Bias in the Mean Reversion Estimator in the Continuous Time Gaussian and Lévy Processes

Transcription

1 Bias in the Mean Reversion Estimator in the Continuous Time Gaussian and Lévy Processes Aman Ullah a, Yun Wang a, Jun Yu b a Department of Economics, University of California, Riverside CA b School of Economics and Sim Kee Boon Institute for Financial Economics Singapore Management University, 7893, Singapore July Abstract This paper considers the bias of the mean reversion estimator (b) in the continuous time Lévy processes. Although an extensive literature has developed methods for estimating the parameters in continuous time di usion models and for approximating the estimation bias, the e ect of nonnormality on the estimation has not been studied. The bias of b is approximated and two bias expressions are obtained for the Lévy-based Ornstein-Uhlenbeck (OU) process. The approximate bias of b under normality is also derived as a special case. The two bias expressions indicate that both the skewness and kurtosis of the Lévy measure a ect the bias when the time span and the sample size is not very large. The initial condition, the long term mean (), and the volatility parameter () also enter the bias expressions. A bias corrected estimator of is proposed. Monte Carlo studies are conducted to compare four di erent estimators of. Simulation results suggest that our proposed estimator of outperforms other bias corrected estimators proposed in the literature. Keywords: Bias, Continuous Time Models, Lévy Process JEL Classi cation: C3, C Address Correspondence to: Aman Ullah, Department of Economics, University of California, Riverside, CA 95-47; aman.ullah@ucr.edu. Yun Wang, Department of Economics, University of California, Riverside, CA 95-47; yun.wang3@ .ucr.edu. Jun Yu, School of Economics, Sinagpore Management University, 9 Stamford Road, Singapore 7893; yujun@smu.edu.sg.

2 Introduction In recent years, an extensive literature has developed on using di usion processes to model the dynamic behavior of nancial securities. For example, Vasicek (977) used the following the Ornstein- Uhlenbeck (OU) process to model the spot interest rate, dx t = ( X t )dt + db t (.) where B t is a standard Brownian motion. This is a Gaussian Markov process and posesses a stationary distribution when >. In this case, is the converge rate to pull the process towards its long term mean. Tang and Chen (9) considered a more general form of a Brownian motion based continuous time model (i.e. di usion process) dx t = ( X t )dt + (X t ; )db t (.) where (X t ; ) is the di usion function of X t at time t. The Vasicek (977) model is a special case of this di usion process, which has the constant di usion function (X t ; ) =. If (X t ; ) = p X t, the di usion process becomes the CIR model (Cox, Ingersoll, and Ross, 985). A even more general di usion process is given by dx t = (X t ; )dt + (X t ; )db t ; (.3) with a general drift function (X t ; ). An important special case is when (X t ; ) = X t and (X t ; ) = X t. Black and Scholes (973) used it to model the spot price of a stock. All these processes are Brownian-motion based. Under some smoothness condition on the the drift function and the di usion function, the sample path generated from X t is continuous everywhere. In recent years, however, it has been reported strong evidence of in nite activity jumps in nancial variables; see, for example, Jacod and Aït-Sahalia (8). Not surprisingly, continuous time Lévy processes have become increasingly popular, and various Lévy models have been developed in asset pricing literature (see for example, Barndor -Nielsen (998), Madan, Carr and Chang (998), Carr and Wu (3)). In practice, one can only obtain the observations at discrete time points from a nite time span, i.e. T (< ) is the time span, h (> ) the sampling interval, and n (= T=h; ) the number of observations. Based on discrete time observations, di erent methods have been used to estimate the continuous time models. Phillips and Yu (9) provide an overview of some widely used estimation methods. When the drift function is linear and slowly mean reverting, it is found that there is serious estimation bias in the mean reversion parameter () by almost all the methods. Because this parameter is of important implications for asset pricing, risk management and forecasting, how to accurately estimate this parameter has received considerable attention in the literature. For example, Yu (9) approaximates the bias of MLE of when the long run mean is known and the initial condition is the marginal distribution under the Gaussion OU process. Tang and Chen (9)

