Abstract KIM, HYUNJUNG

Transcription

1 Abstract KIM, HYUJUG Unit Root ests in Panel Data: Weighted Symmetric Estimation and Maximum Likelihood Estimation (Under the direction of David A Dickey) here has been much interest in testing nonstationarity of panel data in the econometric literature In the last decade, several tests based on the ordinary least squares and Lagrange multiplier method have been developed In contrast to a unit root test in the univariate case, test statistics in panel data have Gaussian limiting distributions his dissertation considers weighted symmetric estimation and maximum likelihood estimation in the autoregressive model with individual effects he asymptotic distributions have been derived as the number of individuals and time periods become large he power study from Monte Carlo experiments shows that the proposed test statistics perform substantially better than those in previous studies even for small samples As an example, we consider the real Gross Domestic Product per Capita for 1 countries

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3 ii o my eternal beloved and my parents

4 iii Biography Hyunjung Kim was born in Seoul, Korea, to parents Wonji Kim and Jaewoon Park on ovember 9, 197 She earned her Bachelor s degree in Statistics in February 1995 and her Master s degree in Statistics in February 1997 at Ewha Womans University She worked for Korea Economic Research Institute as a research assistant Since August 1997, she has studied for the PhD degree in the Department of Statistics at orth Carolina State University While at CSU, she has participated in several projects as a research assistant Under the direction of Dr David A Dickey, she has worked on unit root tests in panel data

5 iv Acknowledgements I would like to express my gratitude to Dr David A Dickey He has been my mentor and good example to me as a researcher Without his kindly and sincere guidance, I could not complete this dissertation I am also grateful to my committee members, Dr Battacharyya, Dr Gumpertz, and Dr Pantula for their comments on my dissertation Dr Pantula has given me encouragement and valuable advice through my academic years It has been a great pleasure for me to work with Dr Brownie Her supervision as well as financial support helped me to finish my graduate program without any difficulty I will not forget the faculties and staff in our department for their enthusiastic teachings and kindness I also want to show my appreciation to the faculties in the department of statistics at Ewha Womans University Especially, Dr Dongwan Shin had provided me invaluable guidance during my master s program and led me to keep studying for Ph D Recently, I realized that I have known many precious friends Even though I do not mention their names, they are treasures to me that I have obtained through my life and will be remained in my deepest heart I really appreciate all the saints who have spent time with me for last 4 years heir love, fellowship and prayer in the Lord enliven me in every situation I can not say enough thanks and pay their love back to my parents heir ceaseless love and support are the most important resources for me to finish my PhD program Last but not least, this dissertation should be dedicated to my eternal company who became my everything

6 v Contents List of ables List of Figures vi vii 1 Introduction 1 11 Statement of the problem 1 1 Literature Review Simple and Weighted Symmetric Estimation 11 1 AR(1) process 11 Higher Order Processes 9 3 Maximum Likelihood Estimation with Fixed Individual Effects Maximum Likelihood Estimator for the Fixed Individual Effects Model 43 3 Limiting Distribution 49 4 Maximum Likelihood Estimation with Random Individual Effects Maximum Likelihood Estimator for the Random Individual Effects Model 57 4 Sequential Limit Distribution 6 5 Power Study 68 6 Example 75 7 Summary 77 References 79 A Finite Sample Moments in Fixed Effects Case 83 B Finite Sample Moments in Random Effects Case 88 C he Moments of the -limit Random Variables 93

7 vi List of ables 1 Regression table for weighted symmetric estimation in AR(1) process 13 Empirical cumulative distribution of (ˆρ s 1 b s ) 5 3 Empirical cumulative distribution of (ˆρ ws 1 b ws ) 6 4 Empirical cumulative distribution of ˆτ s 7 5 Empirical cumulative distribution of ˆτ ws 8 6 Regression table for weighted symmetric estimation in AR() process 31 7 Empirical cumulative distribution of ˆτ(θ 1 ) s 41 8 Empirical cumulative distribution of ˆτ(θ 1 ) ws 4 31 Empirical cumulative distribution of [ (ˆρ ml,fix 1) + ζ 0 ( ) ] Empirical cumulative distribution of [ (ˆρ ml,ran 1) + 1 ] Size of unit root tests in panel data when ρ = Power of unit root tests in panel data when ρ = Power of unit root tests in panel data when ρ = Power of unit root tests in panel data when ρ = Power of unit root tests in panel data when ρ =085 74

8 vii List of Figures 61 Real GDP per Capita using EKS PPPs (United States=100) 76

9 1 Chapter 1 Introduction 11 Statement of the problem ime series across states, industries, or randomly sampled individuals can be easily found in the macroeconomic and microeconomic literature hese kinds of data are called panel data, cross-section data, or longitudinal data Among econometricians, there have been many studies about panel data with the autoregressive covariance structure over time If it is assumed that there is no correlation across individuals, an autoregressive panel data model of order 1 with individual effects µ i can be written as y it = ρy it 1 +(1 ρ)µ i +e it, i =1,,,;t=1,,,, (11) where e it follows a white noise process with mean zero and variance σ Here, y it can be interpreted as an observation which is obtained from the ith individual at time t he individual effect µ i can be assumed to be either a fixed or random effect which is independent of the white noise process e it If ρ < 1, y it follows a stationary process which has stable mean and variance even for large t On the other hand, if ρ = 1, the variance of y it increases as t becomes large For this reason, as in a univariate time series, that is, when = 1, there has been much interest in detecting a unit root in panel data For testing H 0 : ρ =1,

10 many econometricians have developed test statistics and the corresponding limiting distributions Unlike a univariate time series, we need to consider two dimensions of indices, that is, (,) for deriving a limiting distribution In this thesis, we will develop unit root test statistics based on weighted symmetric estimation and maximum likelihood estimation for model (11) For weighted symmetric estimation, we also discuss higher order processes For maximum likelihood estimation, we will consider random individual effects as well as fixed individual effects Concerning white noise, we will assume the Gaussian distribution with mean zero and variance σ Also, we will derive the limiting distributions for the test statistics under H 0 : ρ =1 as, hrough Monte Carlo simulation study, we will compare the power of our test statistics with some of previous studies As an example, we will consider the real Gross Domestic Product per Capita for 1 countries 1 Literature Review Since the 1960 s, there has been much literature about panel data models [eg Balestra and erlove (1966), Mundlak (1978), W Anderson (1978), W Anderson and Hsiao (198)] Anderson and Hsiao (198) present several estimation methods for various regression models with the autoregressive order 1 covariance structure Specifically, they focus on maximum likelihood estimation for the following stationary autoregressive panel data model where y it = ρy it 1 + δ z i + γ x it + α i + e it, i =1,,;t=1,,, (1) ρ < 1,E(α i )=E(e it )=0, Var(e it )=σ,var(α i )=λσ for λ>0,

