Problem set 1October 16, 20131. (Covariance and correlation) Suppose that the annual revenues of the

Problem set 1October 16, 20131. (Covariance and correlation) Suppose that the annual revenues of thear stued dvi y reaC souou rcrs eeH wer aso.comworld’s two top oil producers have a covariance of 1,735,492.(a) Based on the covariance, the claim is made that the revenues arevery strongly positively related. Evaluate the claim.(b) Suppose instead that, again based on the covariance, the claim ismade that the revenues are positively related. Evaluate the claim.(c) Suppose you learn that the revenues have a correlation of 0.93. Inlight of that new information, reevaluate the claims in parts a and b.2. ( Properties of loss function) State whether the following potential lossfunctions meet the criteria introduced in the text and, if so, whether theyare symmetric or asymmetric:(a)L(e) = e2 + e(b)L(e) = e4 + 2e2(c)L(e) = 3e2 + 1√e, e > 0L(e) =|e|, e ≤ 0(d)3. (Calculating forecasts from trend models) You work for the InternationalMonetary Fund in Washington, D.C., monitoring Singapore’s real con-issumption expenditures. Using a sample of real consumption data ( mea-yt ,t= 1990:Q1,…,2006:Q4, youyt = β0 + β1 T IM Et + εt ,ˆestimates β0 = 0.51, β1 = 2.30, andwhere2tσ = 16.∼ N (0, σ 2 ),obtaining theBased on your estimated trend model, construct feasible point ,interval, and density forecasts for 2010:Q1.shThsured in billions of 2005 Singapore dollars),estimate the linear consumption trend model,4. (Selecting forecasting models involving calendar eects) You’re sure that aseries you want to forecast is trending and that a linear trend is adequate,but you’re not sure whether seasonality is important . To be safe, you ta forecasting model with both trend and seasonal dummies,syt = β1 T IM Et +γi Dit + εti=11https:// The hypothesis of no seasonality, in which case you could drop theseasonal dummies, corresponds to equal seasonal coecients acrossseasons, which is a set ofs−1linear restrictions:γ1 = γ2 , γ3 = γ4 , …, γs−1 = γsHow would you perform an F-test of the hypothesis? What assumptions are you implicitly making about the regression’s disturbanceterm?(b) Alternatively, how would you use forecast model selection criteria todecide whether to include the seasonal dummies?ar stued dvi y reaC souou rcrs eeH wer aso.com(c) What would you do in the event that the results of the hypothesistesting and model selection approaches disagree?(d) How, if at all, would your answers change if instead of consideringwhether to include seasonal dummies you were considering whetherto include holiday dummies? Trading-day dummies?5. (Interpreting dummy variables) You t a purely seasonal model with afull set of standard monthly dummy variables to a monthly series of em-ployee hours worked. Discuss how the estimated dummy variable coe-cients γ1 , γ2 , …would change if you change the rst dummy variableD1 =ˆ ˆ(1, 0, 0, 0, …) ( with all the other dummy variables remaining the same) to(a)D1 = (2, 0, 0, 0, …)(b)D1 = (−10, 0, 0, 0, …)(c)D1 = (1, 1, 0, 0, …)6. (Lag operator expression1) Rewrite the following expressions without using the lag operator.(a)(Lτ )yt =(b)yt = ( 2+5L+0.8L )L−0.6L3is(c)t2yt = 2(1 +tL3L ) tTh7. (Lag operator expressions 2) Rewrite the following expressions in lag operator form.yt +yt−1 +…+yt−N = α +εt +εt−1 +…+(b)yt = εt−2 + εt−1 +sh(a)t−N , whereα is a constantt8. (Autocorrelation functions of covariance stationary series) While interviewing at a top investment bank, your interviewer is impressed by thefact that you have taken a course on time series forecasting. She decidesto test your knowledge of the autocovariance structure of covariance stationary series and lists four autocovariance functions:(a)γ(t, τ ) = α2https:// τ ) = e−τ(c)γ(t, τ ) = ατ(d)γ(t, τ ) =Whereαατis a positive constant.Which autocovariance functions(s) areconsistent with covariance stationarity, and which are not? Why?9. ( Conditional and unconditional means) As head of sales of the leadingtechnology and innovation magazine publisher TECGIT, your bonus isdependent on the rm’s revenue. Revenue changes from season to season,as subscriptions and advertising deals are entered or renewed. From yourexperience in the publishing business, you know that the revenue in aar stued dvi y reaC souou rcrs eeH wer aso.comseason is a function of the number of magazines sold in the previous seasonand can be described asresidualsyεt ∼ N (0, 1000),is revenue andxyt = 1000 + 0.9xt−1 + εt ,with uncorrelatedwhereis the number of magazines sold.(a) What is the expected revenue for next season conditional on totalsales of 6340 this season?(b) What is unconditionally expected revenue if unconditionally expectedsales are 8500?(c) A rival publisher oers you a contract identical to your current contract ( same base pay and bonus). Based on a condential interview,you know that the same revenue model with identical coecients isappropriate for your rival.The rival has sold and average of 900magazines in previous seasons but only 5650 this season. Will youaccept the oer? Why or Why not?10. (Empirical questions) Reproduce all the gures and tables in part 4 ofChapter6, Application: Forecasting Housing Starts.(Please refer to PageshThis104, Elements of forecast by Francis x. Diebold)3https:// by TCPDF (