Økonomisk Kandidateksamen 2005(II) Econometrics 2. Solution

Transcription

1 Økonomisk Kandidaeksamen 2005(II) Economerics 2 Soluion his is he proposed soluion for he exam in Economerics 2. For compleeness he soluion gives formal answers o mos of he quesions alhough his is no always required for a good mark. For sub-quesions formulaed using erms such as derive, show, orsae, some degree of precision is required for a op mark, e.g. by using formal argumens. For sub-quesions using formulaions such as discuss, explain, orpresen, hesoluioncanbe verbal wihou use of mahemaical reasoning. Noe, however, ha in mos cases he quesions are posed wih reference o a given model or equaion, and he soluion should answer he precise quesion. Soluion o Quesion 1 (a) he quesion considers a linear regression model, y = x 0 1β 1 + x 2 β 2 +, (1.1) for =1, 2,...,where is i.i.d. wih mean zero and variance σ 2, x 1 is a K 1 vecor of variables, and x 2 is a single variable. We assume ha x 1 conains exogenous variables whereas x 2 is endogenous, i.e. E[x 1 ] = 0 (1.2) E[x 2 ] 6= 0. (1.3) For laer use we define he K 1 vecors Ã! x = x 1 x 2 and β = Ã β 1 β 2!. he idea of a mehod of momens (MM) esimaion is o derive he esimaor as he soluion o he equaions corresponding o a se of momen condiions, E[f(y,x,β)] = 0, 1

2 where f( ) is a funcion of he parameers of ineres. For a given sample, he mahemaical expecaion is replaced by he sample average: g (β) = 1 X f(y,x,β)=0, (1.4) and he MM esimaor is he unique soluion o he equaions in (1.4). As an example, recall ha he minimum assumpion for consisency of he OLS esimaor is ha x is uncorrelaed wih,i.e. E[x ]=E[x y x 0 β ]=0. Replacing expecaions wih sample averages yields g (β) = 1 X x y x 0 β =0. his is a sysem of K equaions wih K unknown, and he soluion is given by 1 X x y = 1 X x x 0 β Ã! 1 Ã! 1 X bβ MM = x x 0 1 X x y = β b OLS, provided ha 1 P x x 0 is inverible. he condiions in (1.2) and (1.3) are no sufficien for consrucing a consisen MM esimaor of β. he inuiive reason is ha (1.2) gives only K 1 equaions, which do no give a unique soluion for he K parameers in β. We say ha he parameer is no idenified by he model. Now assume ha we wan o use an insrumenal variables (IV) esimaor of β based on a single new insrumen, z 2. he firs condiion needed for z 2 is ha i is uncorrelaed wih,i.e. E[z 2 ]=0. (1.5) ogeher wih (1.2) his gives he K momen condiions E[z ]=0, where he vecor of insrumens, z =(x 0 1,z 2) 0, conains he predeermined variables augmened wih he new insrumen. he sample momen condiions are given by which gives he esimaor bβ MM = g (β) = 1 Ã 1 X z y x 0 β =0, (1.6)! 1 Ã X z x 0 1 2! X z y = β b IV,

3 wherewehaveoassumeha 1 P z x 0 is inverible, so ha he new insrumen is informaive. (b) Now assume ha we have R variables, z 1,z 2,...,z R, ha could be useful as insrumens for x 2. In his case here are more equaions in (1.6) han parameers, and in general here is no soluion. echnically, 1 P z x 0 is recangular and no inverible. An alernaive is a generalized mehod of momens (GMM) esimaion. he GMM esimaor is derived from minimizing he disance beween g (β) and zero, where he disance is defined in erms of he weighing marix W (β), i.e. corresponding o minimizing he crieria funcion Q (β) =g (β) 0 W (β)g (β). he GMM esimaor, β b GMM (W ), is consisen for any posiive definie weigh marix; and he efficien GMM esimaor is defined by he opimal weigh marix, W op, which is he inverse of he variance of he momens. Formally, g (β) N(0,S), and plimw op (β) =S 1. he inuiion is ha momens wih a high variance should have low weighs and vice versa. Also, he weigh marix should ake ino accoun he correlaion of he momens. his is in spiri equivalen o weighed leas squares, bu applied o momens raher han observaions. (c) he opimal weigh marix is he inverse of he asympoic variance of g (β). For he example in quesion (b), he sample momens are given by g (β) = 1 X z = 1 Z0, where Z = z1 0 z2 0. and = 1 2. z 0 are he sacked marices. Under he assumpions on giveninquesion(a),iholds ha E[ 0 ]=Ω = σ 2 I, and a naural esimaor of S is µ ³ 1 bs = V g (β) = V Z0 = 1 Z0 ΩZ b 1 = bσ2 Z 0 Z, where bσ 2 is a consisen esimaor of σ 2. his weigh marix, which does no depend on β, will give a GMM esimaor equivalen o he Generalized IV esimaor (GIVE) and he wo sage leas squares (2SLS) esimaor. 3

