What are the conditions for valid extrapolation of statistical predictions? Answer I.

Thanks to Martin Ringo for this answer.

Let me offer a little of the terminology of forecasting, which, I hope, will make the question clearer. When you are forecasting from some kind of structural model, say Y = f(X1, …, Xk), there is a difference in whether you have to forecast the Xs as well as the Y. If you don’t, it is an unconditional forecast; if you do, it is conditional. For an unconditional forecast, the inference is a pretty straightforward exercise of the classical linear model, at least if you structure relationship is so estimated and a nonlinear version for nonlinear estimations. For a conditional forecast, life can be messy since you have to take into account the distribution of the exogenous, the Xs, as well as the error term.

I recently reviewed a forecast where the analyst treated a conditional forecast like an unconditional one, and consequently underestimated the forecast error by over a factor of two.

Anyway, a forecast (or prediction if you like) is presumably “valid” if it is within an _% confidence interval of an unbiased prediction. This is just the same thing as saying that the sample average is a valid estimator of the mean for most distributions while the number 5 — for a univariate distribution, 5 is an estimate — or the minimum value of the sample is often quite a poor estimator. So go ahead and use your classical inference. Just remember the confidence interval grows by the square once the forecast is out of sample, and conditional forecast confidence intervals usually have to be estimated by Monte Carlo means, especially if one has lagged dependent variables in the structural model.

Pindyck and Rubinfeld [Econometric Models and Economic Forecasts — one can ignore the economics and still learn from the statistics, but the authors need to sell the book to undergraduate econometric classes, hence the title] give a nice and not very technical explanation of the forecasting problem in the context of the classical linear regression model.

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