Brooks Introductory Econometrics for Finance

Solutions to the Review Questions at the End of Chapter 8

1. (a). A number of stylised features of financial data have been suggested at the start of Chapter 8 and in other places throughout the book:

- Frequency: Stock market prices are measured every time there is a trade or somebody posts a new quote, so often the frequency of the data is very high

- Non-stationarity: Financial data (asset prices) are covariance non-stationary; but if we assume that we are talking about returns from here on, then we can validly consider them to be stationary.

- Linear Independence: They typically have little evidence of linear (autoregressive) dependence, especially at low frequency.

- Non-normality: They are not normally distributed – they are fat-tailed.

- Volatility pooling and asymmetries in volatility: The returns exhibit volatility clustering and leverage effects.

Of these, we can allow for the non-stationarity within the linear (ARIMA) framework, and we can use whatever frequency of data we like to form the models, but we cannot hope to capture the other features using a linear model with Gaussian disturbances.

(b) GARCH models are designed to capture the volatility clustering effects in the returns (GARCH(1,1) can model the dependence in the squared returns, or squared residuals), and they can also capture some of the unconditional leptokurtosis, so that even if the residuals of a linear model of the form given by the first part of the equation in part (e), the [pic]’s, are leptokurtic, the standardised residuals from the GARCH estimation are likely to be less leptokurtic. Standard GARCH models cannot, however, account for leverage effects.

(c) This is essentially a “which disadvantages of ARCH are overcome by GARCH” question. The disadvantages of ARCH(q) are:

- How do we decide on q?

- The required value of q might be very large

- Non-negativity constraints might be violated.

When we estimate an ARCH model, we require...