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`EstMdl = estimate(Mdl,y)`

[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y)

[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y,Name,Value)

uses
maximum likelihood to estimate the parameters of the regression model
with ARIMA time series errors, `EstMdl`

= estimate(`Mdl`

,`y`

)`Mdl`

, given the
response series `y`

. `EstMdl`

is
a `regARIMA`

model
that stores the results.

`[`

additionally
returns `EstMdl`

,`EstParamCov`

,`logL`

,`info`

]
= estimate(`Mdl`

,`y`

)`EstParamCov`

, the variance-covariance matrix
associated with estimated parameters, `logL`

, the
optimized loglikelihood objective function, and `info`

,
a data structure of summary information.

`[`

estimates
the model using additional options specified by one or more `EstMdl`

,`EstParamCov`

,`logL`

,`info`

]
= estimate(`Mdl`

,`y`

,`Name,Value`

)`Name,Value`

pair
arguments.

Suppose `EstParamCov`

is an estimated parameter
covariance matrix returned by `estimate`

. The software
sets the variances and covariances of parameters fixed during estimation
to `0`

. Enter this command to count the number of
free parameters (`numParams`

) in a fitted model.

numParams = sum(any(EstParamCov))

This command counts the number of columns (or equivalently, rows) with any nonzero values.

`estimate`

estimates the parameters as follows:

Infer the unconditional disturbances from the regression model.

Infer the residuals of the ARIMA error model.

Use the distribution of the innovations to build the likelihood function.

Maximize the loglikelihood function with respect to the parameters using

`fmincon`

.

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. *Time
Series Analysis: Forecasting and Control*. 3rd ed. Englewood
Cliffs, NJ: Prentice Hall, 1994.

[2] Davidson, R., and J. G. MacKinnon. *Econometric
Theory and Methods*. Oxford, UK: Oxford University Press,
2004.

[3] Enders, W. *Applied Econometric Time Series*.
Hoboken, NJ: John Wiley & Sons, Inc., 1995.

[4] Hamilton, J. D. *Time Series Analysis*.
Princeton, NJ: Princeton University Press, 1994.

[5] Pankratz, A. *Forecasting with Dynamic Regression
Models.* John Wiley & Sons, Inc., 1991.

[6] Tsay, R. S. *Analysis of Financial Time Series*.
2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.

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