Computes least-square estimates of parameters for an ARMA model.
For a list of all members of this type, see ARMA Members.
System.Object
Imsl.Stat.ARMA
Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.
Class ARMA
computes estimates of parameters for a nonseasonal ARMA model given a sample of observations, , for , where n = z.Length
. There are two methods, method of moments and least squares, from which to choose. The default is method of moments.
Two methods of parameter estimation, method of moments and least squares, are provided. The user can choose a method using the Method
property. If the user wishes to use the least-squares algorithm, the preliminary estimates are the method of moments estimates by default. Otherwise, the user can input initial estimates by using the SetInitialEstimates
method. The following table lists the appropriate methods and properties for both the method of moments and least-squares algorithm:
Least Squares | Both Method of Moment and Least Squares |
---|---|
Center | |
ARLags | Method |
MALags | RelativeError |
Backcasting | MaxIterations |
ConvergenceTolerance |
Mean |
SetInitialEstimates |
Mean |
Residual | AutoCovariance |
SSResidual | Variance |
ParamEstimatesCovariance |
Constant |
AR | |
MA |
Method of Moments Estimation
Suppose the time series is generated by an ARMA (p, q) model of the form
Let be the estimate of the mean of the time series , where equals the following:
The autocovariance function is estimated by
for , where K = p + q. Note that is an estimate of the sample variance.
Given the sample autocovariances, the function computes the method of moments estimates of the autoregressive parameters using the extended Yule-Walker equations as follows:
where
The overall constant is estimated by the following:
The moving average parameters are estimated based on a system of nonlinear equations given K = p + q + 1 autocovariances, , and p autoregressive parameters for .
Let . The autocovariances of the derived moving average process are estimated by the following relation:
The iterative procedure for determining the moving average parameters is based on the relation
where denotes the autocovariance function of the original process.
Let and , where
and
Then, the value of at the (i + 1)-th iteration is determined by the following:
The estimation procedure begins with the initial value
and terminates at iteration i when either is less than RelativeError
or i equals MaxIterations
. The moving average parameter estimates are obtained from the final estimate of by setting
The random shock variance is estimated by the following:
See Box and Jenkins (1976, pp. 498-500) for a description of a function that performs similar computations.
Least-squares Estimation
Suppose the time series is generated by a nonseasonal ARMA model of the form,
where B is the backward shift operator, is the mean of , and
with p autoregressive and q moving average parameters. Without loss of generality, the following is assumed:
so that the nonseasonal ARMA model is of order , where and . Note that the usual hierarchical model assumes the following:
Consider the sum-of-squares function
where
and T is the backward origin. The random shocks are assumed to be independent and identically distributed
random variables. Hence, the log-likelihood function is given by
where is a function of .
For T = 0, the log-likelihood function is conditional on the past values of both and required to initialize the model. The method of selecting these initial values usually introduces transient bias into the model (Box and Jenkins 1976, pp. 210-211). For , this dependency vanishes, and estimation problem concerns maximization of the unconditional log-likelihood function. Box and Jenkins (1976, p. 213) argue that
dominates
The parameter estimates that minimize the sum-of-squares function are called least-squares estimates. For large n, the unconditional least-squares estimates are approximately equal to the maximum likelihood-estimates.
In practice, a finite value of T will enable sufficient approximation of the unconditional sum-of-squares function. The values of needed to compute the unconditional sum of squares are computed iteratively with initial values of obtained by back forecasting. The residuals (including backcasts), estimate of random shock variance, and covariance matrix of the final parameter estimates also are computed. ARIMA parameters can be computed by using Difference
with ARMA
.
Forecasting
The Box-Jenkins forecasts and their associated probability limits for a nonseasonal ARMA model are computed given a sample of n = z.Length
, for .
Suppose the time series is generated by a nonseasonal ARMA model of the form
for , where B is the backward shift operator, is the constant, and
with p autoregressive and q moving average parameters. Without loss of generality, the following is assumed:
so that the nonseasonal ARMA model is of order , where and . Note that the usual hierarchical model assumes the following:
The Box-Jenkins forecast at origin t for lead time l of is defined in terms of the difference equation
where the following is true:
The percent probability limits for are given by
where is the percentile of the standard normal distribution
and
are the parameters of the random shock form of the difference equation. Note that the forecasts are computed for lead times at origins , where and .
The Box-Jenkins forecasts minimize the mean-square error
Also, the forecasts can be easily updated according to the following equation:
This approach and others are discussed in Chapter 5 of Box and Jenkins (1976).
Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll)
ARMA Members | Imsl.Stat Namespace | Example 1 | Example 2 | Example 3