Muth (1960)
Phillip Cagan, Milton Friedman and others used the ad hoc updating rule which they labeled adaptive expectations to forecast the hidden state y* (e.g., permanent income). In his 1960 paper Muth answered the question for what stochastic process for y will adaptive expectations as postulated by Cagan and Friedman be the optimal forecast of y*. Muth's approach to find recursive optimal linear forecast of a “hidden” state vector, x, given an “observer”, y is very similar to the Kalman filter, presented by Rudolf Kalman in his paper from the same year.
In his paper "Optimal Properties of Exponentially Weighted Forecasts", which was published in the Journal of the American Statistical Association in 1960, Muth rationalized Friedman’s adaptive expectations model for permanent income. He did this by reverse engineering a stochastic process for income for which Cagan’s expectation formula equals a mathematical expectation of future values conditioned on the infinite history of past incomes. Among Muth’s insights was that the stochastic process being forecast should dictate both the distributed lag and the conditioning variables that people use to forecast the future.
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