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We believe that the population relation is described by the following equation:


y​​ =​​ β0​​ +​​ β1x​​ +​​ E(1)


Imagine​​ that​​ we​​ believe​​ that​​ Assumption​​ 4​​ (the​​ unconditional​​ mean​​ zero​​ assumption)​​ is​​ not​​ satisfied. Show that one can find an equation that also describes the relation in the population between​​ y​​ and​​ x​​ and that does satisfy Assumption​​ 4.

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We have the following population model:

y​​ =​​ β0​​ +​​ β1x​​ +​​ E

We​​ have​​ to decide between two alternative estimators. The density functions of the two estima- tors are displayed​​ below.​​ Discuss​​ briefly​​ which estimator you​​ prefer.


We​​ have​​ studied​​ the​​ effect​​ of​​ the​​ NTV​​ TV​​ station​​ coverage​​ on​​ the​​ percentage​​ of​​ votes​​ that​​ the​​ party​​ supporting​​ Vladimir​​ Putin​​ received​​ in​​ the​​ 1999​​ Duma​​ elections.​​ We​​ obtained​​ the STATA output​​ below:


Compute​​ the​​ slope​​ coefficient​​ that​​ we​​ would​​ have​​ obtained​​ if​​ we​​ had​​ run​​ a​​ simple​​ regression​​ of

votesPutin​​ on​​ NTV.


Question​​ 7:​​ We​​ believe that,​​ in​​ the​​ population,​​ the​​ relation​​ between y​​ and​​ x​​ is​​ negative.​​ Unfortunately,​​ our​​ sample​​ was​​ not​​ obtained​​ randomly,​​ but​​ instead​​ it​​ contains​​ only​​ observations​​ with​​ a​​ value​​ of​​ y​​ below​​ a​​ certain​​ threshold.​​ Explain​​ graphically​​ whether​​ the​​ error​​ term​​ in​​ our model​​ is​​ positively​​ or​​ negatively​​ correlated​​ with​​ our​​ independent​​ variable x.



  • X​​ and​​ Y​​ are independent random variables.​​ E(X) = 2, V(X) = 2, E(Y ) = 3 and V(Y ) = 5.


(a) [15%] E(2X - Y )

(b) [20%] V(2X + 4)

(c) [25%] V(2X - 4Y )

(d)​​ [40%] E(2X2 - 4Y )


We wanted to understand the relation between journal demand and the lightness of the journal covers. In order to do that, we ran the following regression:


​​  (0.44)(.10)


where we have standard errors in parentheses. Our sample size is 180. Test the null hypothesis that​​ the​​ effect​​ is​​ zero​​ against the​​ alternative hypothesis​​ that​​ the​​ effect​​ is​​ negative​​ at​​ the​​ 5%​​ level.


Question 9:​​ We wanted to estimate the returns to education separately for every state in the US. Our population model is:


For every state we used a sample size equal to 1/100,000 of the state population. The table below displays the estimated coefficients for ten states: