"Frontmatter". In: Analysis of Financial Time Series

138 NONLINEAR TIME SERIESperiod and an expansion period are approximately 3.69 and 11.31 quarters. Thus, onaverage, a contraction in the U.S. economy lasts about a year, whereas an expansioncan last for 3 years. Finally, the estimated AR coefficients of x t−2 differ substantiallybetween the two states, indicating that the dynamics of the U.S. economy aredifferent between expansion and contraction periods.4.1.5 Nonparametric MethodsIn some financial applications, we may not have sufficient knowledge to pre-specifythe nonlinear structure between two variables Y and X. In other applications, wemay wish to take advantage of the advances in computing facilities and computationalmethods to explore the functional relationship between Y and X. These considerationslead to the use of nonparametric methods and techniques. Nonparametricmethods, however, are not without any cost. They are highly data-dependent and caneasily result in overfitting. Our goal here is to introduce some nonparametric methodsfor financial applications and some nonlinear models that make use of nonparametricmethods and techniques. The nonparametric methods discussed include kernelregression, local least squares estimation, and neural network.The essence of nonparametric methods is smoothing. Consider two financial variablesY and X, which are related byY t = m(X t ) + a t , (4.17)where m(.) is an arbitrary, smooth, but unknown function and {a t } is a white noisesequence. We wish to estimate the nonlinear function m(.) from the data. For simplicity,consider the problem of estimating m(.) at a particular date for which X = x.That is, we are interested in estimating m(x). Suppose that at X = x we haverepeated independent observations y 1 ,...,y T . Then the data becomey t = m(x) + a t ,t = 1,...,T.Taking the average of the data, we have∑ Tt=1y tT∑ Tt=1a t= m(x) + .TBy the Law of Large Number, the average of the shocks converges to zero as Tincreases. Therefore, the average ȳ = ∑ Tt=1 y t /T is a consistent estimate of m(x).That the average ȳ provides a consistent estimate of m(x) or, alternatively, that theaverage of shocks converges to zero shows the power of smoothing.In financial time series, we do not have repeated observations available at X = x.What we observed are {(y t , x t )} for t = 1,...,T . But if the function m(.) is sufficientlysmooth, then the value of Y t for which X t ≈ x continues to provide accurateapproximation of m(x). The value of Y t for which X t is far away from x providesless accurate approximation for m(x). As a compromise, one can use a weighted