On the optimality of adaptive expectations: Muth revisited

Muth (1960, J. American Statistical Association 55, 299-306) showed that adaptive expectations are optimal, in the sense that they minimise the mean squared forecast error of an infinite-history random walk series observed with noise. This paper derives an explicit formula for the optimal forecasting weights for the IMA(1,1) series analysed by Muth when the series has a finite history. We find that the optimality of adaptive expectations is a very robust result for the IMA(1,1) process; this will hold even in very small samples, unless the ratio of the innovation variance of the random walk component to the variance of the noise component is very small. The exact relationship between our explicit forecasting weights and the weights derived from a recursive updating scheme is also explained in the paper.