Bread for Sandwiches

Description

Extract the estimator for the bread of sandwhiches (see bread).

Usage

## S3 method for class 'systemfit'
bread( x, ... )

Arguments

Value

Quadratic symmetric matrix, which is an estimator for the expectation of the negative derivative of the estimating function (see estfun.systemfit).

Warnings

The sandwich package must be loaded before this method can be used.

This method might not be suitable for specific formulas for 3SLS estimations in case of unbalanced systems or different instruments for different equations.

Author(s)

Arne Henningsen

See Also

bread, systemfit.

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )
inst <- ~ income + farmPrice + trend

## OLS estimation
fitols <- systemfit( system, "OLS", data = Kmenta )

## obtain the bread
library( "sandwich" )
bread( fitols )

## this is only true for OLS models
all.equal( bread( fitols ),
   solve( crossprod( model.matrix( fitols ) ) / 40 ) )


## 2SLS estimation
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )

## obtain the bread
bread( fit2sls )

## this is only true for 2SLS models
all.equal( bread( fit2sls ),
   solve( crossprod( model.matrix( fit2sls, which = "xHat" ) ) / 40 ) )


## iterated SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta, maxit = 100 )

## obtain the bread
bread( fitsur )

## this should be true for SUR and WLS models
all.equal( bread( fitsur ),
   solve( t( model.matrix( fitsur ) ) %*%
      (  ( solve( fitsur$residCovEst ) %x% diag( nrow( Kmenta ) ) ) %*% 
      model.matrix( fitsur ) ) / 40 ), check.attributes = FALSE )


## 3SLS estimation
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )

## obtain the bread
bread( fit3sls )

## this should be true for 3SLS and W2SLS models
all.equal( bread( fit3sls ),
   solve( t( model.matrix( fit3sls, which = "xHat" ) ) %*%
      (  ( solve( fit3sls$residCovEst ) %x% diag( nrow( Kmenta ) ) ) %*% 
      model.matrix( fit3sls, which = "xHat" ) ) / 40 ), check.attributes = FALSE )

Coefficients of systemfit object

Description

These functions extract the coefficients from an object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
coef( object, modified.regMat = FALSE, ... )

   ## S3 method for class 'systemfit.equation'
coef( object, ... )

   ## S3 method for class 'summary.systemfit'
coef( object, modified.regMat = FALSE, ... )

   ## S3 method for class 'summary.systemfit.equation'
coef( object, ... )

Arguments

Value

coef.systemfit returns a vector of all estimated coefficients.

coef.systemfit.equation returns a vector of the estimated coefficients of a single equation.

coef.summary.systemfit returns a matrix of all estimated coefficients, their standard errors, t-values, and p-values.

coef.summary.systemfit.equation returns a matrix of the estimated coefficients of a single equation, their standard errors, t-values, and p-values.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, coef

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## all coefficients
coef( fitols )
coef( summary( fitols ) )

## coefficients of the first equation
coef( fitols$eq[[1]] )
coef( summary( fitols$eq[[1]] ) )

## coefficients of the second equation
coef( fitols$eq[[2]] )
coef( summary( fitols$eq[[2]] ) )

## estimation with restriction by modifying the regressor matrix
modReg <- matrix( 0, 7, 6 )
colnames( modReg ) <- c( "demIntercept", "demPrice", "demIncome",
   "supIntercept", "supPrice2", "supTrend" )
modReg[ 1, "demIntercept" ] <- 1
modReg[ 2, "demPrice" ]     <- 1
modReg[ 3, "demIncome" ]    <- 1
modReg[ 4, "supIntercept" ] <- 1
modReg[ 5, "supPrice2" ]    <- 1
modReg[ 6, "supPrice2" ]    <- 1
modReg[ 7, "supTrend" ]     <- 1
fitols3 <- systemfit( system, data = Kmenta, restrict.regMat = modReg )
coef( fitols3, modified.regMat  = TRUE )
coef( fitols3 )

Confidence intervals of coefficients

Description

These functions calculate the confidence intervals of the coefficients from an object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
confint( object, parm = NULL, level = 0.95,
      useDfSys = NULL, ... )

   ## S3 method for class 'systemfit.equation'
confint( object, parm, level = 0.95,
      useDfSys = NULL, ... )

Arguments

Value

An object of class confint.systemfit, which is a matrix with columns giving lower and upper confidence limits for each estimated coefficient. These will be labelled as (1-level)/2 and 1 - (1-level)/2 in % (by default 2.5% and 97.5%).

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, print.confint.systemfit, confint

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## confidence intervals of all coefficients
confint( fitols )

## confidence intervals of the coefficients of the first equation
confint( fitols$eq[[1]] )

## confidence intervals of the coefficients of the second equation
confint( fitols$eq[[2]] )

Correlation between Predictions from Equation i and j

Description

correlation returns a vector of the correlations between the predictions of two equations in a set of equations. The correlation between the predictions is defined as,

r_{ijk} = \frac{x'_{ik}C_{ij}x_{jk}}{\sqrt{(x'_{ik}C_{ii}x_{ik})(x'_{jk}C_{jj}x_{jk})}}

where r_{ijk} is the correlation between the predicted values of equation i and j and C_{ij} is the cross-equation variance-covariance matrix between equations i and j.

Usage

correlation.systemfit( results, eqni, eqnj )

Arguments

Value

correlation returns a vector of the correlations between the predicted values in equation i and equation j.

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

References

Greene, W. H. (1993) Econometric Analysis, Second Edition, Macmillan.

Hasenauer, H; Monserud, R and T. Gregoire. (1998) Using Simultansous Regression Techniques with Individual-Tree Growth Models. Forest Science. 44(1):87-95

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing

See Also

systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
inst <- ~ income + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform 2SLS on each of the equations in the system
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
print( fit2sls )
print( fit2sls$rcov )

## perform the 3SLS
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )
print( fit3sls )
print( "covariance of residuals used for estimation (from 2sls)" )
print( fit3sls$rcovest )
print( "covariance of residuals" )
print( fit3sls$rcov )

## examine the correlation between the predicted values
## of suppy and demand by plotting the correlation over
## the value of q
r12 <- correlation.systemfit( fit3sls, 1, 2 )
plot( Kmenta$consump, r12, main="correlation between predictions from supply and demand" )

Create a Model for systemfit

Description

This function creates a model that can be estimated by systemfit. The data, disturbances, and — if not provided by the user — the coefficients as well as the disturbance covariance matrix are generated by random numbers.

