Bottom-up leading macroeconomic indicators: An application to non-financial corporate defaults using machine learning

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Bottom-up leading macroeconomic indicators: An application to non-financial corporate defaults using machine learning Tyler Pike, Horacio Sapriza, and Tom Zimmermann 2019-070 Please cite this paper as: Pike, Tyler, Horacio Sapriza, and Tom Zimmermann (2019). “Bottom-up leading macroeconomicindicators: Anapplicationtonon-financialcorporatedefaultsusingmachinelearning,” Finance and Economics Discussion Series 2019-070. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2019.070. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Bottom-up leading macroeconomic indicators: An application to non-financial corporate defaults using ∗ machine learning Tyler Pike Horacio Sapriza Tom Zimmermann Federal Reserve Board Federal Reserve Board University of Cologne August 30, 2019 Abstract Thispaperconstructsaleadingmacroeconomicindicatorfrommicroeconomicdatausing recentmachinelearningtechniques. Usingtree-basedmethods,weestimateprobabilitiesof default for publicly traded non-financial firms in the United States. We then use the crosssection of out-of-sample predicted default probabilities to construct a leading indicator of non-financialcorporatehealth. TheindexpredictsrealeconomicoutcomessuchasGDPgrowth andemploymentuptoeightquartersahead. Impulseresponsesvalidatetheinterpretationof theindexasameasureoffinancialstress. Keywords: EarlyWarningIndicators,CorporateDefaults,MachineLearning,EconomicActivity JELClassification: C53,E32,G33 ∗TheviewspresentedherearesolelythoseoftheauthorsanddonotnecessarilyrepresentthoseoftheFederal ReserveSystem.

1 1 Introduction Accurate forecasts of macroeconomic activity are central to the decisions of both private firms and public policymakers, yet forecasting aggregate outcomes such as real Gross Domestic Product has been proven to be notoriously difficult (e.g. Faust and Wright (2009)) using aggregate time series dataalone. While avast amountofgranular individualor firm-levelinformationcouldpotentially aideforecasting,itisunclearhowtobestincorporateinformationfromsurveys,incomestatements or stock returns into forecasts of aggregate activity. In this study, we build a bottom-up indicator of corporate health based on detailed firmlevel information and a trained machine-learning model that predicts firm-level default up to eight quarters ahead. We find that our index effectively forecasts future real macroeconomic activity, both in- and out-of-sample, better than well known and widely used aggregate measures of financial conditions, financial stress, and investor sentiment. In addition, our index can be interpreted as a measure of corporate health, where negative shocks to the index are associated with depressed future macroeconomic activity. Overall, our study shows how a large amount of granular information can effectively be used to improve forecasts for macroeconomic aggregates, when such granular information is first summarized by machine learning methods. Inthefirstpartofthestudy,weuseincomestatement,balancesheet,andstockmarketvariables to estimate the probability of default of each publicly listed non-financial US firm. We compare several modeling approaches, from traditional logistic regressions to more recent approaches such as random forests and gradient boosted trees. We find that the random forest model, an approach that averages predictions from several regression trees (a method that can capture non-linearities and interactions between predictor variables quite well) has the highest accuracy to predict firm defaults out-of-sample. Unlike the widely used logistic regression approach, the random forest model performance is also very stable for predicting defaults at different horizons. The estimated model gives rise to a cross-sectional probability distribution of defaults that we map into an index of non-financial corporate health using different moments of the estimated distribution. We consider information not only from central tendency moments, but also from higher-order moments of the distribution. In particular, our index is a combination of the mean, variance and skewness of the cross-sectional distribution of estimated default probabilities. We

2 find that the index correlates well with NBER-dated recessions in the data. Applying our index to macroeconomic forecasting, we find that it accurately forecasts future real macroeconomic activity. We test its forecasting performance both in- and out-of-sample, and against a set of other measures of financial conditions, financial stress, and investor sentiment widelyusedintheliterature,includingtheKansasCityFedFSI,theSt. LouisFedFSI,theChicago Fed NFCI, the Goldman Sachs FCI, and the excess bond premium (EBP). In a final exercise, we show that our index of corporate health is indeed a reasonable measure of future macroeconomic health: The impulse response of measures of real macroeconomic activity (such as GDP and employment) to shocks to the index is negative. Our paper connects to two strands of the literature. First, a large literature tries to predict adverse corporate events such as defaults at the firm-level. Early studies started with the analysis of simple ratios and yielded corporate health measures such as the Altman (1968) Z-score, Ohlson (1980) O-score. Methodologically, the literature has gone through a few distinct phases, moving from logistic regressions to predict defaults (Martin (1977)), to neural networks in the early 1990s (Odom and Sharda (1990), Wilson and Sharda (1994), and for a thorough literature survey see Atiya (2001)), to support vector machines in the early 2000s (Härdle et al. (2007), Shin et al. (2005), and see Ravi Kumar and Ravi (2007) for a survey of literature at the time), to tree-based methods more recently. Jones et al. (2015) conduct a thorough review and evaluation of classification techniques used to predict bankruptcy and credit downgrades, and find that tree methods, such as tree-based gradient boosting and random forest, are the best predictors of firm credit changes. The first part of our study is in the spirit of Jones et al. (2015), but while these authors explore a larger number of binary classifier techniques, we focus on five algorithms that are representative of the different waves in the default prediction literature. Moreover, while Jones et al. (2015) only compare algorithms based on their performance in a single cross-section analysis, we evaluate algorithms recursively through time (1989 through 2018) and across several forecasting horizons (1 through 8 quarters ahead), leading to a more rigorous comparison of techniques. Second, another body of the literature considers the implications of corporate default risk for macroeconomic activity. For instance, several authors have analyzed the implications of corporate distress and the probability of default for explaining risk premiums in stock and bond returns (see for example Fama and French (1996) or Merton (1973)). However, much evidence suggests

3 that deteriorating firm health, as measured as the probability of default, results in higher than average stock return volatility and lower excess returns (see Campbell et al. (2008) or Chava and Purnanandam (2010)). In addition, recent studies have taken an interest in exploring the impacts of corporate bond market crises on macroeconomic activity, finding that broad corporate debt defaults historically do not have the same real effects as banking sector crises (see Giesecke et al. (2012)). Unlike these studies, we concretely link our improvements in measuring firm defaults to future macroeconomic conditions. The rest of the paper is organized as follows. Section 2 describes the data used in this study, Section 4 describes the index construction from bottom-up machine learning models, Section 5 evaluates the forecasting abilities of the constructed index and investigates the reaction of real macroeconomic outcomes to structural shocks to the index. Section 6 concludes. 2 Data 2.1 Firm-level data We estimate the probability of firm-level default using three sets of historical data that track non-financial defaults and bankruptcies. Prior to 1987:Q1 we use the UCLA-LoPucki bankruptcy research database and NYU’s Altman Default Database. Post 1987:Q1 we use the Mergent corporate FISD daily feed. A default may occur at the time a company declares bankruptcy in court filings, or if it simply ceases to pay interest due on its corporate securities (i.e., bonds). All three databases denote a default event as the filing date when a company enters Chapter 7 or Chapter 11 court proceedings. These databases combined provide default coverage from 1984:Q1 through2018:Q11. Werestrictouranalysistodefaultsofpubliclytradedcompanies,asourmodels require inputs from firm balance sheet data. Figure 1, panel (a), shows that the total number of defaults covered in this time period is 407, an average of 12 defaults per year. The maximum number of defaults in a given quarter occurs in 2001:Q2, approximately 30 defaults. Our panel data tracks all publicly traded firms in US stock markets. Panel (b) of Figure 1 shows that the number of public firms in the US increases steadily 1WehaveadditionallyconductedallanalysesusingonlytheMergentdatastartingin1987,andwefindthatallof ourpresentedresultsarerobust.

