What Categories Do Robust Standard Errors Commonly Refer to in Empirical Work? When Is Each Applicable? How Should You Choose and Justify Your Choice?

1. Opening: One Regression, Three Software Outputs, Three Standard Errors—Which One Should You Report?

You run a simple regression—using firm panel data to study "the impact of R&D investment on firm productivity." You write three lines of code in Stata and obtain three very different sets of t-values:

* Version 1: Default OLS standard errors
reg productivity rd_intensity size leverage, robust
* → t = 3.82 for rd_intensity, p = 0.000
 
* Version 2: Clustered at the firm level
reg productivity rd_intensity size leverage, cluster(firm_id)
* → t = 2.14 for rd_intensity, p = 0.033
 
* Version 3: Clustered at industry × year
reg productivity rd_intensity size leverage, cluster(industry_year)
* → t = 1.76 for rd_intensity, p = 0.079

The same coefficient β̂ = 0.34—unchanged. But the standard error changed three times, and the t-value fell from "significant at the 0.1% level" to "not significant even at the 10% level." Your conclusion—"R&D investment significantly improves productivity"—depends on which number appears in the parentheses you report.

A reviewer will inevitably ask: Which type of robust standard error did you use, and why? If you answer "because others do it too"—the reviewer will ask you to revise. If you answer "it's Stata's default"—the reviewer will reject. This article is designed to help you answer both questions.

Core message: In empirical econometrics, "robust standard errors" are a family, not a single entity. They all stem from the same mathematical framework—the "sandwich estimator"—but diverge into different versions under different assumptions about the error correlation structure. The three most common categories are: (1) Heteroskedasticity-consistent (HC) standard errors (also called White/Huber standard errors)—applicable to heteroskedasticity in cross-sectional data, assuming errors are independent across observations but not necessarily identically distributed; (2) Cluster-robust standard errors—applicable to panel data or any data with a grouped structure, allowing arbitrary correlation of errors within the same group but assuming independence across groups; (3) Heteroskedasticity-and-autocorrelation-consistent (HAC, Newey-West) standard errors—applicable to pure time series, allowing errors to be correlated over time with correlation decaying as the time interval grows. In contemporary micro panel applications (large N, small T), cluster-robust standard errors are the absolute mainstream—because they simultaneously address the two layers of panel data problems (within-unit serial correlation + arbitrary forms of heteroskedasticity) without requiring you to assume a specific serial correlation structure for the errors. The core question in choosing which standard error to use is: In your data, which observations' error terms are likely to be correlated?—This is both a statistical question and a question of economic theory and research design. Justifying your choice is not about reporting the p-value of a test—it is about explaining clearly "what the error correlation structure is in economic terms and why your clustering level corresponds to that structure."

2. Why Do We Need Robust Standard Errors?—OLS's Spherical Error Assumption and Its Two Violations

2.1 What Does OLS's Default Standard Error Assume?

In the classical linear regression $Y_i = X_i \beta + \varepsilon_i$, the OLS variance-covariance matrix formula is:

$\text{Var}_{\text{OLS}}(\hat{\beta}) = \sigma^2 (X'X)^{-1}$

This formula requires the spherical errors assumption:

$\text{Var}(\varepsilon \mid X) = \sigma^2 I_n = \begin{pmatrix} \sigma^2 & 0 & \cdots & 0 \\ 0 & \sigma^2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma^2 \end{pmatrix}$

Translated into two statements:

1. All diagonal elements are identical → $\text{Var}(\varepsilon_i) = \sigma^2$ for all i → homoskedasticity.
2. All off-diagonal elements are zero → $\text{Cov}(\varepsilon_i, \varepsilon_j) = 0$ for all i ≠ j → independent errors.

This is the origin of "spherical": the error covariance matrix is a sphere—uniform on the diagonal, zero off the diagonal.

2.2 Real-World Errors Are Rarely "Spherical"

Real-world errors typically deviate in two ways:

• Heteroskedasticity: diagonal elements are not all equal—the income volatility of the rich exceeds that of the poor, and profit volatility of large firms exceeds that of small firms.
• Autocorrelation/cluster correlation: off-diagonal elements are not all zero—the error of the same firm this year is correlated with its error next year, the errors of students in the same class are correlated, and the errors of different firms in the same year are correlated (common shocks).