Usage

createSystemfitModel( nEq, nRegEq, nObs, coef = NULL, sigma = NULL )

Arguments

Value

createSystemfitModel returns a list with following elements:

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit

Examples

## create a model by random numbers
systemfitModel <- createSystemfitModel( 3, 4, 100 )

## estimate this model by "SUR"
fitsur <- systemfit( systemfitModel$formula, "SUR", data = systemfitModel$data )

## compare the "true" and the estimated coefficients
cbind( systemfitModel$coef, coef( fitsur ) )

Extract Gradients of the Objective Function at each Observation

Description

Extract the gradients of the objective function with respect to the coefficients evaluated at each observation (‘Empirical Estimating Function’, see estfun).

Usage

## S3 method for class 'systemfit'
estfun( x, residFit = TRUE, ... )

Arguments

Value

Matrix of gradients of the objective function with respect to the coefficients evaluated at each observation.

Warnings

The sandwich package must be loaded before this method can be used.

In specific estimations with the 3SLS method, not all columns of the matrix returned by the estfun method sum up to zero, which indicates that an inappropriate estimating function is returned. This can be either with argument residFit set to TRUE or with this argument set to FALSE or even in both cases. This problem depends on the formula used for the 3SLS estimation and seems to be related to unbalanced systems and systems where different instruments are used in different equations.

Author(s)

Arne Henningsen

See Also

estfun, systemfit.

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )
inst <- ~ income + farmPrice + trend

## OLS estimation
fitols <- systemfit( system, "OLS", data = Kmenta )

## obtain the estimation function
library( "sandwich" )
estfun( fitols )

## this is only true for OLS models
all.equal( estfun( fitols ),
   unlist( residuals( fitols ) ) * model.matrix( fitols ) )

# each column should sum up to (approximately) zero
colSums( estfun( fitols ) )


## 2SLS estimation
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )

## obtain the estimation function
estfun( fit2sls )

## this is only true for 2SLS models
all.equal( estfun( fit2sls ),
   drop( rep( Kmenta$consump, 2 ) -  model.matrix( fit2sls, which = "xHat" ) %*% 
   coef( fit2sls ) ) * model.matrix( fit2sls, which = "xHat" ) )
all.equal( estfun( fit2sls, residFit = FALSE ),
   unlist( residuals( fit2sls ) ) * model.matrix( fit2sls, which = "xHat" ) )

# each column should sum up to (approximately) zero
colSums( estfun( fit2sls ) )
colSums( estfun( fit2sls, residFit = FALSE ) )


## iterated SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta, maxit = 100 )

## obtain the estimation function
estfun( fitsur )

## this should be true for SUR and WLS models
all.equal( estfun( fitsur ),
   unlist( residuals( fitsur ) ) * 
   ( ( solve( fitsur$residCovEst ) %x% diag( nrow( Kmenta ) ) ) %*% 
      model.matrix( fitsur ) ), check.attributes = FALSE )

# each column should sum up to (approximately) zero
colSums( estfun( fitsur ) )


## 3SLS estimation
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )

## obtain the estimation function
estfun( fit3sls )
estfun( fit3sls, residFit = FALSE )

## this should be true for 3SLS and W2SLS models
all.equal( estfun( fit3sls ),
   drop( rep( Kmenta$consump, 2 ) -  
   model.matrix( fit2sls, which = "xHat" ) %*% coef( fit3sls ) ) * 
   ( ( solve( fit3sls$residCovEst ) %x% diag( nrow( Kmenta ) ) ) %*% 
      model.matrix( fit3sls, which = "xHat" ) ), check.attributes = FALSE )
all.equal( estfun( fit3sls, residFit = FALSE ),
   unlist( residuals( fit3sls ) ) * 
   ( ( solve( fit3sls$residCovEst ) %x% diag( nrow( Kmenta ) ) ) %*% 
      model.matrix( fit3sls, which = "xHat" ) ), check.attributes = FALSE )

# each column should sum up to (approximately) zero
colSums( estfun( fit3sls ) )
colSums( estfun( fit3sls, residFit = FALSE ) )

Fitted values

Description

These functions extract the fitted values from an object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
fitted( object, ... )

   ## S3 method for class 'systemfit.equation'
fitted( object, na.rm = FALSE, ... )

Arguments

Value

fitted.systemfit returns a data.frame of all fitted values, where each column contains the fitted values of one equation.

fitted.systemfit.equation returns a vector of the fitted values of a single equation.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, fitted

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## all fitted values
fitted( fitols )

## fitted values of the first equation
fitted( fitols$eq[[1]] )

## fitted values of the second equation
fitted( fitols$eq[[2]] )

Model Formulae of systemfit Objects

Description

This method extracts the model formulae from fitted objects returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
formula( x, ... )
   ## S3 method for class 'systemfit.equation'
formula( x, ... )

Arguments

Value

formula.systemfit.equation returns the formula of a single equation of a systemfit object. formula.systemfit.equation returns a list of formulae: one formula object for each equation of the systemfit object.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, formula

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform a SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta )

## formula of the second equation
formula( fitsur$eq[[2]] )

## all formulae of the system
formula( fitsur )

Grunfeld Data as published by Greene (2003)

Description

Panel data on 5 US firms for the years 1935-1954.

Usage

   data("GrunfeldGreene")

Format

A data frame containing 20 annual observations on 3 variables for 5 firms.

invest

gross investment.

value

market value of the firm (at the end of the previous year).

capital

capital stock of the firm (at the end of the previous year).

firm

name of the firm ("General Motors", "Chrysler", "General Electric", "Westinghouse" or "US Steel").

year

year.

Details

There exist several different versions of this data set, and this version is considered incorrect (see https://web.archive.org/web/20170426034143/http://web.stanford.edu/~clint/bench/grunfeld.htm for details). However, we provide this incorrect version to replicate the results published in Theil (1971) and Greene (2003). A correct version of this data set with 5 additional firms is available in the Ecdat package (data set Grunfeld).

Source

Greene (2003), Appendix F, Data Sets Used in Applications, Table F13.1. https://pages.stern.nyu.edu/~wgreene/Text/econometricanalysis.htm (a subset of this data set is available in Theil (1971), p. 296).

References

Greene, W.H. (2003). Econometric Analysis, 5th edition. Prentice Hall, Upper Saddle River (NJ).

