LASSOPACK

## When would you want to use lassopack? #

lassopack is a suite of programs for regularized regression methods suitable for the high-dimensional setting where the number of predictors, $$p$$ , may be large and possibly greater than the number of observations, $$N$$ .

### 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 fundamental problem in 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.