A Comprehensive Guide to Fractional Response Regressions with fracreg
Sulman Olieko Owili
2026-07-06
fracreg_tutorial.RmdIntroduction
The fracreg package is the most
comprehensive tool available in R for estimating, diagnosing, and
interpreting fractional response models. It supports:
- One-part models (1P): A single process governs the entire distribution of .
- Two-part hurdle models (2P): Separately models the binary hurdle () and the continuous fractional part ().
- Three-part double-inflated models (3P): Handles inflation at both 0 and 1.
-
Panel data models (
fracregpd): CRE, QML, and GMM approaches for longitudinal data. -
Endogeneity correction (
fracreghet): Control function and IV-GMM approaches.
This vignette walks you through comprehensive empirical examples (using the 401(k) dataset) and simulated examples for each estimator.
1. Data Description and Preparation
We use the built-in fracreg_k401k dataset, which is the
canonical firm-level 401(k) plan participation data used in
Papke and Wooldridge (1996) (“Econometric methods
for fractional response variables with an application to 401(k) plan
participation rates”, Journal of Applied Econometrics). The dataset
contains 1,534 observations of 401(k) plans.
The primary variables we use are: - prate: The plan
participation rate (the fraction of eligible employees who are active
participants). It strictly falls in the
interval and is our dependent variable
().
- mrate: The firm’s matching rate (the firm’s contribution
per $1 of employee contribution). - age: The age of the
401(k) plan. - totemp: Total number of employees at the
firm. - sole: A binary indicator equal to 1 if the 401(k)
plan is the sole retirement plan offered by the firm.
2. Cross-Sectional Fractional Models (fracreg)
The core fracreg() function is designed for univariate
models where the dependent variable is bounded between 0 and 1 inclusive
().
The examples below demonstrate both empirical 401(k) plan participation
data and general simulated boundaries.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
# 1P Model
fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.316e-01 8.408e-02 1.108e+01 7.668e-01 1.096 < 2e-16
#> mrate 9.531e-01 1.371e-01 6.951e+00 6.843e-01 1.222 3.62e-12
#> age 2.791e-02 4.877e-03 5.723e+00 1.835e-02 0.037 1.05e-08
#> totemp -8.182e-06 3.061e-06 -2.673e+00 -1.418e-05 0.000 0.00751
#> sole 3.405e-01 8.066e-02 4.222e+00 1.824e-01 0.499 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
# 1P Model reporting odds ratios and 99% confidence intervals
fracreg(y, X, type="1P", linkfrac="logit", or=TRUE, level=0.99)Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Odds Ratio Robust Std.Err. z value [99% Conf. Interval] Pr(>|z|)
#> Constant 2.539e+00 2.134e-01 1.108e+01 2.044e+00 3.153 < 2e-16
#> mrate 2.594e+00 3.556e-01 6.951e+00 1.822e+00 3.692 3.62e-12
#> age 1.028e+00 5.015e-03 5.723e+00 1.015e+00 1.041 1.05e-08
#> totemp 1.000e+00 3.061e-06 -2.673e+00 1.000e+00 1.000 0.00751
#> sole 1.406e+00 1.134e-01 4.222e+00 1.142e+00 1.730 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
# 2P Model (modelling mass at 1)
fracreg(y, X, type="2P", inflation=1, linkbin="logit", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 1: Binary logit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.1485
#> Log-likelihood: -938.1759
#> Wald chi2(4): 173.5169
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -1.396e+00 1.270e-01 -1.099e+01 -1.645e+00 -1.147 <2e-16 ***
#> mrate 9.053e-01 9.699e-02 9.334e+00 7.152e-01 1.095 <2e-16 ***
#> age 1.156e-02 6.218e-03 1.858e+00 -6.312e-04 0.024 0.0631 .
#> totemp -1.418e-05 6.324e-06 -2.242e+00 -2.657e-05 0.000 0.0249 *
#> sole 8.651e-01 1.131e-01 7.651e+00 6.435e-01 1.087 2e-14 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 2: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 852
#> Pseudo R-squared: 0.10004
#> Log pseudolikelihood: -450.8391
#> Wald chi2(4): 65.4063
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 7.460e-01 6.850e-02 1.089e+01 6.118e-01 0.880 < 2e-16
#> mrate 3.877e-01 9.725e-02 3.987e+00 1.971e-01 0.578 6.69e-05
#> age 2.562e-02 4.010e-03 6.390e+00 1.777e-02 0.033 1.66e-10
#> totemp -4.061e-06 3.073e-06 -1.322e+00 -1.008e-05 0.000 0.186
#> sole -1.510e-02 6.556e-02 -2.303e-01 -1.436e-01 0.113 0.818
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp
#> sole
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Two-part regression - binary logit + fractional logit
#> --------------------------------------------------------------------------------
#> Data type: Cross-sectional
#> Pseudo R-squared: 0.11243
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
# 3P Model (inject artificial 0s for demonstration)
y_3p <- y; y_3p[1:50] <- 0
fracreg(y_3p, X, type="3P", linkbin=c("logit","logit"), linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 1: Binary logit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.00324
#> Log-likelihood: -216.6222
#> Wald chi2(4): 3.7679
#> Prob > chi2: 0.4383
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 2.867e+00 3.157e-01 9.080e+00 2.248e+00 3.486 <2e-16 ***
#> mrate 1.147e-01 2.131e-01 5.381e-01 -3.030e-01 0.532 0.590
#> age 8.375e-03 1.765e-02 4.745e-01 -2.622e-02 0.043 0.635
#> totemp 1.036e-04 6.959e-05 1.489e+00 -3.275e-05 0.000 0.136
#> sole 3.371e-01 2.983e-01 1.130e+00 -2.476e-01 0.922 0.259
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 2: Binary logit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1484
#> Pseudo R-squared: 0.15199
#> Log-likelihood: -903.5457
#> Wald chi2(4): 172.201
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -1.435e+00 1.306e-01 -1.099e+01 -1.691e+00 -1.179 < 2e-16 ***
#> mrate 9.244e-01 9.877e-02 9.360e+00 7.309e-01 1.118 < 2e-16 ***
#> age 1.198e-02 6.339e-03 1.890e+00 -4.453e-04 0.024 0.0588 .
