M-GARCH, Intercepts of Mean Equations = Sample means?

[garchrookie]

I have two series here: y1 and y2.

Before I run the CCC-MGARCH, I calculate the summary statistics.

As shown in the RATS output, both y1 and y2 have a positive mean.

Statistics on Series LGRY1
Observations 4995
Sample Mean 0.000213 Variance 0.000130
Standard Error 0.011407 of Sample Mean 0.000161
t-Statistic (Mean=0) 1.321191 Signif Level 0.186498
Skewness -0.247157 Signif Level (Sk=0) 0.000000
Kurtosis (excess) 9.732904 Signif Level (Ku=0) 0.000000
Jarque-Bera 19766.415021 Signif Level (JB=0) 0.000000

Statistics on Series LGRY2
Observations 4995
Sample Mean 0.000194 Variance 0.003382
Standard Error 0.058156 of Sample Mean 0.000823
t-Statistic (Mean=0) 0.236064 Signif Level 0.813393
Skewness 0.624373 Signif Level (Sk=0) 0.000000
Kurtosis (excess) 4.521144 Signif Level (Ku=0) 0.000000
Jarque-Bera 4578.773622 Signif Level (JB=0) 0.000000

However, when I run the CCC-GARCH, the y2 mean equation’s intercept is negative.


MV_GARCH, CC - Estimation by BFGS
Convergence in 34 Iterations. Final criterion was 0.0000026 <= 0.0000100
Daily(5) Data From 1990:01:03 To 2009:02:24
Usable Observations 4995
Log Likelihood 25474.51984640

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Mean(1) 5.0002e-04 1.0964e-04 4.56081 0.00000510
2. Mean(2) -5.6011e-04 7.6312e-04 -0.73397 0.46296459
3. C(1) 8.6192e-07 1.5316e-07 5.62770 0.00000002
4. C(2) 3.1710e-04 5.4461e-05 5.82244 0.00000001
5. A(1) 0.0580 4.7308e-03 12.25240 0.00000000
6. A(2) 0.0727 9.6523e-03 7.53485 0.00000000
7. B(1) 0.9344 5.6701e-03 164.79332 0.00000000
8. B(2) 0.8314 0.0240 34.68299 0.00000000
9. R(2,1) -0.7232 6.3492e-03 -113.89604 0.00000000

For the two mean equations, I only have the constants, so I expect the two intercepts of the mean equations should be very similar to the sample means of the two series.

What is wrong here?

[TomDoan]

The means in the GARCH are estimates of the means of the processes, but, unlike the means calculated in the STATS instruction, they're done using (in effect) weighted least squares. Your second series has a positive sample mean, but not significantly so. The GARCH variance is obviously lower in general on data points with negative values, which means the weights are higher, and the weighted mean is negative. Note, though, that it is also not significant.