4 until 1998, and declines steadily afterwards. The number of firms in our dataset never drops below 6000, and as a result, the percent of firms in our set that are labeled as defaulting is less thanone-percentforanygivenquarter. While407defaultsisnotthepopulationofdefaultsduring this time period, Figure 1 suggests that there are sufficiently many defaults to clearly discern business cycles in the data, with peaks occurring in 2001 and 2008. For our default models, we use a set of 26 firm-level variables. As it is not the goal of this paper to identify a new set of variables for predicting non-financial defaults, we use variables previously identified in the corporate finance literature. Our explanatory variables are: Tobin’s Q (Tobin and Brainard (1976)), the Kaplan-Zingales Index (Kaplan and Zingales (1997)), firm age and the HP index (Hadlock and Pierce (2010)), a dividend dummy variable and the Whited-Wu index (Whited and Wu (2005)), Altman’s Z-score (Altman (1968)), measures of net income, total liabilities, cash, stock price, market equity (Campbell, Hilscher and Szilagyi (2008)), measures of working capital, current ratio, asset turnover, return on equity, EBIT cover, and CAPEX (Jones, Johnstone,andWilson(2015)),aswellasexcessreturns,averageexcessreturns,3-digitsicindustry dummy variable, industry sales, and industry sales growth. We provide a full description and summary statistics of all variables in tables 3 and 4 in the Appendix A. All balance sheet and incomestatementdataarefromCompustat,whilestockmarketdatafortheexcessreturnvariables are from CRSP. 2.2 Macroeconomic data When evaluating our index of non-financial corporate health, we will relate it to several measures of real macroeconomic activity: Nonfarm payroll employment and the U3 unemployment rate, both from the US Bureau Labor Statistics, an index of Industrial Production from the Federal ReserveBoard(accessibleonlineviatheFREDwebsite)andRealGrossDomesticProductfromthe US Bureau of Economic Analysis. To compare the forecasting performance of our index to various financial variables and financial conditions and stress indicators, we use two sets of variables: The first set includes the three-month minus ten-year Treasury bond yield spread, with both yields calculated according to Dahlquist and Svensson (1996), and Moody’s seasoned Baa ten-year corporate bond yield minus the ten-year Treasury bond yield spread. The second set of measures

5 includes well-known financial conditions and financial stress indexes, specifically the Chicago Fed NFCI, the St. Louis FSI, the Kansas City Fed FSI, and the EBP. All financial data was accessed through FRED, save for the term spread and the EBP which were provided by the Federal Reserve Board. 3 Firm-level probability of default Weassumethattheprobabilityofafirmdefaulteventbetweentimetandt+htakesthefunctional form: P(I = 1) = f(X ;θ) (1) i,t,t+h i,t where I is the indicator function that is 1 if firm i defaults between times t and t+h, f is i,t,t+h a linear or non-linear function of the firm-level variables contained in X , and θ is a vector of i,t model parameters that needs to be estimated. The classical approach to estimating equation (1) is via a logistic regression. However, there are two primary drawbacks to doing the classic approach in this setting. First, covariates typically enter linearly into a logistic regression, imposing a linear structure on what may be a non-linear relationship. Non-selectively including many non-linearities and interactions, on the other hand, easily leads to a model that fits too closely to the estimation data and that does not generalize well, leading for instance to poor out-of-sample forecasts. Second, logistic regressions are not very robust to outliers and to sparse data. These two drawbacks may be acceptable in some settings and may be traded off against a very interpretable model, but it is clear that the determinants of firm health arise as several non-linear relationships, and as we are working with firm-level data, there are unavoidable data restrictions and outliers. The literature on default probabilities, and more generally the studies on firm stress-events, have primarily relied upon the logistic regression to estimate default probabilities, although some attempts were made in the 1990s and early 2000s to use methods from the machine learning literature (such as neural networks or support vector machines). Given the historical context, and motivated by recent findings (see e.g. Jones et al. (2015)) on the benefits of using machine learning techniques to estimate equation (1) that may circumvent these pitfalls, we estimate equation (1)

6 by both logistic regression and several modern machine learning algorithms, including artificial neural networks, support vector machines, random forests, and tree-based gradient boosting machines. We run two empirical exercises. The first exercise is an in-sample estimation of firm-level defaults within h quarters. The second exercise is a recursive estimation of the defaults, where we estimate the model with data up to some period t and then we estimate the probability of default within the next h quarters. We evaluate the predictions on the areaunderthereceiveroperatingcurve, a standard measure to assess the quality of different classification models (see e.g. Drehmann and Juselius (2014)). Our evaluation of these techniques through time and across different forecast horizons is novel to the default probability literature, and further details regarding the exercises and techniques are outlined in appendix B. When we fit the models with all the data, in-sample, we find that the random forest dominates all other classification techniques. Figure 2 shows the random forest achieves an in-sample AUC greater than 0.99 for all forecast horizons tested and appears to increase as the forecast horizon increases. In comparison, all other techniques, except for the support vector machine, appear to decrease their AUC as the forecast horizon increases. Further, we find that when examining the variables of importance in each model, both the logistic regression and random forest put most weight on measures of leverage, in other words, financial conditions. When the models are evaluated recursively, out-of-sample, we again find that the random forest dominates all other classification techniques2. Figure 2 shows that the random forest achieves an in-sample AUC greater than 0.96 for all the forecast horizons being tested, and a maximum AUC of approximately 0.985. However, as we evaluate the random forest in real time, we find that its AUC peaks at forecast horizons between two and five quarters-ahead. We find that the random forest algorithm outperforms all other techniques at all forecast horizons, both in- and out-of-sample. While the tree-based gradient boosting machine (a similar algorithm to the random forest) is somewhat worse than the random forest, it outperforms the logistic regression, artificial neural network, and support vector machine algorithms. Therefore, we construct our index using predictions from the random forest algorithm, our best forecasting 2The out-of-sample logistic regression exercise used data winsorized at the 5th and 95th percentiles due to the extremelypoorperformanceofthemodelwithoutthedatapre-processing.

7 model. 4 The index of non-financial corporate sector health Our index is based on a bottom-up approach. We first estimate firm-level defaults, and then we aggregate the out-of-sample predictions to create an index of non-financial corporate sector health. We discuss each step below. We use the cross-section of default probabilities predicted with the random forest algorithm to construct our measure of non-financial corporate sector health. We construct the index from three components that reflect different moments of the cross-sectional distribution of predicted defaults: We use asset-weighted observations to derive the weighted mean, so that defaults affecting larger corporations are considered to be more informative about the health of the sector. We allow for the possibility that the shape of the cross-sectional distribution contains additional relevant information, so we also consider the standard deviation and the skewness of the distribution. The currentprevailingviewintheforecastcombinationliteratureisthatasimpleunweightedaveraging of forecasts will outperform more complicated forecast combination techniques (for a textbook treatment of the topic, see Elliott and Timmermann (2016), and for an applied demonstration, see Stock and Watson (2004)). Following the forecast combination literature, we create an aggregate index of corporate health by taking the unweighted average of the sub-indexes, i.e., at each time t, our index is the simple average of the weighted mean, the unweighted standard deviation and the unweighted skewness of the predicted default probabilities distribution. Starting with the weighted mean, we construct the first component of the index as NFCHm = 1 ∑ nt w p (2) t,h n i,t i,t,h t i=1 w = ∑ nt A i,t i,t A j=1 j,t where i indexes the firm, t the time, n is the number of firms in the sample at time t, w is the i,t firm’s cross-sectional weight determined by total assets, A, and p is the firm’s probability of i,t,h default within h quarters from time t.

