Time-Changed Residual Based Tests for Asset Pricing Models in Continuous Time

Transcription

1 Time-Changed Residual Based Tests for Asset Pricing Models in Continuous Time Chi M. Nguyen 1 Department of Economics Indiana University Abstract In this paper, we propose specification tests for a conditional mean of a general asset pricing model in continuous time which is applicable for both stationary and nonstationary processes of asset returns. The testing approach is based on the combination of time change and Khmaladze martingale transformation. Due to the time change, our task of testing for the specification of the conditional mean basically changes from testing for a possibly time-varying and unknown true distribution of the fitted error process, which is obtained from the postulated conditional mean regression, into testing for a standard normal distribution of the time-changed residuals. Moreover, the time change applied on the fitted error process effectively accounts for various types of volatilities, including but not limited to fat tail, time-varying or stochastic volatilities. Due to the Khmaladze martingale transformation, the tests are asymptotically distribution free. Hence, once the critical values of the limit distributions are tabulated, they will be used for testing any parametric form with unknown true parameters of the conditional mean. The simulation results of testing for a conditional mean of an Ornstein-Uhlenbeck process in finite samples show that our tests have satisfactory size performances, and possess discriminatory powers under different alternatives. This version: October 1, 21 Keywords and phrases: Specification testing; Time change; Residual based tests; Conditional mean models; Continuous time; Martingale estimation; CvM-type statistic, KS-type statistic; Khmaladze martingale transformation; Empirical process; Ornstein-Uhlenbeck process; Asset pricing models; Expected excess return 1 I am very grateful to my advisors Yoosoon Chang and Joon Y. Park for their constant support with many constructive suggestions and comments, and to Daehee Jeong for many useful discussions. 1

2 1. Introduction It is widely perceived that model misspecification may cause inconsistent estimators which in turn usually lead to invalid statistical inferences. The literature of specification testing has long history starting from the introduction of Fisher s axiom of correct specification in 1937, and therefore yields many important contributions. Virtually, all current testing approaches are using either the methodology of empirical process or the methodology of nonparametric smoothing or the combination of the two. For an overview of specification testing, the readers are referred to the guest editorial of Journal of Econometrics ). In a framework of continuous time, however, specification testing is still an uncultivated research field with only few specification tests for stationary diffusion models notably initiated by Aït-Sahalia 1996). And unfortunately, there is no existing test for a conditional mean of a general asset pricing model in continuous time which is applicable for both stationary and nonstationary processes of asset returns. Continuous time models have been widely used to capture dynamics of asset returns. Needless to say, the conditional mean of the asset pricing model, i.e. the expected excess return, is crucial not only in finance and financial economics theories but also in financial practices. First, it conveys much useful information about an economy and an economic agent s behavior, such as market prices of risks, individual s utility preference and his risk tolerance toward a portfolio choice. Second, the expected asset return is obviously the predictable component of the excess return, and thus can be used as references for policymakers regulation and surveillance. Third, in financial practices, traders make profit based on the abnormal return which is the difference between the excess return and its expected value. Accordingly, many estimation methods are developed and implemented with high frequency data. Specification testing for the conditional mean of the asset pricing model therefore is an indispensable task to check whether the underlying theories for the expected excess return are supported by the observed data. This specification testing does not only help to achieve a more reliable estimates of the expected return, but also shed more light on the future abnormal return since it provides a dependable specification of the expected excess return for predicting the future volatility of the excess return. Our development of specification tests for asset pricing models in continuous time utilizes the same idea of testing for a continuous martingale proposed by Park and Vasudev 26), Peters and de Vilder 26), Jeong and Park 29). However, while their researches put emphasis on testing any continuous process against a martingale, 2 ours fully takes care of the specification of the conditional mean, thereby testing the fitted continuous error process against a martingale. The idea roughly is: if the conditional mean is correctly specified, the fitted error process acquired from the postulated conditional mean regression should be a martingale. Thus, this error process after being indexed by a time change, which is serving as a new clock and set as inverse of constant increases in the process s quadratic variation, should essentially become a standard Brownian motion BM) by the well-known Dambis, Dubins and Schwartz DDS) theorem. To distinguish our time-changed error process from the original one, we name the normalized increment of the time-changed error process as the 2 Peters and de Vilder 26) presume the conditional mean as zero or constant over one-year period of time. 2

