Chapter 8 Control Variables

8.3.1 Directed Acyclical Graph (or DAG)

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In the remainder of this chapter, we are interested in testing the relation between x and y and we have to make a decision whether we should control for z. The first reason to control for a variable is because it affects the outcome of interest. This is often the main reason given to control for a variable, it affects the outcome.

Figure 8.3: Measurement error

8.3.2 Simulation

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We can illustrate the problem with a simulation. We are interested in the effect of x on y and there is another factor z with a large effect on z. We print the regression of y on x with and without z as a control variable with the stargazer package (Hlavac 2018).

set.seed(12345)
obs = 500
x <- rnorm(n = obs); z <- rnorm(n = obs)
y <- rnorm(n = obs, mean = 1 * x + 20 * z, sd = 1)
d <- tibble(y = y, x = x, z = z)
yx <- lm(y ~ x, data = d)
yxz <- lm(y ~ x + z, data = d)
stargazer(yx, yxz, type = tab_format, 
          label = "measurement_error",
          omit = c("Constant"), digits = 2,
          intercept.bottom = FALSE,
          star.cutoffs = c(0.05, 0.01, 0.001),
          keep.stat = c("n", "rsq"))

The estimate of the effect of x regressions is close to the true value, 1, but the standard error is much smaller when we include z as a control variable. The reason to control for z is to make sure that we have a precise estimate of the effect of interest, however the estimate itself is not going to change when z only has an effect on y.

8.3.3 Real Data

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We can turn to the CEO data to illustrate the idea of controlling for other factors that might effect the variable of interest. We investigate the relationship between CEO ownership and market value of the company in Figure 8.4.

Figure 8.4: Relation between CEO ownership and market value for SP500 firms (2011-2018).

meas_data <-  select(combined, market_value, shares, ipo) %>%
              filter(complete.cases(.))
ownership <-
  ggplot(data = meas_data,
      aes(x = market_value/1000, y = shares + 0.001)) +
  geom_point() +
  scale_x_continuous(trans = "log",
    breaks = scales::log_breaks(n = 5, base = 10),
    labels = function(x) prettyNum(x, dig = 2)) +
  scale_y_continuous(trans = "log",
    breaks = scales::log_breaks(n = 5, base = 10),
    labels = function(x) prettyNum(x, dig = 2),
    limits = c(NA, 50)) +
  labs(y = "CEO ownership", x = "Market Value in $ Billion")
print(ownership)

## Warning: Removed 4 rows containing missing values (geom_point).

We hypothesised back in Chapter 7 that CEOs who are founders will have higher ownership for reasons that have nothing to do with the optimal incentives. We cannot directly measure whether CEOs are founders but we know wether they are already CEO (ipo_ceo) or employee (ipo_employee) at the time of the initial public offering of the company. We include those two variables as indicator variables in the regression because it is more likely that these CEOs are also the founders of the company5656 I will ignore the fact that we lose observations in calculating the new variables. That is not necessarily a good idea..

form <- log(shares + 0.001) ~ log(market_value/1000)
base <- lm(form, data = combined)
form_error <- update(form, . ~ . + ipo_ceo + ipo_employee)
error <- lm(form_error, data = combined)
stargazer(base, error, type = tab_format, 
          label = "ownership",
          digits = 2, intercept.bottom = FALSE,
          star.cutoffs = c(0.05, 0.01, 0.001),
          keep.stat = c("n", "rsq"))

Unfortunately, controlling for ipo_ceo and ipo_employee does not really improve the precision of our estimates for the effect of market value on ownership. The standard error for the estimate stays about the same (i.e. \(0.2\)).

8.3.4 Interpretation of Regression Outcome (or Economic Significance)

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This is a bit of disgression of the main storay and you skip this section. Nevertheless, I would encourage you to go over the example as it gives a good example of how to interpret the numbers in a regression. Something, you will have to do for your thesis as well.