4 Soluion o Quesion 2 (a) he quesion considers he regression model y = x 0 β +, =1, 2,...,, (2.1) where x is a K dimensional vecor of predeermined explanaory variables, and is an error erm. An LM es for no auocorrelaion of order p is based on he auxiliary regression, b = x 0 γ + α 1 b 1 + α 2 b α p b p + residual, (2.2) where b denoes he esimaed residual from equaion (2.1). Under he null of no auocorrelaion, he lagged residuals have no explanaory power in (2.2) and he es saisic is LM = R 2, where R 2 is he coefficien of deerminaion in (2.2). Under he null, LM is disribued as a χ 2 (p). he good soluion noes ha he iniial values for he esimaed residuals are mos ofen se o zero. Nowweassumehawefind auocorrelaion of firs order, i.e. = ρ 1 + η, where η is a whie noise process. Using he lag-operaor, his may be wrien as (1 ρl) = η. Since η is i.i.d., he appropriae ransformaion corresponds o muliplying (2.1) wih (1 ρl), i.e. (1 ρl) y = (1 ρl) x 0 β +(1 ρl) y ρy 1 = x 0 β x 0 1ρβ + η or equivalenly y = ρy 1 + x 0 β x 0 1ρβ + η. his is an ADL(1,1) model bu subjec o a resricion imposed by he common facor (1 ρl). In paricular, noe ha he explanaory variables are y 1, x,and x 1, bu we have only wo parameers: β and ρ. Removing he common facor resricion yields he unresriced ADL(1,1) model y = θ 1 y 1 + x 0 φ 0 + x 0 1φ 1 + η. Residual auocorrelaion could indicae general misspecificaion, and i is ofen preferable o coninue wih he unresriced ADL model. 4

5 (b) Now we consider again he equaion in (2.1) and we assume ha here is no auocorrelaion in. Auoregressive condiional heeroscedasiciy (ARCH) is mean o capure changing volailiy, i.e. he idea ha in some periods he uncerainy on a marke is large whereas in oher periods he uncerainy is smaller. his is ofen relevan for modelling financial markes. Besides he equaion for he condiional mean (2.1) and ARCH model also specifies an equaion for he condiional variance, e.g. he ARCH(1) model σ 2 = + α 2 1, (2.3) where σ 2 = E[ 2 I 1 ] denoes he variance of he error erm condiional on he informaion se a ime 1. he inuiion is ha if he shock a ime 1 is large (measured as a large residual, 1 ) hen he variance a ime is high. his implies a large probabiliy of a large shock, eiher posiive or negaive. Graphically, he 95% confidence bound for changes wih he magniude of he previous shock. (c) Now we le y be he price of a paricular asse on a financial marke. A marke paricipan has he idea ha he volailiy σ 2 affecs he marke price y,which creaes a link beween equaion (2.3) and (2.1). One way o model his feaure is o include a funcion of σ 2 in (2.1), e.g. y = x 0 β + δσ 2 +. (2.4) his is denoed he ARCH-in-Mean model. One inerpreaion is ha he raders require a higher average reurns o compensae a higher risk. o es he null hypohesis of no ARCH-in-Mean, we can esimae he ARCH-in-Mean model, (2.4) and (2.3), and es he hypohesis δ =0. If he hypohesis is no rejeced, here is no much suppor for he conjecure of he marke paricipan. Soluion o Quesion 3 (a) his quesion considers an MA(2) model, Y = µ + + α α 2 2, (3.1) where is a whie noise process wih zero mean and variance σ 2. I is given in he quesion ha Y is a saionary process. o derive he mean we ake expecaions in equaion (3.1), i.e. E[Y ]=µ + E[ ]+α 1 E[ 1 ]+α 2 E[ 2 ]=µ. 5