8 Figure3depictsthefour-quartersimplemovingaverageofthe NFCHm constructedtoforecast events up to eight-quarters ahead3. We use an 8-quarters ahead time horizon because 8 quarters provides an earlier warning sign of adverse business conditions than any other tested timehorizon4. The NFCHm has the desirable quality of rising prior to the 2001 and 2008 recessions. Also, there is a greater increase in the index prior to the 2001 rather than the 2008 recession. This mayrespondtothefactthattherun-uptothe2001recessionwascharacterizedbyover-investment in the information technology sector, giving rise to a slew of weak firms entering the sector, while the 2008 recession was brought on by a crisis in the housing and financial sector, not weakness of non-financial corporate firms. Note that by weighting a firm’s probability of default by the firm’s share of assets, we largely mute the effects of small firm distress. However, given that small firms may be more capital constrained and therefore less able to weather deteriorating economic conditions, the probability of default of small firms may provide valuable information to forecast real economic activity. Furthermore, while we note that the mean is the most efficient measure of central tendency (for the normal distribution), its does not capture information concerning the dispersion or symmetry of the distribution. However, it is documented that higher-order moments of firm stock returns vary predictably over the business cycle (for example Alles and Kling (1994)), suggesting that momentsdescribingthetailsofadistribution,i.e.,standarddeviationandskew,maylendvaluable information regarding the current and future state of economic activity. With these motivations in mind, we next construct an index by calculating the mean of the quarter-over-quarter5 difference in the unweighted standard deviation and quarter-over-quarter difference in the unweighted skew at time t: 1 NFCHs = (γ +σ) (3) t 2 t t where t is the time index, γ is the quarter-over-quarter difference of the cross-sectional skew, and σ is the quarter-over-quarter difference of the cross-sectional standard deviation. 3We find our results are generally robust to smoothing or not smoothing the series, we choose to present the smoothedseries,asitreducesnoise,makingbusinesscyclesmorediscernibleandmaximizesitsin-andour-of-sample forecastingability. 4Weonlypresentindexesconstructedusingh=8,andasaconstant,itwillbedroppedfromfurtherindexnotation. 5Wefindourresultstoberobusttoconstructingthe NFCHs withthefirstdifferenceofthemoments,however,as withthe NFCHm,wefindthatusingthefirstdifferenceofthemomentsmakesbusinesscyclesmorediscernibleand maximizestheforecastingabilityoftheindex.

9 Figure 4 shows the dispersion index NFCHs. The second index moves procyclically, rising prior to all three recessions, 1991, 2001, and 2008, with a maximum achieved one quarter prior to the 2008 recession. Such behavior suggests that this index may contain useful information for forecasting real economic activity. To capitalize on the information content of both indexes, we take the average of the two and construct an ensemble index as: 1 NFCH = (NFCHm+NFCHs). (4) t 2 t t Figure 5 shows the NFCH, standardized with mean zero and standard deviation one. As shown in the figure, the index experiences large increases prior to recessions, especially for the 2001 and 2008 episodes. When the index rises two standard deviations above the historical mean, there is a recession within two quarters. The fact that the NFCH distinctly rises prior recessions suggests that it can be a valuable tool for measuring stress in the economy. In the next section we investigate the ability of the index to forecast future real economic activity more generally. 5 Applications to real macroeconomic activity We next validate the use of our bottom-up non-financial corporate health index as a leading indicator of real macroeconomic activity through in- and out-of-sample forecasting exercises considering a number of widely used measures of real economic activity, and by performing several impulse response exercises with these measures following the direct projection approach by Jordà (2005). 5.1 In-sample forecasting ability To determine the efficacy of our non-financial corporate health index as a leading indicator of economic activity, we run an in-sample forecasting test, as in Gilchrist and Zakrajšek (2012), considering payroll employment, real GDP, industrial production, and the unemployment rate as

10 measures of economic activity. The forecast specification is: 4 ∇hY = α+ ∑ β∇1Y +δ TS +δ RFF +δ NFCH(t,h)+(cid:101) (5) t+h t−i 1 t 2 t 4 t+h i=1 where t indexes time, h is the forecast horizon, NFCH(t,h) is our index of interest, ∇hY := t+h 400 log(Y t+h), Y is the measure of real economic activity. In addition to lagged growth values of h+1 Yt−i thedependentvariable, wecontrolforthestance ofmonetarypolicy byincludingthe termspread, TS, between the constant maturity three-month and ten-year Treasury yield, and the real federal funds rate, RFF. We measure the forecasting accuracy in terms of adjusted R2. Furthermore, we compare our index to the Excess Bond Premium, EBP, an information rich sentiment or risk appetite indicator constructedinGilchristandZakrajšek(2012),basedonthecreditspreadofnon-financialcorporate firms, and a financial conditions index, the Chicago Fed NFCI (see Brave and Butters (2011)). Table 1 displays the in-sample forecast exercise results. The NFCH is statistically significant at the one-percent level in all model specifications. Moreover, the NFCH has a larger adjusted R2 than any other index when forecasting real GDP and industrial production. All the forecast coefficientsassociatedwiththeNFCHhavetheexpectedsign,i.e.,positiveforpayrollemployment, real GDP, and industrial production, and negative for unemployment. Finally, when all four indexes are included in the forecasting exercises, the NFCH remains highly significant, while the Chicago Fed NFCI becomes insignificant across all exercises. The EBP becomes statistically insignificant when forecasting real GDP. 5.2 Out-of-sample forecasting ability TodeterminetheefficacyoftheNFCHasaleadingindicatorofUSeconomicconditionsandassess its potential value for practitioners and policymakers, we supplement our in-sample forecasting tests with an out-of-sample forecasting exercise. To assess the ability of the NFCH to forecast real macroeconomic activity in real-time, we compare the forecast errors generated by a baseline autoregressive (AR) model and a similar index-augmentedARmodel. Weuseamodelspecificationsimilartothatofourin-sampleexercise, but we now drop the term-spread and real federal funds rate as controls, yielding the baseline

11 model:6 4 ∑ Y t+h −Y t−1 = α+ β(Y t−i −Y t−i−1 )+(cid:101) t+h (6) i=1 and the index-augmented model: 4 ∑ Y t+h −Y t−1 = α+ β(Y t−i −Y t−i−1 )+γIndex+(cid:101) t+h (7) i=1 We use a four-quarter forecast horizon. All data start in 1989 so that all indexes produce the same number of forecast estimates. The forecasts are built using an expanding, recursive, window, with the first forecast estimation using 5 years of data. We test the Term Spread, Corporate Bond Spread (Baa - 10-year Treasury yield), Chicago Fed NFCI, EBP, Kansas City Fed FSI, St. Louis Fed FSI, Goldman Sachs FCI, and our NFCH. As the firm-level default models heavily load on measures of corporate leverage (see appendix B), one can think of the NFCH as a function of corporate financing conditions. The Term Spread, Corporate Bond Spread, Chicago Fed NFCI, EBP, Kansas City Fed FSI, St. Louis Fed FSI, and Goldman Sachs FCI, are all indexes that attempt to characterize financing conditions, stress, or investors’ sentiment. Hence, we consider it appropriate to compare the NFCH to these indexes. Each index’s ability to improve forecasts of real macroeconomic activity is measured by constructing the ratio of forecast errors such that the RMSE of equation (7) is divided by the RMSE of equation (6). Therefore, a forecast error ratio less than one is interpreted as indicating that an index improves forecasting ability. To further evaluate the difference between the two sets of errors, we also compute the Diebold and Mariano (1995) forecast error statistic and present the tests’ p-values. Additionally, given the sample bias present in testing nested models, we also include significance levels according to Clark and West (2007). Table 2 presents the out-of-sample forecasting results. Ratios less than one (indicating the tested index improves forecasting ability) are in bold, while Diebold-Mariano p-values significant at the ten-percent confidence level (indicating the index augmented model produces smaller absolute forecast errors) are starred. It is clear that the NFCH has a forecast-error ratio of less than one for all measures of real macroeconomic activity. That is, the NFCH helps to forecast 6Wedroptheterm-spreadandrealfederalfundsratebecausewewanttousetheterm-spreadonitsownasan indicatoroffuturerealactivity,andseveralofthefinancialconditionsindexesthatweevaluateintheexercisearebuilt usingboththeterm-spreadandrealfederalfundsrate.

