* Analysis of Proportions Data
* References:
* William Greene, Econometric Analysis, Fourth Edition,
* Chapter 19.4.6, pp. 834-837.
* Gujarati, Basic Econometrics, Third Edition,
* Chapter 16, pp. 556-561.
SAMPLE 1 10
* Model of home ownership
* Data set from Gujarati, Table 16.4, p. 557.
* X = family income - thousands of $
* NTOT = number of families in the group
* N = number of families owning a house.
READ X NTOT N
6 40 8
8 50 12
10 60 18
13 80 28
15 100 45
20 70 36
25 65 39
30 50 33
35 40 30
40 25 20
* Calculate the proportion of families that own a house in
* each income group.
GENR P=N/NTOT
* Calculate the odds ratio in favor of owning a house.
GENR ODDS=P/(1-P)
* Calculate the logit -- the proportions are in the interval (0,1)
* but the logit has the range on the real number line.
GENR L=LOG(ODDS)
* METHOD A: WLS (see Gujarati)
* Consider heteroskedastic errors - calculate the weights
GENR W=NTOT*P*(1-P)
* Weighted Least Squares
OLS L X / WEIGHT=W NONORM NOMULSIGSQ
* Estimate the percent change in the weighted odds in favour of owning
* a house for a unit increase in weighted income (X).
* With the TEST command, this is reported as the TEST VALUE.
TEST 100*(EXP(X)-1)
* METHOD B: 2-step WLS (see Greene)
* Use a 2-step estimation procedure
* Step 1: OLS
OLS L X / PREDICT=YHAT COEF=BETA
* Calculate weights for the logit model
GENR PHAT=EXP(YHAT)/(1+EXP(YHAT))
GENR W=NTOT*PHAT*(1-PHAT)
* Step 2: Weighted Least Squares
* The UT option is used to obtain PREDICTed values that are
* UnTransformed (calculated with the unweighted data).
OLS L X / WEIGHT=W NONORM NOMULSIGSQ UT PREDICT=LHAT COEF=BHAT
TEST 100*(EXP(X)-1)
* Predict the probability of owning a house at various income levels.
GENR PHAT=EXP(LHAT)/(1+EXP(LHAT))
* Calculate marginal effects - the change in the probability of
* owning a house per unit change in income.
GENR MARGINAL=(BHAT:1)*PHAT*(1-PHAT)
FORMAT(F12.0,2F12.3)
PRINT X PHAT MARGINAL / FORMAT
* METHOD C: Maximum Likelihood Estimation - Probit Model
* Specify the log-likelihood function (see Greene, p. 836).
FBX: NCDF(B1*X+B0)
NL 1 / NCOEF=2 LOGDEN START=BETA
EQ NTOT*(P*LOG([FBX])+(1-P)*LOG(1-[FBX]))
END
TEST 100*(EXP(B1)-1)
STOP