Computes least-square estimates of parameters for an ARMA model.

The type double function is imsls_d_arma.

Pointer to an array of length 1 + p + q with the estimated constant, AR, and MA parameters. If IMSLS_NO_CONSTANT is specified, the 0-th element of this array is 0.0.

Function imsls_f_arma computes estimates of parameters for a nonseasonal ARMA model given a sample of observations, {Wt}, for t = 1, 2, ..., n, where n = n_observations. 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 the method of moments algorithm with the optional argument IMSLS_METHOD_OF_MOMENTS. The least-squares algorithm is used if the user specifies IMSLS_LEAST_SQUARES. 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 specifying optional argument IMSLS_INITIAL_ESTIMATES. The following table lists the appropriate optional arguments for both the method of moments and least-squares algorithm:

The method of moments assumes that the stationary time series {Zt} can be described by a nonseasonal ARMA model of the form

where B is the backward shift operator, μ is the mean of Zt, and

with p autoregressive and q moving average parameters.

Function imsls_f_arma first orders the AR and MA lags in strictly increasing order. Without loss of generality, it therefore can be assumed that

so that the nonseasonal ARMA model is of order (p’, q’), where p’ = lθ (p) and q’ = lθ (q).

In order to keep the notation simple, the following explanations assume the standard ARMA (p, q) model with lφ(i) = i, i = 1,...,p, and lθ (i) = i, i = 1,...,q.

Let μ = z_mean be the estimate of the mean μ of the time series{Zt}, where μ equals the following:

The autocovariance function is estimated by

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 θ0 is estimated by the following:

The moving average parameters are estimated based on a system of nonlinear equations given K = p + q + 1 autocovariances, σ(k) for k = 1, ..., K, and p autoregressive parameters φi for i = 1, ..., p.

Let Zʹt = φ(B)Zt. The autocovariances of the derived moving average process Zʹt = θ(B)At are estimated by the following relation:

The iterative procedure for determining the moving average parameters is based on the relation

where σ(k) denotes the autocovariance function of the original Zt process.

and

The nonlinear system

where

is a square matrix of order q + 1 with entries

The estimation procedure begins with the initial value

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.

Suppose the time series {Zt} is generated by a nonseasonal ARMA model of the form,

where B is the backward shift operator, μ is the mean of Zt, 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 (p’, q’), where p’ = lφ (p) and q’ = lθ (q). 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 {At} are assumed to be independent and identically distributed

random variables. Hence, the log-likelihood function is given by

where f (μ, φ, θ) is a function of μ, φ, and θ.

For T = 0, the log-likelihood function is conditional on the past values of both Zt and At 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 T = ∞, 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 [AT] needed to compute the unconditional sum of squares are computed iteratively with initial values of Zt 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 imsls_f_difference with imsls_f_arma.

Consider the Wolfer Sunspot Data (Anderson 1971, p. 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. The method of moments estimates

for the ARMA(2, 1) model

The data for this example are the same as that for the initial example. Preliminary method of moments estimates are computed by default, and the method of least squares is used to find the final estimates.