k,i k k k { } − { ≤ } − where F () is the cumulative distribution function for m . By definition, G θ0 =G θ0 =0. k · k,i 1 1 2 2 To prove the uniform convergence of G ( ), it is sufficient to show that¡ ¢ ¡ ¢ 1,n,T · n n 1 1 sup 1 m θ Pr(m θ ) = sup 1 m θ E[1 m θ ] 0 θ1∈ Θ1 ¯ ¯ n X i=1 { 1,i,T ≤ 1 }− 1,i ≤ 1 ¯ ¯ θ1∈ Θ1 ¯ ¯ n X i=1 { 1,i,T ≤ 1 }− { 1,i ≤ 1 }¯ ¯ → p ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ uniformly in θ , as (n,T ). Consider, for a fixed θ , 1 1 →∞ T 1 1 c 1 m θ =1 β +y iy + β +y (cid:18) TΛˆ θ . { 1,i,T ≤ 1 } (Ωˆ i T X t=1h¡ i,0 i,t − 1 ¢ T i,t − 1 ¡ i,0 i,t − 1 ¢ i,t − i i ≤ 1 ) For fixed i, by the continuous mapping theorem for almost surely continuous functions, 1 1 1 m θ 1 m θ =1 c J (r)2dr+ J (r)dW (r) θ , { 1,i,T ≤ 1 }⇒ { 1,i ≤ 1 } i i,ci i,ci i ≤ 1 ½ Z0 Z0 ¾ asT ,sinceΛˆ Λ andΩˆ Ω . IftheconditionsofCorollary1ofPhillipsandMoon(1999) i p i i p i →∞ → → are satisfied, it then follows that n 1 1 m θ E[1 m θ ] n { 1,i,T ≤ 1 }→ p { 1,i ≤ 1 } i=1 X as(n,T )forfixedθ . Since1 m θ isuniformlybounded, 1 m θ isuniformly 1 1,i,T 1 1,i,T 1 →∞ { ≤ } || { ≤ }|| integrableinT foralli(Billingsley,1995)andtheotherconditionsofCorollary1ofPhillipsandMoon (1999) hold trivially. Pointwise convergence, for fixed θ , as (n,T ), is thus established. 1 →∞ Since Θ is a compact space, to establish uniform convergence one only needs to show that n 1 X (θ )= (1 m θ E[1 m θ ]) 1,n,T 1 n { 1,i,T ≤ 1 }− { 1,i ≤ 1 } i=1 X 15

isstochasticallyequicontinuous. Thisfollowsbystandardargumentsandtheproofisnotdetailedhere. The same arguments can be applied to G ( ) and will not be repeated. Thus, as (n,T ), 2,n,T · →∞ G (θ ) 1 2Pr(m θ ), k,n,T 1 p k,i k → − ≤ uniformly in θ and the desired result follows. k Proof of Theorem 2. Observe first, that for fixed n, as T , →∞ √n G 1,n,T θ0 1 1 n 1 m 1,i >θ0 1 − 1 m 1,i ≤ θ0 1 . ⎛ G 2,n,T ¡θ0 2 ¢ ⎞⇒ √n X i=1 ⎛ £1©m 2,i >θ0 2 ª − 1©m 2,i ≤ θ0 2 ª¤ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ¡ ¢ ⎠ ⎝ £ © ª © ª¤ ⎠ By the Lindeberg-Feller central limit theorem (CLT), as n , →∞ 1 n 1 m 1,i >θ0 1 − 1 m 1,i ≤ θ0 1 N(0,V), √n X i=1 ⎛ £1©m 2,i >θ0 2 ª − 1©m 2,i ≤ θ0 2 ª¤ ⎞⇒ ⎜ ⎟ ⎝ £ © ª © ª¤ ⎠ where 1 E sign m θ0 sign m θ0 V = 1,i − 1 2,i − 2 . ⎛ ⎞ E sign m θ0 sign m θ0 £ ¡ 1¢ ¡ ¢¤ 1,i − 1 2,i − 2 ⎜ ⎟ ⎝ £ ¡ ¢ ¡ ¢¤ ⎠ Thus, as (T,n ) , →∞ seq G θ0 1,n,T 1 √n N(0,V). ⎛ G ¡θ0¢ ⎞⇒ 2,n,T 2 ⎜ ⎟ ⎝ ¡ ¢ ⎠ Next, for k =1,2, for fixed n, as T , →∞ ν (θ)=√n(G (θ) G (θ)) √n(G (θ) G (θ)). k,n,T k,n,T k k,n k − ⇒ − By standard arguments for LAD estimators, √n(G (θ) G (θ)) is stochastically equicontious as k,n k − n . It follows that ν (θ) is stochastically equicontious as (T,n ) . →∞ k,n,T →∞ seq From the expressions of F (θ ) and F (θ ), derived in Appendix A.1, it is obvious that they 1 1 2 2 are both differentiable, and, hence, so are G (θ ) and G (θ ). Having established the asymptotic 1 1 2 2 normality of G θ0 , and the stochastic equicontinuity of the normalized process ν (θ), the k,n,T k k,n,T asymptoticnormality¡of¢ ˆθ ,ˆθ 0nowfollowsfromstandardresultsforextremumestimatorswithnon- 1 2 ³ ´ 16

