Narrow Results

The main purpose of this work is to contribute to the study of set-valued random variables by providing a kind of Wold decomposition theorem for interval-valued processes. As the set of set-valued random variables is not a vector space, the Wold decomposition theorem as established in 1938 by Herman Wold is not applicable for them. So, a notion of pseudovector space is introduced and used to establish a generalization of the Wold decomposition theorem that works for interval-valued covariance stationary time series processes. Before this, set-valued autoregressive moving-average (S-ARMA) time series process is defined by taking into account an arithmetical difference between random sets and random real variables.

Many time series models have been used extensively in modeling economic and financial data. However, it is difficult to determine the functional forms of the conditional mean and conditional variance in these models. In this paper, a test statistic based on the squared conditional residuals is proposed for testing these functional forms, and the asymptotic distribution of the test statistic is obtained. The test statistic is applicable not only to the family of GARCH models but also to other nonlinear time series models. Simulation results show that the proposed tests are powerful and have reasonable sizes. Two real examples are also given to illustrate our theory.

Conditional probability distribution models have been widely used in economics and finance. In this chapter, we introduce two closely related popular methods to estimate conditional distribution models—Maximum Likelihood Estimation (MLE) and Quasi-MLE (QMLE). MLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is correctly specified, and QMLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is misspecified. Because the score function is an MDS and the dynamic Information Matrix (IM) equality holds when a conditional distribution model is correctly specified, the asymptotic properties of MLE is analogous to those of the OLS estimator when the regression disturbance is an MDS with conditional homoskedasticity, and we can use the Wald test, LM test and Likelihood Ratio (LR) test for hypothesis testing, where the LR test is analogous to the J · F test statistic. On the other hand, when the conditional distribution model is misspecified, the score function has mean zero, but it may no longer be an MDS and the dynamic IM equality may fail. As a result, the asymptotic properties of QMLE are analogous to those of the OLS estimator when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. Robust Wald tests and LM tests can be constructed for hypothesis testing, but the LR test can no longer be used, for a reason similar to the failure of the F-test statistic when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. We discuss methods to test the MDS property of the score function, and the dynamic IM equality, and correct specification of a conditional distribution model.