Limit Theorems for Nonlinear Cointegrating Regression

The following sections are included:

• Preface
• Contents

Consider a nonlinear regression model:

where f : R → R is an unknown function, x_t and u_t are regressors and regression errors, respectively. Let K(x) be a kernel function and h ≡ h_n → 0 be a bandwidth. The conventional kernel estimator of f(x) is given by …

Consider an array {X_nk}_k≥1,n≥1 constructed from some underlying nonstationary time series and assume that there is a continuous limiting process X = {X_t}_t≥0 to which X_n,[nt] converges weakly. Typically, where X_k is a nonstationary time series such as an I(1) process and d_n is an appropriate standardization factor. In many applications, we encounter a functional of the form

, where g is a real integrable function on R and c_n is a certain sequence of positive constants satisfying that c_n → ∞ and n/c_n → ∞. For instance, such functionals arise in nonlinear regression with integrated time series and nonparametric estimation in relation to nonlinear cointegrationmodels. See Chapter 4 for more details. In such cases, g may be a kernel function K or a squared kernel function K², and the sequence c_n depends on the bandwidth used in the nonparametric regression…

Let (S_n,Z_n)_n≥1 be a sequence of random vectors. It is an important issue to consider the joint convergence:

In many situations, we may prove S_n →_D V N, where V > 0 a.s. is a random variable independent of N ~ N (0, 1), that is, S_n converges to a mixture of normal distributions. This is a typical limit theorem in the framework of semimartingales. Usually, the distribution of V is unknown, making the limit theorem useless for statistical purpose. However, if we can prove (3.1) with Z = V , the continuous mapping theorem show that

having a big advantage in statistical applications. This chapter investigates the convergence to a mixture of normal distributions that enables (3.2).

Consider a random array {x_nk, y_nk}k≥1,n≥1 constructed from some underlying nonstationary time series and assume that there is a vector process {X(t), Y (t), 0 ≤ t ≤ 1}, to which {x_n,[nt], y_n,[nt]} converges weakly in Skorokhod topology on D_R² [0, 1], where [a] denotes the integer part of a. A common functional of interest S_n of {x_nk, y_nk} is defined by the sample quantity

, where ∊_nk = y_nk − y_n,k−1 and f is a real function on R. This chapter investigates the sufficient conditions in such a way that the distribution of S_n converges to a stochastic integral, namely, .

Except explicitly mentioned, concept of stochastic processes is defined as in Appendix A. In particular, a (local, semi-, etc.) martingale is always assumed to be cádlág, i.e., its paths are right continuous and admit left-hand limits.

A typical nonlinear cointegrating regression model has the form:

where u_t is a zero mean equilibrium error, x_t is a nonstationary regressor and f(·) is an unknown real function on R. In the presence of more explicit prior information, the regression function f(·) may be specified in parametric form as where g(·, θ) represents a parametric family of functions with unknown true parametric value θ₀ ∈ Θ, where Θ ⊂ R^m for some m ≥ 1 is a compact set

Using the limit theorems developed in previous chapters, this chapter investigates estimation and inference theory for both nonparametric model (5.1) and parametric model (5.2).

The following sections are included:

• Appendix A Concepts of stochastic processes
• Appendix B Metric space
• Appendix C Convergence of probability measure

The following sections are included:

• Bibliography
• Index