#> totemp -1.371e-05 6.317e-06 -2.170e+00 -2.609e-05 0.000 0.0300 *
#> sole 8.852e-01 1.154e-01 7.674e+00 6.591e-01 1.111 1.67e-14 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 3: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 826
#> Pseudo R-squared: 0.09937
#> Log pseudolikelihood: -437.3715
#> Wald chi2(4): 62.3685
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 7.388e-01 7.039e-02 1.050e+01 6.008e-01 0.877 < 2e-16
#> mrate 3.960e-01 1.019e-01 3.885e+00 1.962e-01 0.596 0.000102
#> age 2.531e-02 4.068e-03 6.223e+00 1.734e-02 0.033 4.88e-10
#> totemp -3.817e-06 3.088e-06 -1.236e+00 -9.869e-06 0.000 0.216472
#> sole -4.483e-03 6.672e-02 -6.719e-02 -1.353e-01 0.126 0.946434
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp
#> sole
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Three-part regression - binary logit , binary logit + fractional logit
#> --------------------------------------------------------------------------------
#> Data type: Cross-sectional
#> Pseudo R-squared: 0.07934
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
### Simulated Examples
set.seed(123)
N <- 1000
x1 <- rnorm(N)
x2 <- runif(N)
# Generating a fractional dependent variable with inflation at 0 and 1
XB <- -0.5 + 0.8 * x1 + 1.2 * x2 + rnorm(N)
y_latent <- exp(XB) / (1 + exp(XB))
y <- y_latent
# Inflate at boundaries
y[y_latent < 0.2] <- 0
y[y_latent > 0.8] <- 1
X <- cbind(x1 = x1, x2 = x2)
# fracreg estimation of a logit fractional response model
fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Pseudo R-squared: 0.3903
#> Log pseudolikelihood: -614.973
#> Wald chi2(2): 472.2453
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.56969 0.06750 -8.43960 -0.70199 -0.437 <2e-16 ***
#> x1 0.78822 0.04015 19.63424 0.70954 0.867 <2e-16 ***
#> x2 1.35611 0.11899 11.39668 1.12289 1.589 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
# fracreg estimation of the binary logit component of the two-part fractional
# regression model with y=0 as the relevant boundary value
fracreg(y, X, type="2Pbin", inflation=0, linkbin="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 1: Binary logit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Pseudo R-squared: 0.14527
#> Log-likelihood: -295.2068
#> Wald chi2(2): 102.5171
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 1.4638 0.1933 7.5732 1.0849 1.843 3.64e-14 ***
#> x1 1.1857 0.1287 9.2126 0.9335 1.438 < 2e-16 ***
#> x2 2.2279 0.3932 5.6662 1.4572 2.999 1.46e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
# fracreg estimation of the fractional component of the two-part fractional
# regression model with y=0 as the relevant boundary value and using a
# probit link function
fracreg(y, X, type="2Pfrac", inflation=0, linkfrac="probit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 2: Fractional probit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 881
#> Pseudo R-squared: 0.32474
#> Log pseudolikelihood: -555.3304
#> Wald chi2(2): 391.6944
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.10334 0.03566 -2.89793 -0.17323 -0.033 0.00376 **
#> x1 0.37512 0.02129 17.62293 0.33340 0.417 < 2e-16 ***
#> x2 0.61326 0.06529 9.39237 0.48529 0.741 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
# fracreg estimation of both components of a two-part fractional response model
# with y=0 as the relevant boundary value and using a cloglog binary link
# function and a logit fractional link function
fracreg(y, X, type="2P", inflation=0, linkbin="cloglog", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 1: Binary cloglog regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Pseudo R-squared: 0.14837
#> Log-likelihood: -291.7986
#> Wald chi2(2): 100.5683
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 0.43428 0.08825 4.92113 0.26132 0.607 8.60e-07 ***
#> x1 0.55192 0.06009 9.18429 0.43414 0.670 < 2e-16 ***
#> x2 1.05359 0.17403 6.05409 0.71250 1.395 1.41e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 2: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 881
#> Pseudo R-squared: 0.32412
#> Log pseudolikelihood: -555.4465
#> Wald chi2(2): 368.0384
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.17323 0.05786 -2.99384 -0.28664 -0.060 0.00275 **
#> x1 0.61205 0.03554 17.22078 0.54239 0.682 < 2e-16 ***
#> x2 1.00509 0.10703 9.39074 0.79531 1.215 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Two-part regression - binary cloglog + fractional logit
#> --------------------------------------------------------------------------------
#> Data type: Cross-sectional
#> Pseudo R-squared: 0.38829
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
# Three-part double-inflated model (y has both 0s and 1s)
fracreg(y, X, type="3P", linkbin=c("logit","probit"), linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Part 1: Binary logit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Pseudo R-squared: 0.14527
#> Log-likelihood: -295.2068
#> Wald chi2(2): 102.5171
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 1.4638 0.1933 7.5732 1.0849 1.843 3.64e-14 ***
#> x1 1.1857 0.1287 9.2126 0.9335 1.438 < 2e-16 ***