3 time-changed residuals. Hence, under a correct specification, the time-changed residuals are iid standard normals. In testing for the specification of the conditional mean, we use classical types of the omnibus Komolgorov -Smirnov and Cramér-von Mises statistics as the goodness-of-fit tests for iid normal distribution of the time-changed residuals, together with Khmaladze martingale transformation 1981) to obtain statistics that are asymptotically distribution free. The martingale transformation effectively transforms the empirical process implied by the parametric models into a martingale whose limit distribution is simply a standard BM. This convenient property of the limit distribution has made the transformation quite popular in literature with thorough investigations by Khmaladze 1982, 1988, 1993), Khmaladze and Koul 24), Haywood and Khmaladze 28). In economics application to testing for conditional distributions, Bai 23), and Bai and Chen 28) use an one-dimensional transformation, and Delgado and Stute 28) employ a two-dimensional transformation in a dependent model. 3 To maintain the simplicity of our tests in this paper, we apply the one-dimensional Khmaladze transformation to testing for normality of the time-changed residuals. The beauty of our approach is that it absorbs all desirable properties resulted from both the time change and the Khmaladze transformation. Due to the time change, there is no requirement on the specification of the volatility component of the stochastic process. Thus, the tests accommodate various types of volatilities, including but not limited to fat tail, time-varying or stochastic volatilities. What is particularly striking is that our distributional testing approach does not have to concern about the unknown true distribution of the error process, but simply takes advantage of using standard normals as the true distribution of the time-changed residuals, and thereby being applicable for general asset pricing models. Due to the Khmaladze transformation, the tests are asymptotically distribution free. This property allows us to relax the assumption of known true parameters of the conditional mean, which is of course not always realistic, in tabulating the limit distribution of the test statistics. As the limit distribution of the tests are invariant to the postulated conditional mean, once the critical values are tabulated, they will be used for testing any parametric form of the conditional mean, including linear, regime-switching or more general nonlinear functions of the conditional means. Our tests therefore are very practical for many applications in a continuous time framework. They can potentially become convenient tools for checking various types of asset pricing models that are built upon different assumptions about utility, risk preference and uncertainty of economic entities in an attempt to better explain economic phenomenon. For example, our approach can be used to test for the conditional mean of the asset pricing equations in Jeong, Kim and Park 21) derived from the continuous single-prior and multi-prior recursive utility models of Duffie and Epstein 1992), and Chen and Epstein 22) respectively, where the true parameters related to risk and uncertainty, such as the relative risk aversion and the 3 Bai and Chen 28) consider the multivariate as a collection of independent univariates, and then apply an one-dimensional Khmaladze transformation for each univariate with appropriate modifications. The price of this procedure may be a loss of full consistency in detecting some certain alternatives, and less power as the dimension increases. Delgado and Stute 28) apply the Khmaladze transformation directly to a bivariate X, Y ) where Y is dependent on X. Without knowledge about the direction of dependency, different ordering decisions may result in different transformations, hence different statistics. 3

4 ambiguity about the state of the economy, are naturally unknown to the modeler. The tests are also useful in evaluating different continuous models that offer a resolution to the forward premium anomaly such as the model of foreign exchange rates by Jacewitz, Kim and Park 28). Our tests, however, have not addressed testing for serial uncorrelatedness amongst the fitted time-changed residuals, i.e. the independence of a BM increments. The tests can be used as the first stage of testing for the normality. Once we fail to reject the normality, we may implement the second stage of testing for the first-order serial uncorrelation. 4 Another option is using a two-dimensional Khmaladze transformation to test for the normality and independence simultaneously by testing the two-dimensional empirical distribution of the timechanged residuals against a bivariate standard normal distribution. In this paper, to inspect the size and power performances of our tests for normality in finite samples, we conduct a set of simulations testing for a conditional mean of an Ornstein-Uhlenbeck OU) process. The results show that these tests have satisfactory size performances with rejection probabilities close to the nominal sizes, and possess discriminatory powers at different magnitudes under different alternative hypotheses. The rest of the paper is organized as follows. Section 2 models a conditional mean of a general asset pricing model, and presents equivalent testing hypotheses from two methods of time index, i.e. the conventional fixed interval time index and the time change index. Section 3 discusses the time change estimation using a discrete sample at high frequency, and introduces the concept of the normalized increment or the time-changed residual. Then we construct the time-changed residual based tests, and formally show their limit distributions. To illustrate the performance of our tests, Section 4 provides simulations for the case of an OU process with daily data and a time span of 5 years. Finally, we conclude in Section 5. The practical and numerical procedures to implement the time change and the Khmaladze martingale transformation, as well as some mathematical proofs are explained in the Appendix. 2. Model and Hypotheses A correct specification of an expected excess return of a risky asset is important in both financial theory and practices. If we denote a process P t ) as market price of the risky asset and r f t ) as return of the riskless asset, then no matter where the excess return is derived from, i.e. either from market equilibrium conditions or from a no-arbitrage condition, 5 the pricing equation has a reduced form dp t P t r f t dt = µ tdt + du t, 4 Note that, the first-order serial un-correlation amongst the standard normal variates implies their mutual independence. 5 In capital asset pricing models, the expected excess return is derived from individual utility maximization and market equilibrium conditions. In arbitrage pricing theory models, it is derived directly from no-arbitrage condition without considering utility maximization. The APT model can be seen as a supply-side one. See Eichberger and Harper 1997) for more detail. 4

5 of which U t ) is a continuous martingale process, and µ t ) represents a rate of an instantaneous change in the conditional mean of the excess return. 6 The process µ t ) usually contains meaningful economic information. For instance, in a capital asset pricing model of a certain utility function, we can extract parameters of relative risk aversion, elasticity of inter-temporal substitution and other market prices of risks from the process µ t ). In addition, if we let π t ) be the state-price deflator, i.e. π t = exp t ) rs f ds D t, where D t is the Radon-Nykodym derivative of the equivalent martingale measure with respect to the true probability, then under no-arbitrage condition of E dπ t P t ) F t ) =, the term µ t )dt actually is µ t dt = dπ t π t dp t P t, which moves in the opposite direction to the product of the instantaneous riskless return and the return of risky asset. In this paper, we consider continuous stochastic processes X, Y ) R m R on a measure space Ω, F), and model a conditional mean of the pricing equation in a general form as dy t = µx t, θ )dt + du t, or 1) Y t Y = t µx s, θ )ds + U t, where X t, Y t ) are adapted to a filtration F t ), and µx t, θ ) is a measurable function defined on R m Θ where Θ R k is a compact set of parameters. Note that the model includes the conditional mean of a diffusion process as a special case where dy t = dx t in equation 1). Our specification test aims at testing whether the parametric conditional mean µx t, θ) is correctly specified with an unknown true vector of parameters θ. In practice, to construct the test, consistent estimators ˆθ under the null hypothesis are used. If µx t, θ) is correctly specified, the error process U t θ)) adapted to the filtration F t ) should be a continuous martingale. Assuming the continuity of U t θ)), now testing for a correct specification of µx t, θ) becomes equivalent to testing the error process against a martingale versus the alternative H : U t θ) is a martingale for some θ Θ 2) H 1 : U t θ) is not a martingale for any θ Θ. 3) As a practical matter, testing for martingale properties of a continuous process directly is quite difficult to implement. 6 The realization of U t) is considered as the abnormal return, and the expectation of the realization of integrals of µ t) with respect to time is the expected excess return. 5

