--- title: "Exercise template for 'Instrumental variables'" author: Your name format: html bibliography: book.bib --- ```{r} #| label: setup #| include: true #| cache: false #| message: false library(dplyr, warn.conflicts = FALSE) library(modelsummary) library(fixest) ``` ## Discussion questions 1. In the 1960s and early 1970s, American men were at risk of being drafted for military service. Each year from 1970 to 1972, draft eligibility was prioritized based on the results of a lottery over birthdays. A random number was assigned to the birthdays of 19-year-olds. Those whose birthday was associated with a random number below the cutoff were draft-eligible; the other 19-year-olds were not draft-eligible. @Angrist:2008vk point out that, "in practice, many draft-eligible men were still exempted from service for health or other reasons, while many men who were draft-exempt nevertheless volunteered for service." Using an indicator for draft lottery number being below the cutoff as an instrumental variable, @Angrist:1990dk finds that those who served in Vietnam earned significantly less than their peers even in 1981. Try to draw the causal diagram for this analysis and apply the three questions outlined above to this setting. What assumptions are needed for the answer to each question to be "yes"? How might these assumptions be violated? 2. @Ahern:2012kca study a rule requiring firms in Norway to appoint female directors and "use the pre-quota cross-sectional variation in female board representation to instrument for exogenous changes to corporate boards following the quota." What is the significance of the discussion suggesting that "the quota was implemented without the consent of business leaders" [-@Ahern:2012kca, p. 145]? What about the discussion [-@Ahern:2012kca, p. 155] suggesting that the rule was not anticipated? 3. You and a co-author are interested in studying the effect of independent directors on financial reporting quality. You conjecture that independent directors demand better financial reporting, as they are less likely to benefit from obfuscation of performance than non-independent directors, who are often employees of the firm. However, you are concerned that causation may run in the opposite direction; firms with better financial reporting quality may have more independent directors on their boards. Provide some arguments for the existence of reverse causation. How persuasive do you find these arguments? 4. In response to a request by the SEC in February 2002, the major US exchanges (NYSE and Nasdaq) proposed changes to listing standards that would require firms to have a majority of independent directors. In 2003, the SEC approved these changes and required that firms comply by the earlier of (i) the first annual shareholder meeting after January 15, 2004 and (ii) October 31, 2004. Your co-author argues that this change was "exogenous", as it was not driven by the decisions of individual firms, and this fact allows you to estimate a causal effect. Does "exogeneity" as used by your co-author mean the same thing as exogeneity for the purposes of econometric analysis? Does the former notion imply the latter? Think of "examples" to support your arguments (an example could be a simple model, a numerical argument, or a verbal description of a real or imagined scenario). More specifically, your co-author argues that the change in the number of independent directors imposed on the firm by the new rules is exogenous and thus could be used as an instrument for your study. @Armstrong:2014aa examine the changes in listing standards above and use "the minimum required percentage change in independent directors, `Min % change ID`, as an instrument" in studying the effect of independent directors on firm transparency. What are the parallels between the setting of @Armstrong:2014aa and that of @Ahern:2012kca? Do the two papers use the same basic approach, or are there important differences between their approaches? 