Macroeconometrics 1: The Basic Building Blocks
宏观计量经济学作业代写 These lecture notes closely and sometimes literally follow sections from: Dowling, Edward T. Introduction to Mathematical Economics.
1.Find the product A⋅B using 宏观计量经济学作业代写
and
Also, does the product B ⋅ A exist? Please explain
2.Consider the matrix
*These lecture notes closely and sometimes literally follow sections from: Dowling, Edward T. Introduction to Mathematical Economics. Shaum’s 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, andPeter J. Hammond. Mathematics for Economic Analysis. Prentice Hall, 1995.
Let
N = det [Z] .
What is N equal to?
3.Consider the matrix
Let
W =Z–1.
What is Z–1 equal to? (Please show your work.)
4.Consider the matrix
Find the eigenvalues X.
5.Let X be any n $ k matrix. Show that X0 X is a symmetric matrix.
6.Use the definition of inverse to prove the following: if A and B are n $ n nonsingular matrices, then (A⋅B)–1 = B–1A–1 .
7.Let A be an n × n symmetric, positive definite matrix. Show that if P is any n×n nonsingular matrix, then P’⋅A⋅P is positive definite. Note: a symmetric matrix A is said to be ìpositive definite if x’⋅A⋅x > 0 for all n × 1 vectors x except x = 0.
8.Let 
be the OLS estimate from the regression of yt on Xt. Let At be a k × k non-singular matrix and define: 宏观计量经济学作业代写
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 Zt denote the OLS estimate from a regression of yt on Zt. Show that 
=At-1.
9.This problem is a continuation of the preceding one. Let be the OLS estimate from the regression of yt on Xt. Let At be a k × k non-singular matrix and define:
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. Show that 
for all 1,2,…, n. 宏观计量经济学作业代写
10.Consider a simple model of “lagged income determination.” In this model we assume that consumption is a function of the previous period’s income, and the economy is closed and there is no government sector. As such,
Ct = Co + cYt-1
and 宏观计量经济学作业代写
Yt ≡ Ct + It.
Furthermore, let It = I0 8t. Combining these two mathematical statements yields:
Yt = Co + Io + cYt-1.
Assume Yt = Yo at t = 0. What is the definite solution for Yt? Also, describe the stability condition for this equation.
