A Semi-parametric Test for Drift Speci cation in the Di usion Model

Transcription

1 A Semi-arametric est for Drift Seci cation in the Di usion Model Lin hu Indiana University Aril 3, 29 Abstract In this aer, we roose a misseci cation test for the drift coe cient in a semi-arametric di usion model. Our test is based on the score marked emirical rocess whose asymtotic behavior will be distorted by the estimation of the drift arameters. We use martingale transformtion to take away the estimation e ects which makes our test asymtotic distribution-free. he limit theory relies on both "in- ll" and "long-san" asymtotics. he size and ower roerties are examined via simulation studies and a emirical work is imlemented for testing the mean-reverting sot interest rate model. Keywords and Phrases: Score marked emirical rocess; Semi-arametric estimation; Martingale transformation; Asymtotic distribution-free test. Jel classification: 1. INRODUCION Since Merton s seminal work around 197s, continuous-time models have been widely used in economics and nance, for examle, asset ricing, derivative valuation and term structure theory. he most commonly used continuous-time models are di usion rocesses, which are generally described by means of stochastic di erential equations of the form dx t = (X t )dt + (X t )dw t ; t (1) I am extremely thankful to my advisor Prof. Juan Carlos Escanciano for his numerous guidance and hel. All errors are my own. Deartment of Economics, Indiana University; 1 S. Woodlawn, Wylie Hall; Bloomington, IN ; USA, lz2@indiana.edu 1

2 where W t is the standard Wiener rocess, the drift coe cient () and the di usion coe cient () are both assumed to be continuous and may deend on unknown arameters. In the emirical literature, many arametric models have been roosed in favor of di erent uroses and most of them are mutually exclusive. For examle, in modeling the sot interest rate, di erent authors secify di erent di usion functions () (see Aït-Sahalia (1996b) for a summary), however, almost all these models have seci ed the same mean-reverting drift function. he mean-reverting behavior can be viewed as a common belief when modeling the dynamics of the instantaneous sot interest rate desite di erent beliefs on the di usion function. Hence, in order to reduce the risk of misseci cation, we can work on the semi-arametric model where only the drift function is arametrized and the di usion art is left unseci ed. Another motivation for semi-arametric model comes from the urose of identifying the source of misseci cation. Imagine that, when we jointly test the drift and di usion functions, even we can reject the null, we still have no idea about which art is misseci ed: the drift, the di usion, or both? hus, it would be more aroriate to consider the semi-arametric model by only arametrizing the drift function if we are interested in testing the correct seci cation of the drift. Furthermore, by observing that in (1) the drift is of order dt, and the di usion is of order dt, therefore, with in nitesimal time change, nonarametric identi cation becomes much easier for the di usion term than the drift term, see Jiang and Knight (1997). Hence, with arametric seci cation for drift, it is more recise to do estimation. Comared with extensive studies on estimating and testing the arametric di usion models, there are quite few aers on semi-arametric models, among which Arais and Gao (26), Kristensen (28a,b) and Park (28) are most relevent. Arais and Gao (26) roosed a semi-arametric aroach for testing the linearity of the drift function in the sot intestest rate model based on discretized version of the di usion model. Kristensen (28a,b) develoed a framework for estimating and testing a semi-arametric di usion model with either drift function or di usion function seci ed. Park (28) alied time change method to estimate the arameters in the drift art. In this aer, we modify the otimal martingale estimating equation method to make it alicable to the semi-arametric di usion model and the resulting estimator of the unknown arameters in drift function is -consistent resorting to both "in- ll" and "long-san" asymtotics framework. Our test is based on the score marked emirical rocess roosed in Negri and Nishiyama (28) for di usion models. However, two roablems make their test not alicable to our testing framework: (1) their test is only for simle hyothesis, that is, under the null, both drift and di usion are exlicitly given including the arameters, but in this aer, we want to test the correct seci cation of the functional form of drift function with di usion function unseci ed; (2) their test is not feasible in ractice in the sense that continuous observations are not obtainable for nancial or economic 2

