Problem Set 2.4 — Robust, Clustered, and HAC Standard Errors

Covers Ch 2.4. Methods allowed: everything through Ch 2.4 — the classical variance formula \hat\sigma^2(\mathbf{X}'\mathbf{X})^{-1}, the full sandwich (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\Omega\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}, the White/HC estimator and its HC0–HC3 flavors, the cluster-robust sandwich and the Moulton-style inflation factor 1+(n-1)\rho_x\rho_\varepsilon, the few-clusters caveat, HAC / Newey–West, and the Petersen (2009) panel taxonomy. You may treat \hat\beta as already computed — every problem here is about its standard error, never about the estimate itself. Show your reasoning; a boxed number with no argument earns no credit.

Total: 100 points. Six problems, escalating. Problem 1 is conceptual (why heteroskedasticity hits the SE but not the unbiasedness of \hat\beta); Problems 2–3 are pencil-and-paper sandwich arithmetic (classical vs robust; HC0–HC3); Problem 4 is the clustering / Moulton calculation; Problem 5 is HAC / serial correlation; Problem 6 is a Petersen-style “which clustering dimension?” decision problem on a panel.

A reminder on the one idea that runs through every problem, because it is the whole chapter. OLS hands you two separate things: a point estimate \hat\beta and a claim about how much to trust it (its standard error). The first comes from \hat{\boldsymbol\beta}=\boldsymbol\beta+(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon and is unbiased whenever \mathbb{E}[\boldsymbol\varepsilon\mid\mathbf{X}]=\mathbf{0}, regardless of what the errors’ variances and covariances look like. The second comes from \operatorname{Var}(\hat{\boldsymbol\beta}\mid\mathbf{X})=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\Omega\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1} and depends entirely on the shape of \boldsymbol\Omega=\operatorname{Var}(\boldsymbol\varepsilon\mid\mathbf{X}). When a problem asks “which standard error?”, it is asking you to describe \boldsymbol\Omega — diagonal-and-constant, diagonal-but-varying, block-diagonal, or banded — and then read the right filling for the sandwich off that description. The estimate never moves; only the filling changes.

Problem 1 — Why \boldsymbol\Omega breaks the SE but not \hat\beta (12 points)

Maya regresses the interest rate on a pool of installment loans on a ZIP-code-tier indicator plus controls, and gets \hat\beta_1=0.42 with a classical standard error of 0.07 (t = 6.0). Her mentor warns that the errors are almost certainly heteroskedastic — bigger loans have far less predictable rates than tiny ones — and possibly correlated within lender. Maya panics that her whole result, the 0.42, is now suspect.

(a) (4 pts) Start from the identity \hat{\boldsymbol\beta}=\boldsymbol\beta+(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon from the chapter. Take the conditional expectation of both sides given \mathbf{X}, and use it to argue that \hat{\boldsymbol\beta} is unbiased. Which assumption did you need — and, crucially, did you anywhere have to assume anything about \operatorname{Var}(\boldsymbol\varepsilon\mid\mathbf{X})=\boldsymbol\Omega?

(b) (4 pts) Now write down the general sandwich formula for \operatorname{Var}(\hat{\boldsymbol\beta}\mid\mathbf{X}) and circle (in words) the one piece of it that changes when \boldsymbol\Omega stops being \sigma^2\mathbf{I}. Explain in one or two sentences why this means heteroskedasticity contaminates the standard error while leaving the estimate alone.

(c) (2 pts) Reassure Maya in one sentence: is her \hat\beta_1=0.42 biased by the heteroskedasticity? What exactly is the thing that is wrong?

(d) (2 pts) Maya’s classmate says, “If \hat\beta is still unbiased, then who cares about the standard error — the estimate is right.” Give the one-sentence rebuttal that names the practical thing she loses if she trusts the 0.07.

Problem 2 — The sandwich by hand: classical vs robust (18 points)

Work the smallest case that still shows the mechanism: a single regressor through the origin, y_i=\beta x_i+\varepsilon_i, so that \mathbf{X}'\mathbf{X}=\sum_i x_i^2 is just a number and the heteroskedasticity-robust variance reduces to the scalar formula from §2.1 of the chapter, \widehat{\operatorname{Var}}(\hat\beta)_{\text{HC0}}=\frac{\sum_i x_i^2\,\hat\varepsilon_i^2}{\left(\sum_i x_i^2\right)^2}, \qquad \widehat{\operatorname{Var}}(\hat\beta)_{\text{classical}}=\frac{\hat\sigma^2}{\sum_i x_i^2}, \quad \hat\sigma^2=\frac{1}{N-K}\sum_i\hat\varepsilon_i^2 . Priya has N=5 wildfire claims with regressor (insured value, in $M) and residuals already computed for you (K=1):

i	x_i	\hat\varepsilon_i
1	1	+1
2	2	-2
3	3	+1
4	4	-4
5	5	+5

(a) (3 pts) Compute \sum_i x_i^2 and \sum_i \hat\varepsilon_i^2.