smoothcriterionfunctions(e.g. Theorem7.1. inNeweyandMcfadden,1994). Thelimitingcovariance matrix is given by 1 1 d G (θ ) − d G (θ ) − 1 v12(θ1,θ2) dθ0 1 1 V dθ0 1 1 = f1(θ1)2 f1(θ1)f2(θ2) . ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ d G (θ ) d G (θ ) v12(θ1,θ2) 1 ⎜ dθ0 2 2 ⎟ ⎜ dθ0 2 2 ⎟ ⎜ f1(θ1)f2(θ2) f2(θ2)2 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ The final result for ˆθ /ˆθ folllows from the delta method. 1 2 17

References [1] Billingsley, P., 1995. Probability and Measure, Third Edition (Wiley, New York). [2] Campbell, J., and M. Yogo, 2003. Efficient Tests of Stock Return Predictability, Working Paper, Harvard University. [3] Cavanagh,C.,G.Elliot,andJ.Stock,1995.Inferenceinmodelswithnearlyintegratedregressors, Econometric Theory 11, 1131-1147. [4] Choi, I., 2001. Unit Root Tests for Panel Data, Journal of International Money and Finance 20, 249-272. [5] Elliot,G.,T.J.Rothenberg,andJ.H.Stock,1996.EfficientTestsforanAutoregressiveUnitRoot, Econometrica 64, 813-836. [6] LevinA.,F.Lin,andC.Chu,2002.UnitRootTestsinPanelData: AsymptoticandFinite-Sample Properties, Journal of Econometrics 108, 1-24. [7] Maddala, G.S., and S. Wu, 1999. A Comparative Study of Unit Root Tests with Panel Data and a New Simple Test, Oxford Bulletin of Economics and Statistics 61, 631-651. [8] MoonH.R.,andB.Perron,2003.TestingforaUnitRootinPanelswithDynamicFactors,CLEO Working Paper, USC. [9] Moon H.R., B. Perron, and P.C.B Phillips, 2003. Incidental Trends and the Power of Panel Unit Root Tests, Cowles Foundation Discussion Paper 1435. [10] Moon,H.R.,andP.C.B.Phillips,1999.MaximumLikelihoodEstimationinPanelswithIncidental Trends, Oxford Bulletin of Economics and Statistics 61, 711-748. [11] Moon,H.R.,andP.C.B.Phillips,2000.EstimationofAutoregressiveRootsnearUnityusingPanel Data, Econometric Theory 16, 927-998. [12] Moon, H.R., and P.C.B. Phillips, 2004. GMM Estimation of Autoregressive Roots Near Unity with Panel Data, Econometrica 72, 467-522. [13] Newey, W.K., D. McFadden, 1994. Large sample estimation and hypothesis testing, in Engle, R.F., and D.L. McFadden, eds., Handbook of Econometric, Vol. IV (North-Holland, Amsterdam) 2111-2245. [14] Pagan, A., and A. Ullah, 1999. Nonparametric Econometrics, Cambridge University Press. [15] Phillips, P.C.B, 1987. Towards a Unified Asymptotic Theory