Grunfeld, Y. (1958). The Determinants of Corporate Investment, Unpublished Ph.D. Dissertation, University of Chicago.

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

Examples

## Repeating the OLS and SUR estimations in Greene (2003, pp. 351)
data( "GrunfeldGreene" )
if( requireNamespace( 'plm', quietly = TRUE ) ) {
library( "plm" )
GGPanel <- pdata.frame( GrunfeldGreene, c( "firm", "year" ) )
formulaGrunfeld <- invest ~ value + capital
# OLS
greeneOls <- systemfit( formulaGrunfeld, "OLS",
   data = GGPanel )
summary( greeneOls )
sapply( greeneOls$eq, function(x){return(summary(x)$ssr/20)} ) # sigma^2
# OLS Pooled
greeneOlsPooled <- systemfit( formulaGrunfeld, "OLS",
   data = GGPanel, pooled = TRUE )
summary( greeneOlsPooled )
sum( sapply( greeneOlsPooled$eq, function(x){return(summary(x)$ssr)}) )/97 # sigma^2
# SUR
greeneSur <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, methodResidCov = "noDfCor" )
summary( greeneSur )
# SUR Pooled
greeneSurPooled <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, pooled = TRUE, methodResidCov = "noDfCor",
   residCovWeighted = TRUE )
summary( greeneSurPooled )

## Repeating the OLS and SUR estimations in Theil (1971, pp. 295, 300)
GrunfeldTheil <- subset( GrunfeldGreene,
   firm %in% c( "General Electric", "Westinghouse" ) )
GTPanel <- pdata.frame( GrunfeldTheil, c( "firm", "year" ) )
formulaGrunfeld <- invest ~ value + capital
# OLS
theilOls <- systemfit( formulaGrunfeld, "OLS",
   data = GTPanel )
summary( theilOls )
# SUR
theilSur <- systemfit( formulaGrunfeld, "SUR",
   data = GTPanel, methodResidCov = "noDfCor" )
summary( theilSur )
}

Hausman Test

Description

hausman.systemfit returns the Hausman statistic for a specification test.

Usage

   hausman.systemfit( results2sls, results3sls )

Arguments

Details

The null hypotheses of the test is that all exogenous variables are uncorrelated with all disturbance terms. Under this hypothesis both the 2SLS and the 3SLS estimator are consistent but only the 3SLS estimator is (asymptotically) efficient. Under the alternative hypothesis the 2SLS estimator is consistent but the 3SLS estimator is inconsistent.

The Hausman test statistic is

m = ( b_2 - b_3 )' ( V_2 - V_3 ) ( b_2 - b_3 )

where $b_2$ and $V_2$ are the estimated coefficients and their variance covariance matrix of a 2SLS estimation and $b_3$ and $V_3$ are the estimated coefficients and their variance covariance matrix of a 3SLS estimation.

Value

hausman.systemfit returns a list of the class htest that contains following elements:

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com,
Arne Henningsen arne.henningsen@googlemail.com

References

Greene, W. H. (1993) Econometric Analysis, Fifth Edition, Macmillan.

Hausman, J. A. (1978) Specification Tests in Econometrics. Econometrica. 46:1251-1271.

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing

See Also

systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
inst <- ~ income + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform the estimations
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )

## perform the Hausman test
h <- hausman.systemfit( fit2sls, fit3sls )
print( h )

Klein Model I

Description

Data for Klein's (1950) Model I of the US economy.

Usage

data("KleinI")

Format

A data frame containing annual observations from 1920 to 1941

year

Year.

consump

Consumption.

corpProf

Corporate profits.

corpProfLag

Corporate profits of the previous year.

privWage

Private wage bill.

invest

Investment.

capitalLag

Capital stock of the previous year.

gnp

Gross national product.

gnpLag

Gross national product of the previous year.

govWage

Government wage bill.

govExp

Government spending.

taxes

Taxes.

wages

Sum of private and government wage bill.

trend

time trend measured as years from 1931.

Source

Greene (2003), Appendix F, Data Sets Used in Applications, Table F15.1.

https://pages.stern.nyu.edu/~wgreene/Text/econometricanalysis.htm

References

Greene, W.H. (2003). Econometric Analysis, 5th edition. Prentice Hall, Upper Saddle River (NJ).

Klein, L. (1950). Economic Fluctuations in the United States, 1921–1941. John Wiley, New York.

Examples

## Repeating the estimations of Klein's (1950) Model I
## in Greene (2003, pp. 381 and 412)
data( "KleinI" )
eqConsump  <- consump ~ corpProf + corpProfLag + wages
eqInvest   <- invest ~ corpProf + corpProfLag + capitalLag
eqPrivWage <- privWage ~ gnp + gnpLag + trend
inst <- ~ govExp + taxes + govWage + trend + capitalLag + corpProfLag + gnpLag
system <- list( Consumption = eqConsump, Investment = eqInvest,
   PrivateWages = eqPrivWage )
# OLS
kleinOls <- systemfit( system, data = KleinI )
summary( kleinOls )
# 2SLS
klein2sls <- systemfit( system, "2SLS", inst = inst, data = KleinI,
   methodResidCov = "noDfCor" )
summary( klein2sls )
# 3SLS
klein3sls <- systemfit( system, "3SLS", inst = inst, data = KleinI,
   methodResidCov = "noDfCor" )
summary( klein3sls )
# I3SLS
kleinI3sls <- systemfit( system, "3SLS", inst = inst, data = KleinI,
   methodResidCov = "noDfCor", maxit = 500 )
summary( kleinI3sls )

Partly Artificial Data on the U. S. Economy

Description

These are partly contrived data from Kmenta (1986), constructed to illustrate estimation of a simultaneous-equation model.

Usage

   data("Kmenta")

Format

This data frame contains 20 annual observations of 5 variables:

consump

food consumption per capita.

price

ratio of food prices to general consumer prices.

income

disposable income in constant dollars.

farmPrice

ratio of preceding year's prices received by farmers to general consumer prices.

trend

time trend in years.

Details

The exogenous variables income, farmPrice, and trend are based on real data; the endogenous variables price and consump were generated by simulation.

Source

Kmenta (1986), Table 13-1, p. 687.

References

Kmenta, J. (1986). Elements of Econometrics, Second Edition, Macmillan, New York.

