Dynamic Econometric Models: Y t = + 0 X t + 1 X t X t k X t-k + e t. A. Autoregressive Model:

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1 Dynamic Economeric Models: A. Auoregressive Model: Y = + 0 X 1 Y Y -2 + k Y -k + e (Wih lagged dependen variable(s) on he RHS) B. Disribued-lag Model: Y = + 0 X + 1 X X k X -k + e (Wihou lagged dependen variables on he RHS) Where 0 is known as he shor run muliplier, or impac muliplier, because i gives he change in he mean value of Y following a uni change of X in he same ime period. If he change of X is mainained a he same level hereafer, hen, ( ) gives he change in he mean value of Y in he nex period, ( ) in he following period, and so on. These parial sums are called inerim, or inermediae, muliplier. Finally, afer k periods, ha is k i 0... = B, herefore i is called he long run muliplier or i k oal muliplier, or disribued-lag muliplier. If define he sandardized i * = i / i,, hen i gives he proporion of he long run, or oal, impac fel by a cerain period of ime. In order for he disribued lag model o make sense, he lag coefficiens mus end o zero as k. This is no o say ha 2 is smaller han 1 ; i only means ha he impac of X -k on Y mus evenually become small as k ges large. For example: a consumpion funcion regression is wrien as Y = + 0.4X X X X -3 + e Then he effec of a uni change of X a ime on Y and is subsequen ime periods can be shown as he follow diagram: Effec on Y 0 X 1 X -1 2 X -2 3 X -3 0 = = = Time 1

2 The muliplier round of he Invesmen injecion $160 (k=1/(1-mpc) Period I Y C (if mpc = 0.5) 1 $160 $160 $ => Oher Examples: Money creaion process, inflaion process due o money supply, produciviy growh due o expendiure or invesmen. Period New Deposis New Loans Required Reserve (10%) 1 $1000 $900 $ Toal $10,000 $9,000 $1,000 In general: Suppose he disribued-lag model is Y = + 0 X + 1 X X k X -k + e The basic idea: The long run responses of Y o a change in X are differen from he immediae and shor-run responses. And suppose he expec value in differen periods is A. Then E(X ) = E(X -1 ) = E(X -2 ) = A E(Y ) = + 0 E(X ) + 1 E(X -1 ) + 2 E(X -2 ) E(e ) = + 0 A + 1 A + 2 A = + i A This gives he consan long run corresponding o X = A, and E(Y) will remain a his level uni when X changes again. 2

3 Suppose look for one period ahead (+1), and X +1 = A+1 E(Y +1 ) = + 0 E(X +1 ) + 1 E(X ) + 2 E(X -1 ) + + E(e +1 ) = + 0 (A+1) + 1 A + 2 A = A + 1 A + 2 A + = i A E(Y +2 ) = + 0 E(X +2 ) + 1 E(X +1 ) + 2 E(X ) + +E(e +2 ) = + 0 (A+1) + 1 (A+1) + 2 A = A + 1 A + 2 A + = i A E(Y +3 ) = i A E(Y +j ) = j + i A For all periods afer +3, E(Y) remains unchanged a he level given by E(Y +3 ), given no furher change in X. Thus, E(Y) = + i X Reasons for Lags: Psychological reason: due o habi or ineria naure, people will no reac fully o changing facors, e.g. Income, price level, money supply ec. Informaion reason: because imperfec informaion makes people hesiae on heir full response o changing facors. Insiuional reason: people canno reac o change because of conracual obligaion. Ad Hoc Esimaion of Disribued-Lag Models Esimaion mehod: Firs regress Y on X, hen regress Y on X and X -1, hen regress Y on X, X -1 and X -2, and so on. Y = + 0 X Y = + 0 X + 1 X -1 Y = + 0 X + 1 X X -2 Y = + 0 X + 1 X X X -3 Y = + 0 X + 1 X X X -3 + This sequenial procedure sops when he regression coefficiens of he lagged variables sar becoming saisically insignifican and / or he coefficien of a leas one of he variables change signs, which deviae from our expecaion. 3

4 Significan -saisics of each coefficien The sign of i does no change Highes R 2 and 2 R Min. values of BIC and/or AIC Drawbacks: No a priori guide as o wha is he maximum lengh of he lag. If many lags are included, hen fewer degree of freedom lef, and i makes saisical inference somewha shaky. Successive lags may suffer from muli-collineariy, which lead o imprecise esimaion. (When Cov(X -i, X -j ) is high). Need long enough daa o consruc he disribued-lag model. 4

