Package 'prais' reference manual

Title:	Prais-Winsten Estimator for AR(1) Serial Correlation
Description:	The Prais-Winsten estimator (Prais & Winsten, 1954) 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 attained.
Authors:	Franz X. Mohr [aut, cre] (ORCID: <https://orcid.org/0009-0003-8890-7781>), Michael Škvrňák [ctb] (ORCID: <https://orcid.org/0000-0003-4647-4095>)
Maintainer:	Franz X. Mohr <[email protected]>
License:	GPL (>= 2)
Version:	1.3.0.9000
Built:	2026-09-20 18:59:46 UTC
Source:	https://github.com/franzmohr/prais

Augment Data with Information from a Prais-Winsten Object

Description

Adds the fitted values and residuals of a model of class "prais" to the data it was estimated from, or the predictions for new data. It is a method for the generic augment of package broom, which has to be installed to use it.

Usage

## S3 method for class 'prais'
augment(x, data = x$model, newdata = NULL, ...)
## S3 method for class 'prais'
augment(x, data = x$model, newdata = NULL, ...)

Arguments

x

an object of class "prais", usually, a result of a call to prais_winsten.

data

a data frame with the observations the model was estimated from. Defaults to the model frame of the object, which contains the observations in the order of argument index and without the observations that were dropped because of missing values.

newdata

an optional data frame in which to look for variables with which to predict. If it is specified, the residuals are omitted from the result, because the response is not required for a prediction.

...

not used.

Details

The fitted values and residuals are on the scale of the original data, i.e. they are those of the estimated object and not those of the Prais-Winsten transformed model, which summary.prais reports. The predictions for newdata are obtained with predict.prais and do not contain a forecast of the serially correlated part of the error term.

Value

A tibble that contains data, or newdata, and the columns

.fitted

the fitted values, or the predictions for newdata.

.resid

(only if newdata is not specified) the residuals on the scale of the original data.

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")

# Add fitted values and residuals
if (requireNamespace("broom", quietly = TRUE)) {
  broom::augment(pw)
}

# 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")

# Add fitted values and residuals
if (requireNamespace("broom", quietly = TRUE)) {
  broom::augment(pw)
}

Barium

Description

Monthly microeconomic data from the U.S. chemical industry from February 1978 to December 1988 as used in Krupp and Pollard (1996) and subsequently re-used by Wooldridge (2000) as a textbook dataset. Raw data was obtained from http://fmwww.bc.edu/ec-p/data/wooldridge/barium.dta.

Usage

data("barium")
data("barium")

Format

A data frame with 131 rows and 31 variables:

chnimp: Chinese imports, bar. chl.
bchlimp: Total imports, bar. chl.
befile6: Dummy variable, which is 1 for all six months before filing.
affile6: Dummy variable, which is 1 for all six months after filing.
afdec6: Dummy variable, which is 1 for all six months after decision
befile12: Dummy variable, which is 1 for all twelve months before filing.
affile12: Dummy variable, which is 1 for all twelve months after filing.
afdec12: Dummy variable, which is 1 for all twelve months after decision.
chempi: Chemical production index.
gas: Gasoline production.
rtwex: Exchange rate index.
spr: Dummy variable, which is 1 for spring months.
sum: Dummy variable, which is 1 for summer months.
fall: Dummy variable, which is 1 for fall months.
lchnimp: Log of chnimp.
lgas: Log of gas.
lrtwex: Log of rtwex.
lchempi: Log of chempi.
t: Time trend.
feb: Dummy variable for February.
mar: Dummy variable for March.
apr: Dummy variable for April.
may: Dummy variable for May.
jun: Dummy variable for June.
jul: Dummy variable for July.
aug: Dummy variable for August.
sep: Dummy variable for September.
oct: Dummy variable for October.
nov: Dummy variable for November.
dec: Dummy variable for December.
percchn: Percent of imports from China.

References

Krupp, C.M., & Pollard, P.S., (1996). Market responses to antidumping laws: Some evidence from the U.S. chemical industry. Canadian Journal of Economics 29(1), 199–227. doi:10.2307/136159

Wooldridge, J., (2000). Instructional Stata datasets for econometrics. [barium]. Boston College Department of Economics.