(b) (4 pts) Compute the classical variance and standard error of \hat\beta. (Use \hat\sigma^2=\frac{1}{N-K}\sum_i\hat\varepsilon_i^2 with N-K=4.)

(c) (5 pts) Compute the HC0 robust variance and standard error of \hat\beta by evaluating \sum_i x_i^2\,\hat\varepsilon_i^2 in the numerator. Show the per-observation terms.

(d) (3 pts) Report the ratio robust-SE / classical-SE. Which is larger, and what feature of the data (look at how \hat\varepsilon_i^2 lines up with x_i) makes it so?

(e) (3 pts) In one or two sentences: if the residuals had instead been constant in magnitude across all x_i (say all \hat\varepsilon_i^2=9), what would the ratio in (d) become, and why? You do not need to redo the full arithmetic — argue it.

Problem 3 — HC0, HC1, HC2, HC3: which correction, and when (16 points)

The four robust flavors differ only in how each squared residual is inflated before it goes into the middle of the sandwich: \text{HC0: } \hat\varepsilon_i^2,\quad \text{HC1: } \tfrac{N}{N-K}\hat\varepsilon_i^2,\quad \text{HC2: } \tfrac{\hat\varepsilon_i^2}{1-h_{ii}},\quad \text{HC3: } \tfrac{\hat\varepsilon_i^2}{(1-h_{ii})^2}, where h_{ii} is the leverage (the i-th diagonal of the hat matrix).

(a) (3 pts) Priya has N=60 observations and K=4 regressors. Compute the HC1 inflation factor N/(N-K) and state what kind of error it corrects (and what it ignores).

(b) (5 pts) A single high-leverage point — Priya’s one $5M estate among $200K cabins — has leverage h_{ii}=0.5. For that observation, compute the multiplier that HC0, HC2, and HC3 apply to its \hat\varepsilon_i^2. (HC0’s multiplier is 1 by definition.) Rank the three by how aggressively they inflate this point’s contribution.

(c) (4 pts) Explain why OLS residuals are systematically too small, and why that makes raw HC0 standard errors too small in finite samples. Then explain why the leverage correction in HC2/HC3 targets exactly the points that are shrunk the most by fitting.

(d) (4 pts) Give the practical decision rule: for N=9{,}000 loans with no extreme leverage, do the four flavors matter? For N=25 observations with one big outlier in \mathbf{X}, which flavor should you default to, and why is it the “safe” pick?

Problem 4 — Clustering and the Moulton factor (20 points)

Maya’s pool of N=9{,}000 loans is split into G=45 lenders (“clusters”) of equal size n=N/G=200. Within a lender, errors share a common pricing shock, and her key regressor (the tier indicator) is also correlated within lender. The chapter’s special-case Moulton result says the true variance of \hat\beta_1 exceeds the naive (non-clustered) variance by approximately \text{true Var} \approx \text{naive Var}\times\big[\,1+(n-1)\,\rho_x\,\rho_\varepsilon\,\big].

(a) (3 pts) Explain in two or three sentences why ignoring within-cluster correlation makes the naive standard error too small — i.e., why 9{,}000 correlated loans are not 9{,}000 independent pieces of evidence. Use the “polling one household nine times” framing or your own.

(b) (5 pts) Take \rho_x=\rho_\varepsilon=0.20 and n=200. Compute the bracketed inflation factor. By what factor does the true standard error exceed the naive one? If Maya’s naive t-stat is 6.0, what is her honest (clustered) t-stat, approximately?

(c) (4 pts) Maya wants to halve the gap between her naive and honest standard errors. Her advisor offers two options: (i) collect 9,000 more loans from the same 45 lenders, or (ii) add 45 new lenders. Using the structure of the inflation factor, explain which one actually buys her statistical evidence and why the other essentially does not.

