prais_winsten {prais}

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.

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

If \rho takes a value above 1 during the estimation process, the Prais-Winsten transformation cannot be applied to the first observations, because (1 - \rho^2)^{(1 / 2)} is not real. These observations are dropped during the respective iteration and the estimator effectively becomes the Cochrane-Orcutt estimator.

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.

index

a character specifying the ID and time variables.

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 Cengage.

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)