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Should Bank Stress Tests Be Fair?

Regulatory stress tests have become the primary tool for setting capital requirements at the largest U.S. banks. The Federal Reserve uses confidential models to evaluate bank-specific outcomes for bank-specific portfolios in shared stress scenarios. As a matter of policy, the same models are used for all banks, despite considerable heterogeneity across institutions; individual banks have contended that some models are not suited to their businesses. Motivated by this debate, we ask, what is a fair aggregation of individually tailored models into a common model? We argue that simply pooling data across banks treats banks equally but is subject to two deficiencies: it may distort the impact of legitimate portfolio features, and it is vulnerable to implicit misdirection of legitimate information to infer bank identity. We compare various notions of regression fairness to address these deficiencies, considering both forecast accuracy and equal treatment. In the setting of linear models, we argue for estimating and then discarding centered bank fixed effects as preferable to simply ignoring differences across banks. We present evidence that the overall impact can be material. We also discuss extensions to nonlinear models.

Introduction
In the aftermath of the 2008 financial crisis, U.S. banking regulators adopted stress testing as the primary tool for monitoring the capital adequacy of the largest banks. For each round of annual stress tests, the Federal Reserve announces a "severely adverse stress scenario," defined by a hypothetical path of economic variables over the next several quarters. A typical path includes an increase in unemployment, a decline in GDP, and projections for the level and volatility of the stock market, among other variables. The largest banks provide the Fed with detailed information about their loan portfolios and other assets. The Fed then applies internally developed models to project revenues and losses for each bank through the stress scenario. Banks are required to have sufficient capital to weather the projected losses.
The Fed does not disclose details of the models it uses to project revenues and losses.
The Fed makes clear that to ensure consistent treatment for different banks it uses "industry models," as opposed to models tailored to individual banks. As a matter of policy, the same models are applied to all banks. Quoting Board of Governors [9] (p. 3), "two firms with the same portfolio receive the same results for that portfolio." Banks have countered that the Fed's models fail to capture bank-specific features that should lower projected losses. They have made these arguments in requests for reconsideration of stress test results. Of course, banks are not objective critics of the Fed's supervision; but significant heterogeneity among the largest banks is indisputable. The banks subject to annual stress testing include universal banks, investment banks, large regional banks, the U.S. subsidiaries of certain foreign banks, and a variety of more specialized financial firms. It is certainly possible that bank-specific models would produce more accurate forecasts than a single industry model, in which case using a single model entails a trade-off between forecast accuracy and consistency across banks.
The heterogeneity among large banks motivates the questions we study: What is the best way to aggregate bank-specific models into an industry model? Is simply ignoring bank identity in estimating and applying models the best way to achieve fairness? To what extent is fairness at odds with accuracy? Although the heterogeneity of large banks is widely recognized, we know of no prior work that seeks to address this property within the constraints of the Fed's policy of equal treatment of banks. We will argue that addressing heterogeneity is preferable to ignoring it.
The question of fairness in algorithms and models has received a great deal of renewed interest in recent years, in some cases reviving earlier debates over fairness in testing and related policies that were not explicitly "algorithmic;" see, for example, the overviews in Barocas, Hardt, and Narayanan [6] and Hutchinson and Mitchell [28]. We draw on this literature, but our setting differs in important ways from most discussions of fairness.
Algorithmic fairness is usually concerned with ensuring that certain protected attributesrace or gender, for example -do not influence outcomes -such as hiring decisions or loan approvals. Different methods can be compared based on alternative measures of influence and the degree to which sensitive attributes are indeed protected.
The counterpart of a protected attribute in our setting is a bank's identity; but this attribute is not so much protected (in the sense that race and gender are) as inadmissible for the Fed's purpose. In stating that "two firms with the same portfolio receive the same results for that portfolio," the Fed is stating that bank identity is not a legitimate predictor of losses. Perhaps, then, fairness is achieved as long as the Fed uses the same model for all banks. In other words, perhaps "fairness through unawareness," paraphrasing Dwork et al. [15], is sufficient in this setting. Moreover, in questioning whether the Fed's models apply to them, banks are not claiming discrimination; on the contrary, they are asking for discrimination -asking that the Fed change its models to recognize ways in which an individual bank differs from other banks.
To investigate these issues, we focus primarily on a simple setting in which the "true" loss rate for each bank is described by a bank-specific regression on portfolio features and scenario features. The regulator's goal is to aggregate these bank-specific models into a single model. A natural interpretation of an "industry" model in this setting is a pooled regression based on combining results across banks. The pooled model treats banks equally, but we show that it has at least two significant deficiencies: when applied to heterogeneous banks, it can produce poor measures of the marginal impact of individual features, even resulting in the wrong sign; and it implicitly misdirects legitimate information in portfolio features to infer bank identity in forecasting losses. The second of these deficiencies works against the spirit of equal treatment of banks, even if bank identity is not explicitly used in the model.
We then investigate the application of ideas from algorithmic fairness in our setting. The fairness literature has mainly focused on classification problems (hiring decisions and credit approvals, for example), with regression problems getting somewhat less attention. Chzhen et al. [12] and Le Gouic et al. [35] developed a method of particular importance for regression that Le Gouic et al. [35] call "projection to fairness." This method produces optimal forecasts (in the least-squares sense) subject to a fairness constraint known as demographic parity. We examine the application of this approach in our setting and conclude that it goes too far in leveling results across banks.
The pooled method ignores fairness and the projection method goes too far in imposing fairness, so we seek an intermediate solution. Johnson, Foster, and Stine [30] introduce a variety of methods for introducing fairness considerations in regression. These include methods they call "full equality of opportunity" (FEO) and "substantive equality of opportunity" (SEO). We examine these methods in our setting and conclude that the FEO method provides an attractive solution. In particular, we show that it addresses the two deficiencies of the pooled method highlighted above: it removes the distortion in the pooled coefficients that results from bank heterogeneity, and it prevents the misdirection of legitimate information to infer bank identity.
Indeed, we show that the only way to achieve lower forecast errors than the FEO method is through such misdirection.
Moreover, the method is easy to interpret and implement: fit a pooled model with centered bank fixed effects, and then discard the centered fixed effects to forecast losses. Including the fixed effects prevents misdirection of legitimate information; discarding them is necessary to treat banks equally; centering ensures that the overall mean forecast remains unchanged.
Although we mainly work with linear models, we show that these ideas can be extended to nonlinear models as well.
We also investigate the empirical relevance of these considerations. We regress the loss rates of loan portfolios (credit cards, first lien mortgages, commercial real estate, and corporate and industrial loans) on measures of portfolio quality (past-due rates and allowances for losses) and macroeconomic variables. We document significant heterogeneity across banks in their estimated coefficients, and we show that the difference in pooled and FEO estimates can be material. This investigation is limited by the information banks make public -the Federal Reserve has access to far more granular data in forecasting losses. We cannot claim to approximate the Fed's forecasts; our goal is to provide evidence of the importance of heterogeneity.
We provide additional background on the Federal Reserve's stress tests in Section 2. Section 3 lays out our modeling framework and analyzes the pooled industry model within this framework. Section 4 analyzes various ways to introduce fairness considerations, including the projection-to-fairness and FEO methods. Section 5 presents our empirical results. Section 6 investigates cross-bank externalities that arise when bank-specific models are aggregated into a common model: a change at one bank can impact loss forecasts at other banks. Effects of this type are inevitable under model aggregation, but we argue that the effects are more transparent and less objectionable under FEO than under a pooled model. Section 7 considers nonlinear models. Additional supporting material is included in appendices.
We conclude this introduction with a brief discussion of some other research on bank stress tests. Covas, Rump, and Zakrajsek [14], Kapinos and Mitnik [32], and Kupiec [34] find strong evidence of heterogeneity in banks' responses to macroeconomic shocks, and Kapinos and Mitnik [32] argue that ignoring heterogeneity can substantially underestimate projected capital requirements. The related models of Hirtle et al. [29] and Guerrieri and Welch [26] forecast aggregate results and are therefore not concerned with differences among banks. Heterogeneity in the accuracy of the Fed's models for different banks is suggested by the comparisons in Agarwal et al. [1], Bassett and Berrospide [7], and Flannery, Hirtle, and Kovner [18] [34] and Philippon, Pessarossi, and Camara [39]). Flannery [17] discusses just how much information the Fed should disclose about stress testing procedures and outcomes. For perspectives on the effectiveness of the Fed's stress tests, see Kohn and Liang [33] and Schuermann [42].