OLS default standard errors assume a perfect sphere—but the real errors you face are usually a misshapen ellipsoid.

2.3 The Sandwich Estimator—The Unified Framework for All Robust Standard Errors

The common ancestor of all robust standard errors is the sandwich estimator:

$\text{Var}_{\text{Robust}}(\hat{\beta}) = \underbrace{(X'X)^{-1}}_{\text{bread}} \cdot \underbrace{\hat{\Omega}}_{\text{"meat"—error covariance structure}} \cdot \underbrace{(X'X)^{-1}}_{\text{bread}}$

OLS default standard errors assume $\hat{\Omega} = \hat{\sigma}^2 I$ (the meat is a uniform sphere). The difference between different types of robust standard errors lies solely in how the "meat" ($\hat{\Omega}$) in the middle is estimated—that is, what structural assumptions you impose on the error covariance matrix.

3. Five Types of Robust Standard Errors—Same Framework, Different $\hat{\Omega}$

3.1 Heteroskedasticity-Consistent (HC) Standard Errors—The Most "Basic" but by No Means the Most "Common"

Applicable scenarios: Cross-sectional data where errors are independent across observations but have different variances (pure heteroskedasticity, no autocorrelation).

$\hat{\Omega}_{\text{HC}} = \text{diag}\left(\frac{\hat{\varepsilon}_1^2}{w_1}, \frac{\hat{\varepsilon}_2^2}{w_2}, \dots, \frac{\hat{\varepsilon}_n^2}{w_n}\right)$

All off-diagonal elements are zero (independent errors); each diagonal element is a scaled squared residual.

HC has four variants—differing only in the adjustment made to $w_i$ (small-sample degrees-of-freedom corrections):

reg y x1 x2, robust            // Stata default uses HC1
reg y x1 x2, vce(hc2)          // HC2
reg y x1 x2, vce(hc3)          // HC3 (recommended for N < 250)

Practical recommendations:

• N > 500: The difference between HC1 and HC3 is usually negligible.
• N < 250: HC3 is recommended—it is the most conservative of the four variants and has the best finite-sample properties.
• Cross-sectional data (e.g., a single wave of CGSS or CFPS) → HC is your default choice (if you are confident that errors across different individuals are independent).

Limitations of HC: It assumes errors are independent—all off-diagonal elements are zero. If your data have any grouping or time structure, HC cannot handle it—it will severely underestimate the true standard errors.

3.2 Cluster-Robust Standard Errors—The King of Contemporary Micro Panel Applications

Applicable scenarios: Panel data (multiple periods for the same unit), grouped data (students nested in classes, patients nested in hospitals), experimental data (treatment randomly assigned at the group level).

Core idea: Partition the data into G non-overlapping "clusters." Allow arbitrary correlation of errors within the same cluster (without assuming a serial correlation structure or imposing a parametric form), but assume errors are independent across different clusters.

$\hat{\Omega}_{\text{Cluster}} = \text{blockdiag}(\hat{\Omega}_1, \hat{\Omega}_2, \dots, \hat{\Omega}_G)$

Each $\hat{\Omega}_g$ is a $T_g \times T_g$ block whose elements are $\hat{\varepsilon}_{it} \cdot \hat{\varepsilon}_{is}$—that is, products of errors from different periods within the same cluster. Everything outside the blocks (across different clusters) is zero.

* Clustering at the individual level (most common in panel data)
reg y x1 x2, cluster(id)
xtreg y x1 x2, fe cluster(id)    // FE + clustering
reghdfe y x1 x2, absorb(id year) vce(cluster id)
 
* Clustering at a higher level (e.g., industry × year, city, class)
reg y x1 x2, cluster(industry)
reg y x1 x2, cluster(city)

Choice of clustering level—This is the most important and error-prone decision in using clustered standard errors:

• Individual-level clustering: The most conservative and most common. Suitable for panel data—errors for the same firm in different years may be correlated (persistence of firm-specific shocks).
• Higher-level clustering: If errors are correlated not only within individuals but also across individuals in the same group—e.g., "firms in the same industry face the same macro shocks in the same year" → cluster at the industry level or the industry × year level.
• Clustering at the treatment assignment level: In RCTs or DID designs—if treatment is assigned at the group level (e.g., village-level randomization), you must cluster at the unit of treatment assignment (village), not at the individual level.

