Exam Formula Sheet (Lec 1–10)

Part of: Econometrics · Glossary: _Econometrics Concepts Scope: the whole Sem-2 causal-inference toolkit. Each section = one lecture: key terms → formulas → how to read them → exam tips. Default everywhere: 5% significance, two-sided; reject if $|t|>1.96$ (or $p<0.05$). Standard errors are robust/clustered unless told otherwise.

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0. Method picker — "which tool does this question want?"

First move on every applied question

(1) Is the outcome binary? (2) Is the key regressor exogenous, or is there a confounder / reverse causality / selection? (3) What's the data structure — cross-section, time series, or panel? Those three answers pin down the method.

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1. Foundations recap (assumed from Intro Metrics)

Classical assumptions — A1–A5

A1 linear in parameters · A2 random sampling · A3 no perfect collinearity · A4 zero conditional mean $\mathbb{E}[u\mid X]=0$ (the causal one) · A5 homoskedasticity $\text{Var}(u\mid X)=\sigma^2$. Under A1–A5, OLS is BLUE.

Course convention (varies by source): A3 = independent errors / A4 = $\mathbb E[u\mid x]=0$ / A5 = homoskedastic. Read the question's numbering — the content is what matters.

t-test (single coefficient)

Reject $H_0$ if $|\hat t|>t_{crit}\approx1.96$ (large $n$, 5%).

F-test (joint / nested models) — F-test

$q$ = number of restrictions (regressors dropped). Large $F$ / small $p$ ⇒ reject that the dropped variables are jointly zero. In R: anova(unrestricted, restricted).

Confidence interval

Functional-form interpretation (quick table)

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2. Lecture 1 — Causal inference & treatment effects

Goal: read $\hat\beta_1$ as causal, not just correlational. Requires ruling out confounders, reverse causality, selection.

• Data types: cross-section · time series · panel · repeated cross-section.
• Exogenous (X) → endogenous (Y): the arrow is causal only if X varies independently of everything else affecting Y.
• Continuous vs dummy regressor: with few discrete values, use dummies (one per level, minus a reference). Dummies are flexible (each level free); a continuous slope forces one straight line — test it with a joint F-test on the dummies.
• Panel breaks A3/A4: repeated obs per unit ⇒ errors correlated within unit ⇒ classical SEs too small. Quick fix: subsample one obs/unit. Proper fix: fixed effects + clustered SEs (L8).

Exam tip

"Is this coefficient causal?" → name the specific confounder / reverse-causality / selection story, state $\text{Cov}(x,u)\neq0$, then say which tool (control, FE, IV, RDD, DiD) shuts that path.

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3. Lecture 2 — Linear Probability Model (LPM)

Binary $y\in\{0,1\}$ run through OLS. Under A2, the fit is a probability:

• Coefficient = marginal effect in percentage points (constant). $\beta_1=\partial P(y=1)/\partial x$. Dummy coef = difference in $P(y=1)$ vs base group.
• Drawback 1 — unbounded: $\hat y$ can fall below 0 or exceed 1 (nonsense probabilities). → motivates logit/probit.
• Drawback 2 — built-in heteroskedasticity (always):

• Fix for drawback 2: robust (HC) standard errors — mandatory for any LPM. $\hat\beta$ stays unbiased; only SEs were wrong. R: feols(y ~ x, se="hetero").

Exam tip

If the outcome is 0/1, every lm/feols is an LPM — interpret coefficients in pp and expect robust/clustered SEs. Saying "$\hat y$ can leave [0,1]" + "errors are heteroskedastic by construction" earns the two standard marks.

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4. Lecture 3 — Logit & Probit (discrete choice)

Wrap the linear index in a CDF $G$ so $P\in(0,1)$:

$G$ is increasing, S-shaped, → 0/1 at the tails, steepest at $\mathbf x'\beta=0$ ($P=0.5$), symmetric $G(z)=1-G(-z)$.

Marginal effects (the key formula)

Non-constant — depends where you sit on the curve. Evaluate at sample means $g(\bar{\mathbf x}'\hat{\boldsymbol\beta})\cdot\hat\beta_j$ for "the average person."