#> x2 2.2279 0.3932 5.6662 1.4572 2.999 1.46e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 2: Binary probit regression
#> --------------------------------------------------------------------------------
#> Estimator: ML
#> Data type: Cross-sectional
#> Number of observations: 881
#> Pseudo R-squared: 0.18634
#> Log-likelihood: -348.8644
#> Wald chi2(2): 123.5731
#> Prob > chi2: 0.0000
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient EIM Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -1.71537 0.12933 -13.26354 -1.96885 -1.462 < 2e-16 ***
#> x1 0.64810 0.06323 10.25024 0.52418 0.772 < 2e-16 ***
#> x2 1.07752 0.19174 5.61978 0.70172 1.453 1.91e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Part 3: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 715
#> Pseudo R-squared: 0.24292
#> Log pseudolikelihood: -484.9654
#> Wald chi2(2): 261.436
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.26763 0.04539 -5.89578 -0.35660 -0.179 3.73e-09 ***
#> x1 0.36198 0.02454 14.75144 0.31389 0.410 < 2e-16 ***
#> x2 0.58505 0.07969 7.34179 0.42886 0.741 2.11e-13 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#>
#>
#> --------------------------------------------------------------------------------
#> Three-part regression - binary logit , binary probit + fractional logit
#> --------------------------------------------------------------------------------
#> Data type: Cross-sectional
#> Pseudo R-squared: 0.38917
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------1.1 Partial Effects (fracreg.pe)
Raw coefficients from fractional models are not directly
interpretable as marginal effects. Use fracreg.pe() to
compute the Average Partial Effects (APE) and Conditional Partial
Effects (CPE) using the analytical delta method.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
m <- fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.316e-01 8.408e-02 1.108e+01 7.668e-01 1.096 < 2e-16
#> mrate 9.531e-01 1.371e-01 6.951e+00 6.843e-01 1.222 3.62e-12
#> age 2.791e-02 4.877e-03 5.723e+00 1.835e-02 0.037 1.05e-08
#> totemp -8.182e-06 3.061e-06 -2.673e+00 -1.418e-05 0.000 0.00751
#> sole 3.405e-01 8.066e-02 4.222e+00 1.824e-01 0.499 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
fracreg.pe(m) Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> mrate 1.018e-01 1.456e-02 6.989 2.77e-12 ***
#> age 2.980e-03 5.293e-04 5.630 1.80e-08 ***
#> totemp -8.736e-07 3.279e-07 -2.665 0.00771 **
#> sole 3.635e-02 8.515e-03 4.270 1.96e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
ym <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
y <- rbeta(N,ym*20,20*(1-ym))
y[y > 0.9] <- 1
#Computing average partial effects for a logit fractional response model
res <- fracreg(y,X,linkfrac="logit",table=FALSE)
fracreg.pe(res) Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X1 0.161653 0.009584 16.87 <2e-16 ***
#> X2 0.165141 0.012853 12.85 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
#Computing average partial effects for a binary logit + fractional probit
#two-part model
res <- fracreg(y,X,linkbin="logit",linkfrac="probit",type="2P",inf=1,table=FALSE)
fracreg.pe(res) Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects
#> --------------------------------------------------------------------------------
#> Binary logit + Fractional probit two-part regression
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X1 0.08479 0.01192 7.115 1.12e-12 ***
#> X2 0.09068 0.01249 7.259 3.89e-13 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
#Computing conditional partial effects for X2 in the logit component
#of a two-part fractional response model, with the covariates evaluated
#at their median values
res <- fracreg(y,X,linkfrac="logit",type="2Pfrac",inf=1,table=FALSE)
fracreg.pe(res,APE=FALSE,CPE=TRUE,at="median",which.x="X2") Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Conditional partial effects
#> --------------------------------------------------------------------------------
#> Fractional logit component of a two-part regression
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X2 0.17106 0.01895 9.026 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:41
#> --------------------------------------------------------------------------------
#>
#> Note: covariates evaluated at median (or mode, for dummies) values
#Computing average partial effects for a three-part double-inflated model
y3p <- y
y3p[1:20] <- 0
y3p[21:40] <- 1
res3p <- fracreg(y3p,X,linkbin=c("logit","probit"),linkfrac="logit",type="3P",table=FALSE)
fracreg.pe(res3p) Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects
#> --------------------------------------------------------------------------------
#> Three-part regression - binary logit , binary probit + fractional logit
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X1 0.15535 0.01440 10.790 < 2e-16 ***
#> X2 0.11405 0.01854 6.153 7.62e-10 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------3. Hypothesis Testing and Specification Diagnostics
fracreg includes state-of-the-art specification tests to
validate your model’s functional form and link function assumptions.
