Sources and References

Primary textbook — Wooldridge, Jeffrey M. (2025). Introductory Econometrics: A Modern Approach, 8th ed. Cengage Learning. (Chapters 10–15, 17.)

Supplementary texts — Angrist, J. D. & Pischke, J.-S. (2009). Mostly Harmless Econometrics. Princeton UP; Cameron, A. C. & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge UP.

Online resources — MIT OCW 14.32 (Econometrics); Wooldridge companion datasets (R package wooldridge); Nick Huntington-Klein, The Effect (free at theeffectbook.net).

Chapter 1: Review of OLS and Introduction to the Course

1.1 The Core Regression Setup

ECON 323 extends the OLS framework of ECON 322 to settings where the standard assumptions break down: data observed over time, panel structures, endogenous regressors, and non-continuous outcomes. A brief review of the multiple regression model sets the stage.

The population model is \( y = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k + u \), with OLS estimator:

Key properties: unbiasedness under MLR.1–MLR.4, efficiency (BLUE) under MLR.1–MLR.5, and asymptotic normality in large samples.

1.2 Endogeneity: The Central Problem

The zero conditional mean assumption \( E[u \mid \mathbf{x}] = 0 \) fails — making OLS biased and inconsistent — in three main situations:

1. Omitted variables: A variable correlated with both \( y \) and at least one included \( x \) is left out.
2. Measurement error in a regressor: Classical errors-in-variables attenuate the coefficient toward zero.
3. Simultaneity: \( y \) and \( x \) are jointly determined (supply and demand, for instance).

All three generate endogeneity: \( \text{Cov}(x_j, u) \neq 0 \). The response is to find instruments or exploit quasi-experimental variation.

Chapter 2: Time Series Regression

2.1 Basic Concepts

Time series data are observations on one unit (a country, firm, or price index) collected at equally-spaced intervals: \( \{y_t, x_{1t}, \ldots, x_{kt}\}_{t=1}^T \). The ordering of observations is informative, unlike cross-sectional data.

Under the time-series Gauss-Markov assumptions (TS.1–TS.5), OLS is unbiased and consistent for static time-series regression.

2.2 Finite Distributed Lag Models

A finite distributed lag (FDL) model relates \( y_t \) to current and lagged values of \( x_t \):

The impact propensity is \( \delta_0 \) (immediate effect of a unit change in \( x \)). The long-run propensity (LRP) is the cumulative effect once all dynamics have played out:

2.3 Serial Correlation

Serial correlation (autocorrelation) is the violation of the time-series assumption that errors are uncorrelated across periods: \( \text{Cov}(u_t, u_s) \neq 0 \) for \( t \neq s \). The most common form is first-order autocorrelation:

Consequences for OLS parallel heteroskedasticity: estimates remain unbiased and consistent (under strict exogeneity), but OLS is no longer efficient and conventional standard errors are invalid.

The Durbin-Watson statistic tests \( H_0: \rho = 0 \) against \( H_1: \rho > 0 \):

Values near 2 indicate no serial correlation; values near 0 (or 4) indicate positive (or negative) autocorrelation. The Breusch-Godfrey LM test is more general and allows for testing higher-order autocorrelation.

2.4 Serial-Correlation-Robust and HAC Standard Errors

The Newey-West HAC (Heteroskedasticity and Autocorrelation Consistent) estimator corrects OLS standard errors for both heteroskedasticity and serial correlation:

where \( \hat{\boldsymbol{\Omega}}_{NW} \) is the Newey-West kernel estimator with bandwidth \( M \) (often chosen as \( \lfloor 4(T/100)^{2/9} \rfloor \)). HAC standard errors are the time-series analog of Eicker-Huber-White robust standard errors.

2.5 Spurious Regression and Trending Variables

When two variables share a deterministic trend, OLS may find a statistically significant relationship between them even though they are causally unrelated — spurious regression. Including a time trend \( t = 1, 2, \ldots, T \) controls for common deterministic trends.

