Question

A) Why is the conditional variance a good measure of uncertainty?

B) Outline the GARCH model and GARCH-M model.

C) Outline one of the extensions to the basic GARCH family of models

Answer 1

A)  Statisticians have long viewed the quest for more information, for example through the acquisition of additional data, as being central to the goal of reducing uncertainty about some aspect of the world. This paper explores that objective through the variance, a common way of quantifying uncertainty. In particular, it examines the relationship between information and uncertainty. Surprisingly it shows that increasing the amount of information can in some cases increase the variance while in others it can decrease it. Which of these occurs is not explained by the seductive thesis that it depends simply on whether that uncertainty is merely aleatory-due to chance alone-or epistemic-due to lack of knowledge.

For a joint distribution function over a two-variable (two-dimensional) space, the marginal variance for data sampled along a particular reference direction (indicated by a double-headed arrow) can be understood as a shadow cast from the joint uncertainty region in the orthogonal direction (indicated by the corresponding perpendicular pair of dashed lines). On the other hand, the conditional variance is directly obtained by slicing the joint uncertainty region about its center along the same reference direction. The marginal uncertainty region is the region defined by pairs of points bounding the marginal variances in all directions, which is similar to how the conditional uncertainty region is related to the conditional variances.

B)  The generalized autoregressive conditional heteroscedasticity (GARCH) model of Bollerslev (1986) is an important type of time series model for heteroscedastic data. It explicitly models a time-varying conditional variance as a linear function of past squared residuals and of its past values. The GARCH process has been widely used to model economic and financial time-series data.

The risk-return tradeoff has typically been examined by means of the GARCH-in-mean (GARCH-M) model originally proposed by Engle, Lilien and Robins (1987). The general idea of the GARCH-M model is that the conditional variance is included in the conditional
mean equation and its coefficient is interpreted as to measure the strength of risk
aversion.

In finance, the return of a security may depend on its volatility. To model such a phenomenon, one may consider the GARCH-M model, where M stands for GARCH in the mean. A simple GARCH(1,1)-M model can be written as

where ? and c are constants. The parameter c is called the risk premium parameter. A positive c indicates that the return is positively related to its volatility. Other specifications of risk premium have also been used in the literature, including rt = ? + c?t + at and .

C)  The most popular models in modelling volatility are
GARCH type models because they can account excess kurtosis and asymmetric effects of
financial time series. Since standard GARCH(1,1) model usually indicate high persistence in the
conditional variance, the empirical researches turned to GJR-GARCH model and reveal its
superiority in fitting the asymmetric heteroscedasticity in the data. In order to capture both
asymmetry and nonlinearity in data, the goal of this paper is to develop a parsimonious NN
model as an extension to GJR-GARCH model and to determine if GJR-GARCH-NN outperforms
the GJR-GARCH model.

The objective of this paper is to develop a parsimonious NN model as an extension to GJR-
GARCH model which will capture the nonlinear relationship between past return innovations and
conditional variance. The second objective of this paper is to determine if GJR-GARCH-NN
model outperforms GJR-GARCH models when there is high persistence of the conditional
variance.