3 data. Hence, we have to ll u the gas by taking into account of the estimation e ects and discreteness of the data. We utilize the martingale transformation method, roosed in Khmaladze (1981), to take away the estimation e ects which makes our test asymtotic distribution-free. he remainder of this aer is organized as follows. In Section 2 we state the hyothesis of interest and introduce our test statistic. In Section 3 the asymtotic roerties of the test are discussed. he size and ower erformance of the test are examined by Monte Carlo study in Section 4, and we will also test the hyothesis of linear mean-reverting drift in the sot interest rate. Section 5 concludes. he roofs are rovided in the Aendix. 2. NULL HYPOHESIS AND ES SAISIC hroughout this aer, we consider a semiarametric scalar di usion model of this form dx t = (X t ; )dt + (X t )dw t (2) where (; ) is a arametric function known uto the nite dimensional arameter 2 R d for a ositive integer d, and () is unknown but su ciently smooth. Under some regularity conditions, the above stochastic di erential equation has a unique strong solution fx t ; t > g which is stationary and ergodic, see Durrett (1996). he stationary marginal density is given by (x; ) = () x 2 (x) ex (u; ) 2 x 2 (u) du where the rocess is distributed on D = (x; x) with of articular interest in nance). (3) 1 x < x 1 (for examle, x = ; x = 1 is he lower bound x is irrelevant because () is chosen to ensure that the density integrates to one. Notice that since () is unknown, we don t have an exlicit form for the marginal density. he observations are obtained at discrete-time level, and we assume the observations are equisaced to kee exosition simle although the results can be easily extended to non-equisaced case, however, our method is not alicable to randomly samled data due to additional unknown structure in conditional moments indued by random samling (see Du e and Glynn (24)). Denote the data as fx i : i = ; 1; :::; N; and N = g, where is the observation horizon. As will be seen later, our asymtotic theory will deend on both "in- ll" and "long-san" asymtotics, that is,! and! 1. In this aer, as ointed out in the introduction, we are interested in testing the correct seci - cation of the drift coe cient. Formally, our null and alternative hyotheses are H : 9 some 2 ; such that () = (; ) (4) H 1 : () 6= (; ) for any 2 3

4 where () is the true drift function. Our test statistic is based on the score marked emirical rocess roosed by Negri and Nishiyama (28) which is tailored for testing di usion models. In Negri and Nishiyama (28), the null hyothesis is simle with known arameter in drift function, and di usion function is also known under both the null and the alternative. However, almost all existing arametric models only secify the functional form and we need to estimate the arameters in the model. As we will show later, it turns out that the estimation e ect is not negligible. Moreover, their test statistic is not feasible in the sense that they assume data are continuously observed, thus in this aer, we will overcome this roblem by using discretely observed data. Also notice that the underlying assumtion for our testing roblem is that the observations come from a stationary and ergodic di usion rocess. Before introducing our test statistic, let s rst take a look at how to estimate the arameters in the drift coe cient. For the unknown di usion function (), we use Nadaraya-Watson kernel estimator due to its simlicity and good nite-samle erfermance for small. o estimate the drift arameter, Kristensen (28a) rovided a N-consistent Pseudo-MLE estimator with small but xed by imosing stronger conditions on the di usion rocess. In this aer, since we are working in the "in- ll" asymtotic framework, we can take another aroach, otimal martingale estimating equation method, as in Bibby, Jacobsen and Sørensen (24): G N () = NX i=1 (X (i 1) ; ) b 2 (X (i 1) ) Xi X (i 1) (X (i 1) ; ) where (X (i 1) ; ) is the gradient of (X (i 1) ; ) w.r.t. and b 2 () is the Nadaraya-Watson kernel estimator. We get our estimator b by solving the equation G N () =. As discussed in Bibby, Jacobsen and Sørensen (24) and Kessler (1997), this estimator is N(or )-consistent and asymtotically normal when! and N 2!. From now on, we assume that a reliminary -consistent estimator of and a consistent estimator of () are available. o make the exosition clear and easy to follow, let s assume at this moment that the data are continuously observed and () is known. marked emirical rocesses with true arameter and with estimated arameter b : ev (x) = V (x) = 1 1 = V (x) We rst de ne two score 1 ( 1;x] (X t ) 1 (X t )(dx t (X t ; )dt) (5) 1 ( 1;x] (X t ) 1 (X t )(dx t (X t ; b )dt) (6) 1 1 ( 1;x] (X t ) 1 (X t )((X t ; b ) (X t ; ))dt then as in Negri and Nishiyama (28), fv (x) : x 2 Rg converges weakly to the Gaussian rocess fb F (x) : x 2 Rg with mean and covariance function given by Cov(B F (x); B F (y)) = F (x^y) 4

