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Maximizing Equity Market Sector Predictability in a Bayesian Time Varying Parameter Model* - page 6 / 46

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  • t |t 1

is the variance of the prediction error and Q is the diagonal matrix of the variances of the

s h o c k s t o t h e f a c t o r l o a d i n g s , 2 k v , f o r k = 1 t o K . I f ) , ( 2 Q t S u a r e k n o w n , w e c a n m a k e

i n f e r e n c e s a b o u t t h e b e h a v i o r o f t h e s t a t e v e c t o r . I f u n k n o w n , ) , ( 2 Q t S u c a n b e e s t i m a t e d u s i n g

Bayesian inference as described in Carter and Kohn (1994) and Kim and Nelson (1999). Bayesian parameter estimation and model selection criteria are described in greater detail in Section 3.

While some conditional pricing models use lagged macroeconomic variables to model variation in economic risk premia on sorted portfolio attributes or the market portfolio, we are interested in investigating a related but different question. Given the strong evidence for the importance of lagged economic information in pricing equities, we would like to measure time variation in equity risk sensitivities to macroeconomic factors directly.

The case for persistent time varying second moments in returns on equity market portfolios has been discussed in French, Schwert and Stambaugh (1987) and Schwert and Seguin (1990). To address this property of equity returns, a Markov-switching process in the variance of the portfolio return error terms is also provided for in the model. Attention is limited to two discrete states over a state variable, St , where a high variance state exists when St 1and a low variance

state prevails when St 0. The error term in (1) follows the distribution:

ut 2 u 2 u1 S t ~ N(0,u 2 0 1 ( 2 0 u . 2 u S t

) St )

2 u 1

St

(4)

As such, ut will be heteroskedastic with conditional variance determined by the unobserved state

variable St . The state variable St evolves based on the following transition probabilities:

P(St 1| St1

1) p

P(St 0 | St1

1) 1p

P(St 0 | St1

0) q

P(St 1| St1

0) 1q.

5

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