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# 2. Advanced topics in OLS regression
Statistics and Econometrics
for Business II
Fall 2014
Instructor: Maksym Obrizan
Lecture notes III
# 3. Working with natural logs
Suppose that we regress log(salary) of CEOs on
log(sales) of their firms
# 4. Quadratic functions are also used quite
often in applied economics to capture
decreasing or increasing marginal effects.
For example, consider how wage y depends on
experience x
When interpreting the effect we need 2 terms
# 5. Consider the effects of experience on wage
# 6.
The first year of experience brings about 30 ¢
In going from 10 to 11 years of experience, wage
is predicted to increase by
Thus, exper has diminishing returns on wage
# 7. Sometimes the partial effect of the
dependent depends on the magnitude of
another explanatory variable
# 8. R-squared can never fall when a new
independent variable is added to regression
Thus, adjusted R-squared is often used because
it imposes a penalty for adding additional
independent variables
# 9. Recall F test for joint significance of a group
of variables
# 10. Some of the variables take only 2 values (male
of female) – they are called binary
How do we incorporate binary variables into
regression models?
But what if models are nonnested when neither
equation is a special case of the other?
# 11. Example
If we compare a woman and a man with the same
levels of education, experience, and tenure, the
woman earns, on average, $1.81 less per hour
than the man.
For example,
# 12. Suppose we estimate a model that allows
for wage differences among four groups:
married men, married women, single men, and
single women.
How many dummies can we include (if we also
have an intercept)?
# 13. The linear probability model (LPM)
Sometimes the dependent variable is also binary
(employed or unemployed)
Linear Probability Model (LPM) estimates the
response probability as linear in the
parameters
# 15. Heteroskedasticity
In the presence of heteroskedasticity many OLS
test statistics are no longer valid
Heteroskedasticity-robust procedures are used in
this case
# 14. Probability of “being in labor force”
For example, 10 more years of education
increases the probability of being in the labor
force by 0.038(10) = 0.38
# 16. Example (robust standard errors are in
parenthesis)
# 17. How to test for heteroskedasticity
The Breusch-Pagan test for heteroskedasticity
(BP test)
# 18. Economists are often interested in policy
analysis
Does job training improve chances of becoming
employed?
Will the construction of incinerator affect house
prices?
# 19. Quote from Wooldridge: “Kiel and McClain
(1995) studied the effect that a new garbage
incinerator had on housing values in North
Andover, Massachusetts.
The rumor that a new incinerator would be built
in North Andover began after 1978, and
construction began in 1981.
We will use data on prices of houses that sold in
1978 and another sample on those that sold in
1981.”
# 20. A naïve analysis would be to use data for
1981
where nearinc is dummy (=1 if a house is near
incinerator, 0 – otherwise)
# 21. However, the data for 1978 (prior to rumors
about construction) shows
# 22. Did building of a new incinerator depress
housing values?
The key is to compare the coefficient on nearinc
changed between 1978 and 1981.
Use difference-in-differences estimator
using the data pooled over both years
# 23. Interpreting the results
# 24. The parameter we are interested is on the
interaction term y81·nearinc
# 25. Omitted variable bias
Suppose that the true relationship is
# 26. Thus, the estimator of w will biased (not
equal to the correct one) and inconsistent (not
converging to the true one as the sample size
increases)
but ability is not observed so we estimate
# 27. Instrumental variables (IV)
Suppose that education is correlated with the
error term u (because it contains ability)
# 28. Stata example
Use the data on married working women in
MROZ.RAW to estimate the return to
education
In addition, let z be such a variable that is
uncorrelated with u but correlated with x
OLS results first
# 29. IV estimation
Suppose that father education is a good
instrument for educ
# 31. Criticisms of IV estimation
Observe that OLS estimate is included in 95%
interval for IV estimate
# 30.
# 32. Multicollinearity
Example of perfect collinearity –
constant+female+male
Thus, the difference is not statistically significant
Example of multicollinearity
# 33. Consequences of multicollinearity
- OLS estimators are still BLUE but would have
large covariances
# 34. What to do in the case of multicollinearity?
- Sometimes no choice (data deficiency) – so
do nothing
Detecting multicollinearity
# 35. Micronumerosity – the problem of small
sample size
This is a related problem to multicollinearity
# 36. Including irrelevant variables in the OLS
regression
Including an irrelevant variable will not lead to
unbiasedness of the intercept and other slope
estimators