Background
This section provides background on the Federal Reserve's stress testing process and on the heterogeneity of the participating banks.

Regulatory Bank Stress Tests
In early 2009, in the depths of the Global Financial Crisis, the Federal Reserve launched a stress test of the 19 largest U.S. bank holding companies to gauge how much more capital they would need if economic conditions continued to worsen. The results of the stress test were made public, and the transparency and credibility of the process have been credited with restoring public confidence and helping to end the crisis.
The Dodd-Frank Act, the package of reforms that followed the crisis, codified the use of stress testing for bank supervision. The number of banks subject to DFAST (Dodd-Frank Act Stress Tests) has varied over time. The current requirement applies annually to banks with over $250 billion in assets and every other year to banks with assets between $100 billion and $250 billion. The 2021 DFAST covered 23 banks, down from a peak of 35 in 2018. We refer to the participating firms as "banks," but they are more precisely holding companies, including the U.S. subsidiaries of some foreign banks.
The inputs to the stress test analysis are the stress scenario, which is common to all banks, and bank-specific balance sheet information. A scenario is specified through a hypothetical path of economic variables over the next nine quarters. The 2021 DFAST specified paths for 28 variables, including GDP, inflation, unemployment, stock market and real estate indexes, interest rates, exchange rates, and measures of overseas economic activity. Each bank submits detailed information on its loans and other assets.
The Fed uses 21 models to integrate the stress scenarios with bank-level information to make bank-level projections. For example, one model applies to commercial and industrial loans, one to credit cards, one to commercial real estate loans, and another to first lien residential mortgages. These models project losses in each of these portfolios. Some other models project revenues.
The Fed does not disclose details of its models, either to banks or the general public. But it does describe its general modeling approach in public documents. At a high level, a model assigns a loss rate to a set of bank-specific loan portfolio features x and a common set of scenario variables z through a function f (x, z). The function f is estimated from past observations of the macro variables and the bank-specific variables for multiple banks. Thus, f is estimated as an industry-wide model and then applied individually to each bank. This approach is described, for example, on p.3 of Board of Governors [9], where we read, "The Federal Reserve generally develops its models under an industry-level approach calibrated using data from many financial institutions.. . . The Federal Reserve models the response of specific portfolios and instruments to variations in macroeconomic and financial scenario variables such that differences across firms are driven by differences in firm-specific input data, as opposed to differences in model parameters and specifications. As a result, two firms with the same portfolio receive the same results for that portfolio in the supervisory stress test, facilitating the comparability of results." We will refer to the principle that banks with the same portfolio receive the same results as equal treatment.

Bank Heterogeneity
The appropriateness of equal treatment seems incontrovertible. But the right notion of consistency across firms becomes less clear when portfolios vary widely. The largest U.S. banks are a highly heterogeneous group. They include universal banks (like JPMorgan Chase and Bank of America), investment banks (Goldman Sachs and Morgan Stanley), custodians (BNY Mellon and State Street), regional banks (like US Bancorp and PNC Financial), specialized banks (like American Express), and the U.S. subsidiaries of large foreign banks (such as TD Group and HSBC North America). We may not expect a regional bank to have an investment bank's skill in the capital markets, nor do we expect the investment bank to have the regional bank's skill in making single-family residential loans.
Heterogeneity among the eight U.S. Global Systemically Important Banks (G-SIBs) is illustrated in Figure 2.1. The left panel shows variation in bank size, with JPMorgan Chase (JPM) more than ten times larger than State Street (STT), as measured by total assets. The left panel also shows significant variation in the proportion of assets made up by loans. The right panel shows heterogeneity in the fractions of loans the banks hold in each of four categories.
For example, for Wells Fargo (WFC) first lien mortgages are a relatively large fraction of its loans, whereas for Citigroup (C), credit cards make up a relatively large fraction.
Beginning in 2020, the Federal Reserve allowed banks to submit requests for reconsideration of the stress capital buffer set by the Fed through the stress testing process. (The capital buffer is set through the Comprehensive Capital Analysis and Review, or CCAR, process, which accompanies the stress test.) The banks' requests are confidential, but the Fed's responses to these requests are public. The responses show that the banks were arguing for reconsideration Five firms requested reconsideration in 2020, and all five requests were rejected. In its response 1 to Goldman Sachs, the Fed wrote, "the Board has determined that it will follow its published principles for stress testing, including the principle of creating industry-level models, and not modify the existing results of these models. In particular, models used in the supervisory stress test are generally developed according to an industry-level approach, calibrated using data from many institutions." Similar statements appear in all five rejections. These exchanges point to a debate in which the banks highlight their heterogeneity and the Fed asserts the importance of consistency.

