Theoretical Exercises
- Consider the following simple regression model . Please derive and
Answer:
the first order condition is
thus
We can also derive it using
when
where
thus
The result is the same as above using F.O.C.
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- is an OLS estimator. What is the variance formula under homoskedasticity?
Answer:
For regression model
or
where
the OLS estimator is
thus
By the WLLN,
where
By Slutsky theorem
where . For homoskedasticity, i.e.
thus
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- Exercise 1 in Wooldridge Chapter 15
Answer:
(i)
The error term contains family income, which has a positive effect on GPA and is also very likely to be correlated with PC owernship.
(ii)
Denote parentsโ annual income by faminc. Since , it is not a good IV for PC.
(iii)
Define a dummy variable, grant, which equal to one if the student received a grant, and zero if not. If grant was randomly assigned, it is uncorrelated with . Besides, grant should be correlated with PC since the probability of owning a PC should be significantly higher for student receiving grants. Incidentally, if the university gave grant priority to low-income students, grant would be negatively correlated with , and IV would be inconsistent.
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- Exercise 3 in Wooldridge Chapter 15
Answer:
According to 15.10
The numerator
where is the number of observations with ; , the average of the over the with .
Rewrite as a weighted average of the averages over two subgroups:
where . Therefore,
thus the numberator can be wirtten as .
Replacing with , the denominator in can be expressed as , leaving
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- Exercise 5 in Wooldridge Chapter 15
Answer:
(i)
where is the IV estimator. So the asymptotic bias is .5.
(ii)
where is the OLS estimator. So we would have to have before the asymptotic bias in OLS exceeds that of IV.
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- Exercise 1 in Wooldridge Chapter 16
Answer:
(i)
โ , the right-hand-side depends only on the exogenous variable and the error term
โ , the right-hand-side depends only on the exogenous variable and the error term
โ plug into the first equation and solve for :
or
Dividing by (because ) gives
where .
(ii)
Multiply the second equation by and subtract it from the first equation, we obtain
Since , , thus
where
A reduced form does exist for , as can be seen by subtracting the second equation from the first:
because , we can rearrange and divide by to obtain the reduced form.
(iii)
In supply and demand examples, is reasonable. If the first equation is the supply function, we generally expect , and if the second equation is the demand function, . The reduced forms can exist even in cases where the supply function is not upward sloping and the demand function is not downward sloping, but we might question the usefulness of such models.
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- Exercise 4 in Wooldridge Chapter 16
Answer:
The first equation is identified provided . The second equation is not identified because there are no exogenous variables appearing in the first equation that are not also in the second equation. is an exogenous variable and can be used as an IV for in estimating the first equation by 2SLS.
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- Exercise 9.5 in Stock and Watson
Answer:
(a)
Substitute into , we have
thus
Substitute into , we obtain
(b)
Since ,
and
(c)
Since and are mutually uncorrelated
and
As for the covariance between
(d)
(i) According to the equation 4.7 and 4.8 in Stock and Watson book:
(ii) since supply curves slope up and demand curves slope down, and , thus
That is, the estimated slope is too large.
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- Exercise 11.1 in Stock and Watson
Answer:
(a)
The t-stat for the coefficient on Schooling is 0.272/0.029=9.38, which is significant at the 1% level.
(b)
;
(c)
;
(d)
; . This is unlikely to be accurate because the sample did not include anyone with more than 24 years of shcooling.
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- Exercise 11.2 in Stock and Watson
Answer:
t-stat for the coefficient on Schooling is 0.551/0.062=8.89, which is significant at the 1% level.
;
;
The probit and logit models are similar.
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- Exercise 11.3 in Stock and Watson
Answer:
t-stat for the coefficient on Schooling is 0.035/0.003=11.67, which is significant at the 1% level.
The linear probability is not appropriate here because when Schooling is smaller than 4.91, the probability is smaller than zero which is not reasonable.
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- Exercise 11.4 in Stock and Watson
Answer:
Probit:
;
;
Logit:
;
;
Linear:
These three models are not different because results of them are the same.
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- Exercise 11.5 in Stock and Watson
Answer:
a.
b.
c. The t-stat on the interaction term is -0.344/0.096=-3.58, which is not significant at the 10% level. Therefore, the effect of the years of schooling on employment in the government does not depend on gender.
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- Exercise 12.6 in Stock and Watson
Answer:
model
:
When
since , we cannot reject under the 10% level, that is, we cannot reject is a weak instrument.
when
since , we cannot reject under the 10% level, that is, we cannot reject is a weak instrument.
Computer Exercises
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