Testing linear hypothesis on the coefficients of a system of equations by an Ftest or Waldtest.
1 2 3 4 
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' ( 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45  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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