A Primer of Nonparametric Econometrics and Their Applications to Economics and Finance

Transcription

1 A Primer of Nonparametric Econometrics and Their Applications to Economics and Finance Zongwu Cai University of North Carolina at Charlotte, USA and Xiamen University, China

2 WHY DO WE NEED TO STUDY NONPARAMETRIC ECONOMETRICS? 1

3 Contents Part I: Method and Theory 1. Review of Nonparametric Methods Nonparametric density Estimation Nonparametric Regression Estimation 2. Computational Issues 3. Nonparametric Estimation for Non-Stationary Variables Part II: Applications Nonparametric Diffusion Models Nonparametric Predictive Models for Stock Returns Nonparametric Inflation Forecasting Models (???) Nonparametric Pricing Kernel Models (???) 2

4 A nonlinear time series model for a forecasting can be expressed Y t = g(x t ) + ε t, (1) where Y t is the forecasting variable(s) and X t is a vector of predictor variables. In the last three decades, various forms of model (1) has been explored. See the books by Tong (1990), Granger and Teräsvirta (1993), and Fan and Yao (2003). In the literature, most of studies can be classified into nonlinear parametric form and purely nonparametric approach. Also, it is commonly assumed that all variables (particularly X t ) in model (1) are stationary, denoted by I(0). 3

5 Part I: Review of Nonparametric Methods Cai, Z. (2007). Lecture Notes on Nonparametric Econometrics. WISE. Cai, Z. and Y. Hong (2009). Some Recent Developments in Nonparametric Finance. Forthcoming in Advances in Econometrics. Cai, Z., J. Gu and Q. Li (2009). Some Recent Developments on Nonparametric Econometrics. Forthcoming in Advances in Econometrics. 4

6 Part I: Review of Nonparametric Methods Nonparametric Density Estimate: Let {X i } be a random sample with an unknown marginal distribution F( ) (CDF) and its probability density function f( ) (PDF). The question is how to estimate f( ) and F( ). Since F(x) = P(X i x) = E[I(X i x)] = and if F( ) is differentiable, x f(u)du, f(x) = lim h 0 F(x + h) F(x h) 2 h F(x + h) F(x h) 2 h 5

7 Nonparametric Density Estimate if h is very small, by the method of moment estimation (MME), F(x) can be estimated by F n (x) = 1 n n i=1 I(X i x), which is called the empirical cumulative distribution function (ecdf), so that f(x) can be estimated by f n (x) = F n(x + h) F n (x h) 2 h = 1 n n i=1 K h (X i x), where K(u) = I( u 1)/2 and K h (u) = K(u/h)/h. 6

8 Nonparametric Density Estimate Indeed, the kernel function K(u) can be taken to be any symmetric density function. Here, h is called the bandwidth. f n (x) was proposed initially by Rosenblatt (1956) and Parzen (1962) explored its properties in detail. Therefore, it is called the Rosenblatt-Parzen density estimate. Roussas (1968, 1969) applied this method in applications. Since then, this approach has been studied extensively and applied to many applied areas such as economics and finance, for example, estimation of the drift and diffusion function in the diffusion model, estimation of nonparametric pricing kernel, and calculation of value-at-risk, and so on. 7

9 Nonparametric Density Estimate Asymptotic Properties for ECDF If {X i } is stationary, then E[F n (x)] = F(x) and n var(f n (x)) σf(x) 2 F(x)[1 F(x)] + 2 Cov(I(X 1 x), I(X i x)) } i=2 {{} This term is called A d. Therefore, n var(f n (x)) σ 2 F(x). (2) 8

10 Nonparametric Density Estimate One can show based on the mixing theory that n [Fn (x) F(x)] N ( 0, σ 2 F(x) ) (3) if {X i } is stationary and mixing with the mixing coefficient satisfying a certain condition. It is clear that if {X i } are independent, A d = 0 and σf 2 (x) = F(x)[1 F(x)]. 9

11 Nonparametric Density Estimate If A d 0, the question is how to estimate it. We can use the HC estimator by White (1980) or the HAC estimator by Newey and West (1987), or the kernel method by Andrew (1991). To implement the HC or HAC estimator in a computer software, we can use the package sandwich in the computing software R which can be downloaded from and the commands are vcovhc() or vcovhac() or meathac(). 10