6 Fortunately, the well-known Dambis, Dubins and Schwartz DDS) theorem offers us an approach to tackle this problem. The theorem essentially states that any continuous martingale process U t ) can be transformed into a standard Brownian motion BM) by a time change T t ) which is used as a new clock and set as an inverse of a proportional increase in the quadratic variation of U t ). The original idea of using this transformation to test for martingales and semi-martingales in continuous time is due to Park and Vasudev 26), Peters and de Vilder 26), and Jeong and Park 29). Their tests, however, are based on either an assumption of known true parameters of the conditional mean as in Park and Vasudev 26) and Jeong and Park 21), or a presumptions of a zero and a piece-wise linear drift term with a constant slope over one-year period as in Peters and de Vilder 26). In the latter case, the presumption is fairly restrictive if not unrealistic, and consequently may cause an invalid statistical inference if the drift term is misspecified. In this paper, we utilize the idea with full consideration for the specification of the drift term, and interestingly relax the assumption of known true parameters. Our tests therefore are very practical for many applications. More particularly, if we denote the quadratic variation of U t ) over a time interval [, t] as [U] t, and define a time change to be a collection of non-decreasing stopping times T t ) t adapted to the filtration F t ) such that T t = inf{s [U] s > t}, then the process U Tt ) indexed by the new clock T t ) becomes a standard BM. As a result of the time change, the testing hypotheses defined in 2) and 3) can be written as H : U Tt θ) is a standard BM for some θ Θ, 4) H 1 : U Tt θ) is not a standard BM for any θ Θ. 5) Testing for a BM is of course simpler than testing for a martingale in continuous time. This is because we only need to check whether U Tt θ)) satisfies two properties of the BM, i.e. normality and independence of the increments. Notice that the error process U t ) in equation 1) is totally unrestricted, hence under a correct specification we can present U t ) as a martingale in a general form as du t = σ t dw t, where W t is a standard BM, and σ t is a general volatility function that is adapted to F t ) and locally bounded in L 2 for t [, T ]. 7 As it is clearly seen, without being indexed by the time change, the increments of the error process U t ) could have a distribution changing over time and far away from being normal since σ t could be any unknown time-varying or stochastic processes that are quite persistent and endogenous. However, after the time change is implemented, the normalized increments of U Tt ), i.e. the time-changed residuals, are iid standard normals. Therefore, the task of testing for the specification of the conditional mean basically changes from testing for an unknown true distribution of the error process U t ) into testing for a well-known standard normal distribution of the time-changed residuals. This natural simplification benefited from the time change makes our testing approach valid for general continuous-time stochastic processes. We shall discuss the issue more formally in the next sections. 7 See Representation of Martingales, Revuz and Yor 1999), page

7 3. Time-Changed Residual Based Tests 3.1 Time Change Estimation and Time-Changed Residuals To implement our approach, we use discrete samples at relatively high frequencies. This section therefore discusses time change approximation using discrete data, and shows that the approximation errors asymptotically disappear as data are collected at a frequency high enough. Then, we introduce the normalized increment of the time-changed error process obtained from the postulated regression, and define the time-changed residual. Hereinafter, for brevity, all variables and processes constructed upon the discrete sample are distinguished from those constructed upon the continuous sample path by the superscript of the sampling interval δ. For the model in equation 1), as the drift component has zero quadratic variation because of its bounded variation property, the quadratic variation of the continuous process Y t ) equals the quadratic variation of the error process U t ), i.e. [Y ] t = [U] t. For the actual data collected at the sampling interval δ, let n be the number of observations, then we record the observable variables X iδ, Y iδ ) for i = 1,...n, and the sampling time horizon T = nδ. Note that the corresponding errors U iδ ) are unobservable. Now let [t/δ] [Y ] δ ) 2 t = Yiδ Y i 1)δ, i=1 be the realized variance of Y t ) over the time interval [, t], where [t/δ] denotes the integer part of t/δ, we can estimate [U] t using [Y ] δ t under some mild assumptions introduced below. Assumption 1 The error process U t ) has a.s. continuous sample path. Assumption 2 For all s t T, a T t s) [U] t [U] s b T t s), where a T, b T > are some constants depending only upon T. The continuity condition in Assumption 1 ensures that the time change is applicable for U t ). However, it can be relaxed by allowing finite jumps with a suitable treatment as suggested by Park 21), Jeong and Park 29). Assumption 2 holds for a broad class of continuous martingales. For a general martingale process U t = σ t W t, then a T inf t T σt 2 and sup t T σt 2 b T will satisfy the condition. Assumptions 1 and 2 are the same as those used in Park 21), and Jeong and Park 29). Let [U] δ t be the realized variance of the error process U t ) over the time interval [, t], then these assumptions guarantee that [U] δ t converges to the true quadratic variation [U] t uniformly in L 2 for all t [, T ] at the rate of O b T T δ) 1/2), i.e. E sup t T [U] δ t [U] t ) 2 = O b 2 T T δ ). 6) 7