5 I. Koyck Approach o Disribued-Lag Models (Geomerical lag Model): Y = + 0 X + 0 X X e or Y = X X X e Where i 0 i and 0 < < 1 (A priori resricion on i by assumpion ha he i s follow a sysemaic paern.) k k 0 ( 1...) 0 ( ) which is he long run muliplier 1 Then Y -1 = + 0 X X X e -1 Y -1 = + 0 X X X e -1 and (Y - Y -1 ) = ( - ) X + ( 0 X -1-0 X -1 ) + ( 0 2 X X -2 ) + ( 0 3 X X -3 ) + +(e e -1 ) By Koyck ransformaion (from a disribued-lag model ransformed ino an auoregressive model): Y Y -1 (1- ) + 0 X + (e e -1 ) Y = (1- ) + 0 X + Y -1 + v where v ~ iid(0, 2 ) Y = X + Y -1 + v The Y -1 is a shor run dynamic erm and is buil ino he auoregressive model. The imporan of his auoregressive model gives he long-run muliplier ha implied by he disribued lags model. The long-run muliplier can be obained from he auoregressive model by calculaing 0 (1/(1- )). 5

6 II. Raionalizaion of he Koyck Model: Adapive Expecaion Model Y = X* +e (Long run funcion, and X* is unobserved expeced level) Go back one period and muliply (1- ), i becomes: (1- )Y -1 =( (1- )X* -1 + (1- )e -1 (o be subraced laer) By posulaing: X* - X* -1 = (X - X* -1 ) where 0< <1 and is he coefficien of expecaion X* = X -+ (1- ) X* -1 Y = { X -+ (1- ) X* -1 } +e Y = X -+ 1 (1- ) X* -1 +e By subracing: Y - (1- )Y -1 = X + [e -(1- )e -1 ] Y X +(1- )Y -1 + v (shor run dynamic funcion) Y X + 2 Y -1 + v A shor run dynamic funcion is used o derive for long run funcion. 6

7 III. Raionalizaion of he Koyck Model: Parial Adjusmen Model Y* = X +e (Long run funcion and Y* is unobserved and desired level) By Y -Y -1 = (Y* -Y -1 ) where 0< <1 and he is he coefficien of adjusmen Y = Y* +(1- )Y -1 By subsiuing: Y = [ X +e ] +(1- )Y -1 Y = X +(1- )Y -1 + e (shor run funcion) Y = X + (1- )Y -1 + v where v ~ iid(0, 2 ) Use he shor run funcion o derive long run funcion. IV. Combinaion of Adapive Expecaion and Parial adjusmen Y* = X* +e where Y* and X* are he unobserved and desired level Since he posulaions of adapive expecaion and parial adjusmen are Y -Y -1 = (Y* -Y -1 ); Y = Y* +(1- )Y -1 X* - X* -1 = (X - X* -1 ); X* X -+ (1- ) X* -1 Y = X* +e ] +(1- )Y -1 Go back one period and muliply (1- ) on boh side, i becomes: (1- )Y -1 = (1- )[ X* -1 +e -1 ] +(1- )(1- )Y -2 (1- )Y -1 = (1-0 + (1-1 X* -1 +(1- )(1- )Y -2 + (1- )e -1 (o be subraced laer) Y = X* +(1- )Y -1 + e Y = X -+ (1- ) X* -1 ] +(1- )Y -1 + e 7

8 Y = X -+ (1-1 X* -1 +(1- )Y -1 + e By subracing: Y -(1- )Y -1 = 0 1 X +(1- )Y -1 -(1- )(1- )Y -2 e (1- )e -1 Y = 0 1 X +{(1- )+ (1- )}Y -1 -(1- )(1- )Y -2 +{ e (1- )e -1 } Y 0 1 X 2 Y -1 3 Y -2 +v Deecing Auocorrelaion in Auoregressive Models: Durbin-h es: DW h ( 1 ) 2 n 1 n Var( ) 2 Where var( 2 ) = Variance of coefficien of he lagged dependen variable,y -1 If h >1.96, rejec Ho. There is posiive firs order auocorrelaion If h <-1.96, rejec Ho. There is negaive firs order auocorrelaion If 1.96 <h <1.96, do no rejec Ho. There is no firs order auocorrelaion In general, Lagrange Muliplier es is preferred o he h-es because i comparaively suis for large as well as small sample size. The seps of Lagrange Muliplier Tes: (1) Obain he residuals from he esimaed equaion: vˆ Y Yˆ Y ˆ (2) Run an auxiliary regression: 0 ˆ ˆ ˆ 1X 2Y 1 3 Y 2 vˆ ' ' X 0 1 ' Y 2 Y v ' (3) Tes he null hypohesis ha =0 wih he following es saisic: LM = nr 2 ~ 2 LM has a chi-square disribuion wih degrees of freedom equal o he number of resricions in he null hypohesis (in his case is one). If LM is greaer he criical chi-square value, hen we rejec he null hypohesis and conclude ha here is indeed auocorrelaion exis in he original auoregressive equaion. 8