• Raw coef → sign & significance only, NOT magnitude. Never compare raw coef magnitudes across LPM/logit/probit; do compare marginal effects.
• Estimation = MLE (model is non-linear, OLS invalid). Log-likelihood:

Joint testing = Likelihood Ratio test (replaces F)

$q$ = # restrictions. Reject if $LR>\chi^2_{crit}(q)$. R: lrtest(ur, r).

Exam tip

Recipe for a marginal effect: (1) get $\hat\beta$; (2) compute $\bar{\mathbf x}'\hat\beta$ at the means; (3) multiply by $g(\cdot)$ — dnorm (probit) / dlogis (logit). If only sign asked, just read the coefficient sign + $p$/$z$.

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5. Lecture 4 — Instrumental Variables & 2SLS

Problem: endogeneity $\text{Cov}(x,u)\neq0$ (OVB, reverse causality, simultaneity) ⇒ OLS biased & inconsistent (more data doesn't help).

Fix: an instrument $z$ that moves $x$ but not $y$ directly. Two conditions:

2SLS

• First stage: $x=\gamma_0+\gamma_1 z+(\text{controls})+\varepsilon\Rightarrow\hat x$
• Second stage: $y=\beta_0+\beta_1\hat x+(\text{controls})+u$
• Simple IV estimator: $\hat\beta_1^{IV}=\dfrac{\text{Cov}(y,z)}{\text{Cov}(x,z)}$

R (correct SEs): feols(y ~ controls | x ~ z). Never run two manual lms (wrong second-stage SEs).

• # instruments ≥ # endogenous regressors (need ≥2 instruments for 2 endogenous vars).
• Weak instrument (F<10): tiny denominator amplifies bias — can be worse than OLS.
• IV estimates LATE, not ATE — the effect for compliers (those whose $x$ moves with $z$).
• IV is less precise than OLS (larger SEs) — the price of consistency.
• Sargan/overid test (only if overidentified): $H_0$ = all instruments valid. Reject ⇒ ≥1 invalid.
• Wu–Hausman: $H_0$ = regressor exogenous (OLS fine). Reject ⇒ IV needed.

Exam tip

To defend an instrument, write both conditions explicitly with the covariance, show relevance via first-stage F, then give a one-sentence story for why $z$ can't reach $y$ except through $x$ — and a sentence on how it could fail. Validity can never be "proven from a table."

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6. Lecture 5 — Sample selection & Heckman

Consistency vs unbiasedness

• Unbiased: $\mathbb E[\hat\beta]=\beta$ (right on average, any $n$).
• Consistent: $\hat\beta\to\beta$ as $n\to\infty$.
• OLS with endogenous $x$: biased AND inconsistent. 2SLS: biased in small samples but consistent.

Sample selection — you only observe $y$ when $s_i=1$

Mincer running example

Heckman two-step ("Heckit")

• Step 1 (full sample, probit): $P(s_i=1\mid z)=\Phi(z_i'\gamma)$. Need an exclusion restriction — a variable in $z$ that affects selection but not the outcome (e.g. kids under 6 → labour-force participation, not the wage). Compute inverse Mills ratio:

• Step 2 (selected sample, OLS): add $\hat\lambda_i$ as a regressor:

• Test for selection bias: $H_0:\rho=0$. Reject ⇒ endogenous selection present, correction matters.

Exam tip

Heckman mirrors 2SLS: step 1 builds a correction term, step 2 adds it as a control. Always state the exclusion-restriction variable and what testing $\rho=0$ tells you.

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7. Lecture 6 — Simultaneous equations & time series

Simultaneous equations (supply & demand)

Price & quantity jointly determined ⇒ $\text{Cov}(P,u)\neq0$ in both equations ⇒ OLS biased. Fix: a shifter of one curve instruments price to trace the other.

• Weather shifts supply → identifies the demand curve.
• Day of week shifts demand → identifies the supply curve.

Serial correlation

$\text{Cov}(u_t,u_{t-1})\neq0$. OLS stays unbiased & consistent but not efficient and SEs are wrong (too small) ⇒ invalid $t$/$F$.

