policymakers demonstrate

Big Data in Economics: Evolution or Revolution? 617
Box 14.2: The Ridge Regression Estimator
The ridge regression estimator is given by
[ ]
ˆβ ridge = argmin β ‖Y − Xβ‖
2
2
+ λ 2 ‖β‖ 2 2
= (X ′ X + λ 2 I) −1 X ′ (14.5)
Y
where I is the identity matrix and λ 2 > 0 is the so-called ‘regularization
parameter’, which, as seen from (14.5), reduces the impact of the smallest
eigenvalues of X ′ X, at the origin of the instability of the OLS estimator.
An alternative to quadratic penalties that allows for variable selection by
enforcing sparsity, that is, the presence of zeroes in the vector β of the regression
coefficients, has been popularized in the statistics and machine-learning
literature under the name of ‘Lasso regression’ by Tibshirani (1996). It consists
in replacing the L 2 -norm penalty used in ridge regression by a penalty
proportional to the L 1 -norm of β (see Box 14.3).
In the case of orthonormal regressors, it is easily seen that the Lasso penalty
provides a nonlinear shrinkage of the components of ˆβ ols , which are shrunk
differently according to their magnitude, as well as sparsity, since the jth coefficient
[ ˆβ lasso ] j = 0if|[X ′ Y] j |