In econometrics, Prais–Winsten estimation is a procedure meant to take care of the serial correlation of type AR(1) in a linear model. It is a modification of Cochrane–Orcutt estimation in the sense that it does not lose the first observation and leads to more efficiency as a result.
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 a white noise. In addition to the Cochrane–Orcutt procedure transformation, which is
for t=2,3,...,T, Prais-Winsten procedure makes a reasonable transformation for t=1 in the following form
Then the usual least squares estimation is done.
To do the estimation in a compact way it is directive to look at the auto-covariance function of the error term considered in the model above:
Now is easy to see that the variance-covariance, , of the model is
Now having (or an estimate of it), we see that,
where is a matrix of observations on the independent variable (Xt, t = 1, 2, ..., T) including a vector of ones, is a vector stacking the observations on the dependent variable (Xt, t = 1, 2, ..., T) and includes the model parameters.
To see why the initial observation assumption stated by Prais-Winsten (1954) is reasonable, considering the mechanics of general 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.
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