5 where B() is a standard Brownian motion and F () is the invariant distribution function of the di usion rocess under the null; notice the weak convergence takes lace in (C B (R); B), the sace of continuous and bounded functions on R with Borel -algebra B induced by the uniform norm, as! 1, and we denote the weak convergence as V =) B F hereafter. After we relace by its estimator b under the null, it turns out that the estimation e ect is not negligible, that is, the second term in (6) will distort the asymtotic distribution of e V such that its asymtotic distribution deends on unknown arameters. Basically, there are two ways to tackle this roblem: we can either use bootstra method (or subsamling) to aroximate its nite-samle distribution, or we can take away the estimation e ects by alying a roer transformation to V e (x) such that the transformed rocess is asymtotic distribution-free (ADF hereafter). he second aroach is known as martingale transformation method roosed by Khmaladze (1981), later Stute, hies and hu (1998) alied this method to model checks for regression model with unknown heterogeneity and Koul and Stute (1999) utilized it for time series models. Brie y, the martingale transformation is an isometry in the sace orthogonal to the score function determined by the model. It su ces to make sure that the martingale tranformation reserves the distribution of B F and takes away the asymtotic distortion from estimating the unknown arameters. In our roblem, the tranformation is de ned as ( ')(x) = '(x) x 1 (y) (y; )A 1 (y) 1 x y 1 (u) (u; )d'(u) df (y) (7) where A(y) = R 1 y 2 (u) (u; ) (u; )df (u) is assumed to be ositive de nite for all y 2 D: Notice that involves a set of unknown things: ; () and F (). o get a feasible transformation, we need to relace all unknown factors by their corresonding estimates, and we denote the feasible transformation as b : ( b ')(x) = '(x) x b 1 (y) (y; b ) b A 1 (y) 1 x y b 1 (u) (u; b )d'(u) d b F (y) (8) where A(y) b = R 1 b 2 (u) (u; b ) (u; b )df b (u). y Since all nancial and economic data are discretely observed, any feasible estimation method or testing rocedure for economic models should take the discreteness into consideration. Hence, with discrete observations fx i : i = ; 1; :::; N; with N = g, we de ne two score marked emirical rocesses here, one with true di usion term () and another with estimate b() bv N;(x) = bv 1 N;(x) = N 1 X1 1 ( 1;x] (X i ) 1 (X i )(X (i+1) X i (X i ; b ) 4): (9) N i= N 1 X1 1 ( 1;x] (X i )b 1 (X i )(X (i+1) X i (X i ; b ) 4): (1) N i= 5

