232 ANSWERS CHAPTER 6
6.7.a. The data can be generated in EViews by means of the following program.
create exer6_7 u 1 200 ’ create workfile
genr x = @trend(0) ’ generate series x, ystar and y
genr ystar = -10 + 0.1*x + nrnd
genr y = (ystar >= 0)
b. The theoretical odds ratios and the odds ratios in the sample can be computed as follows.
vector(5) tx ’ stores the 5 values of x
vector(5) tratio_b ’ theoretical odds ratios
vector(5) ratio_b ’ odds ratios in sample
tx.fill 60, 80, 100, 120, 140
for !i = 1 to 5 ’ compute odds ratios
tratio_b(!i) = @cnorm(-10+0.1*tx(!i))/(1-@cnorm(-10+0.1*tx(!i)))
!s1 = (35+!i*20) ’ begin of sample
!s2 = (45+!i*20) ’ end of sample
smpl !s1 !s2 ’ adjust sample
if @mean(y) <> 1 then
ratio_b(!i) = @mean(y)/(1-@mean(y))
else
ratio_b(!i) = na
endif
next ’ end of for loop
smpl 1 200
scat x ystar ’ scatter diagram
The theoretical and sample odds ratios are shown in Table S 6.2. The sample odds ratio for
95 ≤xi≤105 (1.2) is quite close to the theoretical value of 1 for x= 100. For 55 ≤xi≤65
and 75 ≤xi≤85 all observed values of yiare zero, so the sample odds ratio is also zero. Note
that both intervals contain 11 observations, so that these outcomes could be expected on
the basis of the theoretical odds ratios (of zero and 0.023 respectively). Further, the sample
odds ratios for the intervals 115 ≤xi≤125 and 135 ≤xi≤145 can not be determined as all
yihave the value 1 in these two subsamples (as could be expected from the large theoretical
odds ratios for x= 120 and x= 140).
Figure S 6.3 shows the scatter diagram of y∗against x. Clearly, around x= 60 and x= 80
there are no values y∗>0, so that y= 0 in these intervals. For instance, for x= 80 there
holds y∗=−10+8+εiso that P[y∗>0] = P[εi>2] = 0.023. For similar reasons, around
x= 120 and x= 140 there are no values y∗<0 (although in our simulation for xi= 127 the
corresponding value y∗
i= 0.479 comes quite close to zero), so that y= 1 in these intervals.
c. The results of OLS are shown in Table S 6.3, and the corresponding estimated odds ratios
are given in the third column of Table S 6.4. The OLS estimates of the odds ratios are very
imprecise. For small values of xthe odds ratio is much overestimated, whereas for large
values of xit is much underestimated. That is, the estimated effect of xon the odds ratio
is much smaller than the actual effect.
