# Likelihood Ratio test for Equation Systems

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### Description

Testing linear hypothesis on the coefficients of a system of equations by a Likelihood Ratio test.

### Usage

 1 2  ## S3 method for class 'systemfit' lrtest( object, ... ) 

### Arguments

 object a fitted model object of class systemfit. ... further fitted model objects of class systemfit.

### Details

lrtest.systemfit consecutively compares the fitted model object object with the models passed in ....

The LR-statistic for sytems of equations is

LR = T \cdot ≤ft( log ≤ft| \hat{ \hat{ Σ } }_r \right| - log ≤ft| \hat{ \hat{ Σ } }_u \right| \right)

where T is the number of observations per equation, and \hat{\hat{Σ}}_r and \hat{\hat{Σ}}_u are the residual covariance matrices calculated by formula "0" (see systemfit) of the restricted and unrestricted estimation, respectively. Asymptotically, LR has a χ^2 distribution with j degrees of freedom under the null hypothesis (Green, 2003, p. 349).

### Value

An object of class anova, which contains the log-likelihood value, degrees of freedom, the difference in degrees of freedom, likelihood ratio Chi-squared statistic and corresponding p value. See documentation of lrtest in package "lmtest".

### References

Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.

systemfit, lrtest (package "lmtest"), linearHypothesis.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 data( "Kmenta" ) eqDemand <- consump ~ price + income eqSupply <- consump ~ price + farmPrice + trend system <- list( demand = eqDemand, supply = eqSupply ) ## unconstrained SUR estimation fitsur <- systemfit( system, "SUR", data = Kmenta ) # create restriction matrix to impose \eqn{beta_2 = \beta_6} R1 <- matrix( 0, nrow = 1, ncol = 7 ) R1[ 1, 2 ] <- 1 R1[ 1, 6 ] <- -1 ## constrained SUR estimation fitsur1 <- systemfit( system, "SUR", data = Kmenta, restrict.matrix = R1 ) ## perform LR-test lrTest1 <- lrtest( fitsur1, fitsur ) print( lrTest1 ) # rejected # create restriction matrix to impose \eqn{beta_2 = - \beta_6} R2 <- matrix( 0, nrow = 1, ncol = 7 ) R2[ 1, 2 ] <- 1 R2[ 1, 6 ] <- 1 ## constrained SUR estimation fitsur2 <- systemfit( system, "SUR", data = Kmenta, restrict.matrix = R2 ) ## perform LR-test lrTest2 <- lrtest( fitsur2, fitsur ) print( lrTest2 ) # accepted