• Test: regress $\hat u_t=\rho\hat u_{t-1}+e_t$; $H_0:\rho=0$.
• Fix: HAC (Newey–West) SEs — coefficients unchanged, SEs widened. R: vcov="newey_west".

Time-series OLS assumptions

• TS.2 Strict exogeneity $\mathbb E[u_t\mid\mathbf X]=0$ for all $t$ (past, present, future) → unbiasedness.
• Contemporaneous exogeneity $\mathbb E[u_t\mid x_t]=0$ → consistency only.
• TS.3 homoskedastic · TS.4 no serial corr · TS.5 normal.

Model zoo

• DL: $\delta_0$ = impact effect; $\sum_{k}\delta_k$ = long-run multiplier. Lags are collinear ⇒ individual $\delta_k$ imprecise but the sum can be well-identified.
• AR: $y_{t-1}$ correlates with $e_{t-1}$ ⇒ strict exogeneity fails ⇒ consistent but biased in finite samples.
• Seasonality: add seasonal dummies (11 months or 3 quarters, one omitted as reference). Skip if data already seasonally adjusted.

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8. Lecture 7 — Time trends & event studies

Deterministic trends

• Linear: $y_t=\alpha_0+\alpha_1 t+e_t$ — $\alpha_1$ = change per period.
• Exponential: $\log y_t=\alpha_0+\alpha_1 t+e_t$ — $\alpha_1\approx$ growth rate per period.

Spurious regression

Two trending series ⇒ huge $R^2$ + "significant" coef even if unrelated (OLS picks up shared drift). Two equivalent fixes (same $\hat\beta_1$ by Frisch–Waugh–Lovell):

1. Add $t$ as a regressor: $y_t=\beta_0+\beta_1x_t+\alpha t+u_t$.
2. Detrend both: regress $y$ and $x$ on $t$, keep residuals $\ddot y,\ddot x$, then regress $\ddot y$ on $\ddot x$.

Event study

• Estimation period (pre-event): fit a "normal" model, e.g. $R_t^{\text{stock}}=\beta_0+\beta_1R_t^{\text{market}}+u_t$.
• Abnormal return: $AR_t=R_t^{\text{actual}}-\hat R_t^{\text{predicted}}$.
• Dummy version: add day.before, day.of, day.after dummies; their coefficients $\hat\gamma$ are the ARs. Significant $\hat\gamma_0$ = event effect; significant $\hat\gamma_{-1}$ = anticipation/leak.
• Use trading days, not calendar days. Watch for confounding events near the window.

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9. Lecture 8 — Fixed effects in panel data

Panel = same units over time ⇒ compare a unit to itself, controlling for all its time-invariant traits.

$a_i$ = individual fixed effect — absorbs everything stable about $i$ (observed or not).

• Between variation (across-unit means) = confounded. Within variation (unit vs its own mean) = clean.
• Within estimator / demeaning: subtract each unit's mean, $\ddot y_{it}=y_{it}-\bar y_i$, then OLS:

Equivalent to adding a dummy per unit (FWL theorem). Time-invariant variables drop out (can't be estimated).

• Time FE: dummy per period (absorbs shocks common to all units). Two-way FE: unit + time. Interacted (city×month) absorbs cell-specific effects.
• R: feols(y ~ x | unit) · feols(y ~ x | city + month) · feols(y ~ x | city^month).

Standard-error chooser

Exam tip

"Why FE over OLS?" → name the time-invariant confounder, show $a_i$ absorbs it by demeaning (anything constant within a unit subtracts to 0), and cluster SEs at the FE level. Put the FE at the level where the confounder is constant.

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10. Lecture 9 — Regression Discontinuity (RDD)

Treatment switches at a cutoff of a running variable. Units just on either side are comparable ⇒ the jump in the outcome at the cutoff = the effect.

• Continuity: absent treatment, the outcome would pass smoothly through the cutoff (credible because units can't perfectly sort).
• Gives a LATE at the threshold: strong internal, weak external validity.

Sharp RDD regression (center the running variable!)

Treatment effect = $\hat\beta_2$ (coefficient on the Treated dummy = the gap at the cutoff). Centering at $c$ makes $\beta_0$ and $\beta_0+\beta_2$ the two lines' heights at the threshold. Quadratic version adds $(\text R-c)^2$ terms — jump is still the coef on Treated.

