A Wavelet-Based Nonparametric Estimator of the Variance Function
Zuohong Pan and Xiaodi Wang - Western Connecticut State University
The need to estimate the variance function has been of crucial importance in
two areas of economic applications: (1) to estimate the variance covariance
matrix in the face of heteroskedasticity of some form; (2) to proxy the risk
premium in the financial market when the risk is not perceived constant.
Standard estimation method has been parametric in that some form of the
variance function is specified and a two-step Generalized Least Square or
iterative GLS (Maximum Likelihood) procedure is used. In the financial time
series analysis, the increasingly popular ARCH model (Engle, 1982) is an
alternative parametric specification for the variance function where the
variance of the errors is a function of its own past squared.
Parametric estimation has long been criticized for its arbitrariness in the form of the variance function. Another line of approach has been to apply nonparametric estimation methods, following the works of Singh and Tracy (1977), Rose (1978), Carroll (1982), Carroll et al. (1986), Robinson (1983, 1987), Bunke (1986), Muller and Stadtmuller (1987). And more recently, see Hall and Carroll (1989), Hall and Marron (1990), Neumann (1994). In the context of estimating risk premium using the variance estimation, see Pagan and Ullah (1988), Pagan and Hong (1991), Antoniadis and Lavergne (1995). Most of the work in this field use the kernel method to estimate the unknown density function. The theoretical framework on the properties of density function estimator has been well explored in the literature, see Izenman (1991) for a comprehensive review.
While previous studies have shown that the efficiency of the estimator is little affected by the choice of the kernel, and the choice of the smoothing parameter (window width) plays a significant role, it is also shown that the convergence rate of Mean Square Errors can be quite different for some carefully selected high-order kernels. e.g. if k is an order s kernel, then the fastest asymptotic rate of MSE convergence of $\hat f$ to f is O(n^{-2s/(2s+1)}), which can be faster than the best rate for nonnegative kernels, O(n^{-4/5}) (Gasser, et al, 1985). However, this is achieved by paying extra costs -- in the form of larger sample size requirement and negative kernel values (Wand and Jones, 1995). The window width choice still remains a challenging problem.
Other estimators are also suggested in variance estimation literature, e.g. Hall et al (1990)'s difference-based estimator, Gasser et al (1986)'s local linear fitting estimator. Generally these estimators all require strong conditions to be consistent (e.g. no multiple measures at any data point) that its practical applications could be limited. Although orthogonal series has been used in the general density estimation literature, little work has been done to apply the orthogonal series to the estimation of the variance function. This motivates the present study.
The most popular orthogonal series estimators are the Hermite series and the Fourier series in the context of density estimation. However, the Hermite estimator, as well as the Fourier estimator, is not unbiased (Walter, 1994). Also Fourier series because of their periodicity are appropriate only when the density has compact support. For the influence of periodicity and the Gibbs phenomenon on the Fourier estimators, see Wahba, (1975a,b, 1981) and Hall (1981).
The purpose of the present study is to introduce a nonparametric estimator of the variance function based on wavelets. It will be shown that the wavelet-based estimator is asymptotically unbiased, consistent and the convergence rate of Integrated MSE could be as close as to $O(n^{-1})$. It also avoids the window width choice problem in the kernel approach. By the use of Fast Wavelet Transform, the computation could be potentially faster. The application of the estimator is illustrated in the following two scenarios:
$$\eqalignno{y_t &= {\bf x}_t'{\bf b} + e_t,\cr
{\rm Var}(e_t\vert{\bf z}) &= M({\bf z}_t) + \nu_t,\cr
E(e_t) &= 0,\cr
E(\nu_t) &= 0,\cr
{\bf x} &\in {\bf z}\quad {\rm information\ set,}\cr
M({\bf z}_t) &= \sum_{k\in {\bf z}}a_{j0,k}\thinspace\varphi_{j0,k}({\bf z}_t) +
\sum_{j\geq j0\atop k\in{\bf x}}\beta_{j,k}\thinspace\psi_{j,k}({\bf z})
&(1) }$$
$$\eqalignno{R_t &= \delta\sigma_t^2 + e_t,\cr
\sigma^2_t &= M({\bf z}_t) +\nu_t,\cr
E(e_t) &= 0,\cr
E(\nu_t) &= 0,\cr
R &\quad{\rm risk\ premimum\ of\ an\ asset},\cr
{\bf z} &\quad {\rm information\ set},\cr
M({\bf z}_t) &= \sum_{k\in {\bf z}}a_{j0,k}\thinspace\varphi_{j0,k}({\bf z}_t) +
\sum_{j\geq j0\atop k\in{\bf x}}\beta_{j,k}\thinspace\psi_{j,k}({\bf z})
&(2)}$$
The small sample properties of the new estimator will also be examined particularly in comparison with that of the popular ARCH model in Monte Carlo study.
Scheduled for Session 1.1 Computation And Econometrics - I