Computes least‑squares estimates of parameters for a nonseasonal ARMA model.

Routine NSLSE computes least‑squares estimates of parameters for a nonseasonal ARMA model given a sample of n = NOBS observations {Wt} for t = 1, 2, …, n.

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

where B is the backward shift operator, μ is the mean of Wt,

with p = NPAR and q = NPMA. Without loss of generality, we assume

so that the nonseasonal ARMA model is of order (pʹ, qʹ) where pʹ = lɸ(p) and qʹ = lθ(q). Note that the usual hierarchal model assumes

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,ɸ,θ) is a function of μ, ɸ, and θ.

For T = 0, the log‑likelihood function is conditional on the past values of both Wt 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, pages 210–211). For T = ∞, this dependency vanishes, and the estimation problem concerns maximization of the unconditional log‑likelihood function. Box and Jenkins (1976, page 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 Wt obtained by back‑forecasting. The residuals (including backcasts), estimate of random shock variance, and covariance matrix of the final parameter estimates are also computed. Note that application of an appropriate transformation using routine BCTR followed by differencing using routine DIFF allows for fitting of nonseasonal ARIMA models. The algorithm for nonseasonal ARIMA models is developed in Chapter 7 of Box and Jenkins (1976). The extension to multiplicative seasonal ARIMA models is given in Box and Jenkins (1976, pages 500–504).

Consider the Wölfer Sunspot Data (Anderson 1971, page 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. Routine NSPE is first invoked to compute preliminary estimates for an ARMA(2, 1) model. Then, NSLSE is invoked with the preliminary estimates as input in order to compute the least‑squares estimates

for the ARMA(2, 1) model

where the errors At are independently distributed each normal with mean zero and variance

Note at the end of the output a warning error appears. Most of the time this error message can be ignored. There are three general reasons this error can occur.

Examination of the history of iterations using IPRINT = 2 and trying a smaller value for TOLSS can help you determine what caused the error message.