Econometrics Assignment: Gauss-Markov Theorem and Regression

This econometrics assignment solution addresses key concepts in regression analysis. It begins with the Gauss-Markov theorem and its assumptions, discussing the properties of the OLS estimator and the implications of violating these assumptions. The solution then explores hypothesis testing, interpreting coefficients, and evaluating model fit using R-squared. It further delves into heteroskedasticity, explaining its impact, how to detect it using RSS and the Breusch-Pagan test, and methods for addressing it. The assignment analyzes the effects of different variables on house prices and labor demand, comparing model performance and discussing the importance of addressing issues like collinearity and heteroskedasticity to ensure valid and reliable results. The solution also provides the reference from where the content is taken.

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Question 1
a) According to Gauss Markow Theorem, OLS is the Best Linear Unbiased Estimator when the
variance is the least among the forecasted values or calculated values of Regressions and the
actual values of the co-efficients of regressions. (Wooldridge, 2015)
The Gauss Markow is also, true , only if the assumptions that the regression is linear and errors are
homoskedascistic. (Wooldridge, 2015)
b) Statistically, Null Hypothesis is that Beta values are equal and all are equal to zero. The
statistical interpretation is that the Co-efficients of Regression of the log of lot size, the number
of bathrooms, the dummy variable that the house has 3 bedrooms, dummby variables that the
house has 4 bedrooms, and dummy variable that the house has 5 bedrooms is zero. This implies
that the variable are not regressors of the independent variable or they have a negligible impact
of changes in the dependent variable, here lprice is higher. (Lambert, 2013)
In terms of economics, this indicates that the increase in house price is autonomous. This
implies that the house price and an increase in house price is exogenous and is not related to the lot
size, the number of bedrooms or bathrooms. The supply is either artificially fixed or completely
inelastic to these factors.
c) If the null hypothesis is true, then β2 = β3 = β4= 0, then (lprice)= β0 + β1llotsize + β2bathrooms+ β3
bd2 + β4bd3 + β5bd4 +u = δ(lprice)= β0 + β1llotsize + β2bathrooms+u
The partial regression is not equal to the full equation. Hence, the null hypothesis is rejected.
The size of the bedroom has an effect on the price of the house.
d) R-Square measures the Goodness of Fit of regression or how close the estimated values are to
the actual values. The goodness of It is computed as
{n∑xy – (∑x)(∑y)}/ √{(n∑x2 – (∑x)2} {(n∑y2 – (∑y)2} (Wooldridge, 2015)
The R-Square and the Adjusted R-Square for the second model are higher, implying, this model is a
better fit than the first one. Hence, based on the goodness of the fit, the second model is preferred.
e) The results for variable bd2 were possibly eliminated due to the presence of collinearity or
similarity. (Lamber, 2013)
f) The new equation will be:
(lprice)= β0 + β1llotsize + β2 (good)+u
The co-efficient will indicate the extent to which being located in a good area would be a
regressor of house price.
Question 2
a) i) In the given equation, RSS stands for residual Sum of Squares or the Sum of Squared Errors of
Prediction. It is the sum of squared residuals. An Equation is homoskedastic (there is no
heteroskedascity) if the RSS is within a range for all values i.e for observations 1- 100. However,

in this case, the RSS is digressing i.e the RSS increases significantly as the value of X increases.
The RSS of the lowest 33 observations is at lower than the RSs of the highest 33 observations.
Hence, there is a strong evidence against homoskedasticity. (Lambert, Heteroskedasticity
summary, 2013)
ii) In this case, the co-efficient or the β represent the marginal effect of adding one more input (or
one more observation of x)
 In the first equation, the β2 value or the value of the first co-efficient representing the co-
efficient of input x2 is higher, implying that the marginal effect of adding one more unit
of x2is higher, than in the second equation. This means that input x2 is a bigger factor in
equation 1. This means that the output is diminishing or that there are diminishing
returns.
 Similarly, input x3 is a negative regressor, implying that as input 3 or x3 or the total wgae
per worker increases, the labour demand decreases. This co-efficient β3, too increases, as
the wages are higher, implying that the impact of the β3 is higher when the total wages
are higher.
b) I) If there is hetereoskedascity, then the assumption of Gauss Markow Theorem of uniformly
spread residuals will be violated. In this case. The OLS estimator will not be a Best linear
Unbiased Estimator. This implies that there is another linear regression equation which is a
better estimator. This means that there are other variables that must be considered in order to
understand the demand for labour better.
ii) A change in parameters should decrease heteroscedaticity. (Lambert, The Breusch Pagan test
for heteroscedasticity, 2013)
 Sometimes, Heteroskedasciticity could also, be a result of bias in standard errors. The
standard errors could be corrected using some software . White’s Tests or Newey’s test
could be used. (Lambert, 2013)
 Instead of using OLS, Generalized Least Squares could be used as a method to estimated
the model. This method in the sample differently. This will remove heteroscedasticity
and the errors become homoscedasctic. However, this method requires a knowledge of
the structure of the variance of the population. (Lambert, 2013)
c) The Beusch Pagan Test has a null hypothesis is Homoskedascity is present. The test is computed
as
F = (R2/ P)/ {(1- R2)/(N-P-1)
Where
 R 2 = R square of Auxilliary Regression
 P= Number of variables
 N-P- 1 = Number of degrees of Freedom.
If this statistic does not have an F- Distribution, for P degrees of freedom and (N-P-1) degrees of
Freedom for the second input. If the F Statistic is greater than the critical value (usually, 0.05%,
or 0.01 %), then the null hypothesis is accepted and it is confirmed that there are homoskedastic

error. If the F-Statistic is less than the critical value, then null hypothesis is rejected and there is
heteroskedasticity. (Lambert, 2013)
Model 1 is preferred since the R 2 is higher and the residuals are more homskedascitic
comparatively.
Reference
Lambert, B. (2013, June 3). Heteroscedasticity: as a symptom of omitted variable bias - part 1. Retrieved
from YouTube: https://www.youtube.com/watch?v=sFOtuCKQztc
Lambert, B. (2013, June 3). Heteroskedasticity summary. Retrieved from YouTube:
https://www.youtube.com/watch?v=zRklTsY9w9c&t=185s
Lambert, B. (2013, June 18). Interpreting Regression Coefficients in Linear Regression. Retrieved from
YouTube.com: https://www.youtube.com/watch?v=JwGaos2Y9bM
Lambert, B. (2013, June 22). The Breusch Pagan test for heteroscedasticity. Retrieved from YouTube:
https://www.youtube.com/watch?v=wzLADO24CDk
Wooldridge, J. M. (2015). Introductory Econometrics: A Modern Approach : Sixth Edition (pg 75). online:
Cengage Learning.