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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.
- Judge, George G.; Griffiths, William E.; Hill, R. Carter; Lee, Tsoung-Chao (1980). The Theory and Practice of Econometrics. New York: Wiley. pp. 180–183. ISBN 0-471-05938-2.
- Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 302–320. ISBN 0-02-365070-2.
- Prais, S. J.; Winsten, C. B. (1954). "Trend Estimators and Serial Correlation". Cowles Commission Discussion Paper No. 383 (Chicago).
- Wooldridge, J. (2008). Introductory Econometrics: A Modern Approach (4th ed.). South-Western. p. 435. ISBN 0-324-66054-5.