Are the variables in your GARCH model relevant?

GARCH Models in R

Kris Boudt

Professor of finance and econometrics

Example

Case of AR(1) GJR GARCH model with skewed student t innovations

names(coef(flexgarchfit))

"mu"     "ar1"    "omega"  "alpha1" "beta1"  "gamma1" "skew"   "shape"

Can you simplify the model? $\Longleftrightarrow$ Are there parameters zero?
- If the ar1 parameter is zero, you can use a constant mean model.
- If the gamma1 parameter is zero, there is no GARCH-in-mean and you can use a standard GARCH model instead of the GJR.

Challenge

We don't know the true parameter value. It needs to be estimated.
Caveat: even if the true parameter is 0, the estimated parameter will not be 0.
Example: are the ar1 and gamma1 parameters 0?

round(coef(flexgarchfit), 6)

       mu       ar1     omega    alpha1     beta1    gamma1      skew     shape 
-0.000021  0.000150  0.000000  0.034281  0.968688 -0.010093  1.013487  9.139252

Test of statistical significance

Use of statistical tests to decide whether the magnitude of the estimated parameter is significantly large enough to conclude that the true parameter is not zero.
How? By comparing the estimated parameter to its standard error

Standard error is the standard deviation of the parameter estimate

t-statistic

t-statistic = estimated parameter / standard error

Example for MSFT returns

Specify and estimate the model

flexgarchspec <- ugarchspec(mean.model = list(armaOrder = c(1, 0)),
                            variance.model = list(model = "gjrGARCH"),
                            distribution.model = "sstd")
flexgarchfit <- ugarchfit(data = msftret, spec = flexgarchspec)

Print table with parameter estimates, standard errors, and t-statistics using:

round(flexgarchfit@fit$matcoef, 6)

Parameter estimation table

round(flexgarchfit@fit$matcoef, 6)

        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000610    0.000189   3.220843 0.001278
ar1    -0.037799    0.013718  -2.755532 0.005860
omega   0.000002    0.000001   2.617696 0.008853
alpha1  0.034574    0.003395  10.182558 0.000000
beta1   0.935927    0.007163 130.667531 0.000000
gamma1  0.055483    0.009772   5.677857 0.000000
skew    1.059959    0.020676  51.264435 0.000000
shape   4.392327    0.256700  17.110745 0.000000

Note: 3.220843 = 0.000610 / 0.000189, -2.755532 = -0.037799 / 0.013718, ...

Interpretation of t-statistic

t-statistic = estimated parameter / standard error

Its absolute value is a distance measure: It tells you how many standard errors the estimated parameter is separated from 0.
The larger the distance, the more evidence the parameter is not 0.

Rule of thumb for deciding:

If |t-statistic| > 2
Then the estimated parameter is statistically significant
Therefore we conclude that true parameter $\neq$ 0.

Conclusion for MSFT returns using t-statistics

All t-statistics are larger than 2: all parameter estimates are statistically significant.

        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000610    0.000189   3.220843 0.001278
ar1    -0.037799    0.013718  -2.755532 0.005860
omega   0.000002    0.000001   2.617696 0.008853
alpha1  0.034574    0.003395  10.182558 0.000000
beta1   0.935927    0.007163 130.667531 0.000000
gamma1  0.055483    0.009772   5.677857 0.000000
skew    1.059959    0.020676  51.264435 0.000000
shape   4.392327    0.256700  17.110745 0.0000000

Same conclusion can be reached using the p-values in the last column.

Interpretation of p-value

The p-value measures how likely it is that the parameter is zero:
- The lower its value, the more evidence the parameter is not zero

Rule of thumb for deciding:

If p-value < 5%
Then the estimated parameter is statistically significant
Therefore we conclude that true parameter $\neq$ 0

Let's practice your skill to analyze statistical significance.

GARCH Models in R