14.2 Construction of the Estimator

The LP method introduced in the previous section will now be applied under the assumption of a nonparametric autoregressive model of the form (13.1) to estimate the volatility function $ s(x)$ of the process $ (Y_i)$ based on the observations $ Y_0, \dots, Y_n$.

The conditional volatility $ s_i(x)$ and the conditional variance $ v_i(x)$ respectively at time $ i$ is defined by

$\displaystyle v_{i}(x) = s_i^2(x) = {\mathop{\text{\rm\sf E}}}[Y_{i}^2 \,\vert\, Y_{i-1} = x] - {\mathop{\text{\rm\sf E}}}^2 [Y_{i}\,\vert\,Y_{i-1}=x].$ (14.8)

Included in the assumptions of the model (13.1) is the independence from the time index $ i$. An estimate for $ v(x) = s^2(x)$ using the LP Method is based on the fact that the two dimensional marginal distribution $ (Y_{i-1}, Y_i)$ is independent of $ i$. In the following we will see that $ (Y_i)$ approach is a stationary process, with which the following application is justified.

Referring back to the representation (13.8) of the conditional variance $ v(x)$ we search for an estimator $ \hat{v}_{n}$ for $ v$ with the form

$\displaystyle \hat{v}_{n}(x) = \hat{g}_{n}(x) - \hat{f}_{n}^2(x),$ (14.9)

i.e., we are looking for an estimator $ \hat{g}_n(x)$ for $ g(x)=f^2(x)+s^2(x)$ and an estimator $ \hat{f}_n(x)$ for $ f(x)$.

In order to define these two estimators with the LP Method, after applying the steps discussed in the previous section we have to solve both of the following minimization problems:

\begin{displaymath}\begin{array}{l} \bar{c}_{n}(x) = \arg \min_{c \in \mathbb{R}...
...U_{in})^2 K \left( \frac{Y_{i-1}-x}{h_{n}} \right). \end{array}\end{displaymath} (14.10)

Here $ K: \mathbb{R}\longrightarrow \mathbb{R}$ is a kernel and $ \{h_{n}\}$ a series of positive numbers (bandwidth) with $ \lim_{n\rightarrow\infty} h_n = 0$. The vectors $ U_{in}$ from (13.10) are defined by

$\displaystyle U_{in} = F(u_{in}), \quad u_{in} = \frac{Y_{i-1}-x}{h_{n}}.$ (14.11)

with $ \mathbb{R}^{p+1}$ valued function $ F(u) = \{F_0(u), ...,
F_p(u)\}^\top $ given by

$\displaystyle F_k(u) = \frac{u^k}{k!}.$

According to the LP Method we define $ \hat{g}_n$ and $ \hat{f}_n$ with

$\displaystyle \hat{g}_n(x) = \bar{c}_{n}(x)^\top F(0)$   and$\displaystyle \quad
\hat{f}_n(x) = c_{n}(x)^\top F(0) , $

which the above mentioned application ensures that

$\displaystyle \hat{v}_{n}(x) = \bar{c}_{n}(x)^\top F(0) - \big\{ c_{n}(x)^\top F(0) \big\}^2.$ (14.12)

This estimate is a direct modification of the estimator from the local polynomial, nonparametric regression in Tsybakov (1986).

Fig.: DEM/ USD Exchange rate and its returns 23422 SFEdmusrate.xpl
\includegraphics[width=1\defpicwidth]{dmusrate1.ps}

To illustrate the estimator we consider an example. Figure 13.1 above displays the DEM/USD exchange rate from October 1, 1992 to September 30, 1993 in 20 minute intervals (volatility time scale). There are $ n = 25434$ observations. We have calculated the returns of this series (see Figure 13.1 below) and applied the estimator (13.12) to the time series of the returns. Under the model for the geometric Brownian motion for the price, which is based on the Black-Scholes method (see Section 6.1), the returns must follow an ordinary Brownian motion. Their volatilities $ v(x)$ are thus constant and independent of $ x$. The estimated conditional variance functions (see Figure 13.2) show a U shaped structure, which is called a ``smiling face'' structure or smile. The estimated volatility functions $ \hat{s}(x) = \sqrt{\hat{v}(x)}$ appear to be qualitatively analogous. This means that the expected risk of the returns is significantly higher when extreme values were observed the period before.

Fig.: The estimated conditional variance function $ \hat{v}(x)$ of the DEM/ USD returns 23428 SFEdmusvol.xpl
\includegraphics[width=1\defpicwidth]{dmusvol1.ps}

As an alternative to equation (13.9) it is also possible to first determine the sample residuals

$\displaystyle \hat{Z}_i = Y_i - \hat{f}_{n}(Y_{i-1}) , \quad i=1,...,n,.$

They approximate the true residuals $ Z_i = Y_i - f(Y_{i-1})$, which under the assumptions of model (13.1) satisfy

$\displaystyle E[Z_i \vert Y_{i-1} = x] = 0 , \quad E[Z_i^2 \vert Y_{i-1} = x] = v(x). $

The volatility can be estimated as in the previous section directly from the nonparametric regression of $ \hat{Z}_i^2$ on $ Y_{i-1}$. Fan and Yao (1998) have shown that this process has advantages in heteroscedastic regression models. In estimating $ f(x)$ and $ v(x) = s^2(x)$, various bandwidths may be used that do not encounter the danger of the variance estimator taking on negative values. This makes sense when it is expected that the local fluctuations of $ f$ and $ s$ are of very different proportions.