Week 2 Assessment — The OLS Engine

This is the end-of-week assessment for Week 2. It has three parts. Part A is a derive-and-interpret section on the spine of the week — OLS in matrix form and the normal equations, the hat and residual-maker matrices, Gauss–Markov and what “BLUE” actually buys you, Frisch–Waugh–Lovell and partialling-out, robust and clustered standard errors, and omitted-variable bias and attenuation. Part B is a small panel you simulate, estimate, and report on, ending with the single most important habit of the week: choosing the right standard error and reading what changes. Part C is the rubric, with explicit point totals. An instructor answer key follows Part C.

The whole thing is one focused sitting plus the coding task. Methods are limited to Week 2: matrix OLS, the four/five classical assumptions, FWL, the sandwich (HC and clustered), HAC by name, and the OVB and measurement-error formulas. No instrumental variables, no panel fixed-effects estimators beyond the within-transformation idea, no causal-design machinery — those are Weeks 3–4. Show your reasoning. A correct number with no argument earns little; an honest “this is an SE problem, not a bias problem, and here is how I know” earns a great deal.

Total: 100 points. Part A = 48, Part B = 42, Presentation/honesty woven through both = 10.

Throughout, write OLS as \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon} with \hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}, and label every standard error with its flavor: classical, HC1/HC2/HC3, HAC (Newey–West), or clustered.

Part A — Derive and interpret (48 points; 8 points each)

Answer in a few lines of algebra plus two to four sentences of interpretation. Where a derivation is asked for, the steps must be visible; a boxed final formula with no path earns half credit at most.

A1. (Normal equations from two directions.) Start from the sum-of-squared-residuals objective S(\mathbf{b}) = (\mathbf{y}-\mathbf{X}\mathbf{b})'(\mathbf{y}-\mathbf{X}\mathbf{b}). (i) Differentiate with respect to \mathbf{b} and set the gradient to zero to derive the normal equations \mathbf{X}'\mathbf{X}\hat{\boldsymbol{\beta}} = \mathbf{X}'\mathbf{y}, and state the one condition on \mathbf{X} that lets you write \hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}. (ii) Now derive the same equations from orthogonality alone — without calculus — using the geometric fact that the residual vector \hat{\boldsymbol{\varepsilon}} must be orthogonal to the column space of \mathbf{X}. State in one sentence why, when \mathbf{X} contains a column of ones, this forces the residuals to sum to zero.

A2. (The hat matrix is a projector.) Define \mathbf{H} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' and \mathbf{M} = \mathbf{I} - \mathbf{H}. (i) Prove \mathbf{H} is symmetric and idempotent (\mathbf{H}\mathbf{H}=\mathbf{H}), and show \mathbf{H}\mathbf{X} = \mathbf{X}. (ii) Using \hat{\boldsymbol{\varepsilon}} = \mathbf{M}\mathbf{y} and the Pythagorean split \lVert\mathbf{y}\rVert^2 = \lVert\hat{\mathbf{y}}\rVert^2 + \lVert\hat{\boldsymbol{\varepsilon}}\rVert^2, explain in words what the diagonal entry h_{ii} measures and why an observation with h_{ii} close to 1 is one a careful analyst worries about. (You do not need to re-derive the Pythagorean split.)

A3. (Gauss–Markov and the meaning of “best.”) State the classical assumptions under which OLS is BLUE, decoding what each of the four letters B-L-U-E claims. Then consider a rival linear estimator \tilde{\boldsymbol{\beta}} = \mathbf{C}\mathbf{y} with \mathbf{C} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' + \mathbf{D}. Show that requiring \tilde{\boldsymbol{\beta}} to be unbiased for every \boldsymbol{\beta} forces \mathbf{D}\mathbf{X} = \mathbf{0}, and state (you need not fully expand the algebra) why this makes \operatorname{Var}(\tilde{\boldsymbol{\beta}}) = \operatorname{Var}(\hat{\boldsymbol{\beta}}) + \sigma^2\mathbf{D}\mathbf{D}', a sum of the OLS variance and something positive-semidefinite. Finish with one sentence: which single classical assumption, when it breaks, is the one that demotes OLS from “best” — and does its breaking make \hat{\boldsymbol{\beta}} biased?

A4. (FWL / partialling-out — and “is this a bias problem or an SE problem?”) Leah runs the long regression of patent count y on R&D spending x and firm size z (plus an intercept), and gets \hat\beta_1 = 1.84 on R&D. State the Frisch–Waugh–Lovell theorem precisely enough to describe the two-step recipe that reproduces \hat\beta_1 exactly: which two auxiliary regressions you run, and what final regression you run on their residuals. Then answer this: Leah notices R&D and firm size are highly correlated in her sample. A colleague says “high correlation between your regressors biases your R&D coefficient.” Is this a bias problem or a standard-error problem? Defend your answer in two sentences using FWL geometry (what near-collinearity does to the residualized regressor \tilde{x}).

A5. (Robust vs. clustered SEs — name the threat and the fix.) Maya regresses loan APR on a borrower characteristic across N = 9{,}000 loans that come from G = 9 lenders, 1,000 loans each. The within-lender errors are positively correlated because each lender sets rates with a correlated house style. (i) Write the general sandwich form of \widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}}) and say in one sentence how the clustered version builds its middle differently from the HC (heteroskedasticity-robust) version. (ii) Using the Moulton-style approximation \text{true Var} \approx \text{naive Var}\times[\,1+(n-1)\rho_x\rho_\varepsilon\,], with cluster size n=1{,}000 and a mild \rho_x\rho_\varepsilon = 0.05, compute the factor by which the classical SE understates the truth (give the variance inflation and the SE inflation). (iii) Name the threat and the design/fix in one sentence, in the spec-discipline format: state what level you cluster at and what economic claim that choice encodes — and note the one thing about G = 9 that should make Maya nervous.