Examples

## Replicating the estimations in Kmenta (1986), p. 712, Tab 13-2
data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
inst <- ~ income + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## OLS estimation
fitOls <- systemfit( system, data = Kmenta )
summary( fitOls )

## 2SLS estimation
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
summary( fit2sls )

## 3SLS estimation
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )
summary( fit3sls )

## I3LS estimation
fitI3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta,
   maxit = 250 )
summary( fitI3sls )

Test Linear Hypothesis

Description

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

Usage

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

Arguments

Details

Theil's F statistic for sytems of equations is

F = \frac{ ( R \hat{b} - q )' ( R ( X' ( \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 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 \chi^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 \chi^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 arne.henningsen@googlemail.com

References

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

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

See Also

systemfit, linearHypothesis (package "car"), 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( 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" )

Log-Likelihood value of systemfit object

Description

This method calculates the log-likelihood value of a fitted object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
logLik( object, residCovDiag = FALSE, ... )

Arguments

Details

The residual covariance matrix that is used for calculating the log-likelihood value is calculated based on the actually obtained (final) residuals (not correcting for degrees of freedom). In case of systems of equations with unequal numbers of observations, the calculation of the residual covariance matrix is only based on the residuals/observations that are available in all equations.

Value

A numeric scalar (the log-likelihood value) with 2 attributes: nobs (total number of observations in all equations) and df (number of free parameters, i.e. coefficients + elements of the residual covariance matrix).

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, logLik

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform a SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta )

## residuals of all equations
logLik( fitsur )

Likelihood Ratio test for Equation Systems

Description

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

Usage

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

Arguments

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 \left( log \left| \hat{ \hat{ \Sigma } }_r \right| - log \left| \hat{ \hat{ \Sigma } }_u \right| \right)

where T is the number of observations per equation, and \hat{\hat{\Sigma}}_r and \hat{\hat{\Sigma}}_u are the residual covariance matrices calculated by formula "0" (see systemfit) of the restricted and unrestricted estimation, respectively. Asymptotically, LR has a \chi^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".

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

References

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

See Also

systemfit, lrtest (package "lmtest"), linearHypothesis.systemfit

Examples

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

Extracting the Data of a systemfit Object

Description

These functions return the data used by systemfit to estimate a system of equations.

Usage

   ## S3 method for class 'systemfit'
model.frame( formula, ... )

   ## S3 method for class 'systemfit.equation'
model.frame( formula, ... )

Arguments

Value

model.frame.systemfit returns a simple data frame (without a 'terms' attribute) that contains all variables used to estimate the entire system of equations.

model.frame.systemfit.equation returns a model frame (data frame with a 'terms' attribute) that contains all variables used to estimate the respective equation.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, model.frame, and model.matrix.systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS of the system
fitols <- systemfit( system, data = Kmenta )

## data used to estimate the entire system
model.frame( fitols )

## data used to estimate the first equation
model.frame( fitols$eq[[ 1 ]] )

Construct Design Matrices for Systems of Equations

Description

These functions create design matrices from objects returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
model.matrix( object, which = "x", ... )

   ## S3 method for class 'systemfit.equation'
model.matrix( object, which = "x", ... )

Arguments

Value

model.matrix.systemfit returns a design matrix to estimate the specified system of equations.

model.matrix.systemfit.equation returns a design matrix to estimate the specified formula of the respective equation.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, model.matrix, and model.frame.systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS of the system
fitols <- systemfit( system, data = Kmenta )

## design matrix of the entire system
model.matrix( fitols )

## design matrix of the first equation
model.matrix( fitols$eq[[ 1 ]] )

Nonlinear Equation System Estimation

Description

Fits a set of structural nonlinear equations using Ordinary Least Squares (OLS), Seemingly Unrelated Regression (SUR), Two-Stage Least Squares (2SLS), Three-Stage Least Squares (3SLS).

Usage

 nlsystemfit( method="OLS", eqns, startvals,
                     eqnlabels=c(as.character(1:length(eqns))), inst=NULL,
                     data=list(), solvtol=.Machine$double.eps,
                     maxiter=1000, ... )

Arguments

Details

The nlsystemfit function relies on nlm to perform the minimization of the objective functions and the qr set of functions.

A system of nonlinear equations can be written as:

\epsilon_{t} = q( y_t, x_t, \theta )

z_{t} = Z( x_t )

where \epsilon_{t} are the residuals from the y observations and the function evaluated at the coefficient estimates.

The objective functions for the methods are:

where, r is a column vector for the residuals for each equation, S is variance-covariance matrix between the equations (\hat{\sigma}_{ij} = (\hat{e}_i' \hat{e}_j) / \sqrt{(T - k_i)*(T - k_j)}), X is matrix of the partial derivates with respect to the coefficients, W is a matrix of the instrument variables Z(Z'Z)^{-1}Z, Z is a matrix of the instrument variables, and I is an nxn identity matrix.

The SUR and 3SLS methods requires two solutions. The first solution for the SUR is an OLS solution to obtain the variance-covariance matrix. The 3SLS uses the variance-covatiance from a 2SLS solution, then fits all the equations simultaneously.

The user should be aware that the function is VERY sensative to the starting values and the nlm function may not converge. The nlm function will be called with the typsize argument set the absolute values of the starting values for the OLS and 2SLS methods. For the SUR and 3SLS methods, the typsize argument is set to the absolute values of the resulting OLS and 2SLS coefficient estimates from the nlm result structre. In addition, the starting values for the SUR and 3SLS methods are obtained from the OLS and 2SLS coefficient estimates to shorten the number of iterations. The number of iterations reported in the summary are only those used in the last call to nlm, thus the number of iterations in the OLS portion of the SUR fit and the 2SLS portion of the 3SLS fit are not included.

Value

nlsystemfit returns a list of the class nlsystemfit.system and contains all results that belong to the whole system. This list contains one special object: "eq". It is a list and contains one object for each estimated equation. These objects are of the class nlsystemfit.equation and contain the results that belong only to the regarding equation.

The objects of the class nlsystemfit.system and nlsystemfit.equation have the following components (the elements of the latter are marked with an asterisk (*)):

## elements of the class nlsystemfit.eq

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

References

Gallant, R. H. (1987) Nonlinear Equation Estimation, John Wiley and Sons, 610 pp.

SAS Institute (1999) SAS/ETS User's Guide, Version 8, Cary NC: SAS Institute 1546 pp.

