Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y)$ values in the product space $\R^d\bigotimes \R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation.
Deep Dive into Strongly consistent nonparametric forecasting and regression for stationary ergodic sequences.
Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y)$ values in the product space $\R^d\bigotimes \R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation.
Nonparametric regression has been applied to a variety of contexts, in particular to time series modeling and prediction. The present study contributes to the methodology by showing how a regression function can be consistently inferred from time series data under no process assumptions beyond stationarity and ergodicity. (A Lipschitz condition on the regression function itself will be imposed.)
Toward showing how our methodology can impinge on an established research area, we give one substantive application to a practical problem in stochastic finance: Many works, such as the Chapter entitled “Some Recent Developments in Investment Research” of the prominent text [5], argue for the need to move beyond the Black-Scholes stochastic differential equation. This and other studies suggest the so-called ARCH and GARCH extensions as a promising direction. The review of this approach by Bollerslev et al. [6] cites a litany of unresolved issues. Of particular relevance is the discussion of the need to account for persistency of the variance (Sections 2.6 and 3.6). (ARCH and GARCH models can be long-range dependent for certain ranges of parameters. In these cases, statistical analysis is delicate [8].)
The basic idea behind the ARCH/GARCH setup is that one must allow the asset volatility (variance) to change dynamically, and perhaps (GARCH) to depend on current and past volatility values. The review [6] documents (p. 30) that several authors have applied nonparametric and semiparametric regression, with some success, to infer the ARCH functions from data. These methods can fail if fairly stringent mixing conditions are not in force. Masry and Tjostheim [21], because of their rigorous consideration of consistency, sets the stage for appreciating the potential of the present investigation.
They propose that both the asset dynamics and volatility of a nonlinear ARCH series be inferred from nonparametric classes of regression functions. By imposing some fairly severe assumptions, which would be tricky to validate from data, these authors are able to assure that the ARCH process is strongly mixing (with exponentially decreasing parameter) and consequently standard kernel techniques are applicable.
On another avenue toward asset series modelling, decades ago, Mandelbrot suggested that fractal processes should be considered in this context. Fractals have been of interest to theorists and modellers alike in part because they can display persistency. In his 1999 study, “A Multifractal Walk down Wall Street,” [20] Mandelbrot argues that conventional models for portfolio theory ignore soaring volatility, and that is akin to a mariner ignoring the possibility of a typhoon on the basis of the observation that weather is moderate 95% of the time.
Such persistence as exhibited in the models of finance calls into question whether various processes of interest are actually strongly mixing, a consistency requirement for conventional nonparametric regression techniques. We mention parenthetically that telecommunications modelers are increasingly turning toward long-range-dependent processes (e.g., [28] and [37])
As mentioned, the primary contribution of the present paper is an algorithm which is demonstratably consistent without imposition of mixing assumptions. The implication is that process assumptions such as in [21] are not required for our algorithm. The price paid for this flexibility is that convergence rates and asymptotic normality cannot be assured. This avenue is worthy of exploration, nevertheless, because the limits of process inference are clarified, and as a practical matter, future work might lead to methods which are reasonably efficient if the process does satisfy mixing assumptions, but simultaneously assures convergence when mixing fails.
The algorithm is of the series-expansion type. The foundational idea (after Kieffer [17]) is that sometimes it is possible to bound the error of ignoring the series tail, and additionally assure that the leading coefficients are consistently estimated. Specific constructs are given for a partition-type estimator (Section 2) and for a kernel series (Section 3).
We close this introduction with a survey of the literature of nonparametric estimation for stationary series without mixing hypotheses.
Let Y be a real-valued random variable and let X be a d-dimensional random vector (i.e., the observation or co-variate). We do not assume anything about the distribution of X. As is customary in regression and forecasting, the main aim of the analysis here is to minimize the mean-squared error :
over some space of real-valued functions f (•) defined on the range of X. This minimum is achieved by the regression function m(x), which is defined to be the conditional distribution of Y given X:
assuming the expectation is well-defined, i.e., if E|Y | < ∞. For each measurable function
where µ stands for the distribution of the observation X. The second term on the right hand side is c
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