Economics 241B Hyothesis Testing: Large Samle Inference Statistical inference in large-samle theory is base on test statistics whose istributions are nown uner the truth of the null hyothesis. Derivation of these istributions is easier than in nite-samle theory because we are only concerne with the large-samle aroximation to the exact istribution. In what follows we assume that a consistent estimator of S exists, which we term ^S. Recall that S = E (g t gt), 0 where g t = X t U t. Testing Linear Hyotheses Consier testing a hyothesis regaring the -th coe cient. Proosition 2.1, which establishe the asymtotic istribution of the OLS estimator, imlies that uner H 0 : =, n B! N (0; Avar (B )) an \ Avar (B )! Avar (B ) : Here B is the -th element of the OLS estimator B an Avar (B ) is the (; ) element of the K K matrix Avar (B). The ey issue here is that we have not assume conitional homoseasticity, hence \ Avar (B ) = S 1 XX ^S S 1 XX ; which is the (heteroseasticity-consistent) robust asymtotic variance. the Slutsy result (Lemma 2.4c), the resultant robust t-ratio n B t q = B Avar \ SE (B ) (B )! N (0; 1) ; Uner q where the robust stanar error is SE 1 = Avar \(B n ). Note this robust t- ratio is istinct from the t-ratio introuce uner the nite-samle assumtions in earlier lectures. To test H 0 : =, simly follow these stes: Ste 1: Calculate the robust t-ratio Ste 2: Obtain the critical value from the N (0; 1) istribution
Ste 3: Reject the null hyothesis if jt j excees the critical value There are several i erences from the nite-samle test that relies on conitional homoseasticity. The stanar error is calculate in a i erent way, to accommoate conitional heteroseasticity. The normal istribution is use to obtain critical values, rather than the t istribution. The actual (or emirical) size of the test is not necessarily equal to the nominal size. The i erence between the actual size an the nominal size is the size istortion. Because the asymtotic istribution of the robust t ratio is stanar normal, the size istortion shrins to zero as the samle size goes to in nity. To summarize these results, together with the behavior of the Wal statistic let us brie y recall the assumtions require for Proosition 2.1: Assumtion 2.1 (linearity): Y t = X 0 t + U t (t = 1; : : : ; n; ) where X t is a K-imensional vector of regressors, is a K-imensional vector of coe cients an U t is the latent error. Assumtion 2.2 (ergoic stationarity): The (K + 1)-imensional vector stochastic rocess fy t ; X t g is jointly stationary an ergoic. Assumtion 2.3 (reetermine regressors): All regressors are reetermine, in the sense that they are orthogonal to the contemoraneous error: E (X t U t ) = 0 for all t an (= 1; 2; : : : ; K). This can be written as E (g t ) = 0 where g t X t U t : Assumtion 2.4 (ran conition): The K K matrix E (X t Xt) 0 is nonsingular (an hence nite). We enote this matrix by XX. Assumtion 2.5 (g t is a martingale i erence sequence with nite secon moments): fg t g is a martingale i erence sequence (so by e nition E (g t ) = 0). The K K matrix of cross moments, E (g t gt), 0 is nonsingular. Let S enote Avar (g) (the variance of the asymtotic istribution of P ng, where g = 1 n t g t).
By Assumtion 2.2 an the Ergoic Stationary Martingale Di erences CLT, S = E (g t g 0 t). Proosition 2.3 (robust t-ratio an Wal statistic): estimator ^S of S, if Assumtions 2.1 to 2.5 hol, then a) Uner the null hyothesis H 0 : = ; t! N (0; 1) Given a consistent b) Uner the null hyothesis H 0 : R = r, where R is a #r K matrix (where #r, the imension of r, is the number of restrictions) of full row ran, W n (Rb r) 0 n R \ Avar(B) R 0 o 1 (Rb r)! 2 (#r) : Proof: We have alreay establishe art a. Part b is a straightforwar alication of Lemma 2.4(). Write W as W = c 0 nq 1 n c n where c n = n (Rb r) an Q n = R \ Avar(B) R 0 : Uner H 0, c n = R n (b ), so Proosition 2.1 imlies Also by Proosition 2.1 c n! c where c N (0; RAvar (B) R 0 ) : Q n! Q where Q RAvar (B) R 0 : Because R is full row ran an Avar (B) is ositive e nite, Q is invertible. Therefore, Lemma 2.4() imlies W! c 0 Q 1 c: Because c is normally istribute with imension #r, an because Q equals the variance of c, c 0 Q 1 c = 2 (#r). QED The statistic W is a Wal statistic because it is constructe from unrestricte estimators (B an Avar \ (B)) that are not constraine by the null hyothesis. To test H 0 :: R = r, simly follow these stes: Ste 1: Construct W:
Ste 2: To n the critical value for a test with size 5%, n the oint of the 2 (#r) istribution that gives 5% to the uer tail. Ste 3: If W excees the critical value, then reject the null hyothesis. Consistent Test The ower of a test is the robability of rejecting a false null hyothesis. The ower of a test (for a given size) eens on the alternative DGP. For examle, consier any DGP fy t ; X t g that satis es Assumtions 2.1-2.5 but for which 6=. The ower of a test base on the t ratio is ower = Pr (jt j > cv ) ; where cv is the critical value associate with a size of. Because the DGP controls the istribution of t, the ower eens uon the DGP. The test is consistent against a set of DGP s (none of which satisfy the null) if the ower against any member of the set of DGP s aroaches unity as n! 1 (for any assume signi cance level). To see that the test base on the t ratio is consistent, note that for n B t = q Avar \(B ) the enominator converges to Avar (B ) esite the fact that DGP oes not satisfy the null (Proosition 2.1 requires only Assumtions 2.1-2.5 an oes not een on the truth of the null hyothesis). In contrast, the numerator tens to either +1 or 1, because B converges to the value of for the DGP, which i ers from. Hence the ower tens to unity as the samle size tens to in nity, imlying that the test is consistent for all members of the set of DGP s. The same is true for the Wal test. Asymtotic Power The ower of the t test aroaches unity as the samle size increases for any xe alternative DGP. If, however, the DGP is not hel xe but allowe to get closer an closer to the null hyothesis as the samle size increases, then the ower may not converge to unity. A sequence of such DGP s is calle a sequence of local alternatives. For the regression moel with the null of H 0 : =, a sequence of n local alternatives is a sequence of DGP s such that (i) the n-th DGP Y (n) t ; X (n) t (t = 1; 2; : : :) satis es Assumtions 2.1-2.5 an converges in a certain sense to a o
xe DGP fy t ; X t g, an (ii) the value of for the n-th DGP, (n), converges to. Suose, further, that (n) satis es (n) = + n for some given 6= 0. Such a sequence of local alternatives, in which (n) aroaches at the rate of 1= n, is calle a Pitman rift or Pitman sequence. Uner this sequence of local alternatives, the t ratio is n B (n) t = q + q : Avar \(B ) Avar \(B ) If a samle of size n is generate by the n-th DGP of a Pitman sequence, oes t converge to a nontrivial istribution? Because the n-th DGP satis es Assumtions 2.1-2.5, the rst term on the right sie converges in istribution to N (0; 1) by Proosition 2.3(a). By art (c) of Proosition 2.1 ( [Avar! Avar) an the n o fact that Y (n) t ; X (n) t "converges" to a xe DGP, the secon term converges in robability to := Avar (B ) where Avar (B ) is evaluate at the xe DGP. Therefore, t! N (; 1) along this sequence of local alternatives. If the signi cance level is, the ower converges to Pr (jxj > cv) ; where cv is the critical value (i.e. 1.96) an x N (; 1). This robability is terme the asymtotic ower. Eviently, the larger is jj the higher is the asymtotic ower for any given size. By a similar argument, the Wal statistic converges to a non-central chi-square istribution along a Pitman sequence. Testing Nonlinear Hyotheses The Wal test can be generalize to a test of nonlinear restrictions on. Consier a null hyothesis of the form H 0 : a () = 0:
Here a is a vector-value function with continuous rst erivatives. Let #a be the imension of a (so the null hyothesis has #a restrictions) an let A () be the #a K matrix of rst erivatives evaluate at : A () = @a () =@ 0. For the hyothesis to be well e ne, we assume that A () is full row ran (this is the generalization of the requirement for linear hyotheses R = r that R is of full row ran). By Lemma 2.5 (the elta metho) we have n (a (B) a ())! c; c N 0; A () Avar (B) A () 0 : Because B!, Lemma 2.3(a) imlies that A (B)! A (). Because (Proosition 2.1(c)) \ Avar (B)! Avar (B), Lemma 2.3(a) imlies A (B) \ Avar (B)A (B) 0! A () Avar (B) A () 0 = V ar (c) : Because A (B) has full row ran an Avar (B) is ositive e nite, V ar (c) is invertible. Uon noting that, uner the null hyothesis a () = 0, Lemma 2.4() imlies na (B) 0 n A (B) \ Avar (B)A (B) 0 o 1 na (B)! c 0 V ar (c) 1 c 2 (#a) : If we combine the two n s into n we have roven Proosition 2.3 (continue - nonlinear hyotheses): c) Uner the null hyothesis H 0 : a () = 0 (which contains #a restrictions) if A () (the #a K matrix of continuous rst erivatives of a ()) is of full row ran, then W n a (B) 0 n A (B) \ Avar (B)A (B) 0 o 1 a (B)! 2 (#a) : Part (c) is a generalization of art (b), if we set a (B) = RB r, then W reuces to the Wal statistic for linear restrictions. For a given set of restrictions, a () may not be unique. For examle, the restriction 1 2 = 1 coul be written as a () = 0 with a () = 1 2 1 or with a () = 1 1= 2. While art c of the Proosition guarantees that the outcome of the Wal test is the same (for either exression of the restriction) in large samles, the numerical value of W oes een on the way the restriction
is written. Hence, the outcome of the test can een on the exression in nite samles. In this examle, the reresentation a () = 1 1= 2 oes not satisfy the requirement of continuous erivatives at 2 = 0. Inee, a 1985 Monte Carlo stuy by Gregory an Veal reorts that, when 2 is close to zero, the Wal test base on the secon reresentation rejects the null too often in small samles. (Actual size excees nominal size.)