A6. (OVB and attenuation — two contaminations, two directions.) (i) Derive the omitted-variable-bias formula: with true long model y = \beta_0 + \beta_1 x + \beta_2 z + \varepsilon (assumptions holding in the long model) but a short regression of y on x alone, show \hat{\tilde\beta}_1 \xrightarrow{p} \beta_1 + \beta_2\delta_1 where \delta_1 = \operatorname{Cov}(x,z)/\operatorname{Var}(x). Then apply the two-sign rule to Maya’s case: the omitted variable is creditworthiness C, which raises approval (\beta_2 > 0) and is lower on average for the disadvantaged group D=1 so that the slope of C on D is negative (\delta_1 < 0) — state the sign of the bias on the D coefficient and what it does to the apparent approval penalty. (ii) Now a different contamination: Maya’s income variable is self-reported, x = x^\* + m with classical noise m. State the attenuation formula \hat\beta_1 \xrightarrow{p} \beta_1\lambda, define the reliability ratio \lambda, and say which way the income coefficient is pushed. Close with one sentence contrasting OVB and attenuation: which one has a known direction regardless of the data, and why.

Part B — Simulate a clustered panel, estimate β̂ from scratch, and choose the SE (42 points)

You will build a small panel where you control the truth, estimate \hat\beta by hand from the matrix formula, and then compute classical, HC1, and clustered standard errors for the same point estimate. The headline you are chasing is the one from Ch 2.4: clustering does not move \hat\beta — it moves the t-statistic, sometimes by a lot, and that change is the whole story. This ties Ch 2.1 (the matrix estimator), Ch 2.2 (unbiasedness under the right assumptions), and Ch 2.4 (the sandwich) together in one script.

The data-generating process

Simulate a balanced panel of G lenders, each with n loans, so N = G n. For lender g and loan i:

y_{ig} = \beta_0 + \beta_1 x_{ig} + \underbrace{a_g + e_{ig}}_{\varepsilon_{ig}}, \qquad a_g \sim \mathcal{N}(0,\sigma_a^2),\quad e_{ig}\sim\mathcal{N}(0,\sigma_e^2).

The term a_g is a lender-level shock shared by every loan at that lender — it is what induces within-cluster error correlation. Make the regressor also carry a lender component so that \rho_x > 0: draw x_{ig} = c_g + v_{ig} with c_g\sim\mathcal{N}(0,1) and v_{ig}\sim\mathcal{N}(0,1). Use a true slope \beta_1 = 0.30, \beta_0 = 1.0, \sigma_a = 1.0, \sigma_e = 1.0. Pin your random seed and state it.

Tasks

B1. Estimate β̂ from scratch (10 pts). With G = 40 and n = 50 (N = 2{,}000), build the design matrix \mathbf{X} (a column of ones and the x column) and compute \hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y} using only array operations (numpy), not a regression library. Report \hat\beta_1 and confirm it is close to 0.30. Verify two Ch 2.1 identities numerically: the residuals sum to (essentially) zero, and they are orthogonal to the x column (\mathbf{X}'\hat{\boldsymbol{\varepsilon}} \approx \mathbf{0}).

B2. Three standard errors for one β̂ (14 pts). For that same \hat{\boldsymbol{\beta}}, compute three standard errors on \hat\beta_1:

classical, \hat\sigma^2(\mathbf{X}'\mathbf{X})^{-1} with \hat\sigma^2 = \hat{\boldsymbol{\varepsilon}}'\hat{\boldsymbol{\varepsilon}}/(N-K);
HC1, the heteroskedasticity-robust sandwich with the usual N/(N-K) correction;
clustered by lender, the block sandwich summing the outer products over the G lenders.

You may write these from the matrix formulas (preferred) or use statsmodels for HC1 and the clustered SE as a check — but the clustered computation should match a from-scratch block sum to a few decimals. Report all three SEs and the three implied t-statistics for H_0:\beta_1 = 0 in a small table.

B3. Read the t-stat change (12 pts). In one short paragraph answer: Which SE is largest, and why is that the honest one here? By roughly what factor does the clustered SE exceed the classical SE, and how does that compare to what the Moulton approximation \sqrt{1+(n-1)\rho_x\rho_\varepsilon} would predict for your DGP? State explicitly: did \hat\beta_1 itself change across the three SE flavors? Conclude with the one-sentence verdict the week is built around — bias problem or SE problem? — and what that means for whether Maya’s result is “real.”

B4. Shrink the clusters (6 pts). Re-run the clustered SE with G = 6 clusters (keep N roughly the same by raising n). Report the clustered SE and note whether you would trust the clustered t-test at face value, invoking the few-clusters rule of thumb from Ch 2.4. (You do not need to implement a wild cluster bootstrap; naming it as the remedy is enough.)

Deliverables

A single notebook or script that runs end-to-end on a fresh environment with the stated seed, plus the two small tables (B2 and B4) and the B3 paragraph. State your software versions. Same seed must reproduce the numbers.

Optional extension (no extra points): set \sigma_a = 0 so there is no lender shock, and confirm the clustered SE collapses back toward the classical SE — clustering only “costs” you when there is genuine within-cluster correlation to correct for.

Part C — Analytic rubric (point allocations explicit)

Each row is scored at one of four levels. Part-A rows describe how the 48 Part-A points are awarded across the six derive-and-interpret items; Part B is graded by the task allocation above, refined by the criteria here. The Presentation/honesty row spans both parts.

Criterion	Excellent	Proficient	Developing	Missing	Points
Derivation correctness (Part A1–A3)	Normal equations derived both by calculus and by orthogonality with the full-rank condition stated; \mathbf{H} shown symmetric + idempotent and \mathbf{HX}=\mathbf{X}; Gauss–Markov rival argument reaches \mathbf{DX}=\mathbf{0} and the PSD variance gap.	One derivation has a gap (e.g. orthogonality route asserted not shown), rest correct.	Final formulas stated but derivations mostly missing or with sign/transpose errors.	Not attempted or fundamentally wrong.	22
FWL & identification reasoning (A4)	States FWL two-step recipe correctly (residualize y on z, x on z, regress residuals) and correctly classifies near-collinearity as an SE problem, not bias, justified by what it does to \operatorname{Var}(\tilde{x}).	Recipe right; SE-vs-bias call right but thin justification.	Recipe garbled or calls collinearity a bias problem without seeing the unbiasedness point.	Absent or wrong on both.	8
OVB & measurement error (A5–A6)	Moulton inflation computed correctly; clustered-vs-HC middle distinguished; OVB formula derived with correct \delta_1; two-sign rule applied to the right sign; attenuation \lambda defined as reliability ratio with the toward-zero direction; OVB-vs-attenuation direction contrast correct.	One numeric slip or one sign error, logic sound.	Formulas recalled but misapplied (wrong sign, \lambda direction reversed).	Not attempted.	18
Code correctness & reproducibility (Part B)	\hat\beta computed from the matrix formula; residual identities verified; three SEs correct; clustered block sum matches library; seed pinned; runs end-to-end and reproduces.	Runs and is essentially correct; minor non-reproducibility or uses a library where from-scratch was asked.	Logic flaw (e.g. clustered SE smaller than classical, or \hat\beta recomputed per SE flavor); partial output.	Does not run or wrong estimator.	24
SE choice & t-stat interpretation (B3–B4)	Identifies clustered as the honest SE, states \hat\beta did not change, lands the bias-vs-SE verdict, compares the inflation to the Moulton prediction, and applies the few-clusters caution at G=6 (names wild cluster bootstrap).	Mostly correct; one link (Moulton check or few-clusters) missing.	Reports the SEs without interpreting which to trust or why.	Absent or treats the SE change as a change in the effect.	18
Presentation, units, honest threats	Clean prose; specifications named in spec-discipline format (outcome · regressor · clustering · identifying assumption); never says collinearity or clustering “biases” \hat\beta; no “proves”/“accepts the null.”	One stylistic or labeling lapse.	Several lapses; SE flavors unlabeled; verdicts without reasoning.	Unreadable or rife with banned claims.	10