(d) (4 pts) Write the cluster-robust sandwich estimator (the middle summed over clusters g=1,\dots,G) and explain, by comparing it to White’s HC0 middle \sum_i\hat\varepsilon_i^2\,\mathbf{x}_i\mathbf{x}_i', what term the cluster version keeps that White’s throws away. One sentence on the slogan “White SEs are cluster SEs with clusters of size one.”

(e) (4 pts) Now suppose Maya instead clusters by state, and she operates in only G=9 states. Why should she be nervous about the resulting t-statistics even though clustering “more broadly” sounds safer? Name one concrete fix from the chapter.

Problem 5 — HAC / Newey–West and serial correlation (16 points)

Sam regresses his strategy’s monthly return on a market-timing signal over T=120 months and reports classical standard errors. His residuals are positively autocorrelated: each month’s residual is about 0.4-correlated with the previous month’s, fading with the gap.

(a) (4 pts) Describe what \boldsymbol\Omega looks like in Sam’s case — diagonal, block-diagonal, or banded? Where are the nonzero entries, and what happens to them as the time gap \ell grows? Contrast this with the clustering case in Problem 4.

(b) (4 pts) Is Sam’s reported classical t-statistic likely too big or too small? Argue it through the “effective sample size” intuition: with positive autocorrelation, how does the number of genuinely independent monthly draws compare to 120, and how does that bias the standard error?

(c) (4 pts) Which standard-error flavor should Sam switch to, and what is the one knob (and its name) he must set? Using the Newey–West rule of thumb L\approx 4(T/100)^{2/9} with T=120, compute a starting bandwidth (round to the nearest integer). Briefly: what goes wrong if he sets L too small? Too large?

(d) (4 pts) Sam’s colleague trades a high-frequency strategy whose residuals show negative autocorrelation (bid–ask bounce). For that colleague, are naive t-stats too big or too small, and why? State the general principle linking the sign of the autocorrelation to the direction of the naive SE’s error.

Problem 6 — Petersen (2009): which clustering dimension? (18 points)

Maya assembles a panel: many firms i observed over many years t, and regresses a default outcome on a leverage measure. The coefficient is identical in every column below; only the SE flavor changes. She runs Petersen’s own diagnostic — cluster one way, then the other, and watch which moves the SE more.

SE flavor	Std. error	t-stat
Classical (OLS)	0.018	5.0
White (HC1)	0.021	4.3
Clustered by firm	0.044	2.0
Clustered by time	0.024	3.8
Two-way (firm + time)	0.046	1.96

(a) (4 pts) Reading the table with Petersen’s diagnostic: clustering by firm roughly doubles the White SE while clustering by time barely moves it. Which effect dominates Maya’s panel — a persistent firm effect, a common time effect, or both? What is the correct clustering choice, and what does that choice assert about where the off-diagonal blocks of \boldsymbol\Omega live?

(b) (3 pts) Why do White (HC1) standard errors fail to fix Maya’s problem here? Be specific about what White SEs assume about the off-diagonals of \boldsymbol\Omega.

(c) (4 pts) Suppose a different outcome on the same dataset produced firm-clustered SE \approx White SE but time-clustered SE roughly double the White SE. What would that pattern imply, and which clustering would you choose? This is the point that “the right clustering depends on the left-hand-side variable, not just the dataset” — restate it in your own words.

(d) (4 pts) A classmate says: “I already put firm fixed effects in the regression, so I don’t need to cluster by firm — the fixed effects took care of the firm.” Explain why this is wrong by distinguishing the two jobs (what a firm fixed effect removes vs. what clustering by firm handles). Can you want both at once?

(e) (3 pts) Sketch the mechanical recipe for the two-way (firm + time) clustered variance in terms of three sandwiches you already know how to build. Why is one of them subtracted?

End of PS 2.4. Solutions in book/appendices/E-solutions-manual/E-w2-ps2.4-solutions.md. This set looks ahead to Weeks 3–4, where fixed effects and panel methods are developed in full, and to Ch 5.4, the complete reader’s guide to Petersen (2009).

