* Analysis of Nonlinear Consumption Function
*
* Keywords:
*     regression, nonlinear, consumption, function, f-test, lagrange
*
* Description:
*     We illustrate how to estimate a Nonlinear specificatoin of the Consumption
*     Function and perform hypothesis testing and confidence interval estimation, 
*     specifically we do an F-test and a Lagrange Multiplier test
*
* Author(s):
*     Noel Roy
*     Skif Pankov
*
* Source:
*    William H. Greene, Econometric Analysis - 7th Edition
*    Pearson International Edition, Chapter 7, Example 7.4 (page 231)
*
* Reading the datafile and naming the variables, specifying to ignore the
* first line of the file
read (TableF5-2.shd) year qtr gdp c inv g y cpi m1 r / skiplines=1

* Estimating the linear OLS model
ols c y

* Estimating a nonlinear model of consumption on output with 1 equation and 3 
* coefficients to estimate, specifying to print the covariance matrix and save 
* estimated coefficients in a vector x
nl 1 / ncoef=3 pcov coef=x
   eq c=alpha+beta*y**gamma
end

* Testing a coefficient restriction on a previous model 
test gamma=1

* Testing the hypothesis that the mpc in the previous model equals 1
test beta*gamma*6634.9**(gamma-1)=1

* Even if convergence takes place, all that can be assured is convergence
* to a local maximum. It is good practice in non-linear estimation to try 
* several starting values for the coefficients in order to ensure that 
* convergence to a global maximum has occurred. In the present case, 
* since the estimate for gamma is greater than 1, it would be worthwhile to 
* try a starting value less than 1, say 0.5
genr newy=y**0.5
?ols c newy /coef=newb

* An alternative to the coef command is to read starting values into a 
* vector and specify this vector with the start= option. The starting 
* values must be placed in the order they appear in the eq command
dim st 3
gen1 st(1)=newb(2)
gen1 st(2)=newb(1)
gen1 st(3)=0.5
nl 1 / ncoef=3 pcov start=st iter=250
eq c=alpha+beta*y**gamma

end