unconditional moments

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    pt  Et (mt 1 xt 1 )
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    Et 1 ( Et ( xt 1 ))  Et 1 ( xt 1 )
    E[ E ( x | ) | I  ]  E[ x | I ]       （ ）
    E ( Et ( x ))  E ( x )
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    pt  Et (mt 1 xt 1 )     pt  E[ mt 1 xt 1 | I t ]

                       E ( pt )  E (mt 1 xt 1 )           （ ）
•   pt  I t
               pt  E[mt 1 xt 1 | ]
                E[ pt | I  ]  E[mt 1 xt 1 | I  ]
                pt  E[mt 1 xt 1 | I t  t ]
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      pt zt  Et (mt 1 xt 1 zt )      (3)

    E ( pt zt )  E (mt 1 xt 1 zt )   (4)
    E ( pt zt )  E (mt 1 xt 1 zt )
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    E[(mt 1 xt 1  pt ) zt ]  0

     E[(mt 1 xt 1  pt ) | I t ]  0   (5)
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    E ( pt )  E (mt 1 xt 1 )xt 1  X t 1
                                                 (6)
     pt  E (mt 1 xt 1 | It )
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    zt
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    mt 1  1/ RtW1
                 


                      pt  Et (mt 1 xt 1 )  E ( pt )  E (mt 1 xt 1 )
Conditional vs. Unconditional
Factor Models in Discount Factor
Language
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                                                                   (7)



                                                              1
                                                          a  f  bEt ( RtW1 )  
                           
    1  Et (mt 1 RtW1 ) 1  Et [(a  bRtW1 ) RtW1 ] 
                                                   
                                                              Rt
                                                                                (8)
    
    1  Et (mt 1 ) Rt f
                           
                           1  Et (a  bRt 1 ) Rt
                                          W         f
                                                         b  Et ( RtW1 )  Rt f
                                                                      
                                                         
                                                              Rt f  t2 ( RtW1 )
                                                                             
•            Et ( RtW1 )  t ( Rt 1 )
                    
                             2    W
                                         Rt f

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    mt 1  at  bt RtW1
                      
                                                (9)
• This fact means that we can no longer transparently
  condition down. That is:
               1  Et [( at  bt RtW1 ) Rt 1 ]
                                               (10)
• does not imply that we can find constants a and b so
  that

               1  E[( a  bRtW1 ) Rt 1 ]
                                              (11)
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    1  E[(at  bt RtW1 ) Rt 1 ]  E[at Rt 1  bt RtW1Rt 1 ]
                                                     

     E (at ) E ( Rt 1 )  E (bt ) E ( RtW1Rt 1 )  cov(at , Rt 1 )  cov(bt , RtW1Rt 1 )
                                                                                   

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              1  E[( E (at )  E (bt ) RtW1 ) Rt 1 ]
                                          

                E[ E (at ) Rt 1  E (bt ) RtW1Rt 1 ]
                                              

                E (at ) E ( Rt 1 )  E (bt ) E ( RtW1 Rt 1 )
                                                     
         1                       Et ( RtW1 )  Rt f
    at  f  bEt ( Rt 1 ), bt 
                    W                   

        Rt                        Rt  t2 ( RtW1 )
                                     f
                                              



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    1  Et [(a  bRtW1 ) Rt 1 ]  1  E[(a  bRtW1 ) Rt 1 ]
                                                
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                  Et ( Rti1 )  Rt f   ti t       (14)
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                  E ( Rti1 )     i              (15)
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    E ( Rti1 )  E ( Rt f   ti t )                     (16)
     E ( Rt f )  E (  ti ) E (t )  cov(  ti , t )
A Precise Statement
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                                                        mt 1  at  bt' ft 1
           pt  Et (mt 1 xt 1 )        xt 1  X
                                                            mt 1  a  b ' ft 1
           E ( pt )  E (mt 1 xt 1 )      xt 1  X
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Mean-Variance Frontiers
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    

    

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•          mt 1  at＋bt RtW1
                           

    RtW1
      


              mt 1  a＋bRtW1
                           

    RtW1
      
Using the Orthogonal
Decomposition
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          R   mv
                    R R
                     *       e*     (17)


          Rtmv  Rt*1  t Rte1
            1
                               *
                                    (18)


          Rtmv  Rt*1   Rte1
            1
                              *
                                     (19)
    续

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Brute Force and Examples

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         min E ( R 2 ) s.t.E ( R)             (20)
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       min E[ Et ( R 2 )]s.t.E[ Et ( R )]     (21)
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    R f  R*  R f Re*   (22)
                                R*   Re*
Implications: Hansen-Richard
Critique
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8.4 Scaled Factors: a Partial
Solution
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                    z t2
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    mt 1  a( zt )  b( zt ) f t 1
     a0  a1 zt  (b0  b1 zt ) f t 1         (24)
     a0  a1 zt  b0 ft 1  b1 ( zt ft 1 )
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    pt  Et [(a0  a1 zt  b0 ft 1  b1 ( zt f t 1 )) xt 1 ]
                                                                       (25)
     E ( pt )  E[(a0  a1 zt  b0 ft 1  b1 ( zt ft 1 )) xt 1 ]
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    mt 1  at  bt RtW1
                      
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Kronecker Product
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    m  b1 f1  b2 f1 z1  b3 f1 z2     bN 1 f 2  bN  2 f 2 z1 




                                                               (26)
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