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Maximizing Equity Market Sector Predictability in a Bayesian Time Varying Parameter Model* - page 5 / 46

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ui,t ~ N (0,

2 ui,S

t

),

w h e r e f o r e a c h p o r t f o l i o r e t u r n 1 , t i r , 1 t i s a K x 1 v e c t o r o f K - 1 t i m e v a r y i n g f a c t o r l o a d i n g s

a n d a t i m e v a r y i n g i n t e r c e p t t e r m , t f i s a K x 1 v e c t o r o f u n i t y a n d K - 1 l a g g e d m a c r o e c o n o m i c

factors and ui,t1 is a normally distributed disturbance term with conditional variance to allow

for heteroskedasticity. From here forward in this section, for simplified notation, the portfolio subscript i is dropped.

We incorporate time variation in the factor sensitivities by modeling each factor sensitivity, or beta, as a random walk such that each can be expressed as

k ,t k ,t1 vk ,t ) , 0 ( ~ 2 , k v t k N v .

(2)

Assuming the variance parameters are known, the evolution of risk factor sensitivities can be estimated in state space form. Prediction and updating for the state vector of factor loadings using the Kalman filter proceeds as follows:

t|t 1

t1|t1

Pt|t1

Pt1|t1 Q

t |t 1

t r ft1t|t1

t |t 1

ft1Pt|t1 ft1

Prediction:

2 uS

t

(3)

t|t t|t1

t K t|t1

t|t P Pt|t1

Kt ft1Pt|t1

Updating:

where

Kt

Pt|t1

1 | 1 1 t t t f

is

the

Kalman

gain.

In

(3),

  • t|t 1

is the expectation of t

given

information up to time t-1, Pt|t1

is the covariance matrix of t|t1

, t|t1

is the prediction error,

4

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