Question

Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify your answer.

Answer 1

son retum seris iid Ytot +4 +0.25 041 , añ'nlo,ray 1017) all t=1UT @ Mean of n stat.C+1) 70 ELY ) = Elt) + Ecat) +0.25.6(04-) € (94-999042 = 4 + 0 1 0 28 XD 6qAten) = 0 th hto Auto covariance function Ven) = coue Ves Ven) = E(T4-ELVE))(YEth-FLYEth)) = E (lt+Q4 +0.2624-1-)(+th+Ql+ht 0.25Qythi(t+hi)] = E((04+0.250++) (alth +0.26u4th] = € [ 604.04th) +0.25xfaktx Qten)+ 0.25704+ Cutht) +0.45)?(04-1•* Qtthv)] = E(Q.x 04+h) +(.259 (244xQ4+) 7(0.25) El Q4x Cething at +812572 E104 * Alth-1) a ta ?+ (0.25)? ora 2. hao ( 0.25.00 halout o r other use Irena sraz 1+0.25)2) if heo if hal- Otherwisel ang Vin) = VA

since Elf) = t depends upon at , so v ů not weately stationary. Ani ule = PV = 74--ct+ C4 +0.2504-1-(t-1)-971 - 0.25 q-2 W = 1 +94 +0.25-1904,-0.2544-2 WE = 1 +04-0.75044-0125-04-2 Then mean of t is elW ) = 1 +0+0+0 -1 2222222222222222.222.2.2.2.122.22.2.1.1.1111111111 ITS vel)=1 Wt: Jamu and Autocovariance - function. 01h2cou (wy, Wzen) - Elu- €144]).(W4th-E (W4th) ] pen) =€ (04–0.75044–0,2504-2)(4+h-075 Otthy-0-25 Otth-2)] yeh On? (1 +(0-250°+(0.2312) , hao da? (-0,75 + 0-95xa25) h=1,1 1 r2² (-0.25) h=2,-2 Otherwise Po Tylna Oc? (176-75){+ (0-25)?) 3 - 83? (0.1s)• -ra?. x 0.25 hao haling h=2,-2) aus since flwt) =] - construit ut , so we is weakly stationary.