The key criterion: At what level might unobservable shocks be correlated across different observations in your data? The answer depends on your research design and the data-generating process—not just the mathematical features of the data structure.

The "few clusters problem" with clustered standard errors:

• The asymptotic theory for clustered standard errors is built on G → ∞ (the number of clusters tending to infinity).
• If G < 20–30 → clustered standard errors may severely underestimate the true standard errors (downward bias), leading to over-rejection.
• If G < 10 → conventional clustered standard errors are essentially unreliable.

Corrections for few clusters:

• Wild cluster bootstrap (applicable when G ≥ 10)—corrects finite-sample bias through bootstrap resampling.
• Aggregating to fewer dimensions or using the Cameron-Gelbach-Miller wild bootstrap.
• In extreme few-cluster cases (e.g., G = 5): consider wild bootstrap with small-cluster corrections, or honestly acknowledge that "given the limited number of clusters, the standard errors in this paper may be underestimated, and the results should be interpreted with caution."

* Wild cluster bootstrap (using boottest)
boottest rd_intensity, cluster(industry) reps(999) seed(123)

3.3 Heteroskedasticity-and-Autocorrelation-Consistent (HAC / Newey-West) Standard Errors—A Powerful Tool for Time Series

Applicable scenarios: Pure time-series data (T large, N = 1), where errors are correlated over time and the correlation decays as the time interval increases.

$\hat{\Omega}_{\text{HAC}} = \sum_{k=-(T-1)}^{T-1} w_k \cdot \left( \frac{1}{T} \sum_t X_t \hat{\varepsilon}_t X_{t-k}' \hat{\varepsilon}_{t-k} \right)$

where $w_k$ is the kernel weight—for larger lag orders k, $w_k$ is smaller. This ensures that you do not need to estimate all T−1 autocovariances (which would be impossible), but instead smooth the estimation with a decaying weight function.

• Bartlett kernel (Newey-West default): $w_k = 1 - \frac{|k|}{m+1}$ for $|k| \leq m$, and 0 otherwise.
• Truncation parameter m: Controls how long errors are allowed to be correlated. $m = \lfloor 4(T/100)^{2/9} \rfloor$ is a common automatic formula. For T = 100, m ≈ 4.

* Newey-West HAC standard errors
newey y x1 x2, lag(4)     // Manually specify the truncation parameter
ivreg2 y x1 x2, bw(auto)  // Automatically select the lag order

Application of HAC in panels: In micro panels (large N, small T), HAC is less common than clustered standard errors—because the panel structure is better suited to modeling error correlation through "clustering" rather than "time decay." However, in macro panels (large N, large T, e.g., T = 50–60), HAC and clustering may be used simultaneously (or Driscoll-Kraay standard errors that allow for panel autocorrelation may be used).

3.4 Two-Way Clustered Standard Errors—When Errors Are Correlated Along Two Dimensions Simultaneously

Applicable scenarios: Errors are correlated along both the individual and time dimensions—e.g., "errors for the same firm in different years are correlated (individual-dimension clustering)" and "errors for different firms in the same year are correlated (time-dimension clustering, due to common shocks)."

Two-way clustered standard errors (Cameron, Gelbach, and Miller, 2011):

$\hat{\Omega}_{\text{Two-way}} = \hat{\Omega}_{\text{cluster by firm}} + \hat{\Omega}_{\text{cluster by year}} - \hat{\Omega}_{\text{cluster by firm×year}}$

That is, the sum of two one-way clusterings minus their intersection (cross-clustering)—to avoid double-counting the diagonal elements.

