12.3 Hypothesis Testing

Suppose $ (X,Y)$ is defined as in (12.6) and let $ m(x) = \textrm{E}(Y\vert X=x)$ be the conditional mean function, $ f$ be the density of the design points $ X$, and $ \sigma^2(x) = \textrm{Var}(Y\vert X=x)$ be the conditional variance function of $ Y$ given $ X=x\in S$, a closed interval $ S \subset \mathbb{R}$. Suppose that $ \{m_{\theta} \vert \theta \in \Theta\}$ is a parametric model for the mean function $ m$ and that $ \hat{\theta}$ is an estimator of $ \theta $ under this parametric model. The interest is to test the null hypothesis:

$\displaystyle H_0: m(x) = m_{\theta}(x) \quad \hbox{for all} \, x \in S$

against a series of local smooth nonparametric alternatives:

$\displaystyle H_1: m(x) = m_{\theta}(x) + c_n \Delta_n(x),$

where $ c_n$ is a non-random sequence tending to zero as $ n\to \infty$ and $ \Delta_n(x)$ is a sequence of bounded functions.

The problem of testing against a nonparametric alternative is not new for an independent and identically distributed setting, Härdle and Mammen (1993) and Hart (1997). In a time series context the testing procedure has only been considered by Kreiß et al. (1998) as far as we are aware. Also theoretical results on kernel estimators for time series appeared only very recently, Bosq (1998). This is surprising given the interests in time series for financial engineering.

We require a few assumptions to establish the results in this chapter. These assumptions are the following:

(i)
The kernel $ K$ is Lipschitz continuous in $ [-1,1]$, that is $ \vert K(t_1) - K(t_2)\vert \le C \vert\vert t_1 - t_2\vert\vert$ where $ \vert\vert \cdot\vert\vert$ is the Euclidean norm, and $ h = {\mathcal{O}}\{n^{-1/5}\}$;

(ii)
$ f$, $ m$ and $ \sigma^2$ have continuous derivatives up to the second order in $ S$.

(iii)
$ \hat{\theta}$ is a parametric estimator of $ \theta $ within the family of the parametric model, and

$\displaystyle \sup_{x \in S} \vert m_{\hat{\theta}}(x) - m_{\theta}(x) \vert = {\mathcal{O}}_p(n^{-1/2}).$

(iv)
$ \Delta_n(x)$, the local shift in the alternative $ H_1$, is uniformly bounded with respect to $ x$ and $ n$, and $ c_n =n^{-1/2} h^{-1/4}$ which is the order of the difference between $ H_0$ and $ H_1$.

(v)
The process $ \{(X_i,Y_i)\}$ is strictly stationary and $ \alpha$-mixing, i.e.

$\displaystyle \alpha(k) \stackrel{\mathrm{def}}{=}\sup_{ {\cal A} \in {\cal
F}_...
...nfty} } \vert \textrm{P}(A B) -
\textrm{P}(A) \textrm{P}(B) \vert \le a \rho^k $

for some $ a>0$ and $ \rho\in [0,1)$. Here $ {\cal F}_{k}^l$ denotes the $ \sigma$-algebra of events generated by $ \{(X_i,Y_i), k\le i\le l\}$ for $ l \ge k$. For an introduction into $ \alpha$-mixing processes, see Bosq (1998) or Billingsley (1999). As shown by Genon-Catalot et al. (2000) this assumption is fulfilled if $ Z_t$ is an $ \alpha$-mixing process.

(vi)
$ \textrm{E}\{\exp(a_0 \vert Y_1-m(X_1)\vert)\} < \infty$ for some $ a_0 > 0$; The conditional density of $ X$ given $ Y$ and the joint conditional density of $ (X_1, X_l)$ given $ (Y_1,Y_l)$ are bounded for all $ l>1$.

Assumptions (i) and (ii) are standard in nonparametric curve estimation and are satisfied for example for bandwidths selected by cross validation, whereas (iii) and (iv) are common in nonparametric Goodness-of-Fit tests. Assumption (v) means the data are weakly dependent. It is satisfied for a wide class of diffusion processes.