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Macroeconometrics 1: The Basic Building Blocks

Macroeconometrics 1: The Basic Building Blocks
(Graded Homework Problems)
Brendan Epstein, Ph.D.
Johns Hopkins University
1. Let xs;t be the 1 k vector of explanatory variables for observation s such that
Xt =
2
6
6
6
4
x1;t
x2;t
.
.
.
xn;t
3
7
7
7
5
=
2
6
6
6
4
x11;t x12;t    x1k;t
x21;t x22;t    x2k;t
.
.
.
.
.
.
.
.
.
.
.
.
xn1;t xn2;t    xnk;t
3
7
7
7
5
,
so that for s = 1; :::; n
xs;t =

xs1;t xs2;t    xsk;t
,
and
yt =
2
6
6
6
4
y1;t
y2;t
.
.
.
yn;t
3
7
7
7
5
,
where s = 1; 2; :::; n. Show that the OLS estimator ^ can be written as:
^ =
Xn
s=1
x
0
s;txs;t!1 Xn
s=1
x
0
s;tys;t!
.
2. This problem continues the one with the same background as in this Lessonís practice
problems with solutions. Let ^ be the OLS estimate from the regression of yt on Xt
.
Let At be a k k non-singular matrix and deÖne:
zs;t  xs;tAt
for s = 1; :::; n. Therefore, zs;t is 1 k and is a nonsingular linear combination of
xs;t. Let Zt be the n k matrix with rows zs;t. Let ~ denote the OLS estimate
These lecture notes closely and sometimes literally follow sections from: Dowling, Edward T. Introduction to Mathematical Economics. Shaumís Outlines, 3rd ed., McGraw Hill, 2001; Nicholson, Walter.
Microeconomic Theory: Basic Principles and Extensions. South-Western College Pub, 9th ed., 2004; Simon,
Carl P., and Lawrence Blume. Mathematics for Economists. New York: Norton, 1994; Sydsaeter, Knut, and
Peter J. Hammond. Mathematics for Economic Analysis. Prentice Hall, 1995.

from a regression of yt on Zt
. Show that the estimated variance matrix for ~ is
^
2
“A1
t
(X0
tXt)
1

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A1
t
0
.
3. This problem continues the preceding one as well as the one with the same background
as in this Lessonís practice problems with solutions. Let ^ be the OLS estimate from
the regression of yt on Xt
. Let At be a k k non-singular matrix and deÖne:
zs;t  xs;tAt
for s = 1; :::; n. Therefore, zs;t is 1 k and is a nonsingular linear combination of xs;t.
Let Zt be the n k matrix with rows zs;t. Let ~ denote the OLS estimate from a
regression of yt on Zt
. Let ^
j be the OLS estimates from a regression of yt on:
1; x2;t; :::; xk;t;
and, let ~
j be the OLS estimates from a regression of yt on:
1; a2x2;t; :::; akxk;t,
where the as are constants and aj 6= 0 for j = 2; :::; k. State the mathematical relationship between ~
j and ^
j
.
4. Consider the following lagged income determination model:
Ct = 90 + 0:8Yt1;
It = 50 8t;
Y0 = 1200.
In equilibrium, Yt = Ct +It akin to the development in this Lessonís practice problems
with solutions.
(a) Find the time path of national income Yt
.
(b) Characterize the stability of the time path obtained in part (a