Causality

• Causality is when one cause leads to some effect.
• The cause is partly responsible for the effect, and the effect partly depends on the cause.
• A causal relationship tells us what would happen in alternative (counterfactual) worlds.
• Consider a binary treatment X, and outcome Y. We can think of the causal effect \(\tau\) as the difference in potential outcomes: \[ \tau = Y(X=1) - Y(X=0) \]
• In reality, only one outcome is realized, the other is counterfactual. Thus, we have to estimate this missing outcome to learn about the causal effect.
• Average Treatment Effect: the average causal effect is simply the mean of all treatment effects \[ \tau_{ATE} = \mathrm{E}[\tau_i] = \mathrm{E}[Y(1)-Y(0)]= \mathrm{E}[Y(1)]- \mathrm{E}[Y(0)] \]

Identification

• We say an effect is causally identified if we can interpret it causally in our framework and scope.
• If we want to understand the causal impact of studying on income, \(\boldsymbol{y}^{inc} = \boldsymbol{x}^{stu} \beta + \boldsymbol{u}\): Studying → Income
• We likely run into an issue, as you don’t get paid for studying but for your skills: Studying → Skills → Income
• Moreover, ability may confound your effect estimates of studying on income, \(\boldsymbol{y}^{inc} = \boldsymbol{x}^{stu} \beta + \boldsymbol{u}\), affecting studying, skills, and income.
• You cannot identify the causal effect of studying on income.

Ways of Viewing Causal Questions

The Potential Outcomes (PO) framework is one way to view causal questions.

• There is a treatment \(X_i\) that takes on different values for each unit.
• For each possible level of treatment, there is a certain potential outcome \(Y(x)\).
• Only one potential outcome is observed, the others are counterfactuals.

The potential outcomes framework relates very clearly to the notion of a randomized experiment.

A different framework that has its strengths elsewhere: the Directed Acyclic Graphs (DAG) framework.

• It is a graphical framework that helps us identify a causal effect in a network of variables.
• It has its strengths in a world with a large number of (observed) variables, and
• may help people who prefer thinking graphically to understand causal questions.

DAGs for Causal Modeling

Why do we talk about DAGs in an Econometrics class? Because they are really useful for causal modeling.

In the following DAG, nodes represent (random) variables, and edges represent (hypothesized) causal effects.

Missing edges also convey information: the assumption of no causal effect.

Validity

• In order to assess the quality of causal inferences, it helps to think of the validity of a statistical analysis. Different concepts of validity include the following:
• External validity: is the validity of an analysis outside its own context, telling us whether findings can be generalized across situations, people, time, regions etc.
• Internal validity: Internal validity is the validity of an analysis within its own context. It is the extent to which the analysis allows for causal inference.
• Empirical evidence may support various different interpretations.
• We want to be able to credibly eliminate non‐causal interpretations.
• The Principle of Parsimony (Occam’s razor): There may be incomprehensibly many alternatives for each explanation. The idea is to give preference to the simplest explanation (that cannot be refuted), i.e., the one with the fewest parameters and/or assumptions.

Exogeneity

• The exogeneity assumption \(\mathbb{E}[\boldsymbol{u} \mid X] = 0\) is sometimes substituted with weak exogeneity \(\mathrm{Cov}(X,\boldsymbol{u}) = 0\)
• This guarantees consistency, but not unbiasedness of the estimator.
• A failure of exogeneity is called endogeneity
• It causes bias and inconsistency by confounding the effects of our regressors X and the true errors u on y

Omitted Variable Bias

• A confounder is an additional variable that drives both the cause and effect
• If we don’t account for that variable we can’t provide a causal effect explanation for what we observe.
• Bias from a confounder is also called omitted variable bias. It occurs when:
  • The omitted variable is correlated with the regressors ( \(\mathrm{Cov}(x_1,x_2) \neq 0\))
  • It is also a determinant of y (\(\beta_2 \neq 0\))
• Many (potentially) omitted variables cannot be observed.
• To solve this we might be able to use proxy variables
• To use a proxy variable to identify a causal effect, it must:
  1. Correlate with the omitted variable: \(\theta_1 \neq 0\)
  2. Not correlate with other explanatory variables \(Cov(\boldsymbol{X}, \boldsymbol{e}) = 0\)
  3. have no direct impact on the dependent variable \(Cov(\boldsymbol{z}, \boldsymbol{u}) = 0\)