Confidence Intervals for the Coefficients of the Prais-Winsten Estimator

Description

Computes confidence intervals for the coefficients of a model of class "prais".

Usage

## S3 method for class 'prais'
confint(object, parm, level = 0.95, ...)
## S3 method for class 'prais'
confint(object, parm, level = 0.95, ...)

Arguments

object

an object of class "prais", usually, a result of a call to prais_winsten.

parm

a specification of the coefficients for which the intervals are required. Either a vector of names or a vector of positions. If it is omitted, the intervals of all coefficients are computed.

level

the confidence level of the intervals. Defaults to .95.

...

not used.

Details

The intervals are based on the t distribution with the residual degrees of freedom of the model and the standard errors of summary.prais, so they agree with the p-values that the summary reports. Without this method the intervals would be obtained by confint.default, which uses the quantiles of the normal distribution and is therefore too narrow in small samples.

Coefficients of linearly dependent variables are NA and are omitted, as in vcov.prais.

Value

A matrix with a row per coefficient and the lower and upper bound of the interval in its columns, which are named after the corresponding quantiles.

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")

# Confidence intervals
confint(pw)
confint(pw, parm = "x", level = .9)

# 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")

# Confidence intervals
confint(pw)
confint(pw, parm = "x", level = .9)

Glance at a Prais-Winsten Object

Description

Summarises a model of class "prais" in a tidy data frame with a single row of goodness-of-fit statistics. It is a method for the generic glance of package broom, which has to be installed to use it.

Usage

## S3 method for class 'prais'
glance(x, ...)
## S3 method for class 'prais'
glance(x, ...)

Arguments

x

an object of class "prais", usually, a result of a call to prais_winsten.

...

not used.

Details

The statistics are those of summary.prais. In addition to the columns that are reported for a linear model, the result contains the AR(1) coefficient and the Durbin-Watson statistics. Both are NA if the model was estimated with panelwise = TRUE, because there is one coefficient per panel in that case. The full set of estimates is available as rho of the estimated object. Statistics that are not defined for a model, such as the F statistic of a model without regressors, are NA.

Value

A tibble with a single row and the columns

r.squared

R^2, the fraction of variance explained by the model.

adj.r.squared

R^2 adjusted for the number of coefficients.

sigma

the square root of the estimated variance of the random error.

statistic

the F statistic of the model.

p.value

the p-value of the F statistic.

df

the numerator degrees of freedom of the F statistic.

df.residual

the residual degrees of freedom.

nobs

the number of observations that were used for the estimation.

rho

the final estimate of the AR(1) coefficient $\rho$ .

dw.original

the Durbin-Watson statistic of the original linear model.

dw.transformed

the Durbin-Watson statistic of the Prais-Winsten estimator.

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")

# Goodness of fit
if (requireNamespace("broom", quietly = TRUE)) {
  broom::glance(pw)
}

# 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")

# Goodness of fit
if (requireNamespace("broom", quietly = TRUE)) {
  broom::glance(pw)
}

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 = NULL,
  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), ...)
prais_winsten(
  formula,
  data,
  index = NULL,
  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. Alternatively, a fitted panel model of class "plm", from which the formula, the data and the index are taken. See 'Details'.

data

a data frame containing the variables in the model. If panel data is used, it must also contain the ID and time variables. Must not be specified if formula is a model of class "plm".

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. Must not be specified if formula is a model of class "plm". If it is NULL, which is the default, the observations are taken in the order in which they appear in data.

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 formula is a fitted panel model of class "plm", the formula, the data and the index of the panel are taken from it, and the arguments data and index must not be specified. Since the Prais-Winsten estimator obtains all its estimates by ordinary least squares, only models that were estimated with model = "pooling" are supported. The within and the random effects estimator apply a transformation of their own, which does not leave the AR(1) structure of the errors intact. Fixed effects can be estimated by adding the dummy variables to the formula, as in y ~ x + factor(id). The data are obtained by evaluating the data argument of the call of the model, so the object it refers to must still be available. The model frame of the fitted model cannot be used instead, because its columns are named after the terms of the formula, so that a term such as factor(id) or log(x) could not be evaluated again.

