Threshold models are set up so that there is a switch between regimes for the parameters of an unobserved components model. When Gaussianity is assumed, the model is handled by the Kalman filter. The switching depends on a component crossing a boundary, and, because the component is not observed directly, the error in its estimation leads naturally to a smooth transition mechanism. A prominent example motivating thresholds is that of a cyclical time series characterized by a downturn that is more, or less, rapid than the upturn. The situation is illustrated by fitting a model with three potentially asymmetric cycles, each with its own threshold, to observations on ice volume in Antarctica since 799,000 BCE. The model is able to produce multi-step forecasts with associated prediction intervals. A second example shows how a hidden threshold model is able to deal with the asymmetric cycle in monthly US unemployment.
Distributional shifts occur frequently, with detrimental impacts on forecast accuracy. Almost all econometric forecasting models are equilibrium correction so are susceptible to systematic forecast failure after equilibrium-mean shifts, which could be direct or induced. Unanticipated out-of-sample shifts are a well-known cause of forecast failure, but they later become in-sample shifts so require handling. Previous forecast-error taxonomies are extended to include shifts both in-sample and after the forecast origin. The taxonomy reveals which shifts do and do not lead to forecast failure facing both in-sample and post forecast-origin shifts, also highlighting what features are amenable to rapid correction.
A historically structured survey of panel data econometrics is provided, tracing its evolution from the error components models of the 1960s to the modern integration of causal inference, machine learning, and high-dimensional methods. The survey emphasizes the conceptual breakthroughs that shaped the field, including approaches to unobserved heterogeneity, the development of dynamic panel GMM estimators, nonlinear and limited dependent variable models, cross-sectional dependence, nonstationary panels, and recent advances in causal inference and policy evaluation.
For linear factor pricing models with high-dimensional assets, a novel Fisher combination test is proposed by combining the p-values of a maximum-type test and a sum-type test to assess whether a portfolio of candidate factors is mean–variance efficient. It is shown that the proposed combination test converges asymptotically to the chi-square distribution with 4 degrees of freedom without imposing a normality assumption. The proposed test has two main characteristics compared with existing methods. First, it combines the merits of both the sum-type and maximum-type tests, making it powerful under a broader set of alternatives than either test alone. This is because sum-type tests tend to have low power under sparse alternatives, while maximum-type tests are particularly powerful under sparse alternatives. Second, it is applicable to non-normal data. Hence, the proposed test is suitable for financial data, for which asset returns are unlikely to be normally distributed. The finite sample performance and a practical application of the proposed test are illustrated via simulation studies and data collected from the Standard & Poor’s 500 (S&P 500) index, respectively.