8 Thus, the sampling interval δ should decrease to zero faster than the time horizon T and the upper bound b T increase, such that b 2 T T δ. For instance, in the case of an Ornstein- Uhlenbeck OU) process with b T = 1, the sampling interval should be δ = ot 1 ). Assumption 3 We assume that E sup t T sup µx t, θ) θ Θ ) 4 = Oc 2 T ) for some constant c T depending only upon T. The requirement on a maximal growth rate of the instantaneous conditional mean function µx t, θ) in Assumption 3 is satisfied for all stochastic processes commonly used in practice. For example, the condition holds for all bounded processes if c T = 1, for the OU process if c T = T ε with any ε >, for the BM if c T = T p, and for the BM with drift if c T = T 2p with some p >. The readers are referred to Park 21) for more detailed explanation. Park 21) also proves that Assumptions 1 to 3 are sufficient to ensure the uniform consistency of the realized variance [Y ] δ t of the return process Y t ) for the true quadratic variation [U] t of the error process U t ) as soon as δ fast enough. In particular, the convergence rate is given explicitly as E sup t T [Y ] δ t [U] t ) 2 = O δt b 2 T ) + O δt ) 2 c 2 T ) + O δt 2 b T c T ) ). 7) The decreasing rate of the sampling interval δ now has to offset the increasing rates of not only T and b T, but also the maximal growth rate c T introduced in Assumption 3. As a result of equation 7), we require δ = min ot 1 b 2 T ), ot 1 c 1 T ), ot 2 b T c T ) 1 ) ), which is stricter than the condition implied by equation 6). Apparently, to make the approximation errors of using [Y ] δ t for [U] δ t asymptotically negligible, the faster the instantaneous changing rate of the conditional mean µx t, θ) grows, the higher frequency with the smaller sampling interval δ at which should data be collected. In the case of OU process, δ = ot 2 ε ) for any ε >. For brevity, here or elsewhere, following notations are used R δ,t = sup t [Y ] δ t [U] t and q = max δt b 2 T, δt ) 2 c 2 T, δt 2 b T c T ) ). The consistency of the time change approximation based on the discrete data follows immediately from equation 7). Denoting S = [U] T, and approximating the time change Tt δ ) based on the realized variance [Y ] δ t for all t as Tt δ = inf{s [Y ] δ s > t}, then Park 21) shows that ) 2 E sup Tt δ T t = E a 2 ) T R2 δ,t = Oa 2 T q). 8) t S 8

9 As can be seen, for ensuring the errors of the time change approximation to asymptotically disappear, the above condition is involved with the lower bound a T introduced in Assumption 2 such that a 2 T q. This does not necessarily imply a stricter condition than the one in equation 7) as long as a T has an order of magnitude at least as big as O1). Under the time change T t ), the continuous stochastic process Y t ) becomes a stochastic process indexed by T t ), i.e. dy Tt = µx Tt, θ) dt t + du Tt, 9) of which U Tt is a DDS BM if µx t, θ) is correctly specified. For any strictly positive incremental level of the quadratic variation [U] t, hereinafter called as a quadratic-variation time scale, 8 we can construct a collection of stopping times { } T i = inf s [U] s > i for i = 1, 2,..., [S/ ], where [S/ ] denotes the integer part of S/ representing the number of random time intervals collected using the time scale and the unobservable [U] t. Based on this time change, now we may rewrite equation 9) as Ti U Ti U Ti 1) = Y Ti Y Ti 1) ) µx s, θ)ds, T i 1) 1) and define the normalized increment of U Ti ), i.e. Z i θ) = 1/2 U Ti U Ti 1) ) 11) as the time-changed residual from the regression in equation 1) with the time scale. Corresponding to the infeasible time change T i ), we have its discrete sample counterpart } Ti {s δ = inf [Y ] δ s > i as the estimates of the time change T i ) for i = 1, 2,..., N, where N is the integer part of [Y ] δ T /, and thereby being the number of sample points collected at the incremental level of the realized variance [Y ] δ t in the actual implementation. Thus, we call N the time-changed sample size. Note that, given the time horizon T, we record observations at two different levels of frequencies: the original high frequency based on the fixed sampling interval δ with n observations, and the time changed frequency based on the quadratic variation time scale with N observations randomly collected at the estimated time change Ti δ ). For the discrete sample counterpart of Z i θ) in equation 11), we have Zi δ θ) = 1/2 Y T δ Y i T δ δ i 1) n i j=n i 1 +1 µx jδ, θ), 12) 8 The readers are referred to Park 21) for more detailed discussions on the optimal choice of the time scale. 9

10 as the time-changed residuals based on the discrete sample with n i = Ti δ /δ. The approximation Ti δ µx Ti 1) δ t, θ) dt δ n i j=n i 1 +1 µx jδ, θ) is valid for small and strictly positive δ if µx t, θ) is Riemann integrable on the interval [Ti 1) δ, T i δ ], and satisfies Assumption 4 For all s t T, for some constant d T depending only upon T. E sup µx t, θ) µx s, θ) d T t s) 1/2 θ Θ Assumption 4 holds for a wide class of diffusion processes X t ) if µx t, θ) is Lipschitz continuous uniformly in θ Θ, according to Park 21). In addition, the following relations among the constants introduced throughout Assumptions 1 to 4 are needed. Assumption 5 We assume that c T /b T = OT ), c T /a 2 T b T ) = OT ), d 2 T /a2 T b2 T c T ) = OT ), and ) 1 δ = O T 4+ε b 3 T c T for some ε >. Many processes used in practical applications satisfy the order conditions on the constant a T, b T and c T in Assumption 5. With high frequency data, the required order of the sampling interval δ is easily met. Under Assumptions 1 to 5, Park 21) proves that the error incurred by using Zi δ to approximate Z i is uniformly negligible for all i = 1,..., N and any θ Θ, i.e. E sup sup Zi δ θ) Z i θ) = on 1/2 ). 13) 1 i N θ Θ More conveniently, the convergence rate depends only on the time-changed sample size N. As it will prevail later, this rate is the same as a standard convergence rate of parameter estimators in many common estimation methods. 3.2 Time-Changed Residual Based Tests To introduce the time-changed residual based tests, let the fitted time-changed residuals be the discrete sample counterpart of the time-changed residuals based on the estimates ˆθ, i.e. Zi δ ˆθ) = 1/2 Y T δ Y i T δ δ i 1) n i j=n i 1 +1 µx jδ, ˆθ). 14) In testing for normality of the fitted time-changed residuals Z δ i ˆθ), we consider the empirical process of the fitted time-changed residuals, which measures the discrepancy between the empirical distribution of the fitted time-changed residuals and the standard normal distribution. 1

