OLS and IV Estimands

Observe that:

1. If \(X\) is exogenous, \(\text{Cov}(X, U) \neq 0\), then \(\beta_{OLS} \neq \beta\).
2. If \(Z\) is exogenous, \(\text{Cov}(Z,U) = 0\), and relevant, \(\text{Cov}(Z, X) \neq 0\), then \(\beta_{IV} = \beta\).
3. If \(Z\) is endogenous, the weaker its correlation with \(X\), the farther \(\beta_{IV}\) is from \(\beta\).

Decomposing the IV Estimand

\[ \beta_{IV} \equiv \frac{\text{Cov}(Z,Y)}{\text{Cov}(Z,X)} = \frac{\text{Cov}(Z,Y)/\text{Var}(Z)}{\text{Cov}(Z,X)/\text{Var}(Z)} \equiv \frac{\gamma_1}{\pi_1} = \frac{\text{Reduced Form}}{\text{First Stage}} \] Reduced Form: Regress \(Y\) on \(Z\) \[ \gamma_0 \equiv \mathbb{E}(Y) - \gamma_1 \mathbb{E}(Z); \quad \gamma_1 \equiv \frac{\text{Cov}(Z,Y)}{\text{Var}(Z)}; \quad \epsilon \equiv Y - \gamma_0 - \gamma_1 Z \] \[ \implies Y = \gamma_0 + \gamma_1 Z + \epsilon; \quad \mathbb{E}(\epsilon) = \text{Cov}(Z, \epsilon) = 0 \] First Stage: Regress \(X\) on \(Z\) \[ \pi_0 \equiv \mathbb{E}(X) - \pi_1 \mathbb{E}(Z); \quad \pi_1 \equiv \frac{\text{Cov}(Z,X)}{\text{Var}(Z)}; \quad V \equiv X - \pi_0 - \pi_1 Z \] \[ \implies X = \pi_0 + \pi_1 Z + V; \quad \mathbb{E}(V) = \text{Cov}(Z, V) = 0 \]

💪 Exercise A - (20 min)

1. Set the seed to 1234 and then generate 5000 iid standard normal draws to serve as your instrument z. Next, use rmvnorm() to generate 5000 iid draws from a bivariate standard normal distribution with correlation \(\rho = 0.5\), independently of z.
2. Use the draws you made in the preceding part to generate x and y according to the IV model with \(\pi_0 = 0.5\), \(\pi_1 = 0.8\), \(\alpha = -0.3\), and \(\beta = 1\).
3. Using the formulas from the preceding slides, predict the slope coefficient that you would obtain if you ran a linear regression of y on x. Run this regression to check.
4. Using the formulas from the preceding slides, predict the slope coefficient that you would obtain if you ran a linear regression of y on z. Run this regression to check.
5. Use the formulas from the preceding slides to calculate the IV estimate. How does it compare to the true causal effect in your simulation?
6. Try running a regression of y on both x and z. What is your estimate of the slope coefficient on x? How does it compare to the OLS and IV estimates? What gives?
7. Which will give the most accurate predictions of \(Y\) in this example: OLS or IV?

A More Elaborate Example

\[ \begin{align*} Y &= \beta_0 + \beta_1 X + \beta_2 W + U\\ X &= \pi_0 + \pi_1 Z_1 + \pi_2 Z_2 + \pi_3 W + V\\ \end{align*} \]

• The \(Y\) equation is a linear causal model
  • Goal is to learn the causal effect \(\beta_1\) of \(X\).
  • \(X\) is endogenous: \(\text{Cov}(X, U) \neq 0\)
  • Exogenous “control” regressor \(W\): \(\text{Cov}(W, U) = 0\).
• The \(X\) equation is a population linear regression
  • \(\text{Cov}(Z_1, V) = \text{Cov}(Z_2, V) = \text{Cov}(W, V) = 0\) by construction.
• Instrumental Variables Assumptions:
  • Relevance: at least one of \(\pi_0\) and \(\pi_1\) is non-zero
  • Exogeneity: \(\text{Cov}(Z_1, U) = \text{Cov}(Z_2, U) = 0\).

Two-Stage Least Squares (TSLS)

\[ \begin{align*} Y &= \beta_0 + \beta_1 X + \beta_2 W + U\\ X &= \pi_0 + \pi_1 Z_1 + \pi_2 Z_2 + \pi_3 W + V\\ \end{align*} \]

1. Regress \(X\) on \(Z_1, Z_2\), and \(W\). Save the fitted values: \(\hat{X}\).
2. Regress \(Y\) on \(\hat{X}\) and \(W\). The coefficient on \(\hat{X}\) is our estimate of \(\beta_1\).

AER::ivreg()

• The ivreg() function from the AER package carries out TSLS estimation and calculates the correct standard errors. (Today: assume homoskedasticity.)
• tidy(), augment(), and glance() from broom work with ivreg(). See here.
• ivreg() syntax: ivreg([FORMULA_HERE]), data = [DATAFRAME_HERE])
• Formula syntax: [CAUSAL_MODEL_FORMULA] | [FIRST_STAGE_FORMULA]
  • First-stage formula is “one-sided” i.e. there is no LHS variable or ~
  • Our running example: y ~ x + w | z1 + z2 + w

💪 Exercise B - (20 min)

1. Set the seed to 1234 and then generate 10000 draws of \((Z_1, Z_2, W)\) from a trivariate standard normal distribution in which each pair of RVs has correlation \(0.3\). Then generate \((U, V)\) independently of \((Z_1, Z_2, W)\) as in Exercise A above.
2. Use the draws you made in the preceding part to generate x and y according to the IV model from above with coefficients \((\pi_0, \pi_1, \pi_2, \pi_3) = (0.5, 0.2, -0.15, 0.25)\) for the first-stage and \((\beta_0, \beta_1, \beta_2) = (-0.3, 1, -0.7)\) for the causal model.
3. Run TSLS “by hand” by carrying out two regressions with lm(). Compare your estimated coefficients and standard errors to those from AER::ivreg().
4. Run TSLS “by hand” but this time omit w from your first-stage regression, including it only in your second-stage regression. What happens? Why?
5. What happens if you drop \(Z_1\) from your TSLS regression in ivreg()? Explain.
6. What happens if you omit w from both your first-stage and causal model formulas in ivreg()? Are there any situations in which this would work? Explain.

Data Dictionary

💪 Exercise C - (10 min)

In this exercise you will need to work with the columns critics, order, and ranking from the qe tibble.

1. Add a variable called first to qe that takes on the value TRUE if order equals one. You will need this variable in the following parts.
2. Do musicians who perform first receive different average rankings from the jury than other musicians? Discuss briefly.
3. Is it possible to estimate the causal effect of performing first on subsequent ratings by critics using this dataset? If so how and what is the effect?
4. Estimate the causal effect of ranking in the competition on future success as measured by critics’ rating two ways: via OLS and via IV using first to instrument for ranking. Discuss your findings.