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o implies the following model for the excess market returns:
ttftmrrελσ+=−2,
whereλis the market price of risk (see Lectures 2, 6 and Cuthbertson and Nitzsche pp 137-138 and pp 659-661). The predictions of CAPM for the excess market returns are that the intercept is zero (no abnormal returns) and 0>λ(implying a positive risk/return trade-off). In this version of the GARCH model the conditional variance enters the mean equation directly. This is known generally as a GARCH-M model. Specifically, based on the asymmetry analysis, we will estimate EGARCH-M and TARCH-M models to test CAPM.
2. Test for ARCH effects in the excess market returns
i) Generate the excess market returns ftmrr−,
Click Genr on the workfile toolbar and enter:
ex_ret=dlog(ftse_all)-rf_daily
As always, reporting line graphs and summary statistics constitute an important prelude to the main analysis. Comment on the following output:
Figure 1: Line graph of ex_ret -.06-.04-.02.00.02.04.0603M0103M0704M0104M0705M0105M0706M01EX_RET
Table 1: Summary statistics for ex_ret 020406080100120140-0.0250.0000.0250.050Series: EX_RETSample 1/01/2003 1/19/2006Observations 776Mean 0.000514Median 0.000743Maximum 0.050803Minimum -0.041116Std. Dev. 0.007912Skewness 0.093422Kurtosis 7.539827Jarque-Bera 667.5198Probability 0.000000
The excess returns display periods of turbulence and tranquility. This suggests there is volatility clustering.
The excess returns have an unconditional non-normal distribution: the distribution is leptokurtic (fat-tailed) – explain why.
Now test for ARCH effects using an ARCH-LM test:
On the main toolbar click Quick/Estimate Equation and enter:
ex_ret c
On the Equation toolbar click View/Residual Tests/ARCH LM Test (Choose 5 lags)
ARCH Test:F-statistic49.24828 Prob. F(5,765)0Obs*R-squared187.7416 Prob. Chi-Square(5)0Test Equation:Dependent Variable: RESID^2Method: Least SquaresDate: 02/18/07 Time: 09:47Sample (adjusted): 1/09/2003 1/19/2006Included observations: 771 after adjustmentsVariableCoefficientStd. Errort-StatisticProb. C2.33E-055.81E-064.0139720.0001RESID^2(-1)0.3955640.03535611.187950RESID^2(-2)0.0193610.0377550.5128120.6082RESID^2(-3)0.1477260.0373733.9527960.0001RESID^2(-4)-0.1489940.037682-3.9539820.0001RESID^2(-5)0.2068170.0350145.9067220R-squared0.243504 Mean dependent var6.19E-05Adjusted R-squa0.23856 S.D. dependent var0.000159S.E. of regressio0.000139 Akaike info criterion-14.91586Sum squared re1.48E-05 Schwarz criterion-14.87969Log likelihood5756.064 F-statistic49.24828Durbin-Watson s2.047884 Prob(F-statistic)0
Check whether or not inferences on ARCH effects are sensitive to the chosen lag: vary the lag from one day up to 20 days (1 month).
We begin by assuming a constant risk premium and will relax this assumption later.
Alternatively, an ARMA model (identified using the ACF/PACF of ex_ret) could be used to estimate the conditional mean (see Seminar 4).
The ARCH-LM statistic is significant at the 5% level suggesting the presence of ARCH effects. This result provides justification for the next stage in the analysis which involves estimating the conditional variance using a GARCH(1,1) model.
3. Estimate and test a GARCH(1,1) model for the conditional variance.
i) Estimate a GARCH(1,1) model
On the main toolbar click Quick/Estimate Equation (Method=ARCH)
In the Mean equation enter
ex_ret c
The default settings for the conditional variance equation are GARCH(1,1