11 3 α i and e it are independent of each other, z i is a vector of time-invariant exogenous variables and x it is a vector of time-varying exogenous variables For maximum likelihood estimation, the underlying distributions are Gaussian hey assume that the initial value y i0 may be fixed or random For both cases, they allow that y i0 may be related to α i or independent of α i hey present consistency results for ˆρ and ˆδ when for fixed or for fixed Due to the fact that there exist nonstationary time series in many economic data sets, Quah (1994) introduces a unit root test for panel data He considers the simplest panel data model, y it = ρy it 1 + e it, (13) where y i0 is a given random variable with ( + δ)th finite moment and the e it s are independent identically distributed with mean zero, variance σ, and have bounded (4 + δ)th moment for δ>0 He assumes the cross-section dimension and the time dimension to be the same order of magnitude, that is, = ( )=O() For model (13), he considers the ordinary least squares estimator ˆρ ols of ρ hat is, ˆρ ols = [ ( ) Under H 0 : ρ = 1, he shows t=1 y ] 1 [ ( ) it 1 ] y it y it 1 (14) t=1 as, where ( ) (ˆρ ols 1 µ σ 3/ ) = (0, D 1) (15) ( t=1 e )] µ = lim E[ y i0 it His result tells us that the unit root test statistic in panel data under H 0 : ρ = 1 a Gaussian limiting distribution unlike its behavior in a univariate time series has

12 4 Levin and Lin (199) generalize model (13) to have individual-specific effects and a time trend in their model he model with individual-specific effects can be written as y it = η i + ρy it 1 + e it, (16) where the e it are independent and identically distributed with mean 0 and variance σ Also, E e it +δ <, for δ>0 hey consider the ordinary least squares estimator of ρ and the corresponding t- statistic in model (16) hese are defined as ˆρ ols = [ (y it ȳ i ) ] 1 [ (y it ȳ i )(y it 1 ȳ i ) ], (17) t=1 t=1 where t ρ = [ˆσ (y it ȳ i ) ] 1/ [ˆρ ols 1 ], (18) ols t=1 and ȳ i = 1 y it, t=1 ˆσ ols = Under H 0 : ρ =1, 1 [ yit ȳ i ˆρ ols (y it 1 ȳ i ) ] 1t=1 t ρ has the following limiting distribution, based on the central limit theorem for triangular arrays (Billingsley 1986) as and, 15 [ tρ µ1 µ ] D = (0, 1), (19)

13 5 where µ 1 = E [ 1 ] 1 (y σ it 1 ȳ i )e it = t=1 1, and µ = E [ 1 (y σ it 1 ȳ i ) ] = 1 t= If 0, then 15tρ D = (0, 1) (110) In their 1993 paper, they allow the degree of persistence to vary across individuals In other words, their individual-specific effects model (16) is generalized to the following model y it = η i + δ i y it 1 + ɛ it (111) Here, ɛ it is distributed independently across individuals and is a stationary invertible ARMA process as in Said and Dickey (1984) Also, this disturbance has finite fourth moments and bounded variation at frequency zero, that is, E(ɛ it)+ E(ɛ it ɛ it j ) <M,for M>0 j=1 In the model (111), testing for a unit root is equivalent to testing δ i = 0 for all i o remove time-specific aggregate effects, they subtract the cross-section average 1 y it from the original data at time t After this procedure, they perform the augmented Dickey-Fuller regression (1981) for each individual i In other words, they consider the following regression model for each i

14 6 p i y it = η i + δ i y it 1 + θ ij y it j + e it (11) j=1 Let û it be the residual from the regression of y it on η i and p i j=1 θ ij y it j Also let ˆv it 1 be the residual from the regression of y it 1 on η i and p i j=1 θ ij y it j By using these residuals, they consider the following regression model for each i û it = δ iˆv it 1 + e it (113) Dividing û it and ˆv it 1 by the regression standard error in the model (113), they obtain standardized residuals ũ it and ṽ it 1 hey compute a combined estimator ˆδ and the corresponding t-test statistic from these standardized residuals he definitions are and t=+pi ṽ it 1 ũ it ˆδ = t=+pi, (114) ṽit 1 where t δ = ˆδ SE(ˆδ), (115) [ SE(ˆδ)=ˆσ e ṽit 1 t=+p i ] 1/, and ˆσ e = [ 1 t=+p i (ũ it ˆδṽ it 1 ) ] 1/, = 1 p i 1

15 7 Since t δ in (115) goes to for the individual-specific effects model, they introduce the following adjusted t-test statistic t δ = t δ Ŝˆσ e SE(ˆδ)µ 1, (116) σ 1 where the average standard deviation ratio Ŝ = 1 ŝ i and ŝ i is the ratio of the long-run standard deviation to the innovation standard deviation for each i he mean and variance adjustment factors, µ 1,σ can be found in Levin and Lin (1993) hese 1 are presented in a table computed by Monte Carlo simulation for various (,) If 0 then, t δ D = (0, 1) (117) Im, Pesaran, and Shin (1997) propose the LM-bar test and t-bar test which are based on the averages of the Lagrange multiplier statistics and t-statistics, respectively hey consider the models (16) and (111) in Levin and Lin (199 and 1993) However, the alternative hypothesis in this paper is that the fraction of the individual processes that are stationary is non-zero, whereas the alternative hypothesis in Levin and Lin (1993) is that all the individual processes are stationary When errors in regressions are serially uncorrelated, that is, under the model (16), the LM-bar statistic is where LM = 1 LM i (118) LM i = y i P i y i y i M y i, (119) 1 = [ 1, 1,,1 ] 1,

16 8 M = I 1 (1 1 ) 1 1, P i = M y i, 1 (y i, 1M y i, 1 ) 1 y i, 1M, y i, 1 = [ y i0,y i1,,y i 1 ], and y i = [ y i1, y i,, y i ] hen their standardized LM-bar statistic is Γ LM = [ LM E(LM i ) ] Var(LM i ), (10) where E(LM i ) and Var(LM i ) are the mean and the variance of LM i H 0 : ρ =1 heir t-bar statistic is under t = 1 t i, (11) where t i is the t-statistic which corresponds to the estimator of δ i from the augmented Dickey-Fuller regression model y it = η i + δ i y it 1 + e it for each i For testing δ i =0 for every i, they present the standardized t-bar statistic which is Γ t = [ t E(t δ i =0) ] Var(t δ i =0) (1)

17 9 hey also construct the standardized LM-bar and t-bar statistics under the model (111) Under the model assumption of (16), they prove that Γ LM and Γ t have standard Gaussian limiting distributions as and is fixed Also, they show that the standardized LM-bar and t-bar statistics which are constructed under the model (111) have standard Gaussian limiting distributions as and become large with a restriction of k, a finite positive constant Karlsson and Löthgren (000) compare power of Levin and Lin (199) s t- tests (hereafter LL tests) in (19) and (110) with Im, Peseran, and Shin (1997) s tests (hereafter IPS tests) in (10) and (1) In their Monte Carlo experiments, they allow the proportion of stationary series in the panel to vary he results show that IPS tests perform better than LL tests for large Also, the power increases as increases, increases, and the proportion of stationary series in the panel increases Especially, is a critical factor in determining power Harris and zavalis (1999) also consider the unit root test statistics based on the pooled ordinary least squares estimator In their paper, they derive limiting distributions for the normalized least squares estimators when becomes large and is finite Under the model (16) and H 0 : ρ =1, (ˆρ 1 b) D = (0,C), (13) as with fixed, where ˆρ is an ordinary least squares estimator of ρ under the model (16), and b = 3 +1,