3.1 Generalised Goodness-Of-Functional-Form
(fracreg.ggoff)
The GGOFF test tests whether the chosen link function is adequate for the data. A significant result suggests the link may be misspecified.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
m <- fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.316e-01 8.408e-02 1.108e+01 7.668e-01 1.096 < 2e-16
#> mrate 9.531e-01 1.371e-01 6.951e+00 6.843e-01 1.222 3.62e-12
#> age 2.791e-02 4.877e-03 5.723e+00 1.835e-02 0.037 1.05e-08
#> totemp -8.182e-06 3.061e-06 -2.673e+00 -1.418e-05 0.000 0.00751
#> sole 3.405e-01 8.066e-02 4.222e+00 1.824e-01 0.499 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> GGOFF test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> GOFF1 - LM 8.838 0.00295 **
#> GOFF2 - LM 9.828 0.00172 **
#> GGOFF - LM 10.351 0.00565 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
ym <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
y <- rbeta(N,ym*20,20*(1-ym))
y[y > 0.9] <- 1
#Testing the logit specification of a standard fractional response model
#using LM and Wald versions of the GGOFF test, based on 1 or 2 fitted powers of
#the linear predictor
res <- fracreg(y,X,linkfrac="logit",table=FALSE)
fracreg.ggoff(res,c("Wald","LM")) Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> GGOFF test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> GOFF1 - LM 1.256 0.262
#> GOFF1 - Wald 1.274 0.259
#> GOFF2 - LM 1.513 0.219
#> GOFF2 - Wald 1.401 0.237
#> GGOFF - LM 1.612 0.447
#> GGOFF - Wald 1.336 0.513
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
#Testing the probit specification of the binary component of a two-part fractional
#regression model using a LR-based GGOFF test
res <- fracreg(y,X,linkbin="probit",type="2Pbin",inf=1,table=FALSE)
fracreg.ggoff(res,"LR") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> GGOFF test
#> --------------------------------------------------------------------------------
#> H0: Binary probit component of a two-part regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> GOFF1 - LR 0.012 0.914
#> GOFF2 - LR 0.017 0.895
#> GGOFF - LR 0.024 0.988
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------3.2 RESET Test (fracreg.reset)
The RESET test detects general functional form misspecification by testing whether powers of the fitted values have explanatory power. Testing provides a robust diagnostic check.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
m <- fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.316e-01 8.408e-02 1.108e+01 7.668e-01 1.096 < 2e-16
#> mrate 9.531e-01 1.371e-01 6.951e+00 6.843e-01 1.222 3.62e-12
#> age 2.791e-02 4.877e-03 5.723e+00 1.835e-02 0.037 1.05e-08
#> totemp -8.182e-06 3.061e-06 -2.673e+00 -1.418e-05 0.000 0.00751
#> sole 3.405e-01 8.066e-02 4.222e+00 1.824e-01 0.499 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> RESET test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LM(3) 10.29 0.00583 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
ym <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
y <- rbeta(N,ym*20,20*(1-ym))
y[y > 0.9] <- 1
#Testing the logit specification of a standard fractional response model
#using LM and Wald versions of the RESET test, based on 1 or 2 fitted powers of
#the linear predictor
res <- fracreg(y,X,linkfrac="logit",table=FALSE)
fracreg.reset(res,2:3,c("Wald","LM")) Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> RESET test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LM(2) 4.832 0.0279 *
#> Wald(2) 5.816 0.0159 *
#> LM(3) 4.866 0.0878 .
#> Wald(3) 6.870 0.0322 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
#Testing the probit specification of the binary component of a two-part fractional
#regression model using LR-based RESET tests with quadratic and cubic fitted
#powers of the linear predictor
res <- fracreg(y,X,linkbin="probit",type="2Pbin",inf=1,table=FALSE)
fracreg.reset(res,3,"LR")
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurredToggle to see the output
#>
#> --------------------------------------------------------------------------------
#> RESET test
#> --------------------------------------------------------------------------------
#> H0: Binary probit component of a two-part regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LR(3) 0.976 0.614
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------3.3 P-Test for Non-Nested Models (fracreg.ptest)
You can compare non-nested models (e.g., logit
vs. cloglog link) using the Davidson-MacKinnon (1981)
P-test.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
m1 <- fracreg(y, X, type="1P", linkfrac="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14667
#> Log pseudolikelihood: -553.1626
#> Wald chi2(4): 147.3049
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.316e-01 8.408e-02 1.108e+01 7.668e-01 1.096 < 2e-16
#> mrate 9.531e-01 1.371e-01 6.951e+00 6.843e-01 1.222 3.62e-12
#> age 2.791e-02 4.877e-03 5.723e+00 1.835e-02 0.037 1.05e-08
#> totemp -8.182e-06 3.061e-06 -2.673e+00 -1.418e-05 0.000 0.00751
#> sole 3.405e-01 8.066e-02 4.222e+00 1.824e-01 0.499 2.43e-05
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
m2 <- fracreg(y, X, type="1P", linkfrac="probit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional probit regression
#> --------------------------------------------------------------------------------
#> Estimator: QML
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Pseudo R-squared: 0.14069
#> Log pseudolikelihood: -554.2692
#> Wald chi2(4): 148.7649
#> Prob > chi2: 0.0000
#> Standard errors: robust
#> Small sample correction: FALSE
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 6.374e-01 4.517e-02 1.411e+01 5.489e-01 0.726 < 2e-16
#> mrate 4.190e-01 6.820e-02 6.143e+00 2.853e-01 0.553 8.07e-10
#> age 1.483e-02 2.536e-03 5.846e+00 9.855e-03 0.020 5.04e-09
#> totemp -4.546e-06 1.732e-06 -2.625e+00 -7.940e-06 0.000 0.00866
#> sole 2.000e-01 4.311e-02 4.639e+00 1.155e-01 0.284 3.50e-06
#>
#> Constant ***
#> mrate ***
#> age ***
#> totemp **
#> sole ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
fracreg.ptest(m1, m2) Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> P test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> H1: Fractional probit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> Wald -1.483 0.138
#> --------------------------------------------------------------------------------
#> H0: Fractional probit regression
#> H1: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> Wald 2.754 0.00595 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
