Inference in Heteroscedastic Nonlinear Time Series Under Long Memory With Applications to Finance

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PROJECT ABSTRACT:In physical sciences, economics, and finance many realizations of discrete time series exhibit long memory, i.e., their autocovariances as a function of lag decrease to zero at a hyperbolic rate as the lag approaches to infinity. Such processes have unbounded spectral densities at the origin. A part of this proposal is concerned with developing asymptotically optimal and robust estimators for heteroscedastic, non-smooth, non-linear time series models in the presence of regression or explanatory covariates that may have long memory, in a semi-parametric setting. In particular, it is planned to obtain the limits of the experiments generated by the non-smooth autoregressive models when there are long memory explanatory variables present in these models and when the error distributions are unknown. In the second part, the PI/Co-PI propose to develop asymptotically distribution free tests for fitting a parametric autoregressive mean and/or quantile function to a heteroscedastic stationary ergodic time series. These tests are expected to be functions of certain martingale transforms of a partial sum processes that do notinvolve nonparametric curve estimation. PI/Co-PI also plan to carry out a comparative study with some of the existing tests. The results obtained will be used to estimate parameters of interest and test theories relevant to problems in financial economics.A data set is said to have long memory if an association between distant observations is slowly decaying but persistent, as the distance between observations increases. A data set observed over a period of time is called a time series. A heteroscedastic time series is one where the conditional variability of an observation at the current time, given the past, depends on the past. Such data often arises in economics, finance, and physical sciences. In particular, an important example of long memory heteroscedastic time series is the volatility process in spot returns. It is also known that this volatility increases with bank interventions in currency markets. This intervention process is highly non-smooth time series since it is zero most of the times with certain bursts over some times. Part of the emphasis of the proposal is on developing optimal inferential procedures in a class of non-smooth non-linear heteroscedastic time series models. Another part emphasizes application of the results obtained to develop new tests of market efficiency and estimates of time dependent risk premium in financial economics and high frequency data mentioned in the proposal pertaining to German Mark and Swiss Frank vs. US Dollar exchange rates and commodity prices.