12 Nonparametric Density Estimate The results in (3) can used to construct a test statistic to test the null hypothesis H 0 : F(x) = F 0 (x) versus H a : F(x) (>)(<)F 0 (x). This test statistic is the well-known Kolmogorov-Smirnov test, defined as D n = sup F n (x) F 0 (x) <x< for the two-sided test, and D + n = sup <x< [F n (x) F 0 (x)] and D n = sup <x< [F 0 (x) F n (x)] for one-sided tests. 11

13 Nonparametric Density Estimate One can show (see Serfling (1980)) that under some regularity conditions, P( n D n d) 1 2 ( 1) j+1 exp( 2j 2 d 2 ) j=1 and P( n D + n d) = P( n D n d) 1 exp( 2d 2 ). In R, there is a built-in command for the Kolmogorov-Smirnov test, which is ks.test(). 12

14 Nonparametric Density Estimate Asymptotic Properties for Density Estimation Theorem 1: Under regularity conditions, we can show that ] n h [f n (x) f(x) h2 2 µ 2(K) f (x) + o p (h 2 ) N (0, ν 0 (K) f(x)), where the term h2 2 µ 2(K) f (x) is called the asymptotic bias and µ 2 (K) = u 2 K(u)du and ν 0 (K) = K 2 (u)du. Note that the asymptotic variance here is ν 0 (K) f(x) is the same as that for iid data. WHY? 13

15 Nonparametric Density Estimate Importance of Bandwidth: Example 1: Let us examine how importance the choice of bandwidth is. The data {X i } n i=1 are generated from N(0,1) (iid) and n = 300. The grid points are taken to be [ 4,4] with an increment = 0.1. Bandwidth is taken to be 0.25, 0.5 and 1.0, respectively, and the kernel can be the Epanechnikov kernel K(u) = 0.75(1 u 2 )I( u 1) or Gaussian kernel. Comparisons are given in Figure 1. 14

16 True h=0.25 h=0.5 h=1 h=h_o Figure 1: Bandwidth is taken to be 0.25, 0.5, 1.0 and the optimal one denoted by h o (see later) with the Epanechnikov kernel. 15

17 Nonparametric Density Estimate A Real Example: Example 2: Next, we apply the kernel density estimation to estimate the density of the weekly 3-month Treasury bill from January 2, 1970 to December 26, Figure 2 displays the ACF and PACF plots for the original data (top panel) and the first difference (middle panel) and the estimated density of the differencing series together with the true standard normal density: the bottom left panel is for the built-in function density() and the bottom right panel is for own code. 16

18 ACF ACF Lag Lag Lag Density of 3mtb (Buind in) Lag Density of 3mtb Estimated Standard Estimated Standard Figure 2: Partial ACF Partial ACF

19 Nonparametric Density Estimate From the above graphs, what can we see? Typical stylized facts about financial data. Note that R has a built-in function density() for computing the nonparametric density estimation. Also, you can use the command plot(density()) to plot the estimated density. Furthermore, R has a built-in function ecdf() for computing the empirical cumulative distribution function estimation and plot(ecdf()) for plotting the step function. Finally, R has a generic function qqnorm() the default method of which produces a normal Quantile-Quantile plot of the given values. 18

20 Nonparametric Density Estimate Issues: 1: Optimal Kernel: The nonnegative probability density function K minimizing the asymptotic mean integrated squares error (AMISE) is a re-scaling of the Epanechnikov kernel: for any a > 0. K opt (u) = 3 4 a (1 u2 /a 2 ) + 19

21 Nonparametric Density Estimate 2: Optimal Bandwidth: The optimal bandwidth minimizing the AMISE is h opt = [ ν 0 (K)/µ 2 2(K) ] 1/5 f 2/5 2 n 1/5. 20