Sources of Heterogeneity
Stepping back from the specifics of this debate, it is helpful to consider sources of heterogeneity through the lens of a linear regression of a loan portfolio's loss rate on portfolio characteristics and macro variables. In this setting, we have at least four sources of heterogeneity: the distribution of portfolio characteristics, differences in intercepts, differences in slopes, and differences in error distributions. Differences in errors mainly affect the precision of forecasts rather than the forecasts themselves, so we focus on the first three items.
1 https://www.federalreserve.gov/supervisionreg/files/goldman-sachs-group-inc-20200904.pdf Differences in portfolio characteristics are evident. Banks differ in their mix of corporate and retail business, their mix of regional, national, and international clients, and their focus on higher-or lower-risk borrowers. (In its 2020 annual report, Capital One reported that 31% of its credit card receivables were due from customers with FICO scores of 660 or lower. For Citigroup the proportion was less than 15% on its Citi-branded cards.) Potential differences in slopes admit at least two possible interpretations. One possibility is that banks differ in unobserved portfolio characteristics that influence loss rates. If these unobserved characteristics are correlated with observed features, then omitting them changes the coefficients on the observed features. Unobserved portfolio characteristics could thus create differences among otherwise identical banks. A second possibility is that banks differ in their skill in managing loans, perhaps through more effective monitoring of borrowers. Differences in skill are then reflected in different coefficients linking portfolio features to loss rates. The first interpretation suggests that one bank's B-rated borrowers may be better than another bank's; the second interpretation suggests that one bank is simply better at securing repayment from B-rated borrowers.
Differences in intercepts -bank fixed effects -are commonly included in empirical work on banking. They are typically included to absorb unobserved characteristics that persist across time. As in our discussion of slopes, it is possible that some banks are better managed than others and that these differences generate predictable differences in loss rates. Regulators themselves seem to support this idea by emphasizing the importance of establishing a strong "culture" in banks. 2 However, we take the view that such differences should not confound loss forecasts in the design of an industry model for quantitative stress-testing. Equal treatment requires that stress-results be based on observable characteristics. Other parts of the overall bank supervision process (outside of stress testing) address issues like the quality of internal governance, business models, and controls. Within the Basel framework, these considerations are part of the Pillar 2 supervisory process, as described in BCBS [8].
In running the stress tests, the Fed collects highly granular data from each bank, and it has a great deal of power to collect the data it needs. It also designs the models that use the banks' data. With these considerations in mind, and recalling the statements on Fed policy quoted earlier, our analysis proceeds on the following premises: (i) The Fed considers the portfolio features used in its models to be legitimate information for forecasting losses; (ii) no obviously relevant portfolio features are excluded from the models; (iii) bank identity is not a legitimate feature for forecasting losses; (iv) the Fed seeks the "best" forecasts available based on the 2 See, for example, "Enhancing Financial Stability by Improving Culture in the Financial Services Industry," a speech given by then president of the Federal Reserve Bank of New York, William C. Dudley, on October 20, 2014, https://www.newyorkfed.org/newsevents/speeches/2014/dud141020a.html. direct impact of legitimate features; (v) legitimate features should not be used indirectly to identify banks; and (vi) the overall average level of losses projected by the Fed's industry model should be consistent with the average that would be obtained using bank-specific models.
The rest of this paper will make precise and further develop these ideas. We will investigate ways to aggregate bank-specific models into a single industry model, premised on these ideas and recognizing the heterogeneity across banks. Doing so entails balancing the goal of overall accuracy with concerns for consistent treatment of banks.

Basic Model
To capture bank heterogeneity, we consider a market with multiple banks, indexed by s = 1, . . . ,S. The loss rate (or net charge-off rate) Y s for bank s is given by with α s ∈ R and β s ∈ R d . Here, X s is a d-dimensional vector of predictive variables; at this point, we do not distinguish between portfolio characteristics and macro variables. The portfolio characteristics include information about a bank's borrowers and loan terms. We use a linear specification in (1) because it offers the simplest setting to explore the interaction of heterogeneity and fairness; we discuss nonlinear extensions in Section 7. We take (1) to be the true relationship between the loss rate Y s for bank s over the forecast horizon and characteristics X s known at the date the forecast is made. Loss rates are normalized by loan balances to make values of Y s comparable across banks of different sizes.
We think of X s as a draw from some distribution with The randomness in X s can be interpreted as reflecting the variation in the characteristics for bank s (and the macro variables) over time. We assume throughout that each Σ s is nonsingular.
The error s in (1) is assumed to satisfy, for each s, The regulator's problem is to choose a model g that forecasts loss rate g(x, s) for bank s if the bank's portfolio characteristic vector is x. The forecasts should satisfy the following property, which prohibits the regulator from applying different models to different banks: As the true relationship for each bank is linear in (1), we mainly focus on the case of a linear industry-wide model. The regulator's problem is then to choose a single α ∈ R and β ∈ R d that it will use to form a forecastŶ given portfolio characteristics x. The forecast (4) satisfies equal treatment because it has no functional dependence on bank identity s. The parameters of the industry model (4) may depend on the bank-specific parameters (α s , β s ) and on the mean and variance in (2), but they should not depend on the realized features X s .
The regulator would like the forecast lossŶ (X s ) to be close to the actual loss Y s in (1) for every bank s. To aggregate errors across banks, we introduce a random variable S that picks a bank according to a distribution with the probabilities p s summing to 1. In the simplest case, all banks get equal weight, and the p s are all equal; but the p s could also reflect relative asset sizes or other weighting schemes.
When we replace a bank label s with the random variable S, we get a mixture over banks. In particular, we can combine the bank-specific models (1) into a mixture or hierarchical model by writing In choosing parameters α and β in (4), the regulator would like to make the forecast errors small for all banks. A natural way to aggregate forecast errors across banks is to consider the average squared error, in which case the regulator's problem becomes choosing α and β in (4) to solve The objective in (7) averages squared forecast errors over banks. It can also be written as We emphasize that the problem posed by (7), like the more general problem of choosing industry parameters in (4), is one of characterizing ideal coefficients α and β. This is a question of choosing the correct targets of estimation, rather than a question of choosing estimators. In particular, α and β are population quantities rather than sample quantities. In practice, the regulator would have a panel of time series of observations of (1) across banks. Estimation methods for panel data ordinarily focus on coefficients that are common to all units and exploit the panel structure to estimate these shared values. Our concern is precisely with the case of heterogeneous coefficients, where we need to identify suitable targets before we can consider their estimation. It would not be meaningful to refer to an estimator as unbiased or consistent until we have clearly identified the estimation target.
For the solution to (7), writē and with Similarly, Proposition 3.1. Problem (7) is solved by and Loss forecasts using α P ooled and β P ooled in (4) provide fairness through unawareness, in that they ignore bank identity. They satisfy the equal treatment principle articulated by the Federal Reserve that "two firms with the same portfolio receive the same results for that portfolio." Given our starting point (1), problem (7) would seem to be the most direct interpretation of the Fed's policy of developing an "industry-level approach calibrated using data from many financial institutions." However, the solution in (11) is not a satisfactory target. Indeed, (11) shows where heterogeneity is most problematic. If the intercepts α s covary with the means µ s , this effect can distort β P ooled . As an extreme example, consider the case that β s = 0 for all s; in other words, none of the features in X s is predictive of losses for any of the banks. The regulator's model (4) using β P ooled would nevertheless forecast losses based on these features if cov[α S , µ S ] is nonzero.
This covariation would create the illusion of predictability. In applying (11), we would be forecasting losses based on irrelevant features, purely as a consequence of the way we aggregated the bank-specific models.
Even in a less extreme setting in which the β s are nonzero, the presence of the cov[α S , µ S ] term in (11) reflects an indirect influence of bank identity on loss forecasts. If the bank-level mean characteristics µ s positively covary with the bank-level intercepts α s , then in the pooled model this covariance will lead to a higher loss forecast for a bank with a higher value of X s . This is arguably unfair, in the sense that the loss forecast is not based on the legitimate influence of the feature X s . We will formalize the idea that the pooled method misdirects legitimate information in Sections 4.2 and 4.5.
This effect is reminiscent of the bias incurred in panel regressions when fixed effects are present in the data but omitted from a model. As we emphasized above, in our setting the primary objective is to define the appropriate target of estimation, given the heterogeneity in the coefficients. We cannot say the term cov[α S , µ S ] introduces bias until we have decided what we are trying to estimate.