12 all tested measures of real macroeconomic activity out-of-sample. Note as well that the NFCH especially improves the Payroll Employment forecasts by approximately six-percent, with the improvement being statistically significant at the ten-percent level according to the Diebold- Mariano test, and at the five-percent level according to the Clark-West test. The EBP improves the forecast-error ratio for every measure of real macroeconomic activity, but less so than the NFCH for industrial production, and the EBP’s forecast improvement for payroll employment is not statistically significant. The Term Spread has a forecast error ratio less than one for the unemployment rate and payroll employment, but it does not improve our ability to forecast GDP or industrial production. The only other indexes with a forecast error ratio less than one are the Kansas City Fed and St. Louis Fed FSIs, but these only help forecast the unemployment rate. Overall,wefindthattheNFCHeffectivelyforecastspayrollemployment,industrialproduction, GDP, and the unemployment rate, out-of-sample. Finally, we test several well-known indicators of financial conditions, stress, and investor sentiment, and we find that the only indicator that effectively forecasts all the measures of real macroeconomic activity that the NFCH can forecast, is the EBP. 5.3 Impulse responses To further evaluate the macroeconomic implications of a given rise or fall in the NFCH and the effects of real economy shocks on the NFCH, we construct impulse response functions (IRFs) via the local projections method outlined in Jordà (2005). Impulse responses are estimated via direct local projection: 8 ∑ Y t+h −Y t−1 = α h + M t−i ρ i (8) i=1 where t indexes time, h is the horizon index, and M is a matrix of real GDP, industrial production, unemployment rate, real federal funds rate, and the NFCH7. Note that every column in M has been standardized to ease the interpretation of the impulse response functions. First we simulate a positive shock to the NFCH and review its effects on the four measures of economic activity used in the previous exercises. Figure 6 shows the impulse responses to a one standard deviation shock to the NFCH. 7Weuseannualizedquarter-over-quarterchangesofrealGDPandindustrialproduction

13 An increase in the NFCH (a decrease in non-financial corporate health) leads to a decrease in the real federal funds rate. This result is intuitive, as it suggest that as firms’ health deteriorates, a recession or slowdown may become more likely, and policy-makers may decrease interest rates to stimulate activity. A similar effect is visible with Real GDP growth. Within eight quarters after an increase to the NFCH, real GDP growth decreases by one standard deviation, and the decline is statistically significant at the ten-percent level. The unemployment rate increases by approximately one standard deviation, which is statistically significant at the ten-percent level. Lastly, industrial production growth falls, reaches its minimum of approximately -1.25 standard deviations six quarters after a shock to the NFCH, and then begins to return to its historical mean. The results of the impulse response functions suggest that the NFCH is procyclical, and that it is an effective leading indicator of stress in the real economy. Reversing our exercise, we simulate a positive shock to each of the individual measures of real economicactivityandinvestigatetheireffectsonthe NFCH. Figure7showstheimpulseresponse functions of the NFCH to a positive one standard deviation shock to measures of real economic activity. If there is a positive shock to interest rates, then borrowing costs increase and consumers and businesses spend less, so corporate health tends to weaken. This is consistent with the impulse response function, which shows that an increase in interest rates increases the NFCH growth rate by approximately 2 standard deviations. For the rest of the variables, real GDP growth, the unemployment rate, and the industrial production growth rate, the NFCH remains relatively unchanged, suggesting that it takes a broad economic shock to deteriorate overall firms’ health. That is, a weaker labor market alone will not deteriorate firm health by a statistically significant margin, and a large increase in GDP growth alone may not act as a powerful enough tail wind to significantly improve the health of non-financial corporate firms. Rather, it may take several factors together to generate a persistent change in overall non-financial corporate health. 6 Conclusion In this paper we built a bottom-up indicator of non-financial corporate financial stress using machine learning techniques. We find that these techniques more accurately measure the probability

14 of non-financial firm default than traditional models, and we identify moments of the distribution of predicted default probabilities to build an aggregate early warning indicator of corporate stress and macroeconomic conditions. There are two primary contributions of our study. First, we addressed the various weaknesses of the logistic regression in predicting non-financial firm defaults, both its restrictive linear structureandinabilitytoadapttosparsedataandoutliers,byidentifyinganumberofnonstandard statistical techniques that accurately predict future non-financial firm defaults. Using a set of standard balance sheet and income statement variables, we find the random forest to be the most accurate technique, both in and out-of-sample. Second,weleveragedtheimprovedaccuracyinmeasuringdefaultprobabilitiesbyconstructing an information rich macroeconomic index. The macroeconomic index is an aggregation of micro-level default probabilities, specifically utilizing the first three moments of the firm default probabilitydistribution. Theindicatorishighlyeffectiveinpredictingfutureindustrialproduction, payroll employment, real GDP, and the unemployment rate, outperforming the Chicago NFCI and Excess Bond Premium, both in- and out-of-sample. We further validate our index as being an accurate and useful measure of US non-financial corporate health by conducting direct projection impulse response exercises, where positive shocks to the non-financial corporate stress index elicit future downturns in macroeconomic conditions.

15 Figure1: Firm-leveldataovertime Number of Defaults in dataset 30 Quarterly 25 20 15 10 5 0 −5 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 , Number of firms in dataset 15000 Quarterly 12000 9000 6000 3000 0 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Percent of firms who default Basis points 0.30 Quarterly 0.25 0.20 0.15 0.10 0.05 0.00 −0.05 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Alldataisquarterly. Afirmisconsideredtohavedefaultedwhenitfilesforchapter7orchapter11bankruptcy. All bankruptcyfilingdatescomefromacombinationoftheUCLA-LoPuckibankruptcyresearchdatabase,NYU’sAltman defaultdatabase,andMergent’scorporateFISDdailyfeed. TheuniverseoffirmsistheCRSPandCOMPUSTATset ofUSpublicfirms.

16 Figure2: AUCbyMachineLearningandLogisticAlgorithms Logit SVM ANN GBM RF 1 2 3 4 5 6 7 8 evruC reveiceR eht rednU aerA In−Sample Out−of−Sample 1.00 0.98 0.96 0.94 0.92 0.90 0.88 1 2 3 4 5 6 7 8 Forecast Horizon (Quarters) Forecast Horizon (Quarters) The area under the receiver curve (AUC) across 1- through 8-quarter ahead horizons, obtained with the machine learningandlogisticalgorithms. Ascoreofonedenotesaperfectclassifier. Theout-of-samplelogisticregressionwas trainedonwinsorizeddata,atthe5thand95thpercentiles,duetoitsextremelypoorperformanceusingtheuncleaned data.

17 Figure3: Levelindex, NFCHm,derivedfrompredictedcorporatedefaultprobabilitydistribution Standard Deviations 4 3 2 1 0 −1 −2 −3 −4 1989 1994 1999 2004 2009 2014 2019 The weighted mean index, NFCHm, is the 4-quarter moving average of the weighed mean of the probabilities of defaultwithin8-quarters,predictedbytheRFalgorithm. Theprobabilityofdefaultofagivenfirmisweightedbythe firm’sshareoftheassetsofallfirmsinthesample. Theindexhasbeenstandardizedtohavemean0andstandard deviation1. Figure4: Dispersionindex, NFCHs,derivedfrompredictedcorporatedefaultprobabilitydistribution Standard Deviations 4 3 2 1 0 −1 −2 −3 −4 1989 1994 1999 2004 2009 2014 2019 TheDispersionIndex, NFCHs,isaverageoftheQoQchangeinskewnessandstandarddeviationoftheprobabilities of default within 8-quarters, predicted by the RF algorithm. The index has been normalized to have mean 0 and standarddeviation1. Figure5: Ensembleindex,NFCH,constructedastheaverageof NFCHm and NFCHs Standard Deviations 4 3 2 1 0 −1 −2 −3 −4 1989 1994 1999 2004 2009 2014 2019 Theensembleindex,NFCH,istheunweightedaveragethe NFCHm and NFCHs. Theindexhasbeennormalizedto havemean0andstandarddeviation1.