22 Nonparametric Density Estimate 3: Boundary Problems: If the density f( ) has a bounded support, say, [0, 1], then, one can show that f n (ch) = f(0+)µ 0,c (K)+hf (0+)[cµ 0,c (K)+µ 1,c (K)]+o p (h). Particularly, if c = 0 and K( ) is symmetric, then E(f n (0)) = f(0)/2 + o(1). There are several methods to deal with the density estimation at boundary points. Possible approaches include the major methods: A. boundary kernel; see Gasser and Müller (1979) and Müller 21

23 (1993), B. reflection; see Schuster (1985) and Hall and Wehrly (1991), C. transformation; see Wand, Marron and Ruppert (1991) and Marron and Ruppert (1994), and D. local polynomial fitting; see Hjort and Jones (1996a) and Loader (1996), and others. If R is used for the local fit for density estimation, please use the function density.lf() in the package localfit. 22

24 Nonparametric Density Estimate 4: Bandwidth Selection: A: normal reference bandwidth selector is 1.06 s n 1/5 for the Gaussian kernel ĥ opt = 2.34 s n 1/5 for the Epanechnikov kernel. B: Edgeworth expansion selector is ( ĥ opt = h opt ˆγ ˆγ ) 1/ ˆγ2 4, where ˆγ 3 and ˆγ 4 are respectively the sample skewness and kurtosis. 23

25 Nonparametric Density Estimate C: plug-in bandwidth selector: This technique relies on finding an estimate of the functional f 2, which can be obtained by using a pilot bandwidth. D: Cross-Validation Method selector is CV(h) = 1 n 2 s,t K h(x s X t ) 2 n(n 1) n t s K h (X s X t ), where K h ( ) is the convolution of K h( ) and K h ( ). minimizer of CV(h) is denoted by ĥcv. The 24

26 Nonparametric Density Estimate Function dpik() in the package KernSmooth in R selects a bandwidth for estimating the kernel density estimation using the plug-in method. Function lscv() in the package locfit in R selects a bandwidth for estimating the kernel density estimation using the least squares cross-validation method. 25

27 Part I: Review of Nonparametric Methods Prediction and Regression Functions: To forecast the future value, say Y t+1 given the information set I t at time t, there are several forecasting criteria available. The general form is m(i t ) = min a E[ρ(Y t+1 a) I t ], where ρ( ) is an objective (loss) function. 26

28 Part I: Nonparametric Regression Estimate (1) If ρ(z) = z 2 is the quadratic function, then, m(i t ) = E(Y t+1 I t ), called the mean regression function. Implicitly, it requires that the distribution of Y t should be symmetric. If the distribution of Y t is skewed, then this is not a good criterion. (2) If ρ τ (y) = y (τ I {y<0} ) called the check function, where τ (0,1) and I A is the indicator function of any set A, then, m(i t ) = F 1 (τ I t ), where F(m(I t ) I t ) is the conditional CDF of Y t+1 given I t. This m(i t ) becomes the conditional quantile or quantile regression, denoted by q τ (I t ), proposed by Koenker and Bassett (1978, 1982). 27

29 For nonparametric quantile regression models, see the paper by Cai and Xu (2008, JASA). Koenker (2005) developed the R module quantreg to make statistical inferences on the linear quantile regression model q τ (I t ) = β T τ X t. To fit a linear quantile regression using R, one can use the command rq() in the package quantreg. For a nonlinear parametric model, the command is nlrq(). For a nonparametric quantile model for univariate case, one can use the command lprq() for implementing the local polynomial estimation. For an additive quantile regression, one can use the commands rqss() and qss(). 28

30 Part I: Nonparametric Regression Estimate (3) If ρ(x) = 1 2 x2 I x M + M( x M/2) I x >M, the so called Huber function in literature, then it is the Huber robust regression. We will not discuss this topic. If you have an interest, please read the book by Rousseeuw and Leroy (1987). In R, the library MASS has the function rlm for robust linear model. Also, the library lqs contains functions for bounded-influence regression. 29

31 Part I: Review of Nonparametric Methods Nonparametric Regression Estimation: How to estimate m(x) nonparametrically? Let us look at the Nadaraya-Watson estimate of the mean regression m(x). The main idea is as follows: m(x) = y f(y x)dy = y f(x, y)dy f(x, y)dy, where f(x, y) is the joint PDF of X t and Y t. To estimate m(x), we can apply the plug-in method. That is, plug the nonparametric kernel density estimate f n (x, y) (product kernel method) into the right hand side of the above equation to 30

