Truncated Regression

Truncated Regression is used when the sample only contains observations satisfying a threshold — units outside the threshold do not appear at all in the data (unlike Tobit, where threshold units are still observed with a piled value).

Truncation biases OLS

Under truncation, the in-sample error distribution no longer has zero mean ⇒ OLS is biased. A truncated (MLE) estimator is required to correct this.

---

Model specification

For lower truncation at $a$, the likelihood is based on the distribution conditional on $Y > a$:

$$f(Y_i \mid Y_i > a) = \frac{\phi\!\left(\frac{Y_i - X_i\beta}{\sigma}\right)}{\sigma \, \Phi\!\left(\frac{X_i\beta - a}{\sigma}\right)}$$

Estimated by MLE.

---

Running in EcoLab

1. Modeling module → Limited dependent variable family → Truncated.
2. Select $Y$, the $X$ variables; declare the truncation threshold and direction (upper/lower).
3. Run; read the corrected coefficients; export the replication code.

---

Replication code

Stata

* ===== Truncated Regression =====
* Lower-truncated at 0 (observations with y ≤ 0 are absent)
truncreg y x1 x2, ll(0)

* Upper-truncated at a threshold — if needed
* truncreg y x1 x2, ul(100)

* Marginal effects
margins, dydx(*)

R

# ===== Truncated Regression =====
library(truncreg)

# Lower-truncated at 0
model <- truncreg(y ~ x1 + x2,
                  data      = df,
                  point     = 0,
                  direction = "left")

summary(model)
coef(model)

Python

# ===== Truncated Regression (MLE) =====
# No built-in command in statsmodels;
# conceptual MLE approach with scipy.

import numpy as np
from scipy.optimize import minimize
from scipy.stats import norm

def trunc_loglik(params, X, y, lower=0):
    beta = params[:-1]
    sigma = np.exp(params[-1])  # ensure sigma > 0
    Xb = X @ beta
    # Log-likelihood for truncated normal (left-truncated at lower)
    ll = np.sum(
        norm.logpdf(y, Xb, sigma)
        - np.log(1 - norm.cdf((lower - Xb) / sigma))
    )
    return -ll

# Usage:
# import statsmodels.api as sm
# X = sm.add_constant(df[["x1", "x2"]]).values
# y = df["y"].values  (only observations where y > 0)
# k = X.shape[1]
# x0 = np.zeros(k + 1)  # beta + log(sigma)
# result = minimize(trunc_loglik, x0, args=(X, y, 0), method="BFGS")
# beta_hat = result.x[:-1]
# sigma_hat = np.exp(result.x[-1])

---

Limitations

• Requires correct knowledge of the threshold and truncation mechanism.
• Sensitive to the normality assumption of the error.

Video tutorial

Video Tutorial: Guide to running Truncated Regression in EcoLab

See also

• Tobit · Heckman · Catalog