Sparse alpha-norm regularization has many data-rich applications in marketing and economics. In contrast to traditional lasso and ridge regularization, the alpha-norm penalty has the property of jumping to a sparse solution. This is an attractive feature for ultra high-dimensional problems that occur in market demand estimation and forecasting. The underlying nonconvex regularization problem is solved via coordinate descent, and a proximal operator. To illustrate our methodology, we study a classic demand forecasting problem of Bajari, Nekipelov, Ryan, and Yang (2015a). On the empirical side, we find many strong sparse predictors, including price, equivalized volume, promotion, flavor scent, and brand effects. Benchmark methods including linear regression, ridge, lasso and elastic net, are used in an out-of-sample forecasting study. In particular, alpha-norm regularization provides accurate estimates for the promotion effects. Finally, we conclude with directions for future research.
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