32 obtain ˆm nw (x) = y fn (x, y)dy fn (x, y)dy = n t=1 W t Y t, where n W t = K h (X t x)/ K h (X t x). t=1 ˆm nw (x) is the well known Nadaraya-Watson (NW) estimator, proposed by Nadaraya (1964) and Watson (1964). Note that the weights {W t } do not depend on {Y t }. Therefore, ˆm nw (x) is called a linear estimator, similar to the least squares estimate (LSE). 31

33 Nonparametric Regression Estimation Let us look at the NW estimator from a different angle. ˆm nw (x) can be re-expressed as the minimizer of the following locally weighted least squares given by ˆm nw (x) = min a n (Y t a) 2 K h (X t x). t=1 This means that when X t is in a neighborhood of x, m(x t ) is approximated by a constant a (local approximation). Indeed, the working model is Y t = m(x t )+ε t a+ε t with the weights {K h (X t x)}. Therefore, the Nadaraya-Watson estimator is also called the local constant estimator. 32

34 Nonparametric Regression Estimation Theorem 2: Under some regularity conditions, we have nh [ ˆmnw (x) m(x) B nw (x) + o p (h 2 ) ] N { 0, σ 2 m(x) }, where σ 2 m(x) = ν 0 (K) var(ε t X t = x)/f(x) and B nw (x) = h2 2 µ 2(K) [m (x) + 2m (x)f (x)/f(x)] is regarded as the asymptotic bias. The bias term involves not only curvatures of m(x) (m (x)) but also the unknown density function f(x) and its derivative f (x) so that the design can not be adaptive. 33

35 Nonparametric Regression Estimation At the boundary point, one can show that the asymptotic bias term is of the order O(h) but the asymptotic variance term has the same order as that for the interior point although the scaling constant is different. To overcome the above shortcomings of local constant estimate, we can use the local polynomial fitting scheme; see the books by Fan and Gijbels (1996), Fan and Yao (2003), and Li and Racine (2007). 34

36 Nonparametric Regression Estimation Assume that the regression function m(x) has (q + 1)th order continuous derivative. For ease notation, assume that p = 1. When X t (x h, x + h), then m(x t ) q j=0 m (j) (x) j! (X t x) j = q j=0 β j (X t x) j, where β j = m (j) (x)/j!. Therefore, when X t (x h, x+h), the model becomes Y t q j=0 β j (X t x) j + ε t. 35

37 Hence, we can apply the locally weighted least squares method. The locally weighted least squares becomes 2 n q Y t β j (X t x) j K h (X t x). (4) t=1 j=0 Minimizing the above with respect to β = (β 0,..., β q ) T to obtain the local polynomial estimate ˆβ; ( 1 ˆβ = X WX) T X T WY, (5) where W = diag{k h (X 1 x),, K h (X n x)}. Therefore, for 1 j q, ˆm (j) (x) = j! ˆβ j. 36

38 Nonparametric Regression Estimation There are several ways of implementing the local polynomial estimator. One way you can do so is to write your own code by using matrix multiplication as in (5) or employing function lm() or glm() with weights {K h (X t x)}. In R, Function dpill() chooses a bandwidth for the local linear (q = 1) regression estimation using the plug-in approach. Function locpoly() is for the local polynomial fitting including a local polynomial estimate of the density of X (or its derivative) if the dependent variable is omitted. 37

39 Nonparametric Regression Estimation Properties: minimax efficiency design adaption no boundary effects. For details, see the books by Fan and Gijbels (1996), Fan and Yao (2003), and Li and Racine (2007). 38

40 Nonparametric Regression Estimation To preserve a certain shape of regression function, Cai (2002, ET) proposed a weighted Nadaraya-Watson estimator by using the empirical likelihood idea, to posses both properties from the Nadaraya-Watson method and local polynomial approach. In particular, this method is very useful to estimate some positive functions like volatility and CDF/PDF; see Phillips and Xu (2008) for conditional variance and Cai and Wang (2008, JoE) for conditional CDF. 39

