In recent years, there has been a significant growth in research focusing on minimum $\ell_2$ norm (ridgeless) interpolation least squares estimators. However, the majority of these analyses have been limited to an unrealistic regression error structure, assuming independent and identically distributed errors with zero mean and common variance. In this paper, we explore prediction risk as well as estimation risk under more general regression error assumptions, highlighting the benefits of overparameterization in a more realistic setting that allows for clustered or serial dependence. Notably, we establish that the estimation difficulties associated with the variance components of both risks can be summarized through the trace of the variance-covariance matrix of the regression errors. Our findings suggest that the benefits of overparameterization can extend to time series, panel and grouped data.

The log odds ratio is a well-established metric for evaluating the association between binary outcome and exposure variables. Despite its widespread use, there has been limited discussion on how to summarize the log odds ratio as a function of confounders through averaging. To address this issue, we propose the Average Adjusted Association (AAA), which is a summary measure of association in a heterogeneous population, adjusted for observed confounders. To facilitate the use of it, we also develop efficient double/debiased machine learning (DML) estimators of the AAA. Our DML estimators use two equivalent forms of the efficient influence function, and are applicable in various sampling scenarios, including random sampling, outcome-based sampling, and exposure-based sampling. Through real data and simulations, we demonstrate the practicality and effectiveness of our proposed estimators in measuring the AAA.

We develop a new method of online inference for a vector of parameters estimated by the Polyak-Ruppert averaging procedure of stochastic gradient descent (SGD) algorithms. We leverage insights from time series regression in econometrics and construct asymptotically pivotal statistics via random scaling. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem. Our proposed inference method has a couple of key advantages over the existing methods. First, the test statistic is computed in an online fashion with only SGD iterates and the critical values can be obtained without any resampling methods, thereby allowing for efficient implementation suitable for massive online data. Second, there is no need to estimate the asymptotic variance and our inference method is shown to be robust to changes in the tuning parameters for SGD algorithms in simulation experiments with synthetic data.

Researchers may perform regressions using a sketch of data of size m instead of the full sample of size n for a variety of reasons. This paper considers the case when the regression errors do not have constant variance and heteroskedasticity robust standard errors would normally be needed for test statistics to provide accurate inference. We show that estimates using data sketched by random projections will behave ’as if’ the errors were homoskedastic. Estimation by random sampling would not have this property. The result arises because the sketched estimates in the case of random projections can be expressed as degenerate U-statistics, and under certain conditions, these statistics are asymptotically normal with homoskedastic variance. We verify that the conditions hold not only in the case of least squares regression when the covariates are exogenous, but also in instrumental variables estimation when the covariates are endogenous. The result implies that inference can be simpler than the full sample case if the sketching scheme is appropriately chosen.