- The paper develops a double Lasso approach that residualizes outcomes and covariates to achieve precise causal estimates in high-dimensional settings.
- It demonstrates how employing Neyman orthogonality minimizes first-order errors, ensuring unbiased parameter estimation.
- Empirical evidence on economic convergence validates the method's superior precision over traditional least squares techniques.
Statistical Inference in High-Dimensional Regression Models: A Double Lasso Approach
Inference with Double Lasso
Double Lasso involves employing Lasso-based methods twice for residualizing outcomes and the target covariate of interest for which the predictive effect is desired. This methodology is particularly effective in high-dimensional settings where the number of regressors (𝑝) exceeds the number of observations (𝑛). The credibility of this technique hinges on the approximate sparsity of the best linear predictors for the outcome and the target covariate. The resulting estimator localizes around the true value within a √(𝑉/𝑛) neighborhood and exhibits approximately normal distribution, enabling the construction of confidence intervals.
Neyman Orthogonality: Ensuring Low Bias in High-Dimensional Settings
Neyman orthogonality ensures that the estimation error in the first-step nuisance parameters does not have a first-order effect on the target parameter 𝛼. This property is crucial for inferring predictive effects in high-dimensional regression models. By guaranteeing that the target parameter's estimation process is locally insensitive to perturbations of nuisance parameters, Neyman orthogonality helps achieve high-quality estimation and inference in settings where traditional methods may falter due to the curse of dimensionality.
Application and Empirical Evidence: Testing the Convergence Hypothesis
An empirical paper was conducted to assess the convergence hypothesis in economic growth rates relative to initial wealth levels across countries, controlling for various institutional and education characteristics. The paper leveraged a sample containing data on 90 countries and around 60 controls. The traditional least squares method yielded noisy estimates for the convergence rate, failing to provide conclusive insights. However, the Double Lasso approach yielded a more precise estimate for the annual convergence rate, substantiating the conditional convergence hypothesis. This example illustrates the potential of Double Lasso in high-dimensional regression analysis, especially in cases where the ratio 𝑝/𝑛 is not negligible.
Methodological Insights and Practical Implications
The Double Lasso technique offers a potent tool for researchers in fields such as econometrics, where high-dimensional data sets are increasingly common. Its reliance on approximate sparsity and the critical role of Neyman orthogonality underscore the importance of carefully selecting regularization parameters and ensuring methodological rigor. Through its application in empirical research, such as testing economic theories like the convergence hypothesis, Double Lasso showcases its capacity to yield reliable and interpretable results, notwithstanding the high-dimensionality challenge.
The development and application of Double Lasso methods in high-dimensional linear regression models mark a significant advance in statistical inference. By addressing the unique challenges posed by high-dimensional data, these methods enable researchers to uncover meaningful predictive and causal relationships that were previously obscured. Future developments in this area are expected to further refine these techniques, broadening their applicability and enhancing their robustness in facing the complexities of modern data analysis.