"Frontmatter". In: Analysis of Financial Time Series

NONSYNCHRONOUS TRADING 177t − k t , t − k t + 1,...,t − 1). Mathematically, the relationship between r t and r o t is⎧0 with probability πr t with probability (1 − π) 2r t + r t−1 with probability (1 − π)⎪⎨2 πrt o r t + r t−1 + r t−2 with probability (1 − π) 2 π 2=..∑ ki=0r t−i with probability (1 − π) 2 π k−1⎪⎩..(5.1)These probabilities are easy to understand. For example, rto = r t if and only if thereare trades at both t and t − 1, rto = r t + r t−1 if and only if there are trades at t andt − 2,butnotradeatt − 1, and rto = r t + r t−1 + r t−2 if and only if there are tradesat t and t − 3, but no trades at t − 1andt − 2, and so on. As expected, the totalprobability is 1 given byπ + (1 − π) 2 [1 + π + π 2 +···]=π + (1 − π) 2 11 − π = π + 1 − π = 1.We are ready to consider the moment equations of the observed return series {rt o}.First, the expectation of rt o isE(r o t ) = (1 − π)2 E(r t ) + (1 − π) 2 π E(r t + r t−1 ) +···= (1 − π) 2 µ + (1 − π) 2 π2µ + (1 − π) 2 π 2 3µ +···= (1 − π) 2 µ[1 + 2π + 3π 2 + 4π 3 +···]= (1 − π) 2 1µ = µ. (5.2)(1 − π) 2In the prior derivation, we use the result 1 + 2π + 3π 2 + 4π 3 +···= 1(1−π) 2 . Next,for the variance of r o t ,weuseVar(r o t ) = E[(r o t )2 ]−[E(r o t )]2 andE(r o t )2 = (1 − π) 2 E[(r t ) 2 ]+(1 − π) 2 π E[(r t + r t−1 ) 2 ]+···= (1 − π) 2 [(σ 2 + µ 2 ) + π(2σ 2 + 4µ 2 ) + π 2 (3σ 2 + 9µ 2 ) +·] (5.3)= (1 − π) 2 {σ 2 [1 + 2π + 3π 2 +···]+µ 2 [1 + 4π + 9π 2 +···]} (5.4)[ ] 2= σ 2 + µ 2 1 − π − 1 . (5.5)InEq.(5.3),weuseE( ) k∑ 2 ( ) k∑r t−i = Var r t−i +i=0i=0[E( )] k∑ 2r t−i = (k + 1)σ 2 +[(k + 1)µ] 2i=0