ym <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
y <- rbeta(N,ym*20,20*(1-ym))
y[y > 0.9] <- 1
#Testing logit versus loglog specifications for standard fractional
#regression models using a LM version of the P test
res1 <- fracreg(y,X,linkfrac="logit",table=FALSE)
res2 <- fracreg(y,X,linkfrac="loglog",table=FALSE)
fracreg.ptest(res1,res2,"LM") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> P test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> H1: Fractional loglog regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LM 0.239 0.625
#> --------------------------------------------------------------------------------
#> H0: Fractional loglog regression
#> H1: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LM 8.502 0.00355 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
#Testing a logit one-part fractional response model versus a binary logit +
#fractional probit two-part model using a Wald version of the P test
res1 <- fracreg(y,X,linkfrac="logit",table=FALSE)
res2 <- fracreg(y,X,linkbin="logit",linkfrac="probit",type="2P",inf=1,table=FALSE)
fracreg.ptest(res1,res2,"Wald") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> P test
#> --------------------------------------------------------------------------------
#> H0: Fractional logit regression
#> H1: Binary logit + Fractional probit two-part regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> Wald 0.207 0.836
#> --------------------------------------------------------------------------------
#> H0: Binary logit + Fractional probit two-part regression
#> H1: Fractional logit regression
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> Wald 13.95 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------4. Endogeneity & Heteroscedasticity
(fracreghet)
When you suspect that one of your covariates is endogenous, or that
the variance is heteroscedastic, fracreghet() provides
instrumental variable correction. It natively supports IV through a
Control Function (CF) approach or GMM
estimation.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X_het <- cbind(mrate = fracreg_k401k$mrate, ltotemp = fracreg_k401k$ltotemp)
# fracreghet estimators do not allow exact 1s or 0s
y_adj <- y
y_adj[y_adj == 1] <- 0.999
# Instrument mrate using age
Z_emp <- cbind(age = fracreg_k401k$age, ltotemp = fracreg_k401k$ltotemp)
fracreghet(y_adj, X_het, Z_emp, var.endog = X_het[, "mrate"], type="QMLxv", link="logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: QMLxv
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Standard errors: robust
#> Wald chi2(6): 1991.8748
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood xv estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.10561 0.75224 -0.14040 -1.57997 1.369 0.888
#> mrate 3.72138 0.68952 5.39703 2.36994 5.073 6.78e-08 ***
#> ltotemp -0.07009 0.05348 -1.31060 -0.17491 0.035 0.190
#> vhat -2.79515 0.70026 -3.99157 -4.16764 -1.423 6.56e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Reduced form:
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Z_INTERCEPT 0.95046 0.09550 9.95211 0.76327 1.138 < 2e-16
#> Z_age 0.01146 0.00231 4.95997 0.00693 0.016 7.05e-07
#> Z_ltotemp -0.05534 0.01421 -3.89528 -0.08318 -0.027 9.81e-05
#>
#> Z_INTERCEPT ***
#> Z_age ***
#> Z_ltotemp ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
# Compute the same QMLxv estimator reporting Odds Ratios with 90% confidence intervals
fracreghet(y_adj, X_het, Z_emp, var.endog = X_het[, "mrate"], type="QMLxv", link="logit", or=TRUE, level=0.90)Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: QMLxv
#> Data type: Cross-sectional
#> Number of observations: 1534
#> Standard errors: robust
#> Wald chi2(6): 2243425.9766
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood xv estimates
#> --------------------------------------------------------------------------------
#> Odds Ratio Robust Std.Err. z value [90% Conf. Interval] Pr(>|z|)
#> Constant 0.89977 0.67684 -0.14040 0.26108 3.101 0.888
#> mrate 41.32157 28.49225 5.39703 13.29273 128.452 6.78e-08 ***
#> ltotemp 0.93231 0.04986 -1.31060 0.85380 1.018 0.190
#> vhat 0.06111 0.04279 -3.99157 0.01931 0.193 6.56e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Reduced form:
#> --------------------------------------------------------------------------------
#> Odds Ratio Robust Std.Err. z value [90% Conf. Interval] Pr(>|z|)
#> Z_INTERCEPT 2.586889 0.247056 9.952107 2.210829 3.027 < 2e-16
#> Z_age 1.011524 0.002337 4.959966 1.007688 1.015 7.05e-07
#> Z_ltotemp 0.946168 0.013441 -3.895285 0.924315 0.969 9.81e-05
#>
#> Z_INTERCEPT ***
#> Z_age ***
#> Z_ltotemp ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
### Simulated Examples
set.seed(123)
N <- 1000
x1 <- rnorm(N)
# Simulating an endogenous variable (var.endog) and an instrument (z1)
z1 <- rnorm(N)
u <- 0.5 * z1 + rnorm(N)
var.endog <- 0.8 * z1 + u
y_endog <- exp(0.5 * x1 + 1.2 * var.endog + u) / (1 + exp(0.5 * x1 + 1.2 * var.endog + u))
# Avoid exact 0 or 1 boundaries for some estimators
y_endog[y_endog <= 0] <- 0.01
y_endog[y_endog >= 1] <- 0.99
X <- cbind(x1 = x1, var.endog = var.endog)
Z <- cbind(x1 = x1, z1 = z1)
# Exogeneity (assuming var.endog is exogenous for comparison), GMMx estimator
fracreghet(y = y_endog, x = X, type = "GMMx", link = "logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: GMMx
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Standard errors: robust
#> Wald chi2(2): 42761.0276
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMMx estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 9.132e-02 1.536e-02 5.947e+00 6.123e-02 0.121 2.73e-09
#> x1 4.608e-01 1.538e-02 2.995e+01 4.306e-01 0.491 < 2e-16
#> var.endog 1.808e+00 9.021e-03 2.004e+02 1.790e+00 1.826 < 2e-16
#>
#> Constant ***
#> x1 ***
#> var.endog ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
# Endogeneity, GMMz estimator (does not require reduced form for endog)
fracreghet(y = y_endog, x = X, z = Z, type = "GMMz", link = "logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: GMMz
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Standard errors: robust
#> Wald chi2(2): 14903.8562
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMMz estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant 0.15277 0.02018 7.56931 0.11321 0.192 3.75e-14
#> x1 0.47947 0.02051 23.37680 0.43927 0.520 < 2e-16
#> var.endog 1.61252 0.01445 111.61802 1.58421 1.641 < 2e-16
#>
#> Constant ***
#> x1 ***
