## I. INTRODUCTION — wp17108

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---

### Motivation and context
- Bank failures and financial crises impose large costs, as experienced in the global recession of 2008–9.
- Stress tests are widely used to examine risks to financial institutions and financial stability (Bookstaber et al, 2013).
- Stress tests supported by credible official backstops can raise confidence in policy makers’ ability to manage risks (Orphanides, 2015).

### Recent focus and gap
- Recent work on stress testing has been mainly oriented towards scenario design: generation of severe, plausible, and coherent stress scenarios consistent with historical crisis episodes and potential regime changes.
- Policy efforts have focused on integrating stress tests into financial sector surveillance and oversight (IMF, 2012; Bookstaber et al, 2013).
- Improving forecasting models has received less attention despite its importance: numerical outcomes of forecasting models influence policy recommendations and business strategies.
- As stress scenarios encompass an increasing number of primary variables, model selection and forecasting become challenging even within the family of linear models.

### Argument for machine learning methods
- Model selection and forecasting in stress tests can be facilitated using techniques borrowed from machine learning.
- Machine learning techniques have not been widely used in econometric and financial applications despite robustness and good performance in fields with large datasets.
- Prior applications in stress tests and default prediction:
  - Kapinos and Mitnik (2015): suggest using the least absolute shrinkage selection operator (Lasso) to link bank performance indicators to macroeconomic variables.
  - Perdeiy (2009): uses Lasso regressions to predict bankruptcy using non-traditional financial indicators as covariates.
- The paper extends prior work by delving deeper into conceptual issues justifying the use of machine learning techniques and discussing them more extensively.

### Conceptual issues highlighted
- Increasing number of primary variables in comprehensive stress scenarios gives rise to a curse of dimensionality that complicates model selection.
- Machine learning techniques are argued preferable to other dimension reduction techniques in addressing the curse of dimensionality.
- The paper discusses subset selection and shrinkage methods, including Lasso, and presents conceptual underpinnings of Lasso estimation.
- Lasso applications in finance, economics, and financial networks are described; a stress test application forecasting probabilities of default is included later.

### Structure of the paper
- Section II: overview of the multi-step process in a standard stress test.
- Section III: how the large number of primary variables in a stress scenario leads to a curse of dimensionality and complicates model selection.
- Section IV: argument that machine learning techniques are preferable to other dimension reduction techniques for the curse of dimensionality.
- Section V: discussion of subset selection and shrinkage methods, including Lasso.
- Section VI: description of Lasso applications in finance, economics, and financial networks and a stress test application forecasting probabilities of default.

---

### Stress tests: a multi-step process
- A standard stress test comprises:
  1. stress scenarios design (choose horizon, select primary variables, specify paths);
  2. forecasting performance indicators using econometric/statistical "satellite models" that include primary variables as covariates;
  3. evaluation of firm weaknesses and system-wide vulnerabilities to guide business strategy or policy.
- Examples of scenario scope in practice:
  - U.S. Federal Reserve CCAR 2015: sixteen domestic economic and financial variables and twelve international variables.
  - Bank of England 2015: more than sixty primary variables.
  - EBA stress tests: government bond yields, equity prices, house price shocks, funding shocks on real GDP growth for twenty-seven EU countries and changes in real GDP growth for twenty countries and regions outside the EU.

### Model selection challenges in stress tests
- Large number of primary variables creates combinatorial explosion: with contemporaneous values and no interactions, CCAR-like example leads to 2^28 possible linear models for a single dependent variable.
- Typical stress-test data limitations:
  - Data often annual or limited time span → dimensionality (p) may exceed sample length (n) → ordinary least squares cannot yield unique coefficient estimates.
- Common problems: multicollinearity, overfitting, difficulty of expert judgment when firm complexity and counterparty exposures are large.
- Dimension reduction caveat: factor analysis/principal components reduce dimensionality but make economic interpretation and scenario specification of factors difficult.
- Recommendation: address curse of dimensionality at the model selection level rather than pre-reducing covariates; machine learning methods excel here.

