Nonlinear Equation System Estimation

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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

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

Arguments

method

the estimation method, one of "OLS", "SUR", "2SLS", "3SLS".

eqns

a list of structural equations to be estimated.

startvals

a list of starting values for the coefficients.

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, 3SLS and GMM estimations).

data

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

solvtol

tolerance for detecting linear dependencies in the columns of X in the qr function calls.

maxiter

the maximum number of iterations for the nlm function.

...

arguments passed to nlm.

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:

ε_{t} = q( y_t, x_t, θ )

z_{t} = Z( x_t )

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

The objective functions for the methods are:

Method Instruments Objective Function Covariance of θ
OLS no r'r (X(diag(S)^{-1}\bigotimes I)X)^{-1}
SUR no r'(diag(S)_{OLS}^{-1}\bigotimes I)r (X(S^{-1}\bigotimes I)X)^{-1}
2SLS yes r'(I \bigotimes W)r (X(diag(S)^{-1}\bigotimes I)X)^{-1}
3SLS yes r'(S_{2SLS}^{-1} \bigotimes W)r (X(diag(S)^{-1}\bigotimes W)X)^{-1}

where, r is a column vector for the residuals for each equation, S is variance-covariance matrix between the equations (\hat{σ}_{ij} = (\hat{e}_i' \hat{e}_j) / √{(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 (*)):

eq

a list object that contains a list object for each equation.

method

estimation method.

resids

an n \times g matrix of the residuals.

g

number of equations.

n

total number of observations.

k

total number of coefficients.

b

vector of all estimated coefficients.

se

estimated standard errors of b.

t

t values for b.

p

p values for b.

bcov

estimated covariance matrix of b.

rcov

estimated residual covariance matrix.

drcov

determinant of rcov.

rcovest

residual covariance matrix used for estimation (only SUR and 3SLS).

rcor

estimated residual correlation matrix.

nlmest

results from the nlm function call

solvetol

tolerance level when inverting a matrix or calculating a determinant.

## elements of the class nlsystemfit.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.

n*

number of observations of the ith equation.

k*

number of coefficients/regressors in the ith 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.

predicted*

vector of predicted 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{σ}^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{σ}) of the ith equation.

r2*

R-squared (coefficient of determination).

adjr2*

adjusted R-squared value.

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

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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 )

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