Leyes de control óptimas

We have proposed a network model linking the volatility regime transitions of institutions, capturing both exposure to and responsibility for the spread of volatility contagion. Em- pirical analysis shows that the network model is a statistically significant improvement over the regime switching model, which is evidence for the existence of volatility contagion in the network. Focusing on the financial sector, we find that patterns of linkages between insti- tutions are dynamic, with increases prior to the subprime mortgage crisis and connections across industries becoming more common after the crisis. Out-of-sample forecasts show that firms that are more susceptible to contagion undergo significantly more losses during sys- temic events, and firms that spread contagion experience significantly lower losses. Overall, we find that the network model captures certain features of systemic risk exposure for the market and for individual institutions for a specific crisis period. The degree and structure of connectivity have dynamic properties that reflect historically observed patterns of systemic risk.

There are several limitations to this study. Most importantly, the model does not reflect any real economic mechanisms explaining linkages between institutions in the marketplace. Because the study is descriptive, we cannot make any causal conclusions with respect to inferred linkages or their relationship to systemic risk exposure. We have addressed this concern by conducting out-of-sample tests illustrating the predictive power of linkages for forecasting crisis period losses of institutions. Nonetheless, more informative network mod- els can be constructed by incorporating counterparty exposure, marketplace transactions between institutions, and other real-time financial information. This information is not pub- licly available, but regulatory organizations have access to this type of data and the mandate

to evaluate and restrict systemic risk in the marketplace.

We motivated the model in reference to the four “L”s of systemic risk. Our model accounts for losses, through the high volatility crisis regime, and linkages, through the net- work. The model is limited by its failure to account for leverage and liquidity of institutions, although we have shown that leverage effects are captured indirectly by the degree of connec- tivity. An extension of our model to account for additional elements of systemic risk exposure would be to introduce covariates in the network model. For example, firms that have less liquid portfolios may be more likely to make connections in the marketplace. Incorporat- ing covariates into the model is straightforward: the log-odds of a connection would simply be linear in these covariates, in addition to the latent distance of institutions. Covariates in the network model may also include financial statement information or data regarding counterparties.

The regime switching component of our model accounts for dynamics of the returns. However, the network structure itself is static over time. As we have discussed throughout the chapter, the nature, type, and number of relationships between institutions, is constantly evolving, especially in the financial sector. Our empirical analysis has avoided this issue to the extent possible by estimating the network on relatively short time windows (four years in the case of the financial sector study). Daily data would allow for even shorter windows. A more comprehensive solution would be to allow for certain elements of the network to also be dynamic over time. This is potentially an area of future research.

In the post-crisis economy, banks continue to be subjected to scrutiny for their contribu- tions and exposure to systemic risk. Regulatory and reporting requirements have increased following the passage of Dodd-Frank, and the Federal Reserve administers stress tests to de- termine whether banks have enough capital to survive a severe economic crisis. Despite the limitations of our study, the results confirm the need for ongoing evaluation of systemic risk, especially in the financial sector. Connections between institutions across industries have increased in the post-crisis economy, relative to the number of connections within industries.

Given the degree of connectivity observed in our model, we would speculate that the risk of volatility contagion and systemic risk remains as great today as in the pre-crisis economy.

Chapter 2

Benchmarking Bayesian Small Area

Estimates

2.1

Introduction

In many surveys, it is of interest to estimate a parameter at different geographic levels, such as at the county, state, and national level. For example, the Small Area Health Insurance Estimates (SAHIE) program at the United States Census Bureau produces estimates of the count of uninsured individuals within states and counties, cross-classified by age, gender, income, and additionally by race for state estimates. The American Community Survey (ACS) is the primary source of data used for the project. Because the ACS is a large-scale survey with over three million households sampled annually, the direct survey estimates of the number of uninsured individuals at the national or state levels are typically very reliable. However, sample sizes in some domains may not be large enough to produce reliable direct estimates for specific small areas. Here, small area refers to any subpopulation of interest, such as a geographic area or socio-demographic group, for which the domain-specific sample is not large enough for reliable direct estimation. In the case of SAHIE, consider the difficulty of estimating the number of uninsured men between the ages of 40 and 64 with income at

or below 138% of the poverty level in Lawrence County, South Dakota, which has a county population size just over 20,000 and an ACS sample size within the county of only a few hundred households. (In 2009 SAHIE estimated that 40.0% of such residents in Lawrence County were uninsured, with standard error of 8.6%).

