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10 questions · 1 formulas

Prediction & residual

Ŷ = b0 + b1X1 + … + bkXk; e = Y − Ŷ

Trap: Residual = actual − predicted (not the reverse).

Regression BasicsE-level★ High priority — official Canvas example

Actual sale price = $2,800,000; predicted price = $1,207,607. Residual?

→ C) $1,592,393

Canvas Example MCQ — Q52

A) $4,392,393
B) $1,207,607
C) $1,592,393
D) $3,602,607
E) None of A-D is close; write the correct answer.

Residual = actual - predicted = 2,800,000 - 1,207,607 = 1,592,393.

Rule: Positive residual means the actual outcome was above the fitted value.

Regression BasicsE-level★ High priority — official Canvas example

How are OLS residuals calculated?

→ B) actual value minus fitted value.

Canvas Example MCQ — Q56

A) using the unobservable population errors.
B) actual value minus fitted value.
C) they are unknown because the population regression function is unknown.
D) they should not be used because they indicate the regression does not pass through every observation.

e_i = Y_i - Ŷ_i. Population errors are unobserved; residuals are observed after estimation.

Rule: Residual = actual - fitted.

Regression BasicsD–A levelVery high priority

The number of observations is_________.

→ C) 89

Old Exam — Dec 2022, Q18

Given: Regression summary: residual df=84; model has 4 regressors plus intercept. SSR=5.1462, TSS=8.0781.

A) 87
B) 88
C) 89
D) Not possible to compute given the information in the summary output.
E) None of the above (A, B, C, D) is close to be correct.

Residual df = n - number_of_parameters. There are 4 slopes + intercept =5 parameters, so n=84+5=89.

Rule: Identify model units and form before calculating. Never confuse coefficient size, significance, R2 and prediction accuracy.

Regression BasicsD–A levelVery high priority

The standard error of the regression is closest to _________.

→ B) 0.1868

Old Exam — Dec 2022, Q20

Given: Regression summary: SSE=8.0781-5.1462=2.9319; residual df=84.

A) 5.1462
B) 0.1868
C) 0.0349
D) Not possible to compute given the information in the summary output.
E) None of the above (A, B, C, D) is close to be correct.

SER = sqrt(SSE/df_resid). SSE=8.0781-5.1462=2.9319; sqrt(2.9319/84)=0.1868.

Rule: Identify model units and form before calculating. Never confuse coefficient size, significance, R2 and prediction accuracy.

Regression BasicsD–A levelVery high priority

The standard error of the regression is closest to _________.

→ C) 0.273

Old Exam — Oct 2022, Q14

Given: Regression output: SSE=TSS-SSR; residual df=735.

A) 0.0745
B) 0.354
C) 0.273
D) Not possible to compute given the information in the summary output.
E) None of the above (A, B, C, D) is close to be correct.

SER=sqrt[(182.1268-127.4041)/735]≈0.2729 -> C.

Rule: Identify model units and form before calculating. Never confuse coefficient size, significance, R2 and prediction accuracy.

Regression BasicsD–A levelVery high priority

The standard error of the regression is closest to _________.

→ A) 0.329

Old Exam — Oct 2021, Q18

Given: Regression output: n=2160, k=5, SSE=TSS-SSR.

A) 0.329
B) 0.799
C) 0.126
D) Not possible to compute given the information in the summary output.

SSE=642.3786-409.7599=232.6187; residual df=2160-6=2154. SER=sqrt(232.6187/2154)≈0.329 -> A.

Rule: Identify model units and form before calculating. Never confuse coefficient size, significance, R2 and prediction accuracy.

Panel DataD–A levelNormal priority

How do you test for endogenous regressors, or correlation between the error term and any regressor in a panel data random effects model?

→ A) Estimate coefficients with RE and FE; then perform a Hausman test.

Old Exam — Oct 2021, Q24

Given: Hausman concept: estimate FE and RE; test whether differences in common coefficients are systematic.

A) Estimate coefficients with RE and FE; then perform a Hausman test.
B) Estimate the model capturing estimated residuals; then regress residuals on all regressors and perform an Chi2-test
C) Estimate RE model capturing estimated residuals; then estimate coefficients of correlation with each regressor.
D) Estimate RE and FE models and perform an F test on each model individually. If the difference between the F statistics is significant, conclude endogeneity.

Hausman compares FE and RE estimates. If differences are systematic, RE is inconsistent and FE is preferred -> A.

Rule: Time-invariant variable -> FE cannot estimate its coefficient. Small Hausman p-value -> reject RE, prefer FE.

Chi-square & Time SeriesD–A levelLegacy / lower priority

The reason you include three augmentation terms is to__________.

→ A) eliminate autocorrelation in the errors.

Old Exam — Oct 2023, Q13

Given: ADF augmentation lags are added to remove serial correlation from the test regression errors.

A) eliminate autocorrelation in the errors.
B) eliminate heteroscedasticity in the errors.
C) strengthen autocorrelations in the errors.
D) to do the Hausman test for random effects model.
E) None of the above (A, B, C, D) is close to be correct.

ADF augmentation lags of delta-y are included to soak up serial correlation in the test-regression residuals -> A.

Rule: Stationarity means stable distributional properties over time. Current relevance is secondary unless reinforced by current Canvas/labs.

Scenario RiskD–A levelVery high priority

Comparing Problem 3 (unlevered: E[IRR]=10.01%, σ=1.62%) to Problem 4 (levered with a $1.5m/10%/15yr loan: E[Equity IRR]=10.14%, σ=5.52%), has adding leverage increased the equity investor's risk?

→ A) Yes — standard deviation rises from 1.62% to 5.52% while expected return rises only slightly, because fixed debt service makes equity a residual claim

Computer Lab 1 — Exercise 3 Problem 4

A) Yes — standard deviation rises from 1.62% to 5.52% while expected return rises only slightly, because fixed debt service makes equity a residual claim
B) No — leverage has almost no effect since expected return barely changed
C) No — standard deviation actually falls with leverage because debt diversifies risk
D) Cannot be determined without knowing the loan's credit rating

Fixed debt service must be paid regardless of NOI outcome, so equity cash flows (the residual) swing proportionally more with property performance. Standard deviation more than triples (1.62%→5.52%) while expected return only rises slightly (10.01%→10.14%).

Rule: Financial leverage magnifies equity risk because debt service is a fixed prior claim; equity absorbs the full variability on a proportionally smaller base.

Scenario RiskD–A levelVery high priority

Comparing unlevered (E[IRR]=10.01%, σ=1.62%) to levered (E[IRR]=10.14%, σ=5.52%) for the same property, has the $1.5m loan increased risk, and why?

→ A) Yes — σ rises from 1.62% to 5.52% because fixed debt service makes equity a residual claim, amplifying the effect of NOI/sale-price changes on equity returns

Computer Lab 3 — Problem 4B

A) Yes — σ rises from 1.62% to 5.52% because fixed debt service makes equity a residual claim, amplifying the effect of NOI/sale-price changes on equity returns
B) No — expected return only rose slightly (10.01%→10.14%), so risk is essentially unchanged
C) No — leverage reduces risk by diversifying the capital structure
D) Risk is unchanged because E[IRR]/σ actually increased with leverage

σ more than triples (1.62%→5.52%). Because debt service is fixed, all variability in property performance (NOI, sale price) flows through to the smaller equity base, magnifying the percentage swings in equity returns.

Rule: Leverage magnifies equity-return volatility because debt holders are paid first (fixed claim) while equity absorbs all residual variability.