# 1 y 2 y t t y t b y t t y t t ² f y ² y y y t y t y

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± " 1/ y 2 @ y t   , t   ± @ y t   b y t   , t   ± @ y t   , t   ² f y   ² y y ± y t   ± @ y t   y t   ± @ y t

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9 6 y t   , t   ± ) " * y t   a y t   , t   ± 6 y t   , t   ² f y   ² y y ± y t   ² 1/2  ¡ @ y t   , t  ¢ 2 ² 2 f y   ² y 2 y ± y t   ± ) y t   " * " @ 2 2 y t   log y tj ' ± log y t , j " 1 ' ² ) y t , j " 1 ' " * " @ 2 2 y t , j " 1 ' ² u tj ' u tj ' L N 0, @ 2 2 y t , t " 1 ' goal: rewrite this to recognize as a problem for which we know p ) , * | @ 2 , y '   analytically log y tj ' ± log y t , j " 1 ' ² ) y t , j " 1 ' " * " @ 2 2 y t , j " 1 ' ² u tj ' u tj ' L N 0, @ 2 2 y t , t " 1 ' solution: multiply by y t , j " 1 '
10 y ´ tj ± ) x ´ tj ² * z ´ tj ² u tj y ´ tj ± y t , j " 1 ' log y tj ' " log y t , j " 1 ' ² @ 2 y t , j " 1 ' x ´ tj ± y t , j " 1 ' z ´ tj ± " y t , j " 1 ' u tj ± y t , j " 1 ' u tj ' L N 0, @ 2 2   y ± ± y ´ t 1 , y ´ t 2 ,..., y ´ Tm   U X ± ± x ´ 11 z ´ 11 B B x ´ TM z ´ TM * ± ) , *   U y ± ± X ± * ² . prior: * | @ 2 L N m , @ 2 M   posterior: * | @ " 2 , y ± , X ± L N m ' , @ 2 2 M '   M ' ± M " 1 ² X ± U X ±   " 1 m ' ± M ' M " 1 m ² X ± U y ±

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11 What about p @ 2 | ) , * , y '   ? This is not a standard problem, because @ 2 appears both as coefficient and variance. Chib et.al. draw this numerically
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• Winter '09
• JamesHamilton
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