## Welcome to the Stata Lasso Page

On this website we introduce three packages for regularized regression in Stata: lassopack, lassologit and pdslasso. The packages include features intended for prediction, model selection and causal inference. Thus, the routines are applicable in a wide range of settings.

• The package lassopack implements lasso (Tibshirani 1996), square-root lasso (Belloni et al. 2011), elastic net (Zou & Hastie 2005), ridge regression (Hoerl & Kennard 1970), adaptive lasso (Zou 2006) and post-estimation OLS.
• lassologit implements the logistic lasso for binary outcome models.
• pdslasso offers methods to facilitate causal inference in structural models. The package allows to select control variables and/or instruments from a large set of variables in a setting where the researcher is interested in estimating the causal impact of one or more (possibly endogenous) causal variables of interest.

## When would you want to use lassopack & lassologit?

### High-dimensional models

The regularized regression methods implemented in lassopack can deal with situations where the number of regressors is large or may even exceed the number of observations under the assumption of sparsity.

High-dimensionality can arise when (see Belloni et al., 2014):

• There are many variables available for each unit of observation. For example, in cross-country regressions the number of observations is naturally limited by the number of countries, whereas the number of potentially relevant explanatory variables is often large.
• There are only few observed variables, but the functional form through which these regressors enter the model is unknown. We can then use a large set of transformations (e.g. dummy variables, interaction terms and polynomials) to approximate the true functional form.

### Model selection

Identifying the true model is a common problem is applied econometrics. A standard approach is to use hypothesis testing to identify the correct model (e.g. general-to-specific approach). However, this is problematic if the number of regressors is large due to many false positives. Furthermore, sequential hypothesis testing induces a pre-test bias.

Lasso, elastic net and square-root lasso set some coefficient estimates to exactly zero, and thus allow for simultaneous estimation and model selection. The adaptive lasso is known to exhibit good properties as a model selector as shown by Zou (2006).

### Prediction

If there are many predictors, OLS is likely to suffer from overfitting: good in-sample fit (large $R^2$), but poor out-of-sample prediction performance. Regularized regression methods tend to outperform OLS in terms of out-of-sample prediction.

Regularization techniques exploit the variance-bias-tradeoff: they reduce the complexity of the model (through shrinkage or by dropping variables). In doing so, they introduce a bias, but also reduce the variance of the prediction, which can result in improved prediction performance.

### Forecasting with time-series or panel data

lassopack can also applied to time-series or panel data. For example, Medeiros & Mendes (2016) prove model selection consistency of the adaptive lasso when applied to time-series data with non-Gaussian, heteroskedastic errors.

## When would you want to use pdslasso?

The purpose of pdslasso is to improve causal inference when the aim is to assess the effect of one or a few (possibly endogenous) regressors on the outcome variable. pdslasso allows to select control variables and/or instruments.

### Many control variables

The primary interest in an econometric analysis often lies in one or a few regressors, for which we want to estimate the causal effect on an outcome variable. However, to allow for a causal interpretation we need to control for confounding factors. Lasso-type techniques can be employed to appropriately select controls and thus improve the robustness of causal inference.

### Many instruments

High-dimensional instruments can arise when there is inherently large number of potentially relevant instruments or when it’s unclear how these instruments should be specified (e.g. dummy variables, interaction effects).