See Also

systemfit, nlm, and qr

Examples

library( systemfit )
data( ppine )

hg.formula <- hg ~ exp( h0 + h1*log(tht) + h2*tht^2 + h3*elev + h4*cr)
dg.formula <- dg ~ exp( d0 + d1*log(dbh) + d2*hg + d3*cr + d4*ba  )
labels <- list( "height.growth", "diameter.growth" )
inst <- ~ tht + dbh + elev + cr + ba
start.values <- c(h0=-0.5, h1=0.5, h2=-0.001, h3=0.0001, h4=0.08,
                  d0=-0.5, d1=0.009, d2=0.25, d3=0.005, d4=-0.02 )
model <- list( hg.formula, dg.formula )

model.ols <- nlsystemfit( "OLS", model, start.values, data=ppine, eqnlabels=labels )
print( model.ols )

model.sur <- nlsystemfit( "SUR", model, start.values, data=ppine, eqnlabels=labels )
print( model.sur )

model.2sls <- nlsystemfit( "2SLS", model, start.values, data=ppine,
   eqnlabels=labels, inst=inst )
print( model.2sls )

model.3sls <- nlsystemfit( "3SLS", model, start.values, data=ppine,
                                    eqnlabels=labels, inst=inst )
print( model.3sls )

Tree Growth Data for Ponderosa Pine

Description

A subset of tree growth observations from a Ponderosa pine growth database.

The ppine data frame has 166 rows and 8 columns.

Usage

data(ppine)

Format

This data frame contains the following columns:

elev

Altitude of the plot, in feet above mean sea level.

smi

Summer moisture index is the ratio of growing season heating degree days to growing season precipitation.

dbh

Diameter of the tree at breast height (4.5 feet).

tht

Total stem height for the tree.

cr

Crown ratio code. The scale is from 1 to 9 where a crown class of one represents a crown ratio between 0 and 15 percent. A crown ratio code of 2 represents a crown ratio value between 16 and 25%,...,8=76-85%, 9 >=85%.

ba

Plot basal area at the beginning of the growth period.

dg

Five-year diameter increment.

hg

Five-year height increment.

Details

The exogenous variables are elev, smi, dbh, tht, cr, and ba; the endogenous variables dg and hg. There are no lagged variables in the dataset and the observations are for a single remeasurement.

The data was provided by the USDA Forest Service Intermountain Research Station.

Source

William R. Wykoff wwykoff@fs.fed.us Rocky Mountain Research Station, 1221 South Main Street, Moscow, ID 83843

Examples

data(ppine)

Predictions from System Estimation

Description

Returns the predicted values, their standard errors and the confidence limits of prediction.

Usage

## S3 method for class 'systemfit'
predict( object, newdata = NULL,
                             se.fit = FALSE, se.pred = FALSE,
                             interval = "none", level=0.95,
                             useDfSys = NULL, ... )

## S3 method for class 'systemfit.equation'
predict( object, newdata = NULL,
                             se.fit = FALSE, se.pred = FALSE,
                             interval = "none", level=0.95,
                             useDfSys = NULL, ... )

Arguments

Details

The variance of the fitted values (used to calculate the standard errors of the fitted values and the "confidence interval") is calculated by Var[E[y^0]-\hat{y}^0]=x^0 \; Var[b] \; {x^0}'
The variances of the predicted values (used to calculate the standard errors of the predicted values and the "prediction intervals") is calculated by Var[y^0-\hat{y}^0]=\hat{\sigma}^2+x^0 \; Var[b] \; {x^0}'

Value

predict.systemfit returns a dataframe that contains for each equation the predicted values ("<eqnLable>.pred") and if requested the standard errors of the fitted values ("<eqnLable>.se.fit"), the standard errors of the prediction ("<eqnLable>.se.pred"), and the lower ("<eqnLable>.lwr") and upper ("<eqnLable>.upr") limits of the confidence or prediction interval(s).

predict.systemfit.equation returns a dataframe that contains the predicted values ("fit") and if requested the standard errors of the fitted values ("se.fit"), the standard errors of the prediction ("se.pred"), and the lower ("lwr") and upper ("upr") limits of the confidence or prediction interval(s).

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

References

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

Gujarati, D. N. (1995) Basic Econometrics, Third Edition, McGraw-Hill.

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing.

See Also

systemfit, predict

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## OLS estimation
fitols <- systemfit( system, data=Kmenta )

## predicted values and limits
predict( fitols )

## predicted values of the first equation
predict( fitols$eq[[1]] )

## predicted values of the second equation
predict( fitols$eq[[2]] )

Print confidence intervals of coefficients

Description

This function prints the confidence intervals of the coefficients of the estimated equation system.

Usage

   ## S3 method for class 'confint.systemfit'
print( x, digits=3, ... )

Arguments

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, confint.systemfit and confint.systemfit.equation

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## calculate and print the confidence intervals
## of all coefficients
ci <- confint( fitols )
print( ci, digits=4 )

## calculate and print the confidence intervals
## of the coefficients of the second equation
ci2 <- confint( fitols$eq[[2]] )
print( ci2, digits=4 )

Print output of nlsystemfit estimation

Description

These functions print a summary of the estimated equation system.

Usage

   ## S3 method for class 'nlsystemfit.system'
print( x, digits=6, ... )

   ## S3 method for class 'nlsystemfit.equation'
print( x, digits=6, ... )

Arguments

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

See Also

nlsystemfit, summary.nlsystemfit.system

Examples

library( systemfit )
data( ppine )

hg.formula <- hg ~ exp( h0 + h1*log(tht) + h2*tht^2 + h3*elev + h4*cr)
dg.formula <- dg ~ exp( d0 + d1*log(dbh) + d2*hg + d3*cr + d4*ba  )
labels <- list( "height.growth", "diameter.growth" )
inst <- ~ tht + dbh + elev + cr + ba
start.values <- c(h0=-0.5, h1=0.5, h2=-0.001, h3=0.0001, h4=0.08,
                  d0=-0.5, d1=0.009, d2=0.25, d3=0.005, d4=-0.02 )
model <- list( hg.formula, dg.formula )

model.ols <- nlsystemfit( "OLS", model, start.values, data=ppine, eqnlabels=labels )
print( model.ols )

model.3sls <- nlsystemfit( "3SLS", model, start.values, data=ppine,
                                    eqnlabels=labels, inst=inst )
print( model.3sls )

Print results of systemfit estimation

Description

These functions print a few results of the estimated equation system.

