Econometrics Study Guide

📖 Core Concepts Econometrics – uses statistical tools on economic data to give empirical meaning to theory. Multiple Linear Regression – the work‑horse model: \(y = \alpha + \beta1 x1 + \dots + \betak xk + \varepsilon\). Ordinary Least Squares (OLS) – estimates coefficients by minimizing \(\sum (yi-\hat yi)^2\). Estimator Properties – Unbiased (E[ \(\hat\theta\) ] = true \(\theta\)), Consistent (converges as \(n\to\infty\)), Efficient (smallest variance among unbiased linear estimators). Gauss‑Markov Theorem – under its assumptions, OLS is the Best Linear Unbiased Estimator (BLUE). Identification – a model is identified when the data provide enough information to recover the structural parameters (e.g., supply‑demand, IV, RD, DiD). Quasi‑experimental Designs – techniques that mimic experiments using observational data (instrumental variables, regression discontinuity, difference‑in‑differences). --- 📌 Must Remember OLS unbiased ⇔ error term \(\varepsilon\) is uncorrelated with every regressor. Omitted Variable Bias occurs when a left‑out variable influences both \(y\) and an included regressor. BLUE holds only if the Gauss‑Markov assumptions (linearity, zero‑mean errors, homoskedasticity, no autocorrelation, exogeneity) are satisfied. When assumptions fail → use MLE, GLS, GMM, or Bayesian methods. Instrumental Variable (IV) must satisfy: (i) relevance (correlated with endogenous regressor) and (ii) exclusion (uncorrelated with the error term). Regression Discontinuity (RD) exploits a known cutoff to create quasi‑random assignment. Difference‑in‑Differences (DiD) isolates causal impact by comparing pre‑/post changes between treated and control groups. --- 🔄 Key Processes OLS Estimation Write the regression equation. Compute residuals \(ei = yi - \hat yi\). Minimize \(\sum ei^2\) → solve normal equations \((X'X)\hat\beta = X'y\). Testing for Omitted Variable Bias Identify suspect omitted variable. Check correlation with included regressor(s). Add the variable (or a proxy) and re‑estimate; compare \(\hat\beta\) changes. IV Estimation (Two‑Stage Least Squares) First stage: regress endogenous regressor on instrument(s) → obtain fitted values \(\hat x\). Second stage: regress \(y\) on \(\hat x\) (and other exogenous controls). Difference‑in‑Differences Compute average outcome for treatment and control pre‑policy. Compute average outcome for treatment and control post‑policy. DiD estimator = (PostTreat – PreTreat) – (PostControl – PreControl). --- 🔍 Key Comparisons OLS vs. IV – OLS requires exogeneity; IV relaxes this by using a valid instrument. MLE vs. OLS – OLS is a special case of MLE when errors are normal, homoskedastic, and independent. GLS vs. OLS – GLS corrects heteroskedasticity or autocorrelation; OLS does not. Frequentist vs. Bayesian Estimation – Frequentist treats parameters as fixed; Bayesian treats them as random with prior distributions. --- ⚠️ Common Misunderstandings Statistical significance ≠ economic importance – a tiny coefficient can be significant in large samples but have negligible real‑world effect. P‑value = proof of causality – significance only reflects correlation; causal inference needs a credible identification strategy. “More covariates always improve the model” – adding irrelevant variables inflates variance; adding endogenous covariates introduces bias. Assuming OLS is unbiased in the presence of measurement error – classical measurement error in regressors creates attenuation bias. --- 🧠 Mental Models / Intuition Error‑term orthogonality – picture the error term as “noise” that must be perpendicular to every regressor for OLS to hit the true slope. Instrument as a “clean pipe” – the instrument carries variation from the endogenous regressor without leaking any of the error term’s “dirty” influence. Difference‑in‑Differences as “double‑differencing” – it removes both time‑invariant group effects and common shocks, leaving the treatment effect. --- 🚩 Exceptions & Edge Cases Heteroskedastic errors → OLS still unbiased but not efficient; use Robust SEs or GLS. Autocorrelated errors (time series) → OLS standard errors are biased; apply Newey‑West or Cochrane‑Orcutt corrections. Simultaneous equations – OLS on each equation yields inconsistent estimates; need Two‑Stage Least Squares or Full‑Information Maximum Likelihood. Small sample size – finite‑sample bias can make OLS less reliable; consider exact inference or Bayesian shrinkage. --- 📍 When to Use Which Start with OLS if you can plausibly argue exogeneity and homoskedasticity. Switch to IV when a regressor is endogenous (e.g., education ↔ ability). Apply RD when a policy rule creates a sharp cutoff (e.g., test score threshold). Use DiD when you have pre‑ and post‑treatment data for treated and control groups. Choose MLE / GMM for non‑linear models, limited‑dependent variables, or when distributional assumptions differ from normal. Adopt Bayesian when prior information is strong or you need full posterior distributions. --- 👀 Patterns to Recognize “Error term correlated with X” → think omitted variable bias or simultaneity → candidate for IV. Sharp change at a threshold → likely a regression discontinuity opportunity. Parallel trends in pre‑periods → a good sign for DiD validity. Large R‑squared but insignificant coefficients → possible multicollinearity or over‑fitting. Changing sign of coefficient after adding a covariate → classic sign of omitted variable bias. --- 🗂️ Exam Traps Choosing OLS when the regressor is endogenous – the answer will look clean but violates exogeneity; the correct choice is IV or another identification strategy. Treating a statistically significant coefficient as “large” – exam may ask about economic magnitude; remember to examine the coefficient’s units (e.g., log‑wage effect). Confusing homoskedasticity with no autocorrelation – both are separate Gauss‑Markov assumptions; a violation of either invalidates BLUE. Selecting a “natural experiment” without checking exclusion restriction – the instrument must not affect the outcome except through the treatment. Assuming “more data = better estimates” – if data are badly specified (e.g., spurious correlation), increasing \(n\) only shrinks bias‑induced variance, not the bias itself.