The regression we have estimated is

\[\begin{equation*} \mathrm{log}(S + 0.001) = \beta_0 + \beta_1 \mathrm{log}(MV) + \beta_2 \mathrm{ipo\_ceo} + \beta_3 \mathrm{ipo\_employee} \end{equation*}\]

and based on the theory, we are mainly interested in the parameter \(\beta_1\). However, something intriguing is going on with the coefficients for the two control variable. The results seem to imply that whether the CEO was an employee at the time of the IPO is a better indicator for share ownership than whether the CEO was already CEO at the time of the IPO.

That conclusion is not correct. There are a number of ways you can see this. When the CEO was already CEO at the time of the IPO, they were also already an employee. That means that the ipo_ceo variable is dependent on the ipo_employee variable. If ipo_employee = 0 then ipo_ceo = 0 and if ipo_ceo = 1 then ipo_employee = 1. You could also see this if you make a descriptive table with the percentage of observations with each theoretical combination of the ipo_ceo and ipo_employee variable.

combined %>% select(ipo_ceo, ipo_employee, year) %>%
  group_by(year) %>%
  summarise(
    ipo_both = mean(ipo_ceo * ipo_employee, na.rm = T),
    ipo_ceo_only = mean(ipo_ceo * (1 - ipo_employee),
                        na.rm = T),
    ipo_empl_only = mean((1 - ipo_ceo) * ipo_employee,
                         na.rm = T),
    ipo_none = mean((1 - ipo_ceo) * (1 - ipo_employee),
                    na.rm = T)
  ) %>% kableExtra::kable(digits = 2, booktabs = TRUE)

The table again shows that it is impossble for a CEO to have been the CEO at the time of the IPO but not an employee5757 ipo_ceo_only equals 0 in every year.. This example shows the importance of using descriptive statistics to better understand the data. Even if you had not realised that there is a relation between ipo_ceo and ipo_employee, the table would have alerted you to it.

It also shows us that we cannot interpret the coefficients separately. The right way to interpret the coefficients is to interpret them together. However, there is a second problem with interpreting our regression outcomes. We modelled log(shares + 0.001) and so the effect are a bit harder to interpret. With some rearrangement of the regression function, we can make the interpreation easier.

\[\begin{align*} \mathrm{log}(S + 0.001) &= \beta_0 + \beta_1 \mathrm{log}(MV) + \beta_2 \mathrm{ipo\_ceo} + \beta_3 \mathrm{ipo\_employee} \\ S + 0.001 &= e^{\beta_0} e^{\beta_1 \mathrm{log}(MV)} e^{\beta_2 \mathrm{ipo\_ceo}} e^{\beta_3 \mathrm{ipo\_employee}} \\ S + 0.001 &= e^{\beta_0} MV^{\beta_1} e^{\beta_2 \mathrm{ipo\_ceo}} e^{\beta_3 \mathrm{ipo\_employee}} \end{align*}\]

This means that the relation between ownership and whether CEO is a (likely) founder is multiplicative. For a given firm size, the CEO owns \(e^{\beta_3}\) times more shares when they were already an employee at time of the IPO, and \(e^{\beta_2 + \beta_3}\) times more shares when they were already CEO (and thus also employee) at the time of the IPO5858 This ignores the \(0.001\) adjustment but it does not really make a difference in this example.. We can easily calculate these values in R.

beta2 <- error$coefficients["ipo_ceoTRUE"]
beta3 <- error$coefficients["ipo_employeeTRUE"]
exp(beta3)

## ipo_employeeTRUE 
##         2.949815

exp(beta2 + beta3)

## ipo_ceoTRUE 
##    3.555428

This means that the CEO who has already an employee at the time of IPO, has 2.9 times higher ownership and a CEO who was already CEO at the time of IPO has 3.6 times more shares than a CEO who was not in their company of similar size at the time of the IPO.

8.3.5 Better model of CEO founders

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