Essays on Forecasting and Volatility Modelling

Abstract

This thesis contributes to four distinct fields on the econometrics literature: forecasting macroeconomic variables using large datasets, volatility modelling, risk premium estimation and iterative estimators. As a research output, this thesis presents a balance of applied econometrics and econometric theory, with the latter one covering the asymptotic theory of iterative estimators under different models and mapping specifications. In Chapter 1 we introduce and motivate the estimation tools for large datasets, the volatility modelling and the use of iterative estimators. In Chapter 2, we address the issue of forecasting macroeconomic variables using medium and large datasets, by adopting vector autoregressive moving average (VARMA) models. We overcome the estimation issue that arises with this class of models by implementing the iterative ordinary least squares (IOLS) estimator. We establish the consistency and asymptotic distribution considering the ARMA(1,1) and we argue these results can be extended to the multivariate case. Monte Carlo results show that IOLS is consistent and feasible for large systems, and outperforms the maximum likelihood (MLE) estimator when sample size is small. Our empirical application shows that VARMA models outperform the AR(1) (autoregressive of order one model) and vector autoregressive (VAR) models, considering different model dimensions. Chapter 3 proposes a new robust estimator for GARCH-type models: the nonlinear iterative least squares (NL-ILS). This estimator is especially useful on specifications where errors have some degree of dependence over time or when the conditional variance is misspecified. We illustrate the NL-ILS estimator by providing algorithms that consider the GARCH(1,1), weak-GARCH(1,1), GARCH(1,1)-in-mean and RealGARCH(1,1)-in-mean models. I establish the consistency and asymptotic distribution of the NLILS estimator, in the case of the GARCH(1,1) model under assumptions that are compatible with the quasi-maximum likelihood (QMLE) estimator. The consistency result is extended to the weak-GARCH(1,1) model and a further extension of the asymptotic results to the GARCH(1,1)-inmean case is also discussed. A Monte Carlo study provides evidences that the NL-ILS estimator is consistent and outperforms the MLE benchmark in a variety of specifications. Moreover, when the conditional variance is misspecified, the MLE estimator delivers biased estimates of the parameters in the mean equation, whereas the NL-ILS estimator does not. The empirical application investigates the risk premium on the CRSP, S&P500 and S&P100 indices. I document the risk premium parameter to be significant only for the CRSP index when using the robust NL-ILS estimator. We argue that this comes from the wider composition of the CRPS index, resembling the market more accurately, when compared to the S&P500 and S&P100 indices. This nding holds on daily, weekly and monthly frequencies and it is corroborated by a series of robustness checks. Chapter 4 assesses the evolution of the risk premium parameter over time. To this purpose, we introduce a new class of volatility-in-mean model, the time-varying GARCH-in-mean (TVGARCH-in-mean) model, that allows the risk premium parameter to evolve stochastically as a random walk process. We show that the kernel based NL-ILS estimator successfully estimates the time-varying risk premium parameter, presenting a good finite sample performance. Regarding the empirical study, we find evidences that the risk premium parameter is time-varying, oscillating over negative and positive values. Chapter 5 concludes pointing the relevance of of the use of iterative estimators rather than the standard MLE framework, as well as the contributions to the applied econometrics, financial econometrics and econometric theory literatures.