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.