Non-normality of standardized returns
1. Markets take the stairs up, but the elevator down
It is not realistic to assume a normal distribution when analyzing stock returns using a GARCH model. The normal distribution is in fact inconsistent with the phenomenon that stock markets take the stairs up, but the elevator down. This behavior leads to crashes and non-normality in the financial returns.2. How?
To account for this, you need to change the argument distribution-dot-model in the ugarchspec function from normal to skewed student t.3. The normal GARCH model
Under the normal model, the return equals its mean value mu t plus a prediction error e t. The prediction error is normally distributed with a zero mean and variance equal to sigma square t. This implies that if you take the return, subtract from it the mean and then divide by the volatility sigma t, you obtain a standardized return z t that is standard normal distributed.4. Let's test
The important caveat is that the normality is an assumption, which may not be correct. Let's test this.5. Estimated standardized returns
To test the assumption, you need to estimate the mean and volatility, and then compute the standardized returns. In R you do this by applying the method residuals to the output of a ugarchfit estimation. Make sure to set the argument standardize equal to TRUE.6. Testing the normality assumption
If you then use the chart.Histogram function in the PerformanceAnalytics package to plot the histogram of the standardized return and compare it with the density function of the normal distribution you will see the non-normality.7. Density plot - Peakedness
In grey you see the histogram. The blue line is the normal distribution, while in red is the actual one. Compared to the normal, the actual return distribution is much more peaked around zero.8. Density plot - Tails
In the tails you see that extreme observations occur and that the normal distribution cannot fit them, as it assigns them a zero density. We therefore say that the return distribution has fat tails. Furthermore, for the S&P 500 returns, the distribution is not symmetric but negatively skewed: the extremes occur more often for negative returns than for positive returns.9. Solution
A realistic distribution for returns needs to accommodate the potential presence of both fat tails and skewness. In the package rugarch you can do this with the skewed student t distribution.10. Parameters of the skewed student t distribution
Compared to the normal distribution, it has two additional parameters. First, the degrees of freedom parameter $nu$. The lower is nu, the fatter are the tails. Second there is a skewness parameter xi. If xi equals one, the distribution is symmetric. If xi is less than one it is negatively skewed. If xi is higher than one, it is positively skewed. In rugarch, the parameter nu is called shape and the parameter xi is called skew. The normal and student t distribution are special cases. The normal and student t distribution are special cases.11. Normal density plot
If nu is infinity and xi equals one, we have the normal distribution.12. Student t density
If xi is 1 we have the student t distribution. As an example you see here in blue the student t distribution with five degrees of freedom. Note the higher peak in the center and the fatter tails.13. Skewed student t
By setting the skewness parameter to 0.6 you obtain a negatively skewed distribution.14. Tails
You can see the differences in the tail more clearly if you zoom in on the observations exceeding -2 and +2.15. GARCH model estimation with skewed student t
The estimation of a GARCH model with skewed student t distribution requires to first set the argument distribution model to sstd. In the estimation output, two extra parameters appear. The parameter xi equals the skew parameter. For the S&P 500 returns, it is less than 1 indicating a negatively skewed distribution. The degrees of freedom are around 6 indicating fat tails.16. Let's practice!
Now let's try some examples.Create Your Free Account
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