7/21/2019 MIT Econometrics

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14.382

MIDTERM

2006

Answerasifyourtrytoexplainthematerialtoyour fellowstudent.

Consider

the

model,

where

Y

=

X

+

,

where

for

each

t,

t (et 1),whereet is standard exponential variable such thatE[et]=1 andVar[et]=1. AssumethatX areindependentof. Supposethat(xt,t)arei.i.d. acrosst.

1. (10)DoGauss-Markovassumptionsholdforthismodel?

2. (10)Considertheleastsquaresestimator. ComputeE[|X]andVar[X].

Is

normally

distributed

in

finite

samples,

conditional

on

X?

|

3. (10)Carefully, butbriefly,explain the labelBLUE. IsOLSBLUE inthisset-up?

4. (10)Considerestimatingthefollowingeffect

E[yt xt =x]E[yt xt =x

] =(x x)| |

Giveaneconomicexamplewheresuchaneffectmightbeofinterest. Is(xx)BLUE forthiseffect? Whyorwhynot?

5. (10)IsOLStheBUE(bestunbiasedestimator)inthismodel? Abriefanswersuffices.

6. (15)Whatisthelargesampledistributionof? Makeanyadditionalprimitiveassumptions you might need. [Note: high level assumptions will receive partialcredit.]

7.

(10)

Construct

a

consistent

estimator

for

the

large

sample

variance

of

.

Proveitsconsistencybymakinganyadditionalassumptionsyouneed.8.

(10)

Suppose

we

want

to

test

the

null

hypothesis

H0 :j =0vsHA :j