18 10 C = 3( ) 5( 1)( +1) 3 Phillips and Moon (1999) provide regression limit theory for nonstationary panel vector autoregressive models hey provide theorems which describe the relationship between sequential limits, where is followed by and joint limits, where both and become large simultaneously Also, they develop the joint probability limit and joint central limit theorems for deriving a limiting distribution as, In their model, each subject has a vector of observations at each time and the goal is to test these for cointegration For their models, they consider no cointegration, a heterogeneous cointegration relation, a homogeneous cointegration relation, and a near-homogeneous relation hey show that for each model, the pooled estimator is consistent and has a Gaussian limiting distribution when properly normalized

19 11 Chapter Simple and Weighted Symmetric Estimation 1 AR(1) process In this chapter, we consider the simple symmetric and the weighted symmetric estimator in testing for a unit root in the panel data model Let y it be the first order autoregressive time series with fixed individual effects y it = η i + ρy it 1 + e it, i =1,,,;t=1,,,, (1) where the e it s are independent and identically distributed as (0,σ ) and y i0 s are fixed Under the stationarity condition, that is, ρ < 1, we can have a backward representation of the autoregressive time series hat is, y it = η i + ρy it+1 + u it, where e it and u it have same covariance structure for each i he discussion follows the logic of the univariate case as laid out by Dickey, Hasza, and Fuller (1984) In their paper, the following univariate seasonal model with seasonal period d is considered d y t = θ i x it + αy t d + e t ; t =1,,,n, ()

20 1 where x it = 1 if t = i(mod d) = 0 otherwise, y d+1,y d+,,y 0 are fixed initial values and the e t s are independent and identically distributed with mean 0 and variance σ he seasonal process in () can be partitioned into d component series such that y dk+i = θ i + αy d(k 1)+i + e dk+i ; i =0,1,,d 1, and each component runs from k = 1tomwhere the total number of observations is n = dm his partition into d component series establishes a useful link between model (1) and () In other words, θ i in the seasonal process corresponds to an individual intercept η i in the panel model whereas the seasonal period d corresponds to the number of panels he weighted symmetric estimator of ρ in the model (1) is the estimator that minimizes w t [y it η i ρy it 1 ] 1 + (1 w t+1 )[y it η i ρy it+1 ] t=1 t=0 If the weight w t 1, the estimator is the ordinary least squares estimator he simple symmetric estimator which is discussed by Dickey, Hasza, and Fuller (1984) can be obtained by setting w t 05 In weighted symmetric estimation for the first order autoregressive process, the weight is defined as follows w t = 0, t =1 t 1, t =,3,, he weighted symmetric estimator of ρ can be obtained from able 1 Let β =[η 1,η, η,ρ] hen the weighted symmetric estimator of β is

21 13 able 1: Regression table for weighted symmetric estimation in AR(1) process Weight Dependent Parameter Variable η 1 η η ρ w 1 y y 10 w y y 11 w y y w y y 1 1 w 1 y y w 1 y y 11 w 1 y y 0 w y y 1 w y y 1 1 w y y 1 w 1 y y 1 1 w 1 y y 1 ˆβ =(X WX) 1 (X WY), where X is the () (+1) matrix that consists of columns corresponding to parameters, Y is the vector of the dependent variables, and W is the () () diagonal matrix of weights hen the simple symmetric estimator of ρ is where [ ˆρ s = Q 1 s it y it 1 t=1y 1 ( y it 1 +05y i +05y i0 ) ] (3) t=

22 14 Q s = ( t=y it 1 +05y i +05y i0 ) 1 ( y it 1 +05y i +05y i0 ) t= he simple symmetric estimator of an individual intercept η i is ˆη i s = 1 ( y it 1 +05y i +05y i0 ) [ ( yjt 1 +05yj +05y Q j0) s t= j=1 t=1 y jt y jt 1 ] j=1 t= On the other hand, the weighted symmetric estimator of ρ is [ ˆρ ws = Q 1 ws it y it 1 t=1y 1 ( 1 ( y it 1 ) + + ( y t= it 1 )y i t= + 1 y i + +1 y i0 y it t= y i y i0 ) ] (4) where Q ws = [ +1 t=y it y ] i 1 [ +1 y it t= y ] i Also, the weighted symmetric estimator of an individual intercept η i is ˆη i ws = 1 ( y it 1 + y i )+ 1 t= y i0+ 1 [ 1 ( y Q ws 3 jt 1 ) j=1 t= + + ( y 3 jt 1 )y j + 1 y j=1 t= 3 j + +1 ( y j=1 jt 1 )y j0 j=1 t= + 1 ] +1 y j y j0 y jt y jt 1 ( y it j=1 j=1 t=1 t= y i ) Based on the estimators in (3) and (4), test statistics for H 0 : ρ = 1 constructed he normalized bias test statistics are can be

23 15 (ˆρs 1 b s )= and 1 [ t=1 (y it y it 1 )y it (y i0 y i ) b s Q si ]/ (5) 1 Q si / where (ˆρws 1 b ws )= 1 [A i b ws Q wsi ]/, (6) 1 Q wsi / b s = 6 +1 = 3 +O(1/ ), b ws = = +O(1/ ), Q si = y it 1 +05(y i + yi0) 1 t= [ y it 1 +05(y i + y i0 )], t= Q wsi = +1 y it t= y i 1 [ +1 y it t= y i ], and A i = + ( +1) ( y 3 it 1 ) ( y t= 3 it 1 )y i + y t= 3 i it y it 1 )y it 1 + yi0 t=1(y 1 y it 1 1 t= y i0(( +1) y it 1 + y i ) t= Also, we consider the following studentized test statistics

24 16 ˆτ s = ˆρ s 1 b s, (7) ˆσ s Q 1 s and ˆτ ws = ˆρ ws 1 b ws ˆσ ws Q 1 ws (8) where and ˆσ s = 1 1 [Y W s Y Y W s X(X W s X) 1 X W s Y], ˆσ ws = 1 1 [Y W ws Y Y W ws X(X W ws X) 1 X W ws Y] Here, W s and W ws are the diagonal matrices of weights for simple symmetric and weighted symmetric estimations, respectively ow, we discuss the joint limiting distributions of the above test statistics as, o derive the joint limiting distributions of test statistics under H 0 : ρ =1 as,, we employ the following joint limit results from Phillips and Moon (1999) In their paper, they present the following theorems for random vectors instead of random variables heorem 1: (Joint Probability Limits) Suppose the random variables Y i are Y i independent across i for all and integrable Let X, = which is square 1 integrable and µ x =lim, E(X, )=lim, E(Y i ) be finite If 1 lim, Var(Y i ) = 0, then X, P µ x as,