Type of Selection Bias

• If the sample is not random, we may speak of selection bias
• It is the idea that some subjects are more or less likely to be selected for our sample than others, distorting statistical insight
• Selection bias is related to sample issues that may plague external validity, but also threatens in-sample inference.
• There are many types of selection bias
• Consider a true \(f\) describing a population of size \(N\), but we only observe \(M (<N)\). Can we learn something using our subset?
• If our \(M\) represents a random subset of the population, then the selection process is ignorable
• Otherwise there may be selection bias
• endogenous sample selection, related to the dependent variable
• exogenous sample selection, related to the explanatory or third variables

Data Issues

• Data may be subject to various issues, due to errors in collection, which may affect our ability to analyse it.
• If there seems to be a pattern to missingness we may have to account for it to avoid bias or to benefit from accounting for it. (Truncation, Censoring)
• Outliers are observations that are very different from the rest
• Note: estimation for limited dependent variables done via ML estimators

Simultaneity and Reverse Causality

The causal effect of interest, i.e., \(X \rightarrow Y\), is not always as straightforward as we would like. Instead, we may encounter:

• Reverse Causality: The issue is determining the direction of causation where \(Y \rightarrow X\)
  1. Happiness and income: being happy increases productivity → income
  2. Health and exercise: poor health reduces ability to exercise → low exercise correlates with poor health
  3. Education and growth: richer countries invest more in education
• Simultaneity: We need to disentangle the effects where \(Y \leftrightarrow X\)
  1. Wage and Employment
  2. Price and quantity: market clears instantly – price and quantity determined together.

Techniques for Causal Identification

• OLS estimates are biased when regressors are correlated with the error term (omitted variables, simultaneity, reverse causality, measurement errors)
• Several strategies allow us to identify causal effects despite these challenges:
• Difference-in-Differences (DiD): compare changes over time between a treatment and control group
  • Requires: parallel trends assumption – groups would have evolved similarly absent treatment
• Instrumental Variables (IV): Use an instrument \(Z\) that affects \(X\) but has no direct effect on \(Y\)
  • Requires: relevance (\(Z\) correlated with \(X\)) and exclusion restriction (\(Z\) uncorrelated with \(u\))

Techniques for Causal Identification (2)

• Regression Discontinuity Design (RDD): exploit a cutoff rule that determines treatment assignment
  • Requires: no other discontinuity at the cutoff (continuity assumption)
• Randomized Controlled Trials (RCT): random assignment of treatment ensures independence from all confounders
  • The “gold standard” – but often infeasible for ethical or practical reasons

The Instrumental Variables (IV) Estimator

• Core problem: \(X_{it}\) is endogenous – correlated with \(u_{it}\) – so OLS is biased
• Idea: find an instrument \(Z\) that isolates exogenous variation in \(X\) – variation that is “as good as random”
• An instrument \(Z\) must satisfy two conditions:
  • Relevance: \(Z\) is correlated with \(X\) – \(\text{Cov}(Z, X) \neq 0\) – testable with an \(F\)-test (rule of thumb: \(F > 10\))
  • Exclusion restriction: \(Z\) affects \(Y\) only through \(X\) – \(\text{Cov}(Z, u) = 0\) – not directly testable

The Instrumental Variables (IV) Estimator (2)

• Two-Stage Least Squares (2SLS):
  • Stage 1: regress \(X\) on \(Z\) → obtain \(\hat{X}\) (the exogenous part of \(X\)) \[X_i = \pi_0 + \pi_1 Z_i + \nu_i\]
  • Stage 2: regress \(Y\) on \(\hat{X}\) → recovers the causal effect of \(X\) on \(Y\) \[Y_i = \beta_0 + \beta_1 \hat{X}_i + u_i\]
• Intuition: IV “throws away” the endogenous variation in \(X\) and uses only the exogenous variation induced by \(Z\)
• Classic example: returns to education – instrument \(Z\) = quarter of birth (Angrist & Krueger, 1991)
  • Quarter of birth affects schooling (compulsory schooling laws) but has no direct effect on wages

Panel Data

• Panel data (longitudinal data) refers to data for \(N\) different entities observed at \(T\) different time periods
• A balanced panel has all its observations; that is, the variables are observed for each entity and each time period
• An unbalanced panel has some missing data for at least one time period for at least one entity
• Panel data is used as a method for controlling for some types of omitted variables without actually observing them
• By studying changes in the dependent variable over time, it is possible to eliminate the effect of omitted variables that differ across entities but are constant over time
• Panel data allows for analysis of dynamics and improves causal inference compared to pure cross-sections