Average Treatment Effects
We can gain additional insight by considering the case of scalar X s . In this case, the pooled coefficient β P ooled in (11) becomes If cov[α S , µ S ] = 0 and if σ 2 s + µ 2 s −μµ s ≥ 0, for all s, then (13) becomes a convex combination of the individual β s . In Appendix B, we state some simple properties that an aggregation of the individual β s into a single industry value should satisfy, and we show that only a convex combination satisfies these properties. Equation (13) thus shows a further potential problem with the pooled method. Even if cov[α S , µ S ] = 0, the coefficient on some β s could be negative, which would mean that a reduction in β s would increase β P ooled ; we return to these cross-bank effects in Section 6.
We will refer to any convex combination of the β s as a weighted average treatment effect or WATE parameter. This terminology is suggested by thinking of a unit increase in a portfolio characteristic X s as a treatment, and β s as the response to that treatment. The (ordinary) average treatment effect is the the expected coefficient, but weighting the individual coefficients allows other combinations. In particular, if the µ s are all equal, the pooled coefficient (13) becomes We will say more about these cases in subsequent sections.
To translate a WATE coefficient into a loss projectionŶ , as in (4), we also need to specify an intercept. Setting ensures that the forecastŝ have zero expected error, in the sense that

Fair Regressions
We have seen that if the regulator's sole objective is to minimize average squared forecast errors subject to equal treatment, then the solution is given by the pooled coefficients in (11) and (12).
However, we have also seen that (11) has consequences that are undesirable and even unfair.
In this section, we turn to methods that expand the squared loss minimization objective (7) to include fairness considerations.

Projection to Fairness
In the literature on fairness in classification methods, demographic parity is among the most widely discussed fairness principles; see, for example, Chapter 3 of Barocas et al. [6]. In the simplest classification setting, the counterpart of our forecast is a binary outcomeŶ ∈ {0, 1}.
For example,Ŷ = 1 may indicate a hiring decision, a loan approval, or a school admission decision. The decision is to be based on certain features of a candidate that are deemed legitimate. Demographic parity requires that the event {Ŷ = 1} be statistically independent of a protected attribute, such as race or gender. This objective is difficult to achieve when legitimate features covary with the protected attribute.
Chzhen et al. [12] and Le Gouic et al. [35] extend the notion of demographic parity to the regression setting by requiring that model predictions be independent of a protected attribute.
These two articles solve the problem of finding the model that minimizes mean squared prediction errors while achieving demographic parity. We will use the term projection to fairness (PTF), coined in Le Gouic et al. [35], for the method in these papers.
Both papers reduce the problem of regression fairness to one of finding the Wasserstein barycenter of a set of distributions, in the sense of Agueh and Carlier [2]. The barycenter is the distribution closest to the set of distributions in an average sense. For a squared error and onedimensional distributions, the barycenter can be described as the distribution whose quantile function is a weighted average of the individual quantile functions. (The quantile function is the inverse of the cumulative distribution function.) In the setting of Section 3.1, the resulting solution can be interpreted as follows. For bank s, the regulator first forms the forecastŶ s (x) = α s + β s x, using the bank-specific coefficients and the realized features X s = x. SupposeŶ s falls at the 80th percentile of the forecast distribution for bank s. The regulator then takes a weighted average of the 80th percentile forecast for all of the bank-specific models. That weighted average becomes the forecast for bank s.
To make this procedure more explicit and to specialize the general framework of Chzhen et al. [12] and Le Gouic et al. [35] to our setting, we consider the case (for this section only) that each feature vector X s has a multivariate normal distribution N (µ s , Σ s ). Write Σ 1/2 s for the symmetric square root of Σ s , and define the standardized feature vectors each Z s has a multivariate standard normal distribution. Write the basic identity (1) using standardized variables as Suppose β o s = 0, for all s, with · denoting the usual Euclidean norm. Consider the model that assigns, to each bank s = 1, . . . ,S, with features X s = x the forecast If there exists a β ∈ R d and scalars a s > 0 for which then we will see that (18) simplifies to the weighted averagê In the case of scalar X s , (19) holds whenever all β s have the same sign.  (18) is the projection-to-fairness of the bank-specific models (1), meaning that among all models (whether linear or not) that satisfy demographic parity. If (19) holds, the projection-to-fairness is given by (20).
We can see from (18) that the PTF model does not satisfy equal treatment: to calculate the loss forecast for a bank, we need to know its identity s. We have included the special case of (20) because it more nearly parallels the type of model we seek in (4). The coefficients in (20) are weighted averages of bank-specific coefficients. The model in (20) satisfies equal treatment with respect to the standardized features Z s , rather than the raw features X s : two banks with the same standardized features will receive the same forecasts. But the means for the two banks could be very different -the standardization is done separately for each bank -indicating that one bank's portfolio may be much riskier than the other bank's. In treating standardized characteristics for different banks as comparable, the PTF model implicitly evaluates the riskiness of each bank relative to the distribution for that bank, not relative to all banks. The suitability of PTF in our setting is therefore questionable.
The root of the problem is that demographic parity is too strong a property for our setting.
Ensuring that a hiring decision is independent of race or gender is important; but forcing loss projections to be independent of bank identity ignores relevant differences in bank's portfolios.
Whereas the pooled model (11)-(12) does too little to address heterogeneity across banks, the PTF model goes too far in leveling differences. The next section provides a better balance.