---

### Machine learning: the interpretability–flexibility tradeoff
- Forecasting models are supervised learning problems mapping covariates to outputs (e.g., non-performing loan ratios or PDs).
- Methods in decreasing order of interpretability: subset selection, lasso regressions, least squares, generalized additive models, trees, support vector machines, ensemble methods (bagging, boosting).
- Bias–variance tradeoff:
  - More flexible methods → lower bias, higher variance.
  - Interpretability matters for communicating stress test results to decision-makers; linear models preferred when possible.
- Linear models with non-linear covariate transformations can capture some non-linearity while preserving interpretability.

---

### Linear models: subset selection and shrinkage methods
- Covariate reduction in linear regression: two main categories
  - Subset selection:
    - Best subset selection: search over 2^p models; practical up to ~30–40 variables.
    - Forward and backward stepwise: evaluate only 1 + p(p+1)/2 models.
  - Shrinkage methods:
    - Ridge regression: L2 penalty → shrinks coefficients but typically keeps all covariates (coefficients not exactly zero).
    - Lasso (Least Absolute Shrinkage Selection Operator): L1 penalty → can set coefficients exactly to zero → performs variable selection and parameter estimation simultaneously.
- Optimization formulations (preserved as in source):
  - Least Squares: min ... (unconstrained)
  - Ridge: min subject to L2 constraint (or Lagrangian with +λ||β||2^2)
  - Lasso: min subject to L1 constraint (or Lagrangian with +λ||β||1)
- Geometric intuition: L1 constraint has corners increasing chance of zero coefficients.
- Computational aspects: Lasso exhibits lower variability and cheaper computational costs vs subset selection in high-dimensional settings.

---

### Lasso regression, cross-validation and consistency
- The tuning parameter controls size and number of coefficients; higher values → greater number of covariates included (text preserved as in source).
- Cross-validation:
  - K-fold cross-validation (typical K = 5 or 10) selects parameter minimizing estimated test error (MSE) → λ-min.
  - One-standard-error rule chooses λ-1se: minimum λ such that K-fold estimate ≤ minimum K-fold estimate + 1 standard error, favoring simpler models.
  - Leave-One-Out is N-fold cross-validation.
- Lasso yields biased coefficient estimates (shrinkage toward zero) → two-step approaches mitigate bias:
  - First step: Lasso selects covariates.
  - Second step: estimate linear model with selected covariates by ordinary least squares or apply Lasso again (relaxed Lasso). Relaxed Lasso used in the stress test application.

---

### Lasso applications in finance, economics, and financial networks
- Use cases and properties:
  - Estimation of stable variance–covariance matrices for asset returns and portfolio optimization; L1-constrained portfolio weights and equivalence to constrained minimum variance solutions.
  - Multivariate time series: Lasso can perform as well as principal components/factor models for collinear variables and generate sparser, interpretable models.
  - VARs and time dependence: group-Lasso, adaptive Lasso variants accommodate time dependence; adaptive Lasso shown to be consistent and asymptotically efficient in VARs.
  - Financial networks: Lasso helps generate sparse correlation matrices and network edges based on forward-looking default correlations or variance-decomposition contributions.
- Caveat: prediction advantages from Lasso may vanish when p is small relative to n (Hansen, 2013); stress-test contexts typically have p ~ n or p > n.

