74 76 78 80 82 84 86 88 90 92 94 Year

FIGURE 22.6 Log of U.S./U.K. exchange rate, 1973-1995 (monthly)

(Continued)

22You might wonder why we do not use the variance of Xt =Y X^/n as a measure of volatility. This is because we want to take into account changing volatility of asset prices over time. If we use the variance of Xt, it will only be a single value for a given data set.

858 PART FOUR: SIMULTANEOUS-EQUATION MODELS

U.S./U.K. EXCHANGE RATE: AN EXAMPLE (Continued) 0.12-

74 76 78 80 82 84 86 88 90 92 94 Year

FIGURE 22.7 Change in the log of U.S./U.K. exchange rate.

Accepting Xf as a measure of volatility, how do we know if it changes over time? Suppose we consider the following AR(1), or ARIMA(1, 0, 0), model:

This model postulates that volatility in the current period is related to its value in the previous period plus a white noise error term. If pf is positive, it suggests that if volatility was high in the previous period, it will continue to be high in the current period, indicating volatility clustering. If p1 is zero, then there is no volatility clustering. The statistical significance of the estimated pf can be judged by the usual ttest.

There is nothing to prevent us from considering an AR(p) model of volatility such that

Xf = fio + P1 X— + p2 Xf_2 + ••• + fipXf_ p + ut (22.10.2)

This model suggests that volatility in the current period is related to volatility in the past p periods, the value of p being an empirical question. This empirical question can be resolved by one or more of the model selection criteria that we discussed in Chapter 13 (e.g., the Akaike information measure). We can test the significance of any individual p coefficient by the t test and the collective significance of two or more coefficients by the usual F test.

Model (ff.10.1) is an example of an ARCH(1) model and (ff.10.2) is called an ARCH(p) model, where p represents the number of autoregressive terms in the model.

Before proceeding further, let us illustrate the ARCH model with the U.S./U.K. exchange rate data. The results of the ARCH(1) model were as follows.

where Xf is as defined before.

CHAPTER TWENTY-TWO: TIME SERIES ECONOMETRICS: FORECASTING 859

U.S./U.K. EXCHANGE RATE: AN EXAMPLE (Continued)

Since the coefficient of the lagged term is highly significant (p value of about 0.005), it seems volatility clustering is present in the present instance. We tried higher-order ARCH models, but only the AR(1) model turned out to be significant.

How would we test for the ARCH effect in a regression model in general that is based on time series data? To be more specific, let us consider the k-variable linear regression model:

Yt = P1 + fa X2t + ••• + fikXkt + u, (22.10.4)

and assume that conditional on the information available at time (t - 1), the disturbance term is distributed as u, ~ N[0, (a0 + a1u2-^ (22.10.5)

that is, ut is normally distributed with zero mean and var( u, ) = («0 + a1u'2-1) (22.10.6)

that is, the variance of ut follows an ARCH(1) process.

The normality of ut is not new to us. What is new is that the variance of u at time t is dependent on the squared disturbance at time (t - 1), thus giving the appearance of serial correlation.23 Of course, the error variance may depend not only on one lagged term of the squared error term but also on several lagged squared terms as follows:

var( ut) = af = «0 + «1 u2-1 + a2uf-2 +-----+ apuf-p (22.10.7)

If there is no autocorrelation in the error variance, we have

in which case var(ut) = a0, and we do not have the ARCH effect.

Since we do not directly observe at2, Engle has shown that running the following regression can easily test the preceding null hypothesis:

uf = «0 + «1 u\-1 + «2^-2 + ••• + <ip&t- p (22.10.9)

where ut, as usual, denote the OLS variance obtained from the original regression model (22.10.4).

One can test the null hypothesis H0 by the usual Ftest, or alternatively, by computing nR2, where R2 is the coefficient of determination from the auxiliary regression (22.10.9). It can be shown that nR2sy ~ X$ (22.10.10)

that is, in large samples nR2 follows the chi-square distribution with df equal to the number of autoregressive terms in the auxiliary regression.

Before we proceed to illustrate, make sure that you do not confuse autocorrelation of the error term as discussed in Chapter 12 and the ARCH model. In the ARCH model it is the (conditional) variance of u that depends on the (squared) previous error terms, thus giving the impression of autocorrelation.

23A technical note: Remember that for our classical linear model the variance of ut was assumed to be a2, which in the present context becomes unconditional variance. If «1 < 1, the stability condition, we can write a2 = ao + «1a2; that is, a2 = ao/(1 - «1)- This shows that the unconditional variance of u does not depend on t, but does depend on the ARCH parameter a1.

860 PART FOUR: SIMULTANEOUS-EQUATION MODELS

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