Observations with missing values in the variables of formula are dropped before estimation, as in lm. The gap that a dropped observation leaves in the time variable is treated like any other gap, which is described below.

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 orders the observations and gives the distance between them. Gaps are allowed: two observations of a panel that lie $k$ periods apart have the correlation $\rho^k$ under an AR(1) process, so such an observation is transformed to

$(1 + \rho^2 + ... + \rho^{2(k - 1)})^{-1 / 2} (x_t - \rho^k x_{t - k}),$

which is the generalised least squares transformation evaluated at the periods that were observed. For consecutive periods this is the usual $x_t - \rho x_{t - 1}$ , so equally spaced data are unaffected. As a gap grows, $\rho^k$ approaches zero and the factor approaches $(1 - \rho^2)^{(1 / 2)}$ , so that the observation after a long gap is transformed like the first observation of a panel.

The distances are counted in steps of the smallest length that divides all differences of the time variable, so that a variable measured in, say, quarters or days is handled without further arguments, and so that the estimates do not depend on the unit the periods are expressed in. Date and POSIXct variables are supported. If the periods are not multiples of a common step, a warning is issued and the observations that surround a gap are treated as if they were consecutive. The same applies without a warning if the time variable is not numeric, such as a character vector, which only orders the observations and carries no distance between them.

The estimate of $\rho$ itself is obtained from the residuals of an observation and its predecessor, whether or not a gap lies between them. Since the correlation across a gap is $\rho^k$ rather than $\rho$ , the estimate is attenuated towards zero if a large share of the observations follows a gap.

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.

qr

the QR decomposition of the model matrix of the Prais-Winsten transformed model, as lm.fit returned it. It is the decomposition the covariance matrices are obtained from, and not one of the original data.

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.

timeid

the periods of the observations, counted in whole steps from the first period of the panel they belong to. NULL if no index was given, or if the periods are not multiples of a common step.

x

the model matrix after the Prais-Winsten transformation. Only added if the data are a single time series.

y

the response after the Prais-Winsten transformation. Only added if the data are a single time series.

Diagnostic tests

The tests of package lmtest, such as dwtest, bgtest and bptest, do not use the residuals of the model they are given. They take the model matrix and the response from the components x and y of the object and re-estimate the model by ordinary least squares. Since those components hold the transformed data, the tests describe the estimated model and not the original one. The residuals they obtain are the residuals of summary.prais, so that dwtest reports the Durbin-Watson statistic of the transformed model together with a p-value.

For panel data the components are not added, because the tests difference the residuals over all observations at once, which mixes the last observation of a panel with the first observation of the next one. The tests then fall back on the model frame, which holds the original data, and their results do not refer to the estimated model. summary.prais reports a Durbin-Watson statistic that respects the panels instead.

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)

# A fitted pooled panel model can be used instead of formula, data and index

if (requireNamespace("plm", quietly = TRUE)) {
  data("Grunfeld", package = "plm")
  pooled <- plm::plm(inv ~ value + capital, data = Grunfeld,
                     index = c("firm", "year"), model = "pooling")
  summary(prais_winsten(pooled))
}


# 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)

# A fitted pooled panel model can be used instead of formula, data and index

if (requireNamespace("plm", quietly = TRUE)) {
  data("Grunfeld", package = "plm")
  pooled <- plm::plm(inv ~ value + capital, data = Grunfeld,
                     index = c("firm", "year"), model = "pooling")
  summary(prais_winsten(pooled))
}

Predict Method for Objects of Class prais

Description

Predicted values and their standard errors based on Prais-Winsten object.