3 approaximates the bias of MLE of when the long run mean is unknown under the Gaussion OU process and the CIR model. To reduce the estimation bias of, Phillips and Yu (5) proposed the jackknife method. While jackknife increases the variance, a carefully designed jackknife procedure can o er substain improvement in reducing the bias, leading to a decrease in the root mean square errors (RMSE). To further reduce RMSE Phillips and Yu (9) proposed the indirect inference method and Tang and Chen (9) proposed a parametric bootstrap method. The latter methods are simulation-based and hence numerically more demanding. The di culty in estimation is not unexpected because it is related to the nite sample bias problem well documented in the discrete time literature; see, for example, Kendall (954). However, the magnitude of the bias in is so large in practially relevant case that the implications for the bias become very important. For example, Phillips and Yu (5) show that the bias of maximum likelihood estimator for in the CIR model can be over % even though 5 years of data were used (regardless the sample frequency). They further report evidence that the bias in the drift term estimation are even worse than that caused by a misspeci cation of the di usion function and that caused by the discretization of the model using the crudest method, such as the Euler scheme. The simulation results of Phillips and Yu (5) and Tang and Chen (9) show that the bias of the long run mean () and parameters in the di usion function are virtually zero. For instance, in the stationary Vasicek model, as Tang and Chen (9) reported, the bias of b is up to O(T ), while the bias of the di usion parameter and the long run mean parameter are O(n ) and O(n ); respectively. While the bias in b is well studied in the continuous time di usion process, to the best of our knowledge, nothing has been reported on the bias in b in the continuous time Lévy process. The objective of this paper is to approximate the bias of ^ under the Lévy measure, then study the e ects of nonnormality on the estimation bias. Quasi maximum likelihood (QML)/OLS is used to estimate which makes it feasible the analytical expression for b: We present the results on the bias under the assumption where the errors term follow a non-normal distribution with nite rst eight moments. It is found that the kurtosis has negative e ect on the bias of the mean reversion estimator ^: The skewness has positive e ect on the bias of ^ if the distribution has negative skewness. Otherwise, the e ect of skewness on the bias of ^ is negatvie. Both skewness and kurtosis do not a ect the bias as!, or h! : In addition, under the Gaussian OU process the inital condition has non-monotonic e ect on the bias of ^, and the bias of ^ is a monotonically increasing function of the di usion parameter. A bias corrected estimator of the mean reversion estimator ^ is proposed. The simulation results show that our proposed estimator generally performs well in terms of bias and mean square error (MSE), especially, when is small. Small values of correspond to the near unit root situation and is empirically relevant for nancial variables such as interest rates and volatility. The structure of this paper is as follows. In section, we introduce a continuous time Lévy process and derive the bias in the estimation of the mean reversion parameter. Section 3 reports the 3

4 simulation results. Section 4 is conclusions. Parameter Estimation for Lévy Processes. Continuous Time Lévy Process As argued before, while the di usion processes are very useful, empirical evidence points to the need to incorprate jumps with ini nite activity. In this paper we extend the Gaussian OU model of Vasicek to a Lévy-based OU model: dx(t) = ( X(t))dt + dl(t) (.) where (L(t)) t is a Lévy process de ned on (; F; ffg; P ) with L() = and satis es the following three properties:. Independent increments: for every increasing sequence of times t ; : : : ; t n the random variables X t ; X t X t ; : : : ; X tn X tn are independent;. Stationary increment: the law of X t+h X t is independent of t; 3. Stochastic continuity: for all " >, lim h! P (jx t+h X t j ") =. For a given t, the probability of seeing a jump at t is zero. In other words, jumps happen at random times. Obvisouly, the Brownian motion is a special case of the Lévy process. As a result, the Vasicek model is a special case of Model (.). Other well known examples include the Poisson process, the gamma process, the variance gamma process, and the -stable process. While the Brownian motion has a continuous sample path, it does not allow for any jumps. The Poisson process allows for jumps. However, the jump is of nite activity. General Lévy processes allow an in nite number of jumps within any time interval. Also, general Lévy processes allow non-normal increments. The exact discrete time model of (.) is given by r exp( h) X ih = exp( h)x (i )h + " i (.) where the distribution of " i depends on the speci cation of the Lévy measure L(t). This is a discrete time AR() model with a possibly non-normal error term. When L(t) is the Brownion motion, " i N(; ). If L(t) is the variance gamma process of Madan and Seneta (99) (i.e. L(t) = B((t; ; )) where (t; ; ) is a gamma distribution with mean and variance ), then " i follows the variance gamma distribition whose density and the moment generate function are given, respectively, by f(x) = Z e g= p e x =(g) g= dg (.3) g (=)= 4

