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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