systemfit (version 0.8-5)

ftest.systemfit: F-test for Equation Systems

Description

F-test for linear parameter restrictions in equation systems.

Usage

ftest.systemfit( object, R.restr,
      q.restr = rep( 0, nrow( R.restr ) ) )
   ## S3 method for class 'ftest.systemfit':
print( x, digits = 4, ... )

Arguments

object

an object of type systemfit.

R.restr

j x k matrix to impose linear restrictions on the parameters by R.restr * $b$ = q.restr (j = number of restrictions, k = number of all parameters, $b$ = vector of all parameters).

q.restr

an optional vector with j elements to impose linear restrictions (see R.restr); default is a vector that contains only zeros.

x

an object of class ftest.systemfit.

digits

number of digits to print.

...

currently not used.

Value

• ftest.systemfit returns a list of class ftest.systemfit that includes following objects:
• statistic
  the empirical F statistic.
• p.value
  the p-value of the F-test.
• nRestr
  number of restrictions ($j$, degrees of freedom of the numerator).
• dfSys
  degrees of freedom of the equation system ($M \cdot T - K$, degrees of freedom of the denominator).

Details

The F-statistic for sytems of equations is $$F = \frac{ ( R \hat{b} - q )' ( R ( X' ( \hat{\Sigma} \otimes I )^{-1} X )^{-1} R' )^{-1} ( R \hat{b} - q ) / j }{ \hat{e}' ( \Sigma \otimes I )^{-1} \hat{e} / ( M \cdot T - K ) }$$ 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 residual covariance matrix used in the estimation. Under the null hypothesis, $F$ has an F-distribution with $j$ and $M \cdot T - K$ degrees of freedom (Theil, 1971, p. 314).

References

Theil, Henri (1971). Principles of Econometrics, John Wiley & Sons, New York.

See Also

systemfit, waldtest.systemfit, 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( "SUR", system, data=Kmenta )

# create restriction matrix to test whether \eqn{beta_2 = \beta_6}
R1 <- matrix( 0, nrow = 1, ncol = 7 )
R1[ 1, 2 ] <- 1
R1[ 1, 6 ] <- -1

## perform F-test
fTest1 <- ftest.systemfit( fitsur, R1 )
print( fTest1 )   # rejected

# create restriction matrix to test whether \eqn{beta_2 = - \beta_6}
R2 <- matrix( 0, nrow = 1, ncol = 7 )
R2[ 1, 2 ] <- 1
R2[ 1, 6 ] <- 1

## perform F-test
fTest2 <- ftest.systemfit( fitsur, R2 )
print( fTest2 )   # accepted