| linear.hypothesis.systemfit {systemfit} | R Documentation |
Testing linear hypothesis on the coefficients of a system of equations by an F-test or Wald-test.
## S3 method for class 'systemfit':
linear.hypothesis( model,
hypothesis.matrix, rhs = NULL, test = c( "FT", "F", "Chisq" ),
vcov. = NULL, ... )
model |
a fitted object of type systemfit. |
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 linear.hypothesis
in package "car" for details). |
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, "FT", "F", or "Chisq",
specifying whether to compute
Theil's finite-sample F test (with approximate F distribution),
the finite-sample Wald test (with approximate F distribution),
or the large-sample Wald test
(with asymptotic Chi-squared distribution). |
vcov. |
a function for estimating the covariance matrix
of the regression coefficients or an estimated covariance matrix
(function vcov is used by default). |
... |
further arguments passed to
linear.hypothesis.default (package "car"). |
Theil's F statistic for sytems of equations is
F = frac{ ( R hat{b} - q )' ( R ( X' ( Σ otimes I )^{-1} X )^{-1} R' )^{-1} ( R hat{b} - q ) / j }{ hat{e}' ( Σ 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 Σ 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 linear.hypothesis
in package "car".
Arne Henningsen ahenningsen@agric-econ.uni-kiel.de
Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.
Theil, Henri (1971) Principles of Econometrics, John Wiley & Sons, New York.
systemfit, linear.hypothesis
(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 linear.hypothesis( fitsur, R1 ) # rejected linear.hypothesis( fitsur, restrict1 ) ## perform Wald test with F statistic linear.hypothesis( fitsur, R1, test = "F" ) # rejected linear.hypothesis( fitsur, restrict1 ) ## perform Wald-test with chi^2 statistic linear.hypothesis( fitsur, R1, test = "Chisq" ) # rejected linear.hypothesis( 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 linear.hypothesis( fitsur, R2 ) # accepted linear.hypothesis( fitsur, restrict2 ) ## perform Wald test with F statistic linear.hypothesis( fitsur, R2, test = "F" ) # accepted linear.hypothesis( fitsur, restrict2 ) ## perform Wald-test with chi^2 statistic linear.hypothesis( fitsur, R2, test = "Chisq" ) # accepted linear.hypothesis( fitsur, restrict2, test = "Chisq" )