Hello everybody,

I have a fairly easy model, but do not seem to recover all parameters correctly. The setting is a two-period diff-in-diff experiment, where we have a group of treated people that self-select into the treatment (e.g., adoption of an app) and a control group. I model the selection endogeneity using a Tobit-5 Heckman-type approach and I am trying to fit the Full Information Maximum Likelihood directly using the likelihood derived in Maddala (1983) and implemented in the etregress function of STATA: etregress.

All parameters except the treatment variable itself are recovered nicely. The treatment (delta1) is over-estimated by quite a bit and the 95% HDI does not contain the true value. I do not know which part is erroneous.

THE DATA (SIMULATED IN R)

```
N <- 1500 #cross-sections
T <- 2 #time points
NT <- N*T #observations
#correlated error terms
rho <- 0.8
sd1 <- 0.8
Sigma <- matrix(c(sd1, rho, rho, 1), nrow=2, ncol=2) #variance-covariance matrix
errors <- mvrnorm(n=N, mu=c(0, 0), Sigma=Sigma)
e1 <- rep(errors[,1], T)
e2 <- errors[,2]
#variables for outcome and selection equations
ys1 <- rnorm(N,0,1)
ys2 <- rnorm(N,0,1)
x1 <- rnorm(N*T,0,1)
x2 <- rnorm(N*T,0,1)
id <- data.frame(id=1:N)
time <- data.frame(t=0:1)
id <- rep(id$id, T)
t_fix <- rep(time$t, each=N)
#parameters
b0 <- 2.5
b1 <- 0
b2 <- 1
delta <- 1.3 #this is the treatment effect
gamma <- 0.4 #this is the post effect
gammadelta <- 0.8 #this is the diff-in-diff effect
w0 <- 1.2
w1 <- -0.8
w2 <- 1.6
#selection eqn
w_star <- c()
w_star <- w0+w1*ys1+w2*ys2+e2
#selection variable
w_sel = as.numeric(w_star>0)
w_out <- rep(w_sel, T)
#outcome variable and eqn
tw_fix <- t_fix*w_out
y <- b0+b1*x1+b2*x2+ gamma*t_fix+ delta*w_out + gammadelta* tw_fix + e1
intercept_sel <- rep(1, N)
intercept_obs <- rep(1, N*T)
#### independent variables that go into selection eqn
zy <- matrix(c(intercept_sel, ys1, ys2), nrow=N, ncol=3) #matrix for the selection equation
zy_columns = ncol(zy) #number of predictors for the treatment including the intercept
#### independent variables that go into outcome eqn
xb <- matrix(c(intercept_obs, x1, x2), nrow=N*T, ncol=3) #all covariates in the outcome equation without the three Diff-in-Diff variables treat, post, and treat x post
xb_columns = ncol(xb)
standata_heckman_diff <- list(
y = y, #outcome vector of full sample (not observed outcomes are denoted 0)
zy = zy, #predictors of selection equation (time-invariant)
xb = xb, #covariates of outcome equation
zy_columns = zy_columns, # number of predictors in selection
xb_columns = xb_columns, # number of covariates in outcome equation
N = N, #number of cross sections
T=T, #number of time points
NT = N*T, #number of observations overall
id=id, #IDs of the participants N
treat = w_out, #vector of outcomes of the selection equation (1/0)
post = t_fix, #indicator for the time point (0 or 1)
treatpost = tw_fix #interaction between time point and treatment indicators
```

AND HERE THE STAN CODE:

```
heckman_diff = "data{
int<lower=1> N; //cross-sections
int<lower=1> T; //time points
int<lower=1> NT; //all observations
int<lower=1> xb_columns; //number of covariates in outome equation
int<lower=1> zy_columns; //number of predictors in selection equation
matrix[NT,xb_columns] xb; //covariate regressor matrix of outcome equation
matrix[N,zy_columns] zy; //regressor matrix of selection equation
vector[NT] y; //vector of outcome variable
int<lower=1> id[NT]; //vector of proband ids
vector[NT] post; //vector of time variable
vector<lower=0, upper=1> [NT] treat; //self-selection vector of outcome equation (observed 1 and 0)
vector<lower=0, upper=1> [NT] treatpost; //interaction effect
}
parameters {
real<lower=-1,upper=1> rho; //covariance between errors
vector[xb_columns] beta_outcome;
vector[zy_columns] beta_selection;
real delta1; //coefficient of selection dummy variable
real gamma1; //time effect
real gammadelta1; //interaction effect
real<lower=0> sd1; //standard deviation of the outcome model
}
transformed parameters{
real<lower=0> lambda;
lambda = sd1 * rho;
}
model{
vector[NT] mu_y;
vector[N] mu_z;
//Priors
beta_outcome ~ normal(0, 10); //substantive betas
beta_selection ~ normal(0, 10); //selection equation betas
delta1 ~ normal(0,1); //coefficient on the selection dummy w
gamma1 ~ normal(0,1); //coefficient on the selection dummy w
gammadelta1 ~ normal(0,1);
sd1 ~ normal(1,10);
for (n in 1:NT) {
mu_y[n] = xb[n,]*beta_outcome+post[n]*gamma1+treat[n]*delta1+treatpost[n]*gammadelta1;
}
for (j in 1:N) {
mu_z[j] = zy[j]*beta_selection; //selection equation
}
// log likelihood for the overall model
for (n in 1:NT) {
if(treat[id[n]]==1) {
target += log(Phi((mu_z[id[n]]+(y[n]-mu_y[n])*rho/sd1)/sqrt(1-rho^2)))
-0.5*((y[n]-mu_y[n])/sd1)^2
-log(sqrt(2*pi())*sd1); //Likelihood for positive w-values;
} else {
target += log(Phi((-mu_z[id[n]]-(y[n]-mu_y[n])*rho/sd1)/sqrt(1-rho^2)))
-0.5*((y[n]-mu_y[n])/sd1)^2
-log(sqrt(2*pi())*sd1); //Likelihood for negative w-values
}
}
//Likelihood for outcome model
y ~ student_t(4,mu_y, sd1);
}"
fit_heckman_diff = stan(model_code = heckman_diff,
data = standata_heckman_diff,
chains = 6,
iter = 1000,
warmup =200,
control = list(max_treedepth = 15))
}
```

The two-step approach using inverse Millâ€™s Ratios works well, but the FIML is off in the treatment parameter *delta1*. Does anybody have a clue what I am doing wrong?

Thank you!