Example

• Individuals tracked over time (e.g. income, employment, education)
• Households surveyed repeatedly (e.g. consumption, demographics)
• Firms observed across years (e.g. profits, productivity, employment)
• Countries over time (e.g. GDP, inequality, policy changes)
• Students followed across grades (e.g. test scores, outcomes)
• Workers within firms over time (matched employer–employee data)
• Regions or cities tracked over time (e.g. crime rates, housing prices)
• Political units (e.g. voting behavior across elections)
• Hospitals or patients tracked over time (health outcomes)

Notation

• Panel data consist of observations on the same \(N\) entities over \(T\) time periods
  
• If the data set includes variables \(X\) and \(Y\), the data can be written as: \[ (X_{it}, Y_{it}), \quad i = 1, \dots, N, \quad t = 1, \dots, T \]
• \(i\) indexes the entity (individual, firm, country, etc.) being observed
• \(t\) indexes time (year, month, etc.) of the observation

Example: Traffic Deaths and Alcohol Taxes

• There are approximately 40,000 highway traffic fatalities each year in the United States. Approximately one-fourth of fatal crashes involve a driver who was drinking, and this fraction rises during peak drinking periods.
• We study how effective various government policies designed to discourage drunk driving actually are in reducing traffic deaths.
• Panel data set: the number of traffic fatalities in each state in each year (1982-1988), the type of drunk driving laws in each state in each year, and the tax on beer in each state
• Fatality rate, which is the number of annual traffic deaths per 10,000 people
• We can start by doing the regression for every year: \[ \text{Fatality\ Rate}_i = \beta_0 + \beta_1 \text{Beer\ Tax}_i + u_i \]

Example: Traffic Deaths and Alcohol Taxes (2)

• 1982 data: \[ \text{Fatality\ Rate}_i = 2.01 + 0.15 \text{Beer\ Tax}_i \]
• 1988 data: \[ \text{Fatality\ Rate}_i = 1.86 + 0.44 \text{Beer\ Tax}_i \]
• Should we conclude that an increase in the tax on beer leads to more traffic deaths? Not necessarily, because these regressions could have substantial omitted variable bias
• One approach would be to collect data on all these variables and add them to the annual cross-sectional regressions
• Unfortunately, some of these variables might be very hard or even impossible to measure

Panel Data with 2 Time Periods

• When data for each state are obtained for \(T = 2\) time periods, we can compare values of the dependent variable in the second period to values in the first period
• By focusing on changes in the dependent variable, this “before and after” comparison holds constant the unobserved factors that differ from one state to the next but do not change over time within the state
• For example: the local cultural attitude toward drinking and driving
• Let \(Z_i\) be a variable that determines the fatality rate in the \(i^{th}\) state but does not change over time (so the \(t\) subscript is omitted)
• The population linear regression relating \(Z_i\) and the real beer tax to the fatality rate: \[ \text{Fatality\ Rate}_{it} = \beta_0 + \beta_1 \text{Beer\ Tax}_{it} + Z_i + u_{it} \]

Before and After Comparison

• Because \(Z_i\) does not change over time, in the regression model it will not produce any change in the fatality rate between 1982 and 1988 \[ \text{Fatality\ Rate}_{i1982} = \beta_0 + \beta_1 \text{Beer\ Tax}_{i1982} + Z_i + u_{i1982} \] \[ \text{Fatality\ Rate}_{i1988} = \beta_0 + \beta_1 \text{Beer\ Tax}_{i1988} + Z_i + u_{i1988} \]
• The influence of \(Z_i\) can be eliminated by analyzing the change in the fatality rate between the two periods. \[ \begin{align} \text{Fatality\ Rate}_{i,1982} - &\text{Fatality\ Rate}_{i,1988} \\ &= (\beta_0-\beta_0) + \beta_1 (\text{Beer\ Tax}_{i,1982} - \text{Beer\ Tax}_{i,1988}) \nonumber\\ &\quad + (Z_i - Z_i) + (u_{i,1982} - u_{i,1988}) \end{align} \]

Example

• This specification has an intuitive interpretation.
• Cultural attitudes toward drinking and driving affect the level of drunk driving and thus the traffic fatality rate in a state.
• If they did not change between 1982 and 1988, then they did not produce any change in fatalities in the state. \[ \begin{align} \text{Fatality\ Rate}_{i,1982} - &\text{Fatality\ Rate}_{i,1988} \\ &= -0.072 - 1.04 (\text{Beer\ Tax}_{i,1982} - \text{Beer\ Tax}_{i,1988}) \nonumber \end{align} \]
• By examining changes in the fatality rate over time, the regression controls for fixed factors such as cultural attitudes toward drinking and driving.
• The “before and after” method does not apply directly when \(T > 2\).

