Skip to content

๐Ÿ“Š Key Diagrams โ€” ECMT5001 Econometrics

Core econometrics diagrams for quick reference and exam revision. All diagrams are inline SVG โ€” zoom in on any device.

1. OLS Scatter Plot and Fitted Line

The fundamental picture of OLS regression: minimising the sum of squared vertical distances from each point to the fitted line.

x y รปแตข = yแตข โˆ’ ลทแตข ลท = ฮฒฬ‚โ‚€ + ฮฒฬ‚โ‚x 0 x (regressor) y (outcome) OLS: Minimising Sum of Squared Residuals Data points (xแตข, yแตข) Fitted line ลท = ฮฒฬ‚โ‚€ + ฮฒฬ‚โ‚x Residuals รปแตข

Key facts: - OLS chooses \(\hat{\beta}_0, \hat{\beta}_1\) to minimise \(\sum_{i=1}^n \hat{u}_i^2\) - Residuals \(\hat{u}_i = y_i - \hat{y}_i\) are the vertical distances (not perpendicular) - The line always passes through \((\bar{x}, \bar{y})\); residuals sum to zero

2. SST = SSE + SSR Decomposition

Total variation in y is decomposed into explained (SSE) and unexplained (SSR) parts. Rยฒ = SSE/SST.

ศณ SSR (residual) = (yแตข โˆ’ ลทแตข)ยฒ SSE (explained) = (ลทแตข โˆ’ ศณ)ยฒ SST (total) = (yแตข โˆ’ ศณ)ยฒ yแตข ลทแตข x y SST = SSE + SSR โ†’ Rยฒ = SSE/SST = 1 โˆ’ SSR/SST SST = SSE + SSR Decomposition
Quantity Formula Meaning
\(\text{SST}\) \(\sum(y_i - \bar{y})^2\) Total variation in \(y\)
\(\text{SSE}\) \(\sum(\hat{y}_i - \bar{y})^2\) Variation explained by the regression
\(\text{SSR}\) \(\sum \hat{u}_i^2 = \sum(y_i - \hat{y}_i)^2\) Residual (unexplained) variation
\(R^2\) \(\text{SSE}/\text{SST} = 1 - \text{SSR}/\text{SST}\) Proportion of variance explained

3. Omitted Variable Bias โ€” Direction Table

The sign and magnitude of OVB depends on two things: the correlation between the omitted and included variable, and the sign of the omitted variable's true coefficient.

OVB formula: \(\tilde{\beta}_1 = \beta_1 + \underbrace{\beta_2 \cdot \delta_{12}}_{\text{bias}}\) where \(\delta_{12} = \text{Cov}(x_2, x_1)/\text{Var}(x_1)\)

Omitted has positive effect (\(\beta_2 > 0\)) Omitted has negative effect (\(\beta_2 < 0\))
Omitted correlated positively with \(x_1\) (\(\delta_{12} > 0\)) โฌ†๏ธ Upward bias โ€” \(\tilde{\beta}_1 > \beta_1\) โฌ‡๏ธ Downward bias โ€” \(\tilde{\beta}_1 < \beta_1\)
Omitted correlated negatively with \(x_1\) (\(\delta_{12} < 0\)) โฌ‡๏ธ Downward bias โ€” \(\tilde{\beta}_1 < \beta_1\) โฌ†๏ธ Upward bias โ€” \(\tilde{\beta}_1 > \beta_1\)

Classic OVB example

Regressing wages on education only, omitting ability:

  • Ability raises wages โ†’ \(\beta_2 > 0\)
  • Ability is positively correlated with education โ†’ \(\delta_{12} > 0\)
  • Bias = upward: the estimated return to education is overstated because it partly captures the return to unobserved ability.

4. Heteroskedasticity: Residual Plots

Left panel: homoskedastic (constant variance โ€” OLS valid). Right panel: heteroskedastic (fan pattern โ€” OLS SEs biased).