5. The objective of @Cohen:2013uc "is to investigate how governance regulations in SOX and the exchanges are associated with chief executive officers' incentives and risk-taking behavior." What is the treatment of interest in @Cohen:2013uc? What are the outcomes of interest? @Cohen:2013uc state that "our dependent variables, namely, investment and executive incentive compensation, are likely to be determined jointly. As such, the parameter estimates from ordinary least squares (OLS) are likely to be biased. Our empirical analyses address the issue by using simultaneous equations models." What "exclusion restrictions" are implied by Panel A of Table 4? What does the implied causal diagram look like? How persuasive do you find the approach of @Cohen:2013uc to be? ```{r} #| label: sneetch-data #| cache: true #| cache.vars: ["n", "max_d", "z", "a", "gamma", "b", "alpha", "beta", "sd_epsilon", "d_opt_fun", "df", "min_d", "df_1"] #| lst-label: lst-sneetches #| lst-cap: Code to generate the Sneetch world # Set parameters n <- 10000; max_d <- 10 set.seed(1); z <- rlnorm(n) a <- 1 gamma <- 3; b <- gamma * z alpha <- 100; beta <- 10; sd_epsilon <- 5 d_opt_fun <- function(a, b, min_d = 0) { v <- function(x) -a * x^2 + b * x opts <- min_d:max_d vals <- sapply(opts, v) d_opt <- opts[vals == max(vals)] d_opt } d_opt_fun <- Vectorize(d_opt_fun) # Calculate firm value, etc., at t = 0 df <- tibble(a, b) |> mutate(epsilon_0 = rnorm(n, sd = sd_epsilon), V_0 = alpha + beta * b + epsilon_0, d_0 = d_opt_fun(a, b)) # Add firm value, etc., at t = 1 min_d <- 3 df_1 <- df |> mutate(epsilon_1 = rnorm(n, sd = sd_epsilon), V_1 = alpha + beta * b + epsilon_1, d_1 = d_opt_fun(a, b, min_d = min_d)) ``` 6. Suppose the existence of a world of Sneetches.^[The interested reader can learn more about Sneetches in Dr Seuss's *The Sneetches and Other Stories*, available on (for example) Amazon: .] There are two kinds of Sneetches: | *Now, the Star-Belly Sneetches* | *Had bellies with stars.* | *The Plain-Belly Sneetches* | *Had none upon thars.* | ---Dr. Seuss In the Sneetch world, there are `r prettyNum(n, big.mark = ",")` (`n`) firms, each with a board of `r max_d` (`max_d`) directors. The number of Plain-Belly directors $d_i$ is set by firm $i$ as $$d_i = {\underset {x\in [\underline{d}, \overline{d}]}{\operatorname {arg\,max} }}\, v_i(x) = -a_i x^2 + b_i x $$ where $a_i = `r a`$ and $b = \gamma z_i$, $\gamma = `r gamma`$ and $z_i$ is log-normally distributed with default parameters of $\mu=0$ and $\sigma=1$. We start with $\underline{d} = 0$ and $\overline{d} = `r max_d`$. The remaining $`r max_d` - d_i$ directors would be Star-Belly Sneetches. The value of firm $i$ at time $t=0$ is given by the equation: $$ V_{i0} = \alpha + \beta \times b_i + \epsilon_0 $$ where $\alpha = `r alpha`$, $\beta = `r beta`$ and $\epsilon \sim N(0, \sigma_{\epsilon}^2)$, where $\sigma_{\epsilon} = `r sd_epsilon`$. \newline \newline The Sneetches then pass new legislation requiring every board to have at least three Plain-Belly directors at time $t=1$. The value of firm $i$ at time $t=1$ is given by the equation: $$ V_{i1} = \alpha + \beta \times b_i + \epsilon_{i1} $$ The code in @lst-sneetches starts by setting the parameters discussed above and generating $z_i$ (`z`). The code then generates the value of each of the `r prettyNum(n, big.mark = ",")` firms at time $t=0$ (`V_0`), as well as the number of Plain-Belly directors chosen by each firm at $t=0$ to maximize $v_i(\cdot)$ (`d_0`). These values are stored in `df`. The code in @lst-sneetches then sets `min_d` to reflect the requirement of the legislation coming into force for $t = 1$. The code then generates the value of each firm at $t=1$ (`V_1`) and the number of Plain-Belly directors chosen by each firm at $t=1$ (`d_1`), which involves maximization of $v_i(\cdot)$ subject to the new constraint that $d_i \geq `r min_d`$. We assume that neither $a_i$ nor $b_i$ is observable to the researcher. a. Expressed in terms of $a_i$ and $b_i$, what value of $x$ maximizes $v_i(x)$ when there is no constraint (as is true at $t = 0$)? b. What issue do we have in using this maximum to select the number of Plain-Belly directors on a board? c. What does `d_opt_fun()` in @lst-sneetches do to handle this issue? d. What does the function `Vectorize()` do in @lst-sneetches? (*Hint*: Use `? Vectorize` in R to learn more about this function.) e. Using OLS and data in `df_1` after running the code in @lst-sneetches, estimate the relationship between firm value at time $t=1$ ($V_1$) and the number of Plain-Belly directors on the board ($d_1$). f. Using the data in `df_1`, estimate an IV regression using the strategy of @Ahern:2012kca. g. Using the data in `df_1`, estimate an IV regression using the strategy of @Armstrong:2014aa (note that rather than `Min % change X`, you might use something like `Min change X`, as the denominator is `r max_d` for all observations). *Hint*: The `pmin()` function in R gives the point-wise minimum of two vectors of values (e.g., `pmin(x, y)` will return the lower of `x` and `y` for each pair.) h. What is the true causal effect of Plain-Belly directors on firm value? Do the strategies of either @Ahern:2012kca or @Armstrong:2014aa correctly estimate this effect?