Prais–Winsten estimation
In econometrics, Prais–Winsten estimation is a procedure meant to take care of the serial correlation of type AR(1) 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 modelwhere 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
First notice thatNoting that for a stationary process, variance is constant over time,
and thus,
Without loss of generality suppose the variance of the white noise is 1. To do the estimation in a compact way one must look at the autocovariance function of the error term considered in the model below:
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 withA pre-multiplication of model in a matrix notation with this matrix gives the transformed model of Prais–Winsten.