Test of be Random Walk Suppose that our universe of stocks consists of N securities indexed by i, each with the return generating process (21) where RMr represents a factor common to all returns(e. g the market) and is assumed to be an independently and identically distributed (i i d )ran dom variable with mean uM and variance of. The eu term represents the idiosyncratic component of security i's return and is also assumed to be i i.d. (over bothi and t), with mean 0 and variance odf. The return-generating process may thus be identified with N securities each with a unit beta such that the theoretical R2 of a market-model regression for each security is Suppose that in each period t there is some chance that security i does ot trade. One simple approach to modeling this phenomenon is to dis. tinguish between the observed returns process and the virtual returns rocess. For example, suppose that security i has traded in period t-1 consider its behavior in period t. If security i does not trade in period t, define its virtual return as Ri [which is given by Equation(21)], whereas its observed return R is zero. If security i then trades at t 1, its observed return r+ is defined to be the sum of its virtual returns Ra and Ru+; hence, ontrading is assumed to cause returns to cumulate. The cumulation of returns over periods of nontrading captures the essence of spuriously induced correlations due to the nontrading lag To calculate the magnitude of the positive serial correlation induced by nontrading, we must specify the probability law governing the nontrading event. For simplicity, we assume that whether or not a security trades may be modeled by a Bernoulli trial, so that in each period and for each security there is a probability p that it trades and a probability 1-p that it does not. It is assumed that these Bernoulli trials are i.i.d. across securities and, for each security, are i i.d. over time. Now consider the observed return Rh at time t of an equal- weighted portfolio R (22) N The observed return R for security i may be expressed as R=X1(O0)Rn+X(1)R-1+X(2)Rn-2+ where X(D,j=1, 2, 3,.. are random variables defined as odels imply that returns for individual securities ll exhibit negative serial correlation but that portfolio returns will be positively autocorrelated