introductory econometrics for finance Chapter8solutions.docx

1、introductory econometrics for finance Chapter8solutionsSolutions to the Review Questions at the End of Chapter 81. (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

2、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 station

3、ary.- 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 l

4、everage 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

5、 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

6、equation in part (e), thes, 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 disadv

7、antages 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 h

8、as 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 th

9、an 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

10、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 ne

11、gative 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 respectivelyAndt2 = 0 + 1+t-12+ut-12It-1where It-1 = 1 if ut-1 0 = 0 otherwiseFor a l

12、everage 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-negativi

13、ty constraints.One form of the GARCH-M model can be writtenyt = +other terms + t-1+ ut , ut N(0,ht) t2 = 0 + 1+t-12so 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 retur

14、n, 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 da

15、ily 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 value

16、s 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 restri

17、ction 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

18、 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 obtaini

19、ng 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 mo

20、dels 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) forecas

21、ts 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 literatur

22、e 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 equation

23、s for times T+1, T+2, and T+3: hT+1 = 0 + 1 + hT (1) hT+2 = 0 + 1 + hT+1 (2) hT+3 = 0 + 1 +hT+2 (3)Let 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, how do we calculate, that is the 2-step ahead

24、 forecast for h made at time T?From (2), we can write= 0 + 1 ET()+ (4)where ET() is the expectation, made at time T, of, which is the squared disturbance term. The model assumes that the series t has zero mean, so we can now writeVar(ut) = E(ut -E(ut)2= E(ut)2. The conditional variance of ut is ht,

25、so ht t = E(ut)2Turning this argument around, and applying it to the problem that we have, ET(uT+1)2 = hT+1but we do not know hT+1 , so we replace it with, so that (4) becomes= 0 + 1 + = 0 + (1+)What about the 3-step ahead forecast?By similar arguments, = ET(0 + 1 + hT+2) = 0 + (1+) = 0 + (1+) 0 + (

26、1+)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 fo

27、recast for the conditional variance could be written (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 ex

28、plode. 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+)1, the forecasts will converge on the unconditional

29、variance as the forecast horizon increases. For (1+) = 1, known as “integrated GARCH” or IGARCH, there is a unit root in the conditional variance, and the forecasts will stay constant as the forecast horizon increases. 2. (a) Maximum likelihood works by finding the most likely values of the paramete

30、rs given the actual data. More specifically, a log-likelihood function is formed, usually based upon a normality assumption for the disturbance terms, and the values of the parameters that maximise it are sought. Maximum likelihood estimation can be employed to find parameter values for both linear

31、and non-linear models.(b) The three hypothesis testing procedures available within the maximum likelihood approach are lagrange multiplier (LM), likelihood ratio (LR) and Wald tests. The differences between them are described in Figure 8.4, and are not defined again here. The Lagrange multiplier tes

32、t involves estimation only under the null hypothesis, the likelihood ratio test involves estimation under both the null and the alternative hypothesis, while the Wald test involves estimation only under the alternative. Given this, it should be evident that the LM test will in many cases be the simplest to compute since the restrictions implied by the null hypothesis will usually lead to some terms cancelling out to give a simplified mode