Binghamton University, SUNY Department of Economics
Course : Econ 466 - Econometrics
Section : B1
Term : Spring 2019
Instructor : Luis D. Chanc A. ()
T.A. : Shi Zuo ()
Problem Set No 2
Due 02/06/2019 , 8:00 p.m.
1 Estimation.
1. Let, ^ = P(Xi X)(Yi Y)=P(Xi X)2. Show that ^ = P(Xi X)Yi=P(Xi X)2
2. Consider the following regression Yi = 0 +ui (regression without any regressor).
Show that the OLS estimate ^ 0 of 0 is Y = PYi=N.
3. Consider the following regression Yi = Xi +ui (there is not intercept).
Show that the OLS estimator of is ^ = (PXiYi)=(PX2i ).
4. Consider the linear model: Yi = 0 + 1(Xi X) +ui.
Find the OLS estimators of 0 and 1. Compare with the OLS estimators of 0 and 1 in the standard
model discussed in class (Yi = 0 + 1Xi +ui).
5. Based on a sample of 10 observations the following results were obtained:P
Yi = 1;110 ; PXi = 1;700 ; PXiYi = 205;500 ; PX2i = 322;000 ; PY2i = 132;100
Show that the OLS estimate ^ 0 and ^ 1 for the model Yi = 0 + 1Xi+ui are ^ 0 = 24:5 and ^ 1 = 0:509.
2 ANOVA.
1. Let, Yi = 0 + 1Xi +ui and ^ 1 = P(Xi X)(Yi Y)=P(Xi X)2.
Show that SSE = P( ^Yi ^Y)2 = ^ 21(PX2i N X2).
2. Consider the following regression Yi = 0 + ui (regression without any regressor), and let ^ 0 be the
OLS estimate of 0. Find the variance of ^ 0 and the SSR.
3. Consider the following regression Yi = Xi +ui (there is not intercept), and let ^ be the OLS estimate
of . Find the variance of ^ .
4. Consider the following regression Yi = Xi +ui (there is not intercept), and let ~ be the OLS estimate
of . Let ^ 1 be the OLS estimate of 1 for the standard model discussed in class (Yi = 0 + 1Xi +ui).
Show that Var( ~ ) Var( ^ 1).
(Hint: for any sample of data, PX2i P(Xi X)2, with strict inequality unless X = 0)
5. Consider the linear model: Yi = 0 + 1(Xi X) +ui. Let ^ 0 and ^ 1 be the OLS estmates. Find the
variance of ^ 0 and ^ 1. Compare with the variance of the OLS estimator of 0 and 1 in the standard
model discussed in class (Yi = 0 + 1Xi +ui).
6. Based on a sample of 10 observations the following results were obtained:P
Yi = 1;110 ; PXi = 1;700 ; PXiYi = 205;500 ; PX2i = 322;000 ; PY2i = 132;100
Show that SSE 8549:67
ECON 466 Pag. 1 out of 2 Spring 2019
Binghamton University, SUNY Department of Economics
7. Let r1 be the coe cient of correlation between n pairs of values (Yi;Xi) and r2 be the coe cient of
correlation between n pairs of values (aYi +b;cXi +d) where a, b, c, and d are constants. Show that
r1 = r2 and hence establish the principle that the coe cient of correlation is invariant with respect to
the change of scale and the change of origin.
Hint:
r1 = rYX = Cov(X;Y)pVar(X)Var(Y) N
PX
iYi
PX
i
PY
ip
[NPX2i (PXi)2][NPY2i (PYi)2]
8. If r, the coe cient of correlation between n pairs of values (Xi;Yi), is positive, then determine whether
each of the following statements is true or false:
(a) r between ( Xi; Yi) is also positive.
(b) r between ( Xi;Yi) and that between (Xi; Yi) can be either positive or negative.
9. Let ^ YX and ^ XY represent the slopes in the regression of Y on X (Y = 0 + YXX + u) and X on
Y (X = 0 + XYY +v), respectively. Show that ^ YX ^ XY = r2.
ECON 466 Pag. 2 out of 2 Spring 2019