By Halbert White and Karl Shell (Auth.)
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Then E(y \y . ) = O, since E{Vt\yt-x) = E(y \S _ \ where S _ = σ(2/,_,) satisfies S\_ C ft' _ and E(y \ft _ ) = 0 is measurable with respect to S' _ . t 2 t x t t t x χ t x t x x t t x t t t x x x PROPOSITION 3 . 6 5 : If 2/ is a random variable and Ζ is a random variable measurable with respect to S such that £]2/|<<» and E\Zy\ < ©ο, then with probability one III Laws of Large Numbers 54 E(zy\s) = ZE(y\s) and E([y - E(y\S)]Z) = 0. Proof: See Doob [ 1 9 5 3 , p. 2 2 ] . X,'£(y,|X,). EXAMPLE 3 .
Z i _ 1 , y f _ 2 r , ( Z i , y i _ 1 ) ) = X ^ o . If this is legitimate, we then have £(y,IS,-i) = X,A. Note that by definition, y, is measurable with respect to g,, so that {y,, 3 , } is an adapted stochastic sequence. ), g,} is a martingale difference sequence. If we let €, = y , - X , Ä , and it is true that E(yt\%t-X) = Χ,&, then et = yt- E{yt\%t_x\ so {€„ 3 , } is a martingale difference sequence.
1 Ζ, and fi„ = E(Zn). , the extent to which the distributions of the Z, may differ across /), and moments are sufficient to allow the conclusion Z„ — E(Zn) 0 to hold. As we shall see, there are sometimes trade-offs among these restrictions; for example, relaxing dependence or heterogeneity restrictions may require strengthening moment restrictions. III. ) random variables. d. random variables. Then Ζ Λ - ^ - μ if and only if £ ] Z , | < » and Ε(Ζ,)=μ THEOREM Proof: See Rao [1973, p. 115]. An interesting feature of this result is that the condition given is sufficient as^well as necessary for ZN μ.