ARCH and GARCH
The basic ARCH(1) model can be expressed as:
yt = в + et 

et1ti ~ N(0, ht) 

ht = ao + aqe2_i 
(14.3) 
a0 > 0, 0 < a1 < 1 
The first equation describes the behavior of the mean of your timeseries. In this case, equation (14.1) indicates that we expect the timeseries to vary randomly about its mean, в. If the mean of your timeseries drifts over time or is explained by other variables, you’d add them to this equation just as you would a regular regression model. The second equation indicates that the error of the regression, et, are normally distributed and heteroskedastic. The variance of the current period’s error depends on information that is revealed in the preceding period, i. e., It_i. The variance of et is given the symbol ht. The final equation describes how the variance behaves. Notice that ht
The ARCH(1) model can be extended to include more lags of the errors, etq. In this case, q refers to the order of the ARCH model. For example, ARCH(2) replaces (14.3) with ht = ao + a1e21 + a2e22. When estimating regression models that have ARCH errors in gretl, you’ll have to specify this order.
ARCH is treated as a special case of a more general model in gretl called GARCH. GARCH stands for generalized autoregressive conditional heteroskedasticity and it adds lagged values of the variance itself, htp, to (14.3). The GARCH(1,1) model is:
yt = в + et
etIt1 ~ N(0, ht)
ht = 5 + aie21 + eihti (14.4)
The difference between ARCH (14.3) and its generalization (14.4) is a term e1ht1, a function of the lagged variance. In higher order GARCH(p, q) model’s, q refers to the number of lags of et and p refers to the number of lags of ht to include in the model of the regression’s variance.
To open the dialog for estimating ARCH and GARCH in gretl choose Model>Time series>GARCH from the main gretl window.1 This reveals the dialog box where you specify the model (Figure 14.1). To estimate the ARCH(1) model, you’ll place the timeseries r into the dependent variable box and set q=1 and p=0. This yields the results:
Model 1: GARCH, using observations 1500
Dependent variable: r
Standard errors based on Hessian
Coefficient 
Std. Error 
z 
pvalue 

const 
1.06394 
0.0399241 
26.6491 
0.0000 
ao a1 
0.642139 0.569347 
0.0648195 0.0913142 
9.9066 6.2350 
0.0000 0.0000 
1.078294 S. D. dependent var 740.7932 Akaike criterion 1506.445 HannanQuinn
Unconditional error variance = 1.49108
The standard errors and tratios often vary a bit, depending on which software and numerical techniques are used. This is the nature of maximum likelihood estimation of the model’s parameters. With maximum likelihood, the model’s parameters are estimated using numerical optimization techniques. All of the techniques usually get you to the same parameter estimates, i. e., those that maximize the likelihood function; but, they do so in different ways. Each numerical algorithm arrives at the solution iteratively based on reasonable starting values and the method used to measure the curvature of the likelihood function at each round of estimates. Once the algorithm finds the maximum of the function, the curvature measure is reused as an estimate of the variance covariance matrix. Since curvature can be measured in slightly different ways, the routine will produce slightly different estimates of standard errors.
Gretl gives you a way to choose which method you like use for estimating the variance – covariance matrix. And, as expected, this choice will produce different standard errors and tratios. The set garch_vcv command allows you to choose among five alternatives: unsetwhich restores
carer.
Arg um ents: p q; depver [ indepvars ]
Options: —robust (robust standard errors)
—verbose (print details of iterations)
—vcv (print covariance matrix)
—no (do not include a constant)
—stdresid (standardize the residuals)
—fcp (use Fiorentini, Calzolari, Panattoni algorithm)
— anuain. it (initial variance parameters from ARM A)
The series are characterized by random, rapid changes and are said to be volatile. The volatility seems to change over time as well. For instance the U. S. stock returns index (NASDAQ) experiences a relatively sedate period from 1992 to 1996. Then, stock returns become much more volatile until early 2004. Volatility increases again at the end of the sample. The other series exhibit similar periods of relative calm followed by increased volatility.
A histogram graphs of the empirical distribution of a variable. In gretl the freq command generates a histogram. A curve from a normal distribution is overlaid using the normal option and the DoornikHansen test for normality is performed. A histogram for the ALLORDS series appears below in Figure 14.1.
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