25 17 Proof: It is sufficient to show that lim, P{ 1 (Y i E(Y i )) >ɛ}=0 ɛ>0 By Chebyshev s inequality, lim, P{ 1 (Y i E(Y i )) >ɛ} 1 lim ɛ, E 1 (Y i E(Y i )) = 1 1 lim E Y ɛ, i E(Y i ) = 1 lim ɛ, = 0 1 Var(Y i ) he first equality comes from the independence If Y i s are independent and identically distributed across i for all, the condition for heorem 1 reduces to the finiteness of lim Var(Y i ) he following theorem gives us the condition for the joint limit Central Limit heorem heorem : (Joint Limit CL) Suppose that Y i are independent and identically distributed with mean 0 and variance σ across i for all Assume the following conditions hold i) Yi are uniformly integrable in ii) lim σ = σ > 0 hen, 1 Y i D = (0,σ ) as, Proof: Phillips and Moon (1999) show that Lindeberg s theorem holds in panel data In Y i other words, let Z i,, = j=1 and ɛ >0, Var(Y j ) lim E[Zi,, 1{ Z i,, >ɛ}]=0,

26 18 hen as,, Z i,, = (0, D 1) herefore we need to prove that ɛ >0, lim E[ Y i, σ 1{ Y i >ɛ σ}]=0 Since Y i are identically distributed across i for all, E[ Y i 1{ Y σ i >ɛ σ}] = E[ Y 1 1{ Y σ 1 >ɛ σ}] (9) By the assumption of uniformly integrability of Y1, (9) goes to 0 as, Since it satisfies the Lindeberg condition, Y i σ D = (0, 1) Since lim, σ σ =1, Y i σ D = (0, 1) ow we apply these joint limit results to the test statistics in (5)-(8) and assum- heorem 3: Given model (1), under the null hypothesis H 0 : ρ = 1 ing η i = y i0 = 0 for every i, then i) (ˆρs 1 b s ) D = (0, 7) as,,

27 19 ii) ˆτ s = ˆρ s 1 b s [ˆσ sq 1 s ] 05 D = (0, 1) as,, iii) (ˆρws 1 b ws ) D = (0, 9) as,, iv) ˆτ ws = ˆρ ws 1 b ws [ˆσ wsq 1 ws] 05 D = (0, 15) as, Proof of i): Under the assumptions, the normalized bias test statistic for H 0 : ρ =1 (ˆρs 1 b s )= 1 [ t= (y it y it 1 )y it 1 05y i b s Q si ]/ (10) Q si / where Q si = t= y it 1 +05y i 1 [ t= y it 1 +05y i ] If we apply the joint probability limit result to the denominator and the joint limit CL result to the numerator in (10), we find that the normalized bias test statistic has a Gaussian limiting distribution as, For convenience, the first and the second finite sample moments of random variables in the test statistics under the null hypothesis are shown in Appendix A ow, E(Q si / )= +1 σ σ o prove that 1 Q si / P 1 6 σ, we need to show lim Var(Q si / ) < Since 1 is Var(Q si / )= σ 4 σ <, from heorem 1,

28 0 1 Q si / P 1 6 σ as, ow, we will prove the numerator in (10) satisfies the conditions for heorem From the first and second moment results in Appendix A, we find that and E [ t= (y it y it 1 )y it 1 05y i b s Q ] si =0, Var [ t= (y it y it 1 )y it 1 05y ] i b s Q si = σ 4 [ ( +1) 0 ( +1)+17 5 ] Defining Z i as [ t= (y it y it 1 )y it 1 05y i b s Q si ]/, we need to show that Z i is uniformly integrable in If we consider the -limit distribution of Z i, Z i D = [ ( 0 1 Bi(t)dt ( B i (t)dt) )] σ 4 def = Zi 0 as, where B i (t) is a Wiener process (Phillips,1987) Since Zi are uniformly integrable in Hence, by heorem, lim E(Z i )=E(Zi)= σ4 5, 1 [ (y it y it 1 )y it 1 05y i b s Q si ]/ = (0, D σ4 5 ) t=

29 1 herefore, (ˆρs 1 b s ) D = (0, 7) as, by Slutsky s heorem Proof of ii): he studentized test statistic from simple symmetric estimation is ˆτ s = 1 [ t= (y it y it 1 )y it 1 05y i b s Q si ]/ 1 ˆσ s Q si / Since ˆσ P s σ and 1 Q si / P 1 6 σ as,, the denominator in the studentized statistic converges to σ 1 in probability Since the numerator converges 6 to (0, σ4 ) in distribution from the Proof of i), 5 D ˆτ s = (0, 1) Proof of iii): his proof is analogous to the proof of i) We apply joint limit results to the normalized bias test statistic in (6) If we look at the moments of the denominator in (6) first, and E(Q wsi / )= σ σ 6 4 6, Var(Q wsi / )= σ 4

30 Since lim Var(Q wsi / )= σ4 45 <, 1 Q wsi / P σ 6 as, he first and second moments of the numerator in (6) are and E([A i b ws Q wsi ]/ )=0, [ Var([A i b ws Q wsi ]/ ) = σ ] 60 4 ( +1) [ 13 9 ] + σ ( +1) As in the proof of (i), let us define P i =[A i b ws Q wsi ]/ o prove that Pi uniformly integrable in, we consider the -limit distribution of Pi is P i as Since D = [ (B i (1) 1) + Bi (t)dt ( B i (t)dt)b i (1) ] σ 4 def = Pi 0 0 lim E(P i )=E(Pi)= σ4 4, P i is uniformly integrable in Since it satisfies the condition of heorem, 1 [A i b ws Q wsi ]/ = (0, D σ4 4 )

31 3 Hence, (ˆρws 1 b ws ) D = (0, 9) as, Proof of iv): By arguments similar to those in (ii), we can prove that ˆτ ws D = (0, 15) as, ow we present the empirical percentiles of the test statistics in (5)-(8) from Monte Carlo simulations We conduct simulations for the following combinations of and (,) = (10, 5), (10, 50), (10, 100), (5, 5), (5, 50), (5, 100), (50, 5), (50, 50), and (50, 100) For each set, we generate y it from the random walk model, that is, y it = y it 1 + e it with 10,000 replicates, where the e it s are generated from (0,1) and y i0 = 0 for each i As a random number generator, we use the RAOR function from SAS he empirical percentiles of the normalized bias test statistics for the simple symmetric and the weighted symmetric estimators are shown in able and able 3, respectively Also, the empirical percentiles of the studentized test statistics are shown in able 4 and able 5, respectively he percentiles for the limiting distribution in each table can be computed from the standard Gaussian distribution table umbers in the tables are unsmoothed However a smoothing surface describes the 5 % critical values reasonably well In order to smooth the 5 % critical values of the studentized test statistics, we compute the deviations of the 5 % empirical percentiles in finite samples from the limiting 5 % percentiles hen we regress these deviations

32 1 on and 1 For simple symmetric estimation, R is 097 and the smoothed percentiles are given by For the weighted symmetric regression, R is 086 and the smoothing function for the empirical percentiles is

33 5 able : Empirical cumulative distribution of (ˆρ s 1 b s ) Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, ) (b s = 6/( +1))

34 6 able 3: Empirical cumulative distribution of (ˆρ ws 1 b ws ) Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, ) (b ws = ( + +)/( 3 +1))

35 7 able 4: Empirical cumulative distribution of ˆτ s Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, )

