Suppose the algorithm chooses a tree that splits on education but not on age. Conditional on this tree, the estimated coefficients are consistent. But that does not imply that treatment effects do not also vary by age, as education may well covary with age; on other draws of the data, in fact, the same procedure could have chosen a tree that split on age instead

hese heterogenous treatment effects can be used to assign treatments; Misra and Dubé (2016) illustrate this on the problem of price targeting, applying Bayesian regularized methods to a large-scale experiment where prices were randomly assigned

Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, and Newey (2016) take care of high-dimensional controls in treatment effect estimation by solving two simultaneous prediction problems, one in the outcome and one in the treatment equation.

These same techniques applied here result in split-sample instrumental variables (Angrist and Krueger 1995) and “jackknife” instrumental variables

Understood this way, the finite-sample biases in instrumental variables are a consequence of overfitting.

Even when we are interested in a parameter β ˆ, the tool we use to recover that parameter may contain (often implicitly) a prediction component. Take the case of linear instrumental variables understood as a two-stage procedure: first regress x = γ′z + δ on the instrument z, then regress y = β′x + ε on the fitted values x ˆ. The first stage is typically handled as an estimation step. But this is effectively a prediction task: only the predictions x ˆ enter the second stage; the coefficients in the first stage are merely a means to these fitted values.

Prediction in the Service of Estimation

New Data

Zhao and Yu (2006) who establish asymptotic model-selection consistency for the LASSO. Besides assuming that the true model is “sparse”—only a few variables are relevant—they also require the “irrepresentable condition” between observables: loosely put, none of the irrelevant covariates can be even moderately related to the set of relevant ones.

First, it encourages the choice of less complex, but wrong models. Even if the best model uses interactions of number of bathrooms with number of rooms, regularization may lead to a choice of a simpler (but worse) model that uses only number of fireplaces. Second, it can bring with it a cousin of omitted variable bias, where we are typically concerned with correlations between observed variables and unobserved ones. Here, when regular-ization excludes some variables, even a correlation between observed variables and other observed (but excluded) ones can create bias in the estimated coefficients.

97the variables are correlated with each other (say the number of rooms of a house and its square-footage), then such variables are substitutes in predicting house prices. Similar predictions can be produced using very different variables. Which variables are actually chosen depends on the specific finite sample.

Through its regularizer, LASSO produces a sparse prediction function, so that many coefficients are zero and are “not used”—in this example, we find that more than half the variables are unused in each run

One obvious problem that arises in making such inferences is the lack of stan-dard errors on the coefficients. Even when machine-learning predictors produce familiar output like linear functions, forming these standard errors can be more complicated than seems at first glance as they would have to account for the model selection itself. In fact, Leeb and Pötscher (2006, 2008) develop conditions under which it is impossible to obtain (uniformly) consistent estimates of the distribution of model parameters after data-driven selection.

First, econometrics can guide design choices, such as the number of folds or the function class.

These choices about how to represent the features will interact with the regularizer and function class: A linear model can reproduce the log base area per room from log base area and log room number easily, while a regression tree would require many splits to do so.

Ta b l e 2Some Machine Learning Algorithms

Picking the prediction func-tion then involves two steps: The first step is, conditional on a level of complexity, to pick the best in-sample loss-minimizing function.8 The second step is to estimate the optimal level of complexity using empirical tuning (as we saw in cross-validating the depth of the tree).

egularization combines with the observability of predic-tion quality to allow us to fit flexible functional forms and still find generalizable structure.

This procedure works because prediction quality is observable: both predic-tions y ˆ and outcomes y are observed. Contrast this with parameter estimation, where typically we must rely on assumptions about the data-generating process to ensure consistency.

In empirical tuning, we create an out-of-sample experiment inside the original sample.

Performance of Different Algorithms in Predicting House Values

We consider 10,000 randomly selected owner-occupied units from the 2011 metropolitan sample of the American Housing Survey. In addition to the values of each unit, we also include 150 variables that contain information about the unit and its location, such as the number of rooms, the base area, and the census region within the United States. To compare different prediction tech-niques, we evaluate how well each approach predicts (log) unit value on a separate hold-out set of 41,808 units from the same sample. All details on the sample and our empirical exercise can be found in an online appendix available with this paper athttp://e-jep.org

Making sense of complex data such as images and text often involves a prediction pre-processing step.

The fundamental insight behind these breakthroughs is as much statis-tical as computational. Machine intelligence became possible once researchers stopped approaching intelligence tasks procedurally and began tackling them empirically.

In another category of applications, the key object of interest is actually a parameter β, but the inference procedures (often implicitly) contain a prediction task. For example, the first stage of a linear instrumental variables regres-sion is effectively prediction. The same is true when estimating heterogeneous treatment effects, testing for effects on multiple outcomes in experiments, and flexibly controlling for observed confounders.

Machine Learning: An Applied Econometric Approach