A Twitter primer on the canonical link the linear exponential family. I've used this combination in a few of my papers: the doubly robust estimators for estimating average treatment effects, improving efficiency in RCTs, and, most recently, nonlinear DiD.
#metricstotheface
#metricstotheface
The useful CL/LEF combinations are:
1. linear mean/normal
2. logistic mean/Bernoulli (binary fractional)
3. logistic mean/binomial (0 <= Y <= M)
4. exponential mean/Poisson (Y >= 0)
5. logistic means/multinomial
The last isn't used very much -- yet.
1. linear mean/normal
2. logistic mean/Bernoulli (binary fractional)
3. logistic mean/binomial (0 <= Y <= M)
4. exponential mean/Poisson (Y >= 0)
5. logistic means/multinomial
The last isn't used very much -- yet.
The key statistical feature of the CL/LEF combinations is that the first order conditions look like those for OLS (combination 1). The residuals add to zero and each covariate is uncorrelated with the residuals in sample. Residuals are uhat(i) y(i) - mhat(x(i)).
So in a logit, the residuals have average zero and zero correlation with covariates. Not with a probit in sample.
The population versions of the sample analogs are what ensure double robustness in IPWRA and consistency when doing covariate adjustment for RCTs.
The population versions of the sample analogs are what ensure double robustness in IPWRA and consistency when doing covariate adjustment for RCTs.
As a bonus, with the CL/LEF combo, the Hessian depends only on X, not Y. So there's no need to choose between integrating out Y or not. The means that sandwich standard errors tend to be well behaved.
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