Total: 100 points. (Part A criteria sum to 48; Part B criteria sum to 42; the Presentation row adds 10, and the rubric is normalized so the maximum awarded is 100.)

A note on the spirit of the SE-choice row: this week rewards students who keep the point estimate and the standard error in separate mental boxes. An answer that says “\hat\beta_1 was 0.297 under all three flavors; only the SE moved, from 0.022 classical to roughly 0.05 clustered, so the t fell from ~13 to ~6 — this is an SE problem, the estimate is unchanged” outscores “clustering changed my result” even when both report the same numbers. Knowing what kind of problem you have — a wrong target (bias) versus an honest target with a lying t (SE) — is the entire point of Week 2.

Instructor answer key / model-answer sketch

A1. (i) Expand S(\mathbf{b}) = \mathbf{y}'\mathbf{y} - 2\mathbf{b}'\mathbf{X}'\mathbf{y} + \mathbf{b}'\mathbf{X}'\mathbf{X}\mathbf{b}; gradient \partial S/\partial\mathbf{b} = -2\mathbf{X}'\mathbf{y} + 2\mathbf{X}'\mathbf{X}\mathbf{b}; setting it to zero gives \mathbf{X}'\mathbf{X}\hat{\boldsymbol{\beta}} = \mathbf{X}'\mathbf{y}. Inverting requires \mathbf{X}'\mathbf{X} nonsingular, i.e. \mathbf{X} has full column rank (no perfect collinearity). (ii) Geometrically, \hat{\mathbf{y}}=\mathbf{X}\hat{\boldsymbol{\beta}} is the projection of \mathbf{y} onto \operatorname{col}(\mathbf{X}), so the residual \hat{\boldsymbol{\varepsilon}} = \mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}} is orthogonal to every column of \mathbf{X}: \mathbf{X}'\hat{\boldsymbol{\varepsilon}} = \mathbf{0}, i.e. \mathbf{X}'(\mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}})=\mathbf{0} — the same normal equations, no calculus. With a column of ones, \mathbf{1}'\hat{\boldsymbol{\varepsilon}}=0, which says the residuals sum to zero. (Full credit needs both routes and the full-rank condition.)

A2. (i) Symmetric: \mathbf{H}' = (\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}')' = \mathbf{H} since (\mathbf{X}'\mathbf{X})^{-1} is symmetric. Idempotent: \mathbf{H}\mathbf{H} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' = \mathbf{H} (the inner \mathbf{X}'\mathbf{X} cancels its inverse). And \mathbf{H}\mathbf{X} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{X} = \mathbf{X} — vectors already in \operatorname{col}(\mathbf{X}) project to themselves. (ii) h_{ii} is the leverage of observation i: how much its own y_i pulls its own fitted value \hat y_i, i.e. how far its x-row sits from the center of the regressor cloud. Leverage near 1 means that point alone nearly determines its fit; a single high-leverage point can swing \hat{\boldsymbol{\beta}} and is exactly where HC2/HC3 corrections (which divide by 1-h_{ii}) matter most. (Credit the projection argument and the “one point controls its fit / influence” reading.)

A3. Classical assumptions: linearity in parameters; no perfect collinearity (full-rank \mathbf{X}); zero conditional mean \mathbb{E}[\boldsymbol{\varepsilon}\mid\mathbf{X}]=\mathbf{0}; homoskedasticity and no autocorrelation \operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}. Best = smallest variance in its class; Linear = a linear function \mathbf{C}\mathbf{y} of the outcome; Unbiased = \mathbb{E}[\hat{\boldsymbol{\beta}}]=\boldsymbol{\beta}; Estimator. For the rival, \mathbb{E}[\tilde{\boldsymbol{\beta}}] = \mathbf{C}\mathbf{X}\boldsymbol{\beta} = [(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'+\mathbf{D}]\mathbf{X}\boldsymbol{\beta} = \boldsymbol{\beta} + \mathbf{D}\mathbf{X}\boldsymbol{\beta}; unbiasedness for every \boldsymbol{\beta} forces \mathbf{D}\mathbf{X}=\mathbf{0}. Then \operatorname{Var}(\tilde{\boldsymbol{\beta}}) = \sigma^2\mathbf{C}\mathbf{C}'; expanding with \mathbf{D}\mathbf{X}=\mathbf{0} kills the cross terms and leaves \sigma^2(\mathbf{X}'\mathbf{X})^{-1} + \sigma^2\mathbf{D}\mathbf{D}' — the OLS variance plus a PSD term, so OLS is best. The assumption whose failure demotes OLS is homoskedasticity/no-autocorrelation (\operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}); its breaking makes OLS no longer minimum-variance but it stays unbiased — so no, it is not a bias problem. (This is the A3 hook for the SE-vs-bias theme.)

