The Predictive Space, or, If x predicts y, what does y tell us about x?
A predictive regression for y(t) and a time series representation of the predictors, x(t), together imply a univariate reduced form for y(t). In this paper we work backwards, and ask: if we observe y(t), what do its univariate properties tell us about any x(t) in the "predictive space" consistent with those properties? We provide a mathematical characterisation of the predictive space and certain of its derived properties. We derive both a lower and an upper bound for the R^2 for any predictive regression for y(t). We also show that for some empirically relevant univariate properties of y(t), the entire predictive space can be very tightly constrained. We illustrate using Stock and Watson's (2007) univariate representation of inflation.
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