Final Review

ECON 480 • Econometrics • Fall 2022

Dr. Ryan Safner
Associate Professor of Economics

Major Models & Extensions

  • Causality
    • Fundamental problem of causal inference, potential outcomes
    • DAGs, front-doors/back-doors, controlling
  • Multivariate OLS
    • Omitted Variable Bias
    • Variance/Multicollinearity

Major Models & Extensions

  • Categorical data
    • Interpreting dummies, group means
    • Using categorical variables as dummies
    • dummy variable trap
    • interaction effects
  • Nonlinear Models & Transforming Variables
    • quadratic model
    • higher-order polynomials
    • logs
    • standardizing variables
    • joint hypothesis (F-tests)

Major Models & Extensions

  • Panel data

    • pooled regression & problems
    • fixed effects
  • Difference-in-differences

  • Instrumental variables

Question 1

What are the two conditions for a variable \(Z\) to cause .shout[omitted variable bias] if it is left out of the regression?

Question 2

\[Wages_i=\beta_0+\beta_1 \, Education_i + \beta_2 \, Age_i + \beta_3 \, Experience_i + u_i\]

Suppose \(Education_i\) and \(Age_i\) are highly correlated

  • Does this bias \(\hat{\beta_1}\) and \(\hat{\beta_2}\)?
  • What will happen to the variance of \(\hat{\beta_1}\) and \(\hat{\beta_2}\)?
    • How can we measure this?

Question 3

\[Cholesterol_i=\beta_0+\beta_1 \, Treated_i+u_i\]

  • \(Treated_i\) is a dummy variable \(= \begin{cases} 1 & \text{if person received treatment}\\ 0 & \text{if person did not receive treatment}\\ \end{cases}\)
  • What is \(\hat{\beta_0}\)?
  • What is \(\hat{\beta_1}\)?
  • What is the average cholesterol level for someone who recieved treatment?

Question 4

\[Y_i=\beta_0+\beta_1 \, Red_i+\beta_2 \, Orange_i+\beta_3 \, Yellow_i+\beta_4 \, Green_i+\beta_5 \, Blue_i\]

Suppose the color of observation \(i\) can be either \(\{\)Red, Orange, Yellow, Green, Blue, Purple \(\}\)

  • What is \(\hat{\beta_0}\)?
  • What is \(\hat{\beta_1}\)?
  • What is the average value of \(Y_i\) for \(Green\) observations?
  • Why can’t we add \(\beta_6 \, Purple_i\)?

Question 5

\[\widehat{Utility}_i=\beta_0+\beta_1 \, Eggs_i+\beta_2 \, Breakfast_i+\beta_3 (Eggs_i \times Breakfast_i)\]

\(Breakfast_i\) is a dummy variable \(= \begin{cases} 1 & \text{if meal i is breakfast}\\ 0 & \text{if meal i is not breakfast}\\ \end{cases}\)

  • What is \(\hat{\beta_1}\)?
  • What is \(\hat{\beta_2}\)?
  • What is \(\hat{\beta_3}\)?
  • We have two regressions (one for Breakfast; one for Not Breakfast)
    • how can we determine if the intercepts are different?
    • how can we determine if the slopes are different?

Question 6

\[\widehat{Utility}_i=2+4\text{ Ice Cream Cones}_i-1\text{ Ice Cream Cones}_i^2\]

  • What is the marginal effect of eating 1 more Ice Cream Cone?
  • What if we start with 1 Ice Cream Cone?
  • What if we start with 4 Ice Cream Cones?
  • What amount of ice cream cones will maximize utility?
  • How would we know if we should add \(\text{Ice Cream Cones}_i^3\)?

Question 7

\[\ln(GDP_i)=10+2\text{ population (in billions)}_i\]

  • Interpret \(\hat{\beta_1}\) in context.

\[\ln(GDP_i)=10+0.1 \, \ln(\text{population}_i)\]

  • Interpret \(\hat{\beta_1}\) in context.

Question 8

  • Explain what an \(F\)-test is used for.
  • Explain how an \(F\)-statistic is estimated (roughly).

Question 9

Consider a two-way fixed effects model:

\[\text{Divorce Rate}_{it}=\beta_1 \text{Divorce Law}_{it}+\alpha_i+\theta_t+\epsilon_{it}\]

for State \(i\) at time \(t\)

  • Why do we need \(\alpha_i\) and \(\theta_t\)?
  • What sorts of things are in \(\alpha_i\)?
  • What sorts of things are in \(\theta_t\)?

Question 10

Suppose Maryland passes a law (and other States do not) that affects crime rates. Consider the following model:

\[\text{Crime Rate}_{it}=\beta_0+\beta_1 \, \text{Maryland}_{i}+\beta_2 \, \text{After}_t+\beta_3 \, (\text{Maryland}_i \times \text{After}_t)\]

for State \(i\) at time \(t\)

  • What must we assume about Maryland over time?
  • What is the average crime rate for other states before the law?
  • What is the average crime rate for Maryland after the law?
  • What is the causal effect of passing the law?

Question 12

  • What are the two conditions required for an instrument to be valid?
    • How is this different from the conditions for omitted variable bias?
  • How can we test each condition?
  • How do we run a two-stage least squares regression?