Formal Equality of Opportunity
Johnson, Foster, and Stine [30] introduce the concept of formal equality of opportunity (FEO) in regression, based on the use of the term in political philosophy, for which they cite the review in Arneson [5]. According to Arneson [5], FEO means that "positions and posts that confer superior advantages should be open to all applicants. Applications are assessed on their merits." In adapting this idea to our setting, it is helpful to make a contrast with the previous section: whereas demographic parity requires that loss forecasts be independent of bank identity, FEO allows bank-dependence, but only through legitimate portfolio characteristics -through the bank's "merits." This notion aligns well with our definition of equal treatment and with the Fed policy, quoted earlier, that "two firms with the same portfolio receive the same results." The objective of FEO in regression, as developed by Johnson et al. [30], is to ensure that a protected attribute (for us, bank identity) has no direct or "causal" impact on a model's predictions. The predictions may be correlated with protected attributes if legitimate features (portfolio characteristics) are correlated with bank identity.
To develop this idea in our setting, we introduce the centered dummy variables We discuss the implications of centering below. For any coefficients α, δ 1 , . . . δS −1 ∈ R and β ∈ R d , and any x ∈ R d , letŶ We have included the bank label s as an argument ofŶ because U i depends on s. Let α F , {δ i , i = 1, . . . ,S − 1}, and β F solve the error minimization problem With the coefficients that minimize (7), (22) becomes the linear projection of Y S onto the span of 1, the U i (S), and X S , evaluated at S = s and X S = x. Now drop the dummy variables U i and defineŶ The FEO loss forecast for bank s isŶ F (X s ).
Steps (22)-(24) result from applying the definition of an impartial estimate (their Definition 2) in Johnson et al. [30]. (More precisely, steps (22)-(24) define a population counterpart of the sample formulation in [30].) The procedure in (22) We will say more about the implications of this approach, but we first show that our setting allows an explicit expression for the FEO coefficients: The FEO coefficients are given by and In particular, in the scalar case, (ii) We also have We encountered (27) in (15) as a special case of the pooled coefficient when the bank means µ s are constant. The general case in (25) similarly coincides with the pooled coefficient in (11) when the means are constant. In other words, introducing the bank-level fixed effects in (22) purges β F of the effect of different feature means across banks; dropping these fixed effects in (24) ensures that the regulator's model has no explicit dependence on bank identity and satisfies equal treatment.
In what sense is this procedure fair? We adapt the interpretation in Johnson et al. [30] is the linear projection of the bank-identity variables U (S) onto the centered portfolio features X S −μ. The relationship between β P ooled and β F can be expressed as follows.
Decomposing U (S) into its projection onto X S −μ and an orthogonal component leads to the following contrast of these forecasts: The term δ [U (s) − Λ (x −μ)] in (33) changes the forecast based on information in bank identity that is orthogonal to the features x. In the terminology of Johnson et al. [30], this would be disparate treatment. Through "unawareness" (meaning that it has no functional dependence on bank identity) the pooled forecast (34) drops this term, but it retains δ Λ (x −μ), as can be seen from (30).
The term δ Λ (x −μ) is the problematic component of the pooled method. Although it does not explicitly use bank identity (it satisfies equal treatment), this term relies on the fact that bank identity is to some extent predictable from portfolio features. Imagine the regulator forming loss forecasts from blinded data -the regulator does not know the identity of the bank. The term Λ (x −μ) is the least-squares prediction of U (s) from x −μ. In the pooled forecast (34), the regulator is implicitly "misusing" the data in the features x −μ to try to identify the bank and then to adjust the forecast based on the inferred identity. The FEO forecast (35) removes this effect and retains only the direct effect of portfolio features on the loss rate.
We will develop this idea further in Section 4.5 and conclude that the FEO forecast is, in a precise sense, the best way to aggregate the bank-specific models into a single regulatory model.
The FEO forecast satisfies the equal-treatment property and thus has no direct dependence on bank identity; but it also removes the indirect dependence that results when bank identity is partly predictable from portfolio features. We discuss other methods for comparison.

Conditional Expectation Model
A similar "misuse" of information occurs if we project the bank-specific models to an industry model in the sense of conditional expectation, rather than least squares. Suppose X s has density g s , and suppose E[ s |X s ] = 0, s = 1, . . . ,S. Then, by Bayes' rule, This model satisfies equal treatment -Ŷ C (x) depends on the portfolio features x but not on a bank's identity. However, the point of the weights p s g s (x) is to infer the identity of the bank from the features. Indeed, as discussed in Section 7, the conditional expectation can be viewed as a nonlinear generalization of the pooled method, with some of the same shortcomings.

Substantive Equality of Opportunity
As discussed in Arneson [5], a system in which admission decisions are made through a competitive exam open to everyone achieves formal equality of opportunity; but if only the wealthy have access to the preparation required for the exam, the system fails to achieve substantive equality of opportunity (SEO). In the regression setting, Johnson et al. [30] interpret SEO to mean that any influence of protected attributes should be removed from other variables included in a regression model. In the analogy with Arneson's [5] example, SEO would seek to remove the effect of economic status from performance on the exam, whereas FEO would accept exam scores as a legitimate basis for decision-making.
To apply these ideas to our setting, define the (S − 1) × d matrix In accordance with Definition 2 of Johnson et al. [30], defineŶ with α F and β F defined by (23). The SEO forecast adjusts the portfolio features x to remove the linear projection onto the centered bank dummy variables U . We can write (38) somewhat more explicitly as follows: so the SEO forecast (38) is given bŷ The SEO forecast is the linear projection of Y S onto a constant and X S − µ S . Under additional conditions, we get a stronger conclusion:

A Unified Perspective: Legitimate Information
All of the methods we have discussed can be seen as ways of choosing forecastsŶ s , s = 1, . . .S, (of the formŶ (X s ) orŶ (X s , s)) to minimize subject to additional considerations. (i)Ŷ s = α + β X s Pooled (11)-(12) (ii)Ŷ s = g(X s , s), some gŶ S independent of S PTF ( [12,35] The constraint in Table 4.1(v) is weak demographic parity. SEO implicitly takes the view that the only legitimate information in forecasting losses for bank s is the deviation X s − µ s .
In contrast, FEO takes the full set of features X s as legitimate information. Through the constraint in Table 4.1(iv), it enforces a requirement we call no misdirection of legitimate information. FEO uses all of X s in projecting losses; but it chooses the coefficient β F to be the coefficient in a regression of Y S on X S − µ S , which is the part of X S orthogonal to bank identity. This condition ensures that the information in X S is not misdirected to infer bank identity.
To make this idea precise, consider any model of the form (4). If we assume the intercept is chosen to match the unconditional mean, we may write the model aŝ for some γ ∈ R d . With γ = 0, we get the FEO forecast (24).
then the forecastŶ γ misdirects legitimate information in the sense that Recall that Λ (X S −μ) is the linear projection of the centered bank identity variables U (S) onto the centered portfolio features X S −μ. The condition in (i) therefore indicates that γ misdirects some of the legitimate information in X S toward inferring bank identity. Thus, deviating from β F in (42) either increases errors or misdirects information.
Property (ii) has a similar interpretation. The term δ U (S) is the direct influence of bank identity on losses Y S . The proposition states that any deviation γ that reduces forecast errors (relative to γ = 0) implicitly picks up some of the information in bank identity.
To illustrate these ideas, consider a simple example in which some component of X s measures exposure to beachfront real estate. Suppose for simplicity that this feature is uncorrelated with other features. In the SEO forecast, the only legitimate information from this exposure is a bank's deviation from its own mean. Years in which a bank had above average exposure would lead to higher loss forecasts, but the bank's average exposure would not directly inform the forecasts. In contrast, FEO treats the bank's total exposure (mean plus deviation) as legitimate information. Like SEO, in evaluating the impact of this exposure -that is, in estimating the coefficient on the exposure -it relies only on the within-bank variation. This ensures that the information in the exposure is not misdirected toward inferring the bank's identity, as could happen in the pooled regression.

