Empirical Methods in Finance Final Midterm Flashcards

Vocabulary flashcards covering empirical methods in finance, econometrics concepts, regression interpretations, endogeneity, fixed effects, and qualitative choice models.

Population, Sample, Parameter & Statistic

Population = full group; sample = observed subset. A parameter describes the population; a statistic is computed from the sample.

Cross-Sectional, Time-Series & Panel Data

Cross-section: many units at one time. Time series: variables over time. Panel: the same units observed repeatedly over time.

Random Variable & Distribution

A random variable has outcomes determined partly by chance. Its probability distribution describes possible values and their probabilities.

Expected Value, Variance & Standard Deviation

Expected value is the probability-weighted mean. Variance measures squared dispersion; standard deviation is its square root in the variable's units.

Covariance, Correlation & Independence

Covariance/correlation measure linear co-movement. Independence implies zero covariance, but zero covariance does not generally imply independence.

Conditional Probability & Conditional Mean

Conditional probability updates probabilities given information. $E(y|X)$ is the expected value of $y$ for given $X$ and is central to regression interpretation.

Normality in Regression

Normality of the error is not required for OLS unbiasedness. It supports exact small-sample $t/F$ inference; large-sample inference can rely on asymptotics.

Regression Model, Error Term & Ceteris Paribus

A regression relates $y_i$ to explanatory variables $X_i$. The error $u_i$ contains unobserved determinants of $y_i$. A slope coefficient is interpreted ceteris paribus: holding the other included regressors fixed.

OLS

OLS chooses $\hat{\beta}_0$ and $\hat{\beta}_1$ to minimize the sum of squared residuals, $\sum \hat{u}_i^2$, where $\hat{u}_i = y_i - \hat{y}_i$. Under the classical assumptions, OLS is BLUE.

Unbiasedness, Consistency, Efficiency & BLUE

Unbiased: $E(\hat{\beta})=\beta$. Consistent: $\hat{\beta}$ converges to $\beta$ as $N$ grows. Efficient: smallest variance in the relevant class. BLUE = Best Linear Unbiased Estimator.

CLRM Assumptions for BLUE

Lecture notation: [A1] linear in parameters; [A2] random sampling; [A3] sample variation/no perfect collinearity; [A4] $E(u_i|X_i)=0$; [A5] $Var(u_i|X_i)= \sigma^2$. Under [A1]-[A5], OLS is BLUE.

Zero Conditional Mean

$E(u_i|X_i)=0$ means the regressors are unrelated to the expected unobserved determinants in $u_i$. It is the key assumption for unbiased OLS estimates.

Endogeneity

A regressor $x$ is endogenous when $Cov(x,u) \neq 0$. OLS is then generally biased/inconsistent and $\hat{\beta}_1$ cannot be given a causal interpretation.

No Perfect Collinearity

No regressor can be an exact linear combination of the others. Otherwise separate coefficients cannot be estimated.

Multicollinearity

Strong but imperfect correlation among regressors does not bias OLS; it raises standard errors and makes individual effects less precisely estimated.

Homoskedasticity vs. Heteroskedasticity

Homoskedasticity means constant error variance. Heteroskedasticity leaves OLS coefficients unchanged but makes conventional standard errors unreliable.

Heteroskedasticity-Robust Standard Errors

Robust SEs correct inference under heteroskedasticity. They change SEs, $t$-statistics and CIs, but not the OLS coefficient estimates.

Breusch-Pagan Test

Tests whether error variance is systematically related to regressors. A small $p$-value provides evidence of heteroskedasticity.

OLS Standard Error

The standard error measures uncertainty/precision of an estimated coefficient. Smaller SE means a more precisely estimated coefficient.

R-squared

Share of sample variation in $y$ explained by the regression. High R-squared means better in-sample fit, not causality or an unbiased model.

Adjusted R-squared

Penalizes adding regressors that do not improve fit enough. Unlike R-squared, it can fall when a variable is added.

Data Scaling

Rescaling $y$ rescales coefficients and SEs together; rescaling $x$ inversely rescales its coefficient and SE. Relevant $t$-statistics remain unchanged.

