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CHAPTER14

SOLUTIONSTOPROBLEMS

2

14.1First,foreacht1,Var(Δu)=Var(u–u)=Var(u)+Var(u)=2σ,whereweuse

ititi,t-1iti,t-1u

theassumptionsofnoserialcorrelationin{ut}andconstantvariance.Next,wefindthe

covariancweenΔuandΔu.Becausetheseeachhaveazeromean,thecovarianceis

iti,t+1

2

E(Δu⋅Δu)=E[(u–u)(u–u)]=E(uu)–E(u)–E(uu)+E(uu)=

iti,t+1iti,t-1i,t+1ititi,t+1iti,t-1i,t+1i,t-1it

22

−E(u)=−σbecauseofthenoserialcorrelationassumption.Becausethevarianceisconstant

itu

acrosst,byProblem11.1,Corr(Δu22

,Δu)=Cov(Δu,Δu)/Var(∆u)=−σ/(2σ)=−.5.

iti,t+1iti,t+1ituu

14.3(i)E(e)=E(v−λv)=E(v)−λE(v)=0becauseE(v)=0forallt.

ititiitiit

2222

(ii)Var(v−λv)=Var(v)+λVar(v)−2λ⋅Co