---

### A stress test application: forecasting probabilities of default (Section VII)
- Objective: forecast median one-year probabilities of default (PD) for ten industrial sectors in an advanced emerging market economy.
- Sectors: basic materials, communications, consumer cyclicals, consumer non-cyclicals, diversified industries, energy, financials, industrials, technology, utilities.
- Data:
  - Monthly median PD series from CRI database (Risk Management Institute, National University of Singapore), accessed on April 30, 2014; covers December 1990 – February 2014.
  - Quarterly data for primary covariates covering 1990 Q1 – 2013 Q4.
  - Primary covariates: thirteen domestic variables (exchange rate vis-à-vis the U.S. dollar, nominal effective exchange rate, domestic policy rate, consumer price index, real GDP growth rate, unemployment rate, total amount of credit, money market rate, 3-month Treasury bill rate, bank deposit rate, bank lending rate, 10-year Treasury bond rate — list preserved as in source) and five international variables (U.S. real GDP, China real GDP, U.S. policy rate, U.S. consumer price index, commodity price index).
- Model specification and estimation:
  - Equation fitted for each sectoral median PD using both Lasso and relaxed Lasso with 10-fold cross-validation:
    - (6) 4, , , 10, log, sector 1 to 10 1 p it ikt it k it PD Xi PD α ε − = = = (equation preserved as in source).
  - Covariates standardized by Z-score (centered on mean, normalized by standard deviation) before Lasso due to penalty scale sensitivity.
  - Number of potential covariates p = ninety excluding intercept; number of observations n is at most one hundred → situation well-suited for Lasso.
  - Relaxed Lasso: two-step where relaxed Lasso only includes covariates with non-zero coefficients from Lasso λ-min specification.
  - Estimation implemented via R coordinate descent algorithm (Friedman, Hastie, Tibshirani, 2010).
- Empirical results and model selection outcomes:
  - Cross-validation outputs: λ-min corresponds to minimum MSE; λ-1se corresponds to parameter yielding MSE exactly 1 standard deviation above minimum with fewer covariates.
  - Dimensionality reduction achieved:
    - Lasso reduces covariates to ranges: "from four to the mid-20s" for the λ-min specification.
    - For λ-1se specification, number ranges "from zero to slightly below twenty".
    - Worst-case retention: Lasso and relaxed Lasso retain about one quarter of potential covariates.
  - Variance vs sparsity:
    - λ-min specification has lower variance than λ-1se due to higher number of non-zero coefficients.
    - λ-1se yields more sparse models (typically four or five covariates, though some sectors may need up to ten; some sectors end up with only intercept).
  - Model parsimony:
    - No more than ten covariates of the original set of eighteen covariates (including up to four lags → p = 90) are useful for predicting median PD in each sector in λ-min specifications.
    - For many sectors fewer than six variables included.
  - Relaxed Lasso behavior:
    - Relaxed Lasso λ-min tends to retain all non-zero coefficients identified by Lasso.
    - Relaxed Lasso coefficients typically larger in magnitude than Lasso coefficients (reduced noise due to fewer covariates).
  - Standard errors and inference:
    - Tables do not report standard errors; computing standard errors (e.g., via nonparametric bootstrap) is possible but the text argues there are strong arguments against calculating them because Lasso biases coefficients and bootstrapped confidence intervals may convey unjustified precision.
    - Suggested practice in low-dimensional problems: use Lasso for model selection only and estimate final forecasting model with traditional statistical techniques.
  - Implementation details:
    - 10-fold cross-validation used.
    - λ-min and λ-1se selection rules described and illustrated in Figure 2 (panels for each sector).
    - Coordinate descent R implementation used.

---

### Conclusions (Section VIII)
- Lasso regressions likely outperform ordinary least squares in forecasting performance indicators required in applied stress testing due to their ability to handle high-dimensional problems where p is of same order as n.
- Regularization methods like Lasso are specifically designed to handle many high-dimensional issues present in finance, economics, and financial networks.
- Lasso-type estimators can be extended beyond linear models to generalized linear models (logistic, multiclass logistic), Cox proportional hazards, support vector machines, semiparametric nonlinear mixed effect models, and nonlinear regressions (references preserved as in source).
- Addressing interpretability concerns:
  - Lasso framework allows specifying upper and lower bounds or linear restrictions on coefficients to enforce sign or other constraints.
  - Lasso may select variables highly collinear with the "right" ones if they exhibit better signal-to-noise ratios, complicating economic narratives — but stress tests prioritize good out-of-sample forecasts.
- Practical implications:
  - Fast computation algorithms for Lasso in Matlab, R, SAS support use in semi- or fully automated stress testing platforms.
  - Example platform combining Lasso-based forecasting with balance sheet stress testing exists (Chan-Lau and Li, 2015 referenced in source).
  - Such platforms can analyze multiple stress scenarios efficiently within enterprise-wide or systemic risk frameworks.

*Source: wp17108 (IMF Working Paper content provided).*

### Introduction ...........................................................................................................

### Introduction

### Motivation and context
- Bank failures and financial crises impose large costs, as experienced in the global recession of 2008–9.
- Stress tests are widely used to examine risks to financial institutions and financial stability (Bookstaber et al, 2013).
- Stress tests supported by credible official backstops can raise confidence in policy makers’ ability to manage risks (Orphanides, 2015).