Usage

## S3 method for class 'prais'
predict(
  object,
  newdata = NULL,
  se.fit = FALSE,
  interval = c("none", "confidence"),
  level = 0.95,
  ...
)
## S3 method for class 'prais'
predict(
  object,
  newdata = NULL,
  se.fit = FALSE,
  interval = c("none", "confidence"),
  level = 0.95,
  ...
)

Arguments

object

an object of class "prais", usually, a result of a call to prais_winsten.

newdata

an optional data frame in which to look for variables with which to predict. It must contain the variables that appear in formula, which do not have to be transformed beforehand. If omitted, the fitted values are used.

se.fit

logical. If TRUE, the standard errors of the predictions are returned as well. Defaults to FALSE.

interval

a character specifying the type of interval that is added to the predictions. Either "none", which is the default, or "confidence" for the confidence interval of the conditional mean.

level

the confidence level of the interval. Defaults to .95.

...

further arguments passed to or from other methods.

Details

The predictions are the conditional mean of the model, i.e. the product of the regressors and the coefficients. The AR(1) structure of the errors is not used, so the result does not contain the forecast of the serially correlated part of the error term.

The standard errors and the intervals describe that conditional mean. They are obtained from the covariance matrix of vcov.prais, which comes from the Prais-Winsten transformed model, and the quantiles of the t distribution with the residual degrees of freedom of the model, so they agree with the standard errors of summary.prais and the intervals of confint.prais.

Prediction intervals for an individual observation are not available, because they would require an assumption about the error of the predicted period, whose variance depends on the serial correlation and, for a genuine forecast, on the distance to the last observed period. Neither is used by the predictions, as described above.

Value

If se.fit is FALSE and interval is "none", a vector of predictions. If an interval is requested, a matrix with the columns "fit", "lwr" and "upr". If se.fit is TRUE, a list containing the following elements:

fit

a vector of predictions or, if an interval was requested, the matrix that is described above.

se.fit

a vector of the standard errors of the predicted means.

df

the residual degrees of freedom of the model.

residual.scale

the square root of the estimated variance of the random error, as in summary.prais.

References

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

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")

# Predict
fcst <- predict(pw)

# Predictions with standard errors and confidence intervals
newdata <- data.frame("x" = c(25, 30, 35))
predict(pw, newdata = newdata, se.fit = TRUE)
predict(pw, newdata = newdata, interval = "confidence")

# 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")

# Predict
fcst <- predict(pw)

# Predictions with standard errors and confidence intervals
newdata <- data.frame("x" = c(25, 30, 35))
predict(pw, newdata = newdata, se.fit = TRUE)
predict(pw, newdata = newdata, interval = "confidence")

Summarising the Prais-Winsten Estimator

Description

Summary method for class "prais".

Usage

## S3 method for class 'prais'
summary(object, ...)

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

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

Arguments

object

an object of class "prais", usually, a result of a call to prais_winsten.

...

further arguments passed to or from other methods.

x

an object of class "summary.prais", usually, a result of a call to summary.prais.

digits

the number of significant digits to use when printing.

signif.stars

logical. If TRUE, 'significance stars' are printed for each coefficient.

Value

summary.prais returns a list of class "summary.prais", which contains the following components:

call

the matched call.

residuals

the residuals of the Prais-Winsten transformed model, i.e. the residuals the reported standard errors are based on. The residuals on the scale of the original data are available as residuals of the estimated object.

coefficients

a named vector of coefficients.

rho

the values of the AR(1) coefficient $\rho$ from all iterations.

sigma

the square root of the estimated variance of the random error.

df

degrees of freedom, a 3-vector (p, n-p, p*), the first being the number of non-aliased coefficients, the last being the total number of coefficients.

r.squared

R^2, the 'fraction of variance explained by the model',

$R^2 = 1 - \frac{\sum {(y_i - \hat{y}_i)^2}}{\sum {(y_i - \overline{y})^2}},$

where $\overline{y}$ is the mean of $y_i$ for $y_i = 1, ..., N$ if there is an intercept and zero otherwise.

adj.r.squared

the above R^2 statistic 'adjusted', penalising for higher p.

fstatistic

(for models including non-intercept terms) a 3-vector with the value of the F-statistic with its numerator and denominator degrees of freedom.

cov.unscaled

a $p \times p$ matrix of (unscaled) covariances of the coef[j], j=1, ..., p.

dw

a named 2-vector with the Durbin-Watson statistic of the original linear model and the Prais-Winsten estimator.

terms

the terms object used.

