Test Linear Hypothesis
Testing linear hypothesis on the coefficients of a system of equations by an F-test or Wald-test.
## 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 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
|
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 χ^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 χ^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 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" )
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