• Abstract: We provide an econometric analysis for the factor models when the latent factors can be explained partially by several observed explanatory proxies. In financial factor models for instance, the unknown factors can be reasonably well predicted by a few observable proxies, such as the Fama-French factors. In diffusion index forecasts, identified factors are strongly related to several directly measurable economic variables such as consumption-wealth variable, financial ratios, and term spread. To incorporate the explanatory power of these observed characteristics, we propose a new two-step estimation procedure: (i) regress the data onto the observables, and (ii) take the principal components of the fitted data to estimate the loadings and factors. The proposed estimator is robust to possibly heavy-tailed distributions, which are encountered by many macroeconomic and financial time series. With those proxies, the factors can be estimated accurately even if the cross-sectional dimension is mild. Empirically, we apply the model to forecast US bond risk premia, and find that the observed macroeconomic characteristics contain strong explanatory powers of the factors. The gain of forecast is more substantial when these characteristics are incorporated to estimate the common factors than directly used for forecasts.

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