6 And our test statistic is of Kolmogorov tye based on the transformed rocess b V b N;, and we have the following weak convergence result: su b V b N; (x) =) su jb(t)j : (11) x2r t1 Because of the ADF roerty of our test statistic and availability of the limit distribution, we can just imlement our test by comaring the test statistic with the critical values at any seci c signi cance level. Intuitively, the null hyothesis will be rejected if the test statistic is large enough. 3. ASYMPOIC PROPERY OF HE ES 3.1 Asymtotic heory In this section, we will study the asymtotic roerties of our test statistic. o study the behavior of the estimation e ects, we need to imose some conditions on the drift function. Following Khmaladze and Koul (24), we assume for 2 : (x; ) (x; ) = (x; )( ) + (x; ; ); < R (x; ) (x; ) df (x) < 1 C := R (x; ) (x; ) df (x) is ositive de nite R su 2 (x; ; ) df (x) = o( 2 ) k k (12) where F () is the invariant stationary distribution of the di usion rocess and the suerscrit stands for transose. hen under the null hyothesis, we have: Proosition 1: Under the condition (12), together with R x x 2 (x)df (x) < +1 and ( b ) = O P (1), it holds that e V (x) = V (x) G (x) ( b ) + o P (1) uniformly in x 2 D as! 1, where G(x) = R x x 1 (x) (x; ) df (x). Hence, the estimation of distorts the asymtotic distribution along the score functions, which also indicates the way we construct the martingale tranformation. Now, with the martingale transformation de ned in section 2, it is not di ucult to see (G ( b )) = because of its linearity, and our next roosition shows that the tranformation also reserves the distribution of B F: Proosition 2: Let be de ned by (7) and V e de ned by (6), then we have V e =) B F as! 1 in (C B (R); B), the sace of continuous and bounded functions endowed with the toology induced by the uniform norm. 6

7 Now with discretely observed data, we rst examine the di erence between V e and V b N; : As we know, the convergence of residual based emirical rocess deends on the martingale roerty of the residual, in which we should know f(x i ) := E(X (i+1) jx i ) uto unknown arameters. Since excet in a few cases, E(X (i+1) jx i ) generally doesn t have closed form exression for di usion models, we have to rely on the aroximation. Here, E(X (i+1) jx i ) = X i +(X i ; )+O P ( 2 ) is the rst order aroximation, see Stanton (1997). Hence, in addition to the "long-san" framework (! 1), our test also has to resort to the "in- ll" asymtotics (! ). However, our nite-samle simulation result in next section indicates that daily data are good enough to imlement our test. Proosition 3: Under the null hyothesis, with estimators ( b ; b()) for (; ()) de ned above, then b V N; (x) = e V (x) + o P (1) uniformly in x 2 D; as! : A consequence of roosition 3 is that b V N; = e V + o (1), which yields the following theorem by combining with roosition 2: heorem 1: Under the null hyothesis, the tranformed score marked emirical rocess b V N; converges weakly to B F in the Skorokhod sace as! ;! 1 and! : heorem 1 is still infeasible for alication because both and V b N; involve unknown quantities like (), and F. Hence, we need to relace them by corresonding estimates, and the resulting score marked emirical rocess V b N; 1 and the tranformation b are de ned in section 2. (Need some conditions on the estimate of ()) hen our main theorem follows: heorem 2: Under condition (12) and some condition regarding the convergence of b(), then under H ; b V b N; 1 =) B F in distribution in the Skorokhod sace as! 1 and! : Based on the transformed emirical rocess, many asymtotic distribution-free test statistics can be formed through some continuous functionals, for examle Cramer-von-Mises test and Kolmogorov- Smirnov test. In this aer, we would like to use the Kolmogorov-Smirnov test, then theorem 2 and continuous maing theorem yield su b V b N; (x) =) su jb F (x)j = su jb(t)j in distribution. xxx t1 x2r It s easy to simulate the distributions of the right-hand side variable. We use the critical values from Bai (23), namely, 1.94, 2.22 and 2.8 corresondingly at the signi cance level 1%, 5% and 1%. 7

