The AUTOREG Procedure

Autocorrelation correction in regression analysis has a long history, and various approaches have been suggested. Moreover, the same method may be referred to by different names.

Pioneering work in the field was done by Cochrane and Orcutt (1949). The Cochrane-Orcutt method refers to a more primitive version of the Yule-Walker method that drops the first observation. The Cochrane-Orcutt method is like the Yule-Walker method for first-order autoregression, except that the Yule-Walker method retains information from the first observation. The iterative Cochrane-Orcutt method is also in use.

The Yule-Walker method used by PROC AUTOREG is also known by other names. Harvey (1981) refers to the Yule-Walker method as the two-step full transform method. The Yule-Walker method can be considered as generalized least squares using the OLS residuals to estimate the covariances across observations, and Judge et al. (1985) use the term estimated generalized least squares (EGLS) for this method. For a first-order AR process, the Yule-Walker estimates are often termed Prais-Winsten estimates (Prais and Winsten; 1954). There are variations to these methods that use different estimators of the autocorrelations or the autoregressive parameters.

The unconditional least squares (ULS) method, which minimizes the error sum of squares for all observations, is referred to as the nonlinear least squares (NLS) method by Spitzer (1979).

The Hildreth-Lu method (Hildreth and Lu; 1960) uses nonlinear least squares to jointly estimate the parameters with an AR(1) model, but it omits the first transformed residual from the sum of squares. Thus, the Hildreth-Lu method is a more primitive version of the ULS method supported by PROC AUTOREG in the same way Cochrane-Orcutt is a more primitive version of Yule-Walker.

The maximum likelihood method is also widely cited in the literature. Although the maximum likelihood method is well defined, some early literature refers to estimators that are called maximum likelihood but are not full unconditional maximum likelihood estimates. The AUTOREG procedure produces full unconditional maximum likelihood estimates.

Harvey (1981) and Judge et al. (1985) summarize the literature on various estimators for the autoregressive error model. Although asymptotically efficient, the various methods have different small sample properties. Several Monte Carlo experiments have been conducted, although usually for the AR(1) model.

Harvey and McAvinchey (1978) found that for a one-variable model, when the independent variable is trending, methods similar to Cochrane-Orcutt are inefficient in estimating the structural parameter. This is not surprising since a pure trend model is well modeled by an autoregressive process with a parameter close to 1.

Harvey and McAvinchey (1978) also made the following conclusions:

• The Yule-Walker method appears to be about as efficient as the maximum likelihood method. Although Spitzer (1979) recommended ML and NLS, the Yule-Walker method (labeled Prais-Winsten) did as well or better in estimating the structural parameter in Spitzer’s Monte Carlo study (table A2 in their article) when the autoregressive parameter was not too large. Maximum likelihood tends to do better when the autoregressive parameter is large.

• For small samples, it is important to use a full transformation (Yule-Walker) rather than the Cochrane-Orcutt method, which loses the first observation. This was also demonstrated by Maeshiro (1976), Chipman (1979), and Park and Mitchell (1980).

• For large samples (Harvey and McAvinchey used 100), losing the first few observations does not make much difference.