More seriously, stochastic trends (unit roots) can generate spurious regressions even after detrending. The correct treatment depends on whether the series are cointegrated (covered in ECON 423).

Chapter 3: Panel Data Methods

3.1 Panel Data Structures

Panel data (longitudinal data) follow \( N \) units (individuals, firms, countries) over \( T \) time periods, yielding \( NT \) observations \( \{y_{it}, \mathbf{x}_{it}\}_{i=1,t=1}^{N,T} \). Panels are balanced if every unit has all \( T \) observations.

The general panel regression model is:

where \( a_i \) is a unit-specific unobserved effect (or fixed effect), constant over time.

3.2 Pooled OLS

Simply stacking all \( NT \) observations and running OLS (ignoring \( a_i \)) is pooled OLS. If \( \text{Cov}(a_i, \mathbf{x}_{it}) = 0 \) and \( E[u_{it} \mid \mathbf{x}_{i1}, \ldots, \mathbf{x}_{iT}, a_i] = 0 \), pooled OLS is consistent but the standard errors must be clustered at the unit level to account for within-unit serial correlation induced by \( a_i \).

3.3 The Fixed Effects Estimator

When \( a_i \) is arbitrarily correlated with \( \mathbf{x}_{it} \) (the usual concern in economics), the Fixed Effects (FE) or within estimator eliminates \( a_i \) by demeaning:

where \( \bar{y}_i = T^{-1}\sum_t y_{it} \). OLS on this demeaned equation is the FE estimator. Because \( a_i \) has been differenced out, FE is consistent even when \( \text{Cov}(a_i, \mathbf{x}_{it}) \neq 0 \).

Limitation: Because FE differences out all within-unit variation, it cannot identify the effect of any time-invariant variable (e.g., sex, race, country characteristics). Such variables are perfectly collinear with the unit dummies.

3.4 First Differences

An alternative elimination strategy is first differencing: subtract the previous period from the current period:

With \( T = 2 \), FE and FD are identical. With \( T > 2 \), FD is more efficient when \( \Delta u_{it} \) is serially uncorrelated (i.e., \( u_{it} \) follows a random walk), while FE is more efficient when \( u_{it} \) is serially uncorrelated.

3.5 Random Effects

If \( \text{Cov}(a_i, \mathbf{x}_{it}) = 0 \), the Random Effects (RE) estimator exploits both within- and between-unit variation, yielding more efficient estimates than FE. RE is a weighted average of the within estimator (FE) and the between estimator (cross-sectional OLS on group means). The RE estimator uses a GLS transformation with weight \( \lambda = 1 - \sqrt{\sigma_u^2 / (\sigma_u^2 + T\sigma_a^2)} \).

Hausman test: Tests \( H_0: \text{Cov}(a_i, \mathbf{x}_{it}) = 0 \) by comparing RE and FE:

Rejection of \( H_0 \) favors FE.

Chapter 4: Instrumental Variables and Two-Stage Least Squares

4.1 Endogeneity and the Need for Instruments

Recall that if \( \text{Cov}(x_1, u) \neq 0 \) (endogeneity), OLS is inconsistent. An instrumental variable (IV) \( z \) satisfies two conditions:

The exclusion restriction is an identifying assumption that cannot be directly tested (since \( u \) is unobservable). Economic theory and institutional knowledge must justify it.

4.2 The IV Estimator

For the simple model \( y = \beta_0 + \beta_1 x_1 + u \) with one instrument \( z \):

Consistency: \( \text{plim}(\hat{\beta}_1^{IV}) = \beta_1 + \text{Cov}(z,u)/\text{Cov}(z,x_1) = \beta_1 \) when \( \text{Cov}(z,u) = 0 \).

