Two-step TV-HTE estimator (Botosaru and Liu 2025)

Fits the linear-Gaussian dynamic-panel model with correlated random coefficients $$Y_{it} = \rho_Y Y_{i,t-1} + \alpha_i + \sum_j D_{it}^j \delta_{ij} + U_{it}$$ $$\delta_{ij} = \rho_\delta \delta_{i,j-1} + \varepsilon_{ij},\; j \geq 1$$ under common treatment timing. The latent lambda_i = (alpha_i, delta_{i0}) is integrated out under a Gaussian working assumption; in step 1 we estimate the common parameters (and the lambda-prior parameters) by QMLE on the resulting marginal multivariate-normal log-likelihood; in step 2 we form the closed-form Gaussian posterior mean of (alpha_i, delta_{i0}) for each unit. The Botosaru-Liu (2025) paper shows that the QMLE remains consistent under misspecification of the Gaussian working assumption on lambda.

Usage

tvhte(
  Y,
  Y0,
  t0,
  J,
  X = NULL,
  init = NULL,
  control = list(maxit = 500),
  compute_se = TRUE
)

Arguments

Y

N x T matrix of post-baseline outcomes.

Y0

Length-N vector of baseline outcomes Y_{i,0}.

t0

Treatment period. Either a scalar (common timing) or a length-nrow(Y) vector with per-unit cohorts (staggered). Use Inf for never-treated units.

J

Maximum event time observed in-window (in 0:(T - t0)).

X

Optional N x T x K array of strictly exogenous covariates. If supplied, the model adds X_{it}'beta to the outcome equation and estimates beta jointly with the other parameters.

init

Optional named list of starting values for the optimizer (overrides any of the defaults computed from the data).

control

Passed to optim.

Value

A list of class "tvhte" with:

theta

Estimated common parameters c(rho_Y, rho_delta, sigma_U, sigma_eps).

prior

Estimated Gaussian prior on lambda_i.

loglik

Maximised marginal log-likelihood.

lambda_hat

N x 2 matrix of posterior means of (alpha_i, delta_{i0}).

delta_path

N x (J+1) matrix of posterior-mean event-time trajectories. Column j+1 is E[delta_{i,j} | data].

convergence

optim's return code; 0 means converged.

References

Botosaru, Irene and Laura Liu (2025). "Time-Varying Heterogeneous Treatment Effects in Event Studies." arXiv:2509.13698.

Examples

# Small panel, common adoption, no covariates
sim <- simulate_tvhte(N = 300, T = 6, t0 = 3, J = 3,
                      rho_Y = 0.5, rho_delta = 0.7, seed = 1)
fit <- tvhte(sim$Y, sim$Y0, t0 = sim$t0, J = sim$J,
             compute_se = FALSE)
print(fit)
#> Time-Varying Heterogeneous Treatment Effects (Botosaru-Liu 2025)
#>   N = 300 units; T = 6 periods; t0 = 3; J = 3
#>   log-likelihood = -2808.536   convergence = 0
#> 
#> Common parameters (theta):
#>   rho_Y      = 0.5217
#>   rho_delta  = 0.7198
#>   sigma_U    = 1.033
#>   sigma_eps  = 0.349
#> Prior on lambda_i = (alpha, delta_0):
#>   mu = (0.02566, 0.9811)
#>   sd = (0.4622, 0.157),  cor = 0.9997
#> 
#> Mean posterior event-time effects (across units):
#>   delta_0 = 0.9811   delta_1 = 0.7084   delta_2 = 0.5116   delta_3 = 0.3640

# With a strictly-exogenous covariate
sim2 <- simulate_tvhte(N = 300, T = 6, t0 = 3, J = 3,
                       beta = 0.5, seed = 2)
fit2 <- tvhte(sim2$Y, sim2$Y0, t0 = sim2$t0, J = sim2$J,
              X = sim2$X, compute_se = FALSE)
fit2$beta
#> [1] 0.4770541

# Staggered adoption with a never-treated comparison group
set.seed(3)
cohorts <- sample(c(3, 5, Inf), 400, replace = TRUE)
sim3 <- simulate_tvhte(N = 400, T = 7, t0 = cohorts, J = 2, seed = 3)
fit3 <- tvhte(sim3$Y, sim3$Y0, t0 = sim3$t0, J = sim3$J,
              compute_se = FALSE)
fit3$cohort_counts
#> t0
#>   3   5 Inf 
#> 140 138 122