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")

# The coefficient table, the AR(1) coefficient and the Durbin-Watson
# statistics of the original and of the transformed model
summary(pw)

# The standard errors are those of 'vcov'
all.equal(sqrt(diag(vcov(pw))),
          summary(pw)$coefficients[, "Std. Error"])

# 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")

# The coefficient table, the AR(1) coefficient and the Durbin-Watson
# statistics of the original and of the transformed model
summary(pw)

# The standard errors are those of 'vcov'
all.equal(sqrt(diag(vcov(pw))),
          summary(pw)$coefficients[, "Std. Error"])

Tidy a Prais-Winsten Object

Description

Summarises the coefficients of a model of class "prais" in a tidy data frame with one row per coefficient. It is a method for the generic tidy of package broom, which has to be installed to use it.

Usage

## S3 method for class 'prais'
tidy(x, conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'prais'
tidy(x, conf.int = FALSE, conf.level = 0.95, ...)

Arguments

x

an object of class "prais", usually, a result of a call to prais_winsten.

conf.int

logical. If TRUE, confidence intervals are added to the result. Defaults to FALSE.

conf.level

the confidence level of the intervals. Defaults to .95.

...

not used.

Details

The estimates, standard errors, test statistics and p-values are those of summary.prais. The confidence intervals are those of confint.prais, which are based on the t distribution with the residual degrees of freedom of the model, so they agree with the reported p-values. Coefficients of linearly dependent variables are NA and are omitted, as in summary.prais.

Value

A tibble with one row per coefficient and the columns

term

the name of the coefficient.

estimate

the estimated value of the coefficient.

std.error

the standard error of the estimate.

statistic

the t statistic of the estimate.

p.value

the two-sided p-value of the t statistic.

conf.low

(if conf.int = TRUE) the lower bound of the confidence interval.

conf.high

(if conf.int = TRUE) the upper bound of the confidence interval.

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")

# Tidy the coefficients
if (requireNamespace("broom", quietly = TRUE)) {
  broom::tidy(pw, conf.int = TRUE)
}

# 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")

# Tidy the coefficients
if (requireNamespace("broom", quietly = TRUE)) {
  broom::tidy(pw, conf.int = TRUE)
}

Covariance Matrix of the Coefficients of the Prais-Winsten Estimator

Description

Returns the estimated covariance matrix of the coefficients of a model of class "prais".

Usage

## S3 method for class 'prais'
vcov(object, ...)
## S3 method for class 'prais'
vcov(object, ...)

Arguments

object

an object of class "prais", usually, a result of a call to prais_winsten.

...

not used.

Details

The covariance matrix is obtained from the Prais-Winsten transformed model. It is the matrix the standard errors of summary.prais are based on and is identical to vcovHC(object, type = "const"). Robust alternatives are available in vcovHC.prais and vcovPC.prais.

Coefficients of linearly dependent variables are NA and are omitted, as in the covariance matrix of an object of class "lm".

Value

An object of class "matrix" containing the estimate of the covariance matrix of the coefficients.

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")

# Covariance matrix and confidence intervals
vcov(pw)
confint(pw)

# 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")

# Covariance matrix and confidence intervals
vcov(pw)
confint(pw)

Semirobust Covariance Matrix Estimators

Description

Semirobust covariance matrix estimators for models of class "prais".

Usage

## S3 method for class 'prais'
vcovHC(x, type = c("const", "HC1", "HC0"), ...)
## S3 method for class 'prais'
vcovHC(x, type = c("const", "HC1", "HC0"), ...)

Arguments

x

an object of class "prais", usually, the result of a call to prais_winsten.

type

a character string specifying the estimation type.

...

not used.

Details

vcovHC is a function for estimating a robust covariance matrix of parameters for the Prais-Winsten estimator. The weighting schemes specified by type are analogous to those in vcovHC in package sandwich with the caveat that only "const", "HC0" and "HC1" are available.

Value

An object of class "matrix" containing the estimate of the asymptotic covariance matrix of coefficients.