#> var.endog ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
# Endogeneity, GMMxv estimator (assumes linear reduced form for var.endog)
fracreghet(y = y_endog, x = X, z = Z, var.endog = var.endog, type = "GMMxv", link = "logit")
#> Warning in dgamma(y, 1/disp, scale = mu * disp, log = TRUE): NaNs producedToggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: GMMxv
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Standard errors: robust
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMMxv estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.01262 0.01859 -0.67857 -0.04905 0.024 0.497
#> x1 0.48705 0.01884 25.85287 0.45012 0.524 <2e-16
#> var.endog 1.59737 0.01339 119.33166 1.57113 1.624 <2e-16
#> vhat 0.60263 0.01339 45.01988 0.57640 0.629 <2e-16
#>
#> Constant
#> x1 ***
#> var.endog ***
#> vhat ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Reduced form:
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Z_INTERCEPT -0.02093 0.03082 -0.67933 -0.08133 0.039 0.497
#> Z_x1 -0.02149 0.03132 -0.68612 -0.08288 0.040 0.493
#> Z_z1 1.32751 0.02949 45.01988 1.26971 1.385 <2e-16
#>
#> Z_INTERCEPT
#> Z_x1
#> Z_z1 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------
# Endogeneity, QMLxv control function approach
fracreghet(y = y_endog, x = X, z = Z, var.endog = var.endog, type = "QMLxv", link = "logit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression with heteroscedasticity/endogeneity
#> --------------------------------------------------------------------------------
#> Estimator: QMLxv
#> Data type: Cross-sectional
#> Number of observations: 1000
#> Standard errors: robust
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final Quasi-Maximum Likelihood xv estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Constant -0.01262 0.01859 -0.67857 -0.04905 0.024 0.497
#> x1 0.48705 0.01884 25.85287 0.45012 0.524 <2e-16
#> var.endog 1.59737 0.01339 119.33166 1.57113 1.624 <2e-16
#> vhat 0.60263 0.01339 45.01988 0.57640 0.629 <2e-16
#>
#> Constant
#> x1 ***
#> var.endog ***
#> vhat ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Reduced form:
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> Z_INTERCEPT -0.02093 0.03082 -0.67933 -0.08133 0.039 0.497
#> Z_x1 -0.02149 0.03132 -0.68612 -0.08288 0.040 0.493
#> Z_z1 1.32751 0.02949 45.01988 1.26971 1.385 <2e-16
#>
#> Z_INTERCEPT
#> Z_x1
#> Z_z1 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:42
#> --------------------------------------------------------------------------------4.1 Partial Effects for Endogenous Models
(fracreghet.pe)
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X_het <- cbind(mrate = fracreg_k401k$mrate, ltotemp = fracreg_k401k$ltotemp)
# fracreghet estimators do not allow exact 1s or 0s
y_adj <- y
y_adj[y_adj == 1] <- 0.999
# Instrument mrate using age
Z_emp <- cbind(age = fracreg_k401k$age, ltotemp = fracreg_k401k$ltotemp)
res_emp <- fracreghet(y_adj, X_het, Z_emp, var.endog = X_het[, "mrate"], type="QMLxv", link="logit", table=FALSE)
fracreghet.pe(res_emp, which.x="mrate")Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects (conditional only on observables, based on the smearing estimator)
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> Estimator: QMLxv
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> mrate 0.39498 0.09827 4.019 5.84e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
Z <- cbind(rnorm(N),rnorm(N),rnorm(N))
dimnames(Z)[[2]] <- c("Z1","Z2","Z3")
y <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
res <- fracreghet(y,X,type="GMMx",table=FALSE)
#Smearing estimator of average partial effects for variable X1
fracreghet.pe(res,which.x="X1") Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Average partial effects (conditional only on observables, based on the smearing estimator)
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> Estimator: GMMx
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X1 0.166732 0.009667 17.25 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
#Naive estimator of conditional partial effects for all covariates,
#which are evaluated at X1=1 and X2=-1
fracreghet.pe(res,smearing=FALSE,APE=FALSE,CPE=TRUE,at=c(1,-1)) Toggle to see the output
#>
#>
#> --------------------------------------------------------------------------------
#> Conditional partial effects (conditional on both observables and unobservables, with error term = 0)
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> Estimator: GMMx
#> --------------------------------------------------------------------------------
#> Estimate Std. Error z value Pr(>|z|)
#> X1 0.22869 0.01337 17.11 <2e-16 ***
#> X2 0.20596 0.01454 14.17 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
#>
#> Note: covariates evaluated at the following values:
#>
#> X1 X2
#> 1 -14.2 RESET Test for Endogenous Models
(fracreghet.reset)
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X_het <- cbind(mrate = fracreg_k401k$mrate, ltotemp = fracreg_k401k$ltotemp)
# fracreghet estimators do not allow exact 1s or 0s
y_adj <- y
y_adj[y_adj == 1] <- 0.999
# Instrument mrate using age
Z_emp <- cbind(age = fracreg_k401k$age, ltotemp = fracreg_k401k$ltotemp)
res_emp <- fracreghet(y_adj, X_het, type="GMMx", link="logit", table=FALSE)
fracreghet.reset(res_emp)Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> RESET test
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> H0: Estimator: GMMx
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> Wald(3) 47.56 4.7e-11 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
### Simulated Examples
N <- 250
u <- rnorm(N)
X <- cbind(rnorm(N),rnorm(N))
dimnames(X)[[2]] <- c("X1","X2")
Z <- cbind(rnorm(N),rnorm(N),rnorm(N))
dimnames(Z)[[2]] <- c("Z1","Z2","Z3")
y <- exp(X[,1]+X[,2]+u)/(1+exp(X[,1]+X[,2]+u))
res <- fracreghet(y,X,type="GMMx",table=FALSE)
#LM and Wald versions of the RESET test, based on 1 or 2 fitted powers of xb
fracreghet.reset(res,2:3,c("Wald","LM")) Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> RESET test
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> H0: Estimator: GMMx
#> --------------------------------------------------------------------------------
#> Statistic p-value
#> LM(2) 0.216 0.6424
#> Wald(2) 0.180 0.6711
#> LM(3) 3.481 0.1754
#> Wald(3) 6.273 0.0434 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------5. Panel Data Fractional Models (fracregpd)
For longitudinal or panel data where unobserved heterogeneity is a
concern, fracregpd() provides fixed-T panel estimators.
