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Consider a problem, when one is interested in finding influence of variable set {X} on Y (assuming reverse causal relation is infeasible), when confounding influences from a set of variables {Z} are present.

I am familiar with two approches to disentangle effect of {Z} on Y from effect of {X}:

  1. Partial correlation / residualization in a "fixed" effect model, when the number of covariates $p_Z =$ |{Z}| is small compared to the number of available samples, $p_Z \ll N$, as by fitting multiple regressions one removes degrees of freedom, and given linearly independent factors Z, the degrees of freedom will be depleted as $p_Z$ approaches $N$.

  2. Expressing correlation structure of confounding effects as "random" effect, which requires $p_Z > N$ to estimate covariance matrix.

Question: What are alternative approaches? Are there approaches that work for $p_Z \approx N$ or all mentioned regimes? Have sparse regression models been applied to this problem?

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Other (non-experimental) approaches are instrumental variables, regression discontinuity and propensity score matching, to name a few. If you can, you should conduct a randomized control trial to estimate the impact of X on Y. You can find a discussion of these approaches in: Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press.

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  • $\begingroup$ my question is not about experiment design; I see that instrumental variable approach directly relates to partial correlation I mentioned, just puts it in causal inference terms. My question is what analysis method to use when the number of covariates is a bit below sample size. $\endgroup$ – Dima Lituiev Jun 3 '16 at 16:25

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