18 Table1: In-SampleForecastResults PayrollEmployment EBP −0.830∗∗∗ −0.612∗∗∗ −0.388∗∗∗ (0.159) (0.131) (0.144) NFCI −0.851∗∗∗ −0.606∗∗∗ −0.033 (0.211) (0.204) (0.169) NFCD −0.693∗∗∗ −0.491∗∗∗ −0.517∗∗∗ −0.369∗∗∗ (0.178) (0.169) (0.142) (0.117) Observations 100 100 100 100 100 100 AdjustedR2 0.692 0.668 0.681 0.765 0.749 0.802 IndustrialProduction EBP −2.944∗∗∗ −1.888∗∗∗ −1.164∗∗∗ (0.707) (0.347) (0.439) NFCI −2.595∗∗ −1.639∗∗ 0.414 (1.002) (0.687) (0.665) NFCD −2.926∗∗∗ −2.373∗∗∗ −2.496∗∗∗ −1.914∗∗∗ (0.880) (0.740) (0.590)) (0.499) Observations 100 100 100 100 100 100 AdjustedR2 0.395 0.347 0.519 0.603 0.586 0.678 RealGDP EBP −0.850∗∗∗ −0.477∗∗∗ −0.153 (0.192) (0.159) (0.169) NFCI −0.940∗∗∗ −0.562∗∗∗ −0.064 (0.212) (0.194) (0.225) NFCD −0.989∗∗∗ −0.820∗∗∗ −0.803∗∗∗ −0.644∗∗∗ (0.246) (0.248) (0.213) (0.176) Observations 100 100 100 100 100 100 AdjustedR2 0.347 0.369 0.501 0.551 0.565 0.613 Unemployment EBP 8.191∗∗∗ 5.467∗∗∗ 3.578∗∗∗ (0.985) (0.844) (1.185) NFCI 8.752∗∗∗ 5.356∗∗∗ −1.517 (2.385) (1.813) (1.615) NFCD 7.651∗∗∗ 5.723∗∗ 6.072∗∗∗ 4.264∗∗∗ (2.082) (2.261) (1.579) (0.900) Observations 100 100 100 100 100 100 AdjustedR2 0.587 0.548 0.624 0.697 0.673 0.767

19 Table2: Out-of-SampleForecastResults RealActivity Index ForecastErrorRatio DieboldMarianoP-value Clark-WestP-value PayrollEmployment KansasFSI 1.004 0.518 PayrollEmployment ChicagoNFCI 1.05 0.648 PayrollEmployment St. LouisFSI 1.031 0.694 PayrollEmployment GoldmanSachsFCI 1.047 0.995 PayrollEmployment TermSpread 0.986 0.240 0.10 PayrollEmployment Corp. Spread 1.053 0.952 PayrollEmployment EBP 0.935 0.104 0.05 PayrollEmployment NFCD 0.944 0.084* 0.05 GDP KansasFSI 1.016 0.595 GDP ChicagoNFCI 1.01 0.548 GDP St. LouisFSI 1.07 0.874 GDP GoldmanSachsFCI 1.058 0.952 GDP TermSpread 1.015 0.938 GDP Corp. Spread 1.015 0.722 GDP EBP 0.956 0.171 0.10 GDP NFCD 0.982 0.298 0.10 IndustrialProduction KansasFSI 1.075 0.788 IndustrialProduction ChicagoNFCI 1.105 0.785 IndustrialProduction St. LouisFSI 1.05 0.904 IndustrialProduction GoldmanSachsFCI 1.049 0.969 IndustrialProduction TermSpread 1.012 0.820 IndustrialProduction Corp. Spread 1.042 0.799 IndustrialProduction EBP 0.997 0.483 0.05 IndustrialProduction NFCD 0.947 0.155 0.10 UnemploymentRate KansasFSI 0.976 0.383 UnemploymentRate ChicagoNFCI 1.003 0.512 UnemploymentRate St. LouisFSI 0.966 0.208 UnemploymentRate GoldmanSachsFCI 1.038 0.998 UnemploymentRate TermSpread 0.994 0.366 0.05 UnemploymentRate Corp. Spread 1.045 0.911 UnemploymentRate EBP 0.922 0.057* 0.05 UnemploymentRate NFCD 0.972 0.145 0.05 TheForecastErrorRatio(column3)isdefinedastherootmeansquarederror(RMSE)oftheindex-augmentedmodel (equation 7) divided by the baseline AR (equation 6) RMSE. Ratios less than one, indicating the index improves forecastingability,arebolded. Diebold-Mariano(1995)P-values(Column4)statisticallysignificantattheten-percent level(suggestingtheindex-augementedRMSEislessthanthebaselineRMSE)aredenotedbyanastrics. Clark-West (2007)P-values(Column5)arenon-normallydistributedandonlyidentifiedatthe5and10%significancelevel.

20 Figure6: ImpulseResponseFunctions: RealactivityresponsestoNFCHshocks Monetary Policy Response to NFCD Shock Standard Deviations 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 1 2 3 4 5 6 7 8 Horizon (Quarters) Real GDP Growth Response to NFCD Shock Standard Deviations 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 1 2 3 4 5 6 7 8 Horizon (Quarters) Unemployment Response to NFCD Shock Standard Deviations 1.5 1.0 0.5 0.0 −0.5 −1.0 1 2 3 4 5 6 7 8 Horizon (Quarters) IP Response to NFCD Shock Standard Deviations 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 1 2 3 4 5 6 7 8 Horizon (Quarters) Jordà(2005)styledirectprojectionImpulseResponseFunctions. Allinputshavebeenstandardizedwithmeanzero andstandarddeviationone. EachIRFisameasureoftheresponseofrealeconomicactivitytoaonestandarddeviation shocktotheNFCHindex. EachregressioncontainseightlagsofeachofthefourresponsevariablesandtheNFCH index. RealGDPandindustrialproductionarebothquarter-over-quarterdifferences,andmonetarypolicyrefersto therealfederalfundsrate.

21 Figure7: ImpulseResponseFunctions: NFCHresponsetorealactivityshocks NFCD Response to Monetary Policy Shock Standard Deviations 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 1 2 3 4 5 6 7 8 Horizon (Quarters) NFCD Response to Real GDP Growth Shock Standard Deviations 0.50 0.25 0.00 −0.25 −0.50 1 2 3 4 5 6 7 8 Horizon (Quarters) NFCD Response to Unemployment Shock Standard Deviations 4 3 2 1 0 −1 −2 −3 1 2 3 4 5 6 7 8 Horizon (Quarters) NFCD Response to IP Shock Standard Deviations 1.0 0.5 0.0 −0.5 −1.0 1 2 3 4 5 6 7 8 Horizon (Quarters) Jordà(2005)styledirectprojectionImpulseResponseFunctions. Allinputshavebeenstandardizedwithmeanzero andstandarddeviationone. EachIRFistheresponseoftheensembleNFCHindextoaonestandarddeviationshock tosomemeasureofrealeconomicactivity. Eachregressioncontainseightlagsofeachofthefourresponsevariables andtheNFCHindex. RealGDPandindustrialproductionarebothquarter-over-quarterdifferences,andmonetary policyreferstotherealfederalfundsrates.

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25 A Additional Data Tables Table3: Descriptivestatisticsforpredictorvariables Variable mean sd min max range EXRET 0.00 0.99 -4.22 4.51 8.72 EXRETAVG 0.01 0.99 -4.12 3.22 7.33 INDSALES -0.00 1.00 -0.99 5.35 6.34 INDSALESGR -0.00 1.00 -7.40 12.01 19.41 AGE 10.79 7.12 0.00 31.00 31.00 NITA 0.03 0.96 -28.42 6.85 35.27 TLTA 0.01 0.99 -2.22 27.36 29.58 NIMTA 0.02 0.97 -12.09 5.05 17.14 TLMTA 0.01 0.99 -1.84 3.41 5.24 CASHMTA -0.01 0.99 -1.00 7.27 8.27 PRICE 0.03 0.98 -6.17 0.98 7.15 MB -0.02 0.98 -1.46 5.84 7.31 NIMTAAVG 0.02 0.97 -10.34 3.42 13.77 Q -0.02 0.96 -1.07 61.62 62.69 DIVDUMMY 0.29 0.46 0.00 1.00 1.00 KZ -0.01 0.98 -10.55 46.11 56.66 HP 0.02 1.00 -3.14 2.61 5.75 WW 0.01 1.01 -1.96 13.76 15.72 Z -0.02 0.96 -36.18 19.96 56.14 WCTA 0.01 0.96 -72.68 1.25 73.93 CURRENTRATIO -0.01 0.97 -1.19 13.57 14.76 ASSETTURNOVER 0.01 1.01 -0.46 9.24 9.69 ROE 0.01 0.96 -16.12 20.97 37.09 EBITCOVER 0.01 1.00 -10.59 12.05 22.64 CAPEXTA 0.00 0.99 -7.15 9.22 16.37 Twolagsofeachvariableareusedandallmissingobservationshavebeenremoved. Allcontinuousvariableshavebeen standardizedwithmeanzeroandvarianceone.