* Two-way clustering (individual + year)
reghdfe y x1 x2, absorb(id year) vce(cluster id year)
* Or using ivreg2
ivreg2 y x1 x2, cluster(id year)

When to use:

• You suspect unobserved common annual shocks (e.g., the 2008 financial crisis affected all firms).
• Your regression already includes year fixed effects—but if the effect of common shocks is heterogeneous across individuals (different firms respond differently to the same macro shock), year fixed effects cannot fully absorb it.
• In provincial panels and cross-country panels, two-way clustering (country + year) is increasingly common.

3.5 Bootstrap Standard Errors—When You Do Not Trust Any Parametric Assumption

Applicable scenarios: You are uncertain about the error structure, the sample size is small, or existing analytical standard error formulas are unreliable for your data.

Principle: Repeatedly draw samples from the original data with replacement (resampling), run the same regression on each resampled sample, and use the standard deviation of the empirical distribution of $\hat{\beta}$ across the resampled samples as the estimate of the standard error.

* Residual bootstrap
bootstrap, reps(1000) seed(123): reg y x1 x2
 
* Wild bootstrap—more suitable for small samples with heteroskedasticity
boottest y, cluster(id) reps(999)

Two main types of bootstrap:

• Pairs bootstrap (case resampling): Resample observations (rows)—preserving the original relationship structure between X and Y in the data. Robust to heteroskedasticity and arbitrary error correlation—because the resampling unit can be any "cluster" you choose (e.g., resampling individuals with replacement rather than individual-year observations).

• Residual bootstrap: First estimate the model → obtain residuals → resample the residuals → generate new Y → re-estimate. Requires residuals to be independently and identically distributed—which may not hold in panel and heteroskedastic data.

• Wild bootstrap: Multiplies residuals by a random variable (e.g., the Rademacher distribution—±1 with 50% probability each) to generate new Y. Does not assume residual independence and performs best in small samples and few-cluster settings.

Cost of bootstrapping: Computationally intensive (requires running 1,000+ regressions); standard error estimates are still biased with few clusters (but less so than analytical clustered standard errors). Not commonly used in large-N panels—because clustered standard errors are already sufficiently reliable when G is large.

3.6 Comparison of the Five Types of Standard Errors at a Glance

4. How to Choose in Empirical Research?—A Decision Framework

4.1 Step 1: Identify the "Correlated Units" in Your Data

This is the most fundamental question, and it is what you should justify in your paper:

In your data, which observations' error terms are likely to be correlated in economic terms?

• Different years of the same firm → cluster at the firm level. Management culture, technological accumulation, and market position—these time-invariant factors make a firm's shock this year correlated with its shock next year.
• Students in the same class → cluster at the class level. Shared teacher quality, class atmosphere, and peer effects—these make the errors of students in the same class correlated.
• Different firms in the same year → cluster at the year level. Macro shocks (GDP growth, policy changes, financial crises) affect all firms simultaneously.
• Firms in the same industry × year → cluster at the industry × year level. Industry-specific cyclical fluctuations (e.g., "the tourism industry was hit by the pandemic in 2020").

The answer depends on your research design, not the mathematical features of the data structure. If your dependent variable is at the firm level but your key explanatory variable is an industry-level policy change—then you must cluster at the industry level (or at least report industry-level clustering as a robustness check), because the variation in the policy shock comes from the industry, and errors across firms within the same industry are necessarily correlated.

4.2 Step 2: Check the Number of Clusters

• If you cluster at the individual level and G = N (number of individuals) is large (e.g., N > 1000) → the asymptotic properties of clustered standard errors are good.
• If you cluster at the industry level and G = number of industries = 20 → you are in the critical zone—consider reporting wild bootstrap results as well.
• If you cluster at the province level and G = number of provinces = 30 → acceptable, but consider two-way clustering (province + year).

4.3 Step 3: Conduct "Clustering-Level Robustness Checks"

An increasingly common practice in empirical papers is to report your most conservative clustered standard errors in the main regression and then report results at other clustering levels in the robustness checks—demonstrating that your conclusions do not depend on the choice of clustering level.

