Testing the Random Walk Model. i.i.d. ( ) r

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1 he random walk heory saes: esing he Random Walk Model µ ε () np = + np + Momen Condiions where where ε ~ i.i.d he idea here is o es direcly he resricions imposed by momen condiions. lnp lnp µ ( lnp lnp µ ) m ( lnp lnp µ ) m E 4, r = ( lnp lnp µ ) m4 ( ) m r ( lnp lnp µ ) f (, m,..., m r) m r ( ) r = E ε We now illusrae he above es for he case r = by choosing one mean resricions and wo variance resricions

2 g lnp lnp µ m = lnp lnp m < k ( µ, ) ( µ ) = ( lnp lnpk kµ ) km () In esing he model we se a weighed combinaion of he momen condiions equal o zero. Hansen has shown ha he opimal weighed combinaion is given by: DWg ( θ ) where W = he inverse of he variance-covariance marix, S. D g θ ( θ ) = E We derive he analyical covariance marix S in he Appendix. I urns ou o be: m m km = 4 + ( ) 4 + (, ) km km4 + B ( k, ) m k m4 + A( k ) m S m m A m km B k m (4) ( i) i(4i) where Ai ( ) = ( ) ( )( ) k + + ( ) (4 ) B( k, ) = ( ) Given our derivaion of he analyical variance-covariance marix in (4), we can obain he opimal GMM esimaors and ˆm by solving he sysem of equaions ( m ) DWg µ, = (5) where D = k Specifically, invering (4) and hen subsiuing his marix back ino equaion sysem (5) resuls in he following esimaors:

3 ˆ µ = = [ lnp lnp ] (6) mˆ = ( ) [ ] ˆ k * k ( ) ˆ lnp lnp lnp lnp µ k µ k + ( ) k = = (7) Hansen (98) shows ha he esimaor θ is asympoically disribued as mulivariae normal wih mean θ and variance-covariance marix [ DWD ]. Hence, he esimaors ˆµ and m are disribued asympoically normal wih means µ and m respecively and wih variance-covariance marix VAR ( ˆ, µ m ) m m = m m4 + C( k, ) m (8) where C ( k, ) ( )( ) k + + k+ ( + ) 4 (8 4 9) ( ) = k ( k + ) Using he opimal esimaors given in equaions (6) and (7) we can also calculae he value of he chi-square saisic analyically ( g W g [ ˆ ] ( ) k = k k ( k + ) m { lnp lnp kµ k lnp lnp ˆ µ } = = In order o relae o he previous lieraure, consider he es proposed by Lo- Mackinlay (987) in secion. of heir paper. We claim heir es is a special case of () wih = and k allowed o vary. o see his, Lo-Mackinlay show ha under he addiional assumpion ha he є are normally disribued (9) [ ˆ] a k lnp lnp ( )( ) k kµ k k = ( N, ) [ lnp ˆ lnp µ ] k = ()

4 Noe we can rewrie equaion () above as [ lnp ˆ ˆ lnp ] [ ] k kµ k lnp lnp µ = = k [ lnp ˆ lnp µ ] = () A squared normal random variable weighed by is variance follows a From equaion (), ha is, lnp ˆ ˆ a lnp k kµ k lnp lnp µ = = χ k( k )(k ) ( [ ˆ lnp lnp µ ] ) = { [ ] [ ] } ( ~ ) Of course, his is simply equaion (9) wih m replaced by ˆm and =. χ disribuion. In pracice, we employ assumpions much weaker han ε i.i.d. Specifically, we assume. E εε =.. 4. E εε ε k = E εε = m E εε =, k Cross-Momens Here we employ he independence of he ε s more explicily. Specifically, if he ε s are independen hen any funcion of hese ε s is also independen. In his case equaion implies he following es: ( ) ( E[ f ( ε )*...* f ( ε )] E[ f ε ]*...* E[ f ( ε )] = ). () 4

5 A specific example of he above is: f ( ε) ε E εg ( ε ) =, f ( ) and g(.) f ( ε) ( ε) E[ f( ε)] E[ ( ε)] g g Fama-French (987) es wheher he following condiion holds: ( lnp+ lnp) ( lnp lnp) cov, = var( lnp lnp ) () (4) Under he random walk model given by equaion (), he covariance erm in equaion (4) is equivalen o ( ε i µ i + ε i + i µ = = ) E + + = I is clear from equaion () ha his es can be viewed as a special case of more general of more general IID ess. Consider Fama-French s regression es of he condiion saed in equaion (4) above. his is paricularly easy o pu in a GMM framework. Specifically, he momen condiions can be given by. β ( ) ( ) lnp+ lnp a lnp lnp g( αβ, ) = lnp+ lnp a βlnp lnp lnp lnp = hese are he normal OLS esimaes for αand β. he analyical variancecovariance marix associaed wih his regression can be explicily derived under he null hypohesis of a random walk. he derivaion of he variance- covariance marix is provided for in Appendix. Accouning for overlapping observaions his marix is given by S m µ m = ( + ) m 4 um + µ m (5) 5

6 Under hese assumpions, i is also sraighforward o calculae he analyical derivaive marix: D µ = µ ( m + µ ) (6) Using equaions (5) and (6), he esimaes ˆα and ˆβ have an asympoic variancecovariance marix given by ( + ) ( + ) µ + m µ DWD = ( + ) µ + independen of populaion parameer (7) herefore, under he null hypohesis a ˆ + β ~ N, Noe ha when = we ge he familiar resul ha he sandard error of he auocorrelaion coefficien is. Again, i should be noed ha he required assumpions are weaker han hose of model (), namely. E εε =.. E εε ε k =, k E εε = m 6