26 Table4: Predictorvariableconstruction Variable Description Formula(CRSPandCompustatMnuemonics) EXRET ExcessReturns log((1+ret)/100)-log(1+sprtrn) EXRETAVG AverageExcessReturns (1−θ) φ(EXRET)+∑12 θi−1φ(EXRET); 1−θ12 i i=2 i θ =2(− 3 1); φ islagoperator INDSALES IndustrySales TotalsalesofSIC3-digitgroup INDSALESGR IndustrySalesGrowth AverageQoQsalesgrowthinSIC3-digitgroup AGE FirmAge (datadate-min(datadate))/365 niq NITA NetIncome/AdjustedTotalAssets ; atq+.1(me−be) me=abs(prccq)cshoq; be= χ +.1(me- χ); χ =seqq+txditcq-pstkq ltq+mibq TLTA TotalLiabilities/TotalAssets atq+.1∗(me−be) niq NIMTA NetIncome/AdjustedTotalAssets (me+ltq+mibq ltq+mibq TLMTA TotalLiabilities/AdjustedTotalAssets (me+ltq+mibq cheq CASHMTA Cash/AdjustedTotalAssets (me+ltq+mibq PRICE Price log(min(abs(prccq),15)) MB MarketEquity/BookEquity me/be NIMTAAVG NetIncome/(AverageAdjustedTotalAssets) 1−θ3 φ (NIMTA)+ 1−θ12 1 +θ3φ (NIMTA)+θ6φ (NIMTA)+θ9φ (MINTA) 2 3 4 Q Tobin’sQ (atq+cshoq(prccq)-ceqq-txdbq)/atq DIVDUMMY DividendDummy 1ifdividendsarepaidattime t KZ Kaplan-ZingalesIndex -1.002*CASHFLOW/atq+0.283*Q+3.319TOTDEBT/(TOTDEBT+seqq)- -39.368*DIVIDENDS/atq-1.315*cheq/atq; CASHFLOW=ibq+dpq;TOTDEBT=dlcq+dlttq HP HPIndex 0.737*log(pmin(atq,4500))+0.043*log(pmin(atq,4500))2- -0.040*pmin(AGE,37) WW Whited-WuIndex -0.091*(ibq+dpq)/atq+0.062*(DIVDUMMY)+0.021*dlttq/atq Z AltmanZ-score 1.2*(actq-lctq)/atq+1.4*req/atq+3.3*(niq+xintq+txtq)/atq+ +0.6*(cshoq*prccq)/ltq+0.999*saleq/atq) WCTA WorkingCapital/TotalAssets (actq-lctq)/actq CURRENTRATIO CurrentRatio actq/lctq ASSETTURNOVER AssetTurnover saleq ROE ReturnonEquity niq/ceqq EBITCOVER EbitCover ifelse(xintq==0,EBIT/.01,EBIT/xintq); EBIT=niq+xintq+txtq CAPEXTA Capex/TotalAssets capxq/atq Notethatwecalculateallupper-casevariables,whilealllower-casevariablesarerawCRSPandCompustatvariables.

27 B Classification Techniques and Evaluation B.1 Classification techniques We estimate equation (1) by various machine learning algorithms that we describe below. We test a classical single layer artificial neural network trained with and stochastic gradient descent via standard backward propagation, emblematic of the networks used in the second wave of default prediction literature. Further, we train a support vector machine with Gaussian radial basis function kernel, emblematic of the third wave of default prediction literature (for a discussion of this technique, see Scholkopf et al. (1997)). These two techniques are well rooted in statistics, economic, and default prediction literature, so we focus our discussion on the more recently developed, and to-be-proven more effective, tree-methods. Both the tree-based gradient boosting machine and random forest are augmentations of the simpledecisiontree. Asingledecisiontreeisconstructedasfollows: startingwithallobservations, the method finds the variable and associated threshold value that best splits the observations in two groups, as measured by some objective loss function, such as the Gini index or absolute error. Each group is then split again into two groups by the same processes, however it is important to note that the splitting predictor variable does not have to be the same. This processes continues until each group has only one observation in it, or until some predefined number of splits is achieved. A random forest is simply a collection of several decision trees, returning the mean estimation of the single trees (Breiman (2001)). The random forest is outlined in algorithm 1. Conversely, a tree-based gradient boosting machine is initialized with a single decision tree, then decision tree m is trained on the residuals of decision tree m−1. The final output of the gradient boostingmachineisthemeanprobabilityestimateofall M trees(Friedman(2002)). Thetree-based gradient boosting machine is outlined in algorithm 2.

28 Algorithm 1: In-sample construction of Random Forest for b = 1 to B do Draw a bootstrap sample Z∗ of size N from the training data; Grow a decision tree T to the bootstrapped data, by recursively repeating the following b steps for each terminal node of the tree, until the minimum node size n is reached; min while n > n do min Select m variables at random from the p variables.; Pick the best variable/split point among the m.; Split the node into two daughter nodes.; end end Output the ensemble of trees T B; bi Algorithm 2: In-sample construction of tree-based stochastic gradient boosting machine Choose loss function Ψ(y, f), learning rate λ, and tree depth L with five-fold cross validation; Instantiate simple decision tree f(x)(0); for iteration m = 1 ... K do ∂ Ψ(y , f(x )m−1) Compute the gradient y˜ = −( i i ) for all observations i; i ∂f(x )m−1 i Sample from training data without replacement ; (m) Train a tree model h of depth L on the random subset using the gradient as the i outcome ; Update the model f (m) = f (m−1) +λh (m) ; i i i end Instantiate trained model f(K) ;

29 B.2 Classification exercises and evaluation Weconductestimationexercisesintwostages. Thefirststageisanin-samplefittingexerciseofvarious classification algorithms on the entire data set. This exercise mainly serves to illustrate which variables different algorithms consider important in making predictions for firm defaults. The second stage is running the classification techniques recursively out-of-sample to get predictions that could be used in real-time. During the first stage, the in-sample fit and testing, our data begins in 1985:Q1 and ends in 2018:Q3. Since defaults are rare events, representing less than one-percent of the cross-sectional observations each quarter, we mitigate the effect of this severe class imbalance by implementing a down-sample procedure. Specifically, rather than using all the data for training the models, we use all of the incidents of default and then sample from the set of non-defaulted firms to ensure a 90-10 split between the events. An alternative to this approach is to specify a high cost for missclassifying firms that default. In the second stage, the out-of-sample fit and testing data begins in 1985:Q1, and we begin our estimationin1989:Q1,givingourfirstiteration20monthsoftrainingdata. Thedatasetthengrows to include the new observations at time t, as an expanding window. This recursive forecasting process is illustrated in the appendix. Additionally note, that as in the out-of-sample exercise, we down sample to a 90-10 split as we did in the in-sample exercise. We train our models using monthly data. The balance sheet and income statement data are quarterly and are carried forward until new data is available. Both default dates and equity data are available at the daily frequency, and are updated monthly in our exercises. To assess the accuracy of our various model specifications, we use a standard measure of binary-classification techniques, the area under the receiver operating characteristic curve (AUC) (see Drehmann and Juselius (2014)). As we estimate predicted probabilities of defaults, bounded between 0 and 1, we need a cutoff value c for which predicted values above are classified as a default prediction. Given the specific cutoff value c, one can calculate the proportion of true positives and false positives (that is, the number of firms which are predicted to default and do not, and vice versa) that occur. The choice between a high or low value of c implies a trade-off between the proportion of true and false positives. The receiver operating characteristic curve