### Recent focus and gap
- Recent work on stress testing has been mainly oriented towards scenario design, including generation of severe, plausible, and coherent stress scenarios consistent with historical crisis episodes and potential regime changes.
- Policy efforts have focused on integrating stress tests into financial sector surveillance and oversight to enhance regulatory and supervisory guidance (IMF, 2012; Bookstaber et al, 2013).
- Improving forecasting models has received less attention despite its importance: numerical outcomes of forecasting models influence policy recommendations and business strategies.
- As stress scenarios encompass an increasing number of primary variables, model selection and forecasting become challenging even within the family of linear models.

### Argument for machine learning methods
- The paper argues that model selection and forecasting in stress tests can be facilitated using techniques borrowed from machine learning.
- Machine learning techniques have not been widely used in econometric and financial applications despite robustness and good performance in fields with large datasets.
- Notable prior applications in stress tests and default prediction include Kapinos and Mitnik (2015) and Perdeiy (2009):
  - Kapinos and Mitnik suggest using the least absolute shrinkage selection operator (Lasso) to link bank performance indicators to macroeconomic variables.
  - Perdeiy uses Lasso regressions to predict bankruptcy using non-traditional financial indicators as covariates.
- The paper extends prior work by delving in more depth on conceptual issues justifying the use of machine learning techniques and discussing them more extensively.

### Conceptual issues highlighted
- The increasing number of primary variables in comprehensive stress scenarios gives rise to a curse of dimensionality problem that complicates model selection.
- Machine learning techniques are argued to be preferable to other dimension reduction techniques in addressing the curse of dimensionality.
- The paper discusses subset selection and shrinkage methods, including Lasso, and presents conceptual underpinnings of Lasso estimation.
- Lasso applications in finance, economics, and financial networks are described; a stress test application forecasting probabilities of default is included later in the paper.

### Structure of the paper (as organized in the source)
- Section II: overview of the multi-step process in a standard stress test.
- Section III: how the large number of primary variables in a stress scenario leads to a curse of dimensionality and complicates model selection.
- Section IV: argument that machine learning techniques are preferable to other dimension reduction techniques for the curse of dimensionality.
- Section V: discussion of subset selection and shrinkage methods, including Lasso.
- Section VI: description of Lasso applications in finance, economics, and financial networks and a stress test application forecasting probabilities of default.

*I. INTRODUCTION — wp17108.*

### Section VII illustrates the use of Lasso estimation in forecasting probabilities of default in an

### wp17108 - Section VII illustrates the use of Lasso estimation in forecasting probabilities of default in an

### Stress tests: a multi-step process
- A standard stress test comprises: (1) stress scenarios design (choose horizon, select primary variables, specify paths); (2) forecasting performance indicators using econometric/statistical "satellite models" that include primary variables as covariates; (3) evaluation of firm weaknesses and system-wide vulnerabilities to guide business strategy or policy.
- Examples of scenario scope in practice:
  - U.S. Federal Reserve CCAR 2015: sixteen domestic economic and financial variables and twelve international variables.
  - Bank of England 2015: more than sixty primary variables.
  - EBA stress tests: government bond yields, equity prices, house price shocks, funding shocks on real GDP growth for twenty-seven EU countries and changes in real GDP growth for twenty countries and regions outside the EU.

### Model selection challenges in stress tests
- Large number of primary variables creates combinatorial explosion: with contemporaneous values and no interactions, CCAR-like example leads to 2^28 possible linear models for a single dependent variable.
- Typical stress-test data limitations:
  - Data often annual or limited time span → dimensionality (p) may exceed sample length (n) → ordinary least squares cannot yield unique coefficient estimates.
- Common problems: multicollinearity, overfitting, difficulty of expert judgment when firm complexity and counterparty exposures are large.
- Dimension reduction caveat: factor analysis/principal components reduce dimensionality but make economic interpretation and scenario specification of factors difficult.
- Recommendation: address curse of dimensionality at the model selection level rather than pre-reducing covariates; machine learning methods excel here.