Fuzzy RDD = IV at the cutoff

Cutoff only shifts the probability of treatment ⇒ "above cutoff" instruments "treated." Wald estimator:

Validity checks

• McCrary density test: is the count of units smooth at the cutoff? A jump ⇒ manipulation/sorting.
• Placebo test: do predetermined covariates jump at the cutoff? They shouldn't.
• Bandwidth & functional form: narrow window so a line fits locally; check estimate is stable across bandwidths. Avoid high-order polynomials. R: rdrobust(y, x, c=0), rdbwselect.

Exam tip

Read the jump off the Treated coefficient. Always state continuity, that it's a local effect, and the two diagnostic tests (McCrary = data density; placebo = covariates). For fuzzy, report first-stage strength.

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11. Lecture 10 — Difference-in-Differences (DiD)

Two groups (treatment/control) × two periods (before/after). Works on repeated cross-sections (same individuals not needed). Estimand = ATT.

Estimator (difference of differences)

The control's change = the counterfactual common trend you subtract off.

As a regression (gives SEs + controls)

$d2$ = after dummy, $dT$ = treatment dummy. DiD effect = $\delta_1$ (the interaction). Maps to the 2×2 table: $\beta_0$ = control-before, $\delta_0$ = time trend, $\beta_1$ = fixed group gap, $\delta_1$ = effect. With many groups/periods this is two-way FE.

Validity

1. Parallel trends (the big one) — absent treatment, both groups move together. Can't prove; inspect pre-trends.
2. No coincident shock to the control group.
3. Groups comparable. Adding controls/FE ⇒ conditional parallel trends.

Exam tip

Compute $\delta_1$ from the 2×2 means (either difference order — same answer). If significance asked, it's the interaction coef in the regression. Always name parallel trends as the identifying assumption.

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12. Standard-error & test quick-reference

Critical values (5%, large $n$): $t,z\approx1.96$ · $\chi^2_1=3.84$, $\chi^2_2=5.99$, $\chi^2_3=7.81$, $\chi^2_4=9.49$ · weak-instrument first-stage F > 10.

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13. Exam question playbook

"Interpret this coefficient." Identify the model first. LPM → pp change in $P(y=1)$. Log-dep → $\approx100\beta\%$. Logit/probit raw → sign & significance only (magnitude needs $g(\cdot)\beta$). State significance: $|t|>1.96$ or $p<0.05$.

"Is $\hat\beta$ causal / what's wrong with OLS here?" Name the mechanism — confounder (OVB), reverse causality, simultaneity, or selection — write $\text{Cov}(x,u)\neq0$, then prescribe the fix (control / FE / IV / Heckman / RDD / DiD).

"Draw / update the causal diagram." Put the confounder as a node with arrows into both the regressor and the outcome; that backdoor path is the bias. Unobserved confounder sits in $u$.

"Evaluate this instrument." Relevance $\text{Cov}(z,x)\neq0$ (first-stage F>10, testable) + validity $\text{Cov}(z,u)=0$ (argue, untestable). Give one failure story.

"Compute the DiD / RDD effect." DiD = difference of the two changes (= interaction $\delta_1$). RDD = coefficient on the Treated dummy (centered running var). Fuzzy RDD = outcome jump ÷ treatment-prob jump.

"Which SEs and why?" Heteroskedasticity → robust; time series → Newey–West; panel/FE → clustered. Default homoskedastic SEs are almost never right in this course.

"State the identifying assumption." IV → exclusion restriction. FE → confounder is time-invariant. RDD → continuity (no sorting). DiD → parallel trends. Heckman → exclusion variable in selection eqn.

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The 6 estimators in one breath

LPM = OLS on 0/1 (pp, robust SE). Logit/Probit = CDF squashes to (0,1), ME = $g(\cdot)\beta$, MLE + LR test. IV/2SLS = borrow exogenous variation, relevance + validity, LATE. Heckman = probit selection → add IMR. Fixed effects = demean to kill time-invariant confounders, cluster SEs. RDD = jump at a cutoff (continuity). DiD = difference of differences (parallel trends).