Abstract
This dissertation aims to deal with three issues on testing in econometrics: Inference under heteroskedasticity of unknown form for nonlinear and linear models respectively, and usefulness tests of forecasts. The first two discuss a common problem in empirical econometric studies. That is the presence of heteroskedasticity in the error terms - the variance in many cases varies with the regressors, which violates the famed classical assumptions. It is well known that, while heteroskedasticity in linear regression models does not affect the consistency of the underlying estimator, the finite-sample properties and efficiency are not satisfying. We thus try to develop procedures to make some improvements. As for the issue on forecasting, we make a complement to the literature by extending the usefulness measure of forecasts proposed in Ashley (1983) to be a testable hypothesis. First, previous studies concentrate on improving the finite-sample properties of test in linear models under heteroskedasticity, however, nonlinear specifications are also popular in econometrics. Considering both consistency and efficiency for inference on nonlinear models under heteroskedasticity, one of the most commonly used methods is using an efficient estimation, i.e., GMM, and replacing the covariance matrix with a diagnal matrix using squared residuals just like White (1980). In this aspect, our focus is placed on the finite-sample problems caused by the heteroskedasticity in nonlinear models. Specifically, within the framework of nonlinear models, we propose a straightforward approach by extending the applicability of HC2--HC5 for GMM. The small sample refinements of the estimation and hypothesis testing, using GMM with the modified version of HCCME (heteroskedasticity consistent covariance matrix estimator), are demonstrated via our Monte Carlo experiments. In addition, we also provide Bayesian interpretations for the modified versions of HCCME under the setting of nonlinear model, which is an extension of Poirier's (2010) Bayesian explorations for HCCME from a linear model without instrument variables to the GMM framework allowing nonlinear specification and instrument variables to be considered. Second, even though we can still make asymptotically valid inference using the ordinay least squares estimator (OLS) with HCCME, it is asymptotically less efficient than some alternative estimators such as feasible generalized least squares estimator (FGLS) or Cragg (1983) estimator. Flachaire (2005) studies the efficient tests robust to heteroskedasticity which is based on the combination of Cragg estimator and wild bootstrap test proposed by Davidson and Flachaire (2001). Motivated by the good finite sample performance from simulation experiments in Flachaire (2005), we would like to explore an alternative class of testing procedures combining the semi-parametric estimators and wild bootstrap test. In this aspect, we study the small sample properties via comparing the size distortions and testing powers of the test statistic. Our simulation experiments provide evidences that the proposed testing procedure may have potential to complement the existing efficient tests. Regarding the issue of forecasting, Ashley (1983) proposes a criterion (known as Ashley's index) to judge whether the external macroeconomic variables are well forecasted to serve as explanatory variables in forecasting models, which is crucial for policy makers. In this study, we try to extend Ashley's work by providing three testing procedures, including a ratio-based test, a difference-based test, and the Bayesian approach. The Bayesian approach has the advantage of allowing the flexibility of adapting all possible information content within a decision-making environment such as the change of variable's definition due to the evolving system of national accounts. We demonstrate the proposed methods by applying six macroeconomic forecasts in the Survey of Professional Forecasters. Researchers or practitioners can thus formally test whether the external information is helpful.