Fixed Effects Regression

• To analyze all the observations in our panel data set, we use the method of fixed effects regression
• Fixed effects regression is a method for controlling for omitted variables in panel data when the omitted variables vary either only across time or observations
• For instance, a variable could vary across entities (states) but does not change over time
• The fixed effects regression model has \(N\) different intercepts, one for each entity.
• These intercepts can be represented by a set of binary (or indicator) variables
• These binary variables absorb the influences of all omitted variables that differ from one entity to the next but are constant over time

The Fixed Effects Regression Model

• Consider the regression model with FatalityRate and BeerTax denoted as \(Y_{it}\) and \(X_{it}\), respectively: \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 Z_i + u_{it} \]
• \(Z_i\) is an unobserved variable that varies from one state to the next but does not change over time (e.g., cultural attitudes toward drinking and driving)
• We want to estimate \(\beta_1\), the effect on \(Y\) of \(X\), holding constant the unobserved state characteristics \(Z\)
• Since \(Z_i\) is constant over time, our regression can be interpreted as having \(N\) intercepts – one per state: \(\alpha_i = \beta_0 + \beta_2 Z_i\)
• This gives us the fixed effects model: \[ Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it} \]

Entity Fixed Effects

• In \(Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it}\), the terms \(\alpha_1, \ldots, \alpha_N\) are treated as unknown intercepts to be estimated – one for each state
• The population regression line for the \(i^{th}\) state is \(\alpha_i + \beta_1 X_{it}\):
  • The slope \(\beta_1\) is the same for all states
  • The intercept \(\alpha_i\) varies from one state to the next
• Because \(\alpha_i\) can be thought of as the “effect” of being in entity \(i\), the terms \(\alpha_1, \ldots, \alpha_N\) are known as entity fixed effects
• The variation in entity fixed effects comes from omitted variables that, like \(Z_i\), vary across entities but not over time

Fixed Effects via Binary Variables

• The state-specific intercepts in the fixed effects regression model can be expressed using binary variables to denote the individual states.
• Define binary variables \(D1_i, D2_i, \ldots, DN_i\) where \(Dj_i = 1\) when \(i = j\) and \(0\) otherwise
• We cannot include all \(N\) binary variables plus a common intercept – this causes perfect multicollinearity (the dummy variable trap)
• Solution: omit \(D1_i\) for the first entity, giving the equivalent model: \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \gamma_2 D2_i + \gamma_3 D3_i + \cdots + \gamma_N DN_i + u_{it} \]
• The link between the two representations:
  • First state: regression line is \(\beta_0 + \beta_1 X_{it}\), so \(\alpha_1 = \beta_0\)
  • All other states (\(i \geq 2\)): regression line is \(\beta_0 + \beta_1 X_{it} + \gamma_i\), so \(\alpha_i = \beta_0 + \gamma_i\)
• Thus there are two equivalent ways to write the fixed effects regression model

The Fixed Effects Regression Model: General Form

• The fixed effects regression model with \(k\) regressors is: \[ Y_{it} = \beta_1 X_{1,it} + \cdots + \beta_k X_{k,it} + \alpha_i + u_{it} \]
• Where \(i = 1, \ldots, N\); \(t = 1, \ldots, T\); and \(\alpha_1, \ldots, \alpha_N\) are entity-specific intercepts
• Equivalently, using a common intercept and \(N-1\) binary variables (omitting the first entity): \[ Y_{it} = \beta_0 + \beta_1 X_{1,it} + \cdots + \beta_k X_{k,it} + \gamma_2 D2_i + \gamma_3 D3_i + \cdots + \gamma_N DN_i + u_{it} \]
• Where \(D2_i = 1\) if \(i = 2\) and \(0\) otherwise, and so forth
• These two representations are equivalent – both control for all time-invariant entity characteristics
• If there are other observed determinants of \(Y\) that are correlated with \(X\) and that change over time, then these should also be included in the regression to avoid omitted variable bias

