## 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 `Mdl`. 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 `Mdl.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 zeros.