A4. FWL: the long-model coefficient \hat\beta_1 on x equals the slope from (1) regress y on z (and intercept), keep residuals \tilde y; (2) regress x on z, keep residuals \tilde x; (3) regress \tilde y on \tilde x — the slope is exactly 1.84, and the residualized SE matches the long-model SE once the degrees of freedom are corrected. It is a standard-error problem, not a bias problem. Near-collinearity means firm size explains most of R&D’s variation, so the residualized \tilde x (the part of R&D orthogonal to size) has small variance; the slope of \tilde y on a low-variance regressor is still unbiased for \beta_1, but its variance \sigma^2/\sum\tilde x^2 blows up. The estimate is imprecise, not wrong. (Full credit requires the unbiasedness point plus the shrinking-\operatorname{Var}(\tilde x) mechanism.)

A5. (i) Sandwich: \widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}'\mathbf{X})^{-1}\big(\mathbf{X}'\hat{\boldsymbol{\Omega}}\mathbf{X}\big)(\mathbf{X}'\mathbf{X})^{-1}. HC builds the middle from individual squared residuals (diagonal \hat{\boldsymbol{\Omega}}, one entry per observation); the clustered version builds it from a sum of block outer products \sum_g \mathbf{X}_g'\hat{\boldsymbol{\varepsilon}}_g\hat{\boldsymbol{\varepsilon}}_g'\mathbf{X}_g, keeping the within-cluster covariances HC throws away. (ii) Factor = 1+(n-1)\rho_x\rho_\varepsilon = 1 + 999(0.05) = 1 + 49.95 = 50.95. So the true variance is about 51\times the naive one, and the SE is understated by \sqrt{50.95}\approx 7.1\times. (A classical t of, say, 14 is really about 2.) (iii) Spec-discipline: “outcome APR · key regressor borrower characteristic · cluster at the lender level — encoding the claim that errors are correlated within a lender but independent across lenders · sample 9,000 loans.” The worry: G=9 clusters is far below the 30–50 rule of thumb, so even the clustered SE is unreliable; reach for a wild cluster bootstrap. (Credit the \approx 51\times / \approx 7\times split and naming G=9 as the red flag.)

A6. (i) Substitute the long model into the short-slope probability limit: \hat{\tilde\beta}_1 \xrightarrow{p} \dfrac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)} = \dfrac{\operatorname{Cov}(x,\ \beta_1 x + \beta_2 z + \varepsilon)}{\operatorname{Var}(x)} = \beta_1 + \beta_2\dfrac{\operatorname{Cov}(x,z)}{\operatorname{Var}(x)} = \beta_1 + \beta_2\delta_1, the last \operatorname{Cov}(x,\varepsilon)=0 term vanishing because the long model is well-specified. Maya: \beta_2>0, \delta_1<0 \Rightarrow bias =\beta_2\delta_1 < 0, i.e. the D coefficient is biased downward (more negative), so it overstates the approval penalty against D=1 — part of the apparent penalty is omitted creditworthiness loaded onto D. (ii) \hat\beta_1 \xrightarrow{p} \beta_1\lambda with \lambda = \sigma_{x^\*}^2/(\sigma_{x^\*}^2+\sigma_m^2)\in(0,1), the reliability ratio = share of the measured regressor’s variance that is real signal. The income coefficient is pushed toward zero (attenuated) — understated. Contrast: attenuation has a known direction (always toward zero) regardless of the data, because \lambda is a ratio of variances and variances are non-negative; OVB’s direction depends on the unknown signs of \beta_2 and \delta_1, so it can go either way. (Credit correct OVB sign via two-sign rule and the toward-zero attenuation direction.)

Part B expected results (for grading).

B1: \hat\beta_1 should land near 0.30 (typically 0.27–0.33 depending on seed). The two identities are near-machine-zero: \sum\hat\varepsilon_i \approx 0 and \mathbf{X}'\hat{\boldsymbol{\varepsilon}}\approx\mathbf{0}. Computing \hat\beta with a library when the task said “from scratch” is a Proficient-level miss, not full credit.
B2: The point estimate is identical across all three SE flavors — graders should confirm the student did not recompute \hat\beta per flavor (a common conceptual error). Ordering: classical < HC1 \approx classical (heteroskedasticity is mild here) \ll clustered. The clustered SE should be several times the classical SE because a_g induces real within-cluster correlation and x also has a cluster component.
B3: Largest = clustered, and it is honest because the DGP has a genuine lender shock a_g; the classical formula assumes \operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}, which is false here. The inflation factor should be roughly in line with \sqrt{1+(n-1)\rho_x\rho_\varepsilon} for the realized \rho’s (order-of-magnitude agreement is enough; exact match is not expected because the Moulton formula is an equal-cluster approximation). The required verdict: \hat\beta_1 did not change; this is an SE problem, not a bias problem — OLS is still unbiased, the classical t was lying. “Real” means: judged against the honest clustered SE, is the t still large? (With G=40 it usually still is.)
B4: With G=6, the clustered SE is itself noisy and not trustworthy at face value — below the 30–50 rule of thumb. The remedy to name is the wild cluster bootstrap (Cameron, Gelbach & Miller 2008). Students should not conclude “the result got weaker”; they should conclude “I can no longer trust this particular SE.”

Quick grading heuristic. The two highest-signal items are A4/A3 (do they correctly call collinearity and heteroskedasticity SE problems, with OLS still unbiased?) and B2/B3 (did the point estimate stay fixed while only the SE moved, and did they read that as an SE problem?). A student who keeps “wrong target” (bias: OVB, attenuation) cleanly separate from “honest target, lying t” (SE: heteroskedasticity, clustering, collinearity) has the Week-2 mindset; the rest is execution.

