Fitting Fractional Response Regressions
fracreg.Rdfracreg is used to fit fractional response models, which are appropriate for responses that are proportions, percentages, or fractions restricted to the [0, 1] interval. It supports standard one-part models, two-part hurdle models for modelling boundary values at 0 or 1, and three-part models for double inflation at both 0 and 1.
Usage
fracreg(y, x, x2 = x, linkbin, linkfrac, type = "1P", inflation = 0,
intercept = TRUE, table = TRUE, variance = TRUE, var.type = "default",
var.eim = TRUE, var.cluster, dfc = FALSE, offset = NULL,
or = FALSE, level = 0.95, ...)Arguments
- y
a numeric vector containing the values of the response variable.
- x
a numeric matrix, with column names, containing the values of the covariates.
- x2
a numeric matrix, with column names, containing the values of the covariates in the fractional component of two-part models if option
type = "2P"is defined. Defaults tox.- linkbin
a description of the link function to use in the binary component of a two-part fractional response model, or a vector of two link functions for the two binary components of a three-part model (e.g.
c("logit", "probit")). Available options:logit,probit,cauchit,loglog,cloglog.- linkfrac
a description of the link function to use in standard fractional response models or in the fractional component of a two-part fractional response model. Available options:
logit,probit,cauchit,loglog,cloglog.- type
a description of the model to estimate: a standard one-part model (
1P, the default), a two-part model (2P), the binary component of a two-part model (2Pbin), the fractional component of a two-part model (2Pfrac), or a three-part model (3P) for double boundary inflation.- inflation
a numeric value indicating which of the extreme values of
0(the default) or1is the relevant boundary value for defining two-part fractional response models.- intercept
a logical value indicating whether the model should include a constant term or not.
- table
a logical value indicating whether a summary table with the regression results should be printed.
- variance
a logical value indicating whether the variance of the estimated parameters should be calculated. Defaults to
TRUEwhenevertable = TRUE.- var.type
a description of the type of variance of the estimated parameters to be calculated. Options are
standard(recommended for models estimated by maximum likelihood, such as the binary component of two-part models),robust(recommended for models estimated by quasi-maximum likelihood, such as standard fractional response models or the fractional component of a two-part fractional response model),cluster(recommended in the case of panel data) anddefault(implements thestandardorrobustversions as appropriate).- var.eim
a logical value indicating whether the expected information matrix should be used in the calculation of the variance. When false, the observation information matrix will be used. Defaults to
TRUE.- var.cluster
a numeric vector containing the values of the variable that specifies to which cluster each observation belongs.
- dfc
a logical value indicating whether a degrees of freedom correction should be applied to the covariance matrix. Defaults to
FALSE.- offset
an optional numeric vector containing an offset. It must be of the same dimension as the response variable. It specifies that the variable should be included in the model with its coefficient constrained to 1.
- or
a logical value indicating whether to report odds ratios. Only valid when the link function is
"logit". Defaults toFALSE.- level
a numeric value between 0 and 1 indicating the confidence level for the confidence intervals. Defaults to
0.95.- ...
Arguments to pass to glm.
Details
fracreg estimates one-part, two-part hurdle, and three-part double-inflated fractional response models; see Ramalho, Ramalho and Murteira (2011) and Fang and Ma (2013) for details on those models.
One-Part Fractional Response Regressions (type = "1P"):
The standard one-part model assumes that the conditional expectation of the fractional response \(y_i \in [0,1]\) is given by:
$$E(y_i|x_i) = G(x_i \beta)$$
where \(G(\cdot)\) is a known non-linear link function mapping the linear predictor to the unit interval (e.g., logit, probit). The parameters \(\beta\) are estimated by maximising the Bernoulli-based quasi-log-likelihood function:
$$\ln L_i(\beta) = y_i \ln[G(x_i \beta)] + (1 - y_i) \ln[1 - G(x_i \beta)]$$
This estimator requires only the correct specification of the conditional mean to yield consistent parameter estimates (Papke and Wooldridge, 1996).
Two-Part Hurdle Models (type = "2P"):
When the data exhibits a boundary mass (e.g., at \(y_i = 0\)), the two-part hurdle model handles the boundary values separately from the interior fractional values. Let \(y_i^*\) be a binary indicator such that \(y_i^* = 1\) if \(y_i > 0\) and \(y_i^* = 0\) otherwise. The probability of observing a boundary value is modelled as:
$$P(y_i = 0 | x_{1i}) = 1 - F(x_{1i} \gamma_1)$$
$$P(y_i > 0 | x_{1i}) = F(x_{1i} \gamma_1)$$
where \(F(\cdot)\) is a binary link function. Conditional on observing an interior fractional value, the response is modelled as:
$$E(y_i | x_{2i}, y_i > 0) = G(x_{2i} \beta_2)$$
The unconditional mean of the response is therefore:
$$E(y_i|x_i) = F(x_{1i} \gamma_1) \times G(x_{2i} \beta_2)$$
Three-Part Double Inflated Models (type = "3P"):
For data containing boundary mass at both \(0\) and \(1\), the three-part model estimates two separate binary mechanisms for each boundary and a fractional component for the interior values \((0, 1)\), extending the two-part logic to double inflation (Fang and Ma, 2013).