5 and mgf(u) = u = = (.4) where is the gamma function. The variance gamma distribution is a normal conditional on a variance that is distributed as a gamma variate whose mean is and variance is. It is known that moments of all orders exist with the mean, the variance, and the kurtosis 3+3. Since the excess kurtosis is determined by the parameter, it measures the degree of tail thickness. When L(t) is the Brownian motion the initial condition X() = x : the exact discrete time model is r e X ih = X (i )h + ( e h h ) + " i ; " i N(; ); X = x (.5) where = e h. As! ; the above discrete time AR() process will have a unit root in the limit. To simplify notation, we write X ih as X i. Equation (.5) indicates that the transition densition is X i j X i N X i e h + ( e h ); ( e h ) : (.6) Since the conditional distribution is known, it is easy to obtain the maximum likelihood estimator (MLE) or ordinary least square (OLS) estimator of where ^ = n b = P n i= XiXi n P n P n i= Xi i= Xi n P n i= X i n ( P n i= Xi ) ln ^ h By taking a Talyer Expansion to the second order, we obtain (.7) b = b = E(b) = = ln h h ( ^ h ( ^ ) + h ( ^ ) + h ( ^ ) + o(t ) ) + o(t ) h E(^ ) + h E(^ ) + o(t ) h E(^ ) + h (V ar(^) + (E(^ )) ) + o(t ) (.8) For general Lévy processes, the transition density is not normal any more. As a result, ^ and hence b is not a ML estimate. However, ^ is a quasi maximum likelihood estimate (QMLE) and can be obtained by OLS. So is b. While QMLE/OLS is not as e cient as MLE, it is analytically more tractable than MLE. To approximate the bias of b; we follow Bao and Ullah (9) and make the some assumptions about " i. In particular, we assume " i is i:i:d and " i follows a distribution with eight moments: m = ; m = ; m 3 = ; m 4 = + 3 (.9) m 5 = 3 + ; m 6 = m 7 = m 8 =

6 where and are the Pearson s measures of skewness and kurtosis of the distribution and ; : : : ; 6 can be regarded as measures for deviation from normality. For a normal distribution, ; : : : ; 6 all equals :. Bias approximation when the long run mean is known Now assume that = and it is known, the exact discrete time model of the Lévy process can be written as r e h X ih = X (i )h + " i (.) Bao and Ullah (7, 9) and Bao (7) give the approximate bias and MSE of the OLS estimator for the AR() model without intercept and without assuming normally distributed error terms: B(^) = MSE(^) = n n + o(n ) + n 4 ( )x 4r( ) 3 r ( ) + o(n ); where x is xed. In normal case " t ~iidn(; );by using above results, we have, for xed x E(b ) = h ( n ) + h n 4 = T + T + nt = eh + 3 T + nt when! ; E(b ) T + 3 T n when h! (n! ); E(b ) T : + 4 x n + o(t ) x + o(t ) 4 e h e h x + o(t ) If we consider the nonnormality, i.e., " t ~iid(; ); the skewness and excess kurtosis coe cients matter for the approximate MSE up to O(T be obtained as ):Therefore, the formula of the bias, for xed x ; can E(b ) = h ( n ) + h f 4r ( ) n + n [4 3 r ( )]g + o(t ) x = eh T nt [4 e h x eh 4re h ( e h )( + e h ) e h + e h + r (e h )] + o(t ) when! ; E(b ) T + 3 T n when h! (n! ); E(b ) T : 6

7 We summarize the above results in Theorem.. Theorem. Under Model (.) with a known mean, nonnormal error term with moments given in (.9), and xed x, the approximation to the Bias of the Mean Reversion Estimator is given as follows: E[b j x ] = eh T nt [4 e h x eh 4re h ( e h )( + e h ) e h + e h r (e h )] + o(t ) (.) + Furthermore, when! When h! (n! ) E[b j x ] T + 3 T n (.) E[b j x ] T : (.3) Corollary. Under the Lévy process model (.) with a known mean, nonnormal error term with moments given in (.9), and random nonnormal x with mean and varinace = (), the approximation to the Bias of the Mean Reversion Estimator as follows: E(b ) = eh T nt [4 4re h ( e h )( + e h ) eh e h + e h + r (e h )] + o(t ) (.4) Furthermore, when! When h! (n! ) E(b ) T + 6 T n (.5) E(b ) T : (.6) In Theorem. the result on Bias(b) is obtained conditional on x ;that is E[b j x ]:When x is assumed to be random with mean and varinace = (), unconditionally E(b ) = E x [E(b ) j x ]:The result in the Corollary. then follows. Corollary.3 Under the Lévy process Model (.) with a known mean, normal error term (r = ; and r = ); and xed x, the approximation to the Bias of the Mean Reversion Estimator is given as follows: Furthermore, when! When h! (n! ) E[b j x ] = eh + 3 T + 4 e h e h x nt + o(t ) (.7) E[b j x ] T + 3 T n (.8) E[b j x ] T : (.9) 7

