Prais–Winsten estimation

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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. Conceived by Sigbert Prais and Christopher Winsten in 1954,[1] 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.[2]


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.

Estimation procedure[edit]

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 matrix, , 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 generalized least square estimation procedure sketched above is helpful. The inverse of can be decomposed as with[3]

A pre-multiplication of model in a matrix notation with this matrix gives the transformed model of Prais–Winsten.


The error term is still restricted to be of an AR(1) type. If is not known, a recursive procedure may be used to make the estimation feasible. See Cochrane–Orcutt estimation.


  1. ^ Prais, S. J.; Winsten, C. B. (1954). "Trend Estimators and Serial Correlation" (PDF). Cowles Commission Discussion Paper No. 383. Chicago. 
  2. ^ Johnston, John (1972). Econometric Methods (2nd ed.). New York: McGraw-Hill. pp. 259–265. 
  3. ^ Kadiyala, Koteswara Rao (1968). "A Transformation Used to Circumvent the Problem of Autocorrelation". Econometrica. 36 (1): 93–96. JSTOR 1909605. 
  • 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.