A. Thiemer, SLDG.mcd, 6..6 FH-Kiel Universiy of Applied Sciences Prof. Dr. Andreas Thiemer e-mail: andreas.hiemer@fh-kiel.de Modeling Economic Time Series wih Sochasic Linear Difference Equaions Summary: Hansen/Sargen (998) creaed a class of models ha merges recursive linear models of dynamic economies wih dynamic economerics. The underlying sochasic process of he vecor of economic sae variables is consruced recursively using an iniial random vecor and a ime invarian law of moion. Some simulaions will show how special cases of his general model are formed by a variey of ime series processes ha have been sudied by economiss.
A. Thiemer, SLDG.mcd, 6..6 The basic law of moion equaion: < > Α. x < > C. w < > x x <> : sequence of random vecors w <> : sequence of "whie-noise"-vecors (maringale difference sequence) A, C : ransiion marices Example : Deerminisic polynomial ime rends Trend funcion: y.. 3.. 4.. Wihou random effecs C = and w =. The ransiion marix A becomes Α Hence x < > Α. x < > Α. x < > wih Α... 6.
A. Thiemer, 3 SLDG.mcd, 6..6 Given he iniial condiion x < >. 3 6. 4 T we obain for.. he sae vecors < > Α. x < > x The firs elemen of x yields he cubic polynom of he rend funcion above. If you inser o he oher elemens of x < >, you will ge a linear or quadraic rend funcion... y x < > y.5. 5 Polynomial rend. 5 5. 4 Polynomial coefficiens: 8 5 3 7 4.6 4 6 8
A. Thiemer, 4 SLDG.mcd, 6..6 Example : Deerminisic seasonals To represen he model y y 4 le n = 4, C =, Α and < > y, y, y, y 3 x T Given a seasonal paern o he iniial vecor x < > T S S S 3 S 4, he sysem is compleed:.. x < > Α. x < >.. y x < > Deerminisic seasonal Seasonal paern: y S.5 S.3 S 3.6 5 5 S 4
A. Thiemer, 5 SLDG.mcd, 6..6 Example 3: "Whie noise" The random variable ε follows a (,)-Gaussian disribuion. Now we generae a random sample for.. T max wih T max : ε. ln( rnd( ) ). cos(. π. rnd( ) ) Box-Muller-Transformaion 4 Whie noise ε 4 4 6 8 Wih w ε and A =, C = his ime series is a special case of our general linear model. We use his random sequence in he examples 4-8 below.
A. Thiemer, 6 SLDG.mcd, 6..6 Example 4: Sochasic seasonals Model: y α. y 4 w α Α and C We use he seasonal paern of example ogeher wih he random sequence of example 3... T max x < > Α. x < > C. w.. T max y x < > Noice: If α = ( uni roo) he sysem ends o display explosive oscillaions. If < α < ( no uni roo) he explosive oscillaions are no longer presen! 4 Sochasic seasonals y α.9 4 4 6 8
A. Thiemer, 7 SLDG.mcd, 6..6 Example 5: Random walk We call y j = w j a random walk (or maringale process) if w is a whie noise variable. If he random walk includes a deerminisic rend i is called a random walk wih drif: y j = w j δ δ. Now we redefine his process as a special case of our general model: Α C x < > δ δ.. T max < > Α. x < > C. w x.. T max y x < > Random walk wih drif y 8 6 δ δ.8 4 4 6 8
A. Thiemer, 8 SLDG.mcd, 6..6 Example 6: Univariae auoregressive process To represen he AR(4)-model y ρ. y ρ. y ρ. 3 y 3 ρ. 4 y 4 w we se: ρ ρ ρ 3 ρ 4 Α C x < > iniial T.. T max x < > Α. x < > C. w.. T max y x < > 5 Auoregressive process iniial ρ. y 5 ρ.3 ρ 3 ρ 4 4 6 8
A. Thiemer, 9 SLDG.mcd, 6..6 Example 7: Growh wih homoskedasic and heeroskedasic noise The firs order auoregression [AR()-process] y ρ. y w wih ρ> describes a process, where he mean level is growing exponenially a rae ρ per period. The endency for he randomness dies ou, in he sense ha he one-sep ahead predicion error variance remains uniy ( homoskedasic noise). Now we modify his process a lile bi: y ρ. y ω wih ω ρ.5 This specificaion makes he variance of ω equal o ρ. Hence his variance is ime dependen ( heeroskedasic noise). Boh processes have he ransiion marices... w ρ Α C For a beer comparison, boh processes should sar wih he same iniial value: < > x iniial T < > x iniial T < >.. T max x < > x < > Α. x C. w < > Α. x C. ω < >.. T max y x y < > x
A. Thiemer, SLDG.mcd, 6..6 5 Sochasic growh iniial y ρ.5 y 5 4 6 8 homoskedasic growh heeroskedasic growh Look how he variance Var ω ρ increases hrough ime: 5 Time dependen variance ρ 5 4 6 8
A. Thiemer, SLDG.mcd, 6..6 Example 8: Univariae auoregressive moving average process Consider he model of an ARMA(,)-process: y ζ. y ξ. w ξ. w wih ζ < and ξ < ξ We define he sae as: x < > y ξ. w The ransiion marices and iniial vecor are: Α ζ C ξ x iniial 3 ξ.. T max x < > Α. x < > C. w.. T max y x < > 4 ARMA[,]-process y iniial 3 ζ.8 ξ.7 ξ. 4 4 6 8
A. Thiemer, SLDG.mcd, 6..6 Example 9: Vecor auoregressive process We wan o simulae a VAR()-process: z a. z a. z b. z b. z w z a. z a. z b. z b. z w A firs we generae w, w T which is a Gaussian disribued vecor whie noise wih ideniy covariance marix:.. T max w. ln( rnd( ) ). cos (. π. rnd( ) ) w. ln( rnd( ) ). cos (. π. rnd( ) ) Box-Muller-Transformaion Second order whie noise 3 w 3 3 3 ime pah sar end w
A. Thiemer, 3 SLDG.mcd, 6..6 Sae vecor : Whie noise vecor: z x < > z z w < > w w z Ener parameers of he ransiion marix: a a b b a.9 a.5 b.5 b. Α a a b b a.4 a.6 b.75 b. C x < >.. T max x < > Α. x < > C. w < >.. T max z x < > z x < >
A. Thiemer, 4 SLDG.mcd, 6..6 VAR()-process z 5 z 5 4 6 8 An impulse response funcion depics he response of he curren and fuure values of z o an imposiion of a random shock w ( = "innovaion"). Here we simulae he response over T max periods of he wo variables z and z o he firs innovaion w. x < > w < >.. T max x < > Α. x < > C. w < >.. T max z x < > z x < >
A. Thiemer, 5 SLDG.mcd, 6..6 Response o firs innovaion z.5 z.5 4 6 8 And now he response o he second innovaion w : < > x.. T max x < > Α. x < > C. w < >.. T max z x < > z x < > Response o second innovaion.8 z.6 z.4.. 4 6 8
A. Thiemer, 6 SLDG.mcd, 6..6 I's Your Turn! Hin: To ge a new random sample of whie noise for he same se of parameers choose "compue workshee" from he MATHCAD menu... Simulae pure random walks (δ δ ). Simulae a MA()-process (ζ ). Lieraure: Hansen, L.P./Sargen, T.J.: Recursive linear models of dynamic economies. Ch.., 998 hp://www.sanford.edu/~sargen/hansen.hml