8 Corollary.4 Under the Lévy process model (.) with a known mean, normal error term, and random normal x with mean and varinace = (), the approximation to the Bias of the Mean Reversion Estimator as follows: E(b ) = eh + 3 T + nt [7 eh ] + o(t ) (.) Furthermore, when! When h! (n! ) E(b ) T + 6 T n (.) E(b ) T : (.) Remark.. Here we consider the bias of AR() coe cient up to O(n ) and MSE of AR() coe cient up to O(n ) to obtain our new results in Theorem. and Corollary. for nonnormality. Corollary.3 and Corollary.4 give the result under normality. In Theorem. and Corollary.3 the results on Bias(b) is obtained conditional on x : In Corollary. and Corollary.4 the results on Bias(b) is obtained unconditional on x : Yu (9) derives result for the case of normality and x N(; = ()). His result is E(b ) = eh + 3 T ( e nh ) T n( e h ) (.3) where the rst term on the right handside is the same as the rst term on the right handside in (.), but the second term in his result is di erent from that in (.). In addition, we consider both OU process with a known mean and with an unknown mean, however, Yu (9) only discusses OU process with a known mean. In addition, an important di erence between (.3) and (.) is that the former goes to as goes to but not the latter. h i Remark.. The second term nt 4 e h e h x in (.7) incorporates the intial condition x ;which implies that the the intial condition will a ect the bias of the mean reversion estimator. Notice that if x > < ; which implies that the bias is a decreasing function of the start data point when x > ; if x < > : Remark..3 Result (.9) gives the bias of the mean reversion estimator when! (! ) near unit root case. The asymptotic result given in (.8) which considers h! (n! );i.e., very high data frequency, shows that the bias of the mean reversion estimator only depends on the bias of the coe cient of the corresponding AR() model as h!. Remark..4 Result (.) shows that, the initial condition x, skewness and excess kurtosis all a ect the bias of b. We note < ; which imply that the bias is a monotonically decreasing function of the excess kurtosis. If x > < ; 8

9 which implies that the bias is a decreasing function of the start data point when x > ; if x < > : If r > < ; if r < > :In > implies that the bias is a monotonically increasing function of the variance of error terms : Remark..5 (.3) is the special case of (.) with r = ; and r = : Comparing both cases with normality and without normality, when!, or h! ; we get same bias. In the case of near unit root or very high frequency data and nonnormality, the mean reversion and start data point do not a ect the bias of b: Remark..6 (.4) considers random x and nonnormality. (.) is the special case of (.4) with r = ; and r = :Comparing both cases with normality and without normality, when!, or h! ; we get same bias. That is, in the cases of near unit root or very high frequency data, nonnormality does not a ect the bias of : Remark..7 Comparing (.) and (.5), as! ; the bias under the process with an unknown mean is bigger than that under the process with a known mean. Comparing (.3) and (.6), as h! ; the bias in model (.5) is twice as high as the one in model (.). Both theorems represent that the bias doesn t disappear when! or h! (n! ) unless T! :.3 Bias approximation when the long run mean is unknown For the discrete time AR() model with an unknown intercept, the second-order bias up to O(n ) of the OLS estimator ^; is B(^) = +3 n given by Bao and Ullah (7):The MSE up to O(n ) given by Bao and Ullah (9) is as follows M(^) = n + n [3 + 4r ( ) + 3 r ( )] + o(n ) ( )x where x is the initial condition, r is the skewness, and r is the excess kurtosis. In a special case when the error term is normal, r = r = and M(^) = n + n "3 + + # ( )x + o(n ): Substituting above results into (), the bias of the mean reversion estimator in the normality case 9

10 is where = ( E(b) = h ( ) + n h f + n n [3 + ( )x g + o(t ) = + 3 T + T + T n f3 + ( )x + q e h e ); = h E(b) = eh + 3 T + eh T = T (eh + e h + 5) + T n g + o(t ) : So we can rewrite the bias in terms of ; h, and : e h e h ( x ) nt + o(t ) 3 + e h e h ( x ) + o(t ): If we consider the nonnormality, i.e., " t ~iid(; ); the skewness and excess kurtosis coe cients matter for the approximate MSE up to O(T b, for xed x ; as follows E(b) = + 3 T + T + T n f3 + 4r ( ) ):Therefore, we can obtain the formula of the bias of 3 r ( )g + o(t ) + ( )x = T (eh + e h + 5) + T n f3 + e h ( x ) eh 4re h ( e h )( + e h ) + e h + e h r (e h )g + o(t ) The result of estimation bias of the mean reversion parameter for Lévy process with unknow mean are given in Theorem., which considers the error term with a nonnormal distribution. Theorem.5 Under the Lévy process model (.5) with an unknown mean, nonnormal error term with moments given in (.9), and the initial condition x, the approximation to the bias of b can be written as follows: E[b j x ] = T (eh + e h + 5) + T n f3 + e h ( x ) eh 4re h ( e h )( + e h ) + e h + e h r (e h )g + o(t ) (.4) Furthermore, when! When h! (n! ) E[b j x ] 4 T + 33 T n (.5) E[b j x ] 4 T : (.6)

