Lecture Notes

By the end of this lecture you should be able to: 1. define the conditional expectation function (CEF) and prove that it is the best predictor of \(Y\) given \(X\) in mean square; 2. define the best linear predictor (linear projection), derive its coefficients, and explain how it relates to the CEF; 3. derive the OLS estimator in matrix form and interpret it as the sample analogue of the projection coefficient; 4. state and prove the algebraic properties of the projection matrices \(P_X\) and \(M_X\), including the Frisch–Waugh–Lovell theorem; 5. prove unbiasedness of OLS and the Gauss–Markov theorem, and state exactly which assumptions each result uses.

By the end of this lecture you should be able to: 1. state the convergence concepts and limit theorems used in econometrics (WLLN, CLT, continuous mapping, Slutsky, delta method); 2. prove consistency and asymptotic normality of OLS for the linear projection model; 3. derive the exact finite-sample distribution of \(t\) and \(F\) statistics in the normal regression model; 4. construct and interpret asymptotic \(t\) tests, Wald tests, and confidence intervals, including for nonlinear functions of \(\beta\); 5. explain size, power, and the duality between tests and confidence sets.

By the end of this lecture you should be able to: 1. interpret coefficients in level, log and mixed specifications exactly, not only approximately; 2. compute partial effects, turning points and their standard errors in models with polynomials and interactions; 3. derive the omitted-variable bias formula in finite samples and in the limit, and sign the bias; 4. derive attenuation bias from classical measurement error, and explain why "bad controls" bias regression coefficients; 5. use specification tests (RESET) and sensitivity analysis responsibly.

By the end of this lecture you should be able to: 1. derive the heteroskedasticity-robust (HC) covariance estimators and explain the finite-sample corrections HC1–HC3; 2. test for heteroskedasticity and decide when weighted least squares is worth using; 3. derive the cluster-robust variance estimator and the Moulton factor, and explain what goes wrong with few clusters; 4. derive the long-run variance and the Newey–West HAC estimator for time-series regressions; 5. choose the level of clustering from the sampling and assignment design.

By the end of this lecture you should be able to: 1. classify sources of endogeneity and show why each makes OLS inconsistent; 2. derive the IV and 2SLS estimators, prove consistency and asymptotic normality, and compute correct standard errors; 3. diagnose weak instruments and use weak-instrument-robust inference (Anderson–Rubin); 4. test overidentifying restrictions and endogeneity, and state what each test can and cannot establish; 5. interpret IV estimates with heterogeneous effects as local average treatment effects (LATE).

By the end of this lecture you should be able to: 1. write a system of simultaneous equations in structural and reduced form, and derive simultaneity bias; 2. state and apply the order and rank conditions for identification of a single equation; 3. estimate an identified equation by 2SLS and LIML, and a full system by 3SLS, and compare their properties; 4. explain identification through covariance restrictions and heteroskedasticity; 5. connect the identification problem in simultaneous equations to structural VARs (Lecture 11).

By the end of this lecture you should be able to: 1. write the unobserved-effects model and explain what panel data can and cannot identify; 2. derive the fixed-effects (within) and first-difference estimators and state the strict-exogeneity assumption each needs; 3. derive the random-effects GLS estimator as quasi-demeaning and explain when it is consistent; 4. use the Mundlak (correlated random effects) device and a robust Hausman test to choose between FE and RE; 5. conduct inference with clustered standard errors, and recognize the limits of two-way fixed effects.

By the end of this lecture you should be able to: 1. derive the Nickell bias of the within estimator in a dynamic panel with short \(T\); 2. state the generalized method of moments (GMM) framework, the efficient weight matrix, and the \(J\) test; 3. derive the Anderson–Hsiao and Arellano–Bond (difference GMM) moment conditions; 4. derive the additional Blundell–Bond (system GMM) moment conditions and the initial-conditions assumption they require; 5. diagnose dynamic-panel GMM with AR(1)/AR(2) tests, Hansen tests, and instrument-count checks, and recognize instrument proliferation.

By the end of this lecture you should be able to: 1. define strict and weak stationarity, ergodicity and martingale differences, and explain why they replace i.i.d. sampling; 2. derive the moments, autocorrelation function and stationarity conditions of AR, MA and ARMA processes; 3. state the Wold decomposition and use ACF/PACF patterns and information criteria to choose a model; 4. estimate autoregressions by OLS and know when the usual asymptotics apply, including the inconsistency caused by a lagged dependent variable with serially correlated errors; 5. construct optimal forecasts, forecast intervals, and tests of relative forecast accuracy.

By the end of this lecture you should be able to: 1. distinguish trend-stationary from difference-stationary (integrated) processes, and explain why the distinction matters for inference and policy; 2. derive the Dickey–Fuller regression, state its non-standard limiting distribution, and choose the correct deterministic terms and critical values; 3. use ADF, DF-GLS and KPSS tests together, and understand their size and power trade-offs; 4. test for structural breaks with known and unknown break dates (Chow, Andrews sup-\(F\), Bai–Perron); 5. explain why ignored breaks bias unit-root tests toward non-rejection, and how break-robust tests respond.

By the end of this lecture you should be able to: 1. specify and estimate a reduced-form VAR, check stability, choose lag length, and test Granger non-causality; 2. compute impulse responses and forecast-error variance decompositions from the VAR's moving-average representation; 3. state the structural identification problem precisely and count the restrictions needed; 4. implement and critique recursive (Cholesky), long-run, sign-restriction and external-instrument (proxy) identification; 5. compare VAR impulse responses with local projections, and construct valid confidence bands.

By the end of this lecture you should be able to: 1. define cointegration and explain why a regression in levels of \(I(1)\) variables can be meaningful; 2. state the Granger representation theorem and write a vector error-correction model (VECM); 3. implement the Engle–Granger and Johansen procedures, with the correct critical values and deterministic specification; 4. conduct valid inference on cointegrating vectors (DOLS, FMOLS, Johansen MLE) and test weak exogeneity; 5. use the ARDL bounds test, and explain when it is and is not the right tool.

By the end of this lecture you should be able to: 1. define causal estimands (ATE, ATT, LATE, CATE) in the potential-outcomes framework, and decompose a naive comparison into effect and selection bias; 2. estimate effects under unconfoundedness by regression adjustment, inverse probability weighting and doubly robust methods, and check overlap; 3. set up difference-in-differences, test its assumptions with event studies, and explain why two-way fixed effects fails with staggered adoption and heterogeneous effects; 4. describe the synthetic control method and its inference; 5. estimate sharp and fuzzy regression discontinuity designs with local polynomials, optimal bandwidths and robust bias-corrected inference, and validate them.

By the end of this lecture you should be able to: 1. move from a research question to a precisely stated estimand, an identification argument and an estimator; 2. design, run and report a Monte Carlo study, including Monte Carlo standard errors; 3. use the bootstrap correctly (pairs, wild, block, cluster), and know when it fails; 4. control for multiple testing (Holm, Benjamini–Hochberg, Romano–Wolf) and report specification robustness transparently; 5. build a replication package that meets current journal standards.