# Week 2 Assessment — The OLS Engine This is the end-of-week assessment for Week 2. It has three parts. **Part A** is a derive-and-interpret section on the spine of the week — OLS in matrix form and the normal equations, the hat and residual-maker matrices, Gauss–Markov and what "BLUE" actually buys you, Frisch–Waugh–Lovell and partialling-out, robust and clustered standard errors, and omitted-variable bias and attenuation. **Part B** is a small panel you simulate, estimate, and report on, ending with the single most important habit of the week: choosing the right standard error and reading what changes. **Part C** is the rubric, with explicit point totals. An instructor answer key follows Part C. The whole thing is one focused sitting plus the coding task. Methods are limited to Week 2: matrix OLS, the four/five classical assumptions, FWL, the sandwich (HC and clustered), HAC by name, and the OVB and measurement-error formulas. No instrumental variables, no panel fixed-effects estimators beyond the within-transformation idea, no causal-design machinery — those are Weeks 3–4. Show your reasoning. A correct number with no argument earns little; an honest "this is an SE problem, not a bias problem, and here is how I know" earns a great deal. **Total: 100 points.** Part A = 48, Part B = 42, Presentation/honesty woven through both = 10. Throughout, write OLS as $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}$ with $\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}$, and label every standard error with its flavor: classical, HC1/HC2/HC3, HAC (Newey–West), or clustered. --- ## Part A — Derive and interpret (48 points; 8 points each) Answer in a few lines of algebra plus two to four sentences of interpretation. Where a derivation is asked for, the steps must be visible; a boxed final formula with no path earns half credit at most. **A1. (Normal equations from two directions.)** Start from the sum-of-squared-residuals objective $S(\mathbf{b}) = (\mathbf{y}-\mathbf{X}\mathbf{b})'(\mathbf{y}-\mathbf{X}\mathbf{b})$. (i) Differentiate with respect to $\mathbf{b}$ and set the gradient to zero to derive the normal equations $\mathbf{X}'\mathbf{X}\hat{\boldsymbol{\beta}} = \mathbf{X}'\mathbf{y}$, and state the one condition on $\mathbf{X}$ that lets you write $\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}$. (ii) Now derive the *same* equations from orthogonality alone — without calculus — using the geometric fact that the residual vector $\hat{\boldsymbol{\varepsilon}}$ must be orthogonal to the column space of $\mathbf{X}$. State in one sentence why, when $\mathbf{X}$ contains a column of ones, this forces the residuals to sum to zero. **A2. (The hat matrix is a projector.)** Define $\mathbf{H} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'$ and $\mathbf{M} = \mathbf{I} - \mathbf{H}$. (i) Prove $\mathbf{H}$ is symmetric and idempotent ($\mathbf{H}\mathbf{H}=\mathbf{H}$), and show $\mathbf{H}\mathbf{X} = \mathbf{X}$. (ii) Using $\hat{\boldsymbol{\varepsilon}} = \mathbf{M}\mathbf{y}$ and the Pythagorean split $\lVert\mathbf{y}\rVert^2 = \lVert\hat{\mathbf{y}}\rVert^2 + \lVert\hat{\boldsymbol{\varepsilon}}\rVert^2$, explain in words what the diagonal entry $h_{ii}$ measures and why an observation with $h_{ii}$ close to 1 is one a careful analyst worries about. (You do not need to re-derive the Pythagorean split.) **A3. (Gauss–Markov and the meaning of "best.")** State the classical assumptions under which OLS is BLUE, decoding what each of the four letters B-L-U-E claims. Then consider a rival linear estimator $\tilde{\boldsymbol{\beta}} = \mathbf{C}\mathbf{y}$ with $\mathbf{C} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' + \mathbf{D}$. Show that requiring $\tilde{\boldsymbol{\beta}}$ to be unbiased for *every* $\boldsymbol{\beta}$ forces $\mathbf{D}\mathbf{X} = \mathbf{0}$, and state (you need not fully expand the algebra) why this makes $\operatorname{Var}(\tilde{\boldsymbol{\beta}}) = \operatorname{Var}(\hat{\boldsymbol{\beta}}) + \sigma^2\mathbf{D}\mathbf{D}'$, a sum of the OLS variance and something positive-semidefinite. Finish with one sentence: which single classical assumption, when it breaks, is the one that *demotes* OLS from "best" — and does its breaking make $\hat{\boldsymbol{\beta}}$ biased? **A4. (FWL / partialling-out — and "is this a bias problem or an SE problem?")** Leah runs the long regression of patent count $y$ on R&D spending $x$ and firm size $z$ (plus an intercept), and gets $\hat\beta_1 = 1.84$ on R&D. State the Frisch–Waugh–Lovell theorem precisely enough to describe the two-step recipe that reproduces $\hat\beta_1$ exactly: which two auxiliary regressions you run, and what final regression you run on their residuals. Then answer this: Leah notices R&D and firm size are highly correlated in her sample. A colleague says "high correlation between your regressors *biases* your R&D coefficient." **Is this a bias problem or a standard-error problem?** Defend your answer in two sentences using FWL geometry (what near-collinearity does to the residualized regressor $\tilde{x}$). **A5. (Robust vs. clustered SEs — name the threat and the fix.)** Maya regresses loan APR on a borrower characteristic across $N = 9{,}000$ loans that come from $G = 9$ lenders, 1,000 loans each. The within-lender errors are positively correlated because each lender sets rates with a correlated house style. (i) Write the general sandwich form of $\widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}})$ and say in one sentence how the *clustered* version builds its middle differently from the *HC* (heteroskedasticity-robust) version. (ii) Using the Moulton-style approximation $\text{true Var} \approx \text{naive Var}\times[\,1+(n-1)\rho_x\rho_\varepsilon\,]$, with cluster size $n=1{,}000$ and a mild $\rho_x\rho_\varepsilon = 0.05$, compute the factor by which the classical SE understates the truth (give the variance inflation and the SE inflation). (iii) **Name the threat and the design/fix in one sentence**, in the spec-discipline format: state what level you cluster at and what economic claim that choice encodes — and note the one thing about $G = 9$ that should make Maya nervous. **A6. (OVB and attenuation — two contaminations, two directions.)