ETW3510 - Applied Econometric Methods Assignment 1 - University 2019

This document provides a comprehensive solution to an Applied Econometric Methods assignment, addressing key concepts and techniques in econometrics. The assignment covers topics such as the variance of a variable, unconditional expectation, and the variance-covariance matrix. It delves into the order condition of simultaneous equations, reduced form equations, and their applications. The solution also examines the order conditions for demand and supply functions, along with the appropriate methods for estimating their parameters, including the indirect least squares (ILS) method. The assignment aims to assess the student's understanding of econometric theory and their ability to apply it to real-world research, particularly in the context of simultaneous equation systems. The solution demonstrates the application of econometric models to analyze economic relationships, including demand and supply for money, while also discussing the identifiability of structural equations.

Running head: APPLIED ECONOMETRIC METHOD
Applied Econometric Method
Name of the Student:
Name of the University:
Author Note:

1APPLIED ECONOMETRIC METHOD
Table of Contents
Answer 1....................................................................................................................................2
Variance of y..........................................................................................................................2
Answer 2....................................................................................................................................2
Unconditional Expectation of y.............................................................................................2
Variance-Covariance Matrix of y..........................................................................................2
Answer 3....................................................................................................................................3
Order Condition of Simultaneous Equation...............................................................................3
Answer 4....................................................................................................................................4
Reduced Form Equation.........................................................................................................4
Order Condition.....................................................................................................................5
Answer 5....................................................................................................................................6
Order Condition of Demand Function...................................................................................6
Order Condition of Supply Function......................................................................................6
Appropriate Method to Estimate the Parameter of Demand Function...................................6
Appropriate Method to Estimate the Parameter of New Supply Equation............................7
Reference:..................................................................................................................................8

2APPLIED ECONOMETRIC METHOD
Answer 1
Variance of y
Var ( y ) =E [ y −E ( y ) ]2
Var ( y )=E [ X β +u−E ( X β +u ) ]2
Var ( y )=E [X β +u−E ( X β ) −E(u)]2
Var ( y )=E [X β +u−X β−0]2
Var ( y ) =E [u¿¿ 2]¿
Var ( y ) =E(uu ¿¿ ')¿
Var ( y )=I σ2
Answer 2
Unconditional Expectation of ^y
E ( ^y ) =E ( X ^β )
E ( ^y )= XE( ^β)
E ( ^y )= X β
Variance-Covariance Matrix of ^y
cov ( ^y )=E ( ^y− y )( ^y− y )'
cov ( ^y )=E [{ X ( X' X ¿¿¿−1 X' }( X β+u)−X β ][{X ( X' X ¿¿¿−1 X' }(X β +u)−X β ]'
cov ( ^y )=E ¿
cov ( ^y ) =E [ X ( X' X ¿¿¿−1 X' u)][ X ( X' X ¿¿ ¿−1 X ' u)]'

3APPLIED ECONOMETRIC METHOD
cov ( ^y )=E ¿
cov ( ^y ) = X ¿
cov ( ^y ) = X ¿
cov ( ^y )=I σ2 X ¿
cov ( ^y )=I σ2 X ¿
Answer 3
Order Condition of Simultaneous Equation
By the definition 1, is the number of excluded variable is R then it must be equal or
greater than M-1. The number of excluded endogenous variable is (M-m) where “M”
represents number of endogenous variables in a model and “m” represents the number of
endogenous variables present in the equation. The number of excluded predetermined
variable is (K-k) where “K” represents the number of predetermined variables in a model and
the “k” represents the number of predetermined variables in the equation (Wooldridge 2015).
That means:
R ≥ ( M −1 )
( M −m ) +( K−k) ≥ ( M −1 )
Now, the new equation after subtracting (M-m) from the both sides of the above
equation is:
( K −k )≥ ( m−1 )
The above equation is nothing but the equation form of the definition 2. Now, adding
k in both sides of the above equation gives the following result:

4APPLIED ECONOMETRIC METHOD
K ≥ ( m+k−1 )
The above equation is the alternative order condition of identifiability.
Answer 4
Reduced Form Equation
Pt =α0+ α1 ( Qt
Nt )+α 2 ( Y t
Nt )+α3 Ft +u1 t
Qt =β0+ β1 ( Pt
Wt ) +β2 Pt −1+ β3 Ct −1+ β4 T t −1 +u2 t
Reduced form of demand function:
Pt =π0 +π 1 ( Y t
Nt )+ π2 Ft + π3 Pt−1 +π 4 Ct −1+π5 T t−1 +vt
Where
π0= α 0 +α1 β0
1−α1 β1
π1= α2
1−α 1 β1
π2= α3
1−α 1 β1
π3= α1 β2
1−α 1 β1
π4 = α1 β3
1−α1 β1
π5= α1 β4
1−α 1 β1

5APPLIED ECONOMETRIC METHOD
vt= u1 t
1−α1 β1
Reduced form of supply function:
Qt =π6 +π 7 ( Y t
N t )+ π8 Ft + π 9 Pt −1+ π10 Ct −1+π11 T t −1 + wt
Where
π6= β0 +α0 β1
1−α1 β1
π7= α2 β1
1−α1 β1
π8= α3 β1
1−α1 β1
π9= β2
1−α1 β1
π10= β3
1−α1 β1
π11= β4
1−α 1 β1
wt= u2 t
1−α1 β1
Order Condition
The model has 2 endogenous variables and 7 predetermined variables.
For the demand equation (m+k-1) = (1+6-1) = 6 which is less than K (=7).
For the demand equation (m+k-1) = (1+6-1) = 6 which is less than K (=7).

6APPLIED ECONOMETRIC METHOD
So both the equations are over identified.
Answer 5
Demand for money:Dt =β0 + β1 Pt + β2 Y t + β3 IRt ++ ε1 t
Supply of money: St =α 0 +α1 Y t + ε2 t
Order Condition of Demand Function
For demand function: (m+k-1) = (2+2-1) = 3 and K=3
So, for demand equation order condition is satisfied where the equation is just identified that
is K = (m+k-1).
Order Condition of Supply Function
For supply Function: (m+k-1) = 2+0-1 = 1 and K=3.
So, for supply equation order condition is satisfied where the equation is over identified that
is K > (m+k-1).
Appropriate Method to Estimate the Parameter of Demand Function
For the just identified structural equation, the indirect least squares method is better
to obtain the estimate of the structural coefficients by deriving the reduced form coefficients
by using OLS method on the reduced form equation as applying OLS on the structured
equation can distort the true image of the model. In contrary, ILS estimators carry all the
asymptotic properties of the reduced form estimator (Stock and Watson 2015). So, the
appropriate method is ILS to estimate the parameters of identified equation.
Appropriate Method to Estimate the Parameter of New Supply Equation
For new supply function: (m+k-1) = 4+0-1 = 1 and K=3 where: St −1∧:Y t−1are added
in the equation. So, for supply equation order condition is satisfied where the equation is just
identified that is K = (m+k-1).

7APPLIED ECONOMETRIC METHOD
The new supply equation is just identified and the demand function is also exactly
identified. Though here are two methods to be applied on the equation to estimate the
parameters of the equations ILS is more appropriate than OLS. As the endogenous variables
may depend on the error term in the supply equation.

8APPLIED ECONOMETRIC METHOD
Reference:
Stock, J.H. and Watson, M.W., 2015. Introduction to econometrics.
Wooldridge, J.M., 2015. Introductory econometrics: A modern approach. Nelson Education.