# linear.hypothesis.systemfit: Test Linear Hypothesis In systemfit: Estimating Systems of Simultaneous Equations

## Description

Testing linear hypothesis on the coefficients of a system of equations by an F-test or Wald-test.

## Usage

 1 2 3 4  ## S3 method for class 'systemfit' linearHypothesis( model, hypothesis.matrix, rhs = NULL, test = c( "FT", "F", "Chisq" ), vcov. = NULL, ... ) 

## Arguments

 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 linearHypothesis 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 linearHypothesis.default (package "car").

## Details

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).

## Value

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".

## Author(s)

Arne Henningsen [email protected]

## References

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 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" )