Luke Stein Stanford graduate economics core (2006{7) Reference and review (last updated April 11, 2012) Contents Standard deviation Conditional homoscedasticity 1.11 Decision theory . 14 Covariance Regression model Loss function 1 Econometrics .............. 4 Multivariate covariance Linear regression model with non- Decision rule 1.1 Probability foundations . 4 Correlation stochastic covariates Risk function Mean squared error risk function Basic set theory Skewness Seemingly Unrelated Regressions Disjointness Kurtosis Probit model 1.12 Hypothesis testing . 14 Partition Moment generating function Nonlinear least squares Hypothesis testing Sigma algebra Characteristic function Linear instrumental variables Test statistic Probability function Other generating functions 1.9 Statistics . 11 Type I/Type II error Probability space 1.5 Distributions . 7 Sufficient statistic Power Counting Normal distribution Factorization theorem Size Conditional probability Bivariate normal distribution Minimal sufficient statistic Level Bayes' Rule Multivariate normal distribution Likelihood function Distance function principle Independence of events Chi squared distribution Ancillary statistic p-value Mutual independence of events Student's t distribution Complete statistic Uniformly most powerful test Simple likelihood ratio statistic 1.2 Random variables . 5 Snedecor's F distribution Basu's theorem Neyman-Pearson Lemma Random variable Lognormal distribution Statistics in exponential families Exponential families Monotone likelihood ratio models Random vector 1.10 Point estimation . 11 Unbiased test Measurability Location and Scale families Estimator Stable distribution Uniformly most powerful unbiased Smallest σ-field Extremum estimator test Independence of r.v.s 1.6 Random samples . 9 Analogy principle Likelihood ratio test Independence of random vectors Random sample, iid Consistent estimator Wald statistic Mean independence Statistic Consistency with compact parameter Lagrange multiplier statistic Cumulative distribution function Unbiased estimator space Asymptotic significance Probability mass function Sample mean Consistency without compact param- Asymptotic size Marginal pmf Sample variance eter space Consistent test Conditional pmf Sample covariance Uniform (Weak) Law of Large Num- Probability density function Sample correlation bers 1.13 Time-series concepts . 15 Marginal pdf Order statistic Asymptotically normal estimator Stationarity Conditional pdf Samples from the normal distribution Asymptotic variance Covariance stationarity Borel Paradox Asymptotically efficient estimator White noise 1.7 Convergence of random variables . 9 Stochastic ordering Maximum Likelihood estimator Ergodicity Convergence in probability Support set Consistency for for MLE Martingale Uniform convergence in probability Asymptotic normality for MLE Random walk 1.3 Transformations of random variables . 6 Little o error notation M-estimator Martingale difference sequence Transformation 1 ! 1 Big O error notation R R Asymptotic normality for M- Ergodic stationary martingale differ- Transformation 2 ! 2 Order symbols R R estimator ences CLT Convolution formulae Asymptotic equivalence Method of Moments estimator Lag operator Probability integral transformation Convergence in L p Generalized Method of Moments esti- Moving average process Almost sure convergence 1.4 Properties of random variables . 6 mator Autoregressive process of degree one Convergence in distribution Expected value, mean Consistency for GMM estimators Autoregressive process of degree p Central Limit Theorem for iid samples Conditional expectation Asymptotic normality for GMM esti- ARMA process of degree (p; q) Central Limit Theorem for niid sam- Two-way rule for expectations mator Autoregressive conditional het- ples Law of Iterated Expectations Efficient GMM eroscedastic process Slutsky's Theorem Median Minimum Distance estimator GARCH process Delta Method Mode Asymptotic normality for MD estima- Local level model Symmetric distribution 1.8 Parametric models . 10 tor Estimating number of lags Moment Parametric model Uniformly minimum variance unbi- Estimating AR(p) Variance Parameter ased estimator Maximum likelihood with serial corre- lation Multivariate variance Identification Fisher Information Conditional variance Identification in exponential families Cram´er-RaoInequality 1.14 Ordinary least squares . 17 1 Ordinary least squares model Maximum likelihood for SUR Shutdown Rationalization: h and differentiable Ordinary least squares estimators Limited Information Maximum Like- Nonincreasing returns to scale e Asymptotic properties of OLS estima- lihood Nondecreasing returns to scale Rationalization: differentiable h tor Rationalization: differentiable x 1.18 Unit root processes . 