9 V. Almon Approach o Disribued-Lag Models: - (Polynomial Disribued-Lag model) Y = + 0 X + 1 X X -2 + k X -k + e Y = + i X -i + e Where i = 0 o k Suppose he following model have only wo-lagged erms of X, such as Y = + 0 X + 1 X X -2 + e Almon assumes ha i can be approximaed by a suiable-degree polynomial in i which is he lengh of lags. For example, if here is a second degree of polynomial of i : i.e., i = a 0 +a 1 i + a 2 i 2 Then Y = + (a 0 +a 1 i+a 2 i 2 )X -i + e Y = + a 0 X -i + a 1 ix -i + a 2 i 2 X -i + e Afer ransforming he variables, he disribued-lag model becomes: Y = + a 0 Z 0 + a 1 Z 1 + a 2 Z 2 + e Where Z 0 = X -i = X + X -1 + X -2 Z 1 = ix -i = X X -2 Z 2 = i 2 X -i = X X -2 Afer ransforming he Z 0, Z 1, Z 2, run he regression on Y agains on all Z i o find he esimaed coefficien values of â 0, â 1, â 3, hen deduces he esimaed values of ˆ i 's for he original posulaed model. For example, (i) if i = a 0 + a 1 i + a 2 i 2, i is a second degree of polynomial 0 = a 0 1 = a 0 + a 1 2 = a 0 + 2a 1 + 4a 2 3 = a 0 + 3a 1 + 9a 2 k = a 0 + ka 1 + k 2 a 2 9

10 In general, if here are k lagged erms of X and he degree of he polynomial is 2, hen k = a 0 + ka 1 + k 2 a 2 For example, (ii) if i = a 0 + a 1 i + a 2 i 2 + a 3 i 3, i is a hird degree of polynomial 0 = a 0 1 = a 0 + a 1 + a 2 + a 3 2 = a 0 + 2a 1 + 4a 2 + 8a 2 3 = a 0 + 3a 1 + 9a a 3 k = a 0 + ka 1 + k 2 a 2 + k 3 a 3. In general, if here are k lagged erms of X and he degree of he polynomial is 3, hen k = a 0 + ka 1 + k 2 a 2 + k 3 a 3 Then Y = + (a 0 +a 1 i+a 2 i 2 + i 3 a 3 )X -i + e Y = + a 0 X -i + a 1 ix -i + a 2 i 2 X -i + a 3 i 3 X -i + e Afer ransforming he variables, he disribued-lag model becomes: Y = + a 0 Z 0 + a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + e Where Z 0 = X -i = X + X -1 + X -2 +X -3 Z 1 = ix -i = X X X -3 Z 2 = i 2 X -i = X X -2 +9X -3 Z 3 = i 3 X -i = X X X -3 For example, (iii) if i = a 0 + a 1 i + a 2 i 2 + a 3 i 3 + a 4 i 4, i is a fourh degree of polynomial 0 = a 0 1 = a 0 + a 1 + a 2 + a 3 + a 4 2 = a 0 + 2a 1 + 4a 2 + 8a a 4 3 = a 0 + 3a 1 + 9a a a 4 4 = a 0 + 4a a a a 4 In general, if here are k lagged erms of X and he degree of he polynomial is 4, hen 10

11 k = a 0 + ka 1 + k 2 a 2 + k 3 a 3 + k 4 a 4 Then Y = + (a 0 +a 1 i+a 2 i 2 + i 3 a 3 + i 4 a 4 )X -i + e Y = + a 0 X -i + a 1 ix -i + a 2 i 2 X -i + a 3 i 3 X -i + a 4 i 4 X -i + e Afer ransforming he variables, he disribued-lag model becomes: Y = + a 0 Z 0 + a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 4 Z 4 + e Where Z 0 = X -i = X + X -1 + X -2 +X -3 + X -4 Z 1 = ix -i = X X X X -4 Z 2 = i 2 X -i = X X -2 +9X X -4 Z 3 = i 3 X -i = X X X X -4 Z 4 = i 4 X -i = X X X X -4 11

