BEC 440 Econometrics II Take Home Test

Take Home Test
BEC 440
Econometrics II
Instructions
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Answer all the questions
Due Date: 07/06/2026 by 6PM
Question One
a) Orthogonal Partitioned Regression Theorem
In the multiple linear least squares regression of y on two sets of variables X1 and X2 , if the
two sets of variables are orthogonal, then the separate coefficient vectors can be obtained by
separate regressions of Y on X1 alone and y on X2 alone.
Using the necessary notations provide a convincing proof of this theorem. In your proof, you’re
required to explain the concept of orthogonality in relation to the two sets of variables. [5
marks]
b) Frisch–Waugh (1933)–Lovell (1963) Theorem
In the linear least squares regression of vector Y on two sets of variables, X2 and X1 , the
subvector β̂2 is the set of coefficients obtained when the residuals from a regression of Y on X1
alone are regressed on the set of residuals obtained when each column of X 2 is regressed on
X1 .
Using the necessary notations provide a convincing proof of this theorem [5 marks]
c) [10 marks] Assume a model (DSP) of the structure
𝑌 = 𝑋1 𝛽1 + 𝑋2 𝛽2 + 𝑢𝑖
𝑤ℎ𝑒𝑟𝑒 𝛽̂ = (𝛽̂1′ , 𝛽̂2′ ) = (𝑋 ′ 𝑋)−1 𝑋 ′ 𝑌
Here, 𝑌 is the dependent variable, 𝑋1 and 𝑋2 are the matrices of the first and second set
of regressors respectively, 𝛽1 and 𝛽2 are the coefficients to be estimated and 𝑢𝑖 is the
error term.
1. Show that 𝛽̂1 = (𝑋1′ 𝑀2 𝑋1 )−1 (𝑋1′ 𝑀2 𝑌), where 𝑀2 = 𝐼𝑛 − 𝑋2 (𝑋2′ 𝑋2 )−1 𝑋2′ .
2. Explain the properties of the matrix 𝑀2 and explain how this matrix compares to
another similar matrix 𝑀. What does the matrix 𝑎𝑐𝑐𝑜𝑚𝑝𝑙𝑖𝑠ℎ.
3. How does the matrix 𝑀 come about? Derive this matrix (residual maker). In the process
also derive the projection matrix and explain what the projection matrix accomplishes.
d) Suppose a correctly specified model is given by:
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𝑌 = 𝑋1 𝛽1 + 𝑋2 𝛽2 + 𝑢𝑖
If we by mistake of omission of the key variable we regress 𝑌 on 𝑋1 without 𝑋2 so that
the misspecified model is:
𝑌 = 𝑋1 𝛽1 + 𝑣𝑖
Show how the bias comes up in the estimation of 𝛽1 and indicate the actual bias. Explain
the special condition or circumstance in which the estimator for 𝛽1 would be unbiased
despite the omission of the key variable, 𝑋2
e) [10 marks] Given the following DSP
𝑌 = 𝑋𝛽 + 𝜀
Where, 𝑌 is the 𝑛 𝑏𝑦 1 data on the dependent variable across all the n observations indexed by
𝑖 = 1,2,3, , , , , , , , , , , , , 𝑛, the data matrix for the 𝐾 − 1 independent variables across all the
observations is contained in the 𝐾 𝑏𝑦 𝑛 matrix denoted by 𝑋. The 𝐾 parameters of interest are
contained in the 𝐾 𝑏𝑦 1 vector 𝛽 and 𝜀 is the 𝑛 𝑏𝑦 1 vector of the error term.
1. Show the procedure and derive the Ordinary Least Squares (LS) estimators associated
with this DSP. Here, you are required to derive the estimator of 𝛽.
2. Under the CLRM assumptions, derive the estimator of the variance covariance matrix
for 𝛽. This involves deriving the following;
𝑣𝑎𝑟(𝛽̂) = 𝜎 2 (𝑋 ′ 𝑋)−1
3. List the statistical properties of the LS estimator derived in 2 above. In your answer
show that 𝛽̂ is linear, unbiased and has the smallest variance in the class of all linear
unbiased estimator for 𝛽 (in line with the Gauss Markov theorem).
4. Derive and show the consistency property of the OLS estimator using the concept of
the probability limit.
Question Two
a) [13 marks] Answer the following questions:
You are given the following structural model of income determination:
log 𝑤𝑖 = 𝛽1 + 𝛽2 ℎ𝑖𝑔ℎ𝑒𝑑𝑖 + 𝛽3 𝑦𝑒𝑎𝑟𝑠𝑖 + 𝛽4 𝑦𝑒𝑎𝑟𝑠𝑖2 + 𝛽5 𝐴𝑓𝑟𝑖𝑐𝑎𝑛 + 𝜀𝑖
Where 𝑤 is the wage, ℎ𝑖𝑔ℎ𝑒𝑑 is the number of years of education completed, 𝑦𝑒𝑎𝑟𝑠 is the
number of years of experience and 𝐴𝑓𝑟𝑖𝑐𝑎𝑛 is a dummy variable equal to one if the person is
a black South African. The stochastic error term 𝜀 is assumed to be homoscedastic and normally
distributed.
You are given the following regression estimates of this model:
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1. Interpret the coefficients on “ℎ𝑖𝑔ℎ𝑒𝑑” and “𝐴𝑓𝑟𝑖𝑐𝑎𝑛”.
2. You would like to predict the turning point of the relationship between experience and
(expected log) wages. Generate a consistent estimate of this turning point.
3. Assume that the covariance matrix of the estimators (𝑜𝑟𝑑𝑟𝑒𝑑 𝑎𝑠 𝛽̂2 , 𝛽̂3 , 𝛽̂4 , 𝛽̂5 , 𝛽̂1 ) is
given by:
Now generate standard errors for the turning point by means of the delta method.
4. Test for the joint significance of the “𝑦𝑒𝑎𝑟𝑠” and “𝑦𝑒𝑎𝑟𝑠 2”.
b) You think that the process by which wages are set given
log 𝑤𝑖 = 𝛽1 + 𝛽2 𝐸𝑑𝑢𝑐𝑖 + 𝛽3 𝐸𝑥𝑝𝑒𝑟𝑖 + 𝛽4 𝐸𝑥𝑒𝑟𝑖2 + 𝜀𝑖
Where 𝑤 is the wage rate, 𝐸𝑑𝑢𝑐 is the highest level of education obtained and 𝐸𝑥𝑝𝑒𝑟
is potential experience (in years).
You have the following Stata Output.
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1. Interpret the regression coefficients
2. Which of these coefficients are significant at the five percent level?
3. Estimate the turning point in the relationship between experience and (expected) log
wages.
4. Obtain an estimate of the standard error of the estimator in e (3) by means of the delta
method.
5. Construct a 95 percent confidence interval for the turning point calculated in e (3).
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