Essays on infinite-variance stable errors and robust estimation procedures

Abstract

Gaussian normal error assumption and OLS-based regression estimations form the basis of co-integration testing techniques. However, many studies have found evidence that financial and some important economic time series data such as exchange rate returns and inflation rates are subject to high variability. In particular, their innovations exhibit the features of skewness, excessive peakness around the mean and heavier tails than those of the Gaussian normal distribution. Stable distributions (which are used to model high variability in data and infinite-variance processes) provide more realistic distributional assumptions than the Gaussian distribution for heavy-tailed financial and economic time series. Least Absolute Deviation (LAD) based estimators often yield robust results for heavy-tailed data compared to least squares based estimators. In this dissertation, as a natural extension of the extant studies, we consider a new robust residual-based co-integration test under the assumption of infinite-variance errors which are in the domain of attraction of a stable law. We implement the least absolute deviation (LAD) procedure in our regression estimations.In part I, the new co-integration tests are proposed. The test is parametric: the critical values of the test statistic depend on the stability index of the stable distribution from which the errors are driven. The unit root test statistic we consider under the null of no co-integration is taken from Samarakoon and Knight (2009, Econometric Reviews, 28, 314--334). We find the critical values of these new co-integration tests through Monte Carlo simulations and observe that the null convergence of the test statistic is faster for lighter tails. Size and power comparisons are included to evaluate the performance of the new residual-based tests relative to conventional OLS-based ones which are due to Caner (1998, J. of Econometrics, 86, 155--175). We observe that the LAD-based tests have power advantages over the OLS-based tests as the sample size gets larger and the tails get heavier with infinite-variance error assumption, yet there are more size distortions associated with LAD-based tests especially for small sample sizes compared to OLS-based ones.The new tests are employed to test for forward rate unbiasedness hypothesis (FRUH) with daily frequency data for a sample of eight currencies (Australian dollar, Canadian dollar, French franc, German mark, Italian lira, Japanese yen, Swiss franc and U.K. pound) against the U.S. dollar for 1-month, 3-month, 6-month and 1-year forward contracts. We also run fully-modified ordinary least squares (FM-OLS) and fully-modified least absolute deviation (FM-LAD) estimators on the co-integrating regressions to test the coefficient restrictions which are implied by the FRUH. We observe that tests involving longer maturity forward contracts (6-month and 1-year) and LAD-based co-integration tests mostly provide the evidence that is inconsistent with FRUH.In part II, weak and strong-form purchasing power parity (PPP) relations are re-examined by using LAD-based procedures under infinite-variance error assumption. LAD-based co-integration tests that are proposed in part I are used to test for weak-form PPP. FM-OLS and FM-LAD procedures are used to test for the strong-form PPP hypothesis. The results from LAD-based estimations are compared to their OLS-based counterparts. Monthly exchange rate (per U.S. dollar) and PPI data for a sample of eight countries (Austria, Canada, Denmark, Germany, Japan, Netherlands, Sweden and the U.K.) from 1973:1 to 2009:12 are used for estimation purposes. Neither weak-form nor strong-form PPP relations can be justified empirically regardless of the estimation procedure. Results from the new residual-based co-integration tests give slightly more support of the weak-form PPP.