"Edgeworth's Time Series Model:
Not AR(1), but the same Covariance Structure"
Autoregressive models have been popular choices for modeling time series data since the early work of Yule (1927), especially when a scatter plot of pairs of successive observations indicate a linear trend. Curiously, some 40 years earlier, Edgeworth developed a different model for treating economic time series with random increments and decrements. A version of this process with normal errors has the same covariance structure as an AR(1) process, but is actually a mixture of a very large number of processes, some of which are not stationary. That is, joint distributions of lag 3 or greater are not normal but are fixtures of normals (even though all pairs are bivariate normal). This Edgeworth Process has many additional surprising features, two of which are: (1) it has Markov structure, but is not generated by a one-step transition operator, and (2) the sample paths look very much like an AR(1), but it can be distinguished from an AR(1) about as well as distinguishing a mean difference of nearly 1 standard deviation with normal samples of size 100 or greater. It is widely recognized that model identification and verification are needed to avoid serious errors
in inference. Examples like this one show that standard model-fitting diagnostics (like any ones based on second order properties) can be entirely inadequate and misleading.