The Ljung Box Test

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The Ljung-Box test is a test for serial correlation that tests if the autocorrelation coefficients for a given number of lags are jointly not significantly different from zero. The statistic for this test is where T is the sample size, m is the number of lags and is the estimated autocorrelation coefficient. The null hypothesis for this test is that the coefficients are all jointly zero and has a distribution. The alternative hypothesis is that at least one of the coefficients is not equal to zero and implies the presence of serial correlation.

We can estimate the Ljung-Box statistic in Eviews by creating a correlogram for the series rlsp500. In the below table included with the correlogram we are given the Ljung-Box statistic for each
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| 6 0.076 0.073 12.277 0.056 *|. | *|. | 7 -0.081 -0.091 14.441 0.044 .|. | .|. | 8 0.042 0.048 15.025 0.059 *|. | .|. | 9 -0.089 -0.058 17.622 0.040 .|. | .|. | 10 0.007 -0.041 17.639 0.061 .|. | .|. | 11 -0.031 -0.025 17.959 0.083 .|. | *|. | 12 -0.062 -0.069 19.242 0.083

B) The stylized feature of volatility clustering in financial data cannot be explained using linear models and estimation methods such as OLS. This is because one of the underlying assumptions of OLS is that is a constant and is the basis for measuring volatility. A GARCH(1,1) model is able to model this feature because it estimates a model for , which allows the variance to change over time.

C) To test for ARCH effects in the return series, we would start by estimating the return equation using least squares. We would then square the residuals from the return equation and regress the squared residuals on its lags. The R2 from this regression can be used to construct the test statistic, where T is the number of observations and the degrees of freedom, q, is the number of lagged squared residuals in the test equation. The null hypothesis for this test is that all the coefficients on the lagged squared residuals are equal to zero. The alternative hypothesis is that at least one of the coefficients
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-4.710556
Durbin-Watson stat 2.099147

Since we are talking about asset returns, a standard GARCH model may not be the best choice as we would expect there to be asymmetry in the volatility (Brooks 2008, p. 404). The EGARCH model would allow negative shocks to have a larger effect on the conditional variance than positive ones. As we can see in the below Eviews output, it is the case that negative shocks have a larger effect because the coefficient C(4) is negative. Because we are estimating the log of the conditional variance, unlike the standard GARCH model, it can be more difficult to interpret the exact meaning of all the parameters.

Dependent Variable: RLSP500
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 07/29/12 Time: 20:08
Sample (adjusted): 1/10/2005 1/31/2011
Included observations: 317 after adjustments
Convergence achieved after 35 iterations
Presample variance: backcast (parameter = 0.7)
LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(4) *RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))

Variable Coefficient Std. Error z-Statistic

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