systemfit (version 0.8-0)

systemfit: 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( method, eqns, eqnlabels=names(eqns),
           inst=NULL, data=list(), R.restr=NULL,
           q.restr=matrix(0,max(nrow(R.restr),0),1),
           TX=NULL, maxiter=1, tol=1e-5,
           rcovformula=1, centerResiduals = FALSE, formula3sls="GLS",
           probdfsys=!(is.null(R.restr) & is.null(TX)),
           single.eq.sigma=(is.null(R.restr) & is.null(TX)),
           solvetol=.Machine$double.eps,
           saveMemory=( nrow(data) * length(eqns) > 1000 &&
              length(data) > 0 ) )
systemfitClassic( method, formula, eqnVar, timeVar, data,
                  pooled = FALSE, ... )

Arguments

method

the estimation method, one of "OLS", "WLS", "SUR", "WSUR", "2SLS", "W2SLS", "3SLS", or "W3SLS" (see details); iterated estimation methods can be specified by setting argument maxiter larger than 1 (e.g. 500).

eqns

a list of structural equations to be estimated; a regression constant is implied if not explicitly omitted.

eqnlabels

an optional list of character vectors of names for the equation labels.

inst

one-sided model formula specifying instrumental variables or a list of one-sided model formulas if different instruments should be used for the different equations (only needed for 2SLS, W2SLS and 3SLS estimations).

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment from which systemfit is called.

R.restr

an optional 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 j x 1 matrix to impose linear restrictions (see R.restr); default is a j x 1 matrix that contains only zeros.

TX

an optional matrix to transform the regressor matrix and, hence, also the coefficient vector (see details).

maxiter

maximum number of iterations for WLS, SUR, W2SLS and 3SLS estimations.

tol

tolerance level indicating when to stop the iteration (only WLS, SUR, W2SLS and 3SLS estimations).

rcovformula

formula to calculate the estimated residual covariance matrix (see details).

centerResiduals

logical. Subtract the means from the residuals of each equation before calculating the estimated residual covariance matrix.

formula3sls

formula for calculating the 3SLS estimator, one of "GLS", "IV", "GMM", "Schmidt" or "EViews" (see details).

probdfsys

use the degrees of freedom of the whole system (in place of the degrees of freedom of the single equation) to calculate prob values for the t-test of individual parameters.

single.eq.sigma

use different $\sigma^2$s for each single equation to calculate the covariance matrix and the standard errors of the coefficients (only OLS and 2SLS).

solvetol

tolerance level for detecting linear dependencies when inverting a matrix or calculating a determinant (see solve and det).

saveMemory

logical. Save memory by omitting some calculation that are not crucial for the basic estimation (e.g McElroy's $R^2$)?

formula

formula to be estimated (for each equation).

eqnVar

variable name indicating the equation to which the observation belongs.

timeVar

variable name indicating the time.

pooled

logical, restrict coefficients to be equal in all equations.

...

arguments passed to systemfit.

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 ($*$)):

• method
  estimation method.
• g
  number of equations.
• n
  total number of observations.
• k
  total number of coefficients.
• ki
  total number of linear independent coefficients.
• df
  degrees of freedom of the whole system.
• iter
  number of iteration steps.
• b
  vector of all estimated coefficients.
• bt
  coefficient vector transformed by TX.
• se
  estimated standard errors of b.
• t
  t values for b.
• p
  p values for b.
• bcov
  estimated covariance matrix of b.
• btcov
  covariance matrix of bt.
• rcov
  estimated residual covariance matrix.
• drcov
  determinant of rcov.
• rcovest
  residual covariance matrix used for estimation (only SUR and 3SLS).
• olsr2
  System OLS R-squared value.
• mcelr2
  McElroys R-squared value for the system (only SUR and 3SLS).
• y
  vector of all (stacked) endogenous variables
• x
  matrix of all (diagonally stacked) regressors
• h
  matrix of all (diagonally stacked) instrumental variables (only 2SLS and 3SLS)
• data
  data frame of the whole system (including instruments)
• R.restr
  the restriction matrix.
• q.restr
  the restriction vector.
• TX
  matrix used to transform the regressor matrix.
• maxiter
  maximum number of iterations.
• tol
  tolerance level indicating when to stop the iteration
• rcovformula
  formula to calculate the estimated residual covariance matrix
• formula3sls
  formula for calculating the 3SLS estimator.
• probdfsys
  system degrees of freedom to calculate prob values?.
• single.eq.sigma
  different $\sigma^2$s for each single equation?.
• solvetol
  tolerance level when inverting a matrix or calculating a determinant.
• data.name
  name of the data.frame used for estimation.
• ## elements of the class systemfit.eq
• eq
  a list that contains the results that belong to the individual equations.
• eqnlabel*
  the equation label of the ith equation (from the labels list).
• formula*
  model formula of the ith equation.
• inst*
  instruments of the ith equation (only 2SLS and 3SLS).
• n*
  number of observations of the ith equation.
• k*
  number of coefficients/regressors in the ith equation (including the constant).
• ki*
  number of linear independent coefficients in the ith equation (including the constant differs from k only if there are restrictions that are not cross-equation).
• df*
  degrees of freedom of the ith equation.
• b*
  estimated coefficients of the ith equation.
• se*
  estimated standard errors of b.
• t*
  t values for b.
• p*
  p values for b.
• covb*
  estimated covariance matrix of b.
• y*
  vector of endogenous variable (response values) of the ith equation.
• x*
  matrix of regressors (model matrix) of the ith equation.
• h*
  matrix of instrumental variables of the ith equation (only 2SLS and 3SLS).
• data*
  data frame (including instruments) of the ith equation.
• fitted*
  vector of fitted values of the ith equation.
• residuals*
  vector of residuals of the ith equation.
• ssr*
  sum of squared residuals of the ith equation.
• mse*
  estimated variance of the residuals (mean of squared errors) of the ith equation.
• s2*
  estimated variance of the residuals ($\hat{\sigma}^2$) of the ith equation.
• rmse*
  estimated standard error of the residulas (square root of mse) of the ith equation.
• s*
  estimated standard error of the residuals ($\hat{\sigma}$) of the ith equation.
• r2*
  R-squared (coefficient of determination).
• adjr2*
  adjusted R-squared value.