30 (ROC)attemptstovisualizethistrade-offbyplottingthetruepositiverateagainstthefalsepositive rateforvariousvaluesof c between0and1. TheAUCisthedefiniteintegraloftheROC.AnAUC of 1 indicates a perfect classifier, while an AUC of 0.5 suggests the same accuracy as tossing a fair coin to assign classifications. Our choice of preferred models are driven by which algorithms maximize the AUC. B.3 In-sample estimation results In-sample we estimate the probability of default within h-quarters, such that h ∈ [1,8]. At each time-horizonwefitalogisticregression,supportvectormachine,artificialneuralnetwork,random forest, andtree-basedgradientboostingmachine. Allmodelparametersarechosenusingfive-fold cross validation, maximizing the accuracy of each model. In parameter tuning process, accuracy is measured as the proportion of predicted classes that agree with observed classes. Figure 2 depicts the AUCs of the fitted models across the one through eight quarter time horizons. The random forest dominates all other models across all but one time horizon, as measured by AUC. The tree-based gradient boosting machine dominates all methods, except the random forest, at all time horizons. Both the support vector machine and artificial neural network are dominated by the logistic regression at the one-quarter time horizon. However the support vector machine then dominates both the logistic regression and neural network after the one-quarter horizon. The neural network dominates the logit after the two-quarter horizon. Note that the ability of the classifiersisexactlywhatonewouldexpectbasedontheevolutionofdefaultforecastingliterature. Further, the logistic regression decreases monotonically, suggesting that the linear functional form may poorly approximate the nonlinear relationships as they develop overtime. In contrast, the best classifier, the random forest, increases monotonically, and achieves an AUC greater than 0.99 at all time horizons. Random forest are well known for their ability to capture non-linear relationships and robustness to outlier and missing data (for a greater review of random forest, see Hastie et al. (2009)). A consequence of the logistic regression’s linear structure is its inability to capture the nonlinear relationship between variables and their lags in a mean reverting system (unless quadratic terms are specifically imposed via a basis expansion technique). As Figure 3 shows, nine of the ten

31 most important variables used by the random forest are really three variables with two associated lags each8. Conversely, the logistic regression9 only ranks two pairs of variables and their first lag as being in the top ten most important variables. However, both methods do flag, TLMTA, total liabilities over market equity plus total liabilities (a measure of leverage), as being the most important variable. In fact, more than half of the most important variables flagged by the random forest are a measure of leverage, and the other variable EBIT cover is a measure of a firm’s ability to pay its interest expenses. While these specific variables may be important motivation in a micro-driven structural model of corporate stress, that is beyond the scope of this paper. Despite the dominance of the random forest across all forecasting horizons, there are serious disadvantages to drawing conclusions based on only in-sample model fits when the true design of the project is to predict defaults in real-time. For example, the machine learning techniques may be overfitting in-sample, potentially leading to increased out-of-sample bias (Hastie et al. (2009)). For this reason, we test all models out-of-sample. B.4 Out-of-sample estimation results Outofsample,therandomforestonceagaindominatesalltestedalgorithmsateverytimehorizon, as measured by AUC. In fact, the random forest has an average AUC greater than 0.97. Figure 2 presents the AUC’s of the random forest, tree-based gradient boosted machine, support vector machine, artificial neural network, and logit. The tree-based gradient boosting machine again dominates all other classifiers, excluding the random forest. Both tree-based methods can be seen to increase in efficacy after the one-quarter horizon, sustain a stable AUC from at least the three-quarters through the five-quarters horizons, then decrease to another stable region from 8PutforthbyBreimanetal. (1984),avariable’srelativeimportanceinasingledecisiontreeisgivenbyitssquared relevance N−1 R2 = ∑ i2I(v(t)=λ) λ n n=1 whereλisthevariableofinterest, N−1isthenumberofsplitsinsidethetree,nisanodeinthetree,andi2 isthe maximalmarginalimprovementinaccuracy. Themarginalimprovementi2 iscalculatedbypartitioningtheregion associatedwithnodenbyeachvariableinturn,settingtheregionalresponsevaluetosomeconstant,andmeasuring thechangeinaccuracyfromthetreebeforegeneratingthenode. Theindicatorfunction I(v(t)=λ)ispresenttosay thatonlyi2 generatedbythevariableλarecountedwhencalculatingλ’srelativeimportance. Avariables’relative importanceinarandomforestisthevariable’saverageR2 acrossalltreesintheforest. 9Avariable’simportance,asmeasuredbyalogisticregression,istheabsolutevalueofitscoefficient. Becauseall variableshavebeenpreviouslystandardized,comparingthemagnitudeofbetasisstraightforward.

32 Figure8: VariableImportancebyMachineLearningAlgorithm RandomForest LogisticRegression TLMTA l TLMTA l LAG1TLMTA l EXRETAVG l LAG2TLMTA l EXRET l TLTA l DIVDUMMY l LAG2HP l LAG2EXRET l LAG1TLTA l WW l LAG1HP l HP l HP l PRICE l LAG2TLTA l LAG2EXRETAVG l LAG1EBITCOVER l LAG2DIVDUMMY l 40 60 80 100 70 80 90 100 Importance Importance Thetoptenmostimportantvariablesusedinestimatingtheprobabilityoffirmdefaultbyrandomforestandlogistic regression. Pleaseseefootnotes(3)and(4)forconstructiondetails. Notethattherelativeimportanceofeachvariable isscaledsuchthatthemostimportantvariablehasarelevanceof100. either six or seven-quarters to the eight-quarters horizon. Conversely, the neural network’s AUC monotonically decreases from a maximum of 0.94 to a minimum of approximately 0.91. Further evidence of the logit’s weakness in this exercise is the fact that the logit first needed to have the data winsorized at the 5 and 95% levels before producing the AUC’s presented by Figure 2. When the data is not winsorized, the logit’s one-quarter ahead AUC is 0.83, not 0.87. Additionally, while the general shape of the AUC curve is the same, at the six-quarter ahead horizon, the AUC sporadically drops to approximately 0.9, only to rise back up to .93 at seven-quarters ahead and then monotonically decrease. This marked instability further underscores the logit’s susceptibility to data constraints and its own rigid linear structure. Given that the random forest dominates across all time horizons, its predicted probabilities are used to construct our aggregate indexes of the macroeconomy.

33 C In-sample forecasting results for sub-indices C.1 (In-sample) Forecasting Exercises To determine the efficacy of this index as a leading indicator of US non-financial corporate health, and therefore a leading indicator of real economic activity, we run an in-sample forecasting test, as in Gilchrist and Zakrajšek (2012), with payroll employment, real GDP, industrial production, and the unemployment rate. The forecast specification is: 4 ∇hY = α+ ∑ β∇1Y +δ TS +δ RFF +δ X(t,h)+(cid:101) (9) t+h t−i 1 t 2 t 4 t+h i=1 where t is the time index, h is the forecast horizon, X(t,h) is our index of interest, ∇hY := t+h 400 log(Y t+h), Y is the measure of real economic activity. In addition to lagged growth values of h+1 Yt−i thedependentvariable, wecontrolforthestance ofmonetarypolicy byincludingthe termspread, TS, between the constant maturity three-month and ten-year Treasury yield, and the real federal funds rate, RFF. We measure the forecasting accuracy of our indexes in terms of an adjusted R2. Further, we compare our index to the Excess Bond Premium, EBP, an information rich sentiment or risk appetite indicator constructed in Gilchrest and Zakrajsek (2012), based on the credit spread of non-financial corporate firms, and a financial conditions index, the Chicago NFCI (see Brave and Butters (2011)). We run the forecasting exercise with each indicator individually, then in pairs with our index, and with all four indexes at the same time. We evaluate the merits of each index by comparing adjusted R2’s and levels of statistical significance. C.2 Weighted Mean Index First,weconstructanindexofnon-financialcorporatedefaults(NFCH)bycalculatingtheweighted mean of firm-level probabilities of default between time t and t+h: NFCHm = 1 ∑ nt w p (10) t,h n i,t i,t t i=1