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")

# The covariance matrix of the transformed model, which 'summary' reports
vcovHC(pw, type = "const")

# Heteroskedasticity consistent alternatives
vcovHC(pw, type = "HC0")
sqrt(diag(vcovHC(pw, type = "HC1")))

# 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")

# The covariance matrix of the transformed model, which 'summary' reports
vcovHC(pw, type = "const")

# Heteroskedasticity consistent alternatives
vcovHC(pw, type = "HC0")
sqrt(diag(vcovHC(pw, type = "HC1")))

Extract Panel-Corrected Variance Covariance Matrix

Description

Panel-corrected covariance matrix estimators for models of class "prais".

Usage

## S3 method for class 'prais'
vcovPC(x, pairwise = FALSE, ...)
## S3 method for class 'prais'
vcovPC(x, pairwise = FALSE, ...)

Arguments

x

an object of class "prais", usually, the result of a call to prais_winsten.

pairwise

logical. If FALSE (default), only those residuals from periods that are common to all panels are used to computed the covariances. If TRUE all observations that can be matched by period between two panels are used.

...

not used.

Details

vcovPC is a function for estimating a panel-corrected covariance matrix of parameters for the Prais-Winsten estimator.

Value

An object of class "matrix".

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.

Examples

# Generate an artificial panel with AR(1) errors
set.seed(1234567)
n_id <- 8
n_time <- 25
rho <- .6

panel <- do.call(rbind, lapply(seq_len(n_id), function(i) {
  u <- rnorm(n_time, 0, 2)
  for (j in 2:n_time) {
    u[j] <- u[j] + rho * u[j - 1]
  }
  x <- rnorm(n_time, 10, 3)
  data.frame(id = i, year = seq_len(n_time), x = x, y = 5 + 2 * x + u)
}))

# Estimate
pw <- prais_winsten(y ~ x, data = panel, index = c("id", "year"))

# Panel-corrected standard errors
sqrt(diag(vcovPC(pw)))

# Match the observations of two panels by period instead of using only the
# periods that are common to all panels
sqrt(diag(vcovPC(pw, pairwise = TRUE)))

# Generate an artificial panel with AR(1) errors
set.seed(1234567)
n_id <- 8
n_time <- 25
rho <- .6

panel <- do.call(rbind, lapply(seq_len(n_id), function(i) {
  u <- rnorm(n_time, 0, 2)
  for (j in 2:n_time) {
    u[j] <- u[j] + rho * u[j - 1]
  }
  x <- rnorm(n_time, 10, 3)
  data.frame(id = i, year = seq_len(n_time), x = x, y = 5 + 2 * x + u)
}))

# Estimate
pw <- prais_winsten(y ~ x, data = panel, index = c("id", "year"))

# Panel-corrected standard errors
sqrt(diag(vcovPC(pw)))

# Match the observations of two panels by period instead of using only the
# periods that are common to all panels
sqrt(diag(vcovPC(pw, pairwise = TRUE)))

Package 'prais'

Help Index

Augment Data with Information from a Prais-Winsten Object

Description

Usage

Arguments

Details

Value

See Also

Examples

Barium

Description

Usage

Format

References

Confidence Intervals for the Coefficients of the Prais-Winsten Estimator

Description

Usage

Arguments

Details

Value

See Also

Examples

Glance at a Prais-Winsten Object

Description

Usage

Arguments

Details

Value

See Also

Examples

Prais-Winsten Estimator for AR(1) Serial Correlation

Description

Usage

Arguments

Details

Value

Diagnostic tests

References

Examples

Predict Method for Objects of Class prais

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Summarising the Prais-Winsten Estimator

Description

Usage

Arguments

Value

Examples

Tidy a Prais-Winsten Object

Description

Usage

Arguments

Details

Value

See Also

Examples

Covariance Matrix of the Coefficients of the Prais-Winsten Estimator

Description

Usage

Arguments

Details

Value

See Also

Examples

Semirobust Covariance Matrix Estimators

Description

Usage

Arguments

Details

Value

See Also

Examples

Extract Panel-Corrected Variance Covariance Matrix