12 Causaliy in Economics: Granger Tes The disribued lags model can be applied o es he direcion of causaliy in economic relaionship. Such es is useful when we know he wo economic variables are relaed bu we don know which variable causes he oher o move. The Granger causaliy es assumes ha he informaion relevan o he predicion of he respecive variables, e.g. Y(GNP) and X(Money), is conained solely in he ime series daa on hese variables. k q Y i X i jy j e1 i 1 j 1 X ' m X i i i 1 j 1 n Y j j e 2 Four differen siuaions: 1. Unidirecional causaliy from M o GNP when i 0 and j =0. (i.e., X Y) The -saisics of i are significan The -saisics of i are insignifican 2. Unidirecional causaliy from GNP o M when i =0 and j 0. (i.e., Y X) The -saisics of i are insignifican The -saisics of i are significan 3. Feedback or Bilaeral causaliy from M o GNP and GNP o M when i 0 and j 0 (i.e., X Y) The -saisics of i are significan The -saisics of i are significan 4. Independen beween GNP and M when i =0 and j =0 (i.e., X / Y) The -saisics of i are insignifican The -saisics of i are insignifican 12

13 Seps for Granger Causaliy Tes: 1. Ho: i =0 (i.e. X does no cause Y) and H 1 : i 0 2. Run he resriced regression: Regress curren Y on all lagged Y erms and (oher variables if necessary), bu do no include he lagged X variables in his regression. Y = + j Y -j + e 1 And obain he RSS R. 3. Then run he unresriced regression Y = + i X -i + j Y -j + e 1 And obain he RSS UR. 4. F* = [(RSS R - RSS UR )/m] / [RSS UR /(n-k)] where m is he number of lagged X erms and k is he number of parameers esimaed in unresriced regression. 5. If F* > F c (criical value), hen rejec he Ho. Tha is X causes Y. If F* < F c (criical value), hen no rejec he Ho. Tha is X does no cause Y. 6. For esing GNP causes M, jus simply repeas he above five seps. The limiaions of Granger causaliy: 1. The number of lags includes in he unresriced relaionship can affec he level of significance of F. For example, wih small samples, he choice of one lag of m versus longer lags (e.g., using m of 4 versus m of 8) affecs he F-es. 2. In general, here is no good way o deermine he lag lengh used for independen variables. 3. Economericians have been able o show he Granger es can yield conflicing resuls. 4. Granger causaliy es canno always proof of causaliy, i is confirmaion of he direcion of influence. I does no address he issues of causal closure. Tha is, here may be specificaion errors in he relaionship. Example on hree-variable causaliy es: X1 X2 X3 13

14 Some possible rivariae causaliy relaionships: X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 X1 X2 X3 14

15 Empirical sudy: In order o es he inernaional ineres rae ransmissions, we use he hree marke 1-monh ineres raes (i.e., US s Fed Rae (FFR), London s Euro-Dollar Rae (ERL), and Singapore s Asian-Dollar Rae (ARS)) o idenify he ransmission direcion. The daa are obained from DaaSream from o The Resuls of Granger Causaliy es: H 0 lags ERL / ARS ARS / ERL FFR / ARS ARS / FFR FFR / ERL ERL / FFR Criical F c * * * * * * * * * 3.865* 5.576* * * 4.796* * 2.750* 4.523* * * 6.927* 8.600* 7.240* 2.310* * * 7.792* 8.447* 8.119* 3.233* * * 9.326* 6.637* 4.697* 3.641* * * 9.711* 8.598* 7.938* 4.893* * * 8.620* 7.678* 6.783* 4.707* 1.75 Noe: * means he saisics are larger han criical values, hus he H 0 is rejeced. From he 2 o 3-lags resul, we observe he ransmission relaionships are as following: FFR X X ARS X ERL From he 4-lags resul, we observe he ransmission relaionships of he hree markes are as following: 15

16 FFR ARS ERL The Granger(1969) approach o he quesion of wheher X causes Y is applied o see how much of he curren Y can be explained by he pas values of Y and hen o see wheher adding lagged values of X can improve he explanaion. Y is said o be Granger-caused by X if X helps in he predicion of Y, or equivalenly if he coefficiens of he lagged Xs are saisically significan, however, i is imporan o noe ha he saemen X Granger causes Y does no imply ha Y is he effec or he resuls of X. 16