8 3.2 Consistency of the est (to be done) 4. SIMULAION SUDIES AND DAA ANALYSIS 4.1 Finite Samle Performance his section examines the nite-samle erformance of our test statistic through some Monte Carlo exeriments. We are going to use two sot interest rate models to simulate our data: the CKLS model (also known as CEV model) roosed in Chan, Karolyi, Longsta,and Sanders (1992) and the nonlinear drift model as in Aït-Sahalia (1996b). he CKLS model is dx t = ( X t )dt + X t dw t with arameter values (; ; 2 ; ) = (:88; :972; :52186; 1:46): he Aït-Sahalia (1996b) nonlinear drift model is dx t = ( 1 X 1 t X t + 2 X 2 t )dt + X t dw t ) (13) with arameter values ( 1 ; ; 1 ; 2 ; 2 ; ) = (:17; :517; :877; 4:64; :64754; 1:5): Because there is no closed form for the transition density, we will use Milstein s scheme to simulate data: X t+ = X t + (X t ) + (X t ) " t (X t )(" 2 t 1) (14) where f" t : t = 1; 2; :::g are i.i.d. standard normal r.v. s. he initial value is set to equal the mean interest rate level of the data set in Aït-Sahalia (1996b). o reduce discretization bias, we simulate 1 observations each day and samle the data at daily frequency. hroughout this section, we assume that we only observe daily data but treat the time unit "1" di erently. hat is, if we treat "1" as one year, then the samling frequency is = ; if we treat "1" as one month, then the samling frequency becomes = 1 21 : In rincile, for the same data set, we have di erent ways to secify the samling frequency and the time san, but di erent seci cations will have di erent imacts on the size and ower erformance, and we will see this in the following simulation studies. he intuition is that, the drift roerty is more re ected in the long time san, thus to gain more ower, we should require to be as large as ossible; on the other hand, we also need the samling frequency to be small to get better estimation recision which in turn can imrove size erformance Size of the est. 8

9 o examine the size of our test, we simulate data from the CKLS model (13) which has the mean-reverting drift seci cation (X t ; ) = ( X t ) where is the long run mean and is the seed of mean reversion. For the arameters given above and for each samling frequency, we simulate 1 data sets of fx i : i = 1; :::; N; with N = g for N = 252; 126; 252; 54; 18, resectively. hese samle sizes corresond to 1, 5, 1, 2 and 4 years when = ; and 12, 6, 12, 24 and 48 months when = 1 21 : For each data set, we rst nonarametrically estimate the di usion function via kernel method by using the bandwidth bn 1=5 as suggested in Bosq (1998), where b is the samle standard deviation. hen, we estimate the arameters in the drift term = (; ) via the otimal martingale estimating equation method. With these estimates and by relacing marginal CDF with emirical CDF, we can comute the test statistic su b b V N; (x). However, we cannot take the sureme over the whole real x2r line by observing that we can t estimate A(y) b for large y, instead we calculate su b V b N; (x) x2( 1;x ] where x is the 99% quantile of the dataset. We consider the emirical rejection rates using the asymtotic critical values at the 1%, 5% and 1% levels, reectively. able 1 and table 2 reort the size erfermance with = and = 1 21 resectively. As we can see, with small samling frequency, the size erformance is very good even with very short time san ; with large, the emirical size is extremely distorted for short time san, and it becomes better as gets larger, for examle, the emirical size agrees with norminal size when = 48 months (4 years). able 1 Size of the test with = Daily data with = 1=252 Norminal Szie = number of years = :1 = :5 = :1 = = = = = able 2 Size of the test with =

10 Daily data with = 1=21 Norminal Szie = number of months = :1 = :5 = :1 = = = = = Power of the est. o investigate the ower of our test, we simulate data from the aforementioned nonlinear drift model (14) and test the null hyothesis that the data is generated from a mean-reverting model (linear drift). For each data set, we estimate the mean-reverting model with unknown di usion term, then comute our test statistic as in the revious section. And we take the same design as in the size study. able 3 and 4 reort the ower erfermance with = and = 1 21 resectively. As the simulation results indicate, with small time san (when = ), our test almost has no ower to detect the misseci cation of the drift function; with the same data set, if we increase, hence increase, then we gain much better ower, for examle, at the signi cance level = 5%, we can reject the null hyothesis 62.9% out of 1 simulations when = 48 months (4 years). he simulation studies for size and ower erformance con rms our intuition of the dilemma between selecting samling frequency and time san. able 3 Power of the test with = Daily data with = 1=252 Rejection signi cance level = number of years = :1 = :5 = :1 = = = = 2 = 4 able 4 Power of the test with =