Level-Level Interpretation

For $y_i = \beta_0 + \beta_1 x_i + u_i$, a one-unit increase in $x$ is associated with a $\beta_1$-unit change in $y$, ceteris paribus.

Log-Level Interpretation

For $\ln(y_i) = \beta_0 + \beta_1 x_i + u_i$, a one-unit increase in $x$ is associated with approximately $100\beta_1\%$ change in $y$, ceteris paribus.

Level-Log Interpretation

For $y_i = \beta_0 + \beta_1 \ln(x_i) + u_i$, a $1\%$ increase in $x$ is associated with approximately $\beta_1/100$ units change in $y$.

Log-Log Interpretation / Elasticity

For $\ln(y_i) = \beta_0 + \beta_1 \ln(x_i) + u_i$, $\beta_1$ is an elasticity: a $1\%$ increase in $x$ is associated with a $\beta_1\%$ change in $y$.

Percentage vs. Percentage-Point Change

$10\%$ to $11\%$ is $+1$ percentage point but a $+10\%$ relative change. Keep these interpretations distinct.

Dummy Variable & Reference Category

A dummy coefficient compares $D=1$ with the omitted/reference group $D=0$, holding other regressors fixed.

Dummy Variable in a Log-y Model

With $\ln(y)$ as the dependent variable, a dummy coefficient $\beta_1$ is approximately a $100\beta_1\%$ difference; the exact percentage difference is $100[\exp(\beta_1)-1]\%$.

Interaction Term

An interaction means one variable's effect depends on another. With $x_1 x_2$, the partial effect of $x_1$ depends on $x_2$.

Dummy x Continuous Interaction

With $D \times x$, the interaction coefficient gives the difference in the slope of $x$ between $D=1$ and $D=0$; the dummy can also shift the intercept.

Quadratic Term

With $y = \beta_0 + \beta_1 x + \beta_2 x^2 + u$, the marginal effect of $x$ is $\beta_1 + 2\beta_2 x$. The effect therefore depends on the value of $x$.

Turning Point in a Quadratic Model

The turning point of $y = \beta_0 + \beta_1 x + \beta_2 x^2 + u$ occurs where $\beta_1 + 2\beta_2 x = 0$. It is where the fitted relationship changes direction.

Standardized Coefficient

Expresses an effect in standard-deviation units. In bivariate OLS, the standardized slope equals the correlation between $x$ and $y$.

Statistical vs. Economic Significance

Statistical significance concerns evidence/precision. Economic significance concerns the sign and economically meaningful size of the effect.

t-Test & t-Statistic

For $H_0: \beta=\beta_0$, the t-statistic is $t = (\hat{\beta}-\beta_0)/se(\hat{\beta})$. It measures how many estimated standard errors $\hat{\beta}$ lies from the null value.

One-Sided vs. Two-Sided Tests

Two-sided tests allow deviations in either direction; one-sided tests ask whether the parameter is specifically above or below the null.

p-Value

Assuming $H_0$ is true, the p-value measures how extreme the observed evidence is. Reject $H_0$ when $p$ is below the chosen significance level.

Fail to Reject H0

An insignificant result means insufficient evidence against $H_0$. It does not prove $H_0$ or prove the true effect is zero.

Confidence Interval

A confidence interval reports parameter values compatible with the data at a chosen confidence level; wider intervals mean less precision.

F-Test / Joint Significance

Tests multiple coefficient restrictions jointly. A small $p$-value means the restrictions are jointly inconsistent with the data.

Restricted vs. Unrestricted Model

The restricted model imposes $H_0$; the unrestricted model does not. An F-test asks whether imposing the restrictions worsens fit sufficiently.

Omitted Variable Bias (OVB)

OVB occurs when an omitted determinant of $y$ is correlated with an included regressor, causing the included coefficient to absorb part of its effect.

Direction of OVB

Bias direction depends on the omitted variable's effect on $y$ and its relation with the included $x$. Same signs imply upward bias; opposite signs downward.

Omitting Relevant vs. Including Irrelevant Variables

Omitting a relevant correlated variable can bias coefficients. Including an irrelevant variable does not create bias but can reduce precision.