11 Corollary.6 Under the Lévy process model (.5) with an unknown mean, a nonnormal error term with moments given in (.9), and a random initial condition whose mean is and varinace =(), the approximation to the bias of b is as follows: E(b ) = T (eh + e h + 5) + f3 + eh e h T n 4re h ( e h )( + e h ) + e h + e h r (e h )g + o(t ) (.7) Furthermore, when! When h! (n! ) E(b ) 4 T + 6 T n (.8) E(b ) 4 T : (.9) In Theorem.5 the result on Bias(b) is obtained conditional on x ;that is E[b j x ]:When x is assumed to be random with mean and varinace = (), unconditionally E(b ) = E x [E(b ) j x ]:The result in the Corollary.6 then follows. Corollary.7 Under Lévy process model (.5) with an unknown mean, normal error term (r = ; and r = ), and xed x, the approximation to the bias of b is as follows: E[b j x ] = T (eh + e h + 5) + T n 3 + e h e h ( x ) + o(t ) (.3) Furthermore, when! When h! (n! ) E[b j x ] 4 T + 33 T n (.3) E[b j x ] 4 T : (.3) Corollary.8 Under the Lévy process model (.5) with an unknown mean, normal error term (r = ; and r = ), and random normal x with mean and varinace =, the approximation to the Bias of the Mean Reversion Estimator as follows: E(b ) = T (eh + e h + 5) + T n (3 + eh e h ) + o(t ) (.33) Furthermore, when! When h! (n! ) E(b ) 4 T + 6 T n (.34) E(b ) 4 T : (.35)

12 Remark.. We consider the bias of AR() coe cient up to O(n ); and MSE of AR() coe cient up to O(n ) to obtain our new result in Theorem.5 and Corollary.6 under nonnormality, and Corollary.7 and Corollary.8 under normality. In Theorem.5 and Corollary.7 the results on Bias(b) are obtained conditional on x : In Corollary.6 and Corollary.8 the results on Bias(b) are unconditional. Under normality and random x, Tang and Chen (9) consider both the bias and MSE of AR() coe cient up to O(n ) only: Therefore, their bias result for b is E(b) = T (eh + e h + 5) (.36) which is the rst term in the left handside of (.33). Therefore, our result in Theorem. under Lévy-based OU process with an unknown mean, provides an improved approximation of bias and di ers from Tang and Chen (9). In addition, our paper also derives the results (Theorem. and Corollary.-.4) under Lévy-based OU process with a known mean which are not discussed by Tang and Chen (9). Remark.. We note that the second term T n (3+eh e h ( x ) ) in (.3) incorporates both the mean parameter and the data start point x : If x is xed and > x ; < ; which implies that the higher lowers the bias. If x is xed and < x ; > ; that is, higher gives higher bias. We also notice that the bias is not a monotonic function of starting data point, if > x > ; < : When T n is very large, the e ects of and x on bias are negligible. However, under a special case of stationary distribution, x can be replaced by its mean.in that case x is zero and the bias term becomes free from and : Remark..3 Result (.3) gives the bias of the mean reversion estimator when b! (! ) near unit root case. The result given in (.3) which considers h! (n! );i.e., very high data frequency, shows that the bias of the mean reversion estimator only depends on the bias of the coe cient of the corresponding AR() model, since the rst term actually arises from the bias of ^: Remark..4 Result (.4) shows that not only the mean and the data start point x a ect the bias, but also skewness and excess kurtosis a ect it as well. We note < ; which imply the bias is the monotonically decreasing function of the excess kurtosis. skewness, if r < ; if r > : Remark..5 Corollary.6 is the special case of Theorem.5 with r = ; and r = : Comparing both cases with normality and without normality, when!, or h! ; we get same bias. That is, in the cases of near unit root or very high frequency data, nonnormality, and start data point do not a ect the bias of b: For