Under instrument relevance and exogeneity, the IV estimator is consistent and asymptotically normal, though less efficient than OLS when OLS is consistent. The asymptotic variance of IV exceeds that of OLS by a factor proportional to \( 1/\rho_{zx}^2 \).

4.3 Two-Stage Least Squares (2SLS)

With multiple endogenous regressors and multiple instruments, Two-Stage Least Squares (2SLS) is the standard approach.

Obtain fitted values \( \hat{x}_j \).

Stage 2: Replace endogenous \( x_j \) with \( \hat{x}_j \) in the structural equation and run OLS.

The 2SLS estimator has the closed-form:

where \( \hat{\mathbf{X}} \) contains the first-stage fitted values. Standard errors must be computed using the original \( \mathbf{X} \), not \( \hat{\mathbf{X}} \), to be valid.

Order condition for identification: Number of instruments \( \geq \) number of endogenous variables. If equal, the equation is exactly identified (unique IV); if greater, overidentified (2SLS uses all instruments efficiently).

4.4 Testing Instruments

Weak Instruments: If \( \rho_{zx}^2 \) is small, the first-stage F-statistic will be low. A common rule of thumb: first-stage \( F < 10 \) signals weak instruments, causing IV estimates to be nearly as biased as OLS and inference to be unreliable. Use Stock-Yogo critical values for formal testing.

Overidentification Test (Sargan-Hansen J-test): When there are more instruments than endogenous variables, the excess moment conditions can be tested. Under the joint null that all instruments are exogenous:

where the \( R^2 \) is from regressing 2SLS residuals on all instruments and exogenous regressors. Rejection casts doubt on at least one instrument’s exogeneity.

Chapter 5: Limited Dependent Variable Models

5.1 Binary Outcomes and Maximum Likelihood Estimation

When the dependent variable is binary (\( y \in \{0,1\} \)), the linear probability model provides a starting point (Chapter 6 of ECON 322 notes), but its predictions can exceed \([0,1]\). Probit and Logit models impose proper probability bounds via a link function.

Maximum Likelihood Estimation (MLE) finds the parameter vector that maximizes the likelihood of observing the data. For a binary model with \( P(y_i=1\mid\mathbf{x}_i) = G(\mathbf{x}_i^{\top}\boldsymbol{\beta}) \):

MLE is found by maximizing \( \mathcal{L}(\boldsymbol{\beta}) \) numerically (no closed form).

5.2 Interpreting Probit and Logit Coefficients

Unlike OLS, probit/logit coefficients do not directly measure marginal effects. The marginal effect at the mean (MEM) for a continuous regressor \( x_j \) is:

where \( g = G' \) is the density of the link function (standard normal density for probit; \( \Lambda(1-\Lambda) \) for logit). This must be evaluated at specific values of \( \mathbf{x} \) (e.g., sample means). The average marginal effect (AME) averages across all observations:

The AME is generally preferred over the MEM as it better represents the population average.

For a dummy variable \( x_j \in \{0,1\} \), the discrete change in probability is:

5.3 Model Fit for Binary Models

Standard \( R^2 \) is not meaningful for binary models. Alternative measures:

• McFadden’s pseudo-\( R^2 \): \( 1 - \mathcal{L}_{ur}/\mathcal{L}_0 \), where \( \mathcal{L}_0 \) is the log-likelihood of the intercept-only model.
• Percent correctly predicted (comparing \( \hat{P}(y=1) \) to a threshold of 0.5).
• Log-likelihood ratio test (LR test): \( LR = -2(\mathcal{L}_r - \mathcal{L}_{ur}) \xrightarrow{d} \chi^2(q) \) under \( H_0 \) of \( q \) restrictions.

5.4 Multinomial and Ordered Models

When \( y \) has more than two unordered categories (e.g., transportation mode: car/bus/train), the multinomial logit specifies:

with \( \boldsymbol{\beta}_0 = \mathbf{0} \) (base category normalization).

