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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.

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# Conditional expectation and Least Squares Regression

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