11 Then, we construct the test statistics as one-to-one mappings from the martingale component of the estimated empirical process by employing Khmaladze transformation 1981). As this martingale process converges to a standard BM, our statistics are asymptotically distribution free. Their limit distributions are invariant to the parametric family of unknown true parameters µx t, θ ). To show the validity of our methodology in a more formal way, we first introduce the following additional notations for processes whose arguments include the parameter vector θ. Particularly, for any θ Θ, such as the true parameter θ and the estimates ˆθ, let F N z, θ) be the so-called empirical distribution of Z i θ) F N z, θ) = 1 N 1{Z i θ) z}, i=1 where 1{.} is an indicator function which takes value 1 if the function s argument is true and otherwise. We also let ν N z, θ) be the empirical process that is the normalized difference between the empirical distribution F N z, θ) and its hypothetical distribution Φz) ν N z, θ) = N F N z, θ) Φz)), where Φ is a standard normal cdf. Moreover, if we denote V i θ) = ΦZ i θ)), and r = Φz) [, 1], the above empirical process can be written as ν N r, θ) = ) 1 N 1{V i θ) r} r N i=1 In the sequel, we shall use this representation ν N r, θ) for our analysis. Recall that all variables and processes superscripted by δ refer to the discrete sample counterparts of the corresponding theoretical ones. Thus, the discrete sample counterparts of F N., θ) and ν N., θ) are denoted as FN δ,.θ) and νδ N., θ), respectively. To obtain the consistent estimates ˆθ under the condition that the time-changed residuals are iid standard normals, we employ an one-dimensional martingale estimation MGE) of Park 21) that minimizes a Cramér-von Mises distance between the one-dimensional empirical distribution of the time-changed residuals of equation 12) and the standard normal distribution as follows 2 1 ˆθ = arg min 1{Zi δ θ) z} Φz)) Φdz). 16) θ Θ z R N i=1 Based on the Monte Carlo simulation results of Park 21), MGE estimators are shown to have negligible biases without using any bias correction method. Moreover, compared to the Gaussian maximum likelihood estimation, MGE performs reasonably well even in a modest sampling horizon. We now assume 15) 11

12 Assumption 6 For any θ Θ, the conditional distribution of Z i θ) on Zi δθ) Z iθ)) is absolutely continuous with respect to Lebesgue measure, and having density uniformly bounded in 1 i N, Park 21) shows that under Assumptions 6) and the result in equation 13), the negligible difference between the empirical distribution of Z δ i θ) and the empirical distribution of Z iθ) are of order on 1/2 ), 9 E sup r 1 θ Θ sup FNr, δ θ) F N r, θ) = on 1/2 ). 17) As a direct result, for the estimates ˆθ Θ, the difference between the empirical processes ν N r, ˆθ) and ν δ N r, ˆθ) is also negligible, i.e. E sup νnr, δ ˆθ) ν N r, ˆθ) = o1). 18) Hence, the errors incurred by the time change approximation have almost surely no effect on using ν δ N r, ˆθ) to approximate ν N r, ˆθ). For the sake of simplicity, we take first step to present the limit distribution of the theoretically estimated empirical process ν N r, ˆθ), which does not have errors from the time change approximation. The following conditions on the instantaneous rate of change of the conditional mean µx t, θ) in relation to the estimates ˆθ are needed. Assumption 7 We assume a) The estimates ˆθ satisfy Nˆθ θ ) = O p 1), and b) The function µx t, θ) is continuously differentiable with respect to θ, and E sup t T sup θ Θ where the. denotes Euclidean norm. µx t, θ) θ 2 bt N = O T The required convergence rate of the parameter estimates at N 1/2 in Assumption 7a) is standard and easily met by current estimation procedures such as Gaussian likelihood estimation of Yu and Phillips 21) and MGE of Park 21). Assumption 7b) specifies the maximal rate at which the L 2 distance of the first partial derivative of the drift function with respect to the parameter vector can grow. Recall that N = [U] T = Ob T T ). Hence, the condition holds for many stochastic processes as soon as the time scale is appropriately chosen. For the case of the OU process, µx t, θ) = κα X t ), where κ, α >, b T = 1, and the maximum process 1 sup t T X t grows at the rate of log T ) 1/2, we may let = Olog T ) 1 ) to satisfy the condition. 9 For detail, the readers are referred to Lemma 3.5 of Park 21). 1 See Ward and Glynn 23). ), 12