11 Daily data with = 1=21 Rejection signi cance level = number of months = :1 = :5 = :1 = = = = = Alication to Sot Interest Rate Model (not done yet) As a comarison, we rst use our test to reexamine the mean-reverting sot interest rate using the same data set as in Ai-Sahalia (1996b). he daily data are from June 1973 to February 1995 with a total 555 observations. 5. CONCLUSIONS (to be lled) APPENDIX Proof of Proosition 1: Under (12), we can write (6) as ev (x) = 1 = V (x) = V (x) 1 ( 1;x] (X t ) 1 (X t )(dx t (X t ; b )dt) 1 1 = V (x) I 1 (x) I 2 (x) 1 ( 1;x] (X t ) 1 (X t )((X t ; b ) (X t ; ))dt 1 ( 1;x] (X t ) 1 (X t ) (X t ; )dt ( b ) 1 1 ( 1;x] (X t ) 1 (X t ) (X t ; b ; I 1 (x) = (G (x) + o P (1)) ( b ) = G (x) ( b ) + o P (1) follows from ergodicity and stationarity of the di usion rocess; we also need to show I 2 = o P (1), as! 1: I 2 = ( 1 1 ( 1;x] (X t ) 1 (X t ) (X t ; b ; )dt 1 1 ( 1;x](X t ) 2 (X t )dt) 1=2 ( 2 (X t ; b ; )dt) 2 he condition (12) imlies that for every < k < 1; for any >, there exists such that with robability at least 1 the following holds for all > : " # " E su k k 1=2 k 2 (X t ; ; )dt E su 2 (X t ; ; ) k k 1=2 k # dt = o P ( 1 k 2 ) = o P (k 2 ) 11

12 hus, with b = OP ( 1=2 ); we have I 2 = o P (1) uniformly in x 2 D: Proof of Proosition 2: From Lemma 3.1 in Stute, hies and hu (1998), Cov( (BF )(x); (B F )(y)) = F (x ^ y): And we also have e V (x) V (x) = ( e V (x) V (x)) = ( e V (x) V (x)) + x x x x 1 (y) (y; )A 1 (y) 1 (y) (y; )A 1 (y) 1 = ( G (x) 1=2 ( b ) + o P (1) ) +( x x = o P (1) 1 (y) (y; )A 1 (y) 1 y 1 together with V =) (B F ) = B F, the conclusion follows. y 1 (u) (u; )( e V (du) V (du)) df (y) 1 (y;1) (X t ) 2 (X t ) (X t ; )((X t ; b ) (X t ; ))dt df (y) 2 (u) (u; ) (u; )df (u) df (y) 1=2 ( b ) + o P (1)) Proof of Proosition 3: Let Y t (x) = 1 ( 1;x] (X t ) 1 (X t ), Y i (x) = 1 ( 1;x] (X i ) 1 (X i ) and bv (x; ) : = 1 NX 1 1 ( 1;x] (X i ) 1 (X i )(X (i+1) X i (X i ; ) 4); ev (x; ) = 1 i= 1 ( 1;x] (X t ) 1 (X t )(dx t (X t ; )dt) then for each x; Y t (x) is a caglad rocess, and we have for each (x; ), b V (x; ) P! e V (x; ) as!. By observing that E[ 1 ( 1;x1)(X t ) 1 ( 1;x2)(X t ) ] = F (jx 1 x 2 j) and together with the fact that 2 is comact and b V (x; ) is continuous in, then the uniform convergence (in x and ) follows, thus Pro. 3 holds. heorem 1 just follows from roosition 2 and 3. Proof of heorem 2: REFERENCES Aït-Sahalia, Y. (1996b): "esting continuous-time models of the sot interest rate", Review of Financial Studies 9, Aït-Sahalia, Y. (27): "Estimating Continuous-ime Models Using Discretely Samled Data", Advances in Economics and Econometrics, heory and Alications, Ninth World Congress, 27 Arais, M.and J. Gao, 26: "Emirical comarisons in short-term interest rate models using nonarametric methods", Journal of Financial Econometrics 4,

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