qrme

qrme is the companion R package for

Callaway, B., Li, T., Murtazashvili, I., and Tsyawo, E. S. (2026). Distributional Effects with Two-Sided Measurement Error: An Application to Intergenerational Income Mobility. arXiv:2107.09235

It provides two main estimators:

• qrme() — Quantile regression with measurement error in the outcome variable. This is an EM-based implementation of the estimator developed by Hausman, Liu, Luo, and Palmer (2021, Econometrica), extended to support finite Gaussian mixtures for the measurement error distribution. The pseudo-EM algorithm alternates between a Metropolis–Hastings MCMC step (E-step) to integrate out the latent outcome and a standard quantile regression M-step, iterating until convergence. qrme() is a useful standalone tool for any setting where the dependent variable is measured with classical error.

• tsme() — Two-sided measurement error: joint quantile regression when both the outcome and a continuous regressor are measured with error. Combines two qrme() fits via a copula to recover the joint distribution and a range of distributional parameters such as quantile-on-quantile transition matrices and poverty measures.

Installation

# install.packages("remotes")
remotes::install_github("bcallaway11/qrme")

Usage

Quantile regression with measurement error in the outcome (qrme)

library(qrme)

# Simulate data: Y* = 1 + X + u, Y = Y* + e (measurement error)
set.seed(42)
n  <- 500
X  <- runif(n)
Ys <- 1 + X + rnorm(n)         # latent outcome
Y  <- Ys + rnorm(n, sd = 0.5)  # observed outcome with ME
dd <- data.frame(Y = Y, X = X)

fit <- qrme(
  formula = Y ~ X,
  data    = dd,
  tau     = c(0.25, 0.5, 0.75)
)

# Coefficient matrix: one row per quantile
fit$bet

Key arguments:

Argument	Default	Description
tau	0.5	Quantile(s) to estimate
n_mix	1	Number of Gaussian mixture components for ME distribution
conv_criterion	"params"	Convergence criterion: "params" or "loglik"
tol	1e-2	Convergence tolerance
max_em_iters	100	Maximum EM iterations
mcmc_draws	200	MCMC draws per EM E-step
mcmc_burn_in	100	MCMC burn-in draws
se	FALSE	Compute bootstrap standard errors
verbose	FALSE	Print progress (numeric levels 0–3)

Two-sided measurement error (tsme)

data(nlsy97)
tau      <- seq(0.1, 0.9, by = 0.1)
t_values <- quantile(nlsy97$lpi, probs = c(0.25, 0.5, 0.75))

res <- tsme(
  data      = nlsy97,
  y_formula = lci ~ ageC_97 + ageF,
  t_formula = lpi ~ ageC_97 + ageF,
  tau       = tau,
  t_values  = t_values
)

# Transition matrix (distributional mobility summary)
res$me_tmat

tsme() fits a qrme() model for each variable, then links them with a copula to estimate joint distributions and mobility parameters such as quantile-on-quantile transition matrices and poverty measures.

Tuning parameters

The EM algorithm involves several parameters with different roles. The table below groups them by the nature of their tradeoff.

Parameter	Default	Role
tol	1e-2	Pure speed–accuracy: loosen to converge faster, tighten for more precise estimates
max_em_iters	100	Hard iteration cap; does not affect per-iteration cost
conv_patience	1	Consecutive iterations below tol required; raise to 2 if using conv_criterion = "loglik"
mcmc_draws	200	Fundamental tradeoff: more draws = better E-step but slower per iteration
mcmc_burn_in	100	Burn-in fraction of mcmc_draws; 30–50% is typical
loglik_draws	100	MC draws for log-likelihood convergence check; ignored when conv_criterion = "params"
proposal_sd	adaptive	MH step size; auto-updated each iteration to track the current ME scale
n_mix	1	Number of mixture components — a model choice, not a tuning parameter
start_beta	naive QR	Defaults to QR ignoring ME, which is consistent under symmetric error
start_mu	data-informed	Evenly spaced over ±0.25 sd(Y) for n_mix > 1; 0 for n_mix = 1
start_sigma	data-informed	Backed out so the mixture SD equals 0.5 sd(Y)
start_pi	uniform	Equal weights across components

mcmc_draws and mcmc_burn_in are the most consequential parameters. Too few effective draws (= mcmc_draws − mcmc_burn_in) per observation produces a noisy E-step, which can cause the outer EM to wander or fail to converge — this is a correctness problem, not just a speed problem. The defaults of 200/100 work well in most settings; reduce only for exploratory runs.

proposal_sd controls the Metropolis–Hastings step size and determines the effective sample size of the MCMC chain. By default it is initialised from the starting ME parameters and updated each EM iteration to track the current ME distribution scale. A healthy acceptance rate is roughly 20–70%; the acceptance rate is printed at verbose = 2 and a warning is issued if the final-iteration rate falls outside 10–90%. If the warning fires, pass proposal_sd explicitly to override the adaptive default.

n_mix is selected by model fit rather than tuned. Use logLik(), AIC(), and BIC() on fitted objects to compare across values; n_mix = 1 is the right default for classical additive Gaussian measurement error.

Learn more

• Vignettes — Introduction to qrme() | NLSY97 application with tsme()
• Package website — https://bcallaway11.github.io/qrme

References

• Callaway, B., Li, T., Murtazashvili, I., and Tsyawo, E. S. (2026). Distributional Effects with Two-Sided Measurement Error: An Application to Intergenerational Income Mobility. arXiv:2107.09235.

• Hausman, J., Liu, H., Luo, Y. and Palmer, C. (2021). Errors in the dependent variable of quantile regression models. Econometrica, 89(2), pp. 849-873.