Midterm 2011 Econometrics Havard

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Ec. 1126
Spring 2011
MIDTERM EXAMINATION
Please answer all three parts. Your explanations should be concise and to the point.
PART I. (30 points) (a) We observe Di = (Yi, Zi1, Zi2) for i = 1, . . . , n and assume that
Di is independent and identically distributed according to some unknown distribution F .
Consider the following regression function:
E(Yi |Zi1, Zi2) = θZi1 + g(Zi2),
where the function g(·) is an unknown function, which is not restricted. This regression
function does impose the restriction that Zi1 enters linearly and there is no interaction
between Zi1 and Zi2. The random variable Zi2 is discrete and takes on only the values
δ1, δ2, δ3. Suggest an estimator for θ, based on a least-squares fit of Y on X1, . . . , XK,
where
Y =


Y1
...
Yn

 , Xj =


X1j
...
Xnj

 (j = 1, . . . , K).
Be explicit on how the Xj are constructed from the observations on (Zi1, Zi2) for i =
1, . . . , n.
(b) Provide an omitted variable formula using (population) linear predictors. (You
may use the case with a single omitted variable, but try to allow for several included
variables.) Provide a derivation of the formula.
PART II. (35 points) At the beginning of the course, we introduced the population linear
predictor, for a scalar random variable. In the panel data part of the course, we introduced
a generalized linear predictor, for a vector random variable.
(a) An alternative would be to simply apply our original (scalar) linear predictor to
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each component of the random vector. Explain why this does not, in general, give the
same result. When would it give the same result?
(b) Suppose that we observe
Di = (Yi1, Yi2, Zi1, Zi2)
for i = 1, . . . , n, and assume Di i.i.d. from some unknown distribution F . We do not
impose restrictions on F . Define
Y (t) =


Y1t
...
Ynt

 , Z(t) =


Z1t
...
Znt

 (t = 1, 2),
stack to get
Y s =
(
Y (1)
Y (2)
)
, Xs =
(
1 Z(1)
1 Z(2)
)
(1 = ones(n, 1)), and consider a least-squares fit of Y s on Xs:
b = Xs \ Y s.
Explain how the probability limit of b as n → ∞ relates to a (population) generalized
linear predictor.
PART III. (35 points) Suppose that we have panel data with T = 2 observations on each of
n cross section units. The population model is expressed in terms of the vector of random
variables
Wi = (Yi1, Yi2, Zi1, Zi2, Ai).
Assume that the Wi are independent and identically distributed (i.i.d.) according to some
unknown distribution (for i = 1, . . . , n). The structural regression model is
E(Yi1 |Zi1, Zi2, Ai) = θZi1 + Ai,
E(Yi2 |Zi1, Zi2, Ai) = θZi2 + Ai.
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We have observations on
Di = (Yi1, Yi2, Zi1, Zi2)
for i = 1, . . . , n. Data on Ai are not available.
(a) Consider the estimator
θˆ =
∑n
i=1(Yi2 − Yi1)(Zi2 − Zi1)∑n
i=1(Zi2 − Zi1)
2
.
Are any additional assumptions needed to show that θˆ converges in probability to θ as
n→∞? Sketch the proof (making additional assumptions if necessary).
(b) Now suppose the structural regression model is
E(Yi1 |Zi1, Ai) = θZi1 + Ai,
E(Yi2 |Zi1, Zi2, Ai) = θZi2 + Ai,
with Zi2 = Yi1. As before, we have observations on
Di = (Yi1, Yi2, Zi1, Zi2)
for i = 1, . . . , n. Data on Ai are not available. Show that θˆ is not a consistent estimator
of θ as n→∞.
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