Fractional Response Regressions under Unobserved Heterogeneity - Partial Effects

fracreghet.pe is used to compute average and/or conditional partial effects in fractional response models under unobserved heterogeneity.

Usage

fracreghet.pe(object, smearing = T, APE = T, CPE = F, at = NULL, 
              which.x = NULL, table = T, variance = T)

Arguments

object: an object containing the results of an fracreghet command.
smearing: a logical value indicating whether the smearing correction is to be applied
APE: a logical value indicating whether average partial effects are to be computed.
CPE: a logical value indicating whether conditional partial effects are to be computed.
at: a numeric vector containing the covariates' values at which the conditional partial effects are to be computed or the strings "mean" (the default) or "median", in which cases the covariates are evaluated at their mean or median values (or mode, in case of dummy variables), respectively.
which.x: a vector containing the names of the covariates to which the partial effects are to be computed.
table: a logical value indicating whether a summary table with the results should be printed.
variance: a logical value indicating whether the variance of the estimated partial effects should be calculated. Defaults to TRUE whenever table = TRUE.

Details

fracreghet.pe calculates partial effects for fractional response models estimated via fracreghet. fracreghet.pe may be used to compute average or conditional partial effects. These partial effects may be conditional only on observables, using the smearing estimator, or also on unobservables, setting the error term to zero.

Partial Effects under Unobserved Heterogeneity: When unobserved heterogeneity or endogeneity is present, calculating partial effects requires dealing with the unobserved error $v_i$. Let the conditional mean be $E(y|x, v) = G(x\beta + \gamma v)$. - Conditional on Observables (Smearing): The unobserved heterogeneity is integrated out over its empirical distribution. The average partial effect for a continuous variable $x_k$ is computed as: $$PE_k(x) = \frac{1}{N} \sum_{i=1}^N g(x\beta + \gamma \hat{v}_i) \beta_k$$ - Conditional on Unobservables (Error = 0): The partial effect is evaluated for an individual with the mean level of unobserved heterogeneity ($v = 0$): $$PE_k(x) = g(x\beta) \beta_k$$

For discrete variables, the partial effects are calculated as the discrete differences evaluated using either the smearing approach or setting the error term to zero.

For calculating standard errors, it is taken into account the option that was previously chosen for estimating the model. See Ramalho and Ramalho (2017) for details on the computation of partial effects for fractional response models under unobserved heterogeneity.

Value

fracreghet.pe returns a list with the following element:

PE.p: a named vector of partial effects.

If variance = TRUE or table = TRUE, the previous list also contains the following element:

PE.sd: a named vector of standard errors of the estimated partial effects.

When both average and conditional partial effects are requested, two lists containing the previous elements are returned, indexed by the prefixes ape and cpe.

References

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.

Author

Sulman Olieko Owili <oliekosulman@gmail.com>

Examples

### 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")
#> 
#> 
#> -------------------------------------------------------------------------------- 
#> 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 15:19:38 
#> -------------------------------------------------------------------------------- 
 
### 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")
#> 
#> 
#> -------------------------------------------------------------------------------- 
#> 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 15:19:38 
#> -------------------------------------------------------------------------------- 

#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))
#> 
#> 
#> -------------------------------------------------------------------------------- 
#> 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 15:19:38 
#> -------------------------------------------------------------------------------- 
#> 
#> Note: covariates evaluated at the following values:
#> 
#> X1 X2 
#>  1 -1