Detection and Estimation of Change Point in Volatility Function of Foreign Exchange Rate Returns

Abstract

This work aims at detection and estimation of a change point in conditional variance function of a Nonparametric Auto-Regressive Conditional Heteroscedastic model. The conditional mean and conditional variance functions are not specified a priori but estimated using Nadaraya Watson kernel. This is because inferences based on nonparametric approaches are robust against misspecification of the conditional mean function and the conditional variance function of returns. The squared residuals obtained after estimating the regression function of the returns are used in estimating the conditional variance function. Further, the squared residuals are used in developing a test statistic for unknown abrupt change point in volatility of the exchange rate returns. The test statistic takes into consideration the conditional heteroskedasticity of the disturbances, dependence of the returns, heterogeneity and fourth moment of returns. This does not require prior knowledge of the marginal or the conditional densities of the returns as opposed to maximum likelihood estimation methods. The estimator for change point is considered as the augmented maximum of the test statistic. The consistency of the estimator is stated as a theorem. The asymptotic distribution associated with the test for unknown break points is the Bessel process distribution. The Bessel process distributions have no known simple closed-form expression for the distribution function which makes it difficult to compute exact p-values. Also, the Bessel process distributions depend on two parameters which makes it hard to tabulate the critical values hence one needs to simulate them. After simulating the critical values, hypothesis testing is done in the presence and absence of a change point in volatility of a simulated time series and the test is shown to reject the null hypothesis in the presence of a change point at alpha level of significance. Further, the test fails to reject the null hypothesis in the absence of a change point at alpha level of significance. An application to United States Dollar/Kenya Shilling historical exchange rates returns is made from 1 st January 2010 to 27th November 2020 where the sample size n = 2839 is done. Through binary segmentation method, three change points are detected, estimated and accounted for. A significant improvement in describing a time series is expected if a point in time for volatility change has been detected and estimated.