### Empirical 401(k) Examples
data("fracreg_k401k")
y <- fracreg_k401k$prate
X <- cbind(mrate = fracreg_k401k$mrate, age = fracreg_k401k$age,
totemp = fracreg_k401k$totemp, sole = fracreg_k401k$sole)
# Artificial panel data structure for demonstration
N_emp <- nrow(X)
id_emp <- rep(1:(N_emp/2), each=2)
time_emp <- rep(1:2, times=N_emp/2)
fracregpd(id_emp, time_emp, y, X, type="QMLcre", link="probit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional probit (correlated random effects) regression
#> --------------------------------------------------------------------------------
#> Estimator: QMLcre
#> Data type: Panel
#> Exogeneity: TRUE
#> Use first lag of instruments: FALSE
#> Standard errors: cluster
#> --------------------------------------------------------------------------------
#> Number of observations: 1534
#> Number of groups: 767
#> Obs per group: 2
#> Log pseudolikelihood: -554.0205
#> Wald chi2(9): 3191.0501
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final (Correlated Random Effects) Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Cluster Std.Err. z value [95% Conf. Interval]
#> mrate 3.915e-01 7.007e-02 5.588e+00 2.542e-01 0.529
#> age 1.386e-02 3.543e-03 3.911e+00 6.912e-03 0.021
#> totemp -5.250e-06 2.649e-06 -1.982e+00 -1.044e-05 0.000
#> sole 2.378e-01 6.038e-02 3.938e+00 1.194e-01 0.356
#> INTERCEPT_mean 6.235e-01 5.885e-02 1.060e+01 5.082e-01 0.739
#> mrate_mean 5.726e-02 6.343e-02 9.027e-01 -6.706e-02 0.182
#> age_mean 1.544e-03 4.233e-03 3.649e-01 -6.751e-03 0.010
#> totemp_mean 1.223e-06 2.946e-06 4.152e-01 -4.551e-06 0.000
#> sole_mean -7.157e-02 8.424e-02 -8.495e-01 -2.367e-01 0.094
#> Pr(>|z|)
#> mrate 2.30e-08 ***
#> age 9.20e-05 ***
#> totemp 0.0475 *
#> sole 8.22e-05 ***
#> INTERCEPT_mean < 2e-16 ***
#> mrate_mean 0.3667
#> age_mean 0.7152
#> totemp_mean 0.6780
#> sole_mean 0.3956
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
### Simulated Examples
set.seed(123)
# Simulating Panel Data
N <- 100
T_periods <- 5
id <- rep(1:N, each = T_periods)
time <- rep(1:T_periods, times = N)
x_panel <- rnorm(N * T_periods)
# Unobserved individual effect (CRE)
c_i <- rep(rnorm(N), each = T_periods)
y_panel <- exp(x_panel + c_i) / (1 + exp(x_panel + c_i))
X <- cbind(x_panel = x_panel)
# Endogenous variable and instrument simulation
z_panel <- rnorm(N * T_periods)
u_panel <- 0.5 * z_panel + rnorm(N * T_periods)
var_endog <- 0.8 * z_panel + u_panel
y_endog <- exp(x_panel + 1.2 * var_endog + c_i + u_panel) /
(1 + exp(x_panel + 1.2 * var_endog + c_i + u_panel))
X_endog <- cbind(x_panel = x_panel, var_endog = var_endog)
Z_inst <- cbind(x_panel = x_panel, z_panel = z_panel)
# Estimate a Correlated Random Effects (CRE) Model
fracregpd(id=id, time=time, y=y_panel, x=X, type="QMLcre", link="probit") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional probit (correlated random effects) regression
#> --------------------------------------------------------------------------------
#> Estimator: QMLcre
#> Data type: Panel
#> Exogeneity: TRUE
#> Use first lag of instruments: FALSE
#> Standard errors: cluster
#> --------------------------------------------------------------------------------
#> Number of observations: 500
#> Number of groups: 100
#> Obs per group: 5
#> Log pseudolikelihood: -313.6436
#> Wald chi2(3): 2156.6004
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final (Correlated Random Effects) Quasi-Maximum Likelihood estimates
#> --------------------------------------------------------------------------------
#> Coefficient Cluster Std.Err. z value [95% Conf. Interval]
#> x_panel 0.52902 0.01143 46.29965 0.50662 0.551
#> INTERCEPT_mean -0.01246 0.04901 -0.25433 -0.10851 0.084
#> x_panel_mean -0.17409 0.13491 -1.29040 -0.43850 0.090
#> Pr(>|z|)
#> x_panel <2e-16 ***
#> INTERCEPT_mean 0.799
#> x_panel_mean 0.197
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
# Exogeneity, no lags, no time dummies, clustered standard errors, GMMbgw estimator
fracregpd(id=id, time=time, y=y_panel, x=X, type="GMMbgw") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: GMMbgw
#> Data type: Panel
#> Exogeneity: TRUE
#> Use first lag of instruments: FALSE
#> Standard errors: cluster
#> --------------------------------------------------------------------------------
#> Number of observations: 500
#> Number of groups: 100
#> Obs per group: 5
#> Wald chi2(1): 4.87462988128311e+31
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMM bgw estimates
#> --------------------------------------------------------------------------------
#> Coefficient Cluster Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> x_panel 1.000e+00 1.432e-16 6.982e+15 1.000e+00 1 <2e-16
#>
#> x_panel ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
# Estimate the GMMww estimator with odds ratios and 99% confidence intervals
fracregpd(id=id, time=time, y=y_panel, x=X, type="GMMww", or=TRUE, level=0.99)Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: GMMww
#> Data type: Panel
#> Exogeneity: TRUE
#> Use first lag of instruments: TRUE
#> Standard errors: cluster
#> --------------------------------------------------------------------------------
#> Number of obs (initial): 500
#> Number of observations: 400
#> Number of groups (initial): 100
#> Number of groups: 100
#> Obs per group: 4
#> Wald chi2(1): 8.09654990520078e+32
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMM ww estimates
#> --------------------------------------------------------------------------------
#> Odds Ratio Cluster Std.Err. z value [99% Conf. Interval] Pr(>|z|)
#> x_panel 2.718e+00 2.597e-16 1.047e+16 2.718e+00 2.718 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
