In the Roll model trade prices are pt = mt + cdt where mt is the efficient price (mid-price) and dt = 1 is the trade sign (indicator) which shows if the trade is a buy or sell.

The simple Roll model assumes that trade signs are serially uncorrelated: corr(dt, ds) = 0 for t ?= s. In practice, trade signs are correlated, with positive autocorrelation. Suppose that corr(dt, dt?1) = ? > 0 and corr(dt, dt?k) = 0 for k > 1. Suppose that ? > 0 is known. Also, assume that the trade signs dt are uncorrelated with the increments of the efficient price ?mt = ut.

i) Show that V ar(?pt) = 2c2(1 ? ?) + ?u2, Cov(?pt, ?pt?1) = ?c2(1 ? 2?), Cov(?pt, ?pt?2) = ?c2?, and Cov(?pt, ?pt?k) = 0 for k > 2.

ii) Suppose that 0

Hints. For (i), recall the corresponding expressions in the simple Roll 222

model V ar(?pt) = 2c + ?u and Cov(?pt, ?pt?1) = ?c .

Note that the equations above in (i) reduce to these expressions if ? = 0. Repeat the computations for the simple model allowing for the correlation ? and you should get the results stated.

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