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Lehman Brothers, the fourth largest investment bank on Wall Street, declared bankruptcy on the 15th of September 2008, becoming the largest bankruptcy in U.S. history. The investment house, which was founded in the mid-19th century, had become heavily involved in the U.S. housing bubble in the early 2000s, with its large holdings of toxic mortgage-backed securities (MBS) ultimately causing the bank's downfall. The bank had expanded rapidly following the repeal of the Glass-Steagall Act in 1999, which meant that investment banks could also engage in commercial banking activities. Lehman vertically integrated their mortgage business, buying smaller commercial enterprises that originated housing loans, which allowed the bank to expand its MBS holdings. The downfall of Lehman and the crash of '08 As the U.S. housing market began to slow down in 2006, the default rate on housing loans began to spike, triggering losses for Lehman from their MBS portfolio. Lehman's main competitor in mortgage financing, Bear Stearns, was bought by J.P. Morgan Chase in order to prevent bankruptcy in March 2008, leading investors and lenders to become increasingly concerned about the bank's financial health. As the bank relied on short-term funding on money markets in order to meet its obligations, the news of its huge losses in the third-quarter of 2008 further prevented it from funding itself on financial markets. By September, it was clear that without external assistance, the bank would fail. As its losses from credit default swaps mounted due to the deepening crash in the housing market, Lehman was forced to declare bankruptcy on September 15, as no buyer could be found to save the bank. The collapse of Lehman triggered panic in global financial markets, forcing the U.S. government to step in and bail-out the insurance giant AIG the next day on September 16. The effects of this financial crisis hit the non-financial economy hard, causing a global recession in 2009.

We introduce longitudinal factor analysis (LFA) to extract the common risk-free (CRF) rate from a sample of sovereign bonds of countries in a monetary union. Since LFA exploits the typically very large longitudinal dimension of bond data, it performs better than traditional factor analysis methods that rely on the much smaller cross-sectional dimension. European sovereign bond yields for the period 2006-2011 are decomposed into a CRF rate, a default risk premium and a liquidity risk premium. Our empirical findings suggest that investors chase both credit quality and liquidity, and that they price double default risk on credit default swaps.

Matrix-variate time series data are largely available in applications. However, no attempt has been made to study their conditional heteroscedasticity that is often observed in economic and financial data. To address this gap, we propose a novel matrix generalized autoregressive conditional heteroscedasticity (GARCH) model to capture the dynamics of conditional row and column covariance matrices of matrix time series. The key innovation of the matrix GARCH model is the use of a univariate GARCH specification for the trace of conditional row or column covariance matrix, which allows for the model identification. Moreover, we introduce a quasi-maximum likelihood estimator (QMLE) for model estimation and develop a portmanteau test for model diagnostic checking. Simulation studies are conducted to assess the finite-sample performance of the QMLE and portmanteau test. To handle large dimensional matrix time series, we also propose a matrix factor GARCH model, and establish its theoretical properties. Finally, we demonstrate the superiority of the matrix GARCH and matrix factor GARCH models over existing multivariate GARCH-type models in volatility forecasting and portfolio allocations using three applications on credit default swap prices, global stock sector indices, and future prices. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.