View source: R/linear.hypothesis.R
linearHypothesis.systemfit  R Documentation 
Testing linear hypothesis on the coefficients of a system of equations by an Ftest or Waldtest.
## S3 method for class 'systemfit'
linearHypothesis( model,
hypothesis.matrix, rhs = NULL, test = c( "FT", "F", "Chisq" ),
vcov. = NULL, ... )
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 righthandside 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

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).
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".
Arne Henningsen arne.henningsen@googlemail.com
Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.
Theil, Henri (1971) Principles of Econometrics, John Wiley & Sons, New York.
systemfit
, linearHypothesis
(package "car"),
lrtest.systemfit
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 Waldtest 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 Waldtest with chi^2 statistic
linearHypothesis( fitsur, R2, test = "Chisq" ) # accepted
linearHypothesis( fitsur, restrict2, test = "Chisq" )
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