Main Sources of Endogeneity

Main sources: omitted variables, simultaneity/reverse causality, and measurement error in an explanatory variable.

Simultaneity / Reverse Causality

$x$ affects $y$ while $y$ also affects $x$. This creates correlation between $x$ and $u$ and makes OLS endogenous.

Measurement Error in x

Classical measurement error in an explanatory variable causes endogeneity and typically attenuates the OLS slope toward zero.

Measurement Error in y

Classical measurement error in $y$ does not bias OLS slopes if unrelated to $X$, but it increases noise and reduces precision.

Ramsey RESET

A general functional-form check using nonlinear functions of fitted values. Rejection suggests the model's functional form may be misspecified.

Causal Inference

Causal inference asks what would happen to $y$ if $x$ were changed, not merely whether $x$ and $y$ are correlated.

Instrumental Variables (IV)

IV uses an instrument $z$ to identify the causal effect $\beta_1$ of an endogenous regressor $x$ in $y_i = \beta_0 + \beta_1 x_i + u_i$.

IV Relevance

Instrument relevance: $Cov(z,x) \neq 0$. The instrument must predict the endogenous regressor $x$.

IV Exogeneity / Exclusion Logic

Instrument exogeneity: $Cov(z,u) = 0$. The instrument must be unrelated to the unobserved determinants in $u$; this is difficult to verify directly because $u$ is unobserved.

First Stage & Weak Instruments

First stage: $x_i = \pi_0 + \pi_1 z_i + v_i$. Relevance requires $\pi_1 \neq 0$; the lecture uses the rule of thumb $F \ge 10$. Weak instruments make IV estimates unreliable.

2SLS Interpretation

2SLS: first predict $x$ from $z$ (and exogenous controls), obtaining $\hat{x}$; then estimate $y_i = \beta_0 + \beta_1 \hat{x}_i + u_i$. Identification comes from instrument-generated variation.

Invalid Instrument

A relevant but endogenous instrument does not solve endogeneity. If $z$ is correlated with $u$, IV is inconsistent.

IV vs. OLS Precision

Valid IV is often less precise than OLS because it uses only the variation in $x$ generated by the instrument.

IV with Multiple Endogenous Components

If several regressors or endogenous interactions are endogenous, each needs sufficient valid instrument-generated variation for identification.

Randomized Controlled Trial (RCT)

Random assignment makes treatment independent of potential outcomes on average, providing a benchmark design for causal inference.

Natural Experiment

An external institutional or economic event creates plausibly exogenous treatment variation without researcher-controlled randomization.

Difference-in-Differences (DiD)

DiD compares the treated group's before-after change with the control group's before-after change; the difference is the treatment effect.

Parallel Trends

Without treatment, treated and control groups must have followed the same average outcome trend. This is the key DiD identifying assumption.

Pre-Trends

Similar pre-treatment trends support parallel trends; diverging pre-trends cast doubt on the DiD design.

DiD Regression Interpretation

The Treated x Post coefficient is the DiD effect: the treated group's additional change after treatment relative to the control group.

DiD with Firm and Time Fixed Effects

Firm FE absorb permanent firm differences; time FE absorb common shocks. A treatment indicator varying by firm and time captures the DiD effect.

Regression Discontinuity Design (RDD)

RDD compares units near a treatment cutoff. A discontinuity in $y$ at the threshold identifies a local treatment effect.

RDD Identification Assumption

Absent treatment, potential outcomes must vary smoothly through the cutoff; units must not precisely manipulate treatment assignment around it.

Panel Data

Panel data follow the same units over time, allowing researchers to exploit within-unit variation and control for persistent unobserved heterogeneity.

First Differences (FD)

Panel notation: $y_{it} = a_i + \beta x_{it} + u_{it}$, with units $i=1,\dots,N$ and periods $t=1,\dots,T$. Panel data follow the same units over time.

Fixed Effects (FE)

FD subtracts $t-1$ from $t$: $\Delta y_{it} = \beta \Delta x_{it} + \Delta u_{it}$. The time-invariant unit effect $a_i$ drops out.

