In this section we derive some finitesample properties of the OLS estimator. b is an estimator of β. It is a function of the random sample data.


 Milo Cannon
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1 17 3. OLS Part III I this sectio we derive some fiitesample properties of the OLS estimator. 3.1 The Samplig Distributio of the OLS Estimator y = Xβ + ε ; ε ~ N[0, σ 2 I ] b = (X X) 1 X y = f(y) ε is radom y is radom b is radom b is a estimator of β. It is a fuctio of the radom sample data. b is a statistic. b has a probability distributio called its Samplig Distributio. Iterpretatio of samplig distributio Repeatedly draw all possible samples of size. Calculate values of b each time. Costruct relative frequecy distributio for the b values ad probability of occurrece. It is a hypothetical costruct. Why? Samplig distributio offers oe basis for aswerig the questio: How good is b as a estimator of β? Note: Quality of estimator is beig assessed i terms of performace i repeated samples. Tells us othig about quality of estimator for oe particular sample.
2 18 Let s explore some of the properties of the LS estimator, b, ad build up its samplig distributio. Itroduce some geeral results, ad apply them to our problem. Defiitio: A estimator, θ is a ubiased estimator of the parameter vector, θ, if E[θ ] = θ. That is, E[θ (y)] = θ. That is, θ (y)p(y θ)dy = θ. The quatity, B(θ, y) = E[θ (y) θ], is called the Bias of θ. Example: fiite variace, σ 2. {y 1, y 2,, y } is a radom sample from populatio with a fiite mea, μ, ad a Cosider the statistic y = 1 i=1 y i. i=1 ) The, E[y ] = E [ 1 y i=1 i] = 1 E(y i = 1 μ = i=1 ( 1 μ ) = μ. So, y is a ubiased estimator of the parameter, μ. Here, there are lots of possible ubiased estimators of μ. So, eed to cosider additioal characteristics of estimators to help choose. Retur to our LS problem b = (X X) 1 X y Recall either assume that X is oradom, or coditio o X. We ll assume X is oradom get same result if we coditio o X. The: E(b) = E[(X X) 1 X y] = (X X) 1 X E(y)
3 19 So, E(b) = (X X) 1 X E[Xβ + ε] = (X X) 1 X [Xβ + E(ε)] = (X X) 1 X [Xβ + 0] = (X X) 1 X Xβ = β. The LS estimator of β is Ubiased Defiitio: Ay estimator that is a liear fuctio of the radom sample data is called a Liear Estimator. Example: {y 1, y 2,, y } is a radom sample from populatio with a fiite mea, μ, ad a fiite variace, σ 2. Cosider the statistic y = 1 y i=1 i = 1 [y 1 + y y ]. This statistic is a liear estimator of μ. (Note that the weights are oradom.) Retur to our LS problem b = (X X) 1 X y = Ay (k 1) (k )( 1) Note that, uder our assumptios, A is a oradom matrix. So, b 1 a 11 a 1 y 1 ( ) = [ ] ( ). b k a k1 a k y For example, b 1 = [a 11 y 1 + a 12 y a 1 y ] ; etc.
