Conditional Mean Model Estimation with Equality Constraints

For conditional mean model estimation, estimate requires an arima model and a vector of univariate time series data. The model specifies the parametric form of the conditional mean model that estimate estimates. estimate returns fitted values for any parameters in the input model with NaN values. If you pass a T×r exogenous covariate matrix in the X argument, then estimate returns r regression estimates . 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 without a constant term. Specify 'Constant',0 in the model you pass into estimate. estimate views this non-NaN value as an equality constraint, and does not estimate the constant term. estimate also honors all specified equality constraints while estimating parameters without equality constraints. You can set a subset of regression coefficients to a constant and estimate the rest. For example, suppose your model is called model. If your model has three exogenous covariates, and you want to estimate two of them and set the other to one to 5, then specify model.Beta = [NaN 5 NaN].

estimate optionally returns the variance-covariance matrix for estimated parameters. The parameter order in this matrix is:

  • Constant

  • Nonzero AR coefficients at positive lags (AR)

  • Nonzero seasonal AR coefficients at positive lags (SAR)

  • Nonzero MA coefficients at positive lags (MA)

  • Nonzero seasonal MA coefficients at positive lags (SMA)

  • Regression coefficients (when you specify X)

  • Variance parameters (scalar for constant-variance models, vector of additional parameters otherwise)

  • Degrees of freedom (t innovation distribution only)

If any parameter known to the optimizer has an equality constraint, then the corresponding row and column of the variance-covariance matrix has all 0s.

In addition to user-specified equality constraints, estimate sets any AR or MA coefficient with an estimate less than 1e-12 in magnitude equal to 0.

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