Application to Traffic Deaths

• OLS estimate of the fixed effects model using all 7 years of data (336 observations): \[ \text{Fatality\ Rate} = -0.66 \text{Beer\ Tax} + \text{state fixed effects} \]
• The estimated fixed intercepts are not reported – they are not of primary interest
• The coefficient on Beer Tax is negative: higher beer taxes \(\Rightarrow\) fewer traffic deaths
• This is the opposite of the initial cross-sectional results – entity fixed effects remove omitted variable bias
• Difference from the “before and after” regression: uses only 1982 and 1988 data, now we use all 7 years → smaller standard error

Regression with Time Fixed Effects

• Entity fixed effects control for variables constant over time but varying across entities
• Time fixed effects control for variables constant across entities but evolving over time
• Example: automobile safety improvements are introduced nationally → same value for all states but changes over time
• We can extend the model to make the effect of safety \(S_t\) explicit: \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 Z_i + \beta_3 S_t + u_{it} \]
• \(S_t\) has only a \(t\) subscript → changes over time but is constant across states
• If \(S_t\) is correlated with \(X_{it}\) and omitted from the regression → omitted variable bias

Time Fixed Effects Regression Model

• Although \(S_t\) is unobserved, its influence can be eliminated – just as \(Z_i\) was eliminated by entity fixed effects
• Parallel logic:
  • \(Z_i\) varies across states but not over time → entity fixed effects: each state gets its own intercept
  • \(S_t\) varies over time but not across states → time fixed effects: each time period gets its own intercept

Time Fixed Effects Regression Model (2)

• The time fixed effects regression model with a single \(X\) regressor: \[ Y_{it} = \beta_1 X_{it} + \lambda_t + u_{it} \]
• \(\lambda_t\) is a different intercept for each time period
• \(\lambda_t\) captures the “effect” of year \(t\) on \(Y\), so \(\lambda_1, \ldots, \lambda_T\) are the time fixed effects
• The variation in time fixed effects comes from omitted variables that vary over time but are constant across entities
• Just as the entity fixed effects regression model, it can be represented using \(T - 1\) binary indicators
• In the traffic fatalities regression, the time fixed effects specification allows us to eliminate bias arising from omitted variables that change over time but are the same across states in a given year

Both Entity and Time Fixed Effects

• Some omitted variables are constant over time but vary across states (e.g., cultural norms)
• Others are constant across states but vary over time (e.g., national safety standards)
• When both types are present → include both entity and time fixed effects
• The combined model: \[ Y_{it} = \beta_1 X_{it} + \alpha_i + \lambda_t + u_{it} \]
• Where \(\alpha_i\) is the entity fixed effect and \(\lambda_t\) is the time fixed effect
• Equivalently, using \(N-1\) entity binary indicators and \(T-1\) time binary indicators: \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \gamma_2 D2_i + \cdots + \gamma_N DN_i + \delta_2 B2_t + \cdots + \delta_T BT_t + u_{it} \]
• Where \(\beta_0, \beta_1, \gamma_2, \ldots, \gamma_N\) and \(\delta_2, \ldots, \delta_T\) are unknown coefficients to be estimated
• We call this the Two-Way Fixed Effects Estimator

Caveats of the Fixed Effects Estimator

• Caveat 1: Fixed effects cannot solve reverse causality. If you have simultaneity or reverse causality bias then you would not use panel fixed effects, specifically when it is really strong, as your identification strategy.
• Caveat 2: Fixed effects cannot address time-variant unobserved heterogeneity.
• Using the TWFE is helpful but will not solve your endogeneity issues.
• Nothing substitutes for careful reasoning and economic theory, as they are necessary conditions for good research design

Application to Traffic Deaths: Time Effects Added

• Adding time effects to the state fixed effects model gives: \[ \text{Fatality Rate} = -0.64 \text{Beer Tax} + \text{State Fixed Effects} + \text{Time Fixed Effects} \]
• This specification includes 55 right-hand side variables in total:
  • 1 beer tax regressor
  • 47 state binary variables (state fixed effects)
  • 6 single-year binary variables (time fixed effects)
  • 1 intercept
• State/time binary coefficients and intercept are not reported – not of primary interest
• Including time effects has little impact on the coefficient on the real beer tax

