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EstMdl = estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y,Name,Value)
EstMdl = estimate(Mdl,y) uses maximum likelihood to estimate the parameters of the regression model with ARIMA time series errors, Mdl, given the response series y. EstMdl is a regARIMA model that stores the results.
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y) additionally returns EstParamCov, the variance-covariance matrix associated with estimated parameters, logL, the optimized loglikelihood objective function, and info, a data structure of summary information.
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y,Name,Value) estimates the model using additional options specified by one or more Name,Value pair arguments.
EstMdl |
Model containing the parameter estimates, returned as a regARIMA model. estimate uses maximum likelihood to calculate all parameter estimates not constrained by Mdl (i.e., all parameters in Mdl that you set to NaN). | ||||||||||
EstParamCov |
Variance-covariance matrix of maximum likelihood estimates of the model parameters known to the optimizer, returned as a matrix. The rows and columns contain the covariances of the parameter estimates. The standard errors of the parameter estimates are the square root of the entries along the main diagonal. The rows and columns associated with any parameters held fixed as equality constraints contain 0s. estimate uses the outer product of gradients (OPG) method to perform covariance matrix estimation. estimate orders the parameters in EstParamCov as follows:
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logL |
Optimized loglikelihood objective function, returned as a scalar. | ||||||||||
info |
Summary information, returned as a structure.
For example, you can display the vector of final estimates by typing info.X in the Command Window. |
Suppose EstParamCov is an estimated parameter covariance matrix returned by estimate. Since the software sets the variances and covariances of parameters fixed during estimation to 0, one way to count the number of free parameters (numParams) in a fitted model is to enter the following command.
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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