estimate function for conditional variance
fmincon from Optimization
perform maximum likelihood estimation. This optimization function
requires initial (or, starting) values to begin the optimization process.
If you want to specify your own initial values, use name-value
arguments. For example, specify initial values for GARCH coefficients
using the name-value argument
Alternatively, you can let
default initial values. Default initial values are generated using
standard time series techniques. If you partially specify initial
values (that is, specify initial values for some parameters),
the initial values you do specify, and generates default initial values
for the remaining parameters.
When generating initial values,
any stationarity and positivity constraints for the conditional variance
model being estimated. The techniques
to generate default initial values are as follows:
For the GARCH and GJR models, the model is transformed to an equivalent ARMA model for the squared, offset-adjusted response series. Note that the GJR model is treated like a GARCH model, with all leverage coefficients equal to zero. The initial ARMA values are solved for using the modified Yule-Walker equations as described in Box, Jenkins, and Reinsel . The initial GARCH and ARCH starting values are calculated by transforming the ARMA starting values back to the original GARCH (or GJR) representation.
For the EGARCH model, the initial GARCH coefficient values are found by viewing the model as an equivalent ARMA model for the squared, offset-adjusted log response series. The initial GARCH values are solved for using Yule-Walker equations as described in Box, Jenkins, and Reinsel . For the other coefficients, the first nonzero ARCH coefficient is set to a small positive value, and the first nonzero leverage coefficient is set to a small negative value (consistent with the expected signs of these coefficients).
 Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.