Diagnosing absolute standardized returns
1. Validate your assumptions about the mean and variance
The GARCH model makes strong assumptions about the mean and the variance. It is important to validate those assumptions.2. Check 1: Mean and standard deviation of standardized returns
You can do this by analyzing the standardized returns. They are equal the observed returns minus their predicted mean, and divided by their estimated volatility. These standardized returns need to have zero mean and unit standard deviation.3. Check 2: Time series plot of standardized returns
A second check is to make a time series plot and verify that the GARCH model is successful in generating standardized returns for which the variability is constant over time.4. MSFT returns
Here you see this is the case for the daily Microsoft returns. In the top plot you see the clear volatility clusters. The bottom plots shows the standardized returns for which the variability seems constant.5. EUR/USD returns
For the EUR/USD returns we have same. The returns in the top plot show volatility clustering. Note eg that in the middle we see a burst of volatility due to the 2008 financial crisis. In the bottom plot you can see that the standardization leads to returns with constant variability.6. Check 3: No predictability in the absolute standardized returns
A third model validity check is to verify that the correlations between the past absolute standardized return and the current absolute standardized return are zero. This requirement follows directly from the condition of constant variability. Any deviation should be random and thus correlations between past and current absolute standardized returns need to be zero.7. Autocorrelations
These within time series correlations are called autocorrelations. The autocorrelation of order 1 is the correlation between the current value and its previous value. And similarly for the autocorrelation of order two, three, and so one. They should all be zero. One exception is the autocorrelation of order zero, which is the correlation of the variable with itself and thus equals one.8. acf()
In R, we can use the function acf to compute these autocorrelations. It takes two arguments: namely the time series of absolute standardized returns, and the maximum order of the autocorrelation. Its output is the correlogram showing the autocorrelation.9. Application to MSFT
We illustrate this here for the daily Microsoft returns. First we estimate as usual the GARCH model and compute the standardized returns. Then we compute the autocorrelations up to a horizon of 22 days, which corresponds to one month.10. acf of MSFT absolute (standardized) returns
Here you see the correlogram together with blue dashed lines that can be interpreted as critical values. When the autocorrelation exceeds the line, there is statistical evidence that the autocorrelation is significantly different from zero. For the absolute returns, the autocorrelations are all very high and significant. The variance model does a good job in capturing the volatility dynamics since the autocorrelations of the absolute standardized returns are close to zero.11. Check 4: Ljung-Box test
Besides the graphical analysis of the autocorrelation, you can use the Ljung-Box test to formally test that all the autocorrelations up to a certain order are equal to zero. This is similar to the t-test for statistical significance, but here we want to have 0 autocorrelation in in order to have a valid GARCH model. We reject the validity when the p-value is small. As rule of thumb we compare with 5% and thus reject the validity when the p-value is less than 5%.12. Ljung-Box test in R
The function Box-dot-test allows you to do this in R. It takes as argument the series of absolute standardized return and the maximum order for which all autocorrelations needs to be zero. You also need to set the type of test to Ljung-Box.13. Box.test using absolute standardized MSFT returns
In case of Microsoft, we obtain a p-value of 28-point-55%. This value is larger than 5% indicating that the autocorrelations of the absolute standardized returns are 0 and thus that the GARCH model is valid.14. Let's diagnose the absolute standardized returns.
Now it's time to play doctor yourself and diagnose the absolute standardized returns.Create Your Free Account
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