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 (i >0 ( i=1,2,...,q (since variance cannot be negative)

GARCH(1,1) goes some way to get around these. The GARCH(1,1) model has only three parameters in the conditional variance equation, compared to q+1 for the ARCH(q) model, so it is more parsimonious. Since there are less parameters than a typical qth order ARCH model, it is less likely that the estimated values of one or more of these 3 parameters would be negative than all q+1 parameters. Also, the GARCH(1,1) model can usually still capture all of the significant dependence in the squared returns since it is possible to write the GARCH(1,1) model as an ARCH((), so lags of the squared residuals back into the infinite past help to explain the current value of the conditional variance, ht.

(d) There are a number that you could choose from, and the relevant ones that were discussed in Chapter 8, inlcuding EGARCH, GJR or GARCH-M.

The first two of these are designed to capture leverage effects. These are asymmetries in the response of volatility to positive or negative returns. The standard GARCH model cannot capture these, since we are squaring the lagged error term, and we are therefore losing its sign.

The conditional variance equations for the EGARCH and GJR models are respectively
[pic]

And

(t2 = (0 + (1[pic]+((t-12+(ut-12It-1

where It-1 = 1 if ut-1 ( 0 = 0 otherwise

For a leverage effect, we would see ( > 0 in both models.

The EGARCH model also has the added benefit that the model is expressed in terms of the log of ht, so that even if the parameters are negative, the conditional variance will always be positive. We do not therefore have to artificially impose non-negativity constraints.

One form of the GARCH-M model can be written

yt = ( +other terms + ((t-1+ ut , ut ( N(0,ht)
(t2 = (0 + (1[pic]+((t-12

so that the model allows the lagged value of the conditional variance to affect the return. In other words, our best current estimate of the total risk of the asset influences the return, so that we expect a positive coefficient for (. Note that some authors use ((t (i.e. a contemporaneous term).

(e). Since yt are returns, we would expect their mean value (which will be given by () to be positive and small. We are not told the frequency of the data, but suppose that we had a year of daily returns data, then ( would be the average daily percentage return over the year, which might be, say 0.05 (percent). We would expect the value of (0 again to be small, say 0.0001, or something of that order. The unconditional variance of the disturbances would be given by (0/(1-((1 +(2)). Typical values for (1 and (2 are 0.8 and 0.15 respectively. The important thing is that all three alphas must be positive, and the sum of (1 and (2 would be expected to be less than, but close to, unity, with (2 > (1.

(f) Since the model was estimated using maximum likelihood, it does not seem natural to test this restriction using the F-test via comparisons of residual sums of squares (and a t-test cannot be used since it is a test involving more than one coefficient). Thus we should use one of the approaches to hypothesis testing based on the principles of maximum likelihood (Wald, Lagrange Multiplier, Likelihood Ratio). The easiest one to use would be the likelihood ratio test, which would be computed as follows:

1. Estimate the unrestricted model and obtain the maximised value of the log-likelihood function.

2. Impose the restriction by rearranging the model, and estimate the restricted model, again obtaining the value of the likelihood at the new optimum. Note that this value of the LLF will be likely to be lower than the unconstrained maximum.

3. Then form the likelihood ratio test statistic given by

LR = -2(Lr - Lu) ( (2(m)

where Lr and Lu are the values of the LLF for the restricted and unrestricted models respectively, and m denotes the number of restrictions, which in this case is one.

4. If the value of the test statistic is greater than the critical value, reject the null hypothesis that the restrictions are valid.

(g) In fact, it is possible to produce volatility (conditional variance) forecasts in exactly the same way as forecasts are generated from an ARMA model by iterating through the equations with the conditional expectations operator.

We know all information including that available up to time T. The answer to this question will use the convention from the GARCH modelling literature to denote the conditional variance by ht rather than (t2. What we want to generate are forecasts of hT+1 ((T, hT+2 ((T, ..., hT+s ((T where (T denotes all information available up to and including observation T. Adding 1 then 2 then 3 to each of the time subscripts, we have the conditional variance equations for times T+1, T+2, and T+3: hT+1 = (0 + (1[pic] +( hT (1) hT+2 = (0 + (1[pic] +( hT+1 (2) hT+3 = (0 + (1[pic] +(hT+2 (3)

Let [pic] be the one step ahead forecast for h made at time T. This is easy to calculate since, at time T, we know the values of all the terms on the RHS. Given [pic], how do we calculate [pic], that is the 2-step ahead forecast for h made at time T?

From (2), we can write

[pic]= (0 + (1 ET([pic])+([pic] (4)

where ET([pic]) is the expectation, made at time T, of [pic], which is the squared disturbance term. The model assumes that the series (t has zero mean, so we can now write

Var(ut) = E[(ut -E(ut))2]= E[(ut)2].

The conditional variance of ut is ht, so

ht ( (t = E[(ut)2]

Turning this argument around, and applying it to the problem that we have,

ET[(uT+1)2] = hT+1

but we do not know hT+1 , so we replace it with [pic], so that (4) becomes

[pic]= (0 + (1[pic] +([pic] = (0 + ((1+()[pic]

What about the 3-step ahead forecast?
By similar arguments,

[pic]= ET((0 + (1[pic] +( hT+2) = (0 + ((1+() [pic] = (0 + ((1+()[ (0 + ((1+()[pic]]

And so on. This is the method we could use to forecast the conditional variance of yt. If yt were, say, daily returns on the FTSE, we could use these volatility forecasts as an input in the Black Scholes equation to help determine the appropriate price of FTSE index options.

(h) An s-step ahead forecast for the conditional variance could be written [pic] (x)
For the new value of (, the persistence of shocks to the conditional variance, given by ((1+() is 0.1251+ 0.98 = 1.1051, which is bigger than 1. It is obvious from equation (x), that any value for ((1+() bigger than one will lead the forecasts to explode. The forecasts will keep on increasing and will tend to infinity as the forecast horizon increases (i.e. as s increases). This is obviously an undesirable property of a forecasting model! This is called “non-stationarity in variance”.

For ((1+()