Political Economy Group – Stanford GSB
Econometrics Qualifying Exam Review Sheet
August 2012
Relevant textbooks: for probability and statistics, DeGroot and Schervish or Casella and
Berger; for linear models and time series, Amemiya, Hayashi, Greene, Wooldridge,
Hamilton or Harvey; and for non-linear models, Hayashi, Wooldridge, or Davidson and
Mackinnon.
Probability Theory
1. Basic axioms of probability, counting methods (permutations, combinations, sampling with replacement, multinomial coefficients)
2. Conditional probability, independence, Bayes Theorem, Markov Chains (stochastic processes, transition probability matrix, state occupancy probabilities)
Random Variables and Their Distributions
1. Discrete random variables (Bernoulli trials, binomial distribution, geometric distribution, negative binomial distribution)
2. Continuous random variables (uniform distribution, normal distribution)
3. Distribution functions and probability density functions
4. Truncation, the Quantile Function
5. Multivariate distributions (deriving marginal, joint and conditional distributions; independence)
6. Functions of random variables (the probability integral transformation, maxima and minima)
Estimation: Regression
Basic Linear regression models:
1. Gauss Markov Theorem, exact properties of least squares.
2. Rao-Blackwell and ML estimation
More details:
A. Conditional expectation function, best linear predictor, mean squared error, law of iterated expectations.
B. Classical linear model (CLM): Least squares solution (b s
2
OLS =(X’X) -1 X’y),
=e’e/(n-k) unbiased estimate of  2
, proof of this. Gauss-Markov theorem and its proof (finite sample properties of least squares: BLUE), partitioned regression: b
2
=(X
2
’M
1
X
2
)
-1
X
2
’M
1 y

C. Normal linear model: Maximum likelihood estimates for

and
 2 sample properties of ML (normality, unbiasedness of b
ML
, finite
, efficiency), equivalence of b
ML
and b
OLS
. Information matrix equality, score function.
D. Asymptotics tools: Convergence in probability, in mean squared, in distribution. Multivariate law of large numbers, multivariate central limit theorem.
Definitions of consistency, asymptotic normality and asymptotic efficiency. Delta method.
E. Asymptotic properties of least squares under the assumptions of the classical linear model: Consistency, asymptotic normality (CAN). Proofs of consistency
(LLN) and asymptotic normality (CLT).
F. Asymptotic properties of maximum likelihood in the general case: Consistency, asymptotic normality, asymptotic efficiency (Cramer-Rao lower bound), invariance property.
Statistical Inference
1. Deriving estimates of σ² and of the disturbance covariance matrix.
2. Deriving the t and F statistics under Normality: Standardization and Studentization.
4. Wald, LR and LM Statistics: distribution in the asymptotic case is χ²
5. Asymptotics: CLT, LLN, Slutsky theorem, asymptotic properties of least squares
(consistency, asymptotic normality, efficiency).
GMM, GLS, IV, 2SLS and Simultaneous Equations Models
1.
GLS: Nonspherical disturbances: heteroskedasticity and autocorrelation. Properties of
LS under these assumptions (wrong standard errors), White and Newey-West covariance matrices, GLS and FGLS (including the formulas for the covariance matrix under GLS)
More details on GLS:
A. Assumptions of the generalized linear model (GLM). Heteroskedasticity and autocorrelation. Testing for heteroskedasticity and autocorrelation: White’s test,
Breusch-Pagan test, [ Durbin-Watson test ]
B. Least squares estimation under GLM: Unbiasedness retained, OLS covariance matrix
 2 (X’X) -1
is wrong. Correct OLS covariance matrix is: V(b
OLS )=(X’X) -1 X’ 
X(X’X) -1
; how to rescue OLS estimates: Estimating

(n*n) is hopeless, but estimating X’ 
X (k*k) is not (White’s insight). White-robust standard errors
(heteroskedasticity), Newey-West correction (autocorrelation). Note: Even if we have the right covariance matrix, OLS is not BLUE under GLM. It is inefficient even asymptotically.
C. Generalized least squares: if P’P=  -1
, we can write: Py=PX

+P

. This model satisfies CLM (proof of this). The least squares estimate of

in the transformed model is GLS: b
GLS =(X’  -1
X)
-1 X’  -1 y. All properties of least squares (BLUE,

inference) apply to the transformed model. Problem: GLS is not feasible unless we know

.
D. Feasible generalized least squares: if we can obtain get nice asymptotic properties for FGLS: b
FGLS 
(consistent for

), we can
( '
 
1
X )

1
X '

1 y , namely consistency, asymptotic normality, asymptotic efficency. How do we get ? We need to assume a structure for

and estimate whatever parameter the structure involves (hopefully a small number of them).
2.
Simultaneous equations models: SUR and the Multivariate Regression Model as applications of GLS
3.
IV and 2SLS: Endogeneity bias, omitted variables bias. Finding suitable instruments
(properties of good instruments)
4.
GMM: Formulating moment conditions based on orthogonality properties/ assumptions; solving the sample analogs of population moment conditions; just identified case (MM) and overidentified case (GMM) (analogy between IV (=>MM) and 2SLS (=>GMM))
Limited Dependent Variables
1.
Linear Probability Model.
2.
Probit: Non-linear model describing binary choice; latent variables formulation and how it relates to the probability model.
3.
Logit: Different assumption for the error term distribution, otherwise the same as probit.
4.
Tobit: Censored data.
5.
Heckman’s two step procedure.
Panel Data
1.
Pooled least squares.
2.
Decomposing the OLS coefficient into the between and within variation.
3.
Individual specific effects and omitted variables bias (ability/wage/schooling example).
4.
The fixed effects/within estimator: Differencing and dummy variables approaches to get rid of all of the between variation (which causes bias).
5.
The random effects estimator: Using some of the between variation when the fixed effects are not correlated with the right hand side variables.

Time Series
Stationary Time Series
1. Definitions of Stationarity, AR, MA, ARMA, autocorrelation function (ACF), partial autocorrelation function (PACF) and the Box-Jenkins methodology for analyzing stationary time series, distributed lags.
2. The Wold Representation Theorem for a stationary time series, how to estimate AR
(Mann-Wald theorem), MA and ARMA models, VARs.
Nonstationary Time Series
1. Unit Roots and Integration: Random walks, testing for unit roots using DF and ADF tests, persistence and why asymptotic properties of usual estimators break down in the case of nonstationarity, ARIMA models
2. Cointegration: Spurious regression, cointegrating vector, testing for cointegration (test for the stationarity of the residuals - there are various ways to proceed)