** (i) Derive the omitted-variable-bias formula: with true long model $y = \beta_0 + \beta_1 x + \beta_2 z + \varepsilon$ (assumptions holding in the long model) but a short regression of $y$ on $x$ alone, show $\hat{\tilde\beta}_1 \xrightarrow{p} \beta_1 + \beta_2\delta_1$ where $\delta_1 = \operatorname{Cov}(x,z)/\operatorname{Var}(x)$. Then apply the two-sign rule to Maya's case: the omitted variable is creditworthiness $C$, which raises approval ($\beta_2 > 0$) and is *lower* on average for the disadvantaged group $D=1$ so that the slope of $C$ on $D$ is negative ($\delta_1 < 0$) — state the sign of the bias on the $D$ coefficient and what it does to the apparent approval penalty. (ii) Now a *different* contamination: Maya's income variable is self-reported, $x = x^\* + m$ with classical noise $m$. State the attenuation formula $\hat\beta_1 \xrightarrow{p} \beta_1\lambda$, define the reliability ratio $\lambda$, and say which way the income coefficient is pushed. Close with one sentence contrasting OVB and attenuation: which one has a *known* direction regardless of the data, and why. --- ## Part B — Simulate a clustered panel, estimate β̂ from scratch, and choose the SE (42 points) You will build a small panel where you control the truth, estimate $\hat\beta$ by hand from the matrix formula, and then compute classical, HC1, and clustered standard errors for the *same* point estimate. The headline you are chasing is the one from Ch 2.4: clustering does not move $\hat\beta$ — it moves the *t*-statistic, sometimes by a lot, and that change is the whole story. This ties Ch 2.1 (the matrix estimator), Ch 2.2 (unbiasedness under the right assumptions), and Ch 2.4 (the sandwich) together in one script. ### The data-generating process Simulate a balanced panel of $G$ lenders, each with $n$ loans, so $N = G n$. For lender $g$ and loan $i$: $$ y_{ig} = \beta_0 + \beta_1 x_{ig} + \underbrace{a_g + e_{ig}}_{\varepsilon_{ig}}, \qquad a_g \sim \mathcal{N}(0,\sigma_a^2),\quad e_{ig}\sim\mathcal{N}(0,\sigma_e^2). $$ The term $a_g$ is a **lender-level shock shared by every loan at that lender** — it is what induces within-cluster error correlation. Make the regressor *also* carry a lender component so that $\rho_x > 0$: draw $x_{ig} = c_g + v_{ig}$ with $c_g\sim\mathcal{N}(0,1)$ and $v_{ig}\sim\mathcal{N}(0,1)$. Use a true slope $\beta_1 = 0.30$, $\beta_0 = 1.0$, $\sigma_a = 1.0$, $\sigma_e = 1.0$. Pin your random seed and state it. ### Tasks **B1. Estimate β̂ from scratch (10 pts).** With $G = 40$ and $n = 50$ ($N = 2{,}000$), build the design matrix $\mathbf{X}$ (a column of ones and the $x$ column) and compute $\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}$ using only array operations (`numpy`), *not* a regression library. Report $\hat\beta_1$ and confirm it is close to $0.30$. Verify two Ch 2.1 identities numerically: the residuals sum to (essentially) zero, and they are orthogonal to the $x$ column ($\mathbf{X}'\hat{\boldsymbol{\varepsilon}} \approx \mathbf{0}$). **B2. Three standard errors for one β̂ (14 pts).** For that same $\hat{\boldsymbol{\beta}}$, compute three standard errors on $\hat\beta_1$: - **classical**, $\hat\sigma^2(\mathbf{X}'\mathbf{X})^{-1}$ with $\hat\sigma^2 = \hat{\boldsymbol{\varepsilon}}'\hat{\boldsymbol{\varepsilon}}/(N-K)$; - **HC1**, the heteroskedasticity-robust sandwich with the usual $N/(N-K)$ correction; - **clustered by lender**, the block sandwich summing the outer products over the $G$ lenders. You may write these from the matrix formulas (preferred) or use `statsmodels` for HC1 and the clustered SE *as a check* — but the clustered computation should match a from-scratch block sum to a few decimals. Report all three SEs and the three implied *t*-statistics for $H_0:\beta_1 = 0$ in a small table. **B3. Read the t-stat change (12 pts).** In one short paragraph answer: Which SE is largest, and why is that the *honest* one here? By roughly what factor does the clustered SE exceed the classical SE, and how does that compare to what the Moulton approximation $\sqrt{1+(n-1)\rho_x\rho_\varepsilon}$ would predict for your DGP? State explicitly: did $\hat\beta_1$ itself change across the three SE flavors? Conclude with the one-sentence verdict the week is built around — **bias problem or SE problem?** — and what that means for whether Maya's result is "real." **B4. Shrink the clusters (6 pts).** Re-run the clustered SE with $G = 6$ clusters (keep $N$ roughly the same by raising $n$). Report the clustered SE and note whether you would trust the clustered *t*-test at face value, invoking the few-clusters rule of thumb from Ch 2.4. (You do not need to implement a wild cluster bootstrap; naming it as the remedy is enough.) ### Deliverables A single notebook or script that runs end-to-end on a fresh environment with the stated seed, plus the two small tables (B2 and B4) and the B3 paragraph. State your software versions. Same seed must reproduce the numbers. **Optional extension (no extra points):** set $\sigma_a = 0$ so there is no lender shock, and confirm the clustered SE collapses back toward the classical SE — clustering only "costs" you when there is genuine within-cluster correlation to correct for. --- ## Part C — Analytic rubric (point allocations explicit) Each row is scored at one of four levels. Part-A rows describe how the 48 Part-A points are awarded across the six derive-and-interpret items; Part B is graded by the task allocation above, refined by the criteria here. The Presentation/honesty row spans both parts. | Criterion | Excellent | Proficient | Developing | Missing | Points | |---|---|---|---|---|---| | **Derivation correctness (Part A1–A3)** | Normal equations derived both by calculus and by orthogonality with the full-rank condition stated; $\mathbf{H}$ shown symmetric + idempotent and $\mathbf{HX}=\mathbf{X}$; Gauss–Markov rival argument reaches $\mathbf{DX}=\mathbf{0}$ and the PSD variance gap. | One derivation has a gap (e.g. orthogonality route asserted not shown), rest correct. | Final formulas stated but derivations mostly missing or with sign/transpose errors. | Not attempted or fundamentally wrong. | 22 | | **FWL & identification reasoning (A4)** | States FWL two-step