Which regression model to use for a probability as dependent variable?

Question

My dependent variable is a probability. As such, values lie between 0 and 1. The most common values are 0, 0.5, and 1 each occurring in 20% to 30% of the observations but any value in between is possible and some do occur.

Question 1: Which regression model is best to explain such data?

• Ordinary least squares (OLS, function lm in R’s stats package) is not suitable as it does neither account for the limited interval nor the accumulation at the margins.

• Logit regression (function glm with parameter family="binomial" in R’s stats package) accounts for the accumulation at 0 and 1 but does not allow intermediate values.

• Ordered logit regression (function polr in R’s MASS package) could be applied when I divide the [0, 1] interval in subintervals. However, I lose the continuous nature of the dependent variable.

• For probit and ordered probit regressions, the same applies as for logit and ordered logit.

• Left- and right-censored tobit regression (function tobit with parameters left=0 and right=1 in R’s AER package) might be appropriate. However, I found the following quote: “Some researchers have considered using censored normal regression techniques such as tobit ([R] tobit) on proportions data that contain zeros or ones. However, this is not an appropriate strategy, as the observed data in this case are not censored: values outside the [0, 1] interval are not feasible for proportions data.” (p. 302 in Baum (2008), http://www.stata-journal.com/sjpdf.html?articlenum=st0147).

Below you find a code example

# Load libraries
library(stats, MASS, AER)
# Generate data
set.seed(123)
data <- data.frame(x1 <- runif(60, min = 0, max = 1), x2 <- runif(60, min = 0, max = 1))
data$y  <- -0.7 + data$x1 + 2 * data$x2 + rnorm(60, mean = 0, sd = 0.5)
    data$y  <- ifelse(data$y < 0, 0, data$y)
data$y  <- ifelse(data$y > 0.4 & data$y < 0.6, 0.5, data$y)
data$y  <- ifelse(data$y > 1, 1, data$y)
    data$yCat <- data$y
    data$yCat <- ifelse(data$yCat > 0 & data$yCat < 0.5, 0.25, data$yCat)
    data$yCat <- ifelse(data$yCat > 0.5 & data$yCat < 1, 0.75, data$yCat)
    data$yCat <- as.factor(data$yCat)
    hist(data$y, breaks=101)
# Different regression models
summary(lm(y ~ x1 + x2, data=data)) # OLS
summary(glm(y ~ x1 + x2, data=data, family="binomial")) # Logit
summary(polr(yCat ~ x1 + x2, data=data)) # Ordered logit
summary(tobit(y ~ x1 + x2, data=data, left=0, right=1)) # Tobit

To make matters worse, my data is panel data. I know how to handle individual, time, and mixed effects and random and fixed effects models using plm from R’s plm package and F-test, LM-test, and Hausman test do decide which of these is best.

Question 2: For the dependent variable described above, which panel regression model is best?

Below your find a code example for the data structure. This extends the prior example.

# Load library
library(plm)
# Generate data (builds on prior example)
data$id <- rep( paste( "F", 1:15, sep = "_" ), each = 4)
    data$time <- rep( 1981:1984, 15 )
pData <- pdata.frame(data, c( "id", "time" ))
# Panel regression example
summary(plm(y ~ x1 + x2, data=pData, model="within", effect="twoways")) # Based on OLS