ENH/Design target results statistic, ATE, delta method, extra moment condition

[josef-pkt]

Just parking a vague idea

In several cases we want to estimate additional statistics to help in the interpretation, these are like parameters that don't show up in the null model example

• ATE: estimate a regression or doubly robust model and use GMM to get inference for ATE
• relative risk: common is to use log-binomial model, but instead we can just compute average RR for the sample from any standard model
• test for zero-inflation with extra moment condition that estimates excess zero probability
• maybe: smearing estimate of expected value when endog has been transformed, e.g. we estimate log-linear model using OLS(log(endog), ...) (sounds similar to this case but I haven't looked at it in a long time)
• predict maybe for some parts (averaged `"which")
• ...

in general we can use

• delta method given estimates of params and cov_params
• estimate jointly as an extra moment condition as in GMM

those might be algebraically equivalent in some cases.

extra moment condition only work for fit sample, i.e. we take into account that parameters are estimated from the same sample as we use for the estimating the additional target statistics

qu: what do we do with out-of sample predictions of the target statistic or when we want to use different population weights in the estimation and the computation of the target statistic? (Can we use different freq_weights in the moment condition for the target statistic without changing the weights of the estimating equations for the params?)

We already have some of this in GMM and in conditional moment tests, but the latter are designed for specification testing and don't provide estimates of the target parameters.

variable addition score_tests are doing something similar, i.e. estimate score of extra params using correlation and derivatives (score_obs and hessian for augmented model) I guess that would be similar as a one-step estimator for the augmented model. Maybe not, variable augmentation will also affect the included parameters, and under null model the extras have no effect. There might be different cases depending on whether some off-diagonal submatrices of the derivatives, cross-derivatives, are zero. (derivative matrices are not symmetric) A similar case is mean-variance orthogonality in LEF but not in other models. (scale or dispersion estimate from auxiliary estimation or jointly based on extra moment conditions.)

TBC I'm not sure yet what we can support generically.

[josef-pkt]

one implication is that we have another use for score_obs and we need helper methods for derivatives of those extra moment conditions, similar to what we can do with score_factor and hessian_factor in the variable addition score tests.

or we just use the OPG version

[josef-pkt]

nice application, conditional binary association

• estimate two ordinal models y1 | x1 and y2 | x2
• define some association measures based on conditional probabilities
• combine score functions or moment conditions from two models with association statistic
• compute sandwich variance

Chun Li & Bryan E. Shepherd (2010) Test of Association Between Two Ordinal Variables While Adjusting for Covariates, Journal of the American Statistical Association, 105:490, 612-620, DOI: 10.1198/jasa.2010.tm09386
https://doi.org/10.1198/

same authors as in #7076 (I guess that's why I looked at it)

The following sounds related but I didn't look at it Dungang Liu, Shaobo Li, Yan Yu & Irini Moustaki (2020): Assessing Partial Association Between Ordinal Variables: Quantification, Visualization, and Hypothesis Testing, Journal of the American Statistical Association, DOI: 10.1080/01621459.2020.1796394 To link to this article: https://doi.org/10.1080/01621459.2020.1796394