When \( y \) is ordered (e.g., satisfaction scores 1–5), the ordered probit/logit is appropriate. It assumes a single latent variable \( y^* = \mathbf{x}^{\top}\boldsymbol{\beta} + u \) with cutpoints \( \mu_1 < \mu_2 < \cdots < \mu_{J-1} \) mapping \( y^* \) to the observed \( y \).

5.5 Count Data: Poisson Regression

For non-negative integer outcomes (number of patents, doctor visits, etc.), the Poisson regression model specifies:

The log-link ensures \( \lambda > 0 \). The coefficient \( \beta_j \) is the log incidence rate ratio: a unit increase in \( x_j \) multiplies the expected count by \( e^{\beta_j} \). The log-likelihood is:

Overdispersion (\( \text{Var}(y) > E[y] \)) is common in count data and violates the Poisson assumption. The negative binomial model addresses this by adding a gamma-distributed individual heterogeneity term.

Chapter 6: Sample Selection Models

6.1 The Selection Problem

A sample selection problem arises when the sample is not a random draw from the population of interest, but rather a draw from a selected subgroup. For example, wages are only observed for individuals who work; test scores are only observed for students who enroll.

The fundamental issue: \( E[u_i \mid s_i = 1, \mathbf{x}_i] = \rho\sigma_u \lambda(\mathbf{z}_i^{\top}\hat{\boldsymbol{\gamma}}) \neq 0 \) unless \( \rho = 0 \), where \( \lambda(c) = \phi(c)/\Phi(c) \) is the inverse Mills ratio.

6.2 Heckit Estimation

The two-step (Heckit) procedure:

1. Step 1: Estimate the selection probit using all observations. Compute \( \hat{\lambda}_i = \phi(\mathbf{z}_i^{\top}\hat{\boldsymbol{\gamma}})/\Phi(\mathbf{z}_i^{\top}\hat{\boldsymbol{\gamma}}) \).
2. Step 2: Add \( \hat{\lambda}_i \) as a regressor in the outcome equation and estimate by OLS on the selected sample.

The coefficient on \( \hat{\lambda}_i \) estimates \( \rho\sigma_u \). A significant coefficient indicates selection bias in the naive OLS. Standard errors from step 2 must be corrected (or bootstrapped) because \( \hat{\lambda}_i \) uses estimated parameters.

Exclusion restriction for Heckit: The model is technically identified by functional form, but it is much more credible when \( \mathbf{z}_i \) contains at least one variable affecting selection but not the outcome — analogous to an instrument in IV estimation.

Chapter 7: Implementation in R

7.1 Time Series and Panel Data

library(plm)
# Fixed effects
fe_model <- plm(log(wage) ~ exper + married, data = panel_data,
                index = c("id","year"), model = "within")
summary(fe_model)

# Clustered standard errors
library(lmtest)
coeftest(fe_model, vcov = vcovHC(fe_model, type = "HC1", cluster = "group"))

# Hausman test
re_model <- plm(log(wage) ~ exper + married, data = panel_data,
                index = c("id","year"), model = "random")
phtest(fe_model, re_model)

7.2 Instrumental Variables

library(AER)
# 2SLS: endogenous = educ, instrument = nearc4 (proximity to college)
iv_model <- ivreg(log(wage) ~ educ + exper | nearc4 + exper,
                  data = card_data)
summary(iv_model, diagnostics = TRUE)
# diagnostics = TRUE reports Wu-Hausman test, weak instruments F, and Sargan J-test

7.3 Binary and Count Models

# Probit
probit_model <- glm(inlf ~ nwifeinc + educ + exper + kidslt6,
                    family = binomial(link = "probit"), data = mroz)
# Average marginal effects
library(margins)
summary(margins(probit_model))

# Poisson regression
pois_model <- glm(patents ~ R_D + sales, family = poisson, data = rdchem)
# Incidence rate ratios
exp(coef(pois_model))