13 Under Assumptions 1 to 7, the empirical process ν N r, ˆθ) deviates from the true one ν N r, θ ) by an asymptotically non-negligible component: ν N r, ˆθ) ν N r, θ ) = φφ 1 r)) a N) Nˆθ θ ) + o p 1), 19) where φ is the pdf of the standard normal Φ; and a N is the limit in probability of the gradient vector of the function µx t, θ) with respect to θ over the random time intervals T i ), i.e. a N = plim N 1 1/2 N i=1 Ti µx t, θ ) T i 1) θ whose existence is ensured by Assumption 7b). As shown in Bai 23), this non-negligible component is resulted from Taylor expansion of ν N r, ˆθ) around θ. In particular, the term φφ 1 r)) a N = plim N 1 N i=1 ΦZ i θ)) θ dt, Zi θ )=Φ 1 r) actually is the probability limit of the gradient of Φz, θ) with respect to θ evaluated at r, θ ). Note that as N, the true empirical process νr, θ ) converges to a standard Brownian bridge. For the estimated empirical process νr, ˆθ), it can be clearly seen from equation 19) that the limit distribution of the non-negligible component is not easily tabulated. The main reason is that the a N, which is not dependent upon r, is obviously varying upon each postulated function with unknown true parameters µx t, θ ). The other reason is that the estimation errors of ˆθ are also dependent upon the estimation procedures. For example, if Nˆθ θ ) N, Ω), then the efficiency of ˆθ presented by the positive definite covariance matrix Ω is different under different estimation procedures. Therefore, the so-called Kolmogorov-Smirnov KS) statistic 1 KS N = sup N 1{Z z R i ˆθ) z} Φz)) N = sup ν N r, ˆθ), 2) i=1 and the so-called Cramér-von Mises CvM) statistic CM N = N z R 2 1 1{Z i ˆθ) z} Φz)) Φdz) = N i=1 ν N r, ˆθ)) 2 dr 21) will have limit distributions depend not only on µx t, θ ) but also on the estimation errors of ˆθ. These dependencies are undesirable, because for each postulated conditional mean we must assume the knowledge of true parameters, which is of course not always realistic, to tabulate the tests limit distributions in accordance with the hypothesis and the estimation procedure. To deal with the inconvenience mentioned above, Khmaladze 1981) suggests to transform the empirical process ν N r, ˆθ) into a martingale, and then construct statistics as mappings from this martingale. The transformation is popularly known as Khmaladze martingale transformation. In what follows, we let gr) = r, φφ 1 r)) ), and ġr) = 1, Φ 1 r)) which is 13

14 the fist derivative of gr) with respect to r. Using Khmaladze approach, Bai 23) obtains the following martingale process: r ) ŵ N r) = ν N r, ˆθ) ġs) C 1 s) ġτ) dν N τ, ˆθ) ds, 22) s where Cr) = r ġs)ġs) ds is invariant to µx t, θ ). The subtracted term in the above equation plays a role as a conditional expected value of the semi-martingale process ν N s, ˆθ)) up to r [, 1]. What is desirable about ŵ N r) is that its limit distribution is a standard BM. Thus, any statistics mapping one-to-one from ŵ N r) will convey the same statistical inferences as ŵ N r) does, and be asymptotically distribution free. Particularly, the KS-type statistic based on ŵ N r), instead of being based on ν N r, ˆθ) as in equation 2), has the limit distribution as T 1N = sup ŵ N r) d sup W r), 23) and the CvM-type statistic based on ŵ N r), instead of being based on ν N r, ˆθ) as in equation 21), has the limit distribution as T 2N = ŵ N r)) 2 dr d W 2 r) dr, 24) where W is a standard BM. As a second step, we replace the theoretical martingale ŵ N r) by the feasible martingale process ŵ δ N r) extracted from the empirical process νδ N r, ˆθ) of the fitted time-changed residuals Z δ i ˆθ). After Khmaladze transformation, the martingale component of ν δ N r, ˆθ) is r ŵnr) δ = νnr, δ ˆθ) ġs) C 1 s) s ) ġτ) dνnτ, δ ˆθ) ds. 25) One may expect the difference between ŵ δ N r) and ŵ Nr) to be stochastically negligible by comparing equation 22) with 25), since the difference between ν δ N r, ˆθ) and ν N r, ˆθ) is uniformly negligible. Indeed, it is the case since the vector function ġr) is bounded in probability on [, 1]. We have Lemma Under Assumptions 1 to 7, E sup wnr) δ ŵ N r) = o p 1). According to the lemma, the error in approximating the Khmaladze martingale process ŵ N r) by its discrete sample counterpart ŵn δ r) is uniformly negligible in probability. Now we are able to construct feasible statistics as the discrete sample counterparts of T 1N and T 2N in equations 23) and 24), respectively. Recall that the classical KS and CvM statistics are widely known as the omnibus goodness-of-fit tests for distribution. Moreover, 14

15 under any root-n local alternative hypothesis, the limit distribution of the Khmaladze transformed process ŵ δ r) departs from a BM by a non-negligible term as shown in Bai 23). Thus, the KS-type and CvM-type statistics based on this feasible transformed process are also omnibus. Khmaladze 1982) suggests these statistics in some applications of his martingale theory. Koul and Sakhanenko 25) compare the performance of the Khmaladze transformed KS-type statistic with a variety of bootstrap performances in finite samples, and find that the KS-type statistic and the statistic obtained by Monte Carlo method 11 have size performances comparable and the best among other approaches, and have different discriminatory powers under different alternatives. Subsequently, we let be the feasible KS-type statistic, and T1N δ = sup ŵn, δ 26) T δ 2N = be the feasible CvM-type statistic. Then we obtain ŵ δ N) 2 dr 27) Theorem Under Assumptions 1 to 7, the limit distributions of T δ 1N and T δ 2N are T δ 1N d sup W r) and T2N δ d W r) 2 dr, respectively, where W is a standard Brownian motion. This theorem essentially states that our feasible statistics are asymptotically distribution free since their limit distributions depend on neither the hypothetical parametric form of conditional means µx t, θ ) nor the parameter estimation errors. Thus, once the critical values are tabulated, they will be used for testing any conditional means with unknown true parameters. After all, our testing approach possesses two beautiful characteristics. The striking one is due to the time change transformation that makes the our approach applicable for general conditional mean models without any concern about the unknown true distribution of the error process. The desirable one is due to Khmaladze transformation that makes our tests be asymptotically distribution free. The tests, however, have not checked the serial uncorrelatedness among the fitted time-changed residuals obtained from the error process yet. To supplement this issue, we may proceed the following options. On the one hand, considering these tests to be at the first stage, we may implement the second stage by constructing a test for first-order serial uncorrelatedness among the fitted time-changed residuals. On the other 11 In Monte Carlo method, the true data generating process is assumed to be known. Therefore, the empirical distribution of the test under the null hypothesis can be simulated, and the empirical critical values are used to make statistical inferences. 15