24 Constant returns to scale Rationalization: differentiable e Estimating S Integrated process of order 0 Convexity Biases affecting OLS Integrated process of order d Transformation function Slutsky equation Maximum likelihood estimator for Unit root process Marginal rate of transformation Normal good OLS model Brownian motion Production function Inferior good Best linear predictor Function Central Limit Theorem Marginal rate of technological substi- Regular good Fitted value Dickey-Fuller Test tution Giffen good Projection matrix Profit maximization Substitute goods Annihilator matrix 1.19 Limited dependent variable models . 24 Rationalization: profit maximization Complement goods Standard error of the regression Binary response model functions Gross substitute Sampling error Probit model Loss function Gross complement OLS R2 Logit model Hotelling's Lemma Engle curve Standard error Truncated response model Substitution matrix Offer curve Robust standard error Tobit model Law of Supply Roy's identity OLS t test Sample selection model Rationalization: y(·) and differen- Consumer welfare: price changes OLS robust t test 1.20 Inequalities . 25 tiable π(·) Compensating variation OLS F test Bonferroni's Inequality Rationalization: differentiable y(·) Equivalent variation OLS robust Wald statistic Boole's Inequality Rationalization: differentiable π(·) Marshallian consumer surplus Generalized least squares Chebychev's Inequality Rationalization: general y(·) and π(·) Price index Feasible generalized least squares Markov's Inequality Producer Surplus Formula Aggregating consumer demand Weighted least squares Numerical inequality lemma Single-output case 2.5 Choice under uncertainty . 32 Frisch-Waugh Theorem H¨older'sInequality Cost minimization Lottery Short and long regressions Cauchy-Schwarz Inequality Rationalization: single-output cost Preference axioms under uncertainty 1.15 Linear Generalized Method of Moments 19 Minkowski's Inequality function Expected utility function Linear GMM model Jensen's Inequality Monopoly pricing Bernoulli utility function Instrumental variables estimator 1.21 Thoughts on MaCurdy questions . 25 2.3 Comparative statics . 30 von Neumann-Morgenstern utility Linear GMM estimator Implicit function theorem function GMM hypothesis testing 2 Microeconomics ............ 27 Envelope theorem Risk aversion Efficient GMM Certain equivalent 2.1 Choice Theory . 27 Envelope theorem (integral form) Testing overidentifying restrictions Absolute risk aversion Rational preference relation Increasing differences Two-Stage Least Squares Certain equivalent rate of return Choice rule Supermodularity Durbin-Wu-Hausman test Relative risk aversion Revealed preference Submodularity First-order stochastic dominance 1.16 Linear multiple equation GMM . 21 Houthaker's Axiom of Revealed Pref- Topkis' Theorem Second-order stochastic dominance Linear multiple-equation GMM erences Monotone Selection Theorem Demand for insurance model Weak Axiom of Revealed Preference Milgrom-Shannon Monotonicity The- Portfolio problem Linear multiple-equation GMM esti- Generalized Axiom of Revealed Pref- orem Subjective probabilities mator erence MCS: robustness to objective function Full-Information Instrumental Vari- Utility function perturbation 2.6 General equilibrium . 33 ables Efficient Interpersonal comparison Complement inputs Walrasian model Three-Stage Least Squares Continuous preference Substitute inputs Walrasian equilibrium Multiple equation Two-Stage Least Monotone preference LeChatelier principle Feasible allocation Squares Locally non-satiated preference 2.4 Consumer theory . 31 Pareto optimality Seemingly Unrelated Regressions Convex preference Budget set Edgeworth box Multiple-equation GMM with com- Homothetic preference Utility maximization problem First Welfare Theorem mon coefficients Separable preferences Indirect utility function Second Welfare Theorem Random effects estimator Quasi-linear preferences Marshallian demand correspondence Excess demand Pooled OLS Lexicographic preferences Walras' Law Sonnenschein-Mantel-Debreu Theo- rem 1.17 ML for multiple-equation linear models 23 2.2 Producer theory . 28 Expenditure minimization problem Gross substitutes property Structural form Competitive producer behavior Expenditure function Incomplete markets Reduced form Production plan Hicksian demand correspondence Rational expectations equilibrium Full Information Maximum Likeli- Production set Relating Marshallian and Hicksian de- hood Free disposal mand 2.7 Games . 34 2 Game tree Feasible payoff Imperfect competition: final goods One-sided lack of commitment Perfect recall Individually rational payoff Imperfect competition: intermediate Two-sided lack of commitment Perfect information Folk Theorem goods Bulow-Rogoff defaultable debt model Complete information