Introduction to Econometrics

Units

1. 1

   The simple regression model

   4 lessons

   1. The population regression modelWrite wage = β₀ + β₁ educ + u, say what each piece means, and state E(u | x) = 0, the condition that lets β₁ be read as the effect of one more year of schooling.
   2. Deriving the OLS estimatorChoose the line through a sample by minimizing the sum of squared residuals, derive β̂₁ = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² and β̂₀ = ȳ − β̂₁x̄, and compute them for a scatter of six workers.
   3. Fitted values, residuals and R-squaredCompute fitted values, residuals, SST, SSE, SSR and R² for simple OLS with an intercept; explain what R² measures and how changing units or taking logs affects its interpretation.
   4. Unbiasedness and variance of OLSProve that β̂₁ is unbiased under SLR.1 to SLR.4, derive Var(β̂₁) = σ² / SSTₓ under homoskedasticity, and compute the standard error of the regression and of β̂₁.
2. 2

   Multiple regression

   4 lessons

   1. Multiple regression and ceteris paribusFit and read a multiple regression wage = β̂₀ + β̂₁ educ + β̂₂ exper, interpret each coefficient as a partial effect holding the other regressor fixed, and see why it differs from the simple regression slope through partialling out.
   2. The Gauss-Markov assumptionsState assumptions MLR.1 to MLR.5, say which property each delivers (unbiasedness or minimum variance), and state the Gauss-Markov theorem that under all five OLS is the best linear unbiased estimator, BLUE.
   3. Omitted variable biasDerive the omitted variable bias formula E(β̃₁) = β₁ + β₂δ₁, read its sign from the sign of β₂ and the correlation of x₁ with the omitted x₂, and apply it to ability bias in the return to education.
   4. Multicollinearity and variance of OLSUse Var(β̂ⱼ) = σ² / (SSTⱼ(1 − Rⱼ²)), compute the variance inflation factor VIF = 1 / (1 − Rⱼ²), and weigh the cost of related regressors and irrelevant variables against the bias from omitting relevant ones.
3. 3

   Inference with OLS

   4 lessons

   1. The t test for a single coefficientTest H₀: βⱼ = 0 (or any hypothesized value) with t = (β̂ⱼ − a) / se(β̂ⱼ) against a critical value of the t distribution with n − k − 1 degrees of freedom, one-sided and two-sided, report a p-value, and judge economic as well as statistical significance.
   2. Confidence intervalsBuild the confidence interval β̂ⱼ ± c · se(β̂ⱼ) for a regression coefficient, interpret its confidence level correctly, use it to test H₀: βⱼ = a, and test a linear combination of parameters such as β₁ − β₂ by reparameterizing the model.
   3. The F test for joint restrictionsTest several exclusion restrictions at once with F = [(SSRᵣ − SSRᵤᵣ) / q] / [SSRᵤᵣ / (n − k − 1)], in its R² form, see why a joint test can reject when each t test does not, and test the overall significance of a regression.
   4. Large sample properties of OLSExplains consistency of OLS, the law of large numbers, asymptotic normality via the central limit theorem, and the LM test for large samples.
4. 4

   Functional form and dummy variables

   4 lessons

   1. Logarithmic functional formsInterprets coefficients in level-log, log-level and log-log models as semi-elasticities and elasticities, with percentage change approximations.
   2. Quadratics and interaction termsUses quadratic terms to capture diminishing effects with turning point -b1/(2b2), and interaction terms where one variable's effect depends on another.
   3. Dummy variablesInterprets binary regressors as intercept shifts, the dummy variable trap, multiple categories with a base group, and interactions of dummies for differing slopes.
   4. Testing differences across groupsUses the Chow test and fully interacted models to test whether regression functions differ across groups like men and women.
5. 5

   Heteroskedasticity and specification

   4 lessons

   1. Consequences of heteroskedasticityExplains that heteroskedasticity leaves OLS unbiased but invalidates usual standard errors, and introduces White heteroskedasticity-robust standard errors.
   2. Testing for heteroskedasticityPresents the Breusch-Pagan test and the White test, regressing squared residuals on regressors and computing LM or F statistics.
   3. Weighted least squaresUses weighted least squares and feasible GLS when the form of heteroskedasticity is known or estimated, comparing efficiency with OLS.
   4. Misspecification and measurement errorCovers the RESET test for functional form, proxy variables for omitted factors, and attenuation bias from classical measurement error in an explanatory variable.
6. 6

   Instrumental variables

   4 lessons

   1. Endogeneity and its sourcesDefines endogeneity as correlation between a regressor and the error, from omitted variables, measurement error and simultaneity, using supply and demand as an example.
   2. The IV estimatorDefines instrument relevance and exogeneity, derives the simple IV estimator b1 = cov(z, y) / cov(z, x), with Angrist and Krueger's quarter of birth example.
   3. Two stage least squaresExplains 2SLS with first and second stages, multiple instruments, weak instruments diagnosed by the first-stage F statistic, and tests for overidentification and endogeneity.
   4. IV and causal effectsInterprets IV with heterogeneous effects as a local average treatment effect for compliers, linking to randomized experiments with imperfect compliance.
7. 7

   Panel data methods

   3 lessons

   1. Pooled cross sectionsPools cross sections across time with year dummies to allow intercept shifts, and tests for structural changes over time.
   2. First differencingRemoves unobserved time-invariant effects a_i by first differencing two-period panel data, with assumptions and an example on crime and unemployment.
   3. Fixed effects estimationExplains the within (demeaning) fixed effects estimator, equivalence with dummy variable regression, clustered standard errors, and comparison with random effects and the Hausman test.
8. 8

   Policy evaluation designs

   4 lessons

   1. Potential outcomes and randomized experimentsIntroduces the Rubin potential outcomes framework, average treatment effects, selection bias, and why randomization identifies causal effects.
   2. Difference in differencesEstimates treatment effects with a 2x2 difference-in-differences table and regression with interaction term, with the parallel trends assumption and Card and Krueger's minimum wage study.
   3. Event studies and parallel trends checksExtends difference in differences to many periods with leads and lags (event study), testing pre-trends and noting issues with staggered treatment timing.
   4. Regression discontinuityExplains sharp regression discontinuity design using a cutoff in a running variable, local linear estimation and bandwidth choice, with class size or test score examples.
9. 9

   Time series regression

   3 lessons

   1. Static and distributed lag modelsIntroduces time series data, static models and finite distributed lag models, impact propensity and long-run propensity, and trends and seasonality.
   2. Stationarity and weak dependenceDefines covariance stationarity and weak dependence, AR(1) and MA(1) processes, and random walks with unit roots and spurious regression.
   3. Serial correlationExplains consequences of serial correlation for OLS, testing with Durbin-Watson and Breusch-Godfrey, and Newey-West HAC standard errors.
10. 10

    Limited dependent variable models

    3 lessons

    1. The linear probability modelModels a binary outcome with OLS, interprets coefficients as changes in probability, and notes predicted probabilities outside zero to one and built-in heteroskedasticity.
    2. Logit and probit modelsPresents logit and probit with latent variable motivation, maximum likelihood estimation, and the likelihood ratio test.
    3. Marginal effects and interpretationComputes partial effects at the average and average partial effects for logit and probit, and compares them with linear probability model estimates.