* Main regression: clustered at the individual level
reghdfe y x, absorb(id year) vce(cluster id)
estimates store m1
 
* Robustness 1: Clustered at a higher level (e.g., industry × year)
reghdfe y x, absorb(id year) vce(cluster industry_year)
estimates store m2
 
* Robustness 2: Two-way clustering (individual + year)
reghdfe y x, absorb(id year) vce(cluster id year)
estimates store m3

If the coefficient is significant across different clustering levels → your conclusion is robust to the choice of standard errors—which in itself constitutes a strong argument for the reasonableness of your choice.

4.4 Decision Flowchart

What structure does your data have?
│
├── Pure cross-sectional data (no grouping, no time dimension)
│   └── → HC (heteroskedasticity-robust). Use HC3 for N < 250, HC1 for N > 500.
│
├── Panel data (individual × time)
│   │
│   ├── Small T (≤10), large N (> 500)
│   │   ├── Treatment varies at the individual level → cluster(id)
│   │   ├── Treatment varies at a higher level (e.g., industry policy) → cluster(treatment assignment level)
│   │   └── Common annual shocks exist → two-way clustering cluster(id year)
│   │
│   └── Large T (> 30), large N (macro panel)
│       ├── Focus on within-individual serial correlation → cluster(id)
│       ├── Focus on cross-sectional correlation → Driscoll-Kraay or two-way clustering
│       └── Concerned about both → Driscoll-Kraay (robust to both)
│
├── Pure time series (N = 1, large T)
│   └── → HAC (Newey-West). For T < 50, consider FGLS as an alternative.
│
├── Grouped/nested data (students → classes → schools)
│   └── → Cluster at the "level of independent units." If treatment is at the school level → cluster(school).
│
└── Experimental data (RCT)
    └── → Cluster at the level of treatment assignment—this is the only correct choice. If village-level randomization → cluster(village), not cluster(individual).

5. Clustered Standard Errors vs. Heteroskedasticity-Robust Standard Errors—Which Is More "Common"?

5.1 In Micro Panel Applications—Clustered Standard Errors Are the Absolute Mainstream

Open any issue of AER, JPE, QJE, or Economic Research Journal—among empirical papers using panel data, 99% use cluster-robust standard errors rather than HC heteroskedasticity-robust standard errors.

The reason is not that HC is inherently flawed—but that under the panel data structure, clustered standard errors are logically a better match.

The defining feature of panel data is that observations of the same individual across different periods are not independent. If you use HC in panel data—HC assumes all 500 × 5 = 2,500 observations are independent—you will severely underestimate the standard errors, because you are pretending to have 2,500 independent pieces of information when in reality you only have 500 (one per individual), each measured repeatedly 5 times at different points in time.

Clustered standard errors correct this "pseudo-replication"—they "adjust" your effective sample size from N × T back to G (the number of clusters).

5.2 In Pure Cross-Sectional Data—HC Is the Most Common

If your data are a single cross-sectional wave of CGSS, CFPS, or CHFS from one year, with no natural grouping structure—HC is appropriate. But even in cross-sectional data, if grouping exists (e.g., "communities," "schools," "business groups"), clustering at the group level is necessary.

5.3 In Macro Panels (Large T)—Mixed Usage

In cross-country panels (T = 50–60), clustered standard errors and HAC (or its panel variant, Driscoll-Kraay) are often reported simultaneously. Clustering at the country level handles within-individual serial correlation, while Driscoll-Kraay additionally handles cross-sectional correlation. This is a special requirement of macro panels, because T is so large that cross-sectional correlation can no longer be ignored.

6. How to Justify Your Choice of Standard Errors in a Paper?

Reviewers will not accept "because Stata defaults to this" or "because I saw others do it." You need an argument grounded in theory and research design.

