ARMA Class |
Namespace: Imsl.Stat
The ARMA type exposes the following members.
Name | Description | |
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Compute |
Computes least-square estimates of parameters for an ARMA model.
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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 |
Computes forecasts and their associated probability limits for an
ARMA model.
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GetAR |
Returns the final autoregressive parameter estimates.
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GetAutoCovariance |
Returns the autocovariances of the time series z.
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GetDeviations |
Returns the deviations for each forecast used for calculating the
forecast confidence limits.
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GetForecast | Returns forecasts
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GetHashCode | Serves as a hash function for a particular type. (Inherited from Object.) | |
GetMA |
Returns the final moving average parameter estimates.
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GetNumberOfBackcasts |
Returns the number of backcasts used to calculate the AR coefficients
for the time series z.
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GetParamEstimatesCovariance |
Returns the covariances of parameter estimates.
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GetPsiWeights |
Returns the psi weights of the infinite order moving average form of the
model.
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GetResidual |
Returns the residuals at the final parameter estimate.
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GetType | Gets the Type of the current instance. (Inherited from Object.) | |
MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) | |
SetARLags |
The order of the autoregressive parameters.
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SetARMAInfo |
Sets the ARMAInfo object to previously determined values
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SetBackcasting |
Sets backcasting option.
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SetInitialAREstimates |
Sets preliminary autoregressive estimates.
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SetInitialEstimates |
Sets preliminary estimates.
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SetInitialMAEstimates |
Sets preliminary moving average estimates.
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SetMALags |
Sets the order of the moving average parameters.
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ToString | Returns a string that represents the current object. (Inherited from Object.) |
Name | Description | |
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BackwardOrigin |
The maximum backward origin.
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Center |
The center option.
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Confidence |
The confidence level for calculating confidence
limit deviations returned from GetDeviations.
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Constant |
The constant parameter estimate.
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ConvergenceTolerance |
The tolerance level used to determine convergence of the nonlinear
least-squares algorithm.
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InnovationVariance |
The variance of the random shock.
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MaxIterations |
The maximum number of iterations.
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Mean |
An update of the mean of the time series z.
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Method |
The method used to estimate the autoregressive and moving average
parameters estimates.
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NumberOfProcessors |
Perform the parallel calculations with the maximum possible number of
processors set to NumberOfProcessors.
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RelativeError |
The stopping criterion for use in the nonlinear equation solver.
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SSResidual |
The sum of squares of the random shock.
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Variance |
The variance of the time series z.
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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 |
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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).