The Fixed Effects Regression Assumptions

• The fixed effects model: \(Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it}\), \(i = 1, \ldots, N\), \(t = 1, \ldots, T\), where \(\beta_1\) is the causal effect on \(Y\) of \(X\)
• Assumption 1: \(u_{it}\) has conditional mean zero: \[E(u_{it} | X_{i1}, X_{i2}, \ldots, X_{iT}, \alpha_i) = 0\]
• Assumption 2: \((X_{i1}, X_{i2}, \ldots, X_{iT}, u_{i1}, u_{i2}, \ldots, u_{iT})\), \(i = 1, \ldots, N\), are i.i.d. draws from their joint distribution
• Assumption 3: Large outliers are unlikely: \((X_{it}, u_{it})\) have nonzero finite fourth moments
• Assumption 4: No perfect multicollinearity
• For multiple regressors, \(X_{it}\) is replaced by the full list \(X_{1,it}, X_{2,it}, \ldots, X_{k,it}\)

Fixed Effects Assumptions: Closer Look

• Assumption 1 – conditional mean zero given all \(T\) values of \(X\):
  • Implies no omitted variable bias
  • \(u_{it}\) must not depend on any values of \(X\) for that entity – past, present, or future
  • Violated if current \(u_{it}\) is correlated with past, present, or future values of \(X\)
• Assumption 2 – variables are i.i.d. across entities for \(i = 1, \ldots, N\):
  • Variables for one entity are distributed identically to, but independently of, variables for another entity
  • Holds if entities are selected by simple random sampling from the population
  • Key difference from cross-sectional assumption 2: no restriction within an entity – \(X_{it}\) can be correlated over time within the same entity
• Under all four assumptions, the fixed effects estimator is consistent and normally distributed when \(N\) is large

Autocorrelation in Panel Data

• If \(X_{it}\) is correlated with \(X_{is}\) for \(s \neq t\), then \(X_{it}\) is autocorrelated (or serially correlated)
• Autocorrelation is a pervasive feature of time series data: what happens one year tends to be correlated with what happens the next
• Example: beer tax \(X_{it}\) is autocorrelated – legislatures rarely change tax rates year to year, so a high tax in year \(t\) tends to remain high in year \(t+1\)
• \(u_{it}\) can also be autocorrelated due to persistent omitted factors, for example:
  • A local economic downturn reduces commuting → fewer fatalities for multiple years
  • A road improvement project reduces accidents not just at completion but in future years
• Not all omitted factors cause autocorrelation – e.g., severe winter weather that is independently distributed year to year would be serially uncorrelated
• In general, as long as some omitted factors are autocorrelated, \(u_{it}\) will be autocorrelated

Standard Errors for Fixed Effects Regression

• If errors are autocorrelated, the usual heteroskedasticity-robust standard errors are not valid – they assume no serial correlation
• Solution: use heteroskedasticity- and autocorrelation-robust (HAR) standard errors
• The standard approach in panel data: clustered standard errors
  • Allow arbitrary correlation within a cluster (= entity)
  • Assume errors are uncorrelated across clusters (= across entities)
  • Allow for both heteroskedasticity and autocorrelation within an entity
• Clustered standard errors are valid whether or not there is heteroskedasticity, autocorrelation, or both
• When \(N\) is large, inference proceeds using:
  • Normal critical values for \(t\)-statistics
  • \(F_{q,\infty}\) critical values for \(F\)-statistics testing \(q\) restrictions

Finding Data

• High-quality data is the foundation of credible research
• The choice of data shapes: whether you can answer your research questions, which methodology to use, the validity of results
• Data availability often constrains what we can study, but more data sources exist than we initially suspect
• Different sources offer different strengths and limitations:
• Administrative data: Large scale, high accuracy, often longitudinal, Limited to recorded variables, access can be restricted
• Surveys: Flexible design, can capture attitudes and unobservables, Measurement error, recall bias, smaller samples
• Web / scraped data: Real-time, novel behaviors, large and granular, Selection bias, legal/ethical issues, messy structure
• Experimental data: High causal identification (internal validity), Costly, smaller samples, external validity concerns

Different Types of Data

Example – Demographic and Health Surveys (DHS)

