Economics and non-linear regression model
1.
(a) Consider a non-linear regression model
y =0 + x1 + x
22 + u = (x) + u (1)
where y is a dependent variable, x is a scalar independent (explanatory)
variable, and u is an error such that E(u j x) = 0 and V(u j
x) =
2
: Write the OLS and MM procedures (you may denote
x
2 = z) which will give identical sample estimators ^
0
; ^
1
; ^
2 of population
parameters 0
; 1
; 2
respectively based on data fyi
; xig; i =
1; ; n. Are your estimators BLUE? Also, write an estimator of
(x).
(b) Based on your estimators in Q1 (a), write a sample estimator of the
population economics parameter of interest @y=@x = @(x)=@x =
(x). What is the meaning of this estimator? Is this estimator
unbiased? Write a sample average estimator of population average
E[(x)].
(c) Suppose an econometrician estimates a false linear model y = 0 +
x1 +v, where v = x
22 +u. Show bias in OLS estimator of 1
from
this false model. Under what conditions this bias will be positive?
(d) Write two sided test of linearity using model (1) which you can
implement for a given data. How will you extend linearity test
when you have two variables non-linear model in x1; x2 written as
y = 0+x11+x22+x
2
13+x
2
24+x1x25+u? (Hint: Write F-test.)
Alternatively, write y = 0 + x11 + x22 + ^y
23 + u. Write your
test for linearity in this case, and write your expression for y: ^ (For
t-test and F-test, see 107 Review Notes circulated, pp.9-11 and Ch.4
of textbook.) Which is better to use and why?
2.
(a) Using the dataset WAGE2 do the parametric linear regression model
of log wage on education, experience, and IQ score. Explain the
meaning of the estimates of their coe¢ cients, and test the statistical
signiÖcance of each coe¢ cient at 5% level of signiÖcance. Now
consider the regression of log wage on education only. Is coe¢ cient
of education (return to education) now di§erent from your previous
regression result? If di§erent, is it due to a mis-speciÖcation bias
(exclusion of variables)?
1
(b) For WAGE2 data, run a nonparametric (data based) regression of
log wage on education. Is this linear on non-linear? Calculate and
plot varying return to education coe¢ cient and interpret your result.
Also, calculate average return to education and compare with the
parametric estimate based on linear model in Q2 (a).
3. Using the WAGE2 data, estimate the conditional variance of error (heteroskedasticity)
by nonparametric method, and by parametric method
specifying
V(u j x) =E(u
2
j x)
=exp( 0 + 1x)
or
u
2 =[exp( 0 + 1x)]v
where x is education and v is error with E(v j x) = 1. Plot estimated conditional
variances by nonparametric and parametric methods. Interpret
your plots about the nature of heteroskedasticity (conditional variances)
appearing in log wage.
4. Consider a linear regression model
yi = 0 + 1xi + ui
; i = 1; ; n (2)
where E(ui
j xi) = 0 and V(ui
j xi) =
2
(xi).
(a) Show that the OLS estimator of 1
in Equation (2) is unbiased but
its variance is not the same as when V(ui
j xi) =
2
(homoskedastic).
(b) Show that GLS estimator of 1
is BLUE, but not OLS estimator.
(c) At 5% level of signiÖcance, test the presence of heteroskedasticity,
that is conditional variance in Equation (2) is correct, by BreuschPagan
as well as White tests. Write your null and alternative hypotheses
clearly and indicate your decision rules.