For sample surveys, precision for estimates within small areas can be increased by intro- ducing a small area model (Rao, 2003), which borrows strength by connecting different areas and incorporating auxiliary covariate information. When multiple geographic levels are of interest, conceptually, it makes sense to implement a joint model for the higher and lower levels. From a practical standpoint, this is nearly impossible due to the complexity of the models, the number of parameters to be estimated, and the fact the available data and pa- rameters of interest are not necessarily nested from one level to the next. For this reason, the small area modeling is done separately at distinct geographic levels. These model-based es- timates can differ substantially from direct estimates, especially for areas with small sample sizes. An important consequence is that the model-based estimates for the smaller geo- graphic areas typically do not aggregate to the corresponding direct survey or model-based estimate of a larger geographic area. There are several reasons why this inconsistency is undesirable for survey administrators: government funds may be appropriated on the basis of model estimates (e.g. to counties within states), reporting agencies’ credibility with the general public or legislators may be impacted, and the degree of inconsistency may be a reflection of model misspecification. Benchmarking is the statistical procedure which adjusts model-based estimates to satisfy constraints.

Existing benchmarking procedures have been developed from a frequentist perspective primarily for application to linear random effects models. Random effects models are closely related to the classical small area model introduced by Fay and Herriot (1979). Wang et al. (2008) review benchmarking procedures for random effects models, providing simulation re- sults and a mathematical formulation which unifies several different procedures as minimizers of weighted Mean Square Error (MSE). For more general Bayesian small area models, Datta

et al. (2011) extend the results of Wang et al. (2008) to produce a Bayes rule benchmarked estimator, and Toto and Nandram (2010) illustrate Bayesian benchmarking with sampling weights. Nandram et al. (2011) develop a Bayesian benchmarking procedure applicable in the case of the one-way random effects model.

The approaches described in this chapter stand in contrast to existing methods. Rather than obtaining point estimates by minimizing a loss function subject to a benchmark con- straint for linear random effects models, general Bayesian small area models are benchmarked by modifying the posterior distributions themselves at the different geographic levels. This approach has the advantage of guaranteeing that further characteristics, such as the esti- mated variances at each level of the model also agree.

Additionally, this chapter aims to develop a new perspective for benchmarking estimates which considers a flexible benchmarking constraint. Previously existing benchmarking pro- cedures adjust lower geographic level estimates to be consistent with a higher geographic level estimate, which is held fixed. Because there is often uncertainty associated with the higher level estimate, it is possible that inference can be improved by allowing the higher and lower level estimates to be adjusted simultaneously in the benchmarking procedure. This flexible constraint allows the lower level estimates to inform the adjustment of the estimate of the higher level total, and vice versa. With a fixed constraint, there is no mechanism for the lower level estimates to influence the higher level estimate. In this way, the flexible constraint introduces new information into the benchmarking procedure, creating potential for improved performance.

This chapter illustrates two new benchmarking procedures for modifying the posterior distribution for general Bayesian small area models, either by explicitly describing the form of the modified posterior distributions or through approximations based on MCMC output. It is shown that in both procedures, either a fixed or a flexible benchmarking constraint can be used. Section 2.2 provides background on existing benchmarking results and discusses the links between these procedures. Section 2.3 introduces the new benchmarking methods

for Bayesian small area models with either a flexible or a fixed constraint. In Section 2.4, a simulation study is presented comparing the two methods to each other and to existing benchmarking procedures. Concluding remarks are given in Section 2.5.