36 8 able 5: Empirical cumulative distribution of ˆτ ws Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, )

37 9 Higher Order Processes In this section, we consider the limiting distributions of the simple symmetric and the weighted symmetric estimators for the autoregressive higher order processes If we generalize model (1) to a pth order autoregressive process, p y it = η i α k y it k + e it, i =1,,,;t=1,,,, (11) k=1 where the e it s are independent and identically distributed as (0,σ ) and y i, p+1,y i, p+,,y i0 are fixed We consider the case when p 1 roots of p m p + α k m p k =0 k=1 are less than one and there exists only one unit root We can rewrite model (11) as where p y it = η i + θ 1 y it 1 + θ k y it k+1 + e it (1) k= p θ 1 = α k, k=1 p θ l = α k, k=l l =,3,,p and signifies the first lag difference operator, that is, y it = y it y it 1 In model (1), the null hypothesis for testing a unit root is H 0 : θ 1 = 1 In this section, we derive the simple symmetric and weighted symmetric estimators when p = Extension to general p is straightforward From (1), a second order autoregressive model is

38 30 y it = η i + θ 1 y it 1 + θ y it 1 + e it (13) As in the first order autoregressive process, a backward representation can be valid under the stationarity condition, that is, y it = η i + θ 1 y it+1 θ y it+ + u it, where e it and u it have the same covariance structure for each i For simple symmetric estimation, the regression uses weights w t =05 In weighted symmetric estimation of a pth order process, the weight is defined as follows w t = 0, t =1,,,p, t p p+, t = p+1,p+,, p+1, 1, t = p+, p+3,, Fuller (1996) presents the regression table for weighted symmetric estimation in the univariate case We generalize his table to panel data with fixed individual effects and obtain the weighted symmetric estimators from able 6 Let β be a vector of parameters, that is, [ ] η 1,η,,η,θ 1,θ hen the weighted symmetric estimator for β is ˆβ =(X WX) 1 (X WY), where X is the ( ) ( + ) matrix that consists of columns corresponding to ( + ) parameters, Y is the vector of dependent variables, and W is the ( ) ( ) diagonal matrix of weights he simple symmetric estimator of θ 1 is

39 31 able 6: Regression table for weighted symmetric estimation in AR() process Weight Dependent Parameter Variable η 1 η η θ 1 θ w 3 y y 1 y 1 w 4 y y 13 y 13 w y y 1 1 y w 1 y y 1 1 y 1 1 w y y 1 y w y y 1 y 13 w 3 y y y w 4 y y 3 y 3 w y y 1 y 1 1 w 1 y y 1 y 1 w y y y 1 1 w y y y 3 ˆθ 1,s = 1 [ 1 c s S [( y it 1 ) + 1 t=3 ( y it 1 )( y i y i )] t=3 1 + ( ) S 1 [( y it 1 )( y i y i ) 1 t=3 ( y i y i ) ] + S [ y it y it t=3 (y i1y i y i y i 1 )] S 1 [ y it y it t=3 (y i1y i y i y i 1 )] ], (14)

40 3 where S 11 = y it 1 1 ( y it 1 ), t=3 t=3 S 1 = y it 1 y it (y i y i1 y i y i 1 ) t=3 1 ( y it 1 )( y i y i ), ( ) t=3 S = it 1 ) t=3( y 1 [( y i ) ( y i ) ] 1 ( y i y i ), 4( ) and c s = S 11 S S 1 Also, the simple symmetric estimator of θ is ˆθ,s = 1 [ 1 c s S 1 [( y it 1 ) + 1 t=3 ( y it 1 )( y i y i )] t=3 1 ( ) S 11 [( y it 1 )( y i y i ) 1 t=3 ( y i y i ) ] S 1 [ y it y it t=3 (y i1y i y i y i 1 )] + S 11 [ y it y it t=3 (y i1y i y i y i 1 )] ] (15) On the other hand, the weighted symmetric estimator of θ 1 is

41 33 ˆθ 1,ws = 1 [ 1 c ws ( ) W [ 3 ( y it 1 ) +( y it 1 )(y i1 + y i )] t=3 t=3 ( ) W 1 [( y it 1 )(y i1 + y i )+ 3 t=3 ( y it 1 ) ] t=3 + 1 ( ) W 1 [( y i y i )(y i1 + y i + 3 y it 1 )] t=3 1 + W 1 [(y i1 + y i ) + 3 ( y it 1 )(y i1 + y i )] t=3 + W [y i1 y i + y it y it 1 ] t=3 + W 1 [ 1 y it y it y it y it 1 y i1 y i ] ], (16) t=3 t=3 where W 11 = 1 y ( 1) it 1 ( y t=3 ( ) 3 it 1 ), t=3 1 W 1 = [ y it 1 y it 1 y it 1 y i y i 1 +( 1)y i y i1 ] t=3 t=3 + 1 [ ( ) ( y it 1 ) 1 t=3 ( y it 1 )( y i y i )] t=3 1 ( y ( ) it 1 )(y i1 + y i ), t=3 W = + 1 [( y i ) +( y i ) 1 + ( y it ) ] t=3 1 [4( y ( ) 3 it 1 ) +( 1) ( y i y i ) ] t=3 1 [(y i1 + y i ) 4( 1) ( ) ( y it 1 )( y i y i )] t=3 1 [4( y ( ) it 1 )(y i1 + y i ) ( 1)( y i y i )(y i1 + y i )], t=3

42 34 and c ws = W 11 W W 1 Also, the weighted symmetric estimator of θ is ˆθ,ws = 1 1 c ws ( ) W 1 [ 3 ( y it 1 ) +( y it 1 )(y i1 + y i )] t=3 t=3 + ( ) W 11 [( y it 1 )(y i1 + y i )+ 3 t=3 ( y it 1 ) ] t=3 1 ( ) W 11 [( y i y i )(y i1 + y i + 3 y it 1 )] t=3 1 W 11 [(y i1 + y i ) + 3 ( y it 1 )(y i1 + y i )] t=3 W 1 [y i1 y i + y it y it 1 ] t=3 W 11 [ 1 y it y it y it y it 1 y i1 y i ] ] t=3 t=3 (17) Based on simple symmetric estimators and weighted symmetric estimators in (14)- (17), we construct the studentized test statistics for a unit root In the univariate case, it is known that the studentized test statistics in higher order processes have the same limiting distribution as in the first order autoregressive process he following theorem shows that a similar invariance holds for the sequential limit distributions of the studentized test statistics in panel data heorem 4: Given model (13), under the null hypothesis H 0 : θ 1 = 1 and assuming that η i = y i0 = y i, 1 = 0 for each i, then i) ˆτ(θ 1 ) s = ˆθ 1,s 1 b () s (1 ˆθ,s ) [ˆσ ss /c s ] 05 as is followed by, D = (0, 1)

