QUANTITATIVE METHODS: MULTIPLE REGRESSION CHEAT SHEET

CLRM ASSUMPTIONS & VIOLATIONS

|---|---|---|---|---|---|

---

MODEL FIT & SIGNIFICANCE TESTS

R-squared: SSR/SST = 1 − SSE/SST Adjusted R-squared: 1 − [(1 − R²)(n−1)/(n−k−1)]

*Use to compare models with different k*

F-statistic: (SSR/k) / (SSE/(n−k−1)) = MSR/MSE

*H₀: all slope coefficients = 0*

*Significant F ≠ all variables individually significant (see Q1)*

t-statistic: (b̂ − b₀) / SE(b̂), df = n−k−1

*Significant at 5% if p < 0.05 or |t| > ~2 (large n)*

---

HETEROSKEDASTICITY

|---|---|

---

SERIAL CORRELATION

|---|---|

---

MULTICOLLINEARITY

|---|---|

---

COMMON TRAPS

|---|---|

4. Constant variance	Heteroskedasticity	Unbiased	↓ Reliability of SE, t, p	Breusch-Pagan, residual plot	White (HC) SE, transform
Assumption	Violates	Effect on Coefficients	Effect on Inference	Detection	Fix
1. Linear in parameters	—	—	—	—	—
2. No perfect collinearity	Multicollinearity	Unbiased	↑ Std. errors	VIF, t vs F pattern	Remove/combine variables
3. E(ε) = 0	—	—	—	—	—
5. No serial correlation	Serial Correlation	Unbiased	↓ Reliability of SE, t, p	Durbin-Watson, ACF	Newey-West SE, lag variables
6. Normality of ε	—	Unbiased (large n)	↓ Inference	Q-Q plot, histogram	Large sample, robust methods
Cue	Answer
Definition	Variance of ε varies across observations
Root causes	Conditional on variable size; omitted variables
Consequence	SE typically understated → inflated t-stats → spurious significance
Durbin-Watson?	NO—DW tests serial correlation
Breusch-Pagan?	YES—regress squared residuals on X's; chi-square test
Visual test	Plot residuals or squared residuals vs fitted; fan shape = heteroskedasticity
Fix	White (HC) std. errors; log/root transform; robust regression
Cue	Answer
Definition	Cov(εᵢ, εⱼ) ≠ 0 for i ≠ j; common in time series
Durbin-Watson	DW ≈ 2(1 − r), where r = first-lag correlation
DW = 2	No serial correlation
DW → 0	Positive serial correlation
DW → 4	Negative serial correlation
Consequence	SE typically understated → inflated t-stats
Fix	Newey-West (HAC) SE; include lagged Y or residuals; generalized LS
Cue	Answer
Definition	Two or more X's highly correlated with each other
Effect on coefficients	Unbiased but unstable
Effect on SE	↑↑ Standard errors inflate
Classic pattern	High F-stat + low individual t-stats (see Q1)
VIF formula	VIF_j = 1/(1 − R²_j) where R²_j from regressing X_j on other X's
VIF threshold	VIF > 5 → concern; VIF > 10 → severe
Other signs	Large coefficient swings when variables added/removed; high R² but few sig. t's
Fix	Remove redundant variable; combine into composite; domain knowledge
Mistake	Reality
Adjusted R² < R² means misspecification	Always true by design—not a problem
High F-stat means all variables matter	No—F tests joint hypothesis; individual t's show relevance
Multicollinearity biases coefficients	False—it inflates SE, not coefficients
Heteroskedasticity biases coefficients	False—it distorts SE, t-stats, and p-values
Serial correlation detected by Breusch-Pagan	Wrong test—BP is for heteroskedasticity; use DW or ACF

---

QUICK DECISION TREE

High F, low individual t-stats? → Multicollinearity or irrelevant variable Residuals fan-shaped vs fitted? → Heteroskedasticity (use White SE) DW << 2? → Positive serial correlation (use Newey-West SE) VIF > 5 for a variable? → Consider removing/combining Need to compare two models with different # of predictors? → Use adjusted R², not R²

Quantitative Methods

QUANTITATIVE METHODS: MULTIPLE REGRESSION CHEAT SHEET

CLRM ASSUMPTIONS & VIOLATIONS

MODEL FIT & SIGNIFICANCE TESTS

HETEROSKEDASTICITY

SERIAL CORRELATION

MULTICOLLINEARITY

COMMON TRAPS

QUICK DECISION TREE

Make this cheat sheet yours

Sections in this topic