Details

systemfitClassic is a wrapper function for systemfit that can be applied to panel-like data in long format if the regressors are the same for all equations.

If argument method is "WSUR" or "W3SLS", the "SUR" or "3SLS" estimation uses a residual variance covariance matrix that is calculated from a "WLS" or "W2SLS" estimation, respectively (and not from an "OLS" or "2SLS" estimation as for a standard "SUR" or "3SLS" estimation). The "WSUR" 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, "WSUR" and "W3SLS" generate identical results compared to "SUR" and "3SLS", respectively.

The matrix TX transforms the regressor matrix ($X$) by $X^{*} = X *$ TX. Thus, the vector of coefficients is now $b =$ TX $\cdot b^{*}$ , where $b$ is the original (stacked) vector of all coefficients and $b^{*}$ is the new coefficient vector that is estimated instead. Thus, the elements of vector $b$ are $b_i = \sum_j TX_{ij} \cdot b^{*}_j$ The TX matrix can be used to change the order of the coefficients and also to restrict coefficients (if TX has less columns than it has rows). However restricting coefficients by the TX matrix is less powerfull and flexible than the restriction by providing the R.restr matrix and the q.restr vector. The advantage of restricting the coefficients by the TX matrix is that the matrix that is inverted for estimation gets smaller by this procedure, while it gets larger if the restrictions are imposed by R.restr and q.restr.

If iterated (WLS, SUR, W2SLS or 3SLS estimation with maxit>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 formula to calculate the estimated covariance matrix of the residuals ($\hat{\Sigma}$) can be one of the following (see Judge et al., 1985, p. 469): if rcovformula=0: $$\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j}{T}$$ if rcovformula=1 or rcovformula='geomean': $$\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j} {\sqrt{(T - k_i)*(T - k_j)}}$$ if rcovformula=2 or rcovformula='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 rcovformula=3 or rcovformula='max': $$\hat{\sigma}_{ij} = \frac{\hat{e}_i' \hat{e}_j} {T - \max( k_i, k_j)}$$ If $i = j$, formula 1, 2 and 3 are equal. All these three formulas yield unbiased estimators for the diagonal elements of the residual covariance matrix. If $i \neq j$, only formula 2 yields an unbiased estimator for the residual covariance matrix, but it is not neccessarily positive semidefinit. Thus, it is doubtful whether formula 2 is really superior to formula 1 (Theil, 1971, p. 322).

The formulas to calculate 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).

References

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

Judge, George G.; W. E. Griffiths; R. Carter Hill; Helmut L�tkepohl 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.

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("OLS", system, data=Kmenta )
print( fitols )

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

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

## 2SLS estimation
inst <- ~ income + farmPrice + trend
fit2sls <- systemfit( "2SLS", system, 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( "2SLS", system, inst = instlist, data = Kmenta )
print( fit2sls2 )

## 3SLS estimation with GMM-3SLS formula
inst <- ~ income + farmPrice + trend
fit3sls <- systemfit( "3SLS", system, inst = inst, data = Kmenta,
   formula3sls = "GMM" )
print( fit3sls )


## Examples how to use systemfitClassic()
## Repeating the OLS and SUR estimations in Theil (1971, pp. 295, 300)
data( "GrunfeldTheil" )
formulaGrunfeld <- invest ~ value + capital
# OLS
theilOls <- systemfitClassic( "OLS", formulaGrunfeld, "firm", "year",
   data = GrunfeldTheil )
summary( theilOls )
# SUR
theilSur <- systemfitClassic( "SUR", formulaGrunfeld, "firm", "year",
   data = GrunfeldTheil, rcovformula = 0 )
summary( theilSur )


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