Usage

   ## S3 method for class 'systemfit'
print( x,
      digits = max( 3, getOption("digits") - 1 ), ... )

   ## S3 method for class 'systemfit.equation'
print( x,
      digits = max( 3, getOption("digits") - 1 ), ... )

Arguments

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com,
Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, summary.systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## results of the whole system
print( fitols )

## results of the first equation
print( fitols$eq[[1]] )

## results of the second equation
print( fitols$eq[[2]] )

Residuals of systemfit object

Description

These functions extract the residuals from an object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
residuals( object, ... )

   ## S3 method for class 'systemfit.equation'
residuals( object, na.rm = FALSE, ... )

Arguments

Value

residuals.systemfit returns a data.frame of residuals, where each column contains the residuals of one equation.

residuals.systemfit.equation returns a vector of residuals.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, residuals

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## residuals of all equations
residuals( fitols )

## residuals of the first equation
residuals( fitols$eq[[1]] )

## residuals of the second equation
residuals( fitols$eq[[2]] )

Ratio of the Standard Errors

Description

se.ratio.systemfit returns a vector of the ratios of the standard errors of the predictions for two equations.

Usage

se.ratio.systemfit( resultsi, resultsj, eqni )

Arguments

Value

se.ratio returns a vector of the standard errors of the ratios for the predictions between the predicted values in equation i and equation j.

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

References

Hasenauer, H; Monserud, R and T. Gregoire. (1998) Using Simultaneous Regression Techniques with Individual-Tree Growth Models. Forest Science. 44(1):87-95

See Also

systemfit and correlation.systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
inst <- ~ income + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform 2SLS on each of the equations in the system
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta )

## print the results from the fits
print( fit2sls )
print( fit3sls )
print( "covariance of residuals used for estimation (from 2sls)" )
print( fit3sls$residCovEst )
print( "covariance of residuals" )
print( fit3sls$residCov )

## examine the improvement of 3SLS over 2SLS by computing
## the ratio of the standard errors of the estimates
improve.ratio <- se.ratio.systemfit( fit2sls, fit3sls, 2 )
print( "summary values for the ratio in the std. err. for the predictions" )
print( summary( improve.ratio ) )

Summary of nlsystemfit estimation

Description

These functions print a summary of the estimated equation system.

Usage

   ## S3 method for class 'nlsystemfit.system'
summary( object, ... )

   ## S3 method for class 'nlsystemfit.equation'
summary( object, ... )

Arguments

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com

See Also

nlsystemfit, print.nlsystemfit.system

Examples

library( systemfit )
data( ppine )

hg.formula <- hg ~ exp( h0 + h1*log(tht) + h2*tht^2 + h3*elev + h4*cr)
dg.formula <- dg ~ exp( d0 + d1*log(dbh) + d2*hg + d3*cr + d4*ba  )
labels <- list( "height.growth", "diameter.growth" )
inst <- ~ tht + dbh + elev + cr + ba
start.values <- c(h0=-0.5, h1=0.5, h2=-0.001, h3=0.0001, h4=0.08,
                  d0=-0.5, d1=0.009, d2=0.25, d3=0.005, d4=-0.02 )
model <- list( hg.formula, dg.formula )

model.ols <- nlsystemfit( "OLS", model, start.values, data=ppine, eqnlabels=labels )
print( model.ols )

model.3sls <- nlsystemfit( "3SLS", model, start.values, data=ppine,
                                    eqnlabels=labels, inst=inst )
print( model.3sls )

Summary of systemfit estimation

Description

These functions create and print summary results of the estimated equation system.

Usage

   ## S3 method for class 'systemfit'
summary( object, useDfSys = NULL,
      residCov = TRUE, equations = TRUE, ... )

   ## S3 method for class 'systemfit.equation'
summary( object, useDfSys = NULL, ... )

   ## S3 method for class 'summary.systemfit'
print( x,
      digits = max( 3, getOption("digits") - 1 ),
      residCov = x$printResidCov, equations = x$printEquations, ... )

   ## S3 method for class 'summary.systemfit.equation'
print( x,
      digits = max( 3, getOption("digits") - 1 ), ... )

Arguments

Value

Applying summary on an object of class systemfit returns a list of class summary.systemfit. Applying summary on an object of class systemfit.equation returns a list of class summary.systemfit.equation. An object of class summary.systemfit contains all results that belong to the whole system. This list contains one special object: eq. This is a list and contains objects of class summary.systemfit.equation. These objects contain the results that belong to each of the eatimated equations.

The objects of classes summary.systemfit and summary.systemfit.equation have the following components (elements that are marked with a * are available only in objects of class summary.systemfit; elements that are marked with a + are available only in objects of class summary.systemfit.equation):

Author(s)

Jeff D. Hamann jeff.hamann@forestinformatics.com,
Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, print.systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
inst <- ~ income + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## results of the system
summary( fitols )

## short results of the system
summary( fitols, residCov = FALSE, equations = FALSE )

## results of the first equation
summary( fitols$eq[[1]] )

## results of the second equation
summary( fitols$eq[[2]] )

Linear Equation System Estimation

Description

Fits a set of linear structural equations using Ordinary Least Squares (OLS), Weighted Least Squares (WLS), Seemingly Unrelated Regression (SUR), Two-Stage Least Squares (2SLS), Weighted Two-Stage Least Squares (W2SLS) or Three-Stage Least Squares (3SLS).

Usage

systemfit( formula, method = "OLS",
           inst=NULL, data=list(),
           restrict.matrix = NULL, restrict.rhs = NULL, restrict.regMat = NULL,
           pooled = FALSE, control = systemfit.control( ... ), ... )

Arguments

Details

The estimation of systems of equations with unequal numbers of observations has not been thoroughly tested yet. Currently, systemfit calculates the residual covariance matrix only from the residuals/observations that are available in all equations.

If argument data is of class pdata.frame (created with pdata.frame() and thus, contains panel data in long format), argument formula must be a single equation that is applied to all individuals. In this case, argument pooled specifies whether the coefficients are restricted to be equal for all individuals.

