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EWMA

EWMA (Exponentially Weighted Moving Average) is a very tightly parameterized variance model. There is just a single real parameter (\(\alpha\)) governing the evolution of the variance:

\begin{equation} {\bf{H}}_t = (1 - \alpha ){\bf{H}}_{t - 1} + \alpha \left( {{\bf{u}}_{t - 1} {\bf{u}}_{t - 1}^{\prime} } \right) \end{equation}

This is chosen with MV=EWMA. This is a special case of the DVECH model with the coefficients equal across all components and an "I-GARCH" restriction (without variance intercept). The example of this is GARCHMV.RPF is

garch(p=1,q=1,mv=ewma,distrib=t) / xjpn xfra xsui

which estimates a GARCH with EWMA and Student t errors with an estimated degrees of freedom. Note that the log likelihood is substantially better than the previous models because of this—all the previous examples assumed conditional Normal residuals.

Output

The one common "GARCH" parameter is labeled ALPHA.

MV-GARCH, EWMA - Estimation by BFGS

Convergence in 12 Iterations. Final criterion was 0.0000058 <= 0.0000100

Usable Observations 6236

Log Likelihood -10516.1908

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Mean(XJPN) -0.005311325 0.003704975 -1.43357 0.15169623

2. Mean(XFRA) -0.004245558 0.004184180 -1.01467 0.31026361

3. Mean(XSUI) -0.005260662 0.005173633 -1.01682 0.30923824

4. Alpha 0.049804403 0.001974470 25.22419 0.00000000

5. Shape(t degrees) 5.253165325 0.139848297 37.56331 0.00000000