13 Remark..6 Corollary.6 considers random x and nonnormality. Corollary.8 is the special case of Corollary.6 with r = ; and r = :Both the results in corollaries.6 and.8 are free from x and : Comparing both cases with normality and without normality, when!, or h! ; we get same bias. That is, in the cases of near unit root or very high frequency data, nonnormality does not a ect the bias of b: Remark..7 Theorem.5 also shows that the bias depends on the true value of the mean reversion parameter. When! or h! (n! ); the bias doesn t disappear unless T!, which is consistent with studies that bias still exists for the large sample size. 3 Bias Approximations with Higher Order Bias and MSE This section shows the bias approximation by considering both the Bias and MSE of AR() coe cient up to O(=n ): Bao (7) gave the approximate bias and MSE of the OLS estimator for the AR() model without intercept and with general error term as follows: B(^) = n + n 4 + x MSE(^) = n + n r ( + )x r + r ( )( ) ( )x 4r ( ) 3 r ( ) + o(n ); + o(n ) where x is xed. Also, he gave the approximate bias and MSE of the OLS estimator for the AR() model with intercept and with general error term as follows: " + 3 B(^) = n n ( ) +o(n ) MSE(^) = n +o(n ) + n "3 + + ( )x # + 4r ( ) 3 + r # ( )x 4r( ) 3 r ( ) where x is xed. Along the line of Bao (7), we obtain the bias approximations of LS estimator of the mean reversion parameter for both known mean and unknown mean Lévy processes, which are presented in the following theorems and corollaries. Theorem 3. Under Model (.) with a known mean, nonnormal error term with moments given in (.9), and xed x, the approximation to the Bias of the Mean Reversion Estimator is given as follows: E[b j x ] = eh + 3 T + nt [6 e h x (e h + 3) eh (e h ) 4r (3 + e h + 3e h + e h ) e h + e h (eh + )x q e h k r (e h + 3)] + o((nt ) ) (3.)

14 Furthermore, when! E[b j x ] T + T n [5 4x h + 4x p h 4 3 4r ] (3.) When h! (n! ) E[b j x ] T x T : (3.3) Corollary 3. Under the Lévy process model (.) with a known mean, nonnormal error term with moments given in (.9), and random nonnormal x with mean and varinace = (), the approximation to the Bias of the Mean Reversion Estimator as follows: E(b ) = eh + 3 T + nt [6 e h (e h + 3) eh (e h ) 4r (3 + e h + 3e h + e h ) e h + e h + r (e h + 3)] + o((nt ) ) (3.4) Furthermore, when! E(b )! (3.5) since eh (e h +3) (e h )! : When h! (n! ) E(b ) T T : (3.6) Theorem 3.3 Under the Lévy process model (.5) with an unknown mean, nonnormal error term with moments given in (.9), and the initial condition x, the approximation to the bias of b can be written as follows: E[b j x ] = T (eh + e h + 5) + T n f3 + eh 6e h + 8 e h (e h ) ( x ) ( + e h + 5e h ) 4r ( e h )( + e h ) + e h ( + e h ) e h ( e h ) e h ( + e h + e h ) r (e h + 3)g + o((nt ) ) (3.7) Furthermore, when! E[b j x ]! (3.8) since eh 6e h +8 e h (e h )! :When h! (n! ) E[b j x ] 4 T + 7 T 4( x ) T : (3.9) 4

15 Corollary 3.4 Under the Lévy process model (.5) with an unknown mean, a nonnormal error term with moments given in (.9), and a random initial condition whose mean is and varinace =(), the approximation to the bias of b is as follows: E(b ) = T (eh + e h + 5) + T n f3 + eh 6e h + 8 e h (e h ) ( + e h + 5e h ) 4r ( e h )( + e h ) + e h ( + e h ) e h ( e h ) e h ( + e h + e h ) r (e h + 3)g + o((nt ) ) (3.) Furthermore, when! E(b )! (3.) since eh 6e h +8 e h (e h ) (+e h +5e h ) e h ( e h )! :When h! (n! ) E(b ) 4 T + 5 T : (3.) Remark 3. Here we consider the bias and MSE of AR() coe cient up to O(=n ) to obtain our new results in Theorem 3. and Corollary 3. for the Lévy process with known mean. The bias approximations for OU process with normally distributed error can be straightforward developed by substituting = ; = into above results. Similar with previous results, the initial data point, variance of the error term, skewness and excess kurtosis enter the higher order bias approximations. Compared with Theorem., the second term in (.) is di erent. With higher order of the bias of AR() coe cient, the esitmaiton bias approximation of ^ has a cross product term of x and : In addition, the approximated estimation bias is nonmonotonical function of the initial value and skewness. The excess kurtosis still has negative e ect on the estimation bias, and its negative impact is larger than the results in Theorem.. Notice that the initial value, skewness and excess kurtosis still have impacts on the limit bias approximation as! : And the variance of the error term and initial value also enter the limit bias approximation in (3.3) as h! (n! ): Remark 3. The second term in Corollary 3. is di erent from the one in Corollary.. Corollary 3. shows that the higher order bias approximation is still a non-monotonical function of the skewness, and the kurtosis of the error term distribution has larger negative e ect on the estimation bias than it is in Corollary.. Considering higher order of bias, we nd that the limit bias approximation explodes as! ; and goes to a smaller constant compared to Corollary. as h! (n! ): Remark 3.3 With the higher order of the bias, the second terms in the results obtained for the Lévy process with unknown mean and xed intial data point in Theorem 3.3 di er from those 5