of Autoregression, Biometrika 74, 535-547.. [16] Phillips, P.C.B., and H.R. Moon, 1999. Linear Regression Limit Theory for Nonstationary Panel Data, Econometrica 67, 1057-1111. [17] Phillips, P.C.B., H.R. Moon, and Z. Xiao, 1998. How to estimate autoregressive roots near unity, Cowles Foundation Discussion Paper 1191. [18] Phillips, P.C.B., and P. Perron, 1988. Testing for a Unit Root in Time Series Regression, Biometrika 75, 335-346. [19] Steele, J.M., 2001. Stochastic Calculus and Financial Applications (Springer, New York). [20] Tanaka, K., 1996. Time Series Analysis: Nonstationary and Noinvetible Distribution Theory (Wiley, New York). [21] Quah, D., 1994. Exploiting Cross-Section Variations for Unit Root Inference in Dynamic Panels, Economics Letters 44, 9-19. 18

Table 1: The bias properties of the pooled estimator in the case of normally distributed local-to-unity parameters. PanelAshowsthenumericalvaluesforthelimitfunctionofthepooledestimator,f(c,σ ). c Panel B shows the mean values of the pooled estimates of c, cˆ, from a Monte Carlo simulation with n=100 and T =1,000, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from normal distributions with mean c given by the left most column and standard deviation σ given by the top row. c σ c c 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Panel A. -50.0 -50.0 -50.0 -49.9 -49.8 -49.7 -49.5 -49.3 -48.8 76.3 110.5 148.7 -40.0 -40.0 -40.0 -39.9 -39.8 -39.6 -39.4 -39.1 56.2 86.5 120.7 158.7 -30.0 -30.0 -30.0 -29.9 -29.7 -29.5 -29.1 40.1 66.5 96.7 130.8 168.8 -20.0 -20.0 -20.0 -19.8 -19.5 -19.1 29.0 50.6 76.7 106.8 140.9 178.9 -10.0 -10.0 -9.9 -9.6 -8.7 20.3 38.7 60.8 86.9 116.9 150.9 188.9 -9.0 -9.0 -8.9 -8.5 -6.6 21.4 39.7 61.8 87.9 117.9 151.9 189.9 -8.0 -8.0 -7.9 -7.5 -2.6 22.5 40.8 62.8 88.9 118.9 152.9 191.0 -7.0 -7.0 -6.9 -6.4 3.8 23.6 41.8 63.9 89.9 119.9 153.9 192.0 -6.0 -6.0 -5.8 -5.2 8.5 24.7 42.8 64.9 90.9 120.9 155.0 193.0 -5.0 -5.0 -4.8 -3.8 10.9 25.7 43.8 65.9 91.9 122.0 156.0 194.0 -4.0 -4.0 -3.8 -2.1 12.4 26.8 44.9 66.9 92.9 123.0 157.0 195.0 -3.0 -3.0 -2.7 0.2 13.6 27.8 45.9 67.9 94.0 124.0 158.0 196.0 -2.0 -2.0 -1.6 2.8 14.7 28.9 46.9 69.0 95.0 125.0 159.0 197.0 -1.0 -1.0 -0.4 5.0 15.8 29.9 48.0 70.0 96.0 126.0 160.0 198.0 0.0 0.0 0.9 6.6 16.9 31.0 49.0 71.0 97.0 127.0 161.0 199.0 1.0 1.0 2.2 7.9 18.0 32.0 50.0 72.0 98.0 128.0 162.0 200.0 2.0 2.0 3.4 9.1 19.0 33.0 51.0 73.0 99.0 129.0 163.0 201.0 3.0 3.0 4.5 10.2 20.1 34.1 52.0 74.0 100.0 130.0 164.0 202.0 4.0 4.0 5.6 11.3 21.1 35.1 53.1 75.0 101.0 131.0 165.0 203.0 5.0 5.0 6.7 12.3 22.2 36.1 54.1 76.1 102.0 132.0 166.0 204.0 Panel B. -50.0 -50.0 -50.0 -49.9 -49.9 -49.7 -49.5 -49.3 -49.1 -48.7 -48.4 -47.9 -40.0 -40.0 -40.0 -39.9 -39.8 -39.6 -39.4 -39.1 -38.8 -38.3 -37.8 -37.1 -30.0 -30.0 -30.0 -29.9 -29.7 -29.5 -29.2 -28.8 -28.2 -27.4 -26.0 -23.6 -20.0 -20.0 -20.0 -19.8 -19.6 -19.2 -18.6 -17.5 -15.3 -11.1 -5.5 0.3 -10.0 -10.0 -9.9 -9.6 -8.9 -7.0 -2.5 2.6 6.7 9.7 12.3 14.8 -9.0 -9.0 -8.9 -8.6 -7.7 -5.1 -0.2 4.6 8.0 10.7 13.3 16.0 -8.0 -8.0 -7.9 -7.5 -6.4 -3.0 2.1 6.1 9.1 11.7 14.3 16.9 -7.0 -7.0 -6.9 -6.4 -4.8 -0.7 3.9 7.5 10.2 12.8 15.4 17.9 -6.0 -6.0 -5.9 -5.3 -3.0 1.5 5.6 8.6 11.2 13.8 16.4 18.9 -5.0 -5.0 -4.8 -4.0 -1.1 3.5 6.9 9.7 12.2 14.8 17.3 19.7 -4.0 -4.0 -3.8 -2.6 1.0 5.0 8.1 10.7 13.2 15.8 18.3 20.9 -3.0 -3.0 -2.7 -1.1 2.9 6.3 9.1 11.7 14.2 16.9 19.4 21.8 -2.0 -2.0 -1.6 0.6 4.4 7.4 10.1 12.7 15.3 17.9 20.4 22.9 -1.0 -1.0 -0.4 2.3 5.7 8.5 11.1 13.7 16.3 18.8 21.4 23.9 0.0 0.0 0.8 3.7 6.8 9.5 12.1 14.7 17.3 19.8 22.3 24.9 1.0 1.0 2.0 5.0 7.9 10.6 13.2 15.7 18.3 20.8 23.4 25.8 2.0 2.0 3.2 6.1 8.9 11.5 14.2 16.8 19.3 21.7 24.4 26.8 3.0 3.0 4.3 7.1 9.9 12.6 15.2 17.8 20.3 22.8 25.3 27.9 4.0 4.0 5.4 8.2 10.9 13.6 16.1 18.7 21.3 23.9 26.3 28.9 5.0 5.0 6.4 9.2 11.9 14.6 17.2 19.7 22.2 24.8 27.3 29.8 19

Table2: Thebiaspropertiesofthepooledestimatorinthecaseofuniformlydistributedlocal-to-unity parameters, where c is given by the left most column and c by the top row. Panel A shows the l u numerical values of the function fu(c ,c ). Panel B reports the mean values of the pooled estimates l u of c from a Monte Carlo simulation with n=100 and T =1,000, using 10,000 repetitions. The localto-unityparameters aredrawnfromuniformdistributions withparameters c and c . Thenumbers in l u parentheses are the true values for c. c u c -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 l Panel A. -25.0 -22.4 -19.6 -16.4 -12.6 -6.5 4.2 9.4 -20.0 -17.4 -14.5 -10.9 -5.4 4.2 9.4 -15.0 -12.3 -9.2 -4.3 4.2 9.4 -10.0 -7.2 -3.1 4.2 9.4 -5.0 -1.8 4.3 9.4 0.0 4.3 9.4 5.0 9.4 Panel B. -25.0 -22.4 -19.6 -16.5 -12.6 -6.7 3.8 9.2 (-22.5) (-20.0) (-17.5) (-15.0) (-12.5) (-10.0) (-7.5) -20.0 -17.4 -14.5 -11.0 -5.6 3.9 9.2 (-17.5) (-15.0) (-12.5) (-10.0) (-7.5) (-5.0) -15.0 -12.4 -9.2 -4.5 4.0 9.2 (-12.5) (-10.0) (-7.5) (-5.0) (-2.5) -10.0 -7.3 -3.2 4.1 9.3 (-7.5) (-5.0) (-2.5) (0.0) -5.0 -1.8 4.2 9.3 (-2.5) (0.0) (2.5) 0.0 4.3 9.4 (2.5) (5.0) 5.0 9.4 (7.5) 20

Table 3: The bias properties of the median based estimator for normally distributed local-to-unity parameters. Panel A shows numerical values for the limit function, θ0/θ0, for different combinations 1 2 of c and σ . Panel B shows mean values of the median based estimates of c, cˇ, from a Monte Carlo c simulation with n = 100 and T = 1,000, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from normal distributions with c given by the left most column