6 h o find he variance, γ 0 = V (Y )=E (Y µ) 2i,weuseha γ 0 = E h(y µ) 2i = E [( + α α 2 2 )( + α α 2 2 )] = E[ 2 + α α α α α 1 α α α 1 α α ] = 1+α α 2 2 σ 2, whereweusehae[ 2 ]=σ 2, while all cross-erms are zero, E[ i ]=0for i>0. o derive he covariance, γ 1 = Cov(Y,Y 1 )=E[(Y µ)(y 1 µ)], we use again ha all cross erms are zero and find: Similarly γ 1 = E [(Y µ)(y 1 µ)] = E[( + α α 2 2 )( 1 + α α 2 3 )] = (α 1 + α 1 α 2 ) σ 2. γ 2 = E [(Y µ)(y 2 µ)] = E[( + α α 2 2 )( 2 + α α 2 4 )] = α 2 σ 2, while γ k =0 for k =3, 4,... he auocorrelaion funcion, ACF, isgivenby ρ 1 = γ 1 = (α 1 + α 1 α 2 ) σ 2 γ 0 1+α α 2 2 σ 2 = α 1 + α 1 α 2 1+α α2 2 ρ 2 = γ 2 α 2 σ = 2 γ 0 1+α α 2 2 σ 2 = α 2 1+α α2 2 ρ k = 0, k 3. (b) Assume ha you have esimaed he model (3.1) based on a sample Y 1,Y 2,...,Y, and le bµ, bα 1,andbα 2 denoe he esimaed parameers. his implies ha we can consruc also he esimaed residuals, b 1,b 2,...,b. Wewanoconsrucaseriesof ou of sample forecass by +k = E[Y +k Y 1,...,Y ] for k =1, 2,... We noe ha he bes forecas of a fuure shock is zero, e.g. E[ +k Y 1,...,Y ]=0 for k>0. Leing I denoe he informaion se a ime,wefind he forecass by +1 = E[µ α 1 + α 2 1 I ] = µ + α 1 b + α 2 b 1 by +2 = E[µ α α 2 I ] = µ + α 2 b by +3 = E[µ α α 2 +1 I ] = µ by +4 = E[µ α α 2 +2 I ] = µ 6

7 and so on. he main advanage of a univariae ime series forecas is ha he informaion se, I, conains only he hisory of Y, so i is no necessary o forecas any condiioning variables. If, on he oher hand, forecass are based on condiional model, E[Y X ], hen a forecas X b +k of X +k is needed in order o forecas Y b +k. he main drawback is ha here is no economic conen in he forecass. (c) his quesion is a bi ricky. he correc answer is ha a finie order MA process like (3.1) is saionary for all combinaions of he parameers. he reason is ha i is a combinaion of lagged values of saionary variable,. (Depending on he definiion of saionary we migh wan o impose α 1 and α 2 finie bu his qualificaion is no required for a good mark). o discuss inveribiliy, rewrie he MA process (3.1) as Y = µ + + α α 2 2 Y µ = 1+α 1 L + α 2 L 2 Y µ = α(l), where α(l) is he MA lag-polynomial. We can facorize he polynomial as α(l) = 1+α 1 L + α 2 L 2 =(1 φ 1 L)(1 φ 2 L), where φ 1 and φ 2 are he roos. I holds ha he MA process is inverible if and only if he roos φ 1 and φ 2 are smaller han one in absolue value. Soluion o Quesion 4 (a) his quesion considers he price of organic oranges, P org, regular oranges, P reg, and he price differenial Y = P org P reg, allmeasuredinpenceperlb. A sochasic rend is generaed by a uni roo in a sochasic process, and makes he process non-saionary. A simple example is a uni roo AR(1) process wih drif, X = X 1 + δ +, which (by recursive subsiuion) has he soluion X = X 0 + δ + X i, where X 0 is he iniial value, δ is a deerminisic rend generaed from he consan erm, and P i=1 i is he sochasic rend generaed from he innovaions,. In 7 i=1