recipe correctly (residualize $y$ on $z$, $x$ on $z$, regress residuals) and correctly classifies near-collinearity as an **SE problem, not bias**, justified by what it does to $\operatorname{Var}(\tilde{x})$. | Recipe right; SE-vs-bias call right but thin justification. | Recipe garbled or calls collinearity a bias problem without seeing the unbiasedness point. | Absent or wrong on both. | 8 | | **OVB & measurement error (A5–A6)** | Moulton inflation computed correctly; clustered-vs-HC middle distinguished; OVB formula derived with correct $\delta_1$; two-sign rule applied to the right sign; attenuation $\lambda$ defined as reliability ratio with the toward-zero direction; OVB-vs-attenuation direction contrast correct. | One numeric slip or one sign error, logic sound. | Formulas recalled but misapplied (wrong sign, $\lambda$ direction reversed). | Not attempted. | 18 | | **Code correctness & reproducibility (Part B)** | $\hat\beta$ computed from the matrix formula; residual identities verified; three SEs correct; clustered block sum matches library; seed pinned; runs end-to-end and reproduces. | Runs and is essentially correct; minor non-reproducibility or uses a library where from-scratch was asked. | Logic flaw (e.g. clustered SE smaller than classical, or $\hat\beta$ recomputed per SE flavor); partial output. | Does not run or wrong estimator. | 24 | | **SE choice & t-stat interpretation (B3–B4)** | Identifies clustered as the honest SE, states $\hat\beta$ did **not** change, lands the bias-vs-SE verdict, compares the inflation to the Moulton prediction, and applies the few-clusters caution at $G=6$ (names wild cluster bootstrap). | Mostly correct; one link (Moulton check or few-clusters) missing. | Reports the SEs without interpreting which to trust or why. | Absent or treats the SE change as a change in the effect. | 18 | | **Presentation, units, honest threats** | Clean prose; specifications named in spec-discipline format (outcome · regressor · clustering · identifying assumption); never says collinearity or clustering "biases" $\hat\beta$; no "proves"/"accepts the null." | One stylistic or labeling lapse. | Several lapses; SE flavors unlabeled; verdicts without reasoning. | Unreadable or rife with banned claims. | 10 | **Total: 100 points.** (Part A criteria sum to 48; Part B criteria sum to 42; the Presentation row adds 10, and the rubric is normalized so the maximum awarded is 100.) A note on the spirit of the SE-choice row: this week rewards students who *keep the point estimate and the standard error in separate mental boxes*. An answer that says "$\hat\beta_1$ was 0.297 under all three flavors; only the SE moved, from 0.022 classical to roughly 0.05 clustered, so the *t* fell from ~13 to ~6 — this is an SE problem, the estimate is unchanged" outscores "clustering changed my result" even when both report the same numbers. Knowing *what kind* of problem you have — a wrong target (bias) versus an honest target with a lying *t* (SE) — is the entire point of Week 2. --- ## Instructor answer key / model-answer sketch **A1.** (i) Expand $S(\mathbf{b}) = \mathbf{y}'\mathbf{y} - 2\mathbf{b}'\mathbf{X}'\mathbf{y} + \mathbf{b}'\mathbf{X}'\mathbf{X}\mathbf{b}$; gradient $\partial S/\partial\mathbf{b} = -2\mathbf{X}'\mathbf{y} + 2\mathbf{X}'\mathbf{X}\mathbf{b}$; setting it to zero gives $\mathbf{X}'\mathbf{X}\hat{\boldsymbol{\beta}} = \mathbf{X}'\mathbf{y}$. Inverting requires $\mathbf{X}'\mathbf{X}$ nonsingular, i.e. $\mathbf{X}$ has **full column rank** (no perfect collinearity). (ii) Geometrically, $\hat{\mathbf{y}}=\mathbf{X}\hat{\boldsymbol{\beta}}$ is the projection of $\mathbf{y}$ onto $\operatorname{col}(\mathbf{X})$, so the residual $\hat{\boldsymbol{\varepsilon}} = \mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}}$ is orthogonal to every column of $\mathbf{X}$: $\mathbf{X}'\hat{\boldsymbol{\varepsilon}} = \mathbf{0}$, i.e. $\mathbf{X}'(\mathbf{y}-\mathbf{X}\hat{\boldsymbol{\beta}})=\mathbf{0}$ — the same normal equations, no calculus. With a column of ones, $\mathbf{1}'\hat{\boldsymbol{\varepsilon}}=0$, which says the residuals sum to zero. *(Full credit needs both routes and the full-rank condition.)* **A2.** (i) Symmetric: $\mathbf{H}' = (\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}')' = \mathbf{H}$ since $(\mathbf{X}'\mathbf{X})^{-1}$ is symmetric. Idempotent: $\mathbf{H}\mathbf{H} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}' = \mathbf{H}$ (the inner $\mathbf{X}'\mathbf{X}$ cancels its inverse). And $\mathbf{H}\mathbf{X} = \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{X} = \mathbf{X}$ — vectors already in $\operatorname{col}(\mathbf{X})$ project to themselves. (ii) $h_{ii}$ is the **leverage** of observation $i$: how much its own $y_i$ pulls its own fitted value $\hat y_i$, i.e. how far its $x$-row sits from the center of the regressor cloud. Leverage near 1 means that point alone nearly determines its fit; a single high-leverage point can swing $\hat{\boldsymbol{\beta}}$ and is exactly where HC2/HC3 corrections (which divide by $1-h_{ii}$) matter most. *(Credit the projection argument and the "one point controls its fit / influence" reading.)* **A3.** Classical assumptions: linearity in parameters; no perfect collinearity (full-rank $\mathbf{X}$); zero conditional mean $\mathbb{E}[\boldsymbol{\varepsilon}\mid\mathbf{X}]=\mathbf{0}$; homoskedasticity and no autocorrelation $\operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}$. **B**est = smallest variance in its class; **L**inear = a linear function $\mathbf{C}\mathbf{y}$ of the outcome; **U**nbiased = $\mathbb{E}[\hat{\boldsymbol{\beta}}]=\boldsymbol{\beta}$; **E**stimator. For the rival, $\mathbb{E}[\tilde{\boldsymbol{\beta}}] = \mathbf{C}\mathbf{X}\boldsymbol{\beta} = [(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'+\mathbf{D}]\mathbf{X}\boldsymbol{\beta} = \boldsymbol{\beta} + \mathbf{D}\mathbf{X}\boldsymbol{\beta}$; unbiasedness for *every* $\boldsymbol{\beta}$ forces $\mathbf{D}\mathbf{X}=\mathbf{0}$. Then $\operatorname{Var}(\tilde{\boldsymbol{\beta}}) = \sigma^2\mathbf{C}\mathbf{C}'$; expanding with $\mathbf{D}\mathbf{X}=\mathbf{0}$ kills the cross terms and leaves $\sigma^2(\mathbf{X}'\mathbf{X})^{-1} + \sigma^2\mathbf{D}\mathbf{D}'$ — the OLS variance plus a PSD term, so OLS is best. The assumption whose failure demotes OLS is **homoskedasticity/no-autocorrelation** ($\operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}$); its breaking makes OLS no longer minimum-variance but it stays **unbiased** — so no, it is not a bias problem. *(This is the A3 hook for the SE-vs-bias theme.)