In other words, the constraints strongly encourage, but don’t require, that the solutions are monotone in absolute value. However, this need not be the case, as it is possible for both β ^ j + and β ^ j − to be positive and the | β ^ j | to have some non-monotonicity. Its solution typically has one or both of each pair ( β ^ j +, β ^ j −) equal to zero, in which case | β ^ j | = β ^ j + + β ^ j − and the solutions | β ^ j | are monotone non-increasing in j. The use of positive and negative components (rather than absolute values) makes this a convex problem. The penalty term encourages sparsity in β j + and β j −. Section 6 contains some discussion and directions for future work. Section 5 generalizes the ordered lasso and the strongly ordered lasso to the logistic regression model. We also apply this framework to auto-regressive (AR) time series and compare its performance with both the traditional method for fitting the AR model using least squares with the Akaike information criterion and Bayesian information criterion, and the lasso procedure for fitting the AR model ( Nardi & Rinaldo 2011). We demonstrate the usage of such algorithms on real and simulated data in Sections 3.4 and 3.5. Section 3 contains the detailed algorithms for applying the ordered lasso and the strongly ordered lasso to the time-lagged regression. Section 2 contains motivations and algorithms for solving the ordered lasso and the strongly ordered lasso (which enforces monotonicity in absolute value), as well as results comparing the ordered and the standard lasso on simulated data. Moreover, directly from the monotonicity constraint, a key feature of our procedure is that it automatically determines the most suitable value of K for each predictor. It is also reasonable to assume that personal exemption at t has greater impact on fertility rate at t than personal exemption at pervious time points ( Wooldridge 2009). For example, in a model of estimating fertility rate at t as a function of personal exemption, a reasonable assumption is that personal exemption at t, t − 1, ⋯, t − k all have some effect on fertility rate at t. In this case, it is natural to assume that the coefficients decay as we move farther away from t so that the order (monotonicity) constraint is reasonable. The main application of this idea is to time-lagged regression, where we predict an outcome at time t from features at the previous K time points. We derive an efficient algorithm for solving the resulting problem. In this paper we add an additional order constraint on the coefficients, and we call the resulting procedure the ordered lasso. This problem is convex and yields sparse solutions for sufficiently large values of λ.
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