Knowledge that Transforms

The Problem Many discrete time series are generated by the observation of processes which are most naturally considered to be continuously changing with time, or (almost equivalently) which have a fundamental time interval of evolution which is very much smaller than the sampling interval.The case in which the sampling interval is constant has been studied at some 1ength,1 but for some important applications, the sampling intervals are not evenly spaced, and this factor adds considerable complication to analysis of the data.Consider a continuous random process X(t) which is covariance stationary, that is: E(X(t)'X(t+s)) = R(s) is a function of s only.Further, assume that X(t) is "ergodic"; namelyCondition (1) assures that time averages converge to expectations when ca1culating sample autocovariances. 2 Finally, to help simplify the analysis and notation, assume E[X(t)] = 0 Then the discrete process ... , XeD) , X(l), X(2), .. has mean zero and is stationary and "ergodic" in the sense defined.It can be analyzed by standard statistical methods, although a "simple" X(t) in continuous time may give rise to a more "complicated" process in discrete time. 3

In a recent article in this journal, Professors Sinai and Stokes (1972) presented a very interesting test of the hypothesis that money enters the production function, and they suggest that real balances could be a missing variable that has contributed to the unexplained 'residual' being attributed to technological The theory of induced innovation, as presented by Fellner (1961) and Schmookler (1966), suggests that market conditions affect the demand for innovation and the realized technological changes. Since money may be regarded as a proxy for short-run fluctuations in the aggregate demand, this theory suggests that money affects output and technological changes as a demand factor rather than as a factor of production. In this note, we suggest the appropriate tests to distinguish between the two alternative hypotheses, and present some empirical results.