# Problem Set 2.4 — Robust, Clustered, and HAC Standard Errors *Covers Ch 2.4. Methods allowed: everything through Ch 2.4 — the classical variance formula $\hat\sigma^2(\mathbf{X}'\mathbf{X})^{-1}$, the full sandwich $(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\Omega\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}$, the White/HC estimator and its HC0–HC3 flavors, the cluster-robust sandwich and the Moulton-style inflation factor $1+(n-1)\rho_x\rho_\varepsilon$, the few-clusters caveat, HAC / Newey–West, and the Petersen (2009) panel taxonomy. You may treat $\hat\beta$ as already computed — every problem here is about its standard error, never about the estimate itself. Show your reasoning; a boxed number with no argument earns no credit.* **Total: 100 points.** Six problems, escalating. Problem 1 is conceptual (why heteroskedasticity hits the SE but not the unbiasedness of $\hat\beta$); Problems 2–3 are pencil-and-paper sandwich arithmetic (classical vs robust; HC0–HC3); Problem 4 is the clustering / Moulton calculation; Problem 5 is HAC / serial correlation; Problem 6 is a Petersen-style "which clustering dimension?" decision problem on a panel. A reminder on the one idea that runs through every problem, because it is the whole chapter. OLS hands you two separate things: a point estimate $\hat\beta$ and a claim about how much to trust it (its standard error). The first comes from $\hat{\boldsymbol\beta}=\boldsymbol\beta+(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon$ and is unbiased whenever $\mathbb{E}[\boldsymbol\varepsilon\mid\mathbf{X}]=\mathbf{0}$, regardless of what the errors' variances and covariances look like. The second comes from $\operatorname{Var}(\hat{\boldsymbol\beta}\mid\mathbf{X})=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\Omega\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}$ and depends entirely on the shape of $\boldsymbol\Omega=\operatorname{Var}(\boldsymbol\varepsilon\mid\mathbf{X})$. When a problem asks "which standard error?", it is asking you to describe $\boldsymbol\Omega$ — diagonal-and-constant, diagonal-but-varying, block-diagonal, or banded — and then read the right filling for the sandwich off that description. The estimate never moves; only the filling changes. --- ## Problem 1 — Why $\boldsymbol\Omega$ breaks the SE but not $\hat\beta$ (12 points) Maya regresses the interest rate on a pool of installment loans on a ZIP-code-tier indicator plus controls, and gets $\hat\beta_1=0.42$ with a classical standard error of $0.07$ (t = 6.0). Her mentor warns that the errors are almost certainly heteroskedastic — bigger loans have far less predictable rates than tiny ones — and possibly correlated within lender. Maya panics that her whole result, the $0.42$, is now suspect. **(a) (4 pts)** Start from the identity $\hat{\boldsymbol\beta}=\boldsymbol\beta+(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon$ from the chapter. Take the conditional expectation of both sides given $\mathbf{X}$, and use it to argue that $\hat{\boldsymbol\beta}$ is unbiased. Which assumption did you need — and, crucially, did you anywhere have to assume anything about $\operatorname{Var}(\boldsymbol\varepsilon\mid\mathbf{X})=\boldsymbol\Omega$? **(b) (4 pts)** Now write down the general sandwich formula for $\operatorname{Var}(\hat{\boldsymbol\beta}\mid\mathbf{X})$ and circle (in words) the one piece of it that changes when $\boldsymbol\Omega$ stops being $\sigma^2\mathbf{I}$. Explain in one or two sentences why this means heteroskedasticity contaminates the standard error while leaving the estimate alone. **(c) (2 pts)** Reassure Maya in one sentence: is her $\hat\beta_1=0.42$ biased by the heteroskedasticity? What exactly is the thing that is wrong? **(d) (2 pts)** Maya's classmate says, "If $\hat\beta$ is still unbiased, then who cares about the standard error — the estimate is right." Give the one-sentence rebuttal that names the practical thing she loses if she trusts the $0.07$. --- ## Problem 2 — The sandwich by hand: classical vs robust (18 points) Work the smallest case that still shows the mechanism: a single regressor through the origin, $y_i=\beta x_i+\varepsilon_i$, so that $\mathbf{X}'\mathbf{X}=\sum_i x_i^2$ is just a number and the heteroskedasticity-robust variance reduces to the scalar formula from §2.1 of the chapter, $$ \widehat{\operatorname{Var}}(\hat\beta)_{\text{HC0}}=\frac{\sum_i