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 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 First notice that Noting 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