* **A4.** FWL: the long-model coefficient $\hat\beta_1$ on $x$ equals the slope from (1) regress $y$ on $z$ (and intercept), keep residuals $\tilde y$; (2) regress $x$ on $z$, keep residuals $\tilde x$; (3) regress $\tilde y$ on $\tilde x$ — the slope is exactly $1.84$, and the residualized SE matches the long-model SE once the degrees of freedom are corrected. **It is a standard-error problem, not a bias problem.** Near-collinearity means firm size explains most of R&D's variation, so the residualized $\tilde x$ (the part of R&D orthogonal to size) has *small variance*; the slope of $\tilde y$ on a low-variance regressor is still **unbiased** for $\beta_1$, but its variance $\sigma^2/\sum\tilde x^2$ blows up. The estimate is imprecise, not wrong. *(Full credit requires the unbiasedness point plus the shrinking-$\operatorname{Var}(\tilde x)$ mechanism.)* **A5.** (i) Sandwich: $\widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}'\mathbf{X})^{-1}\big(\mathbf{X}'\hat{\boldsymbol{\Omega}}\mathbf{X}\big)(\mathbf{X}'\mathbf{X})^{-1}$. HC builds the middle from individual squared residuals (diagonal $\hat{\boldsymbol{\Omega}}$, one entry per observation); the **clustered** version builds it from a sum of *block* outer products $\sum_g \mathbf{X}_g'\hat{\boldsymbol{\varepsilon}}_g\hat{\boldsymbol{\varepsilon}}_g'\mathbf{X}_g$, keeping the within-cluster covariances HC throws away. (ii) Factor $= 1+(n-1)\rho_x\rho_\varepsilon = 1 + 999(0.05) = 1 + 49.95 = 50.95$. So the true variance is about $51\times$ the naive one, and the SE is understated by $\sqrt{50.95}\approx 7.1\times$. (A classical $t$ of, say, 14 is really about 2.) (iii) Spec-discipline: "outcome APR · key regressor borrower characteristic · cluster at the **lender** level — encoding the claim that errors are correlated within a lender but independent across lenders · sample 9,000 loans." The worry: $G=9$ clusters is *far* below the 30–50 rule of thumb, so even the clustered SE is unreliable; reach for a wild cluster bootstrap. *(Credit the $\approx 51\times$ / $\approx 7\times$ split and naming $G=9$ as the red flag.)* **A6.** (i) Substitute the long model into the short-slope probability limit: $\hat{\tilde\beta}_1 \xrightarrow{p} \dfrac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)} = \dfrac{\operatorname{Cov}(x,\ \beta_1 x + \beta_2 z + \varepsilon)}{\operatorname{Var}(x)} = \beta_1 + \beta_2\dfrac{\operatorname{Cov}(x,z)}{\operatorname{Var}(x)} = \beta_1 + \beta_2\delta_1$, the last $\operatorname{Cov}(x,\varepsilon)=0$ term vanishing because the long model is well-specified. Maya: $\beta_2>0$, $\delta_1<0 \Rightarrow$ bias $=\beta_2\delta_1 < 0$, i.e. the $D$ coefficient is biased **downward** (more negative), so it **overstates** the approval penalty against $D=1$ — part of the apparent penalty is omitted creditworthiness loaded onto $D$. (ii) $\hat\beta_1 \xrightarrow{p} \beta_1\lambda$ with $\lambda = \sigma_{x^\*}^2/(\sigma_{x^\*}^2+\sigma_m^2)\in(0,1)$, the **reliability ratio** = share of the measured regressor's variance that is real signal. The income coefficient is pushed **toward zero** (attenuated) — understated. Contrast: **attenuation has a known direction (always toward zero)** regardless of the data, because $\lambda$ is a ratio of variances and variances are non-negative; OVB's direction depends on the unknown signs of $\beta_2$ and $\delta_1$, so it can go either way. *(Credit correct OVB sign via two-sign rule and the toward-zero attenuation direction.)* **Part B expected results (for grading).** - *B1:* $\hat\beta_1$ should land near $0.30$ (typically $0.27$–$0.33$ depending on seed). The two identities are near-machine-zero: $\sum\hat\varepsilon_i \approx 0$ and $\mathbf{X}'\hat{\boldsymbol{\varepsilon}}\approx\mathbf{0}$. Computing $\hat\beta$ with a library when the task said "from scratch" is a Proficient-level miss, not full credit. - *B2:* The point estimate is **identical** across all three SE flavors — graders should confirm the student did not recompute $\hat\beta$ per flavor (a common conceptual error). Ordering: classical < HC1 $\approx$ classical (heteroskedasticity is mild here) $\ll$ clustered. The clustered SE should be several times the classical SE because $a_g$ induces real within-cluster correlation and $x$ also has a cluster component. - *B3:* Largest = clustered, and it is honest because the DGP has a genuine lender shock $a_g$; the classical formula assumes $\operatorname{Var}(\boldsymbol{\varepsilon}\mid\mathbf{X})=\sigma^2\mathbf{I}$, which is false here. The inflation factor should be roughly in line with $\sqrt{1+(n-1)\rho_x\rho_\varepsilon}$ for the realized $\rho$'s (order-of-magnitude agreement is enough; exact match is not expected because the Moulton formula is an equal-cluster approximation). The required verdict: $\hat\beta_1$ did **not** change; this is an **SE problem, not a bias problem** — OLS is still unbiased, the classical *t* was lying. "Real" means: judged against the honest clustered SE, is the *t* still large? (With $G=40$ it usually still is.) - *B4:* With $G=6$, the clustered SE is itself noisy and not trustworthy at face value — below the 30–50 rule of thumb. The remedy to name is the **wild cluster bootstrap** (Cameron, Gelbach & Miller 2008). Students should *not* conclude "the result got weaker"; they should conclude "I can no longer trust this particular SE." **Quick grading heuristic.** The two highest-signal items are A4/A3 (do they correctly call collinearity and heteroskedasticity *SE problems*, with OLS still unbiased?) and B2/B3 (did the point estimate stay fixed while only the SE moved, and did they read that as an SE problem?). A student who keeps "wrong target" (bias: OVB, attenuation) cleanly separate from "honest target, lying *t*" (SE: heteroskedasticity, clustering, collinearity) has the Week-2 mindset; the rest is execution.