6.1 A Standard Justification Template

Below is a line of reasoning you can adapt for your own paper (using firm panel data clustered at the firm level as an example):

"This paper uses firm-level cluster-robust standard errors for the following reasons. First, the panel data structure is large N and small T (N = 2,347 firms, T = 6 years), and the error terms of the same firm in different years may exhibit serial correlation—firm-specific shocks (such as management changes, technological upgrades, and market strategy adjustments) are persistent and may span multiple years. Second, the key explanatory variable (R&D investment) varies at the firm level over time—the identifying variation comes primarily from within-firm changes, and serial correlation of errors within the firm will directly affect inference on the key coefficient. Third, clustering at the firm level allows errors to have an arbitrary correlation structure within the firm, without requiring assumptions about the specific parametric form of serial correlation (such as AR(1)), making it more flexible than HAC standard errors in micro panels. As robustness checks, this paper also reports standard errors clustered at the industry × year level, as well as two-way clustering (firm + year) results—the core conclusions remain robust across different clustering levels."

6.2 Key Points for Justification in Different Contexts

6.3 What Not to Do

• ❌ Do not simply say "this paper uses robust standard errors" without specifying which type. HC? Clustered? HAC?—Please be specific.
• ❌ Do not use clustering in data without a grouped structure (e.g., pure cross-sectional data with no a priori correlation across individuals)—this will be seen as mechanical imitation.
• ❌ Do not cluster at a level that conflicts with the source of your identifying variation (e.g., if the key X is at the individual level but you cluster at a higher level—this is of course more conservative, but if the number of clusters is small (e.g., G = 10), conservatism may become "unreliability").
• ❌ Do not use a Hausman test or overidentification test to "choose" the type of standard errors—the choice of standard errors is a research design question, not a statistical testing question.

7. Common Misconceptions

7.1 Misconception 1: "I used robust → my standard errors are correct"

Different robust standard errors correct for different error structures. HC only corrects for heteroskedasticity—if your data have a grouped structure but you use HC, your standard errors are still severely underestimated. "Robust" is relative—you need to be clear about "robust to what": heteroskedasticity? Serial correlation? Cluster correlation? Two-way correlation?

7.2 Misconception 2: "Clustered standard errors are always larger than HC → more conservative → safer"

In most cases, clustered standard errors are indeed larger than HC standard errors—because clustered standard errors acknowledge less information. After the "pseudo-replication problem" is corrected, the effective sample size reverts from N × T to G. But in certain special structures—such as negatively correlated errors within clusters (though rare)—clustered standard errors may be smaller than HC. Moreover, clustered standard errors themselves have a downward finite-sample bias when G is small. Conservative does not mean safe—if the conservative standard error itself is biased.

7.3 Misconception 3: "Year FE + clustering at the individual level → all error correlation has been controlled for"

Individual FE controls for time-invariant individual heterogeneity. Year FE controls for annual shocks common to all individuals. But what neither can control is error correlation that varies over time and differs by individual—for example, "firm A experienced a specific shock in 2018 whose effect persisted into 2019"—this is individual-specific, time-varying serial correlation. Clustered standard errors (clustered at the individual level) are precisely designed to handle this layer.

7.4 Misconception 4: "The higher the clustering level, the better—more conservative means safer"

Raising the clustering level does generally make standard errors larger—but the cost is that the number of clusters G decreases. If G is too small (< 20), the asymptotic theory for clustered standard errors breaks down, potentially producing standard errors smaller than the true values (even if they appear numerically larger than HC). "Conservative" is not truly conservative if it rests on unreliable asymptotic approximations. Choosing a clustering level is not about "higher is better"—it is about balancing "the reasonableness of the cluster correlation structure" against "the sufficiency of the number of clusters."

7.5 Misconception 5: "My clustering has G = 50 → that should be large enough"

G = 50 is generally considered "safe"—but Monte Carlo simulations show that even with G = 50, clustered standard errors can still have a downward bias in finite samples (on the order of 5%–10%). How large G needs to be depends on the degree of heterogeneity in the errors—if the within-cluster error structure is highly uneven across clusters (some clusters are very large, others very small; some clusters have extremely volatile errors, others extremely stable), even G = 100 may perform poorly. Cameron and Miller (2015) suggest that when G < 20–30, alternative methods (wild bootstrap or aggregating to fewer inference units) should be explicitly considered.

8. Summary

Seven core takeaways about robust standard errors:

1. All robust standard errors come from the same mathematical framework—the sandwich estimator. The only difference lies in the "meat" in the middle—your structural assumptions about the error covariance matrix. This assumption is not chosen for you by mathematics—it is chosen by you based on research design and economic theory.

