Comparing coefficients across segmented regression models given aggregated, hete...
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My goal is to compare β1 across 7 models:
Y1t=β0+β1X1t+ϵtY2t=β0+β1X1t+ϵt⋮Y7t=β0+β1X7t+ϵt, where the realizations of YLt is time series data and XLt is a binary variable for L=1,2,…,7, interpreted as a treatment.
I want to compare β1 across models to determine if the treatment had a heterogenous effect on YLt. To be concrete, the response variables are different types of crime.
The problem is the means and variances of Y1,Y2,…,Y7 vary greatly. For the sake of argument, suppose Avg(Y1)=100,Avg(Y2)=4, and ^β1=−25,^β2=−1. In terms of averages, the effect is the "same", but the coefficients are quite different. So a direct comparison becomes difficult if not untenable.
I have tried to identify some approaches to compare the treatment effects:
- Percent change
Using the pre-treatment average of YL, compute percent change PC=β1Avg(YL)⋅100, and compare those.
- Standardize YL
Using overall Avg / Std, or the pre-treatment Avg / Std, estimate the coefficients and make direct comparisons (perhaps using a hypothesis test of their difference).
Z-test for difference of coefficients (or some other test)
Seemingly unrelated regression
Ideally, I want to determine if the difference in effects is statistically significant or not.
If willing to reparametrize the model, you might consider Poisson regression. So for Poisson regression we have:
E[Y1t]=exp(β01+β11X1t) E[Y2t]=exp(β02+β12X2t)
etc. for your stacked equations. Note that exp(β01+β11X1t)=exp(β01)⋅exp(β11X1t), and so in this formulation you get your proportional interpretation of the effects for β11, β12. (But I would still test for equality on the linear scale.)
For this simple of a model with all binary inputs on the right hand side, each of them will estimate the same expected value on the left hand side, it just changes the parameters and the nature of how you generalize across equations.
Stata has a program to do seemingly unrelated regression for glm's, see the suest command. Another way to do it though is to stack the equations, fit a single model, and do a likelihood ratio test for the restricted vs the model allowing the treatment effects to vary across the equations. Example below in R:
# Simulating two crime types
set.seed(10)
n <- 1000
t <- rep(0:1,n/2)
# crime type 1, ~10
l1 <- exp(2 + 0.5*t)
c1 <- rpois(n,l1)
# crime type 2, ~130
l2 <- exp(4.6 + 0.5*t)
c2 <- rpois(n,l2)
# Prepping data
ctype <- rep(c("c1","c2"),each=n)
data <- data.frame(crime=c(c1,c2),treat=c(t,t),c=ctype)
# Linear regression
linmod <- lm(crime ~ c + treat:c - 1, data)
summary(linmod)
# Poisson regression
pmod <- glm(crime ~ c + treat:c - 1, family='poisson', data)
summary(pmod)
# Showing linmod and pmod give the same marginal predictions!
all.equal(predict(linmod),predict(pmod,type='response'))
# Restricted model with no treatment het
pmod_restrict <- glm(crime ~ c + treat - 1, family='poisson', data)
anova(pmod_restrict,pmod,test='LRT') #Fail to reject null
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