4 20 The LS estimator, b, is a liear (& ubiased) estimator of β Now let s cosider the dispersio (variability) of b, as a estimator of β. Defiitio: Suppose we have a ( 1) radom vector, x. The the Covariace Matrix of x is defied as the ( ) matrix: V(x) = E[(x E(x))(x E(x)) ]. Diagoal elemets of V(x) are var. (x 1 ),., var. (x ). Offdiagoal elemets are covar. (x i, x j ) ; i, j = 1,, ; i j. Retur to our LS problem We have a (k 1) radom vector, b, ad we kow that E(b) = β. V(b) = E[(b E(b))(b E(b)) ] Now, b = (X X) 1 X y = (X X) 1 X (Xβ + ε) = (X X) 1 (X X)β + (X X) 1 X ε = Iβ + (X X) 1 X ε. So, (b β) = (X X) 1 X ε. [*] Usig the result, [*], i V(b), we have: V(b) = E{[(X X) 1 X ε][(x X) 1 X ε] } = (X X) 1 X E[εε ]X(X X) 1. We showed, earlier, that because E(ε) = 0, V(ε) = E(εε ) = σ 2 I. (What other assumptios did we use to get this result?) So, we have:
5 21 V(b) = (X X) 1 X E[εε ]X(X X) 1 = (X X) 1 X σ 2 IX(X X) 1 = σ 2 (X X) 1 (X X)(X X) 1 = σ 2 (X X) 1. V(b) = σ 2 (X X) 1 (k k) Iterpret diagoal ad offdiagoal elemets of this matrix. Fially, because the error term, ε is assumed to be Normally distributed, 1. y = Xβ + ε : this implies that y is also Normally distributed. (Why?) 2. b = (X X) 1 X y = Ay : this implies that b is also Normally distributed. So, we ow have the full Samplig Distributio of the LS estimator, b : b ~ N[β, σ 2 (X X) 1 ] Note: This result depeds o our various, rigid, assumptios about the various compoets of the regressio model. The Normal distributio here is a multivariate Normal distributio. (See hadout o Spherical Distributios.) As with estimatio of populatio mea, μ, i previous example, there are lots of other ubiased estimators of β i the model = Xβ + ε. How might we choose betwee these possibilities? Is liearity desirable? We eed to cosider other desirable properties that these ubiased estimators may have. Oe optio is to take accout of estimators' precisios.
6 The Efficiecy of OLS Defiitio: Suppose we have two ubiased estimators, θ 1 ad θ 2, of the (scalar) parameter, θ. The we say that θ 1 is at least as efficiet as θ 2 if var. ( θ 1 ) var. ( θ 2 ). Note: 1. The variace of a estimator is just the variace of its samplig distributio. 2. "Efficiecy" is a relative cocept. 3. What if there are 3 or more ubiased estimators beig compared? What if oe or more of the estimators beig compared is biased? I this case we ca take accout of both variace, ad ay bias, at the same time by usig "mea squared error" (MSE) of the estimators. Defiitio: Suppose we have two ubiased estimators, θ 1 ad θ 2, of the parameter vector, θ. The we say that θ 1 is at least as efficiet as θ 2 if Δ = V(θ 2 ) V( θ 1) is at least positive semidefiite. Takig accout of its liearity, ubiasedess, ad its precisio, i what sese is the LS estimator, b, of β optimal? Theorem (GaussMarkhov): I the "stadard" liear regressio model, y = Xβ + ε, the LS estimator, b, of β is Best Liear Ubiased (BLU). That is, it is Efficiet i the class of all liear ad ubiased estimators of β. 1. Is this a iterestig result? 2. What assumptios about the "stadard" model are we goig to exploit?
7 23 Proof Let b0 be ay other liear estimator of β: b 0 = Cy ; for some oradom C. (k 1) (k )( 1) Now, V(b 0 ) = CV(y)C = C(σ 2 I )C = σ 2 CC (k k) Defie: D = C (X X) 1 X so that Dy = Cy (X X) 1 X y = b 0 b. Now restrict b0 to be ubiased, so that E(b 0 ) = E(Cy) = CXβ = β. This requires that CX = I, which i tur implies that DX = [C (X X) 1 X ]X = CX I = 0 (ad D X = 0) (What assumptios have we used so far?) Now, focus o covariace matrix of b0 : V(b 0 ) = σ 2 [D + (X X) 1 X ][D + (X X) 1 X ] = σ 2 [DD + (X X) 1 X X(X X) 1 ] ; DX = 0 = σ 2 DD + σ 2 (X X) 1 = σ 2 DD + V(b), or, [V(b 0 ) V(b)] = σ 2 DD ; σ 2 > 0 Now we just have to "sig" this (matrix) differece: η (DD )η = (D η) (D η) = v 2 v = i=1 v i 0. So, Δ = [V(b 0 ) V(b)] is a p.s.d. matrix, implyig that b0 is relatively less efficiet tha b.
8 24 Result: The LS estimator is the Best Liear Ubiased estimator of β. What assumptios did we use, ad where? Were there ay stadard assumptios that we did't use? What does this suggest?
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