The constructive alternative to complaining about linear regression is non-parametric regression. There are many ways to do this, but we will focus on the conceptually simplest one, which is smoothing; especially kernel smoothing. All smoothers involve local averaging of the training data. The bias-variance trade-off tells us that there is an optimal amount of smoothing, which depends both on how rough the true regression curve is, and on how much data we have; we should smooth less as we get more information about the true curve. Knowing the truly optimal amount of smoothing is impossible, but we can use cross-validation to select a good degree of smoothing, and adapt to the unknown roughness of the true curve. Detailed examples. Analysis o how quickly kernel regression converges on the truth. Using smoothing to automatically discover interactions. Plots to help interpret multivariate smoothing results. Average predictive comparisons.
Optional readings: Hayfield and Racine, "Nonparametric Econometrics: The np Package"; Gelman and Pardoe, "Average Predictive Comparisons for Models with Nonlinearity, Interactions, and Variance Components" [PDF]
Posted by crshalizi at January 26, 2012 10:30 | permanent link