Supervisor: Nour Meddahi
- Raffaella GIACOMINI, Professor, University College London
- Andrew J. PATTON, Professor, Duke University
- Francis DIEBOLD, Professor, University of Pennsylvania
- Christian GOURIEROUX, Professor, Toulouse School of Economics, UT1 Capitole
- Jihyun KIM, Assistant professor, Toulouse School of Economics, UT1 Capitole
- Nour Meddahi, Director of the TSE doctoral program, Toulouse School of Economics, UT1 Capitole
In this thesis, we investigate the various impacts of model misspecification and examine how to handle a model uncertainty. We analyze the impact of ignoring fat tails on an outcome of forecast comparison tests in the first chapter, and then study the effects of ignoring the dynamics of the risk premium of returns on the amount of capital requirements for banks in the second chapter. The third chapter provides a robust way to determine the capital requirements when facing a model uncertainty, that is, a lack of knowledge of the true data generating process.
In the first chapter, we analyze forecast comparison tests under fat tails. Forecast comparison tests are widely implemented to compare the performances of two or more competing forecasts. The critical value is often obtained by the classical central limit theorem (CLT) or by the stationary bootstrap (Politis and Romano, 1994) with regularity conditions, including the one where the second moment of the loss difference is bounded. However, the heavy-tailed nature of the financial variables can violate this moment condition. We show that if the moment condition is violated, the size of the test using the classical Normal asymptotics can be heavily distorted. The distortion is large especially when the tail of the marginal distribution of the loss differences is heavy. As an alternative approach, we propose to use a subsampling method (Politis, Romano, and Wolf, 1999) that is robust to fat tails. In the empirical study, we analyze several variance forecast tests by Hansen and Lunde (2006) and Bollerslev, Patton, and Quaedvlieg (2016). Examining several tail index estimators, we show that the second moment of the loss difference is likely to be unbounded especially when the popular squared error (SE) function is used as a loss function. We also find that the outcome of the tests may change if the subsampling is used.
The second chapter explores the effect of misspecification in the conditional mean dynamics on the determination of capital requirements for banks. In the Basel II accord (Basel Committee on Banking Supervision, 2010), the capital requirements for market risk are determined based upon a risk measure called Value-at-Risk (VaR). When VaR is computed, it is often assumed that the conditional mean of an asset return is constant over time. However, it is well documented that the predictability of returns increases as the prediction horizon becomes longer. This chapter assesses the impact of ignoring such possible predictability of returns on computing VaR. We calibrate models where the conditional mean of returns is persistent and its variance is much smaller than that of the i.i.d. noise, and then compare the term structure of VaR when the conditional mean of returns is assumed to be: first, time-varying; and second, not time-varying. Simulation studies show that the model with a time-varying conditional mean yields a less biased VaR, even though a constant conditional mean model is not rejected at the level of 5% for over 70% of replications. The contribution of this chapter is to demonstrate the problems of ignoring the conditional mean dynamics when we compute VaR. We find that even though the models with a constant and a time-varying conditional mean may be statistically indistinguishable, the implied VaR can differ. This finding then raises another question on how to produce VaR when we acknowledge the time-variability of the conditional mean but there is an uncertainty of its current value.
The third chapter puts forward a solution to the question raised in the second chapter by examining a robust way to determine the capital requirements when there is an uncertainty in the conditional mean of returns. We focus on Expected Shortfall (ES) rather than Value-at-Risk (VaR), since the capital reserves are now determined by ES in the Basel III accord (Basel Committee on Banking Supervision, 2019). We propose to determine the capital reserves based on the worst-case ES. That is, we choose the maximum value within a set of ES forecasts mapped from the set of models that are pre-selected by the forecaster. In our setting, the forecaster knows the risk-free rate that is assumed to be constant, yet the forecaster is uncertain of the current value of the risk premium. In this case, the set of models is defined by a set of values of the risk premium that the forecaster believes are plausible. With an assumption that the risk premium is believed to be non-negative, we show that the robust ES can in fact be achieved with a model in which the conditional mean is constant and the risk premium is always zero. This finding serves as an answer to the question raised in Chapter 2, and is one justification for assuming a constant conditional mean. We then consider a more general setting in which the forecaster is uncertain not only about the conditional mean but also about other aspects of the conditional distribution, such as the second or higher moments or the tails. There are many ways to define the set of models, and we focus on those defined with respect to the relative entropy, applying the robust control theory of Hansen and Sargent (2001).