16 hand, we could directly extend our approach to testing a two-dimensional empirical distribution of the fitted time-changed residuals versus a bivariate standard normal by employing two-dimensional Khmaladze transformation. These options will be experimented in the near future. 4. Simulations For our illustrative purpose, we conduct a set of simulations testing for the conditional mean of an OU process, which is often used in term structure of interest rates and option pricing models. Considering a stochastic process dy t = µ t dt + du t, we are interested in testing the null hypothesis for some κ, α > against the alternative hypothesis H : µ t = κα Y t ), 28) H 1 : µ t κα Y t ) 29) for any κ, α >. To investigate the size performance of the tests, we generate the OU process under the null hypothesis with the true parameter vector κ, α, σ) =.5,.6,.3) that is chosen to be realistic for US interest rates and used in the Monte Carlo simulation by Aït-Sahalia 22). For evaluating the power performances, four data generating processes DGPs) having different conditional means under different alternatives are generated as H 11 : dy t = 1{ Y t.1}.5.6 Y t ) dt +1{ Y t >.1}.5.6 Y t ) dt +.3 dw t, 3) H 12 : dy t =.3 dw t, 31) H 13 : dy t =.5 dt +.3 dw t, 32) H 14 : dy t = 1{ Y t.1}.5.6 Y t ) dt + 1{ Y t >.1}.5 dt +.3 dw t. 33) The DGP under the alternative H 11 provides a regime-shift in the conditional mean of the mean-reverting process. The parameters of this process are set such that while presenting the same time trend component as that of the postulated OU process, they reflect the return-risk tradeoff in the conditional mean: if the asset return lies within a given bounded range, it has a smaller mean α =.6 in exchange for a smaller stationary long-term) variance with a faster speed of mean reversion κ =.5; once the return is out of the range, it has a bigger mean α =.6 at the cost of a higher variance with a slower speed κ =.5. The DGP with µ t = and µ t =.5 under the alternatives H 12 and H 13 express a BM and a BM with drift, respectively, whose diffusion terms are the same as the diffusion term of the postulated 16

17 OU process. Finally, we consider the DGP under the alternative H 14 to investigate the power performance of our tests against a process that changes its dynamics from the stationary OU process to the nonstationary BM with drift. The data are generated to mimic daily observations of 5 years with the time horizon T = 5 and the sampling interval δ = 1/ We carry out 5, iterations for each hypothesis, and report simulation results of four statistics. The first two ones KS δ N and CM δ N are the discrete sample counterparts of KS N and CM N defined in equations 2) and 21), respectively, i.e. KSN δ = sup νnr, δ ˆθ) and CMN δ = ν δ Nr, ˆθ)) 2 dr. Because the limit distributions of these two statistics depend on the parametric form of µx t, θ ), we assume the true model of the null hypothesis to be known, thereby implementing Monte Carlo simulation with 5, replications to obtain their empirical distributions and critical values for statistical inferences. The third and the fourth statistics are the timechanged residual based tests T1N δ and T 2N δ from the postulated conditional mean of which the true parameters θ are unknown, as defined in equations 26) and 27), respectively. 13 We use the KSN δ and CM N δ as the benchmark to compare and evaluate the performances of our time-changed residual based tests. According to suggestions of Park 21), we may use a monthly average of the quadratic variation of Y t ) as the time scale, and denote it as m for our simulations. Thus, the timechanged sample size N based on m includes 6 random time intervals. To estimate the unknown parameters with the given m, we minimize the objective function MGE, and set the lower bounds of κ and α strictly positive at.1 and.1, correspondingly. In particular, if we let v i θ)) to be the ordered values of Vi δ θ)), the minimization in equation 16) simply is ˆθ = arg min θ Θ i=1 v i θ) 2i 1 ) 2. 2N Recall that these estimates are consistent at the rate of N 1/2. All simulations are done in Matlab. Table 1 reports the size and power performances of all the tests in finite samples for 1%, 5% and 1% nominal sizes. As can be clearly seen, our time-changed residual based tests T1N δ and T2N δ have satisfactory size performances since most of the rejection probabilities are close to their corresponding nominal sizes. Moreover, they are as good as the tests KSN δ and CM N δ of the known true model. Only in one case for the 1% nominal size, T1N δ slightly over-rejects at 2.2%. In term of power performances, the three tests consistently maintain discriminatory powers at different levels under different alternatives. They possess comparable and excellent powers under some certain alternatives. Under both H 13 and H 14 where the generated process working days per year are used. 13 For a detailed implementation of the time change and Khmaladze martingale transformation, see Computation of the Test Statistics, Appendix A.1 and A.2. 17

