| This dissertation uses three essays to introduce and apply the wavelet analysis to a number of important issues in time series econometrics and spectral analysis. Wavelet analysis is a newly developed mathematical tool generalizing traditional Fourier analysis. Wavelets are more effective and powerful than such methods as kernels in capturing nonsmooth features, for example, peaks and spikes in spectral density functions, which can arise due to strong serial dependence, business cycles and seasonalities in economic and financial time series.; In the first essay, I apply the wavelet method to estimate heteroskedasticity and autocorrelation consistent covariance matrices (HAC). HAC is an important issue in time series econometrics, particularly in the context of least squares regression, maximum likelihood estimation, generalized method of moments, and the unit root model. Economic and financial data often display strong serial autocorrelation, which results in a sharp spectral peak at zero frequency. As a local average method, kernels always tend to underestimate the spectral peak, and thus lead to overly narrow confidence interval estimates and liberal tests. Using wavelets as a particularly effective tool in capturing nonsmooth features such as peaks, a class of wavelet-based HAC estimators is proposed. A simulation study shows that wavelet estimators outperform their kernel counterparts when strong serial correlation is present.; In the second essay, I propose a consistent test for unknown forms of serial correlation applying wavelet methods. While detection and inference of serial correlation have been of interest in the time series context for decades, the existing methods, like kernels, perform poorly against the alternatives when nonsmooth spectral densities exist. In such situations, wavelets are particularly suitable for use as a test procedure with good power against the alternatives. Asymptotic null distributions and the consistency for the wavelet-based test are derived. Since the spectral densities of time series that arise in practice usually have unknown smoothnesses, the wavelet-based test is a useful complement to the kernel-based test.; In the final essay, a one-sided test using wavelet methods to test for autoregressive conditional heteroskedasticity (ARCH) is proposed. There has been an increasing interest in hypothesis testing when there are inequality parameter restrictions. ARCH is an important example of this, with parameters of interest equal to zero under the null of no ARCH and are nonnegative under the alternative of the existence of ARCH. Economic and financial data exhibit a spectral peak at frequency zero when there is a persistent ARCH or when the ARCH effect carries over a long distributional lag. A wavelet-based test for ARCH, by a simulation study, is shown to be more powerful than the existing one-sided and two-sided ARCH tests when spectral peaks are present. |