41 Nonparametric Regression Estimation Example 3: We apply the kernel regression estimation and local polynomial fitting methods to estimate the drift and diffusion of the weekly 3-month Treasury bill from January 2, 1970 to December 26, Let x t denote the weekly 3- month Treasury bill. It is often to model x t by assuming that it satisfies the continuous-time stochastic differential equation (Black-Scholes model) dx t = µ(x t ) dt + σ(x t ) dw t, (6) where W t is a Wiener process, µ(x t ) is called the drift function 40

42 and σ(x t ) is called the diffusion function. Our interest is to identify µ(x t ) and σ(x t ). Assume a time series sequence {X t, 1 t n} is observed at equally spaced time points. Using the infinitesimal generator (Øksendal, 1985), the firstorder approximations of moments of x t, a discretized version of the Ito s process, are given by Stanton (1997) x t = µ(x t ) + σ(x t ) ε, (7) where x t = x t+ x t, ε N(0,1), and x t and ε t are independent. (7) is called the Euler approximation of (6). 41

43 Therefore, by (7), and µ(x t ) = lim 0 E[ x t x t ]/ σ 2 (x t ) = lim 0 E [ ( x t ) 2 x t ] /. Hence, estimating µ(x) and σ 2 (x) becomes a nonparametric regression problem. We can use both local constant and local polynomial method to estimate µ(x) and σ 2 (x). As a result, the local constant estimators (red line) together with the lowess() smoothers (black line) and the scatterplots of x t [in (a)], x t [in (b)], and ( x t ) 2 [in (c)] versus x t are presented in Figure 3. 42

44 (a) y(t) vs x(t) Local Constant Estimate x(t 1) (b) y(t) vs x(t) x(t 1) (c) y(t)^2 vs x(t) 0e+00 1e 04 2e 04 3e x(t 1) Figure 3: Scatterplots of x t, x t, and ( x t ) 2 versus x t with the smoothed curves computed using scatter.smooth() and the local constant estimation. 43

45 The local linear estimators (red line) together with the lowess() smoothers (black line) and the scatterplots of x t [in (a)], x t [in (b)], and ( x t ) 2 [in (c)] versus x t are displaced in Figure 4. 44

46 (a) y(t) vs x(t) Local Linear Estimate x(t 1) (b) y(t) vs x(t) x(t 1) (c) y(t)^2 vs x(t) 0e+00 1e 04 2e 04 3e x(t 1) Figure 4: Scatterplots of x t, x t, and ( x t ) 2 versus x t with the smoothed curves computed using scatter.smooth() and the local linear estimation. 45

47 Nonparametric Models and Methods In a sum, the aforementioned models and methods assume that all variables involved (dependent variable and independent variables as well as measurement errors) are stationary. But in economics and finance, some variables might not be stationary. For example, many financial variables are commonly nonstationary, including log dividend-price ratio, log earningprice ratio, the log book-to-market ratio, the dividend yield, the term spread and default premium, and the interest rates, and so on. Another example is that the velocity of money supply is unit root too and it can be used to forecast the inflation rate (see later). 46

48 Nonparametric Models and Methods According to Campbell and Yogo (2006), a linear/nonlinear regression model onto lagged variables and stationary variables has a low predictive power because stationary predictors are extremely noisy and/or highly persistent. By Campbell and Yogo (2006), if noise can be eliminated partially, the predictive power might be improved significantly. One of solutions is to use some less noisy variables such as integrated or nearly integrated variables and to use a model which has an ability of capturing the persistence. 47

49 Nonstationary Variables There are many nonlinear forms which can be explored. Let us look at the following general varying coefficient model Y t = p j=1 β j (Z t ) X jt + ε t, (8) where Z t is a vector of stationary or nonstationary variables or time t and X t is a vector of stationary or nonstationary variables. Model (8) can be regarded as an approximation of a general nonlinear regression function. 48

50 Nonstationary Variables Model (8) covers several scenarios: (I) X t is I(1) and Z t is stationary, (II) X t is stationary and Z t is I(1); (III) Z t = t is a time trend variable and X t is I(1); see Park and Hahn (1999) and Chang and Martinez-Chombo (2003); (IV) both X t and Z t are I(1); (V) ε t is either stationary or I(1); (VI) partially linear models; 49