If argument restrict.regMat is specified, the regressor matrix X is post-multiplied by this matrix: X^{*} = X \cdot restrict.regMat. Then, this modified regressor matrix X^{*} is used for the estimation of the coefficient vector b^{*}. This means that the coefficients of the original regressors (X), vector b, can be represented by b = restrict.regMat \cdot b^{*}. If restrict.regMat is a non-singular quadratic matrix, there are no restrictions on the coefficients imposed, but the coefficients b^{*} are linear combinations of the original coefficients b. If restrict.regMat has less columns than rows, linear restrictions are imposed on the coefficients b. However, imposing linear restrictions by the restrict.regMat matrix is less flexible than by providing the matrix restrict.matrix and the vector restrict.rhs. The advantage of imposing restrictions on the coefficients by the matrix restrict.regMat is that the matrix, which has to be inverted during the estimation, gets smaller by this procedure, while it gets larger if the restrictions are imposed by restrict.matrix and restrict.rhs.

In the context of multi-equation models, the term “weighted” in “weighted least squares” (WLS) and “weighted two-stage least squares” (W2SLS) means that the equations might have different weights and not that the observations have different weights.

It is important to realize the limitations on estimating the residuals covariance matrix imposed by the number of observations T in each equation. With g equations we estimate g*(g+1)/2 elements using T*g observations total. Beck and Katz (1995,1993) discuss the issue and the resulting overconfidence when the ratio of T/g is small (e.g. 3). Even for T/g=5 the estimate is unstable both numerically and statistically and the 95 approximately [0.5*\sigma^2, 3*\sigma^2], which is inadequate precision if the covariance matrix will be used for simulation of asset return paths either for investment or risk management decisions. For a starter on models with large cross-sections see Reichlin (2002). [This paragraph has been provided by Stephen C. Bond – Thanks!]

Value

systemfit returns a list of the class systemfit and contains all results that belong to the whole system. This list contains one special object: "eq". It is a list and contains one object for each estimated equation. These objects are of the class systemfit.equation and contain the results that belong only to the regarding equation.

The objects of the class systemfit and systemfit.equation have the following components (the elements of the latter are marked with an asterisk (*)):

## elements of the class systemfit.eq

Author(s)

Arne Henningsen arne.henningsen@googlemail.com,
Jeff D. Hamann jeff.hamann@forestinformatics.com

References

Beck, N.; J.N. Katz (1995) What to do (and not to do) with Time-Series Cross-Section Data, The American Political Science Review, 89, pp. 634-647.

Beck, N.; J.N. Katz; M.R. Alvarez; G. Garrett; P. Lange (1993) Government Partisanship, Labor Organization, and Macroeconomic Performance: a Corrigendum, American Political Science Review, 87, pp. 945-48.

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

Judge, George G.; W. E. Griffiths; R. Carter Hill; Helmut Luetkepohl and Tsoung-Chao Lee (1985) The Theory and Practice of Econometrics, Second Edition, Wiley.

Kmenta, J. (1997) Elements of Econometrics, Second Edition, University of Michigan Publishing.

Reichlin, L. (2002) Factor models in large cross-sections of time series, Working Paper, ECARES and CEPR.

Schmidt, P. (1990) Three-Stage Least Squares with different Instruments for different equations, Journal of Econometrics 43, p. 389-394.

Theil, H. (1971) Principles of Econometrics, Wiley, New York.

See Also

lm and nlsystemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## OLS estimation
fitols <- systemfit( system, data=Kmenta )
print( fitols )

## OLS estimation with 2 restrictions
Rrestr <- matrix(0,2,7)
Rrestr[1,3] <-  1
Rrestr[1,7] <- -1
Rrestr[2,2] <- -1
Rrestr[2,5] <-  1
qrestr <- c( 0, 0.5 )
fitols2 <- systemfit( system, data = Kmenta,
                      restrict.matrix = Rrestr, restrict.rhs = qrestr )
print( fitols2 )

## OLS estimation with the same 2 restrictions in symbolic form
restrict <- c( "demand_income - supply_trend = 0",
   "- demand_price + supply_price = 0.5" )
fitols2b <- systemfit( system, data = Kmenta, restrict.matrix = restrict )
print( fitols2b )

# test whether both restricted estimators are identical
all.equal( coef( fitols2 ), coef( fitols2b ) )

## OLS with restrictions on the coefficients by modifying the regressor matrix
## with argument restrict.regMat
modReg <- matrix( 0, 7, 6 )
colnames( modReg ) <- c( "demIntercept", "demPrice", "demIncome",
   "supIntercept", "supPrice2", "supTrend" )
modReg[ 1, "demIntercept" ] <- 1
modReg[ 2, "demPrice" ]     <- 1
modReg[ 3, "demIncome" ]    <- 1
modReg[ 4, "supIntercept" ] <- 1
modReg[ 5, "supPrice2" ]    <- 1
modReg[ 6, "supPrice2" ]    <- 1
modReg[ 7, "supTrend" ]     <- 1
fitols3 <- systemfit( system, data = Kmenta, restrict.regMat = modReg )
print( fitols3 )


## iterated SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta, maxit = 100 )
print( fitsur )

## 2SLS estimation
inst <- ~ income + farmPrice + trend
fit2sls <- systemfit( system, "2SLS", inst = inst, data = Kmenta )
print( fit2sls )

## 2SLS estimation with different instruments in each equation
inst1 <- ~ income + farmPrice
inst2 <- ~ income + farmPrice + trend
instlist <- list( inst1, inst2 )
fit2sls2 <- systemfit( system, "2SLS", inst = instlist, data = Kmenta )
print( fit2sls2 )

## 3SLS estimation with GMM-3SLS formula
inst <- ~ income + farmPrice + trend
fit3sls <- systemfit( system, "3SLS", inst = inst, data = Kmenta,
   method3sls = "GMM" )
print( fit3sls )


## Examples how to use systemfit() with panel-like data
## Repeating the SUR estimations in Greene (2003, p. 351)
data( "GrunfeldGreene" )
if( requireNamespace( 'plm', quietly = TRUE ) ) {
library( "plm" )
GGPanel <- pdata.frame( GrunfeldGreene, c( "firm", "year" ) )
formulaGrunfeld <- invest ~ value + capital
# SUR
greeneSur <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, methodResidCov = "noDfCor" )
summary( greeneSur )
# SUR Pooled
greeneSurPooled <- systemfit( formulaGrunfeld, "SUR",
   data = GGPanel, pooled = TRUE, methodResidCov = "noDfCor",
   residCovWeighted = TRUE )
summary( greeneSurPooled )
}

## Further examples are in the documentation to the data sets
## 'KleinI' and 'GrunfeldGreene'.

Internal systemfit functions

Description

Internal systemfit functions

Usage

knls( theta, eqns, data, fitmethod="OLS", parmnames, instr=NULL, S=NULL )

Details

These functions are no the called by the user. The knls is the driver function that is passed to the nlm call in nlsystemfit.