x_i^2\,\hat\varepsilon_i^2}{\left(\sum_i x_i^2\right)^2}, \qquad \widehat{\operatorname{Var}}(\hat\beta)_{\text{classical}}=\frac{\hat\sigma^2}{\sum_i x_i^2}, \quad \hat\sigma^2=\frac{1}{N-K}\sum_i\hat\varepsilon_i^2 . $$ Priya has $N=5$ wildfire claims with regressor (insured value, in \$M) and residuals already computed for you ($K=1$): | $i$ | $x_i$ | $\hat\varepsilon_i$ | |-----|-------|---------------------| | 1 | 1 | $+1$ | | 2 | 2 | $-2$ | | 3 | 3 | $+1$ | | 4 | 4 | $-4$ | | 5 | 5 | $+5$ | **(a) (3 pts)** Compute $\sum_i x_i^2$ and $\sum_i \hat\varepsilon_i^2$. **(b) (4 pts)** Compute the **classical** variance and standard error of $\hat\beta$. (Use $\hat\sigma^2=\frac{1}{N-K}\sum_i\hat\varepsilon_i^2$ with $N-K=4$.) **(c) (5 pts)** Compute the **HC0 robust** variance and standard error of $\hat\beta$ by evaluating $\sum_i x_i^2\,\hat\varepsilon_i^2$ in the numerator. Show the per-observation terms. **(d) (3 pts)** Report the ratio robust-SE / classical-SE. Which is larger, and what feature of the data (look at how $\hat\varepsilon_i^2$ lines up with $x_i$) makes it so? **(e) (3 pts)** In one or two sentences: if the residuals had instead been *constant in magnitude* across all $x_i$ (say all $\hat\varepsilon_i^2=9$), what would the ratio in (d) become, and why? You do not need to redo the full arithmetic — argue it. --- ## Problem 3 — HC0, HC1, HC2, HC3: which correction, and when (16 points) The four robust flavors differ only in how each squared residual is inflated before it goes into the middle of the sandwich: $$ \text{HC0: } \hat\varepsilon_i^2,\quad \text{HC1: } \tfrac{N}{N-K}\hat\varepsilon_i^2,\quad \text{HC2: } \tfrac{\hat\varepsilon_i^2}{1-h_{ii}},\quad \text{HC3: } \tfrac{\hat\varepsilon_i^2}{(1-h_{ii})^2}, $$ where $h_{ii}$ is the leverage (the $i$-th diagonal of the hat matrix). **(a) (3 pts)** Priya has $N=60$ observations and $K=4$ regressors. Compute the HC1 inflation factor $N/(N-K)$ and state what kind of error it corrects (and what it ignores). **(b) (5 pts)** A single high-leverage point — Priya's one \$5M estate among \$200K cabins — has leverage $h_{ii}=0.5$. For *that observation*, compute the multiplier that HC0, HC2, and HC3 apply to its $\hat\varepsilon_i^2$. (HC0's multiplier is 1 by definition.) Rank the three by how aggressively they inflate this point's contribution. **(c) (4 pts)** Explain *why* OLS residuals are systematically too small, and why that makes raw HC0 standard errors too small in finite samples. Then explain why the leverage correction in HC2/HC3 targets exactly the points that are shrunk the most by fitting. **(d) (4 pts)** Give the practical decision rule: for $N=9{,}000$ loans with no extreme leverage, do the four flavors matter? For $N=25$ observations with one big outlier in $\mathbf{X}$, which flavor should you default to, and why is it the "safe" pick? --- ## Problem 4 — Clustering and the Moulton factor (20 points) Maya's pool of $N=9{,}000$ loans is split into $G=45$ lenders ("clusters") of equal size $n=N/G=200$. Within a lender, errors share a common pricing shock, and her key regressor (the tier indicator) is also correlated within lender. The chapter's special-case Moulton result says the *true* variance of $\hat\beta_1$ exceeds the naive (non-clustered) variance by approximately $$ \text{true Var} \approx \text{naive Var}\times\big[\,1+(n-1)\,\rho_x\,\rho_\varepsilon\,\big]. $$ **(a) (3 pts)** Explain in two or three sentences *why* ignoring within-cluster correlation makes the naive standard error too small — i.e., why $9{,}000$ correlated loans are not $9{,}000$ independent pieces of evidence. Use the "polling one household nine times" framing or your own. **(b) (5 pts)** Take $\rho_x=\rho_\varepsilon=0.20$ and $n=200$. Compute the bracketed inflation factor. By what factor does the *true standard error* exceed the naive one? If Maya's naive t-stat is 6.0, what is her honest (clustered) t-stat, approximately? **(c) (4 pts)** Maya wants to halve the gap between her naive and honest standard errors. Her advisor offers two options: (i) collect 9,000 *more* loans from the *same* 45 lenders, or (ii) add 45 *new* lenders. Using the structure of the inflation factor, explain which one actually buys her statistical evidence and why the other essentially does not. **(d) (4 pts)** Write the cluster-robust sandwich estimator (the