### Machine learning: the interpretability-flexibility tradeoff
- Forecasting models are supervised learning problems mapping covariates to outputs (e.g., non-performing loan ratios or PDs).
- Methods in decreasing order of interpretability: subset selection, lasso regressions, least squares, generalized additive models, trees, support vector machines, ensemble methods (bagging, boosting).
- Bias-variance tradeoff summary:
  - More flexible methods → lower bias, higher variance.
  - Interpretability matters for communicating stress test results to decision-makers; linear models preferred when possible.
- Linear models with non-linear covariate transformations can capture some non-linearity while preserving interpretability.

### Linear models: subset selection and shrinkage methods
- Two main categories for covariate reduction in linear regression:
  - Subset selection: best subset selection (search over 2^p models; practical up to ~30–40 variables), forward and backward stepwise (evaluate only 1 + p(p+1)/2 models).
  - Shrinkage methods: ridge regression and Lasso (Least Absolute Shrinkage Selection Operator).
- Key properties:
  - Ridge uses L2 penalty → shrinks coefficients but typically keeps all covariates (coefficients not exactly zero).
  - Lasso uses L1 penalty → can set coefficients exactly to zero → performs variable selection and parameter estimation simultaneously.
- Optimization formulations (as given):
  - Least Squares: min ... (unconstrained)
  - Ridge: min subject to L2 constraint (or Lagrangian with +λ||β||2^2)
  - Lasso: min subject to L1 constraint (or Lagrangian with +λ||β||1)
- Geometric intuition: L1 constraint has corners increasing chance of zero coefficients.
- Computational aspects: Lasso exhibits lower variability and cheaper computational costs vs subset selection in high-dimensional settings.

### Lasso regression, cross-validation and consistency
- The tuning parameter controls size and number of coefficients; higher values → greater number of covariates included (note: text states higher values leading to greater number of covariates — preserved as in source).
- Cross-validation:
  - K-fold cross-validation (typical K = 5 or 10) selects parameter minimizing estimated test error (MSE) → λ-min.
  - One-standard-error rule chooses λ-1se: minimum λ such that K-fold estimate ≤ minimum K-fold estimate + 1 standard error, favoring simpler models.
  - Leave-One-Out is N-fold cross-validation.
- Lasso yields biased coefficient estimates (shrinkage toward zero) → two-step approaches mitigate bias:
  - First step: Lasso selects covariates.
  - Second step: estimate linear model with selected covariates by ordinary least squares or apply Lasso again (relaxed Lasso). Relaxed Lasso used in the stress test application.

### Lasso applications in finance, economics, and financial networks
- Lasso is used to construct sparse models in high-dimensional environments, including:
  - Estimation of stable variance-covariance matrices for asset returns and portfolio optimization; L1-constrained portfolio weights and equivalence to constrained minimum variance solutions.
  - Multivariate time series: Lasso can perform as well as principal components/factor models for collinear variables and generate sparser, interpretable models.
  - VARs and time dependence: group-Lasso, adaptive Lasso variants accommodate time dependence; adaptive Lasso shown to be consistent and asymptotically efficient in VARs.
  - Financial networks: Lasso helps generate sparse correlation matrices and network edges based on forward-looking default correlations or variance-decomposition contributions.
- Caveat: prediction advantages from Lasso may vanish when p is small relative to n (Hansen, 2013); but stress-test contexts typically have p ~ n or p > n.

