Y = beta_0 + beta_1 * X_1 + beta_2 * X_2 + ... + beta_k * X_k + epsilon

OLS minimizes: Sum of squared residuals = Sum(Y_i - Y_hat_i)^2

Linear-linear:  Y = a + bX       → 1 unit increase in X → b unit change in Y
Log-linear:     ln(Y) = a + bX   → 1 unit increase in X → b*100% change in Y
Linear-log:     Y = a + b*ln(X)  → 1% increase in X → b/100 unit change in Y
Log-log:        ln(Y) = a + b*ln(X) → 1% increase in X → b% change in Y (elasticity)

Y_it = beta_0 + beta_1 * X_it + epsilon_it

Ignores panel structure. Inconsistent if entity-specific effects correlated with X.

Y_it = alpha_i + beta * X_it + epsilon_it

alpha_i = entity-specific fixed effect (absorbed / differenced out)

Advantages:
  - Controls for all time-invariant unobserved heterogeneity
  - Consistent even if alpha_i is correlated with X_it
  - Most common panel estimator in applied economics

Limitations:
  - Cannot estimate effects of time-invariant variables (industry, country dummies)
  - Requires within-entity variation for identification
  - Less efficient than RE if RE assumptions hold

Time fixed effects: Y_it = alpha_i + gamma_t + beta * X_it + epsilon_it
  → Also absorbs common shocks affecting all entities in each period
  → Two-way fixed effects (entity + time) is standard practice

Y_it = beta_0 + beta * X_it + (alpha_i + epsilon_it)

alpha_i treated as random, uncorrelated with X_it
More efficient than FE if assumption holds
Can estimate coefficients on time-invariant variables

H0: alpha_i uncorrelated with X_it (RE is consistent and efficient)
H1: alpha_i correlated with X_it (only FE is consistent)

If p-value < 0.05: Reject H0, use FE
If p-value > 0.05: Fail to reject, RE is preferred (more efficient)

In practice: FE is the default choice in most applied work because
the RE assumption is strong and often implausible

Requirements for a valid instrument Z:
  1. Relevance: Cov(Z, X) ≠ 0 (Z predicts X — testable)
  2. Exclusion restriction: Cov(Z, epsilon) = 0 (Z affects Y only through X — NOT testable)

Two-Stage Least Squares (2SLS):
  Stage 1: X_hat = gamma_0 + gamma_1 * Z + v     (regress X on instrument)
  Stage 2: Y = beta_0 + beta_1 * X_hat + epsilon  (regress Y on predicted X)

Diagnostics:
  - First-stage F-statistic: F > 10 indicates instrument is not weak (Staiger-Stock rule)
  - Overidentification test (Sargan/Hansen J): If more instruments than endogenous variables,
    test whether excluded instruments are uncorrelated with error (if rejected, at least one
    instrument is invalid)
  - Weak instruments: IV estimates are biased toward OLS; use LIML instead of 2SLS

Y_t = c + A_1 * Y_{t-1} + A_2 * Y_{t-2} + ... + A_p * Y_{t-p} + epsilon_t

Y_t = vector of endogenous variables (e.g., GDP growth, inflation, interest rate)
A_i = coefficient matrices
p   = lag order (selected by information criteria: AIC, BIC, HQ)
epsilon_t = vector of error terms (serially uncorrelated)

Y_it = beta_0 + beta_1 * Treat_i + beta_2 * Post_t + beta_3 * (Treat_i x Post_t) + epsilon_it

beta_3 = DiD estimator = causal effect of treatment

Key assumption: Parallel trends — absent treatment, treated and control groups
would have followed the same trend

Diagnostics:
  - Pre-treatment trend test: Plot outcomes for treated vs control before treatment
  - Placebo tests: Apply DiD to periods before treatment (should find no effect)
  - Event study specification: Estimate treatment effect at each time period

=== REGRESSION RESULTS ===

Dependent variable: __________    N = ____    R-squared = ____

                    (1) OLS      (2) FE       (3) IV/2SLS
Variable 1          ____         ____          ____
                   (SE)         (SE)          (SE)
Variable 2          ____         ____          ____
                   (SE)         (SE)          (SE)
Variable 3          ____         ____          ____
                   (SE)         (SE)          (SE)
Constant            ____         —             ____
                   (SE)                       (SE)

Entity FE           No           Yes           No
Time FE             No           Yes           No
Clustered SEs       No           Entity        No
First-stage F       —            —             ____
Hausman test p      —            ____          —
Observations        ____         ____          ____
R-squared           ____         ____          ____

Standard errors in parentheses. * p<0.10, ** p<0.05, *** p<0.01

=== ECONOMETRIC DIAGNOSTIC CHECKLIST ===

Model: __________    Estimator: __________

[ ] Heteroskedasticity test (Breusch-Pagan / White):      p = ____
    → If rejected: using robust SEs? [ ] Yes
[ ] Autocorrelation test (Durbin-Watson / Breusch-Godfrey): DW = ____
    → If detected: using Newey-West SEs? [ ] Yes
[ ] Multicollinearity (max VIF):                           VIF = ____
    → If VIF > 10: action taken? ____
[ ] Endogeneity (Hausman test for panel / theory for IV):  p = ____
    → If suspected: IV or FE used? [ ] Yes
[ ] Specification (Ramsey RESET):                          p = ____
    → If rejected: functional form reviewed? [ ] Yes
[ ] Normality of residuals (Jarque-Bera):                  p = ____
    → If rejected and small sample: bootstrap used? [ ] Yes
[ ] Stationarity (ADF test on each variable):              ____
    → If I(1): differenced or cointegration tested? [ ] Yes