Homoskedastic (MLR.5 holds) โœ“ ลท (fitted values) Residual รป Var(รป) constant โ€” OLS SEs valid Heteroskedastic (MLR.5 violated) โœ— ลท (fitted values) Residual รป Var(รป) increases with ลท โ€” fan pattern โš ๏ธ

What to do when you see heteroskedasticity:

  1. Formal tests: Breusch-Pagan (LM = nยทRยฒ_aux ~ ฯ‡ยฒ_k) or White test (more general)
  2. Always report robust (HC/White) standard errors in applied work
  3. If variance function is known: use WLS with weights \(w_i = 1/h_i\)

5. Instrumental Variables โ€” Causal Path Diagram

A valid instrument z must affect y ONLY through x (exclusion restriction), and must be correlated with x (relevance).

z Instrument x Endogenous regressor y Outcome u Confounder/error Relevance Cov(z,x) โ‰  0 ฮฒโ‚ (causal) Cov(x,u)โ‰ 0 (endogeneity!) โœ— Exclusion restriction: Cov(z,u) = 0 z affects y ONLY through x Instrumental Variables: Causal Structure 2SLS Procedure: Stage 1: regress x on z โ†’ get xฬ‚ Stage 2: regress y on xฬ‚
Condition What it means Testable?
Relevance: \(\text{Cov}(z,x) \neq 0\) Instrument must move the endogenous regressor โœ… Yes โ€” first-stage F-stat (rule of thumb: F > 10)
Exclusion: \(\text{Cov}(z,u) = 0\) Instrument affects \(y\) only through \(x\), not directly โŒ No โ€” requires economic argument

DiD estimates the treatment effect by comparing the change in the treated group to the change in the control group over the same period.

Treatment occurs Pre-treatment Post-treatment Control Treated Counterfactual (no treatment) ฮฒฬ‚_DD (treatment effect) Y tโ‚€ tโ‚ tโ‚‚ ฮฒฬ‚_DD = (ศฒโ‚โ‚ โˆ’ ศฒโ‚โ‚€) โˆ’ (ศฒโ‚€โ‚ โˆ’ ศฒโ‚€โ‚€) = Change in treated minus Change in control Parallel Trends Assumption: Counterfactual trend = Control trend Differences-in-Differences: Parallel Trends

The logic of DiD: - First difference (treated group: post โˆ’ pre): removes time-invariant characteristics of the treated group - Second difference (control group: post โˆ’ pre): estimates the counterfactual trend - DiD: treatment effect = \((\bar{Y}_{1,1} - \bar{Y}_{1,0}) - (\bar{Y}_{0,1} - \bar{Y}_{0,0})\)

Regression form: \(y_{it} = \alpha + \gamma D_i^{treat} + \lambda D_t^{post} + \beta(D_i^{treat} \times D_t^{post}) + \varepsilon_{it}\)

The coefficient \(\beta\) on the interaction term is the DiD estimate.

7. Quick Reference: OVB Formula

\[\tilde{\beta}_1 = \underbrace{\beta_1}_{\text{true effect}} + \underbrace{\beta_2 \cdot \frac{\text{Cov}(x_2, x_1)}{\text{Var}(x_1)}}_{\text{omitted variable bias}}\]

where \(x_2\) is the omitted variable, \(\beta_2\) is its true coefficient, and the second term is zero only if \(\text{Cov}(x_2, x_1) = 0\).

8. Key Formulas at a Glance

Concept Formula
OLS slope \(\hat{\beta}_1 = \dfrac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2} = \dfrac{\widehat{\text{Cov}}(x,y)}{\widehat{\text{Var}}(x)}\)
OLS intercept \(\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\)
R-squared \(R^2 = 1 - \text{SSR}/\text{SST}\)
Variance of OLS slope \(\text{Var}(\hat{\beta}_1) = \dfrac{\sigma^2}{\sum(x_i-\bar{x})^2}\)
Error variance estimator \(\hat{\sigma}^2 = \dfrac{\text{SSR}}{n-k-1}\)
t-statistic \(t = \dfrac{\hat{\beta}_j - \beta_{j,0}}{\text{se}(\hat{\beta}_j)} \sim t_{n-k-1}\)
F-statistic \(F = \dfrac{(\text{SSR}_r - \text{SSR}_{ur})/q}{\text{SSR}_{ur}/(n-k-1)} \sim F_{q,\,n-k-1}\)
BP LM test \(LM = n \cdot R^2_{\text{aux}} \sim \chi^2_k\)
IV estimator \(\hat{\beta}_{IV} = \dfrac{\text{Cov}(z,y)}{\text{Cov}(z,x)}\)
DiD estimator \(\hat{\beta}_{DD} = (\bar{Y}_{1,1} - \bar{Y}_{1,0}) - (\bar{Y}_{0,1} - \bar{Y}_{0,0})\)
Durbin-Watson \(DW \approx 2(1-\hat{\rho})\)
Quadratic turning point \(x^* = -\hat{\beta}_1/(2\hat{\beta}_2)\)