and σ given by the top row. The estimates have not been bias c corrected. σ c c 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Panel A. -50.0 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -51.3 -40.0 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -41.3 -30.0 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -31.3 -20.0 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -21.3 -10.0 -11.3 -11.3 -11.3 -11.3 -11.3 -11.3 -11.2 -11.2 -11.2 -11.2 -11.2 -9.0 -10.3 -10.3 -10.3 -10.3 -10.3 -10.3 -10.2 -10.2 -10.2 -10.2 -10.2 -8.0 -9.3 -9.3 -9.3 -9.3 -9.3 -9.2 -9.2 -9.2 -9.2 -9.2 -9.2 -7.0 -8.3 -8.3 -8.3 -8.3 -8.3 -8.2 -8.2 -8.2 -8.2 -8.2 -8.2 -6.0 -7.3 -7.3 -7.3 -7.3 -7.2 -7.2 -7.2 -7.2 -7.2 -7.1 -7.1 -5.0 -6.3 -6.3 -6.3 -6.2 -6.2 -6.2 -6.2 -6.1 -6.1 -6.1 -6.1 -4.0 -5.3 -5.3 -5.2 -5.2 -5.2 -5.2 -5.1 -5.1 -5.1 -5.1 -5.1 -3.0 -4.2 -4.2 -4.2 -4.2 -4.1 -4.1 -4.1 -4.1 -4.1 -4.1 -4.0 -2.0 -3.2 -3.2 -3.2 -3.1 -3.1 -3.1 -3.1 -3.0 -3.0 -3.0 -3.0 -1.0 -2.1 -2.1 -2.1 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 -2.0 0.0 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 1.0 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 2.0 1.7 1.6 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.4 3.0 2.9 2.7 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 4.0 4.0 3.8 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 5.0 5.0 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 Panel B. -50.0 -51.0 -51.0 -51.0 -51.0 -51.0 -51.0 -51.0 -51.0 -51.0 -50.9 -51.0 -40.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -41.0 -30.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -31.0 -20.0 -21.1 -21.1 -21.1 -21.1 -21.0 -21.1 -21.0 -21.0 -21.0 -21.0 -21.0 -10.0 -11.1 -11.1 -11.1 -11.1 -11.1 -11.1 -11.1 -11.1 -11.0 -11.0 -11.0 -9.0 -10.2 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.0 -10.0 -10.0 -8.0 -9.1 -9.2 -9.1 -9.1 -9.1 -9.1 -9.1 -9.1 -9.0 -9.0 -9.0 -7.0 -8.3 -8.2 -8.2 -8.1 -8.1 -8.1 -8.1 -8.1 -8.0 -8.0 -8.0 -6.0 -7.3 -7.2 -7.3 -7.2 -7.1 -7.1 -7.1 -7.1 -7.0 -7.0 -7.0 -5.0 -6.3 -6.2 -6.3 -6.2 -6.1 -6.1 -6.1 -6.0 -6.1 -6.0 -6.0 -4.0 -5.2 -5.2 -5.3 -5.2 -5.1 -5.1 -5.1 -5.1 -5.0 -5.0 -5.0 -3.0 -4.2 -4.2 -4.3 -4.1 -4.1 -4.1 -4.1 -4.1 -4.0 -4.0 -4.0 -2.0 -3.3 -3.2 -3.1 -3.1 -3.1 -3.0 -3.0 -3.0 -3.0 -3.0 -3.0 -1.0 -2.1 -2.1 -2.1 -2.0 -2.0 -2.0 -2.0 -2.0 -1.9 -1.9 -1.9 0.0 -1.0 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.9 -0.8 -0.8 -0.8 1.0 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 2.0 1.7 1.6 1.5 1.4 1.5 1.5 1.5 1.5 1.5 1.5 1.5 3.0 2.9 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 4.0 4.0 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 5.0 5.0 4.8 4.8 4.8 4.8 4.8 4.8 4.9 4.9 4.9 4.9 21

