Gaussian models for Euro high grade government yields

Abstract

This paper tests a¢ ne, quadratic and Black-type Gaussian models on Euro area triple A Government bond yields for maturities up to 30 years. Quadratic Gaussian models beat a¢ ne Gaussian models both in sample and out of sample. A Black-type model best …ts the shortest maturities and the extremely low yields since 2013, but worst …ts the longest maturities. Even for quadratic models we can infer the latent factors from some yields observed without errors, which makes quasi maximum likelihood (QML) estimation feasible. New speci…cations of quadratic models …t the longest maturities better than does the "classic" speci…cation of Ahn-Dittmar-Gallant (2002), but the opposite is true for the shortest maturities. These new speci…cations are more suitable to QML estimation. Overall quadratic models seem preferable to a¢ ne Gaussian models, because of superior empirical performance, and to Black-type models, because of superior tractability. This paper also proposes the vertical method of lines (MOL) to solve numerically partial di¤erential equations (PDE's) for pricing bonds under multiple non-independent stochastic factors. "Splitting" the PDE drastically reduces computations. Vertical MOL can be considerably faster and more accurate than …nite di¤erence methods.

A Shadow Rate or a Quadratic Policy Rule? The Best Way to Enforce the Zero Lower Bound in the United States

“…Graph A. ATSM: 1990-2006Graph B. ATSM: 2009-2016 Graph C. SRM: 1990-2007Graph D. SRM: 2009-2016 Graph E. QTSM: 1990-2007Graph F. QTSM: 1990-2016 3-factor 4-factor 5-factor Data 95% Confidence Interval (data) ending in 2007. The heavy black lines indicate the estimated Campbell-Shiller loadings (φ j ) in the data.…”

“…In addition, we require µ = 0 in the ATSM and SRM, whereas it is sufficient in the QTSM to restrict µ ≥ 0 (as shown by Realdon (2016)). The parameters in h 0 and h x are unrestricted in all 3 models.…”

“…Graph A. ATSM: 1990-2007Graph B. ATSM: 1990-2016 Graph C. SRM: 1990-2007Graph D. SRM: 1990-2016 Graph E. QTSM: 1990-2007Graph F. QTSM: 1990-2016 3-Factor 4-Factor 5-Factor 95% Confidence Interval (3-factor) While the point estimates from the 3-factor QTSM are slightly above 1, the desired value nevertheless falls within the confidence interval at all maturities. We therefore conclude that all of the 3-factor models satisfy the LPY(ii) test when they are estimated on a sample of yields that is away from the ZLB.…”

“…Realdon (2016) studies ZLB-consistent models estimated using euro-area sovereign yields, which may provide a more relevant comparison with U.S. yields. However, the version of the SRM he estimates di¤ers from the speci…cation considered previously in the literature.…”

A Shadow Rate or a Quadratic Policy Rule? The Best Way to Enforce the Zero Lower Bound in the United States

“…3 In addition, we require = 0 in the ATSM and SRM, whereas it is su¢ cient in the QTSM to restrict 0 (as shown by Realdon (2016)). The parameters in h 0 and h x are unrestricted in all three models.…”