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
y t = α + X t β + ε t , {\displaystyle y_{t}=\alpha +X_{t}\beta +\varepsilon _{t},\,}
where y t {\displaystyle y_{t}} is the time series of interest at time t, β {\displaystyle \beta } is a vector of coefficients, X t {\displaystyle X_{t}} is a matrix of explanatory variables, and ε t {\displaystyle \varepsilon _{t}} is the error term. The error term can be serially correlated over time: ε t = ρ ε t − 1 + e t , | ρ | < 1 {\displaystyle \varepsilon _{t}=\rho \varepsilon _{t-1}+e_{t},\ |\rho |<1} and e t {\displaystyle e_{t}} is white noise. In addition to the Cochrane–Orcutt transformation, which is
y t − ρ y t − 1 = α ( 1 − ρ ) + ( X t − ρ X t − 1 ) β + e t , {\displaystyle y_{t}-\rho y_{t-1}=\alpha (1-\rho )+(X_{t}-\rho X_{t-1})\beta +e_{t},\,}
for t = 2,3,...,T, the Prais-Winsten procedure makes a reasonable transformation for t = 1 in the following form:
1 − ρ 2 y 1 = α 1 − ρ 2 + ( 1 − ρ 2 X 1 ) β + 1 − ρ 2 ε 1 . {\displaystyle {\sqrt {1-\rho ^{2}}}y_{1}=\alpha {\sqrt {1-\rho ^{2}}}+\left({\sqrt {1-\rho ^{2}}}X_{1}\right)\beta +{\sqrt {1-\rho ^{2}}}\varepsilon _{1}.\,}
Then the usual least squares estimation is done.
Estimation procedure First notice that
v a r ( ε t ) = v a r ( ρ ε t − 1 + e t ) = ρ 2 v a r ( ε t − 1 ) + v a r ( e t ) {\displaystyle \mathrm {var} (\varepsilon _{t})=\mathrm {var} (\rho \varepsilon _{t-1}+e_{t})=\rho ^{2}\mathrm {var} (\varepsilon _{t-1})+\mathrm {var} (e_{t})}
Noting that for a stationary process, variance is constant over time,
( 1 − ρ 2 ) v a r ( ε t ) = v a r ( e t ) {\displaystyle (1-\rho ^{2})\mathrm {var} (\varepsilon _{t})=\mathrm {var} (e_{t})}
and thus,
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