ARMAMaxLikelihood Class

Computes maximum likelihood estimates of parameters for an ARMA model with p and q autoregressive and moving average terms respectively.

Inheritance Hierarchy

SystemObject
Imsl.StatARMAMaxLikelihood

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class ARMAMaxLikelihood

<SerializableAttribute>
Public Class ARMAMaxLikelihood

[SerializableAttribute]
public ref class ARMAMaxLikelihood

[<SerializableAttribute>]
type ARMAMaxLikelihood =  class end

The ARMAMaxLikelihood type exposes the following members.

Constructors

	Name	Description
	ARMAMaxLikelihood	Constructor for ARMAMaxLikelihood.

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Methods

	Name	Description
	Compute	Computes the exact maximum likelihood estimates for the autoregressive and moving average parameters of an ARMA time series.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	Forecast	Returns forecasts for lead times $l=1,2,\ldots,\rm{nForecast}$ at origins z.Length-BackwardOrigin-1+j where $j=1,\ldots,\rm{BackwardOrigin}+1$ .
	GetAR	Returns the final autoregressive parameter estimates.
	GetDeviations	Returns the deviations for each forecast used for calculating the forecast confidence limits.
	GetForecast	Returns forecasts
	GetGradients	Returns the gradients for the final parameter estimates.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetMA	Returns the final moving average parameter estimates.
	GetPsiWeights	Returns the psi weights used for calculating forecasts from the infinite order moving average form of the ARMA model.
	GetResiduals	The current values of the vector of residuals.
	GetTimeSeries	Returns the time series used to construct ARMAMaxLikelihood.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	IsInvertible	Tests whether the coefficients in ma are invertible.
	IsStationary	Tests whether the coefficients in ar are stationary.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetAR	Sets the initial values for the autoregressive terms to the p values in ar.
	SetMA	Sets the initial values for the moving average terms to the q values in ma.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	BackwardOrigin	The maximum backward origin.
	Confidence	The confidence level for calculating confidence limit deviations returned from GetDeviations.
	Constant	The constant parameter in the ARMA series.
	GradientTolerance	The gradient tolerance for the convergence algorithm.
	InnovationVariance	The estimated innovation variance of this series.
	Likelihood	The final estimate for $-2\ln(L)$ , where is equal to the likelihood function evaluated using the final parameter estimates.
	MaxIterations	The maximum number of iterations.
	Mean	The mean used for centering the series.
	P	The number of autoregressive terms in the ARMA model
	Q	The number of moving average terms in the ARMA model
	Tolerance	The tolerance for the convergence algorithm.

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Remarks

ARMAMaxLikelihood computes estimates of parameters for a nonseasonal ARMA model given a sample of observations, , for $t=1, 2,\ldots, n,$ where z.Length. The class is derived from the maximum likelihood estimation algorithm described by Akaike, Kitagawa, Arahata and Tada (1979), and the XSARMA routine published in the TIMSAC-78 Library.

The stationary time series with mean $\mu$ can be represented by the nonseasonal autoregressive moving average (ARMA) model by the following relationship:

${\phi}(B)(W_t-\mu)=\theta(B)a_t$

where

$t{\in}Z, ~~ Z=\{\ldots,-2,-1,0,1,2,\ldots\}\rm{,}$

B is the backward shift operator defined by $B^kW_t=W_{t-k}\rm{,}$

$\phi(B)=1-{\phi_1}B-{\phi_2}{B^2}-\ldots-{\phi_p}{B^p}, ~ p \ge 0\rm{.}$

and

$\theta(B)=1-{\theta_1}B-{\theta_2}{B^2}-\ldots-{\theta_q}{B^q}, ~ q \ge 0\rm{.}$

The ARMAMaxLikelihood class estimates the AR coefficients $\phi_1,\phi_2,\ldots,\phi_p$ and the MA coefficients $\theta_1,\theta_2,\ldots,\theta_q$ using maximum likelihood estimation.

ARMAMaxLikelihood checks the initial estimates for both the autoregressive and moving average coefficients to ensure that they represent a stationary and invertible series respectively.

${\phi_1},{\phi_2},\ldots,{\phi_p}$

are the initial estimates for a stationary series then all (complex) roots of the following polynomial will fall outside the unit circle:

$1-{\phi_1}z-{\phi_2}{z^2}-\ldots-{\phi_p}z^p\rm{.}$

${\theta_1},{\theta_2},\ldots,{\theta_q}$

are initial estimates for an invertible series then all (complex) roots of the polynomial

$1-{\theta_1}z-{\theta_2}{z^2}-\ldots-{\theta_q}z^q$

will fall outside the unit circle.

By default, the order of the lags for the autoregressive terms is $1, 2, \ldots, p$ and $1, 2, \ldots, q$ for the moving average terms. However, this cannot be overridden.

Initial values for the AR and MA coefficients can be supplied via the SetAR and SetMA methods. Otherwise, initial estimates are computed internally by the method of moments. The class computes the roots of the associated polynomials. If the AR estimates represent a nonstationary series, ARMAMaxLikelihood issues a warning message and replaces the intial AR estimates with initial values that are stationary. If the MA estimates represent a noninvertible series, a terminal error is issued and new initial values must be sought.

ARMAMaxLikelihood also validates the final estimates of the AR coefficients to ensure that they too represent a stationary series. This is done to guard against the possibility that the internal log-likelihood optimizer converged to a nonstationary solution. If nonstationary estimates are encountered, an exception will be thrown.

For model selection, the ARMA model with the minimum value for AIC might be preferred, $\mbox{AIC} = -2\ln(L)+2(p+q)$ , where L is the value of the maximum likelihood function evaluated at the parameter estimates.

ARMAMaxLikelihood can also handle white noise processes, i.e. processes.

Reference

Imsl.Stat Namespace

Other Resources

Example