43 35 ii) ˆτ(θ 1 ) ws = ˆθ 1,ws 1 b () ws(1 ˆθ,ws ) [ˆσ wsw /c ws ] 05 as is followed by, where D = (0, 15) and 6( ) s = ( ) +1, b () b () ws = ( ) +( ) + ( ) 3 +1 Here, b () s and b () ws can be obtained by plugging instead of into b s and b ws in the AR(1) process his idea comes from the fact that we use ( ) observations in the AR() regression table However, this does not affect our limiting distribution results Also, ˆσ s and ˆσ ws are mean squared errors in symmetric and weighted symmetric estimations, respectively Proof: Before we derive the sequential limit distributions of test statistics, we consider the -limit distributions of several random variables which are used in our test statistics Dickey (1976) presents the -limit distribution results in higher order processes When θ 1 = 1 and y i0 = y i, 1 = η i = 0 in model (13), we have the following equation t (1 θ )y it = e ij θ y it j=1 From the above equation and Dickey (1976) s results, we can obtain the following -limit results for each i t=3 y it 1 σ 1 0 B i (t)dt, (1 θ )

44 36 t=3 y it 1 3/ σ 1 0 B i(t)dt, 1 θ y i 1/ σb i(1) 1 θ, and t=3 y it 1 y it 1 t=3 ( y it 1 ) + yi 1 = + O p (1/ ) σ [ 1 1 θ + B i (1) ], (1 θ ) t=3 y it 1 y it = y i t= ( y it ) + O p (1/ ) σ [ Bi (1) (1 θ ) 1 1 θ ] Proof of i): From the above -limit results, we can assert that S 11 σ (1 θ ) [ Bi (t)dt ( B i (t)dt) ], 0 S 1 3/ 0, and S 1 σ (1 θ ), S σ 1 θ

45 37 herefore, 3 c s σ 4 [ 1 0 B i (t)dt ( 1 0 B i(t)dt) ] (1 θ ) (1 θ ) If we look at the studentized test statistic based on the symmetric estimator, ˆτ(θ 1 ) s = [ˆθ 1,s 1 b () s (1 ˆθ,s )] [ˆσ s( 1 S )/( 3 c s )] 05 Let us consider the -limit distribution of [ˆθ 1,s 1 b () s (1 ˆθ,s )] first [ˆθ 1,s 1 b () s (1 ˆθ,s )] = Since ˆθ,s θ as, S 1 + S + S S 1 1 [ S ( t=3 y it 1 )( y i y i ) 3 c s ( ) ( y i y i ) 4( ) t=3 ( y it )y it 1 (y i1 y i y i y i 1 ) t=3 y it y it 1 b () s c s (1 ˆθ 3,s ) ] [ˆθ 1,s 1 b () s (1 ˆθ,s )] (1 θ ) [ 1 +3( 1 0 B i(t)dt ( 1 0 B i(t)dt) )] ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) Also, the denominator part has the following -limit distribution ˆσ s( 1 S )/( 3 c s ) (1 θ ) ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) herefore, the -limit distribution of ˆτ(θ 1 ) s is

46 38 [ˆθ 1,s 1 b () s (1 ˆθ,s )] [ˆσ s( 1 S )/( 3 c s )] 05 From Appendix C, we know that 1/ [ 1 +3( 1 0 B i(t)dt ( 1 0 B i(t)dt) )] 1 ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) E [ 1 +3( Bi(t)dt ( B i (t)dt) )] =0, 0 and Var [ 1 +3( Bi(t)dt ( B i (t)dt) )] = 1 0 5, E [ Bi (t)dt ( B i (t)dt) ] = If we apply the Central Limit heorem and Strong Law of Large umbers to our -limit distribution, we find the same limit distribution as in the AR(1) case: ˆτ(θ 1 ) s as is followed by D = (0, 1) Proof of ii): From the -limit results in the previous pages, we find that W 11 σ (1 θ ) [ Bi (t)dt ( B i (t)dt) ], 0 W 1 3/ 0,

47 39 W 1 σ (1 θ ) + σ [ 1 B i (1) 1 0 B i (t)dt +( 1 0 B i(t)dt) ( 1 0 B i(t)dt)b i (1)] (1 θ ), and W σ 1 θ, 3 c ws σ 4 [ 1 0 B i (t)dt ( 1 0 B i(t)dt) ] (1 θ ) (1 θ ) Using similar arguments to those in the proof of i), we obtain the following result and [ˆθ 1,ws 1 b () ws(1 ˆθ,ws )] (1 θ ) [ 1 ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) B i (1) Bi (t)dt ( B i (t)dt)b i (1) ], 0 0 ˆσ ws( 1 W )/( 3 c ws ) (1 θ ) ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) herefore, the -limit of ˆτ(θ 1 ) ws is ˆτ(θ 1 ) ws 1/ [ 1 (B i (1) 1) B i (t)dt ( 1 0 B i(t)dt)b i (1)] 1 ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) If we consider the moments of the numerator and denominator as in the proof of i), E [ (B i (1) 1) + Bi (t)dt ( B i (t)dt)b i (1) ] =0, 0 0

48 40 Var [ (B i(1) 1) + Bi (t)dt ( B i (t)dt)b i (1) ] = , and E [ Bi (t)dt ( B i (t)dt) ] = By the Central Limit heorem and Strong Law of Large umbers, we again have the same sequential limit distribution as in the AR(1) case: ˆτ(θ 1 ) ws D = (0, 15) as is followed by We find that the studentized test statistics in higher order processes have the same limiting distribution as in the first order autoregressive process Even though we have different bias adjust terms like b () s (1 ˆθ s ) in the test statistics, those do not affect the limiting distributions hrough Monte Carlo simulation, we compute the empirical percentiles of ˆτ(θ 1 ) s and ˆτ(θ 1 ) ws under the model (13) and the unit root null hypothesis We consider the same sets of (,) as in able Data are generated from y it = y it 1 + θ y it 1 + e it with 10,000 replicates where the e it s are generated from (0,1) and we set the value of θ equal to 05 able 7 and able 8 show the empirical percentiles for ˆτ(θ 1 ) s and ˆτ(θ 1 ) ws, respectively For small samples like = 5 and = 50, there are discrepancies between the first order autoregressive process and higher order processes hese discrepancies for small samples are due in part to the fact that we apply the bias adjustment factors from the first order autoregressive process to the test statistics in higher order processes When =100, the empirical percentiles are close to each other

49 41 able 7: Empirical cumulative distribution of ˆτ(θ 1 ) s Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, )

50 4 able 8: Empirical cumulative distribution of ˆτ(θ 1 ) ws Probability of a Smaller Value (,) (10,5) (10,50) (10,100) (5,5) (5,50) (5,100) (50,5) (50,50) (50,100) (, )

51 43 Chapter 3 Maximum Likelihood Estimation with Fixed Individual Effects 31 Maximum Likelihood Estimator for the Fixed Individual Effects Model In this chapter, we consider the maximum likelihood estimation method in testing for H 0 : ρ = 1 in the panel data model with fixed individual effects Let y it be a univariate random variable for the ith individual unit at time t he autoregressive model of order 1 with fixed individual effects can be written as follows y it = µ i + ɛ it ɛ it = ρɛ it 1 + e it, i =1,,,;t=1,,,, (31) where the e it s are independent identically distributed as (0,σ ) We assume that the first observation y i1 is distributed as (µ i, ) for each i Gonzalez and 1 ρ Dickey (1999) present the likelihood function in a single time series, that is, for =1 σ