51 (VII) Nonparametric cointegration issues such as Y t and X t are I(1) but ε t is I(0). How to test nonparametric cointegration? (VIII) How about the case when X t is nearly integrated (like X t = (1 + c/n)x t 1 + e t )? 50

52 Nonstationary Variables For simplicity, I consider the following simple varying coefficient model in scenario I: Y t = } β 1 (Z {{ t ) X t1 } + } β 2 (Z {{ t ) X t2 } +ε t stationary nonstationary = X T t β(z t ) + ε t, 1 t n, (9) where X t1, Z t, and ε t are stationary, X t2 is nonstationary such as an I(1) process, β(z t ) = (β 1 (Z t ), β 2 (Z t )) T, and X t = (X t1, X t2 ) T. 51

53 Nonstationary Variables We apply the local linear fitting scheme to estimate the coefficient functions {β j ( )}. Theorem 3: Under some regularity conditions, nhhn [ ˆβ(z) β(z) 1 ] 2 h2 µ 2 (K)β (z) d MN(Σ β (z)), where H n = diag{1, n} and MN(Σ β ) is a mixed normal distribution with mean of zero and some conditional covariance Σ β involving integrations of a standard Brownian motion. 52

54 Consequences of Theorem 3: Nonstationary Variables If there is no X t2, e.g. there is no nonstationary covariate, the results are the same as those in Cai, Fan and Yao (2000). The asymptotic bias for ˆβ j (z) is h 2 µ 2 (K)β j (z)/2, same as that for stationary case. Convergence rate for ˆβ2 (z) (coefficient function for nonstationary covariate) is faster than that of ˆβ 1 (z) (coefficient function for stationary covariate) by a factor of n 1/2. 53

55 Nonstationary Variables What are the asymptotic results for scenario II: X t is stationary and Z t is I(1)? For this case, the local linear fitting scheme is still applicable here. But the asymptotic result is totally different and the theoretical proof is very involved. Theorem 4: Under some regularity conditions for scenario II, [ n 1/2 h ˆβ(z) β(z) 1 ] 2 h2 µ 2 (K)β d (z) MN(Σ β ), where MN(Σ β ) is a mixed normal distribution with mean zero and conditional covariance Σ β associated with the local time of a standard Brownian motion. 54

56 Nonstationary Variables Main Findings: 1. Asymptotic bias is the same as that for stationary case. 2. The rate of convergence is totally different with an extra factor n 1/4, which is slower than that for stationary case. 3. The optimal bandwidth is h opt = O p (n 1/10 ). 4. Asymptotic behavior at boundary is different from that for stationary case. For details, see the paper by Cai, Li and Park (2009, JoE). 55

57 Nonstationary Variables Curse of Dimensionality It is easy to derive the optimal rate of convergence for MSE which is O(n 4/(4+p) ) where p is the dimension, by trading off the rates between the bias and variance. When p is large, the so called curse of dimensionality exists. To understand this problem quantitatively, let us look at the rate of convergence. To have a comparable performance with onedimensional nonparametric regression with n 1 data points, for p-dimensional nonparametric regression, we need the number 56

58 of data points n p, O(n 4/(4+p) p ) = O(n 4/5 1 ), or n p = O(n (p+4)/5 1 ). Note that here we only emphasize on the rate of convergence for MSE by ignoring the constant part. Table 1: Sample sizes required for p-dimensional nonparametric regression to have comparable performance with that of 1-dimensional nonparametric regression using size 100 dimension sample size ,585 3,982 10,000 25,119 63, ,490 Table 1 shows the result with n 1 = 100. The increase of required sample sizes is exponentially fast. 57

59 Nonstationary Variables Dimension Reduction Methods 1. Additive models 2. Functional coefficient models 3. Index models 4.. See the papers by Cai and Hong (2009) and Cai and Li (2009) for details on methodology and applications in economics and finance or the book by Li and Racine (2007). 58

60 Part II: Applications 59

61 Part II: Applications 1. Nonparametric Estimation in Jump Diffusion Model Cai, Z. and Y. Hong (2009). Some Recent Developments in Nonparametric Finance. Forthcoming in Advances in Econometrics. 60

