14. Non-parametric Concepts for Financial Time Series

With the analysis of (financial) time series one of the most important goals is to produce forecasts. Using past observed data one would like to make some statements about the future mean, the future volatility, etc., i.e., one would like to estimate the expectation and variance of the underlying process conditional on the past. One method to produce such estimates will be introduced in this chapter.

Let $ (Y_t), \, t=0,1,2,\dots,$ be a time series. We consider a nonlinear autoregressive heteroscedastic model of the form

$\displaystyle Y_{t} = f(Y_{t-1}) + s(Y_{t-1}) \xi_{t}, \quad t = 1, 2, \ldots$ (14.1)

Here the innovations $ \xi_{t}$ are i.i.d. random variables with $ {\mathop{\text{\rm\sf E}}}[\xi_{t}]=0$ and $ {\mathop{\text{\rm\sf E}}}[\xi_{t}^2] = 1$, $ f: \mathbb{R}
\longrightarrow \mathbb{R}$ and $ s: \mathbb{R}\longrightarrow (0, \infty)$ are unknown functions, and $ Y_{0}$ is independent of $ (\xi_{t})$. Under these assumptions and according to Theorem 3.1 it holds that

$\displaystyle {\mathop{\text{\rm\sf E}}}[Y_{t}\,\vert\,Y_{t-1}=x] = f(x) + {\ma...
..._{t}\,\vert\,Y_{t-1}=x] = f(x) + s(x) {\mathop{\text{\rm\sf E}}}[\xi_t] = f(x),$    

where in the second to last equation the independence of $ \xi_t$ and $ Y_{t-1}$ is used. A similar calculation gives $ s^2 (x) =
\mathop{\text{\rm Var}}[Y_{t}\,\vert\,Y_{t-1}=x]$. The unknown functions $ f$ and $ s$ describe the conditional mean and the conditional volatility of the process, which we want to estimate.

With the specific choice $ f(x) = \alpha x$ and $ s =\sigma
> 0$ the process $ Y_t$ is an AR(1) process. Every strong ARCH(1) process $ (Y_i)$ satisfies the model (13.1). In this case $ f = 0$, and it holds that $ s(x) = \sqrt{\omega + \alpha
x^2}$ with the parameters $ \omega>0$ and $ \alpha \ge 0$, compare Definition 12.1. With respect to the structure of the conditional mean and the conditional variance, the model above is another broad generalization of the (strong) ARCH models. The advantage of this nonparametric Ansatz is that the model contains no structural assumptions about the functions $ f$ and $ s$, since such assumptions are often not supported by observations in the data.

Closely related to our model is the Qualitative Threshold ARCH model (QTARCH) studied in  Gouriéroux and Monfort (1992), which for the case of one lag (QTARCH(1)) is a special case of  (13.1), where the unknown functions $ f: \mathbb{R}
\longrightarrow \mathbb{R}$ and $ s: \mathbb{R}\longrightarrow (0, \infty)$ take the form of step functions - see (12.31). On the other hand (13.1) can also be described under certain regularity assumptions on $ f$ and $ s$ as a limit model of the QTARCH(1) models when $ J\rightarrow\infty$, in that $ f$ and $ s$ are approximated with elementary functions.

The work of Gouriéroux and Monfort is the first to consider the conditional mean and the conditional variance together in a nonparametric model. The applications of this idea introduced here are taken from Härdle and Tsybakov (1997), and are also considered independently in Franke et al. (2002). In the following we will construct a class of estimators based on the local polynomial regression for the conditional volatility $ v(x) = s^2(x)$ and the conditional mean $ f(x)$ of the time series $ (Y_i)$ under the model assumptions (13.1).

In addition to the model assumptions (13.1) certain regularity assumptions, although no structural assumptions, on $ f$ and $ s$ will be made. As the main result of this chapter we will show that this combined estimation of the conditional expectation and the conditional volatility is asymptotically normally distributed.