11.3 Model Transformation

To appreciate the flexibility of the state-space model, we rewrite some well-known econometric and financial models in state-space form.

11.3.1 CAPM with Time-Varying Coefficients

First, consider the capital asset pricing model (CAPM) with time-varying intercept and slope. The model is

11.29 11.29

where rt is the excess return of an asset, rM, t is the excess return of the market, and the innovations {et, ηt, ϵt} are mutually independent. This CAPM allows for time-varying α and β that evolve as a random walk over time. We can easily rewrite the model as

inline

Thus, the time-varying CAPM is a special case of the state-space model with st = Tt = Rt, = It the 2 × 2 identity matrix, dt = 0, ct = 0, Zt, = (1,r,m,t), inline and inline. Furthermore, in the form of Eq. (11.28), we have εt = 0, ut = ηt,εt,et

inline

If diffuse initialization is used, then

inline

SsfPack/S-Plus Specification of Time-Varying ...

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