| linearHypothesis.systemfit {systemfit} | R Documentation |
Test Linear Hypothesis
Description
Testing linear hypothesis on the coefficients of a system of equations by an F-test or Wald-test.
Usage
## S3 method for class 'systemfit'
linearHypothesis( model,
hypothesis.matrix, rhs = NULL, test = c( "FT", "F", "Chisq" ),
vcov. = NULL, ... )
Arguments
model |
a fitted object of type |
hypothesis.matrix |
matrix (or vector) giving linear combinations
of coefficients by rows,
or a character vector giving the hypothesis in symbolic form
(see documentation of |
rhs |
optional right-hand-side vector for hypothesis, with as many entries as rows in the hypothesis matrix; if omitted, it defaults to a vector of zeroes. |
test |
character string, " |
vcov. |
a function for estimating the covariance matrix
of the regression coefficients or an estimated covariance matrix
(function |
... |
further arguments passed to
|
Details
Theil's F statistic for sytems of equations is
F = \frac{
( R \hat{b} - q )'
( R ( X' ( \Sigma \otimes I )^{-1} X )^{-1} R' )^{-1}
( R \hat{b} - q ) /
j
}{
\hat{e}' ( \Sigma \otimes I )^{-1} \hat{e} /
( M \cdot T - K )
}
where j is the number of restrictions,
M is the number of equations,
T is the number of observations per equation,
K is the total number of estimated coefficients, and
\Sigma is the estimated residual covariance matrix.
Under the null hypothesis, F has an approximate F distribution
with j and M \cdot T - K degrees of freedom
(Theil, 1971, p. 314).
The F statistic for a Wald test is
F = \frac{
( R \hat{b} - q )'
( R \, \widehat{Cov} [ \hat{b} ] R' )^{-1}
( R \hat{b} - q )
}{
j
}
Under the null hypothesis, F has an approximate F distribution
with j and M \cdot T - K degrees of freedom
(Greene, 2003, p. 346).
The \chi^2 statistic for a Wald test is
W =
( R \hat{b} - q )'
( R \widehat{Cov} [ \hat{b} ] R' )^{-1}
( R \hat{b} - q )
Asymptotically, W has a \chi^2
distribution with j degrees of freedom
under the null hypothesis
(Greene, 2003, p. 347).
Value
An object of class anova,
which contains the residual degrees of freedom in the model,
the difference in degrees of freedom,
the test statistic (either F or Wald/Chisq)
and the corresponding p value.
See documentation of linearHypothesis
in package "car".
Author(s)
Arne Henningsen arne.henningsen@googlemail.com
References
Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.
Theil, Henri (1971) Principles of Econometrics, John Wiley & Sons, New York.
See Also
systemfit, linearHypothesis
(package "car"),
lrtest.systemfit
Examples
data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )
## unconstrained SUR estimation
fitsur <- systemfit( system, method = "SUR", data=Kmenta )
# create hypothesis matrix to test whether beta_2 = \beta_6
R1 <- matrix( 0, nrow = 1, ncol = 7 )
R1[ 1, 2 ] <- 1
R1[ 1, 6 ] <- -1
# the same hypothesis in symbolic form
restrict1 <- "demand_price - supply_farmPrice = 0"
## perform Theil's F test
linearHypothesis( fitsur, R1 ) # rejected
linearHypothesis( fitsur, restrict1 )
## perform Wald test with F statistic
linearHypothesis( fitsur, R1, test = "F" ) # rejected
linearHypothesis( fitsur, restrict1 )
## perform Wald-test with chi^2 statistic
linearHypothesis( fitsur, R1, test = "Chisq" ) # rejected
linearHypothesis( fitsur, restrict1, test = "Chisq" )
# create hypothesis matrix to test whether beta_2 = - \beta_6
R2 <- matrix( 0, nrow = 1, ncol = 7 )
R2[ 1, 2 ] <- 1
R2[ 1, 6 ] <- 1
# the same hypothesis in symbolic form
restrict2 <- "demand_price + supply_farmPrice = 0"
## perform Theil's F test
linearHypothesis( fitsur, R2 ) # accepted
linearHypothesis( fitsur, restrict2 )
## perform Wald test with F statistic
linearHypothesis( fitsur, R2, test = "F" ) # accepted
linearHypothesis( fitsur, restrict2 )
## perform Wald-test with chi^2 statistic
linearHypothesis( fitsur, R2, test = "Chisq" ) # accepted
linearHypothesis( fitsur, restrict2, test = "Chisq" )