Create list of control parameters for systemfit

Description

Create a list of control pararameters for function systemfit. All control parameters that are not passed to this function are set to default values.

Usage

systemfit.control(
      maxiter = 1,
      tol = 1e-5,
      methodResidCov = "geomean",
      centerResiduals = FALSE,
      residCovRestricted = TRUE,
      residCovWeighted = FALSE,
      method3sls = "GLS",
      singleEqSigma = NULL,
      useMatrix = TRUE,
      solvetol = .Machine$double.eps,
      model = TRUE,
      x = FALSE,
      y = FALSE,
      z = FALSE )

Arguments

Details

If the estimation is iterated (WLS, SUR, W2SLS or 3SLS estimation with maxiter>1), the convergence criterion is

\sqrt{ \frac{ \sum_i (b_{i,g} - b_{i,g-1})^2 }{ \sum_i b_{i,g-1}^2 }} < \code{tol}

(b_{i,g} is the ith coefficient of the gth iteration step).

The method for calculating the estimated covariance matrix of the residuals (\hat{\Sigma}) can be one of the following (see Judge et al., 1985, p. 469):
if methodResidCov='noDfCor':

\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j}{T}

if methodResidCov='geomean':

\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j} {\sqrt{(T - k_i)*(T - k_j)}}

if methodResidCov='Theil':

\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j}{T - k_i - k_j + tr[X_i(X_i'X_i)^{-1}X_i'X_j(X_j'X_j)^{-1}X_j']}

if methodResidCov='max':

\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j} {T - \max( k_i, k_j)}

If i = j, the formulas 'geomean', 'Theil', and 'max' are equal. All these three formulas yield unbiased estimators for the diagonal elements of the residual covariance matrix. If i \neq j, only formula 'Theil' yields an unbiased estimator for the residual covariance matrix, but it is not neccessarily positive semidefinit. Thus, it is doubtful whether formula 'Theil' is really superior to formula 'noDfCor' (Theil, 1971, p. 322).

The methods for calculating the 3SLS estimator lead to identical results if the same instruments are used in all equations. If different instruments are used in the different equations, only the GMM-3SLS estimator ("GMM") and the 3SLS estimator proposed by Schmidt (1990) ("Schmidt") are consistent, whereas "GMM" is efficient relative to "Schmidt" (see Schmidt, 1990).

If residCovWeighted is TRUE, systemfit does a OLS or 2SLS estimation in a first step. It uses the residuals from the first-step estimation to calculate the residual covariance matrix that is used in a second-step WLS or W2SLS estimation. Then, it uses the residuals from the second-step estimation to calculate the residual covariance matrix that is used in a final SUR or 3SLS estimation. This three-step method is the default method of command "TSCS" in the software LIMDEP that carries out "SUR" estimations in which all coefficient vectors are constrained to be equal (personal information from W.H. Greene, 2006/02/16). If no cross-equation restrictions are imposed, residCovWeighted has no effect on the estimation results.

Value

A list of the above components.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

References

Judge, George G.; W. E. Griffiths; R. Carter Hill; Helmut Luetkepohl and Tsoung-Chao Lee (1985) The Theory and Practice of Econometrics, Second Edition, Wiley.

Schmidt, P. (1990) Three-Stage Least Squares with different Instruments for different equations, Journal of Econometrics 43, p. 389-394.

Theil, H. (1971) Principles of Econometrics, Wiley, New York.

See Also

systemfit

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
eqSystem <- list( demand = eqDemand, supply = eqSupply )

## SUR estimation: calculation of residual covariance
## matrix without correction for degrees of freedom
fitsur <- systemfit( eqSystem, "SUR", data = Kmenta,
    control = systemfit.control( methodResidCov = "noDfCor" ) )
print( fitsur )

Model Terms of systemfit Objects

Description

This method extracts the model terms from fitted objects returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
terms( x, ... )
   ## S3 method for class 'systemfit.equation'
terms( x, ... )

Arguments

Value

terms.systemfit.equation returns the model terms of a single equation of a systemfit object. terms.systemfit.equation returns a list of model terms: one model term object for each equation of the systemfit object.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, terms

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform a SUR estimation
fitsur <- systemfit( system, "SUR", data = Kmenta )

## model terms of the second equation
terms( fitsur$eq[[ 2 ]] )

## all model terms of the system
terms( fitsur )

Variance covariance matrix of coefficients

Description

These functions extract the variance covariance matrix of the coefficients from an object returned by systemfit.

Usage

   ## S3 method for class 'systemfit'
vcov( object, modified.regMat = FALSE, ... )

   ## S3 method for class 'systemfit.equation'
vcov( object, ... )

Arguments

Value

vcov.systemfit returns the variance covariance matrix of all estimated coefficients.

Author(s)

Arne Henningsen arne.henningsen@googlemail.com

See Also

systemfit, vcov

Examples

data( "Kmenta" )
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list( demand = eqDemand, supply = eqSupply )

## perform OLS on each of the equations in the system
fitols <- systemfit( system, data = Kmenta )

## variance covariance matrix of all coefficients
vcov( fitols )

## variance covariance matrix of the coefficients in the first equation
vcov( fitols$eq[[1]] )

## variance covariance matrix of the coefficients in the second equation
vcov( fitols$eq[[2]] )

## estimation with restriction by modifying the regressor matrix
modReg <- matrix( 0, 7, 6 )
colnames( modReg ) <- c( "demIntercept", "demPrice", "demIncome",
   "supIntercept", "supPrice2", "supTrend" )
modReg[ 1, "demIntercept" ] <- 1
modReg[ 2, "demPrice" ]     <- 1
modReg[ 3, "demIncome" ]    <- 1
modReg[ 4, "supIntercept" ] <- 1
modReg[ 5, "supPrice2" ]    <- 1
modReg[ 6, "supPrice2" ]    <- 1
modReg[ 7, "supTrend" ]     <- 1
fitsur <- systemfit( system, "SUR", data = Kmenta, restrict.regMat = modReg )
vcov( fitsur, modified.regMat  = TRUE )
vcov( fitsur )