prais_winsten: Prais-Winsten Estimator for AR(1) Serial Correlation

In prais: Prais-Winsten Estimator for AR(1) Serial Correlation

Prais-Winsten Estimator for AR(1) Serial Correlation

Description

The Prais-Winsten estimator takes into account AR(1) serial correlation of the errors in a linear regression model. The procedure recursively estimates the coefficients and the error autocorrelation of the specified model until sufficient convergence of the AR(1) coefficient is reached. All estimates are obtained by OLS.

Usage

prais_winsten(
  formula,
  data,
  index,
  max_iter = 50L,
  tol = 1e-06,
  twostep = FALSE,
  panelwise = FALSE,
  rhoweight = c("none", "T", "T1"),
  ...
)

## S3 method for class 'prais'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

formula	an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
data	a data frame containing the variables in the model. If panel data is used, it must also contain the ID and time variables.
index	a character vector specifying the ID and time variables. If only one variable is provided, it is assumed to be the time variable and the data will be reordered accordingly. The specified variables must not contain NA values.
max_iter	integer specifying the maximum number of allowed iterations. Default is 50.
tol	numeric specifying the maximum absolute difference between the estimator of rho in the current and the previous iteration that has to be attained to reach convergence. Default is 1e-6.
twostep	logical. If TRUE, the estimation will stop after the first iteration.
panelwise	logical. If TRUE, \rho will be calculated for each panel separately. Default is FALSE. Only used for panel data. See 'Details'.
rhoweight	character specifying how \rho should be calculated if panelwise = TRUE. See 'Details'.
...	arguments passed to lm.
x	an object of class "prais", usually, a result of a call to prais_winsten.
digits	the number of significant digits to use when printing.

Details

Observations with missing values in the variables of formula are dropped before estimation, as in lm. Since the Prais-Winsten transformation uses the previous observation of a panel, the observations that surround a dropped one are treated as if they were consecutive.

Estimates of \rho are bounded to the interval [-1, 1], because the transformation of the first observation of a panel requires (1 - \rho^2)^{(1 / 2)} to be real. If \rho attains one of those bounds, the first observation of each panel becomes zero and does not contribute to the estimates, so that the estimator effectively becomes the Cochrane-Orcutt estimator.

The time variable is only used to order the observations. If it is not equally spaced, a warning is issued, because the observations that surround a gap are treated as if they were consecutive.

If panelwise = TRUE, twostep = FALSE and rhoweight = "none", each individual estimate of rho is re-estimated until convergence is achieved for all coefficients.

If panelwise = TRUE, the calculation of \rho can be further specified in argument rhoweight. If rhoweight = "none", \rho is assumed to be panel-specific. If rhoweight = "T", \rho is calculated as a weighted mean of panel-specific estimates, where the number of available observations per panel, i.e. T_i, is used as weight. If rhoweight = "T1", \rho is calculated as a weighted mean of panel-specific estimates, where the number of available observations per panel minus one, i.e. T_i - 1, is used as weight.

Value

A list of class "prais" containing the following components:

coefficients	a named vector of coefficients.
rho	the values of the AR(1) coefficient \rho from all iterations.
residuals	the residuals, that is the response minus the fitted values.
fitted.values	the fitted mean values.
rank	the numeric rank of the fitted linear model.
df.residual	the residual degrees of freedom.
call	the matched call.
terms	the terms object used.
model	the original model frame, i.e., before the Prais-Winsten transformation.
xlevels	a record of the levels of the factors used in fitting.
contrasts	the contrasts used, if the model contains factors.
index	a character specifying the ID and time variables. Only added if panel data were used.

References

Beck, N. L. and Katz, J. N. (1995): What to do (and not to do) with time-series cross-section data. American Political Science Review 89, 634-647.

Prais, S. J. and Winsten, C. B. (1954): Trend Estimators and Serial Correlation. Cowles Commission Discussion Paper, 383 (Chicago).

Wooldridge, J. M. (2013): Introductory Econometrics. A Modern Approach. 5th ed. Mason, OH: South-Western Cengage Learning.

Examples

# Generate an artificial sample
set.seed(1234567)
n <- 100
x <- sample(20:40, n, replace = TRUE)
rho <- .5

# AR(1) errors
u <- rnorm(n, 0, 5)
for (i in 2:n) {
  u[i] <- u[i] + rho * u[i - 1]
}
pw_sample <- data.frame("x" = x, "y" = 10 + 1.5 * x + u, "time" = 1:n)

# Estimate
pw <- prais_winsten(y ~ x, data = pw_sample, index = "time")
summary(pw)

prais documentation built on Sept. 19, 2026, 1:06 a.m.