10.3 Motivating Example for Count Data
We will use the following article for our motivating example: “Legislative Capacity and Executive Unilateralism” by Alex Bolton and Sharece Thrower, which was published in the American Journal of Political Science in 2015.
Abstract. This article develops a theory of presidential unilateralism in which both ideological divergence with Congress and legislative capacity influence the president’s use of executive orders. We argue that when Congress is less capable of constraining the executive, the president will issue more executive orders during periods of divided government. Conversely, in periods of high legislative capacity, the president is less likely to issue executive orders when faced with an opposed Congress. Based on an examination of institutional changes, we identify years prior to the mid‐1940s as characterized by low congressional capacity and the subsequent period as characterized by high capacity. Testing the theory between 1905 and 2013, we find strong support for these predictions and demonstrate that legislative capacity conditions the role of ideological disagreement in shaping presidential action. Overall, this article deepens our current understanding of the dynamics of separation‐of‐powers politics and the limits of executive power.
The primary research question: Is the president constrained by an ideologically opposed Congress? The authors explore how the number of executive orders made per year varies acording to whether the government is divided or unified.
- Outcome:
allnoncerm_eo, all non-ceremonial executive orders in a year - Key Explanatory variable:
divided, whether or not there was divided government, where the president and the majority party in either the House or Senate are different parties - Other explanatory variables include dummy variables for the president, an indicator if it is war time, measures related to the economy, and whether it is close to the beginning or end of an administration
Let’s load the data and look at the outcome variable.
library(foreign)
bolton <- read.dta("https://github.com/ktmccabe/teachingdata/blob/main/bt.dta?raw=true")
table(bolton$allnoncerm_eo)
20 26 30 31 34 35 37 38 39 40 41 42 43 45 48 49 50 52 53 54
1 2 1 2 2 2 3 1 1 1 3 2 1 4 2 3 1 2 2 2
55 56 57 61 63 64 65 66 68 69 70 71 72 75 76 78 85 92 96 97
1 1 3 1 1 2 1 2 2 1 2 1 1 1 1 1 1 1 1 1
98 112 116 117 120 146 164 172 188 200 212 219 225 232 241 247 250 253 265 267
2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 3 1 1 1 1
273 286 287 303 305 307 309 315 319 328 338 339 345 358 382 393 438 471 473 501
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Often for count variables, it can be useful to visualize them in a histogram. Here is a ggplot version.
library(ggplot2)
ggplot(bolton, aes(allnoncerm_eo))+
geom_histogram(binwidth = 5)+
theme_minimal()
The authors distinguish time pre- and post-1945 based on different levels of Congressional capacity. We can look at how the outcome changed over time and note how there were far more executive orders in the earlier period.
plot(x=bolton$year, y=bolton$allnoncerm_eo, pch =20,
main = "Executive Orders by Year",
cex.main = .8,
ylim = c(0, 500))
abline(v=1945, lty =2) # vertical line at 1945
10.3.1 Fitting Poisson in R
We will investigate the relationship between divided government and executive orders in the first time period.
The authors’ hypothesize, “During periods of low legislative capacity (prior to the mid-1940s), the president issues more executive orders under divided government.
To fit a Poisson model in R, we use the glm function. However, now we have a different family= "poisson" and link = "log". We don’t actually have to explicitly write the link because R will use this link by default.
Let’s fit a regression of our outcome on the key explanatory variables, along with other controls the authors use. Note that because I want the early period, I have to subset the data. I can do this outside, prior to the regression. Or, I can subset in the data argument, as is done in the below code:
fit <- glm(allnoncerm_eo ~ divided + inflation +
spending_percent_gdp + war + lame_duck +
administration_change + trend +
+ tr+ taft + wilson + harding
+ coolidge + hoover,
family = "poisson",
data = subset(bolton, year < 1945))10.3.2 Interpreting regression output
The summary output for Poisson is much nicer than the multinomial output we were working with previously. Let’s extract just the divided coefficient output from the summary.
summary(fit)$coefficients[2,] Estimate Std. Error z value Pr(>|z|)
4.435893e-01 4.195090e-02 1.057401e+01 3.933025e-26 How should we interpret this coefficient?
- Recall \(\log \hat \lambda = \mathbf x_i'\hat \beta\)
For every one-unit change in \(x\), we estimate an average \(\hat \beta\) change in the \(\log\) of the expected executive orders, holding the other covariates constant.
- Note that usually counts are measured given a particular time or space interval. For this reason sometimes these are considered “rates” (i.e., the number of executive orders per year).