LECTURE 13 SPURIOUS REGRESSION, TESTING FOR UNIT ROOT = C (1) C (1) 0! ! uv! 2 v. t=1 X2 t

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1 APRIL 9, 7 Sprios regressio LECTURE 3 SPURIOUS REGRESSION, TESTING FOR UNIT ROOT I this sectio, we cosider the sitatio whe is oe it root process, say Y t is regressed agaist aother it root process, say t while the two processes are related. Assme that t t + t Y t Y t + v t Y t C (L) " t v t f" t g is iid, E" t E" t " t a positive de ite matrix P j j kc j k < C () is o-siglar. We ca also set the iitial vales of ad Y to be di eret from zero o p ad Y o p Let C () C () v v v where ad v are the log-r variaces of t ad v t respectively, ad v is the log-r covariace betwee t ad v t Let b be the OLS regressio coe ciet P b P ty t t Here we cosider the regressio withot a itercept, however, essetially the same reslts ca be obtaied for the regressio with a itercept. Let W (r) [r] W Y (r) Y [r] From the FCLT we have W (r) W Y (r) ) B (r) B Y (r) W (r) W Y (r) where W (r) ad W Y (r) are two idepedet stadard Browia motios. Notice that covergece is joit, which is importat for all sbseqet reslts.

2 Cosider rst 3 t t W + W + W + + W The fctio W (r) is cadlag, costat o the iterval (t Z t t W so that Hece, W W 3 where the last reslt is by the CMT. Next, Now, cosider 3 t t 3 (t ) Z t (t ) Z Z t ( ) (t ) W (r) dr d B (r) dr ) r < t ad, therefore, t drw W (r) dr W (r) dr W (r) dr ( t + t ) W (r) dr W (r) dr 3 t t + o p () d B (r) dr t ( t + t ) t + t t + t t t + t t + o p ()

3 First, Next, ad where E t. Therefore, ad Next, Next, cosider P ty t First, t t W (r) dr d B (r) dr t t t t t t t t t t t W () d B () W (), t t o p () t t d B (r) dr () t Y t t Y t W (r) W Y (r) dr d B (r) B Y (r) dr t Y t ( t + t ) (Y t + v t ) t Y t + t v t + Y t t + t v t Now, P tv t p v where v E t v t, ad, therefore, t v t o p () 3

4 Frther, Similarly, Ths, v Y t t t v t o p () t Y t Y t t v t o p () t Y t t Y t + o p () d B (r) B Y (r) dr () The covergece i distribtio reslts i () ad () are joit, ad it follows that b B (r) B Y (r) dr d B (r) dr The reslt holds eve if f t g ad fv t g are idepedet. Oe cold expect that b wold coverge i probability to zero, however, it coverges i distribtio to a radom variable ad, therefore, is icosistet. The radom variable ca be iterpreted as a regressio coe ciet from the "poplatio" or "cotios time" regressio of the Browia motio B Y agaist B Next, cosider the sal t-statistic for H t b b s P t where s is the sample variace of the tted residals. ad, therefore, s ( ) Y t b t ( ) Y t Y t s d BY (r) dr + b b t t b t Y t t Y t (B Y (r) B (r)) dr B (r) B Y (r) dr 4

5 Lastly, t b b s P d t (B Y (r) B (r)) dr B (r) dr We coclde that Hece, as for ay K > P t b O p tb > K ad the ecoometricia will reject H with the probability approachig. This is despite the fact that the two variables ad Y ca be idepedet. Testig for it root Sppose that the scalar process f t g is geerated satis es the followig assmptios t t + t t C (L) " t f" t g is iid, E" t E" t. P j j c j < C () 6 We are iterested i testig agaist H H jj < Uder the ll, t I () while der the alterative, it is a statioary short memory process. Cosider the regressio of t agaist t b P t t P t From the previos sectio, we kow that where W is a stadard Browia motio, ad P + t t P t t d B (r) dr W (r) dr t t d B () W () W () + 5

6 where Now, der H (b (h) h (h) h ) P t t P t W () + d R W (r) dr W () + W (r) dr I the it root case, the asymptotic distribtio depeds o fctioals of a stadard Browia motio ad the isace parameters, ad The covergece rate of b is faster tha the sal p Next, cosider the t statistic for H T (b ) b P t where Ths, b b t ( t b t ) ( t t (b ) t ) t + (b ) t (b ) t + O p p T (b ) b P t d W () + W (r) dr W () + R W (r) dr t R W (r) dr t 6

7 Agai, the asymptotic distribtio of the statistic depeds o the kow isace parameters ad Phillips (987) ad Phillips ad Perro (988) sggested a adjstmet, which leads to a asymptotic distribtio free of isace parameters. Let b ad b be cosistet estimators of ad, where ca be estimated sig the Newey-West type estimator m b h h m + th+ Notice that a cosistet estimator of the log-r variace is Cosider the followig modi catio of the t statistic. Uder H Z T b b T Z T d b b + b b t b t h b b P t W () R W (r) dr Uder the alterative, jj < ad b coverges i probability to a egative costat. Coseqetly, der the statioary alteratives, T ad Z T diverge to Oe shold reject the ll of it root whe where c is sch that Z T < c P (Z T < c ) H Uder the ll, the distribtio is o-stadard, however, it is parameter free, ad the critical vales ca be simlated as follows. First, oe geerates idepedet N ( ) radom variables r r ad comptes Z Tr P P tr Pt s sr Oe repeats this for r R where R is large. The simlated critical vale c R is the qatile of Z T ZTR While the distribtio of Z T is free of isace parameters, it depeds o the model. For example, i geeral oe wold like to allow for a itercept, t + t + t I this case, b depeds o the demeaed t b P t t P t where t 7

8 Notice that t d W (r) B (r) W (r) fw (r) dr W (r) dr dr B (r) dr dr W (r) dr dr where f W is a demeaed stadard Browia motio Hece, i this case, fw (r) W (r) W (r) dr T (b ) b P t Z T b b T b b P t ad the asymptotic distribtio of Z T der the ll of it root is give by fw () f W (r) dr 8

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

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