34 w = ∑ nt A i,t i,t A j=1 j,t where i is the firm index, t is the time index, n is the number of firms in the sample at time t, w i,t is the firm’s cross-sectional weight determined by assets, A, and p is the firm’s probability of i,t,h default within h quarters from time t. Table 5 shows the in-sample forecast exercise results using the NFCHm as I(t,h) in equation (2). The NFCHm is statistically significant at the one-percent level any time it is either alone or with one other leading indicator. Further, the NFCHm is significant at the five-percent level when in a forecasting regression with all four other leading indicators. In all exercises that include all three indicators, the Chicago NFCI is insignificant, while the Excess Bond Premium is made insignificant in forecasting real GDP. Lastly, the sign of the indicator is as expected, negative, meaning that as the probability of firm default increases, future real economic activity decreases. C.3 Unweighted Dispersion Index Second, we note that by weighting a firm’s probability of default by the firm’s share of assets, we largelymutetheeffectsofsmallfirmdistress. However,giventhatsmallfirmsmaybemorecapital constrained and therefore less able to weather deteriorating economic conditions, the probability of small firm default may provide valuable information to forecast real economic activity. Further, while we note that the mean is the most efficient measure of central tendency, its does not capture information concerning a distributions dispersion. However, it is documented that higher-order momentsoffirmstockreturnsvarypredictablyoverthebusinesscycle(forexample,seeAllesand Kling (1994)), suggesting that moments describing the tails of a distribution, standard deviation and skew, may lend valuable information regarding the current and future state of economic activity. Given these motivations, we next construct an index by calculating the mean of the quarter-over-quarter10 difference in the unweighted standard deviation and quarter-over-quarter 10We find our results to be robust to constructing the NFCHs with differenced moments, however, as with the NFCHm,wefindthatdifferencingthemomentsmakesbusinesscyclesmorediscernibleandmaximizestheindex’s forecastingabilities.

35 difference in the unweighted skew at time t: 1 NFCHs = (γ +σ) (11) t 2 t t where t is the time index, γ is the cross-sectional skew, and σ is the cross-sectional standard deviation. Figure 9 shows the unweighted standard deviation and skew. It is notable that the standard deviation appears countercyclical, rising before and through the 2001 and 2008 recession while falling in economic recoveries. However, the skew does not appear cyclical, but rather characterized by relative calms punctuated by rapid increases and decreases. The skew spikes two years before the 2001 recession, and three years before the 2008 recession. These pre-recession spikes may suggest that the skew is a good leading indicator of recession, however, it does also rise during recessions. Table6presentstheresultsofthein-sampleforecastingexercisesusingequation(2)asthe NFCHs. The NFCHs is statistically significant in all model specifications, except when forecasting the unemployment rate with EBP. The NFCH coefficients are all the expected direction for a pro-cyclical index. However, the adjusted R2s of the exercises using only one indicator suggest that the skew and standard deviation are less informative than other indexes.

36 Figure9: UnweightedMomentsoftheCorporateDefaultProbabilityDistribution Standard Deviation 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 1989 1994 1999 2004 2009 2014 2019 Skew 5.5 5.0 4.5 4.0 3.5 3.0 2.5 1989 1994 1999 2004 2009 2014 2019 The second and third moments generated by the cross-section of unweighted probabilities of firm default within eight-quartersfromtimet,generatedbytheRFalgorithm.

37 Table5: In-SampleForecastResultsusingtheWeightedMeanSubindex PayrollEmployment EBP −0.830∗∗∗ −0.720∗∗∗ −0.342∗∗ (0.159) (0.173) (0.164) NFCI −0.851∗∗∗ −0.786∗∗∗ −0.151 (0.211) (0.223) (0.180) NFCD −0.514∗∗∗ −0.262∗∗∗ −0.434∗∗∗ −0.237∗∗∗ (0.103) (0.088) (0.101) (0.089) Observations 100 100 100 100 100 100 AdjustedR2 0.692 0.668 0.593 0.708 0.725 0.777 IndustrialProduction EBP −2.944∗∗∗ −2.292∗∗∗ −1.123∗∗ (0.707) (0.671) (0.514) NFCI −2.595∗∗ −2.219∗∗ 0.196 (1.002) (1.018) (0.724) NFCD −2.140∗∗∗ −1.405∗∗∗ −1.808∗∗∗ −1.112∗∗∗ (0.335) (0.344) (0.350) (0.305) Observations 100 100 100 100 100 100 AdjustedR2 0.395 0.347 0.337 0.459 0.473 0.590 RealGDP EBP −0.850∗∗∗ −0.641∗∗∗ −0.156 (0.192) (0.223) (0.222) NFCI −0.940∗∗∗ −0.770∗∗∗ −0.126 (0.212) (0.248) (0.235) NFCD −0.664∗∗∗ −0.402∗∗∗ −0.464∗∗∗ −0.264∗ (0.151) (0.144) (0.159) (0.150) Observations 100 100 100 100 100 100 AdjustedR2 0.347 0.369 0.301 0.387 0.434 0.516 Unemployment EBP 8.191∗∗∗ 6.733∗∗∗ 3.458∗∗ (0.985) (0.949) (1.333) NFCI 8.752∗∗∗ 7.728∗∗∗ −0.091 (2.385) (2.191) (1.781) NFCD 5.606∗∗∗ 3.349∗∗∗ 4.680∗∗∗ 2.498∗∗ (0.939) (0.776) (1.068) (0.958) Observations 100 100 100 100 100 100 AdjustedR2 0.587 0.548 0.505 0.621 0.630 0.729

38 Table6: In-SampleForecastResultsusingUnweightedDispersionMeasuresSubindex PayrollEmployment EBP −0.830∗∗∗ −0.744∗∗∗ −0.543∗∗∗ (0.159) (0.135) (0.135) NFCI −0.851∗∗∗ −0.687∗∗∗ 0.168 (0.211) (0.188) (0.169) NFCD −0.563∗∗∗ −0.443∗∗ −0.327∗∗ −0.362∗∗∗ (0.199) (0.180) (0.143) (0.110) Observations 100 100 100 100 100 100 AdjustedR2 0.692 0.668 0.610 0.751 0.693 0.798 IndustrialProduction EBP −2.944∗∗∗ −2.606∗∗∗ −1.801∗∗∗ (0.707) (0.532) (0.448) NFCI −2.595∗∗ −2.024∗∗∗ 1.071∗ (1.002) (0.733) (0.585) NFCD −2.074∗∗ −1.734∗∗ −1.556∗∗ −1.488∗∗∗ (0.886) (0.853) (0.749) (0.546) Observations 100 100 100 100 100 100 AdjustedR2 0.395 0.347 0.331 0.515 0.435 0.636 RealGDP EBP −0.850∗∗∗ −0.755∗∗∗ −0.351 (0.192) (0.174) (0.216) NFCI −0.940∗∗∗ −0.769∗∗∗ 0.110 (0.212) (0.183) (0.304) NFCD −0.683∗∗ −0.586∗∗ −0.491∗∗ −0.474∗∗ (0.300) (0.267) (0.245) (0.190) Observations 100 100 100 100 100 100 AdjustedR2 0.347 0.369 0.308 0.464 0.443 0.572 Unemployment EBP 8.191∗∗∗ 7.270∗∗∗ 4.914∗∗∗ (0.985) (1.079) (1.237) NFCI 8.752∗∗∗ 7.100∗∗∗ −2.995 (2.385) (1.673) (1.807) NFCD 5.176∗∗ 3.763 3.061∗∗∗ 3.484∗∗∗ (2.594) (2.334) (0.950) (0.955) Observations 100 100 100 100 100 100 AdjustedR2 0.587 0.548 0.483 0.635 0.569 0.739