Average Treatment Effect as an Extension of FEO
Recall that the FEO forecast controls for bank fixed effects. One might similarly consider controlling for interactions between bank indicators and components of the feature vectors.
This leads to a family of extensions of FEO that differ in which interactions they include. We will show that with a full set of interactions, the extended FEO model becomes the ATE model (14). Although we do not recommend this choice, for reasons we return to at the end of this section, this analysis clarifies how the ATE forecast fits within a more general framework.
with s uncorrelated with X s and V s .
We extend FEO to the following procedure: If V S is empty, then we know from (28) that these steps do indeed reduce to the original FEO forecast. We have included the interaction US(S)V S in the first step (even though we omitted US(S)) to simplify the derivation of γ F . Including this term means that the coefficients on the interactions U i (S)V S are determined only up to a constant, because U 1 (S)V S +· · ·+US(S)V S = 0.
These coefficients are dropped in the second step, so their value is immaterial. for each s = 1, . . . ,S. Then β F is given by (25) and (28), and γ F =γ = s p s γ s . In particular, if interactions with U (S) are included for all features, the FEO vector of coefficients reduces to the average treatment effect (14).
This result allows us to interpret the ATE forecast as a version of the FEO forecast that removes the effects of certain interactions. As a convex combination of the bank-specific coefficients, the ATE coefficient retains some of the advantages of the FEO coefficient, particularly for the comparisons in Section 6.
However, we do not see a compelling case for controlling for interactions between bank identity and portfolio features. When we control for the bank-identity variables in FEO, we are ensuring that the industry β for legitimate features is not affected by bank fixed-effects.
Extending this idea to include interactions is tantamount to saying that we do not want heterogeneity with respect to one portfolio feature to affect the industry coefficient for another portfolio feature. But the logic for FEO does not extend to this case: here we are dealing with two presumably legitimate features whereas the point of the FEO forecast is to remove the influence of the bank-identity variables, which are not legitimate features under equal treatment.

Empirical Evidence
In this section, we document empirical evidence of heterogeneity in bank-specific models of loss rates, and we examine the implications of this heterogeneity for the choice of an industry-wide model. We find strong evidence of statistically significant differences in model parameters across banks. These differences can lead to material differences between pooled and FEO coefficients in an industry model.

Data
We use two types of data and data sources: loan information for individual banks and historical macroeconomic data.

Loan Information for Individual Banks.
Bank holding companies publicly report financial information quarterly through the Federal Reserve's form Y-9C. We use these filings to collect information on four loan types that are treated separately in the Fed's stress tests: credit cards, first lien mortgages, commercial real estate loans, and commercial and industrial loans. For each category, each bank, and each quarter, we collect loan balances, charge-offs, recoveries, and total amounts past due. For each bank-quarter we also collect the bank's allowances for losses; allowances are not consistently reported separately by loan category, so we use a bank's total allowances across all loan types.
Allowances and amounts past due are our proxy measures of loan portfolio risk.
We collect this data from 2002 to 2021 for the twenty largest banks by total assets (as of December 2020). The banks are listed in Table D This measure is commonly used in stress testing; see, for example, Guerrieri and Welch [26], Hirtle et al. [29], and Kapinos and Mitnik [32]. We similarly normalize the amounts past due and allowances to get a PastDueRate s,t and an AllowanceRate s,t for each bank-quarter, except that the allowance rate is normalized by the total loans in all categories. We remove values less than −100% or greater than 100% of LossRate, PastDueRate and AllowanceRate, and we winsorize PastDueRate at the upper and lower 5% levels.

Macroeconomic Data
We use data on seven of the macro variables used in the Federal Reserve's stress tests: real disposable income growth, real GDP growth, house price index level, inflation rate, unemployment rate, Dow Jones total stock index level, and the Treasury spread. The Federal Reserve provides historical data on its website for all variables used in forming stress scenarios, including these.
We use the values reported by the Fed for these variables in the June 2020 stress test; these values run from 1990 through 2019.
We aggregate these variables into a single macro variable by taking the first principal component of their correlation matrix.

Heterogeneity in Slopes and Intercepts
We use the bank data to approximate our theoretical framework through the specification For each loan category, we want to test for heterogeneity in parameters across banks. When we test for heterogeneity, the null hypothesis states that slopes for all banks are equal, or that the intercepts are equal, The alternative hypothesis in each case states that the indicated parameters are not identical across banks. We will run these tests with different subsets of the variables in (44) included and interpret the coefficients in (45) accordingly.
To test these hypotheses for a particular loan category, let X s be the n s by p data matrix for bank s, where n s is the number of observations for bank s in the loan category, and p =1, 2, or 3, is the number of variables included on the right side of (44). LetX s = (1, X s ) be X s concatenated with a column of 1s, and let X * be the diagonal block matrix X * = diag(X 1 , ..,X S ).
We apply the Wald test to test linear constraints on θ * in (47)  Almost all the tests indicate strong evidence of heterogeneity in the intercepts and in the coefficients for past due rates. The evidence is more mixed for allowance rates and the macro variable. These variables are also less predictive of losses than the past due rates.
Next we examine the impact of heterogeneity. the columns labeled "% diff" show the percentage difference 100%×(β F −β P ooled )/β P ooled . This is a measure of the impact of addressing heterogeneity in choosing an industry model. We also report confidence intervals for these ratios. These are bootstrap confidence intervals, estimated using the cross-sectional method in Kapetanios [31], based on resampling independently from the panel of banks.
The percentage differences between β P ooled and β F vary substantially, and the associated confidence intervals are quite wide. But we see many cases in which the percentage difference in estimated coefficients is roughly 5-10% or even larger. In the context of setting capital requirements for banks, these differences could be material.
Consider, for example, the case of first lien mortgages, with coefficients β P ooled = 0.053 and β F = 0.060 in the top panel of Table 5.5. From