fracreg uses the standard glm command to perform the estimations. Therefore, fracreg is essentially a convenience command, allowing estimation of several alternative fractional response models using the same command. In addition, fracreg provides an R-squared measure for all models (calculated as the square of the correlation coefficient between the actual and fitted values of the dependent variable), calculates the fitted values of the dependent variable in two-part models and stores the information needed to implement some very useful commands for fractional response models: fracreg.reset (RESET test), fracreg.ptest (P test), fracreg.ggoff (GGOFF tests) and fracreg.pe (partial effects).
Value
When type = "1P" or "2Pfrac", fracreg returns a list with the following elements:
- class
"fracreg".
- formula
the model formula.
- type
the name of the estimated model.
- link
the name of the specified link.
- method
estimation method. Currently, "QML" (quasi-maximum likelihood) for fractional components or models and"ML" (maximum likelihood) for the binary component of two-part models.
- p
a named vector of coefficients.
- yhat
the fitted mean values.
- xbhat
the fitted mean values of the linear predictor.
- converged
logical. Was the algorithm judged to have converged?
- x.names
a vector containing the names of the covariates.
If variance = TRUE or table = TRUE, the previous list also contains the following elements:
- p.var
a named covariance matrix.
- var.type
covariance matrix type.
- var.eim
logical. Was the expected information matrix used in the computation of the covariance matrix?
- dfc
logical. Was a degrees of freedom correction used for the computation of the covariance matrix?
If var.type = "cluster", the list also contains the following element:
- var.cluster
the variable that specifies to which cluster each observation belongs.
When type = "2Pbin", fracreg returns a similar list with the following additional element:
- LL
the value of the log-likelihood.
When type = "2P", fracreg returns the previous lists, indexed by the prefixes resBIN and resFRAC, and the following additional elements:
- class
"fracreg".
- type
"2P".
- ybase
a numeric vector containing the values of the response variable.
- x2base
a numeric matrix containing the values of the covariates.
- yhat2P
the overall fitted mean values.
- converged
logical. Were the algorithms judged to have converged in both parts of the model?
When type = "3P", fracreg returns the previous lists, indexed by the prefixes resBIN0, resBIN1, and resFRAC, and the following additional elements:
- class
"fracreg".
- type
"3P".
- ybase
a numeric vector containing the values of the response variable.
- x2base
a numeric matrix containing the values of the covariates.
- yhat3P
the overall fitted mean values.
- converged
logical. Were the algorithms judged to have converged in all parts of the model?
Odds Ratios
When or=TRUE and the fractional link function (linkfrac or link) is "logit", the model additionally computes odds ratios for the coefficients.
Odds Ratios are exponentiated coefficients.
The corresponding standard errors for the odds ratios are calculated using the Delta method.
The confidence intervals for the odds ratios are calculated using the adjusted standard errors and the specified level (defaulting to 95%).
Odds ratios are particularly useful in fractional logit models as they provide a direct multiplicative interpretation of the independent variable on the odds of the fractional outcome.
References
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.
Ramalho, E.A., J.J.S. Ramalho and J.M.R. Murteira (2011), "Alternative estimating and testing empirical strategies for fractional response models", Journal of Economic Surveys, 25(1), 19-68.
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.
See also
fracreg.reset and fracreg.ggoff, for specification tests.fracreg.ptest, for non-nested hypothesis tests.fracreg.pe, for computing partial effects.fracreghet, for fitting cross-sectional fractional response models with unobserved heterogeneity.fracregpd, for fitting panel data fractional response models.
Examples
### 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")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
# 1P Model reporting odds ratios and 99% confidence intervals
fracreg(y, X, type="1P", linkfrac="logit", or=TRUE, level=0.99)
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
# 2P Model (modelling mass at 1)
fracreg(y, X, type="2P", inflation=1, linkbin="logit", linkfrac="logit")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
# 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")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
### 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")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
# 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")
#>
#> --------------------------------------------------------------------------------
#> 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")
#>
#> --------------------------------------------------------------------------------
#> 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")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>
# Three-part double-inflated model (y has both 0s and 1s)
fracreg(y, X, type="3P", linkbin=c("logit","probit"), linkfrac="logit")
#>
#> --------------------------------------------------------------------------------
#> 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 15:19:34
#> --------------------------------------------------------------------------------
#>