16 in Theorem.5. The marginal e ects of the long run mean, initial data point, skewness and kurtosis in former are obviously di erent from those in latter. The squared skewness and kurtosis in former has larger negative impact on the estimation bias compared with the latter. In addition, notice that the limit conditional estimation bias explodes as! ; and the limit conditional estimation bias is a function of the long run mean, initial data point and the variance. Remark 3.4 In comparison with Corollary.6, the results of Corollary 3.4 di er in the second term. Corollary 3.4 which represents the bias approximation for the Lévy process with unknown mean and random initial data point, shows higher marginal impact of squared skewness and kurtosis than the former. And as! ; the limit bias approximation in the latter explodes, while the one in the former is constant. In addition, as h! (n! ), the limit bias approximation in Corollary 3.4 is a function of itself and larger than the result in Corollary.6. 4 Simulation Results In this section, we perform Monte Carlo simulations to illustrate the nite sample performance of our bias correction in comparison with OLS and the estimator corrected by Yu (9) and Tang and Chen (9) in terms of mean, relative bias (r. bias %), mean squared error (MSE), and root mean squared error (RMSE). We consider both Lévy Processes with a known mean and with an unknown mean. All simulation results come from, repetitions. 4. Bias Correction for Lévy Process with a known mean under nonnormality Here we consider four estimators for Lévy process with a known mean under nonnormality: OLS, Yu (9) estimator (Yu) corrected by the bias given in Remark.., the estimator (UWY) corrected by the bias corresponding to our Theorem., and the estimator (UWYH) corrected by the bias corresponding to Theorem 3.. Both xed initial value case and random initial value case are considered. In order to obtain non-normal error terms we rst generate the errors from the gamma distribution with mean and variance ; where v = :5; ;respectively, and second make transformation on the generated errors to satisfy the assumption in (.9), then generate the discrete time observations under the model (.). An extensive literature shows that the securities data generally has unit root or near unit root, so usually has small values. Hence, here we consider four small values of = :; :5; :; 3:: And we set up T = 5; and h = =; =5; =5; respectively. For xed x case, we set the start data point equal. For random x case, we generate x from variance gamma distribution. Figure and plot the true bias, the biases according to Yu (9), Theorem. and Theorem 6

17 3. for Levy processes with a known mean. The red line represents the true bias, the black dashed line is Yu bias, the green line is UWY bias based on Theorem., and the blue dashed line shows UWYH bias based on Theorem 3.. For random x case, gure shows that when is smaller than.5 both Yu and UWYH estimator drop below the true bias. When is greater than.5 both Yu and UWYH bias approximations can match the true bias very well. However, the green line shows that UWY bias approximation is a little above the true bias. As goes larger, all three bias approximations are approaching the true bias more closely. For xed x case, gure shows that all bias approximations have very small discrepency from the true bias, especially, for is greater than.5. Yu bias approximation is closest to the true bias for less than.5. While is greater than.5, our bias approximations match better with the true bias than Yu bias does. Bias Bias Figure Levy Process with a Known Mean and Random x kappa Figure Levy Process with a Known Mean and Fixed x kappa Our ndings in this case according to Tables include: rst, Yu almost has the smallest bias and RMSE among all three estimators when = :; second, when is moderately larger ( = :5; :; 3:); UWYH has smallest RMSE, and Yu estimator provides a slightly smaller bias than our estimator does; third, when = 3:, UWYH has both the smallest bias and lowest RMSE. It is not di cult to nd that Yu and UWYH perform very similarly. When = :; Yu performs slightly better than UWYH in the sense of having lower bias and RMSE. When is moderately larger ( = :5; :; 3:); UWYH performs slightly more e cient than Yu in the sense of having lower RMSE. 7

18 Table 4.. Bias Correction for a Known Mean Lévy Process under Fixed x v = :5 O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, = 3: T = 5, h = / 5, = 3: T = 5, h = / 5, = 3: M e a n r. b ia s ( % ) M S E R M SE

19 Table 4.. Bias Correction for a Known Mean Lévy Process under Fixed x v = : O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =.5 T = 5, h = / 5, =.5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE

20 Table 4..3 Bias Correction for a Known Mean Lévy Process Random x v = :5 O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =.5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE

21 Table 4..4 Bias Correction for a Known Mean Lévy Process under Random x v = : O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H O L S Y u U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE Bias Correction for Lévy Process with an unknown mean under nonnormality In this case, we also consider three estimators: OLS, Tang and Chen (9) estimator (TC) corrected by the bias given in Remark.., the estimators (UWY and UWYH) corrected by the biases corresponding to our Theorem.5 and Theorem 3.3, respectively, under Lévy process with an unknown mean under nonnormality. Same as the previous cases, the error term is rst generated from the gamma distribution with mean and variance ; where v = :5; ;respectively. We set = : and = : For xed initial value case, the start data point is xed at : For random initial value case, the start data point is generated from the gamma distribution with mean and variance ; where v = :5; ;respectively. Tables list the simulation results for this case. Figure 3 and 4 plot the true bias, the biases according to Tang and Chen (9), Theorem.5 and Theorem 3.3 for Levy processes with an unknown mean. We set = : and = :: The red

22 Figure 3 Levy Process with an Unknown Mean and random x.5 Bias kappa Figure 4 Levy Process with an Unknown Mean and fixed x.5 Bias kappa line represents the true bias, the black dashed line is TC bias according to Tang and Chen (9), the green line is UWY bias based on Theorem.5, and the blue dashed line shows UWYH bias based on Theorem 3.3. For random x case, all three bias approximations have some distances from the true bias. Among three bias approximations shown in gure 3, UWYH performs the relatively best, and it can also show the curvature as is small. Figure 4 shows the performance of all three bias approximations for Lévy process with an unknown mean and xed x case. When is close to zero, UWYH bias approximation goes up dramatically. When is greater than.5, UWYH has the smallest distance from the true bias among all three bias approximations. Like Lévy process with a known mean, the simulation results under Lévy process with an unknown mean also provides the support that the estimator based on bias correction under nonnormality for both xed x and random x, UWYH is useful in nite samples. The simulations in Table show that UWYH always has the smallest bias and mean squared error with only one exception in the case of = : under which UWY has the smallest bias and MSE than others. These results are in accordance with gures and. In a word, our estimators UWY and UWYH have improvement over OLS and TC. Especially, when = :5; :; 3:; UWYH is the most e cient estimator in the sense of having the smallest bias and the lowest RMSE.

23 Table 4.. Bias Correction for an Unknown Mean Lévy Process under Fixed x v = :5 O L S T C U W Y U W Y H O L S T C U W Y U W Y H O L S T C U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE

24 Table 4.. Bias Correction for an Unknown Mean Lévy Process under Fixed x v = : O L S T C U W Y U W Y H O L S T C U W Y U W Y H O L S T C U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =.5 M e a n r. b ia s M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s M S E R M SE

25 Table 4..3 Bias Correction for an Unknown Mean Lévy Process under Random x v = :5 O L S T C U W Y U W Y H O L S T C U W Y U W Y H O L S T C U W Y U W Y H T = 5, h = /, = : T = 5, h = / 5, = : T = 5, h = / 5, = : M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE

26 Table 4..4 Bias Correction for an Unknown Mean Lévy Process under Random x v = : O L S T C U W Y U W Y H O L S T C U W Y U W Y H O L S T C U W Y U W Y H T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =.5 T = 5, h = / 5, =. 5 T = 5, h = / 5, =. 5 M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =. T = 5, h = / 5, =. T = 5, h = / 5, =. M e a n r. b ia s ( % ) M S E R M SE T = 5, h = /, =3. T = 5, h = / 5, =3. T = 5, h = / 5, =3. M e a n r. b ia s ( % ) M S E R M SE To sum up, our ndings are as follow: (i) For Lévy processes with a unknown mean, regardless of whether x is xed or random, under non-normality the MSE/RMSE of UWY is always smaller than the MSE/RMSE of TC estimator. And UWYH which considers higher order bias of AR() coe cient has the smallest bias and MSE/RMSE when = :5; :; 3:: (ii) For Lévy processes with a known mean, Yu and UWYH perform similarly. When = :; Yu has slightly smaller bias and MSE/RMSE than UWYH. However, when = :5; :; 3:, UWYH performs slightly better than Yu in the sense of having a little lower MSE. (iv) Figures -4 show that UWYH bias approximation has large distance from the true bias as is very close to. However, as goes larger, UWYH bias approximation gets closer to the true bias with the exception of Levy process with an unknown mean and random x under which all bias approximation has a large discrepancy from the true bias. All simulation results in this seciton illustrate that considering nonnormality and higher order 6

27 bias approximation is useful to improve the e ciency and accuracy of the mean reversion parameter estimation in nite samples. 5 Conclusions This paper considers the nonnormality of error tems under both Lévy processes with a known mean and with an unknown mean. We obtain the bias approximations of the mean reversion parameter estimator under general errors and nd that the skewness ( ) and kurtorsis ( ), the starting data point, the long term mean (); the di usion parameter ( ); and itself (), all a ect the bias of : Monte Carlo simulations provide strong supports that our proposed bias corrected estimator of the mean reversion parameter is e cient in nite samples. 7