8 general, a combinaion of uni roo processes will be non-saionary, because i conains he sochasic rends. Only excepion is called coinegraion, where here exiss a combinaion of he variables ha cancels he sochasic rends o become saionary. For he case of wo variables he sochasic rends should be proporional and here exiss a β so ha Y βx is a saionary process. In Figure 4.1 he individual prices, P org and P reg, look non-saionary, due o eiher a deerminisic or a sochasic rend. In Figure 4.2 he price differenial, Y,looks saionary. If he non-saionariy of P org and P reg is driven by sochasic rends hen P org and P reg appears o be coinegraion wih a coinegraion vecor (1, 1) 0. able 4.1 presens he OLS esimaes of he model Y = δ + c 1 Y 1 + c 2 Y 2 + πy 1 +, (4.1) for =4, 5,...,181. o es he null hypohesis ha he variables do no coinegrae corresponds o a Dickey-Fuller es of a uni roo in Y. As a minimum he soluion should noe ha a uni roo in Y corresponds o π =0. A more horough soluion may derive he es from he characerisic polynomial, A(L), noinghaa(1) = π. he soluion should indicae ha he alernaive o a uni roo, π =0,isa saionary process, 2 <π<0, and ha he Dickey-Fuller es is jus he raio π=0, which follows a DF (and no a sandard normal) disribuion. For he price differenial, Y = P org P reg, he Dickey-Fuller es for a uni roo is π=0 = 6.39 and should be compared wih a 5% criical value of around 2.88 (for = 250). We can herefore easily rejec he null of a uni roo and conclude ha he variables are likely o coinegrae wih coinegraion vecor (1, 1) 0,which is also in line wih he impression from Figure 4.2. (b) Given ha he price differenial, Y, is a saionary process, an esimae of he expeced addiional price for organic oranges, µ = E[Y ]=E[P org P reg ], canbe found from he OLS esimaes in able 4.1. One way o derive he mean is o noe ha in an AR(3) model, he mean is given by Y = δ + θ 1 Y 1 + θ 2 Y 2 + θ 3 Y 3 +, µ = δ 1 θ 1 θ 2 θ 3 = δ A(1) = δ π. From able 4.1 we find bµ = b δ bπ = =20.1. Given Figure 4.2, he repored esimae should be close o bµ 20. he soluion should explain ha wo variables coinegrae if and only if here exis an error correcing equaion for a leas one of he variables. Error correcion 8

9 is he force ha susain he equilibrium defined by he coinegraing relaion. In he presen case he following error correcion equaions are given: P org = (8.55) reg P org P P org 1 P reg 1 + org (1.27) ( 0.97) ( 8.34) + reg P reg = reg P org P (2.97) ( 1.16) ( 0.54) (0.22) P org 1 P reg 1 We noe ha he organic price error correcs very significanly, while he regular price does no error correc. A es for no-coinegraion is he same as a es for no error correcion. he srong error correcion (a raio of 8.3 in he organic orange price equaion which is close o DF disribued) is consisen wih he srong suppor for coinegraion found in quesion (a). (c) A simplified sysem could be approximaed as P org = 20.9 P org 1 P reg 1 + org (4.2) P reg = 0.9+ reg. (4.3) Equaion (4.3) gives he soluion for he price of regular oranges P reg = P reg X i=1 reg i, (4.4) which is simply a random walk wih drif. he organic price is given by P org P org 1 = 20.9 P org 1 P reg 1 + org P org = 20.9+P reg 1 + org, (4.5) which saes ha he organic price is given by he lagged price of regular oranges plus he addiional price for organic oranges, 20.9, plus a random shock, org. Consider he wo scenarios: (i) Posiive shock o reg while org is unchanged. his scenario implies ha P reg increases permanenly as he shock is accumulaed in he random walk in (4.4). In he nex period, +1, he price of organic oranges will follow o he new level. (ii) Posiive shock o org while reg is unchanged. he increase in he price of organic oranges, P org,isransiorysince org is no accumulaed. here is no ransmission o he price of regular oranges and in he nex period, +1, he price of organic oranges is pulled back o he price or regular oranges according o (4.5), so he shock has only effec for one period. 9