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Please answer all of the following questions as best as possible. Points will be deducted for
writing irrelevant statements. Partial credit will be awarded when it is earned. The point value
for each question is in parentheses. The detail of your answer should correspond to the amount of
points the question is worth. There are a total of 16 questions on this exam.
Intuitive: Please answer the following questions trying to use logic and economic reasoning.
I.1. (5) Explain why the linear probability model, for some level of the covariates, will
always deliver estimated probabilities that are either greater than 1 or less than 0.
The linear probability model always has the ability to produce estimated probabilities
that are either greater than one or negative because the model is being t with a line,
which is unbounded. Probabilities, by de nition are bounded between 0 and 1. Thus,
this is a case where we knowingly have model misspeci cation and one of the costs of
this misspeci cation is that we may obtain estimated probabilities that are inconsistent
with theory.
I.2. (10) Why do the coecient estimates in models with nonlinearities in the covariates
lose their ceteris paribus interpretation?
Coecient estimates from models with nonlinear covariates no longer retain their ce-
teris paribus interpretation because the nonlinearities no longer make it feasible to
hold everything else xed. For example, if the conditional mean was speci ed as
0 + 1x + 2×2, we cannot interpret 1 in a ceteris paribus fashion because it makes
no sense to change x while holding x2 xed. Rather, we can interpret 1 directly by
anchoring our change to the point x = 0, in which case, 1 is the change in y given a
change in x, when x = 0.
I.3. (5) Comment on the veracity of the following statement: standard t-statistics are
invalid if heteroskedasticity is present.”
This statement is true. When the error terms are heteroskedastic then the estimate of
the error variance under the assumption of homoskedasticity is invalid, which implies
that the t-statistic is incorrect. This invalid formula suggests that all inference based
o it will be useless.
I.4. (15) De ne an instrument for the model yi = 0 + 1xi + “i. Next, comment on how
having an instrument does not imply that the IV estimator will be accurate (your
answer should discuss both issues of bias and variance).
An instrument for the above model is any variable z such that E[“jz] = 0. Now, if
this variable z existed, we cannot guarantee that the IV estimator will be accurate as
this also depends on the covariance between x (the potentially endogenous variable) and
the instrument. Further, our IV estimator will be less precise than the OLS estimator
because the IV estimator replaces the full variation that occurs in x with reduced form
variation based on the link between x and z. Additionally, our IV estimator will not be
accurate for a given sample due to the bias that is present in the estimator. It is only
for large sample sizes that we can claim the IV estimator is accurate, which in practice
may be untenable.
I.5. (10) Suppose I test H0 : 1 = 0 against a two-sided alternative. If I fail to reject the
null hypothesis at the 10% level does this also mean I would fail to reject the null
hypothesis at the 5% level? What about the 15% level?
If I fail to reject at the 10% level this means that I must have a test statistic with a p
value that is greater than 0.1. Thus, my p-value is also greater than 0.05 so I would fail
to reject the null hypothesis at the 5% level as well. However, without more information
I cannot know the outcome of the test if it were conducted at the 15% level since I only
know that p > 0:1.
I.6. (10) The more variation in one of my explanatory variables means that the variance
of the associated slope coecient estimator is lower. Explain the intuition underlying
this statement.
This statement touches on the implication that variation in a covariate leads to more
accurate estimates of the unknown coecients of the model. In essence, the OLS es-
timator’s variance depends upon the variation in the error term and the proportion of
the variation of the covariate of interest that is not explained by other covariates in
the model. In math this is var(b j) = 2
TSSxj (1