Demeaning / Within Transformation

FE removes the time-invariant unit effect $a_i$ and allows $a_i$ to be correlated with $x_{it}$. $\beta$ is identified from within-unit variation over time.

Time Fixed Effects

Within transformation: subtract each unit's time average, e.g. $y_{it} - \bar{y}_i$. The time-invariant effect $a_i$ disappears.

Two-Way Fixed Effects

Includes both unit and time FE, controlling for permanent unit differences and common time shocks.

Time-Invariant Regressors under Unit FE

A regressor constant within a unit over time is absorbed by unit FE and its separate coefficient cannot be estimated.

FE vs. FD

Both remove time-invariant unit effects. With $T=2$ they coincide; with longer/unbalanced panels their efficiency and data use can differ.

Random Effects (RE)

RE writes the unit intercept as $a_i = \alpha + v_i$, so $y_{it} = \alpha + \beta x_{it} + v_i + u_{it}$. RE requires $v_i$ to be independent of the regressors $x_{it}$.

FE vs. RE

FE allows the unit effect $a_i$ to correlate with $x_{it}$; RE requires $Cov(x_{it},v_i)=0$. RE can be more efficient if this assumption holds.

Hausman Test

Hausman test: assesses whether the FE and RE estimates are statistically close. A significant difference casts doubt on the RE independence assumption.

Clustered Standard Errors

Clustered SEs allow errors to be correlated within clusters. Ignoring such dependence often makes SEs too small and significance look too strong.

Firm vs. Time Clustering

Firm clustering allows dependence over time within firms; time clustering allows dependence across firms within the same period.

Two-Way Clustering

Clustering by both firm and time allows within-firm serial correlation and same-period cross-sectional correlation.

Why Fixed Effects Do Not Replace Clustering

FE remove systematic level differences, not residual dependence. Correlated errors can remain, so clustered SEs may still be necessary.

Binary Outcome: Conditional Mean as Probability

For binary $y_i \in \{0,1\}$, $E(y_i|X_i)=P(y_i=1|X_i)$. The conditional mean is therefore a probability.

Linear Probability Model (LPM)

LPM: $P(y=1|X)=\alpha+X\beta$, estimated by OLS. A coefficient gives a constant change in probability, but fitted probabilities can lie outside $[0,1]$.

Logit and Probit

Logit/Probit: $P(y=1|X)=G(\alpha+X\beta)$. Logit uses $G(z)=\Lambda(z)$, the logistic CDF; Probit uses $G(z)=\Phi(z)$, the standard normal CDF.

Why Logit/Probit Use MLE

Logit and Probit are nonlinear in the parameters through $G(\alpha+X\beta)$, so the lecture estimates them by Maximum Likelihood Estimation (MLE), not OLS.

Logit/Probit Coefficient Interpretation

A Logit/Probit coefficient $\beta_j$ is not itself a probability change. Its sign gives the direction; the probability effect depends on $X$ and is interpreted using marginal effects.

Marginal Effect in Logit/Probit

For a continuous $x_j$, the marginal effect is $\partial G(\alpha+X\beta)/\partial x_j = g(\alpha+X\beta)\beta_j$, where $g$ is $\lambda$ for Logit and $\phi$ for Probit.

Average Marginal Effect (AME)

The Average Marginal Effect (AME) averages the observation-specific marginal effects $g(\alpha+X_i\beta)\beta_j$ across observations.

AME for a Dummy Variable

For a dummy $x_j$, compare predicted $P(y=1|X)$ when $x_j$ changes from 0 to 1 for each observation, then average those probability differences.

LPM vs. Logit/Probit

LPM is directly interpretable but may predict outside $[0,1]$. Logit/Probit respect probability bounds but require marginal effects for probability interpretation.

Proxy Variable

An observed variable used to approximate an important unobserved factor. A useful proxy can reduce omitted variable bias by capturing part of the missing factor.

Individual vs. Joint Significance

Variables can be individually insignificant in t-tests but jointly significant in an F-test. The F-test asks whether their coefficients are jointly equal to zero.

Overcontrolling / Bad Controls

Do not automatically control for every available variable. Controlling for an outcome, mediator, or variable determined by the process being studied can remove part of the effect of interest.