1 (a) Is the MA(2) processet=At-0.1At-1+ 0.21At-2stationary, or invertible?… 1 answer below »


The University of Western AustraliaSchool of Mathematics & StatisticsSTAT7450: Time Series Methods and ApplicationsAssignment 2(2012)Submit solutions by end of Friday 4th May.In these questions, (At) is a white noise process.1 (a) Is the MA(2) processet=At-0.1At-1+ 0.21At-2stationary, or invertible? Givereasons, and find the correlation structure.(b) Letqbe a positive integer andt= (1 +q)-1?qj=0At-j. Using a scale change, expressthis process as an ARMA, identifying its orders. Is it stationary or invertible? Determine itscorrelation structure.(c) Show that the ARMA(2,2) processt= 0.4t-1+ 0.45t-2+At+At-1+ 0.25At-2can be reduced to an ARMA(1,1), and determine the difference equation form of the reducedprocess.2. An MA(6) process has weights?1= 0.5, ?2=-0.25, ?3= 0.125, ?4=-0.0625, ?5=0.03125 &?6=-0.015625. By considering MA(8) representations, find a (very) low orderautoregressive process whose GLP weights are essentially the same as for the?’s. Compare thevariances and correlations of the two processes, i.e., tabulate them for lags`= 1,…,6. [Hint:Look at how the weights relate to each other.] You lose no generality by assumingvA= 1, sodo this.3. Let|f| `).Is the process stationary? Determine circumstances for whichCorr(t,t-`)˜f`.(b) Show that redefining1=A1/v1-f2gives a stationary process.4. There is quite a lot of fairly recent work in the journal literature about models for sta-tionary processes which have discrete marginal laws. Here is one such. Fort= 0,±1,±2,…,1
lett=At-12At-1where theAt’s are independent and standard normal.(a) Show that{t}is strictly stationary and find its correlation structure.Now define the ‘clipped’ processYt={1 ift=0,-1 ift0). Note that strict stationarity of (t)implies that?indeedisindependent oft. Also, you don’t have to evaluate?for this part ofthe question. Is (Yt) stationary?(c) [Hard!] Compute?1(Y) :=Corr(Yt-1,Yt). Now you do need to evaluate?. Refresh yourmemories about the bivariate normal law, in particular the conditional law oftgivent-1=a,say. Explain why?= 2p, wherep=P(tt-1>0 &t-1>0). Let?=Corr(t-1,t),considered arbitrary for now, and leta=-?/v1-?2. Use bivariate normal theory to showthatp=?80F(az)f(z)dz,wherefand F are the standard normal density and distribution functions, respectively, andF = 1-F, the survivor function. By writingF(az) as an integral, expresspas a repeatedintegral, reverse the order of integration, and then change to polar coordinates to obtain asimple explicit evaluation ofp. [Instead, you may be able to use Maple, etc.]Are theYtmutually independent?2

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