Table4: Numericalvaluesforthelimitfunctionofthemedianbasedestimator,θ0/θ0,forthehomoge- 1 2 nous case σ =0. c c θ0/θ0 c θ0/θ0 c θ0/θ0 c θ0/θ0 c θ0/θ0 c θ0/θ0 1 2 1 2 1 2 1 2 1 2 1 2 -50.00 -51.28 -14.00 -15.28 -6.80 -8.07 -3.20 -4.45 0.40 -0.42 4.00 3.98 -49.00 -50.28 -13.00 -14.28 -6.70 -7.97 -3.10 -4.35 0.50 -0.28 4.10 4.08 -48.00 -49.28 -12.00 -13.28 -6.60 -7.87 -3.00 -4.24 0.60 -0.15 4.20 4.18 -47.00 -48.28 -11.00 -12.28 -6.50 -7.77 -2.90 -4.14 0.70 -0.01 4.30 4.29 -46.00 -47.28 -10.00 -11.28 -6.40 -7.67 -2.80 -4.04 0.80 0.13 4.40 4.39 -45.00 -46.28 -9.90 -11.18 -6.30 -7.57 -2.70 -3.94 0.90 0.27 4.50 4.49 -44.00 -45.28 -9.80 -11.08 -6.20 -7.47 -2.60 -3.83 1.00 0.41 4.60 4.59 -43.00 -44.28 -9.70 -10.98 -6.10 -7.37 -2.50 -3.73 1.10 0.54 4.70 4.69 -42.00 -43.28 -9.60 -10.88 -6.00 -7.27 -2.40 -3.63 1.20 0.68 4.80 4.79 -41.00 -42.28 -9.50 -10.78 -5.90 -7.17 -2.30 -3.52 1.30 0.82 4.90 4.90 -40.00 -41.28 -9.40 -10.68 -5.80 -7.07 -2.20 -3.42 1.40 0.96 5.00 5.00 -39.00 -40.28 -9.30 -10.58 -5.70 -6.97 -2.10 -3.32 1.50 1.09 6.00 6.00 -38.00 -39.28 -9.20 -10.48 -5.60 -6.87 -2.00 -3.21 1.60 1.22 7.00 7.00 -37.00 -38.28 -9.10 -10.38 -5.50 -6.77 -1.90 -3.11 1.70 1.36 8.00 8.00 -36.00 -37.28 -9.00 -10.28 -5.40 -6.67 -1.80 -3.00 1.80 1.49 9.00 9.00 -35.00 -36.28 -8.90 -10.18 -5.30 -6.57 -1.70 -2.89 1.90 1.61 10.00 10.00 -34.00 -35.28 -8.80 -10.08 -5.20 -6.47 -1.60 -2.79 2.00 1.74 -33.00 -34.28 -8.70 -9.98 -5.10 -6.37 -1.50 -2.68 2.10 1.87 -32.00 -33.28 -8.60 -9.88 -5.00 -6.27 -1.40 -2.57 2.20 1.99 -31.00 -32.28 -8.50 -9.78 -4.90 -6.17 -1.30 -2.46 2.30 2.11 -30.00 -31.28 -8.40 -9.68 -4.80 -6.07 -1.20 -2.35 2.40 2.23 -29.00 -30.28 -8.30 -9.58 -4.70 -5.97 -1.10 -2.24 2.50 2.35 -28.00 -29.28 -8.20 -9.48 -4.60 -5.87 -1.00 -2.13 2.60 2.47 -27.00 -28.28 -8.10 -9.38 -4.50 -5.76 -0.90 -2.02 2.70 2.58 -26.00 -27.28 -8.00 -9.28 -4.40 -5.66 -0.80 -1.90 2.80 2.69 -25.00 -26.28 -7.90 -9.18 -4.30 -5.56 -0.70 -1.79 2.90 2.81 -24.00 -25.28 -7.80 -9.08 -4.20 -5.46 -0.60 -1.67 3.00 2.92 -23.00 -24.28 -7.70 -8.98 -4.10 -5.36 -0.50 -1.55 3.10 3.03 -22.00 -23.28 -7.60 -8.88 -4.00 -5.26 -0.40 -1.43 3.20 3.14 -21.00 -22.28 -7.50 -8.78 -3.90 -5.16 -0.30 -1.31 3.30 3.24 -20.00 -21.28 -7.40 -8.67 -3.80 -5.06 -0.20 -1.19 3.40 3.35 -19.00 -20.28 -7.30 -8.57 -3.70 -4.96 -0.10 -1.06 3.50 3.46 -18.00 -19.28 -7.20 -8.47 -3.60 -4.86 0.00 -0.94 3.60 3.56 -17.00 -18.28 -7.10 -8.37 -3.50 -4.75 0.10 -0.81 3.70 3.67 -16.00 -17.28 -7.00 -8.27 -3.40 -4.65 0.20 -0.68 3.80 3.77 -15.00 -16.28 -6.90 -8.17 -3.30 -4.55 0.30 -0.55 3.90 3.88 22

Table 5: Mean values of the bias corrected median estimates of c, cˇ+, from a Monte Carlo simulation with n = 20 and T = 100, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from a normal distribution with c given by the left most column and σ given by the top row. c σ c c 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -50.0 -49.1 -49.1 -49.1 -49.1 -49.0 -49.1 -49.0 -49.0 -48.9 -48.9 -48.8 -40.0 -39.3 -39.3 -39.3 -39.3 -39.2 -39.2 -39.2 -39.1 -39.1 -39.1 -39.1 -30.0 -29.4 -29.4 -29.4 -29.4 -29.4 -29.4 -29.3 -29.4 -29.3 -29.2 -29.3 -20.0 -19.5 -19.5 -19.6 -19.5 -19.5 -19.5 -19.5 -19.4 -19.5 -19.4 -19.4 -10.0 -9.7 -9.7 -9.7 -9.7 -9.6 -9.6 -9.5 -9.5 -9.4 -9.4 -9.3 -9.0 -8.7 -8.7 -8.7 -8.7 -8.6 -8.6 -8.5 -8.5 -8.4 -8.4 -8.3 -8.0 -7.7 -7.7 -7.7 -7.7 -7.7 -7.6 -7.5 -7.5 -7.5 -7.4 -7.4 -7.0 -6.8 -6.7 -6.7 -6.7 -6.7 -6.6 -6.5 -6.5 -6.4 -6.4 -6.3 -6.0 -5.8 -5.8 -5.7 -5.7 -5.6 -5.6 -5.5 -5.5 -5.4 -5.5 -5.4 -5.0 -4.8 -4.8 -4.7 -4.7 -4.7 -4.6 -4.6 -4.5 -4.5 -4.5 -4.4 -4.0 -3.8 -3.8 -3.8 -3.7 -3.7 -3.6 -3.6 -3.5 -3.5 -3.5 -3.4 -3.0 -2.9 -2.9 -2.8 -2.7 -2.7 -2.6 -2.6 -2.6 -2.5 -2.5 -2.5 -2.0 -1.9 -1.9 -1.8 -1.8 -1.7 -1.7 -1.7 -1.6 -1.6 -1.5 -1.5 -1.0 -1.0 -0.9 -0.9 -0.9 -0.8 -0.8 -0.8 -0.7 -0.7 -0.6 -0.6 0.0 -0.0 0.0 0.0 0.1 0.1 0.1 0.2 0.3 0.3 0.4 0.5 1.0 1.0 0.9 0.9 1.0 1.0 1.1 1.1 1.2 1.3 1.3 1.4 2.0 2.0 1.9 1.9 1.9 2.0 2.0 2.1 2.2 2.2 2.3 2.4 3.0 3.0 2.9 2.8 2.9 2.9 3.0 3.1 3.1 3.2 3.3 3.4 4.0 4.0 3.8 3.8 3.9 3.9 4.0 4.1 4.2 4.3 4.3 4.3 5.0 5.0 4.9 4.9 4.9 5.0 5.1 5.1 5.2 5.2 5.4 5.4 23

Figure1: Estimatesofthedensityfunctionsofthepooledestimatesofc,cˆ,inaMonteCarlosimulation. The sample size is n= 100 and T =1,000, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from a normal distribution with the mean and variance given above each graph. In the right hand graphs, the dashed line corresponds to σ =5 and the dotted line to σ =10. c c 24

Figure 2: Estimates of the density functions of the bias corrected median estimates of c, cˇ+, in a Monte Carlo simulation. The sample size is n = 100 and T = 1,000, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from normal distributions with the mean and variance given above each graph. In the right hand graphs, the dashed line corresponds to σ =5 and the dotted line to σ =10. c c 25

Figure3: Estimatesofthedensityfunctionsofthebiascorrectedmedianestimatesofc,cˇ+,inaMonte Carlo simulation. The sample size is n = 20 and T = 100, using 10,000 repetitions. The innovations are iid normal with variance equal to one. The local-to-unity parameters are also drawn from normal distributionswiththemeanandvariancegivenaboveeachgraph. Intherighthandgraphs,thedashed line correspond to σ =5 and the dotted line to σ =10. c c 26