18 is either a BM with drift or shifting from the stationary OU to the nonstationary BM with drift, the time-changed residual based tests T1N δ and T 2N δ are substantially powerful being close to 1% for the 1% nominal level, and just slightly more or less powerful than KSN δ and CMN δ. On the other hand, under certain local alternatives, when the power performances of all of the tests start losing strength, the tests T1N δ and T 2N δ perform less effectively than KSδ N and CMN δ. Particularly, under the alternative H 11 and in the presence of the regime-shift between two stationary OU processes that have the same time trend component, while the KSN δ and CMN δ tests have a discriminatory power of 21% for the 1% nominal level, the tests T 1N δ and T2N δ are slightly less powerful at 14.2% and 16.6%, respectively. Under the alternative H 12, KSN δ and CM N δ discriminate the BM from the OU process significantly better than T 1N δ and T2N δ. Note that the BM process under H 12 is a special case of the OU process with κ =. Yet the two tests T1N δ and T 2N δ at least maintain some discriminatory powers. Such loss of powers of these time-changed residual based tests in detecting some local alternatives may be considered as the cost of relaxing the conventional assumption of known true parameters in the conditional mean. Apparently, the amount of this cost varies upon each alternative. Overall, our tests exhibit satisfactory size performances, and maintain their discriminatory powers against different alternative hypotheses in finite samples. 5. Conclusion This paper considers specification testing for the conditional mean of the general asset pricing model in continuous time which is applicable for both stationary and nonstationary processes of asset returns. The basic idea is: if the conditional mean with unknown true parameters is correctly specified, the fitted error process acquired from the postulated conditional mean regression should be a martingale. Thus, by DDS theorem, the fitted error process after being indexed by the time change should essentially become a standard BM. The tests therefore aim at testing the normalized increment of this time-changed error process, i.e. the fitted time changed residuals, against iid standard normals. Furthermore, in order to eliminate the parameter estimation effect and the presence of the unknown true parameters in the tests limit distributions, we apply Khmaladze transformation to extract the martingale component from the empirical process of the fitted time-changed residuals for constructing KS-type and CvM-type statistics. Besides having power against any local alternative, our tests possess several desirable properties. Due to the time change, the time-changed residual based tests are simply applicable for general asset pricing models, and capable of accommodating various types of volatilities, including but not limited to fat tail, time-varying or stochastic volatilities. Due to the Khmaladze martingale transformation, our tests are asymptotically distribution free. However, our tests have not checked the serial uncorrelatedness among the fitted time-changed residuals. To supplement this issue, two solutions are suggested for further research. In this paper, we also provide the simulations of specification testing for a conditional mean of an OU process. The results show that our tests perform reasonably well in term of size, and maintain their discriminatory powers under different alternatives. 18

19 Appendix A. Computation of the Test Statistics A.1 Time Change and Fitted Residuals from MGE As mentioned in Section 1, for any given positive number, we can construct a time change to transform the demeaned stochastic process Y t ) into a standard BM. However, in finite samples, the choice of will affect the MGE performance and our testing results. 14 Once is determined, we estimate the time change T δ t ) by simply minimizing the global distance 15 T δ i = arg min t> [Y ] δ t i, for i = 1,..., N. With the estimated time change Ti δ ), we can easily obtain the time changed processes X T δ, Y i T δ ). i We use MGE as suggested by Park 21) to get the consistent estimates of the unknown parameters θ ˆθ = 1 arg min θ Θ z R N = arg min θ Θ 1 N 2 1{Zi δ θ) z} Φz)) Φdz) 1 i=1 1{V δ i θ) r} r) 2 dr. If we let v i θ)) to be the the ordered values of Vi δ θ)), this minimization simply is ˆθ = arg min θ Θ Thus, the fitted time-changed residual is 1 v i θ) 2i 1 ) 2. 2N Zi δ ˆθ) = 1/2 Y T δ Y i T δ δ i 1) n i j=n i 1 +1 µx jδ, ˆθ) The tests now can be constructed from the empirical process of these fitted time-changed residuals. 14 For more detail of these affects of in finite samples, see Park 21) and Jeong and Park 29). 15 We can also estimate the time change by sequentially minimizing Ti δ = arg min t T δ [Y ] δ t [Y ] δ i 1) T i 1) δ in a recursive manner starting from T δ =. 19

20 A.2 Khmaladze Martingale Transformation Bai 23) shows that an alternative expression for ŵn δ r) in equation 25), which has a simpler computation, is ŵ δ N = N r ĴNr) δ ġs) C 1 s) s ) ) ġτ) dĵ Nτ) δ ds, where Ĵ N δ r) = 1 N N 1 1{V i δˆθ) r}. In this subsection, we use ˆV i to imply Vi δˆθ). Denote by ˆv i the ordered realized values of ˆV i for i = 1,..., N such that ˆv 1 < ˆv 2 <... < ˆv N with additional values ˆv = and ˆv N+1 = 1. If s = ˆv k then the integral s ġτ) dĵ δ Nτ) = 1 N ġˆv k ). Denote D k = N i=k ġˆv k). If we also denote C k = Cˆv k ) then C k = ˆv k ġτ)ġτ) dτ i=k ġˆv i )ġˆv i ) ˆv i+1 ˆv i ). Note that Ĵ N δ ˆv j) = j/n, hence the value of ŵn δ ˆv j) can be approximated as ŵnˆv δ j ) = ) j N N 1 j ġˆv k ) C 1 N k D kˆv k ˆv k 1 ). k=1 i=k While the statistic T1N δ is simply a maxima of the process ŵδ N ˆv j) over j = 1,..., N and ˆv j [, 1], the statistic T2N δ is approximated as the average sum of squares of ŵδ N ˆv j). B. Mathematical Proofs Proof of the Lemma For the proof of equation 18), i.e. E sup ˆν Nr) δ ˆν N r) = o1), readers are referred to the proof of Lemma 3.5 in Park 21). For the proof of equation 22), i.e. r ) ŵ N r) = ˆν N r) ġs) C 1 s) ġτ) dˆν N τ) ds, readers are referred to the proof of Theorem 1 in Bai 23). We rewrite the difference of ŵ δ N r) ŵ Nr) as following s ŵ δ Nr) ŵ N r) = ˆν δ Nr) ˆν N r) r ġs) C 1 s) s ġτ) ) dˆν Nτ) δ dˆν N τ) ds. 34) 2