62 Part II: Jump Diffusion Model The nonparametric jump-diffusion model for single factor can be expressed as dx t = µ(x t, t) dt + σ(x t, t) dw t + dj t, (10) where J t is a compensated jump process with arrival rate λ t = λ(x t ) 0 and the jump size, ξ t, has a fixed distribution Π( ) with mean zero. For example, J t = ξ t P t, where P t is a Poisson process or a binomial distribution with an intensity λ(x t ), and Π( ) can be taken to be normal or uniform. 61

63 Part II: Jump Diffusion Model Chernov, Gallant, Ghysels, and Tauchen (2003) considered J t as a Lévy process. If λ t = 0 or ξ t = 0, the jump-diffusion model becomes the pure diffusion model dx t = µ(x t, t) dt + σ(x t, t) dw t. (11) There are many other types of specification of J t, see Kou (2002) or Tsay (2005, p.275). 62

64 Part II: Jump Diffusion Model One can show (see, e.g, Johannes (2004) and Bandi and Nguyen (2002, 2003)) that µ 1 (X t ) = lim 0 E[{X t+ X t } X t ]/ = µ(x t ) + λ(x t ) E(ξ t ), and σ 2 1(X t ) = lim 0 E ] [{X t+ X t } 2 X t / = σ 2 (X t ) + λ(x t ) E(ξ 2 t) σ 2 (X t ). 63

65 Part II: Jump Diffusion Model This implies that the first two moments are the same as those for a pure diffusion model by using the new diffusion coefficients µ 1 (x) and σ1(x). 2 Note that the jump model can be also used to describe the heavy tail property. 64

66 Part II: Jump Diffusion Model The main difference between pure diffusion model in (6) and jump diffusion model in (11) relies on the higher moments. Using the infinitesimal generator (Øksendal, 1985) of X t, we can compute, j > 2, E µ j (X t ) = lim 0 [{X t+ X t } j X t ] See Duffie, Pan, and Singleton (2000) for details. = λ(x t ) E(ξ j t). 65

67 Part II: Jump Diffusion Model The conditional kurtosis and skewness are given by k(x t ) λ(x t ) E(ξ 4 t) [σ 2 (X t ) + λ(x t ) E(ξ 2 t)] 2, and λ(x t ) E(ξ 3 s(x t ) t) [σ 2 (X t ) + λ(x t ) E(ξt)] 2 3/2. Johannes (2004) used the conditional kurtosis to measure the departures from the pure diffusion model. 66

68 Part II: Jump Diffusion Model The NW estimation of µ j ( ) is considered by Johannes (2004) and Bandi and Nguyen (2000). Moreover, Bandi and Nguyen (2000) provided a general asymptotic theory for the resulting estimators. Further, by specifying a particular form of Π(λ) = Π 0 (λ, θ), say, ξ t N(0, σξ 2 ), Bandi and Nguyen (2000) proposed consistent estimators of λ( ), σξ 2, and σ2 ( ) and derived their asymptotic properties. 67

69 Part II: Jump Diffusion Model Test: A natural question arises how to measure the departures from the pure diffusion model. That is to test H 0 : (10) against (11). Equivalently, it is to test H 0 : λ( ) 0 or ξ = 0. One of the test statistics can be T = ˆµ 2 3(x) w(x) dx, or T = ˆµ 4 (x) w(x) dx, where w( ) is a weighting function. investigation theoretically and empirically. This needs a further 68

70 Part II: Jump Diffusion Model QUESTION: Are these test methods suitable for testing a jump diffusion model? HW for Students: Please do a Monte Carlo simulation as follows. Consider the true model given in (6) with discontinuous µ(x) or σ(x) or both and the working model given by (11) with continuous µ(x) and σ(x). Use the existing testing procedures in literature to test whether there exits a jump and to see what you can obtain. 69

71 Part II: Applications 2. Nonparametric Predictive Models for Stock Returns 3. Nonparametric Inflation Forecasting Model (???) 4. Nonparametric Pricing Kernel Models (???) 70

72 End THANK YOU for COMING! 71