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Conditional Variance Model Estimation with Equality Constraints

For conditional variance model estimation, the required inputs for estimate are a model object and a vector of univariate time series data. The model object specifies the parametric form of the conditional variance model being estimated. estimate returns fitted values for any parameters in the input model object with NaN values. If you specify non-NaN values for any parameters, estimate views these values as equality constraints and honors them during estimation.

For example, suppose you are estimating a model with a mean offset known to be 0.3. To indicate this, specify 'Offset',0.3 in the model object you input to estimate. estimate views this non-NaN value as an equality constraint, and does not estimate the mean offset. estimate also honors all specified equality constraints during estimation of the parameters without equality constraints.

estimate optionally returns the variance-covariance matrix for estimated parameters. The parameters in the variance-covariance matrix are ordered as follows:

  • Constant

  • Nonzero GARCH coefficients at positive lags

  • Nonzero ARCH coefficients at positive lags

  • Nonzero leverage coefficients at positive lags (EGARCH and GJR models only)

  • Degrees of freedom (t innovation distribution only)

  • Offset (models with nonzero offset only)

If any parameter known to the optimizer has an equality constraint, the corresponding row and column of the variance-covariance matrix has all zeros.

In addition to user-specified equality constraints, note that estimate sets any GARCH, ARCH, or leverage coefficient with an estimate less than 1e-12 in magnitude equal to zero.

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