# Lagged covariates and instruments, robust standard errors, GMMww estimator
fracregpd(id=id, time=time, y=y_panel, x=X, lags=TRUE, type="GMMww", var.type="robust") Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit regression
#> --------------------------------------------------------------------------------
#> Estimator: GMMww
#> Data type: Panel
#> Exogeneity: TRUE
#> Use first lag of instruments: TRUE
#> Standard errors: robust
#> --------------------------------------------------------------------------------
#> Number of obs (initial): 500
#> Number of observations: 400
#> Number of groups (initial): 100
#> Number of groups: 100
#> Obs per group: 4
#> Wald chi2(1): 1.27815198429371e+32
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final GMM ww estimates
#> --------------------------------------------------------------------------------
#> Coefficient Robust Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> x_panel 1.000e+00 8.845e-17 1.131e+16 1.000e+00 1 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:43
#> --------------------------------------------------------------------------------
# Endogeneity, time dummies, GMMpfe estimator
fracregpd(id=id, time=time, y=y_endog, x=X_endog, z=Z_inst,
x.exogenous=FALSE, type="GMMpfe", tdummies=TRUE) Toggle to see the output
#>
#> --------------------------------------------------------------------------------
#> Fractional logit (pooled fixed effects) regression
#> --------------------------------------------------------------------------------
#> Estimator: GMMpfe
#> Data type: Panel
#> Exogeneity: FALSE
#> Use first lag of instruments: FALSE
#> Standard errors: cluster
#> --------------------------------------------------------------------------------
#> Number of observations: 500
#> Number of groups: 100
#> Obs per group: 5
#> Wald chi2(6): 7156.2911
#> Prob > chi2: 0.0000
#> Convergence: Successful
#> --------------------------------------------------------------------------------
#> Final (Pooled Fixed Effects) Generalized Method of Moments estimates
#> --------------------------------------------------------------------------------
#> Coefficient Cluster Std.Err. z value [95% Conf. Interval] Pr(>|z|)
#> x_panel 0.99602 0.03117 31.95739 0.93493 1.057 <2e-16
#> var_endog 1.57689 0.02524 62.47420 1.52742 1.626 <2e-16
#> time.2 -0.15363 0.09752 -1.57541 -0.34476 0.038 0.1152
#> time.3 -0.04402 0.09147 -0.48128 -0.22331 0.135 0.6303
#> time.4 -0.10609 0.09379 -1.13110 -0.28992 0.078 0.2580
#> time.5 -0.19163 0.08917 -2.14916 -0.36640 -0.017 0.0316
#>
#> x_panel ***
#> var_endog ***
#> time.2
#> time.3
#> time.4
#> time.5 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Run Date: 2026-07-06 15:19:44
#> --------------------------------------------------------------------------------Conclusion
With fracreg, fracreghet, and
fracregpd, you have a complete, mathematically robust
toolkit for modelling fractional responses bounded between
,
regardless of inflation, endogeneity, or unobserved panel effects.
Acknowledgements
This package builds upon, consolidates, and modernises the fractional
regression frameworks originally implemented in the frm,
frmhet, and frmpd R packages developed by
Joaquim J.S. Ramalho. As those original packages have been deprecated
and removed from the active CRAN repository, fracreg serves
as an actively maintained successor, ensuring these econometric tools
remain available to the R community.
References
- Ramalho, J. J. S. (2022). frm: Fractional Regression Models. R package. Formerly available on CRAN, currently archived.
- Ramalho, J. J. S. (2023). frmhet: Fractional Regression Models under Heterogeneity. R package. Formerly available on CRAN, currently archived.
- Ramalho, J. J. S. (2023). frmpd: Fractional Regression Models for Panel Data. R package. Formerly available on CRAN, currently archived.
- Davidson, R. and MacKinnon, J.G. (1981). “Several tests for model specification in the presence of alternative hypotheses”, Econometrica, 49(3), 781-793.
- Fang, K., & Ma, S. (2013). “Three-part model for fractional response variables with application to Chinese household health insurance coverage”, Journal of Applied Statistics, 40(5), 925-940.
- Papke, L. E. and Wooldridge, J. M. (1996). “Econometric methods for fractional response variables with an application to 401(k) plan participation rates”, Journal of Applied Econometrics, 11(6), 619-632.
- Papke, L. and Wooldridge, J.M. (2008). “Panel data methods for fractional response variables with an application to test pass rates”, Journal of Econometrics, 145(1-2), 121-233.
- Pregibon, D. (1980). “Goodness of Link Tests for Generalized Linear Models”, Journal of the Royal Statistical Society: Series C (Applied Statistics), 29(1), 15-24.
- Ramalho, E. A., & Ramalho, J. J. S. (2017). “Moment-based estimation of nonlinear regression models with boundary outcomes and endogeneity, with applications to nonnegative and fractional responses”, Econometric Reviews, 36(4), 397-420.
- Ramalho, E.A., Ramalho, J.J.S. and Murteira, J.M.R. (2011). “Alternative estimating and testing empirical strategies for fractional response models”, Journal of Economic Surveys, 25(1), 19-68.
- Ramalho, E.A., Ramalho, J.J.S. and Murteira, J.M.R. (2014). “A generalized goodness-of-functional form test for binary and fractional response models”, Manchester School, 82(4), 488-507.
- Ramsey, J.B. (1969). “Tests for Specification Errors in Classical Linear Least-Squares Regression Analysis”, Journal of the Royal Statistical Society: Series B (Methodological), 31(2), 350-371.
For more information, please visit the package website or file an issue on GitHub.
To cite this package in your research:
citation("fracreg")