### A stress test application: forecasting probabilities of default (Section VII)
- Objective: forecast median one-year probabilities of default (PD) for ten industrial sectors in an advanced emerging market economy.
- Sectors: basic materials, communications, consumer cyclicals, consumer non-cyclicals, diversified industries, energy, financials, industrials, technology, utilities.
- Data:
  - Monthly median PD series from CRI database (Risk Management Institute, National University of Singapore), accessed on April 30, 2014; covers December 1990 – February 2014.
  - Quarterly data for primary covariates covering 1990 Q1 – 2013 Q4.
  - Primary covariates: thirteen domestic variables (exchange rate vis-à-vis the U.S. dollar, nominal effective exchange rate, domestic policy rate, consumer price index, real GDP growth rate, unemployment rate, total amount of credit, money market rate, 3-month Treasury bill rate, bank deposit rate, bank lending rate, 10-year Treasury bond rate — list preserved as in source) and five international variables (U.S. real GDP, China real GDP, U.S. policy rate, U.S. consumer price index, commodity price index).
- Model specification and estimation:
  - Equation fitted for each sectoral median PD using both Lasso and relaxed Lasso with 10-fold cross-validation:
    - (6) 4, , , 10, log, sector 1 to 10 1 p it ikt it k it PD Xi PD α ε − = = = (equation preserved as in source).
  - Covariates standardized by Z-score (centered on mean, normalized by standard deviation) before Lasso due to penalty scale sensitivity.
  - Number of potential covariates p = ninety excluding intercept; number of observations n is at most one hundred → situation well-suited for Lasso.
  - Relaxed Lasso: two-step where relaxed Lasso only includes covariates with non-zero coefficients from Lasso λ-min specification.
  - Estimation implemented via R coordinate descent algorithm (Friedman, Hastie, Tibshirani, 2010).
- Empirical results (summary of reported findings):
  - Figures and Tables: Figure 2 and Tables 1 and 2 illustrate Lasso vs relaxed Lasso performance and coefficients.
  - Cross-validation outputs: λ-min corresponds to minimum MSE; λ-1se corresponds to parameter yielding MSE exactly 1 standard deviation above minimum with fewer covariates.
  - Dimensionality reduction achieved:
    - Lasso reduces covariates to ranges: "from four to the mid-20s" for the λ-min specification.
    - For λ-1se specification, number ranges "from zero to slightly below twenty".
    - Worst-case retention: Lasso and relaxed Lasso retain about one quarter of potential covariates.
  - Variance vs sparsity:
    - λ-min specification has lower variance than λ-1se due to higher number of non-zero coefficients.
    - λ-1se yields more sparse models (typically four or five covariates, though some sectors may need up to ten; some sectors end up with only intercept).
  - Model parsimony:
    - No more than ten covariates of the original set of eighteen covariates (including up to four lags → p = 90) are useful for predicting median PD in each sector in λ-min specifications.
    - For many sectors fewer than six variables included.
  - Relaxed Lasso behavior:
    - Relaxed Lasso λ-min tends to retain all non-zero coefficients identified by Lasso.
    - Relaxed Lasso coefficients typically larger in magnitude than Lasso coefficients (reduced noise due to fewer covariates).
  - Standard errors:
    - Tables do not report standard errors; computing standard errors (e.g., via nonparametric bootstrap) is possible but the text argues there are strong arguments against calculating them because Lasso biases coefficients and bootstrapped confidence intervals may convey unjustified precision.
    - Suggested practice in low-dimensional problems: use Lasso for model selection only and estimate final forecasting model with traditional statistical techniques.
  - Implementation details referenced:
    - 10-fold cross-validation used.
    - λ-min and λ-1se selection rules described and illustrated in Figure 2 (panels for each sector).
    - Coordinate descent R implementation used.

### Conclusions (Section VIII)
- Lasso regressions likely outperform ordinary least squares in forecasting performance indicators required in applied stress testing due to their ability to handle high-dimensional problems where p is of same order as n.
- Regularization methods like Lasso are specifically designed to handle many high-dimensional issues present in finance, economics, and financial networks.
- Lasso-type estimators can be extended beyond linear models to generalized linear models (logistic, multiclass logistic), Cox proportional hazards, support vector machines, semiparametric nonlinear mixed effect models, and nonlinear regressions (references preserved as in source).
- Addressing interpretability concerns:
  - Lasso framework allows specifying upper and lower bounds or linear restrictions on coefficients to enforce sign or other constraints.
  - Lasso may select variables highly collinear with the "right" ones if they exhibit better signal-to-noise ratios, complicating economic narratives — but stress tests prioritize good out-of-sample forecasts.
- Practical implications:
  - Fast computation algorithms for Lasso in Matlab, R, SAS support use in semi- or fully automated stress testing platforms.
  - Example platform combining Lasso-based forecasting with balance sheet stress testing exists (Chan-Lau and Li, 2015 referenced in source).
  - Such platforms can analyze multiple stress scenarios efficiently within enterprise-wide or systemic risk frameworks.

*Source: wp17108 - Section VII illustrates the use of Lasso estimation in forecasting probabilities of default in an (IMF Working Paper content provided).*

### References

### References

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*Source: wp17108 - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17108.pdf_