Cross-Bank Parameter Externalities
As a consequence of aggregating bank-specific results into a single industry model, changes at one bank can affect loss forecasts at other banks, and the results are sometimes counterintuitive.
In this section, we argue that these cross-bank externalities are generally more reasonable under FEO forecasts than under the pooled method.
For simplicity, we consider a setting with a single scalar feature x. More generally, we can think of this as a feature that is uncorrelated with all other features. We adopt the convention that this feature is nonnegative, and that higher values of x are associated with higher losses.
Thus, for each bank s we assume µ s ≥ 0 and β s ≥ 0. In reducing µ s , a bank improves its portfolio quality; in reducing β s , a bank improves its ability to manage portfolio risk; and in reducing α s , a bank improves unobserved features to reduces its losses. We examine how these improvements -reductions in µ s , α s , and β s -affect stress test results for bank s and other banks l.
We can write the FEO loss forecast (24) for bank l evaluated at X l = x aŝ (27). The forecast is the same for all banks l because FEO satisfies equal treatment. It is now easy to see that and In (49) we see that if bank s has above-average (relative to β F ) sensitivity to feature x, then reducing its average exposure to that feature µ s reduces loss forecasts for all banks. Equation (A decrease in a forecast corresponds to a positive derivative because we are considering a decrease µ s , α s , or β s .) Counterparts to (49)-(51) continue to hold if we replace β F in (48) with any convex combination of the β s , as in the WATE model. However, the pooled method behaves quite differently.
The pooled forecastŶ P (x) can be written in the same form as (48) but with β F replaced by β P ooled in (11). We now get The sign of the last term is not determined by a simple condition, so the overall directional effect is difficult to predict. The sign of depends on the magnitudes of µ s and x, relative toμ. For the sensitivity to β s , we can write .
Among the most troubling aspects of the pooled model is that the last term could be negative: a reduction in β s could produce an increase in β P ooled . In particular, σ 2 s + µ s (µ s −μ) is negative for a bank with below-average exposure to feature x (so µ s <μ) and low variability σ 2 s in this exposure. Under the pooled model, it is therefore possible for an improvement in risk management at one bank (a reduction in β s ) to produce an increase in loss forecasts at all banks.
The top panel of Table 6.1 shows sufficient conditions for positive sensitivities ofŶ F (x) and Using estimated parameters for the credit card data in Section 5, we find that this derivative is negative when l is Citigroup and s is American Express or JPMorgan Chase. In other words, an improvement at either of these two banks would result in a higher expected loss forecast at Citigroup under the pooled model.
The bias sensitivities in Table 6.1 are more complicated than the other cases because the bias involves the difference between the predicted and actual loss rates. A reduction in the  predicted loss rate can increase or decrease bias, depending on whether the initial forecast is too low or too high.

Nonlinear Models
Most of the ideas developed in previous sections for linear regressions extend to generalized linear models through a transformation of the response variable. For example, rather than work with the loss rate Y s , we could specify a linear model for its logit transformation log(Y s /(1−Y s )).
But we can also extend ideas from previous sections to more fully nonlinear models. Replace the mixture model in (6) with a general representation of the form In other words, the loss for bank s is given by g(s, X s ) + s . We assume that g(S, X S ) and S are square-integrable.
The counterpart of the pooled estimate becomes This rule satisfies equal treatment -it has no functional dependence on S -but we argued earlier (in Section 4.3) that this forecast implicitly uses the information in the portfolio features x to infer bank identity.
To introduce a nonlinear version of the FEO forecast, we will make the relatively modest assumption that (52) admits a decomposition of the form We now introduceŶ as a nonlinear counterpart of the FEO forecast. We justify this interpretation by showing that Y F exhibits properties that are nonlinear counterparts of the key properties of the FEO forecast in Section 4.2 and 4.5. To state the result, consider forecasts of the form for some γ : Proposition 7.1. The nonlinear FEO forecast (56) satisfies ForŶ γ as in (57), if γ reduces errors, in the sense that then it misdirects legitimate information, in the sense that the positive covariance with γ(X S ) thus indicates that γ misdirects some of the information in X S to inferring S.
Proposition 7.1 shows that applying ideas from previous sections to nonlinear models is a computational rather than a conceptual matter. We leave a fuller investigation into the application of this result for future work.
We briefly contrast our FEO forecast in (56) with an alternative approach to extending fairness concerns to complex, nonlinear models. The alternative seeks to strip X S of any protected attributes before a model is estimated. Examples of this general approach include Grűnewalder and Khaleghi [24] and Madras et al. [36]. This approach is primarily concerned with ensuring demographic parity: if a model has no access -not even indirect access -to a protected attribute, its forecasts will be independent of the attribute. But we argued previously that demographic parity is too strong a condition for our setting. Our FEO forecast in (56) treats all the information in X S as legitimate information -even elements that could help infer S -but it ensures that the information is not in fact misdirected to infer S.

Concluding Remarks
The current practice of regulatory stress testing ignores bank heterogeneity as a matter of policy and principle. We have argued that simply pooling banks can distort coefficients on legitimate features and is vulnerable to implicit misdirection of legitimate information to infer bank identity. We have examined various ways of incorporating fairness considerations and shown that estimating and discarding centered bank fixed effects addresses the deficiencies of pooling -and it does so in an optimal sense.
Beyond this specific recommendation, the broader conclusion to be drawn from our analysis is that accuracy and equal treatment can more effectively be addressed by accounting for bank heterogeneity rather than ignoring it. Although we have focused on the stress testing application, our analysis applies more generally to settings requiring the fair aggregation of individually tailored models into a single common model. It follows that Proposition 4.1. By Proposition 3.4 of Chzhen et al. [12] or Theorem 6 of Le Gouic et al. [35], the expected squared error is minimized subject to demographic parity by the rule that assigns to bank s with features x the loss forecast where F s is the cumulative distribution function of α s + β s X s . By construction, F s is then also the cumulative distribution function of α o s + β o s Z s , which is normal with mean α o s and variance β o s 2 . Writing Φ for the standard normal distribution function, we get Making these substitutions in (61) and writing α o for suitable a i . Minimizing (23) over the a i and β yields the same value for β as minimizing (23) using (22)  We may therefore evaluate β F as the coefficient in the regression of which is (28). For the first factor in (28) For the second factor, we similarly have so (25) follows. The optimal α F in (23) ensures that E[Ŷ (X S , S)] = E[Y S ], which yields (26).
The minimization in (23) yields coefficients α F , δ, and β F , with which we can write where the error u has mean zero and is uncorrelated with U (S) and X S . We thus have using the expression for Λ in (29) for the last step.
Next, we evaluate δ. Using (62), we can derive δ as the vector of coefficients in a regression To evaluate var[U (S)] −1 , we first note that and similarly row s of the matrix var[U (S)] −1 cov[U (S), X S ] is given by Combining these terms in (63) yields (31).   (20) and (40) (65) is the α F defined by (23). We have thus shown that the optimal forecast in row (v) iŝ In case (iv), by applying (39)  in (iv) is the coefficient in a linear regression of Y S on X S −M U (S). As noted in the discussion of (v), this is β F , and it follows from E[U (S)] = 0 that the optimal α in (iv) is α F . Proposition 4.6. By construction, the least-squares projection of Y S onto a constant and X S is given by the pooled forecast, so for some orthogonal error P with a variance σ 2 P that does not depend on γ. We therefore have from which (i) follows.
Using the linear projection of Y S onto (1, U (S), X S ) in (32), we can write for some orthogonal error with a variance σ 2 that does not depend on γ. We therefore have where the third equality uses (39), and the last equality uses the orthogonality of X S − µ S and U (S). If this expression is smaller than the corresponding value with γ = 0, then (ii) must hold.
Proposition 4.7. We saw in the proof of Proposition 4.2 that X S − µ S is uncorrelated with the centered indicators U i (S). It is also uncorrelated with V S − ν S because under our assumption that X s and V s are uncorrelated. Similarly, Starting from the representation of (43) as we may write which expresses Y S as the sum of three mutually orthogonal terms. As X S − µ S is uncorrelated with O, andỸ is uncorrelated with X S − µ S , we may calculate the projection of Y S onto the span of X S − µ S and O by projecting β S (X S − µ S ) onto X S − µ S and projectingỸ onto O.
We know from (28) that the projection of β S (X S − µ S ) onto X S − µ S is β F (X S − µ S ); in other words, including V S and the interaction terms does not change β F .
For the projection ofỸ onto O, let a i = α i + β i µ i +γ ν i andā = i p i a i . Then, Thus,Ỹ is in the span of O, and its coefficient on For the second claim, we have The last step uses It now follows that if γ reduces the expected squared forecast error then E[γ(X S )f 1 (S)] > 0, which implies (59) and (60).