International Finance Discussion Papers IFDP Number Titles Author(s) 2006 851 Exchange-Rate Pass-Through in the G-7 Countries Jane E. Ihrig Mario Marazzi Alexander D. Rothenberg 850 The Adjustment of Global External Imbalances: Does Partial Christopher Gust Exchange Rate Pass-Through to Trade Prices Matter? Nathan Sheets 2005 849 Interest Rate Rules, Endogenous Cycles and Chaotic Dynamics Marco Airaudo in Open Economies Luis-Felipe Zanna 848 Fighting Against Currency Depreciation Macroeconomic Luis-Felipe Zanna Instability and Sudden Stops 847 The Baby Boom Predictability in House Prices and Interest Rates Robert F. Martin 846 Explaining the Global Pattern of Current Account Imbalances Joseph W. Gruber Steven B. Kamin 845 DSGE Models of High Exchange-Rate Volatility and Low Giancarlo Corsetti Pass-Through Luca Dedola Sylvain Leduc 844 The Response of Global Equity Indexes to U.S. Monetary Jon Wongswan Policy Announcements 843 Accounting Standards and Information: Inferences from John Ammer Cross-Listed Financial Firms Nathanael Clinton Gregory P. Nini 842 Alternative Procedures for Estimating Vector Autoregressions Lawrence J Christiano Identified with Long-Run Restrictions Martin Eichenbaum Robert J. Vigfusson ________ Please address requests for copies to International Finance Discussion Papers, Publications, Stop 127, Board of Governors of the Federal Reserve System, Washington, DC 20551. Email: publications-bog@frb.gov. Fax (202) 728-5886. 27

International Finance Discussion Papers IFDP Number Titles Author(s) 841 Monetary Policy and House Prices: A Cross-Country Study Alan G. Ahearne John Ammer Brian M. Doyle Linda S. Kole Robert F. Martin 840 International Capital Flows and U.S. Interest Rates Francis E. Warnock Veronica C. Warnock 839 Effects of Financial Autarky and Integration: The Case of the Brahima Coulibaly South Africa Embargo 838 General-to-specific Modeling: An Overview and Selected Julia Campos Bibliography Neil R. Ericsson David F. Hendry 837 Currency Crashes and Bond Yields in Industrial Countries Joseph E. Gagnon 836 Estimating Elasticities for U.S. Trade in Services Jaime Marquez 835 SIGMA: A New Open Economy Model for Policy Analysis Christopher Erceg Luca Guerrieri Christopher Gust 834 Optimal Fiscal and Monetary Policy with Sticky Wages and Sanjay K. Chugh Sticky Prices 833 Exchange Rate Pass-through to U.S. Import Prices: Some New Mario Marazzi Evidence Nathan Sheets Robert J. Vigufsson And Others 832 A Flexible Finite-Horizon Identification of Technology Shocks Neville Francis Michael T. Owyang Jennifer E. Roush 831 Adjusting Chinese Bilateral Trade Data: How Big is China’s John W. Schindler Surplus Dustin H. Beckett 830 Order Flow and Exchange Rate Dynamics in Electronic David W. Berger Brokerage System Data Alain P. Chaboud Sergey V. Chernenko Edward Howorka Raj S. Krishnasami Iyer David Liu Jonathan H. Wright 28