Econometric Problem Solving

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Econometrics Problem-solving
(a) Prove that E(bR) = β1 + Pβ2.
First, it is essential to express bR in terms of ∈:
bR = (X’1 X1) -1 X’1y
= (X’1 X1) -1 X’1 (X1 bI + X2b2 + e)
= (X’1 X1) -1 X’1 X1 bI + (X’1 X1) -1 X’1 X2 β2 + (X’1 X1) -1 X’1 e
=I β1 + (X’1 X1) -1 X’1 X2 β2 + (X’1 X1) -1 X’1 ∈
= β1 + P β2 + (X’1 X1) -1 X’1 ∈
By definition
y =X1 β1 + X2 β1 + ∈
By matrix algebra
(X’1 X1) -1 X’1 X1 = I
(X’1 X1) -1 X’1 X2 = P
Therefore, E(bR) is:
E(bR) = E( β1 + P β2 + (X’1 X1) -1 X’1 ∈)
= β1 + P β2 + (X’1 X1) -1 X’1 E(∈) β, X fixed (A2, A6)
= β1 + P β2 E(∈) = 0 (A3)
(b) Prove that var(bR) = σ 2 (X 0 1X1) −1
First, we define the br – E(br) expression in terms of X1, with part (a) findings.
bR – E(br) = [β1 + Pβ2 + (Xi X1) 1Xie]  – [β1 + Pβ2 ]  by (a)
= (Xi’ X1) 1Xie
Then we establish var (bR):
var (bR): = E([bR – E(bR)] [bR – E(bR)]1 ) by Probability theory
= E( [(X’1 X1) -1 X’1 e] [(X1’ X1 ) -1 X1’ ∈]’)
= E( [(X’1 X1) -1 X’1 e] [∈’(X1’)’((Xi X1)-1)’] ) (X’1 X1)-1 Symmetric matrix
= E( [(X’1 X1) -1 X’1 e] [∈’XI)( X’1 X1)-1] )X, X1 fixed (A2)
= E( (X’1 X1) -1 X’1 ee’ XI (X1’ X1)-1] )α2 =E(e2) (A4, A5)
= (X’1 X1) -1X1’ E(ee’) XI (X’1 X1)-1α2 scalar
= (X’1 X1) -1X1’ α2 I XI (X’1 X1)-1(X’1 X1) -1 X1’ X1 = 1
= α2 (X’1 X1) -1X1’ XI (X’1 X1)-1
= α2 (X’1 X1)-1

(c) Prove that bR = b1 + Pb2.
Determining bR:
bR = (X’1 X1) -1 X’1 y
= (X’1 X1) -1 X’1 (X1 bI + X2b2 + e)
= (X’1 X1) -1 X’1 X1 bI + (X’1 X1) -1 X’1 X2 b2 + (X’1 X1) -1 X’1 e
=Ib1 + (X’1 X1) -1 X’1 X2 b2 + (X’1 X1) -1 X’1 e
= b1 + Pb2 + (X’1 X1) -1 X’1 e
= b1 + Pb2
By definition
Y =X1b1 + X2b2 +e
By matrix algebra
(X’1 X1) -1 X’1 X1 = I
(X’1 X1) -1 X’1 X2 = P
X’1 e = O, due to orthogonality
(d) PP matrix columns
Let us presume the column of the (2 x 32 x 3) matrix PP are attained by regressing each of the variables, which are Age, Educ, and Partime on a continuous basis and the variable Female.
The element of X1X1 and X2X2 matrices, are as follows:
X1X1 is an (nx2nx2) matrix, with 22 columns for the constant term and the ‘Female’
X’1X1’ IS ITS 2×n2×n) transpose.
And X2X2 is a (n×3n×3) matrix, with 33 columns for the ‘Age’, ‘Educ’ and ‘Parttime’ variables.
Thus, (X′1X1)−1(X1′X1)−1 is a (2×22×2) matrix.
and (X′1X1)−1X′1(X1′X1)−1X1′ is a (2×n2×n) matrix.
So, P=(X′1X1)−1X′1X2P=(X1′X1)−1X1′X2 is a (2×32×3) matrix, with columns as such:
Column 1: (X′1X1)−1X′1(Age)(X1′X1)−1X1′(Age); the OLS formula for regressing ‘Age’ on X1X1.
Column 2: (X′1X1)−1X′1(Educ)(X1′X1)−1X1′(Educ); the OLS formula for regressing ‘Educ’ on X1X1.
Column 3: (X′1X1)−1X′1(Parttime)(X1′X1)−1X1′(Parttime); the OLS formula for regressing ‘Parttime’ on X1X1.
(e) PP Matrix Value.
To determine the values of PP from the results based on Lecture 2.1, the following analyses were computed and produced by an R script, as shown in figures below.
Dataset loading in R studio

Then, X1X1 (n×2n×2) matrix was built from the dataset as shown below.

Based on here the matrix is X′1X1′ (2×n2×n) is transposed as shown below.

Building X2X2 (n×3n×3) matrix as shown in figure below.

After building the X2X2 (n×3n×3) matrix, then its transformed to (X′1X1)−1(X1′X1)−1 (2×22×2) inverse as shown in figure below.

Then followed by (X′1X1)−1X′1(X1′X1)−1X1′ (2×n2×n) matrix as displayed in figure below.

Lastly, creating the P=(X′1X1)−1X′1X2P=(X1′X1)−1X1′X2 (2×32×3) matrix, as shown in figure below.

So, PP matrix can be rounded as:
P= [40.05−0.112.26−0.490.20.25]
(f) Part (c) Checking for numerical validity
When checking for the numerical validity of the results generated in part (c), the equation used cannot hold the exact results since the coefficients were rounded to either two or three decimal places. The exactness of the results would have been attained for a higher precision coefficients (Kleiber, Christian, & Achim, 2008). Therefore, in lecture 2.5, the OLS regression equation was calculated as:
Log(Wage)i=3.053-0.041Femalei+0.031Agei+0.233Educi-0.365Partimei+ei
The equation determines the following;

In part (e), PP matric was calculated as shown below;

In part (c) it was shown that: bR=b1+Pb2bR=b1+Pb2, which can be computed as;
bR =b1+Pb2

This resonates with the bR defined in Lecture 2.1 by the “Log(Wage)” OLS regression on the constant and “Female” variable:
Log(Wage)i=4.73-0.25Femalei+ei
Which determined bR as follows:

Basically, minor bR values difference is explained by the reduced precision because of the OLS coefficient’s calculation rounding values (Kleiber, Christian, & Achim, 2008).

Work Cited
Kleiber, Christian, and Achim Zeileis. Applied Econometrics with R. New York: Springer, 2008. Internet resource.