52 44 Under the model (31), the log-likelihood function is lnl = ln(1 ρ ) lnσ 1 ρ lnπ (y σ i1 µ i ) 1 [y σ it µ i ρ(y it 1 µ i )] (3) t= For given ρ < 1, the individual mean estimator, ˆµ i which maximizes the likelihood function (3) is y i1 + y i +(1 ρ) 1 +( )(1 ρ) t= y it, (33) his mean estimator is same as the one in a single time series (Gonzalez and Dickey, 1999) Let ˆρ ml,fix be the maximum likelihood estimator of ρ under the model (31) o find the limiting distribution of (ˆρ ml,fix 1), we reparametrize (1 ρ) asζ he individual mean estimator, ˆµ i can be derived from the following regression table Dependent Parameter Variable µ 1 µ µ ζ ( ζ )y 11 y 1 y 11 + ζ y 11 y 1 y ζ y 1 1 ζ ( ζ ) 0 0 ζ 0 0 ζ 0 0 ζ ( ζ )y y y 1 + ζ y y y 1 + ζ y ζ ( ζ ) ζ ζ

53 45 o reduce the number of unknown parameters, we substitute ˆµ i for µ i in the loglikelihood (3) he resulting profile log-likelihood is lnl = ζ ln( ( ζ )) lnσ lnπ 1 SS(ζ), (34) σ where SS(ζ) is the sum of squares of errors from the previous regression table, that is, SS(ζ) = ( 1)ζ3 ( )ζ 3 ζ y i1 + (y it y it 1 ) ( ζ + ζ) t= + ζ (y it y it 1 )y it 1 + ζ y t= it 1 t= 1 [ ζ ζ( ζ y 1 i + ζ4 ( y )+ ζ 3 it 1 ) t= + ζ (1 ζ ) y i1 y i + ζ3 (1 ζ ) y i1 ( y it 1 ) t= + ζ3 y i ( y it 1 ) ] t= From (34), we know that the maximum likelihood estimator of σ is SS(ζ) By plugging this estimator into (34), we can concentrate on finding the maximum likelihood estimator of ζ, that is, (1 ρ) Since the maximum likelihood estimators of µ i, σ, and ζ are linked to each other, iterative calculation is needed until the estimators converge he corresponding profile log-likelihood becomes Let f (ζ) be 1 hen lnl = ζ ln( ( ζ )) lnss(ζ) lnπ (35) [ ( lnl) ζ ] SS(ζ)

54 46 (1 ζ f (ζ) = ) SS(ζ) ζ( ζ ) + 1 (SS(ζ)) ζ Finding the value ˆζ which makes f (ˆζ) = 0 is equivalent to finding the maximum likelihood estimator of ζ herefore we obtain the maximum likelihood estimator of ζ by solving the equation f (ζ) = 0, where f (ζ) is a polynomial function of ζ y t= i1 Also, we can express f (ζ) as a linear function of, (y it y it 1 ), t= (y it y it 1 )y t= it 1 y it 1 y i (,,, t= y it 1 ), 3 y i1 y i y i1 (, t= y it 1 ) y i (, and t= y it 1 ) hat is, y t= i1 (y it y it 1 ) f (ζ) = c 1 + c t= y it 1 y i + c 4 +c 5 + c 7 y i1 y i + c 8 y i1 ( t= y it 1 ) where the coefficients and their -limits are +c 3 t= (y it y it 1 )y it 1 +c 6 ( t= y it 1 ) 3 +c 9 y i ( t= y it 1 ), c 1 = 0, 1 3 ( + ζ)( ζ + ζ) [ ( 1) ( )ζ 4 + ( )ζ 3 ( )ζ ( )ζ 1 4 ( 1) ] (1 ζ c = ) ζ( ζ 1 ) ζ,

55 47 c 3 = ( )ζ 4 ( 1) (ζ ), c 4 = ( 1)ζ +( 4 )ζ (ζ ) ζ, c 5 = ( )ζ 4 ( )ζ +4 ( 1) ( ζ + ζ) (ζ ) ( + ζ), c 6 = ( )( 1)ζ4 +4( 4 +)ζ 3 +4 (3 1)ζ ( ζ + ζ) (ζ ) ζ (3 + ζ) ( + ζ), c 7 = ( ) ζ 3 4 ( 6 +6)ζ 4 ( 1)ζ +8 3 ( 1) ( ζ + ζ) (ζ ) 0, c 8 = 4( )( 1)ζ4 ( )ζ 3 +4 ( 11 +6)ζ ( ζ + ζ) (ζ ) 8 3 ( 1)ζ + 0, ( ζ + ζ) (ζ ) and c 9 = ( ) ζ 3 +4( 4)( 1)ζ +8 ( 1)ζ ( ζ + ζ) (ζ ) ζ (4 + ζ) ( + ζ) Since f (ζ) is a higher order polynomial of ζ, the Gauss ewton iteration method is appropriate to find ˆζ For this procedure, we need to compute the first derivative of f (ζ) with respect to ζ If we define this first derivative as f (ζ),

56 48 f (ζ) y t= i1 (y it y it 1 ) = d 1 + d t= y it 1 y i + d 4 +d 5 + d 8 y i1 ( t= y it 1 ) where the coefficients and their -limits are t= (y it y it 1 )y it 1 +d 3 ( +d t= y it 1 ) y i1 y i 6 +d 3 7 +d 9 y i ( t= y it 1 ), d 1 = 1 ( ζ + ζ) 3 (ζ ) 3 [ ( 1) ( ) ζ 5 +4 ( 1) ( 7)( )ζ 4 ( )ζ 3 0, 4 3 ( )ζ 4 4 ( )ζ +8 5 ( 5 +5) ] d = (ζ ζ + ) ζ (ζ ) 1 ζ, d 3 = 4 (ζ ) 0, d 4 = ( 1)ζ 8 ( 1)ζ +4 ( 1) (ζ ), d 5 = ( ) ζ 3 +4( 7 +6)ζ ( ζ + ζ) 3 (ζ ) + 4 ( )(5 3)ζ +8 3 ( 5 +1) ( ζ + ζ) 3 (ζ ) 4 ( + ζ), 3

57 49 d 6 = 1 ( ζ + ζ) 3 (ζ ) [ ( 1)( ) ζ 5 +4 ( 1)( )( 7)ζ 4 4 ( )ζ (3 1)ζ ] (ζ +6ζ+ 1)ζ, ( + ζ) 3 d 7 = 8( )ζ3 +16( 3)ζ 8 (5 3 6)ζ ( 1) ( ζ + ζ) 3 (ζ ) 0, and d 8 = 0, 1 ( ζ + ζ) 3 (ζ ) [ 4( 1)( ) ζ 5 8 ( 1)( 7)( )ζ 4 +4 ( )ζ ( )ζ (17 10)ζ 3 5 ( 1) ] d 9 = 1 ( ζ + ζ) 3 (ζ ) [ 4( 4)( )ζ 3 8 (7 1)ζ (5 6)ζ 3 4 ( 1) ] 16 ( + ζ) 3 In the following section, we discuss the joint limit distribution of the test statistic based on ˆζ, ie, (1 ˆρ ml,fix ) under H 0 : ρ =1 3 Limiting Distribution Before we derive the joint limit distribution, let us discuss the sequential limit distribution From the previous section,

