log_to_level: Translate estimated multiplicative (in logs) effects to level...

In michaelpollmann/parTreat: Efficiently Estimate Treatment Effects Based On A Parametric Model For Treatment Effects

Description

After estimating treatment effects with the outcomes in logs, this helper function translates the effect back into a level effect. It also calculates the appropriate standard error for the in-sample effect using the delta method.

Usage

log_to_level(est_log, X_pop, Y_pop)

Arguments

est_log	a list of (at least) the two elements tau and se, for instance as returned by eif_additive or waq. It is assumed that the estimate is the (untransformed) estimate of an additive specification with the outcome variables in logs.
X_pop	numeric vector, the outcomes of the control observations in levels, or alternatively of a super-population of interest.
Y_pop	numeric vector, the outcomes of the treated observations in levels, or alternatively of a super-population of interest.

Details

If one assumes a multiplicative model Y(1) = θ Y(0), the specification is additive after taking the natural logarithm. Hence, one can use the function eif_additive or waq applied to outcomes in logs to estimate log(θ). However, one is often substantively interested in the average effect of the treatment rather than in the multiplicative parameter θ. This function therefore translates the effect into the sample average level effect for a sample of control observations X_pop and treated observations Y_pop.

In the multiplicative model, even if θ is known exactly, estimates of the population average effect, under the multiplicative model

Y(1) - Y(0) = (θ-1) Y(0)

can be noisy because they require estimating the population mean of the control (e.g. by the sample mean). The standard errors here are therefore calculate for a fixed sample of control X and treated Y. Under the assumption of the multiplicative model, outcomes can then be considered fixed, and the only variation comes from estimating θ

Value

list of four elements:

tau_level	the point estimate of the treatment effect in levels
se_level	the standard error for the treatment effect in levels
tau_log	the point estimate of the treatment effect with the outcome in logs (copied from the input list, est_log$tau)
se_log	the standard error for the treatment effect with the outcome in logs (copied from the input list, est_log$se)

Examples

# draw a random sample with multiplicative treatment effect
X <- rexp(n=1000, rate=2)
Y <- 1.1 * rexp(n=200, rate=2)
# estimate the treatment effect with an additive specification in logs, either waq or eif
est_waq_log <- waq(log(X),log(Y))
est_eif_log <- eif_additive(log(X),log(Y))
# translate the effects into level effects
log_to_level(est_waq_log,X,Y)
log_to_level(est_eif_log,X,Y)

michaelpollmann/parTreat documentation built on Dec. 21, 2021, 5:58 p.m.