• Large-scale, nationally representative household surveys in developing countries
• Cover topics such as:
• Fertility, family planning, and maternal health
• Child health and mortality
• HIV/AIDS knowledge and prevalence
• Gender, education, and household characteristics
• Key strengths: High-quality, comparable across countries and time, rich micro-level data (individual & household)
• Key limitations: Self-reported data (measurement error), Cross-sectional (limited panel structure)
• Widely used in development and health economics

Remotely Sensed Data

Example – Luxembourg Income Study (LIS)

• Cross-national microdata database on income, inequality, and poverty
  
• Harmonizes household survey data from multiple countries
  
• Covers: Income, earnings, and redistribution, Wealth (via LWS), Demographics and labor market outcomes
  
• Key strengths: High-quality harmonization for inequality research, widely used in top economic research, strong comparability across countries and over time
  
• Key limitations: Limited variable detail compared to raw surveys, access requires application and approval, remote execution system (no direct microdata download)

Example – Census: IPUMS

• Data harmonization project by IPUMS at the University of Minnesota
• Provides access to: Census and household survey microdata from around the world, U.S. Census and American Community Survey data
• Key feature: Harmonized variables across countries and time, thus enables cross-country and longitudinal comparisons
• Available datasets: IPUMS USA, IPUMS International, IPUMS CPS, IPUMS DHS, etc.
• Key strengths: Large samples, high comparability, easy-to-use extracts
• Key limitations: Some loss of detail due to harmonization, Access restrictions for sensitive data

Example – WVS & ANES: Public Opinion Data

• Cross-national and U.S.-focused surveys on values, attitudes, and political behavior
  
• WVS: Covers 100+ countries over multiple waves, focus on cultural values, beliefs, trust, religion, and social norms
  
• ANES: U.S.-focused, election-centered surveys since 1948, covers voting behavior, political attitudes, and public opinion
  
• Key strengths: rich information on preferences, beliefs, and norms, long time span (especially ANES), cross-country comparability (WVS)
  
• Key limitations: self-reported attitudes (measurement error, social desirability bias), limited causal identification, repeated cross-sections rather than true panels

Example – Text

• Unstructured data from written or spoken language sources
  
• Common sources: news articles, policy documents, transcripts, social media posts, forums, reviews, legal texts, reports, historical archives
  
• Methods: Keyword search, dictionaries, sentiment analysis, topic modeling, machine learning / NLP
  
• Key strengths: Captures beliefs, narratives, and framing, large-scale and often real-time, enables analysis of previously unobservable concepts
  
• Key limitations: Requires substantial preprocessing, measurement depends on method choices, potential bias in text sources (selection, platform effects)

Example – Macro data

• Aggregate data describing the economy at the country or regional level
  
• Common indicators: GDP, inflation, unemployment, trade, government spending, debt, productivity
• Common sources: World Bank, IMF, OECD, Eurostat, FRED, ECB, ILO, UN data
• Key strengths: Broad coverage across countries and long time periods, Standardized and comparable indicators
• Key limitations: Aggregation masks heterogeneity, Limited ability for causal inference

What is a Quasi-Experiment?

Natural Experiments

Cholera

A Natural Experiment

A Natural Experiment

Other Examples for Quasi-Experiments

The Differences Estimator

Two Groups Observed Twice

The Difference-in-Differences Estimator

\[ y_{i t} = \alpha + \text{after}\: \phi + \text{treated} \: \theta + \text{after}\times\text{treated}\: \delta + u_{it} \]

Diff-in-Diff Illustration

Estimating a DiD

The DiD Estimator With Additional Regressors

DiD Using Repeated Cross-Sections

IV Estimation in DiD Settings

Parallel Trends and Event Studies

Event Studies

Event Study Estimation

Event Study Plot

Staggered Treatment

Regression Discontinuity

How to Find a Discontinuity Setting?

Sharp and Fuzzy RDD

How an RDD Works

How Do We Find the Treatment Effect?

Requirements for an RDD

Look, I’ve Found a Discontinuity

Estimating a Sharp RDD

Estimating a Sharp RDD

Plotting a Sharp RDD

Non-Linear Sharp RDD

Networks

Revisiting Graph Theory Basics

Revisiting Graph Theory Basics

Social Networks

Trade Networks

Drawing a Network

Mapping Subway Stations

Centrality

Peers

The Reflection Problem

Space

Autocorrelation, Networks, Peers, and Space

The Linear Model

The Spatial Autoregressive (SAR) Model

The Spatial Lag of X (SLX) Model

The Spatial Durbin Model (SDM)

Spatial Errors