"Frontmatter". In: Analysis of Financial Time Series

178 HIGH-FREQUENCY DATAunder the serial independence assumption of r t . Using techniques similar to that ofEq. (5.2), we can show that the first term of Eq. (5.4) reduces to σ 2 . For the secondterm of Eq. (5.4), we use the identity1 + 4π + 9π 2 + 16π 3 +···=which can be obtained as follows: Let2(1 − π) 3 − 1(1 − π) 2 ,H = 1 + 4π + 9π 2 + 16π 3 +··· and G = 1 + 3π + 5π 2 + 7π 3 +···.Then (1 − π)H = G and(1 − π)G = 1 + 2π + 2π 2 + 2π 3 +···= 2(1 + π + π 2 +···) − 1 =Consequently, from Eqs. (5.2) and (5.5), we have2(1 − π) − 1.Var(r o t ) = σ 2 + µ 2 [ 21 − π − 1 ]− µ 2 = σ 2 + 2πµ21 − π . (5.6)Consider next the lag-1 autocovariance of {rt o}.HereweuseCov(r t o, r t−1 o ) =E(rt or t−1 o ) − E(r t 0)E(r t−1 o ) = E(r t or t−1 o ) − µ2 . The question then reduces to findingE(rt or t−1 o ). Notice that r t or t−1 o is zero if there is no trade at t, notradeatt − 1, or notrade at both t and t − 1. Therefore, we have0 with probability 2π − π⎧⎪ 2r t r t−1 with probability (1 − π) 3r t (r t−1 + r t−2 ) with probability (1 − π) 3 π⎨rt o r t−1 o = r t (r t−1 + r t−2 + r t−3 ) with probability (1 − π) 3 π 2..r t ( ∑ ki=1 r t−i ) with probability (1 − π) 3 π k−1⎪ ⎩..(5.7)Again the total probability is unity. To understand the prior result, notice thatr o t r o t−1 = r tr t−1 if and only if there are three consecutive trades at t − 2, t − 1,and t. Using Eq. (5.7) and the fact that E(r t r t− j ) = E(r t )E(r t− j ) = µ 2 for j > 0,we haveE(rt o r t−1 o ) = (1 − π)3 {E(r t r t−1 ) + π E[r t (r t−1 + r t−2 )][ ( )] 3∑+ π 2 E r t r t−i +···}i=1= (1 − π) 3 µ 2 [1 + 2π + 3π 2 +···]=(1 − π)µ 2 .