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Conditional expectation and Least Squares Regression

  1. Jul 13, 2009 #1
    Hello everybody,

    I have two questions on conditional expectation w.r.t (Polynomial) OLS:
    Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how we can prove the following statements are true/false:

    (feel free to add assumptions)

    1. OLS( f(X_T), Vect_n{X_t} ) -> E( f(X_T) | F_t ), when n-> \infty
    2. Norm_L2{ E( f(X_T) | F_T ) - OLS( (X_T), Vect_n{X_T} ) } >= Norm_L2{ E( f(X_T) | F_t ) - OLS( (X_T), Vect_n{X_t} ) }

    For the first one, suppose X_t is Markov + Stone-Weierstrass + projection, we may have something interesting. But for the second one, I don't have any idea...

    Any help? Thx.
  2. jcsd
  3. Jul 14, 2009 #2
    Can anyone tell me if there is some (or no) relationship between the projection of f(X_T) on Vector space{X_t^0, X_t^1, ..., X_t^m} and the projection of f(X_T) on Vector space{X_T^0, X_T^1, ..., X_T^m}? Where X_t is a markov process.(or any other appropriate process).
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