Prais–Winsten estimation


In econometrics, Prais–Winsten estimation is a procedure meant to take care of the serial correlation of type in a linear model. Conceived by Sigbert Prais and Christopher Winsten in 1954, it is a modification of Cochrane–Orcutt estimation in the sense that it does not lose the first observation, which leads to more efficiency as a result and makes it a special case of feasible generalized least squares.

Theory

Consider the model
where is the time series of interest at time t, is a vector of coefficients, is a matrix of explanatory variables, and is the error term. The error term can be serially correlated over time: and is white noise. In addition to the Cochrane–Orcutt transformation, which is
for t = 2,3,...,T, the Prais-Winsten procedure makes a reasonable transformation for t = 1 in the following form:
Then the usual least squares estimation is done.

Estimation procedure

To do the estimation in a compact way one must look at the autocovariance function of the error term considered in the model blow:
It is easy to see that the variance–covariance matrix,, of the model is
Having , we see that,
where is a matrix of observations on the independent variable including a vector of ones, is a vector stacking the observations on the dependent variable and includes the model parameters.

Note

To see why the initial observation assumption stated by Prais–Winsten is reasonable, considering the mechanics of generalized least square estimation procedure sketched above is helpful. The inverse of can be decomposed as with
A pre-multiplication of model in a matrix notation with this matrix gives the transformed model of Prais–Winsten.

Restrictions

The error term is still restricted to be of an AR type. If is not known, a recursive procedure or grid-search may be used to make the estimation feasible. Alternatively, a full information maximum likelihood procedure that estimates all parameters simultaneously has been suggested by Beach and MacKinnon.