*
* Chapter 4 exercise. System of factor demands
* From pp 317-320
*
open data greene.xls
data(org=columns,format=xls) 1 99 id tc q pl pk pf sl sk
*
* id = firm ID
* tc = total cost
* q  = output in kilowatt hours
* pl = price of labor
* pk = price of capital services
* pf = price of fuel
* sl = labor factor share
* sk = capital factor share
*
* Define sf as residual from labor and capital
*
set sf = 1-sl-sk
*
* (a) Simple statistics
*
table / q sl sk sf
*
* (b) Random-effects estimation.
*
* First do multivariate regression without cross-equation restrictions. Since the
* regressors are the same across equations, multivariate regression w/o
* cross-equation restrictions is equivalent to equation by equation OLS.
*
* Estimation with capital share equation dropped
*
* Define parameters.
*
nonlin al af gll glf gfl gff gly gfy
*
* Define two equation system
*
frml labor sl = al+gll*log(pl/pk)+glf*log(pf/pk)+gly*log(q)
frml fuel  sf = af+gfl*log(pl/pk)+gff*log(pf/pk)+gfy*log(q)
*
* Set starting values for the parameters.
*
compute al=af=gll=glf=gfl=gff=gly=gfy=0
*
* Now do multivariate regression w/o cross-eq. restriction.
*
nlsystem(outsigma=sigma) / labor fuel
*
* Now impose cross-equation restriction on fuel share equation, that is, make
* gfl=glf. There are two ways to do this
*
* Method (1): reparameterize the equations
*
* Define parameters.
*
nonlin al af gll glf gff gly gfy
*
* Impose symmetry
*
frml fuelb sf = af+glf*log(pl/pk)+gff*log(pf/pk)+gfy*log(q)
*
* Multivariate regression with covariance matrix calculated from
* equation-by-equation OLS residuals
*
nlsystem(cv=sigma) / labor fuelb
*
* Method (2): impose the restriction in the parameter set, and use the original
* equations
*
nonlin al af gll glf gfl gff gly gfy glf=gfl
nlsystem(cv=sigma) / labor fuel
*
* (c) Calculate Sargan's statistic
*
* Step (1). Get the residuals
*
set ul = sl-labor
set uf = sf-fuel
*
* Step (2). Compute a cross product matrix including the explanatory variables
* (=instruments), and the residuals. This will include all the pieces that we
* need for computing the Sargan statistic
*
set ploverk = log(pl/pk)
set pfoverk = log(pf/pk)
set logq    = log(q)
*
cmom(print)
# constant ploverk pfoverk logq ul uf
*
* Step (3). Compute the statistic. sxy-delta*sxz is (here) just the sum of u
* kroneker x, divided by n. That will just be the part of the %cmom where the u's
* intersect the x's. x'x will be the top submatrix
*
compute sxu=%xsubmat(%cmom,1,4,5,6)
compute sxx=%xsubmat(%cmom,1,4,1,4)
compute sargan=%qform(%kroneker(inv(sigma),inv(sxx)),%vec(sxu))
cdf(title="Sargan Statistic") chisqr sargan 1
*
* (e) Wald test of symmetry. Go back to the unconstrained estimates
*
nonlin al af gll glf gfl gff gly gfy
nlsystem(cv=sigma,vcv) / labor fuel
*
* Test that the difference between coefficients 4 and 5 is 0
*
restrict(title="Wald Test of Symmetry") 1
# 4 5
# 1 -1 0
*
* (f) Calculate average labor-capital substitution elasticity
*
* Redo the restricted estimates
*
nonlin al af gll glf gfl gff gly gfy glf=gfl
nlsystem(cv=sigma) / labor fuel
*
* Compute the missing parameters
*
compute ak =1-al-af
compute glk=-(gll+glf)
compute gfk=-(gfl+gff)
compute gkk=-(gfk+glk)
compute gky=-(gly+gfy)
*
* Compute the fitted cost shares for labor and capital
*
set slfit = al+gll*log(pl/pf)+glk*log(pk/pf)+gly*log(q)
set skfit = ak+glk*log(pl/pf)+gkk*log(pk/pf)+gky*log(q)
set elk = (glk+slfit*skfit)/(slfit*skfit)
table / slfit skfit elk