2. Choosing the type of standard errors = answering one question: In your data, which observations' errors are correlated? The answer does not come from statistical tests—it comes from your understanding of the data-generating process. Different periods of the same unit are correlated → cluster at the unit level. Firms in the same industry face common shocks → cluster at the industry level. Annual macro shocks → cluster at the year level.

3. In micro panels (large N, small T)—clustered standard errors are the default and are accepted as the default by reviewers. HC is no longer suitable for panel data because it does not handle serial correlation. HAC performs poorly in short panels because it relies on large-T asymptotics. Clustered standard errors handle both heteroskedasticity and arbitrary within-cluster correlation simultaneously—they are the natural match for panel data.

4. The clustering level should match the source of variation in your key explanatory variable. If treatment/policy varies at a higher level—you must at least cluster at that level. Using a clustering level lower than the source of identifying variation—is pretending you have more independent information than you actually do.

5. The number of clusters (G) is the Achilles' heel of clustered standard errors. G < 20–30 → standard errors may be underestimated, leading to over-rejection. G < 10 → conventional clustered standard errors are essentially unreliable. With few clusters, reporting wild cluster bootstrap results is the norm.

6. Justify your choice—write in your paper why this clustering level is reasonable in your study. Do not write "this paper uses cluster-robust standard errors"—write "this paper clusters at the firm level because shocks to the same firm are persistent across years—firm-specific market strategies and management changes affect performance across periods."

7. Conduct clustering-level robustness checks. If your conclusions hold across different clustering levels (individual, industry, year, two-way)—the reasonableness of your choice is self-evident.

One-sentence conclusion:

"Robust standard errors are not a 'magical parenthesis' that makes you safer—they are a modeling statement about your error structure. HC says 'my error variances differ but are independent of each other'; clustering says 'errors within the same group are acquaintances who know each other, while errors across groups are strangers'; HAC says 'my errors' relationships fade over time.' You are not choosing a parenthesis—you are choosing what you believe about how errors are correlated in this world. And in this choice, the data are not the sole judge—your research design and economic theory have at least equal voting power. The best choice is not one that is always correct—it is one where you clearly know what it protects you from and what it exposes you to."

9. Presentation Suggestions for Bilibili/WeChat Official Account

• Bilibili video: Use the "sandwich" as the core visual metaphor. Opening: a sandwich is cut open—two slices of bread ($(X'X)^{-1}$) with a piece of meat ($\hat{\Omega}$) in between. Voiceover: "All robust standard errors are the same sandwich—the two slices of bread are the data, and the meat in the middle is the error structure. OLS defaults assume the meat is a uniform sphere ($\sigma^2 I$). Heteroskedasticity flattens the sphere—the diagonal changes. Autocorrelation turns the off-diagonal elements from zero to nonzero—the sphere becomes an ellipsoid. Clustering says—the meat comes in chunks, and within each chunk, anything goes." Act 1: "Five standard errors, five kinds of meat"—the screen shows the five $\hat{\Omega}$ matrices in sequence: HC (diagonal only), clustered (block-diagonal matrix—several large color blocks), HAC (a banded matrix fading along the diagonal), two-way clustering (blocks crossing in two dimensions), bootstrap (the matrix perturbed across bootstrap resamples—dynamic, assuming no fixed shape). Act 2: "Choosing the clustering level"—a panel data plot: 500 firms, each with 5 years. Voiceover: "Cluster at the firm level—you acknowledge that these 5 points (5 years of the same firm) are correlated, but the 500 firms are independent of each other. You go from 2,500 'independent' observations back to 500—you have less information, but your inference is more honest." Act 3: "The few-clusters problem"—visual: only 10 clusters, each with many data points inside. Conventional clustered standard errors produce confidence intervals that are too narrow—wild bootstrap produces wider confidence intervals. Voiceover: "Too few clusters—conventional clustered standard errors may be unreliable. Wild bootstrap simulates the sampling distribution by repeatedly flipping the signs of residuals—