Table

Below the regression tables for the 10-th, 25-th, 50-th (LAD), 75-th and 90-th quantiles, together with the OLS estimates. It reproduces Table 2. of the paper. Several notes are worth mentioning: i) the point estimates are slightly different -most likely due to both a slightly different cleaning, I have some more observations here, and differences in statistical softwares computations-; ii) the standard errors also differ a bit because I didn’t go for bootstrapped computations (too time-consuming). It is possible to compute the bootstrapped errors if you change the se option by 'boot' in summary().

There are several threat to identification. As highlighted by the authors, there is likely an OVB issue since they can not control for all maternal behavior that are likely correlated with other regressors in the model, like alcohol use or prenatal diet for instance. Another issue is the likely failure of the rank invariance condition: it seems implausible that the babies would remain on the same position in the birthweight distribution, independently of regressors values. The interpretations are thus only descriptive here, but still informative. For instance,

1. If anything, the black-white differential in birthweight at the 10-th quantile is 253 grams. In other words, the 10-th quantile of birthweight for a baby born to a black mother is 253 grams below than the 10-th quantile of birthweight for a baby born to a white mother;

2. If anything, the differential in birthweight between college graduate and “dropouts” shrink completely from 82 grams at the 10-th quantile compared to the 90-th quantile. The effect of college on birthweight is sizeable for low-birthweight babies, while not being significant at the high-end of the distribution;

3. The intercept can be interpreted as the estimated conditional quantile function of the birthweight distribution of a girl born to an unmaried, white mother with less than a high school education, who is 27yo and had a weight gain of 30 pounds, didn’t smoke, and had her first prenatal visit in the first trimester of the pregnancy (the number reflects mean values). In this example, given the high number of covariates, this baseline comparison seems hard to grasp but it might be more informative in simpler settings.

# Regression table output
stargazer(qr_0.1, 
          qr_0.25, 
          qr_0.5,
          qr_0.75, 
          qr_0.9, 
          ols,
          se = c(list(qr_rob_0.1), 
                 list(qr_rob_0.25), 
                 list(qr_rob_0.5), 
                 list(qr_rob_0.75), 
                 list(qr_rob_0.9),
                 list(ols_rob)),
          type = "html",
          dep.var.labels = "Birth weight",
          model.names = FALSE,
          column.labels = c("0.1","0.25","0.5","0.75","0.9", "OLS"),
          covariate.labels = sapply(data, function(x) attr(x, "label"))[-1])

Wald tests

The two tables below respectively test,

1. For equality of the quantile coefficients along the conditional y-distribution \(\longrightarrow\) except for High School Graduate, Second Trimester Prenatal Care, Non-Smoker During Pregnancy and Cigarette Usage, all variables show coefficients that significantly differ across the quantiles;

2. For the interquantile range (are the \(\beta_{0.1}\) and \(\beta_{0.9}\) statistically different?) \(\longrightarrow\) same conclusion as above, with High School Graduate that shows significantly different coefficients across the 10-th and the 90-th quantiles at 10% confidence level now.

# Wald test 
qr_all <- rq(formula_qr, q, data, method = 'fn')
qr_wald1 <- anova(qr_all, test = "Wald", joint = F)$table %>% 
  as.data.frame() %>%
  round(., digits = 3)

qr_wald2 <- anova(qr_0.1, qr_0.9, test = "Wald", joint = F)$table %>% 
  as.data.frame() %>%
  round(., digits = 3)

# Table 1.
colnames(qr_wald1) <- c("DF", "Res.DF", "F-value", "p-values")
row.names(qr_wald1) <- var_label(data)[-1] 
qr_wald1 %>% kable(., booktabs = TRUE,
             caption = "Wald test for QR") %>%
  kable_styling(font_size = 12) %>% 
  scroll_box(width = "100%")

# Table 2.
colnames(qr_wald2) <- c("DF", "Res.DF", "F-value", "p-values")
row.names(qr_wald2) <- var_label(data)[-1] 
qr_wald2 %>% kable(., booktabs = TRUE,
             caption = "Wald test for QR") %>%
  kable_styling(font_size = 12) %>% 
  scroll_box(width = "100%")

Plots

Here are some (more or less customed) plots that can be useful for visualization and comparison of coefficients.

# No customization. Good for first analysis and overview.
summary(qr_all) %>% plot()

# Customized plot for each regressor: see function 'ggplot.qr' in 00.dependencies.R
qr_plots <- list()
for(k in colnames(data)[-1]){
  qr_plots[[k]] <- ggplot.qr(ols, qr_all, q, data, k, "ker")
  qr_plots[[k]]
}
names(qr_plots) <- paste0("qr_plot_", names(qr_plots))
invisible(list2env(qr_plots, envir = .GlobalEnv))

# Examples 
grid.arrange(qr_plot_married, qr_plot_boy, qr_plot_collgrad,
             qr_plot_novisit, qr_plot_black, ncol = 2)

# Some customization w/ comparison between regressor. 
# TYPE "help(plot_models)" FOR MORE INFO!
qr_comp_1to7 <- plot_models(ols, qr_0.5, qr_0.9,
                            spacing = 0.8,
                            show.values = T,
                            rm.terms = colnames(data)[9:15],
                            m.labels = c("OLS", "LAD", "QR 90%"),
                            legend.title = "Specification",
                            colors = c(my_purple, my_blue, my_red)) +
  my_theme +
  theme(legend.position = c(0.1, 0.1)) +
  coord_flip()
qr_comp_1to7

qr_comp_8to15 <- plot_models(ols, qr_0.5, qr_0.9,
                             spacing = 0.8,
                             show.values = T,
                             rm.terms = colnames(data)[2:8],
                             m.labels = c("OLS", "LAD", "QR 90%"),
                             legend.title = "Specification",
                             colors = c(my_purple, my_blue, my_red)) +
  my_theme +
  theme(legend.position = c(0.1, 0.1)) +
  coord_flip()
qr_comp_8to15