B Convex Combinations of Coefficients
Equation (13) that combines bank-specific coefficients β s into an "industry" parameter β * .
Property (i) is needed for the aggregation to perform sensibly under a change of units in the measurement of X s : if we divide each X s by k, each β s increases by a factor of k, and it is natural to require that β * scale accordingly. Properties (ii) and (iii) are also very modest requirements. Property (iv) is harder to motivate but not unreasonable. These properties constrain the aggregation function as follows: Proposition B.1. If (i)-(iv) hold, then f (β 1 , . . . , βS) is a convex combination of its arguments.
Proof. Fix β ∈ RS. Let g(t) = f (tβ). By condition (iv), g (0) = β f (0). Condition (i) and (ii) imply g(t) = tf (β), so g (t) = f (β) for any t. Thus, The scalar FEO coefficient in (27) is a convex combination of the bank-specific coefficients β s , but the pooled coefficient (13) is generally not. This property of the FEO model extends to the multivariate case under additional conditions. If all the bank-specific covariance matrices Σ s , s = 1, . . . ,S, coincide, then in (25) we get β F = E[β S ] = s p s β s . If all Σ s are diagonal (but not necessarily identical), then the representation of the scalar FEO coefficient in (27) applies to each coordinate of β F . If all Σ s have the same eigenvectors, then we can transform the original features X s into uncorrelated features using principal components. Using these transformed features, each coordinate of β F is a convex combination of bank-specific coefficients.

C Sensitivity Analysis
This appendix provides supporting details for Section 6, particularly the conclusions summarized in the middle and bottom panels of Table 6.1. We begin with an analysis of forecast bias that is of independent interest.

C.1 Forecast Bias
If losses at different banks are described by different models, then forecast bias becomes inevitable when we apply a single model to all banks. But the distribution of bias across banks may differ under different choices of the single model. LetŶ s be any of the forecasts for bank s in Table 4.1, and, as in (1), let Y s denote the actual loss rate for bank s. BothŶ s and Y s are evaluated at X s . Define the forecast bias for bank s to be The expectation integrates over the distribution of the error s in (1) and the features X s .
Proposition C.1. For each forecast in Table 4.1, the bias is as follows.
Proof. For (i), we have, using the definition of α P ooled in (12), For the PTF forecast, (17) and (18) yield and the bias in (ii) follows. The expression in (iii) holds by definition. The argument for (iv) is the same as the argument for (i). The bias in (v) follows from (iv) because we see from (38) that the SEO forecast for bank s subtracts β F (µ s −μ) from the FEO forecast.
In every case of Proposition C.1, the average bias s p s bias(s) is zero, but the methods differ in how they distribute bias across banks. We saw previously that the PTF and SEO methods go the farthest in equalizing differences; we now see that the bias for each of these between the average loss rate for all banks and the average for an individual bank.
In light of the discussion in Section 4.2, this difference is the expected disparate impact on bank s of using the pooled model.

C.2 Improvement in Intercept α s
By taking the expectation of (48), we get and the same holds for the expected pooled forecast with β F replaced by β P ooled . It follows that, for any banks s and l, In other words, all expected forecasts decrease following a reduction in α s .
In contrast, for the pooled model we get Bank s benefits from its reduction of α s , in the sense that the derivative with l = s is positive.
For l = s, the sign of (68) does not admit a simple description. In particular, it may be negative when µ s and µ l are on opposite sides ofμ, meaning that one bank's loans are riskier than average and the other bank's loans are less risky than average.
For the bias under FEO we have It is then immediate that The direction of change makes sense. If the bias for a bank is positive, meaning that the industry model overestimates its losses, then improvements at other banks will reduce loss forecasts and thus reduce the bias. The bank's own improvements will increase the bias by reducing the bank's own losses by more than they reduce the model's forecasts. The situation is reversed for a bank with a negative bias.
However, for the pooled regression method, and the direction of change is unclear.

C.3 Improvement in Loan Quality
Now suppose bank s improves the quality of its loan portfolio, resulting in a smaller µ s . This has no effect on β F , which makes sense -changing one bank's loan quality should not change the sensitivity of losses to loan quality. However, it is evident from (11) that β P ooled does change with µ s .
Under FEO, the mean the mean predicted loss rate satisfies which is always positive if l = s. This means that an improvement in bank l's loan quality (a reduction in µ l ) reduces bank l's mean predicted losses. In the pooled model, this expression could be negative, even with l = s, meaning that a bank could be penalized (through a higher mean predicted loss rate) as a result of improving its loan quality.
The sensitivity of the bias under FEO is given by in particular, the bias for bank l moves in opposite directions with respect to changes in µ l and µ s , s = l. Suppose industry model overestimates bank l's losses, in the sense that the bias is positive, and suppose the industry model overestimates bank l's sensitivity to loan quality, in the sense that β F > β l . Then bank l will benefit (in the sense of reducing the bias) from improving its loan quality by reducing µ l .
The sign of this expression does not admit a simple condition.

C.4 Improvement in Loan Management
Now suppose bank s improves its abilities in loan management, resulting in a reduction in β s .
For l = s, the sensitivity of the bias for bank l with respect to β s equals the sensitivity of the mean predicted loss because the actual expected loss E[Y l ] is unaffected by β s . We therefore focus on the case l = s. Under FEO, which is guaranteed to be negative if µ s <μ. Under the pooled model, the sign of ∂bias P (s) ∂β s = (p s − 1)µ s + p s (σ 2 s + µ s (µ s −μ)) i p i (σ 2 i + µ i (µ i −μ)) (µ s −μ) does not admit a simple characterization.  We construct the loss rates, past due rates, and allowance rates using the entries in FR Y-9C forms outlined in Table D.2.

E Robustness Check: Pre-COVID Data
We repeat the analysis of Section 5, limiting the data to 2002-2019. This serves two purposes.
It addresses the possibility that our results are driven by a few extreme values during the COVID period 2020-2021. It also accounts for a change in how how banks measure allowances (the Current Expected Credit Losses methodology) that took effect at the end of 2019. The evidence for heterogeneity and its impact is generally at least as strong using the pre-COVID data as using data through 2021.