58 50 f (ζ) σ [ 1 ζ +ζ ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) (Bi (1) 1) + ( + ζ) ( 1 (B i (1) 1 0 B i(t)dt ( 1 0 B i(t)dt) ) 0 B i(t)dt B i (1)) ] We find that the -limit distribution of f (ζ) can be expressed as a function of averages of -limit random quantities In other words, where f (ζ) σ [ 1 ζ +ζa + B ( + ζ) C ] A = ( 1 0 B i (t)dt ( 1 0 B i(t)dt) ) 1 6, B = (B i (1) 1) (B i (1) 1 0 B i(t)dt ( 1 0 B i(t)dt) ) 1 3, and C = ( 1 0 B i(t)dt B i (1)) 1 3 Let us define this -limit of f (ζ) asm(a,b,c,ζ) If ζ(a,b,c ) denotes a solution of ζ such that M(A,B,C,ζ) = 0, then this would be the maximum likelihood estimator of ζ in the -limit We expand ζ(a,b,c ) in a aylor series about ζ( 1 6, 1 3, 1 3 ) on the surface defined by M(A,B,C,ζ)=0 Since [ A 1 6,B + 1 3,C 3] 1 =Op (1/ ),

59 51 ζ(a,b,c ) = ζ( 1 6, 1 3, 1 ζ )+ (A 1 3 A 6 )+ ζ (B + 1 B 3 ) + ζ (C 1 C 3 )+O p(1/ ) = (A 1 6 ) 1896(B ) +0199(C 1 3 )+O p(1/ ) From the -limit moment results in Appendix C and the Central Limit heorem, [ (ˆρml,fix 1)+37 ] as is followed by D = (0, 6719) ow, we discuss the joint limit distribution of the test statistic based on ˆζ where the order of and does not matter From the previous section, we know that f (ζ) can be expressed as a linear function of random quantities which have the form yi1 of averages across i, for example, From the moment results in Appendix A and Weak Law of Large umber theory, f (ζ) P [ 1 1 c 1 + c + c 4 +c 7 + c c c c ] σ, as Let us define this -probability limit as f(ζ) hen f(ζ) does not have randomness anymore If ζ 0 ( ) denotes the value which satisfies f(ζ 0 ( )) = 0, we find that ζ 0 ( ) depends on Because 1 < ˆρ ml,fix < 1 and ˆζ = (1 ˆρ ml,fix ), we restrict the range of ζ 0 ( ) such that it lies within [ 0, ] for > For several s, we display the values of ζ 0 ( ) hese values are rounded off to have 4 digits

60 5 ζ 0 ( ) From the sequential limit result, we know that ˆζ-ζ 0 ( )iso p (1/ ) herefore, if we consider the aylor series expansion of f (ζ) about ζ 0 ( ), f (ˆζ) =f (ζ 0 ()) + f (ζ 0 ( ))(ˆζ ζ 0 ( )) + O p ( 1 ) Since f (ˆζ) = 0 by definition of ˆζ, [ˆζ ζ0 ( ) ] f (ζ 0 ()) f (ζ (36) 0 ( )) ow we derive the joint limit distribution of [ (ˆρ ml,fix 1) + ζ 0 ( ) ] by using joint probability limit theory and the joint CL from Chapter heorem 31: Given model (31), let us assume that y i0 = 0 for every i hen under H 0 : ρ =1, as, [ (ˆρml,fix 1) + ζ 0 ( ) ] D = (0, 6719), Proof: Moment results used in this theorem are from Appendix A From (36), we can obtain the joint limit distribution of [ (ˆρ ml,fix 1) + ζ 0 ( ) ] by deriving the joint limit distribution f (ζ 0 ()) of f (ζ In other words, we apply joint probability theory to f (ζ 0 ( )) and 0 ( )) the joint CL to f (ζ 0 ())

61 53 ow where f (ζ) = f i (ζ), f t= i (ζ) = d 1 yi1 (y it y it 1 ) t= (y it y it 1 )y it 1 +d + d 3 y i + d 5 + d ( t= y it 1 ) y i1 ( t= y it 1 ) 6 + d 3 7 y i1 y i + d 8 y i ( + d t= y it 1 ) 9 + d 4 t= y it 1 If we take the expectation for f i (ζ 0 ( )), E [ f i (ζ 0 ( )) ] In order to show that f (ζ 0 ( )) P 057 as,, we need to prove lim Var [ f i(ζ 0 ()) ] < From the variancecovariance calculation results in Appendix A, a few values of the variance are Var [ f i(ζ 0 ()) ]

62 54 herefore, from heorem 1 (a joint probability limit theorem), as, ow, if we look at f (ζ), f (ζ 0 ( )) P 057 (37) where f (ζ)= f i (ζ), t= f i (ζ) = c 1 yi1 (y it y it 1 ) t= (y it y it 1 )y it 1 +c + c 3 y i + c 5 + c ( t= y it 1 ) y i1 ( t= y it 1 ) 6 + c 3 7 y i1 y i + c 8 + c 4 t= y it 1 + c 9 y i ( t= y it 1 ) By the definition of ζ 0 ( ), we know that E [ f i (ζ 0 ( )) ] = 0 he variances of f i (ζ 0 ( )) for several s are Var [ f i (ζ 0 ()) ] ow, we need to prove the uniformly integrability of fi (ζ 0 ( )) in If we consider the -limit distribution of fi (ζ 0 ( )), f i (ζ 0 ( )) D = [ 1 ( B i (1) ζ 0 ( ) +ζ 0( ) ( B i (t)dt ( Bi(t)dt ( B i (t)dt) ) +(Bi(1) 1) 0 B i (t)dt) ) ( 1 0 B i(t)dt B i (1)) (ζ 0 ( )+) ] σ 4,

63 55 as, where, as previously stated, ζ 0 ( )=37 From Appendix C, the expectation of the right hand side in the above equation is 1866 Since this value is equal to lim E [ fi (ζ 0 ( )) ], fi (ζ 0 ( )) is uniformly integrable in If we apply Slutsky s theorem to (36) using the result (37), [ (ˆρml,fix 1) + ζ 0 ( ) ] D = (0, 6719), as, We conduct Monte Carlo simulation to obtain the empirical percentiles of [ (ˆρml,fix 1) + ζ 0 ( ) ] hese percentiles are shown in able 31 We use the same sets of (,) as in Chapter For each combination of (,), we generate y it such that y it = y it 1 + e it, where the e it s are generated using the RAOR function in SAS and y i0 = 0 for each i We have 10,000 replicates For the computation of ˆρ ml,fix, we run PROC MIXED in SAS, since PROC MIXED has the likelihood function for the autoregressive model of order 1 We use the same method as in Chapter for smoothing the 5 % empirical percentiles in able 31 he coefficient of determination is over 099 and the square root of the mean squared error is herefore, we can approximate the 5 % critical values with a linear function of and 1, namely critical value