middle summed over clusters $g=1,\dots,G$) and explain, by comparing it to White's HC0 middle $\sum_i\hat\varepsilon_i^2\,\mathbf{x}_i\mathbf{x}_i'$, what term the cluster version keeps that White's throws away. One sentence on the slogan "White SEs are cluster SEs with clusters of size one." **(e) (4 pts)** Now suppose Maya instead clusters by **state**, and she operates in only $G=9$ states. Why should she be nervous about the resulting t-statistics even though clustering "more broadly" sounds safer? Name one concrete fix from the chapter. --- ## Problem 5 — HAC / Newey–West and serial correlation (16 points) Sam regresses his strategy's monthly return on a market-timing signal over $T=120$ months and reports classical standard errors. His residuals are positively autocorrelated: each month's residual is about $0.4$-correlated with the previous month's, fading with the gap. **(a) (4 pts)** Describe what $\boldsymbol\Omega$ looks like in Sam's case — diagonal, block-diagonal, or banded? Where are the nonzero entries, and what happens to them as the time gap $\ell$ grows? Contrast this with the clustering case in Problem 4. **(b) (4 pts)** Is Sam's reported classical t-statistic likely too big or too small? Argue it through the "effective sample size" intuition: with positive autocorrelation, how does the number of genuinely independent monthly draws compare to 120, and how does that bias the standard error? **(c) (4 pts)** Which standard-error flavor should Sam switch to, and what is the one knob (and its name) he must set? Using the Newey–West rule of thumb $L\approx 4(T/100)^{2/9}$ with $T=120$, compute a starting bandwidth (round to the nearest integer). Briefly: what goes wrong if he sets $L$ too small? Too large? **(d) (4 pts)** Sam's colleague trades a high-frequency strategy whose residuals show *negative* autocorrelation (bid–ask bounce). For that colleague, are naive t-stats too big or too small, and why? State the general principle linking the *sign* of the autocorrelation to the direction of the naive SE's error. --- ## Problem 6 — Petersen (2009): which clustering dimension? (18 points) Maya assembles a **panel**: many firms $i$ observed over many years $t$, and regresses a default outcome on a leverage measure. The coefficient is identical in every column below; only the SE flavor changes. She runs Petersen's own diagnostic — cluster one way, then the other, and watch which moves the SE more. | SE flavor | Std. error | t-stat | |-----------|-----------|--------| | Classical (OLS) | 0.018 | 5.0 | | White (HC1) | 0.021 | 4.3 | | Clustered by **firm** | 0.044 | 2.0 | | Clustered by **time** | 0.024 | 3.8 | | Two-way (firm + time) | 0.046 | 1.96 | **(a) (4 pts)** Reading the table with Petersen's diagnostic: clustering by firm roughly *doubles* the White SE while clustering by time barely moves it. Which effect dominates Maya's panel — a persistent firm effect, a common time effect, or both? What is the correct clustering choice, and what does that choice assert about where the off-diagonal blocks of $\boldsymbol\Omega$ live? **(b) (3 pts)** Why do White (HC1) standard errors fail to fix Maya's problem here? Be specific about what White SEs assume about the off-diagonals of $\boldsymbol\Omega$. **(c) (4 pts)** Suppose a different outcome on the *same* dataset produced firm-clustered SE $\approx$ White SE but time-clustered SE roughly double the White SE. What would that pattern imply, and which clustering would you choose? This is the point that "the right clustering depends on the left-hand-side variable, not just the dataset" — restate it in your own words. **(d) (4 pts)** A classmate says: "I already put **firm fixed effects** in the regression, so I don't need to cluster by firm — the fixed effects took care of the firm." Explain why this is wrong by distinguishing the two jobs (what a firm fixed effect removes vs. what clustering by firm handles). Can you want both at once? **(e) (3 pts)** Sketch the mechanical recipe for the **two-way** (firm + time) clustered variance in terms of three sandwiches you already know how to build. Why is one of them *subtracted*? --- *End of PS 2.4. Solutions in `book/appendices/E-solutions-manual/E-w2-ps2.4-solutions.md`. This set looks ahead to Weeks 3–4, where fixed effects and panel methods are developed in full, and to Ch 5.4, the complete reader's guide to Petersen (2009).*