Suppose X, Y are real-valued random vectors. When can we say E [ g ( X , Y ) | X ] is an a.e. continuous function in X? I have mostly convinced myself of this argument, save for one step where I want to argue E [ 1 n ∑ i = 1 n X i X i ′ e i 2 | ( X 1 , . . . , X n ) ′ ] → p E [ E [ X i X i ′ e i 2 ] | ( X 1 , . . . , X n ) ′ ] = E [ X i X i ′ e i 2 ] by appealing to the weak law of large numbers and the continuous mapping theorem.

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Suppose   X,  Y are real-valued random vectors. When can we say   E  [  g  (  X  ,  Y  )      |    X  ] is an a.e. continuous function in   X? I have mostly convinced myself of this argument, save for one step where I want to argue   E      [          1      n              ∑              i        =        1                    n                    X              i                    X              i            ′              e              i                    2                    |        (          X      1        ,    .    .    .    ,          X      n              )      ′        ]        →    p    E      [    E          [              X                  i                            X                  i                ′                    e                  i                          2                    ]              |        (          X      1        ,    .    .    .    ,          X      n              )      ′        ]    =  E      [          X              i                    X              i            ′              e              i                    2              ]   by appealing to the weak law of large numbers and the continuous mapping theorem.

Donavan Scott · Accepted Answer

The error is       e    i    =      Y    i    −      X    i    ′    β and model assumptions include (      X    i    ,      Y    i  ) iid and finite fourth moments of       X    i    ,      Y    i  . Then the claim can be shown without appealing to the continuous mapping theorem since  E      [          1      n              ∑              i        =        1                    n                    X              i                    X              i            ′              e              i                    2                    |        (          X      1        ,    .    .    .    ,          X      n              )      ′        ]    =      1    n        ∑          i      =      1              n            X          i            X          i        ′    E      [          e              i                    2                    |        (          X      1        ,    .    .    .    ,          X      n              )      ′        ]      =      1    n        ∑          i      =      1              n            X          i            X          i        ′    E      [          e              i                    2                    |              X      i        ]      (  (      X    i    ,      Y    i    )   iid  )        →    p    E      [          X              i                    X              i            ′        E          [              e                  i                          2                            |                    X        i            ]        ]      (  WLLN  )    =  E      [    E          [              X                  i                            X                  i                ′                    e                  i                          2                            |                    X        i            ]        ]      =  E      [          X              i                    X              i            ′              e              i                    2              ]      (  LIE  )

Suppose X , Y are real-valued random vectors. When can we say E [ g ( X

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