Linear Dnamic Models and Forecasing Reference aricle: Ineracions beween he muliplier analsis and he principle of acceleraion Ouline. The sae space ssem as an approach o working wih ssems of difference equaions. Analsis of he ssem, focusing on he Samuelson model
. Placing he Samuelson model in sae space form Consumpion funcion c Invesmen funcion = α i = β ( c c ) Implicaion = α( + β ) αβ + g 3 An approach o analzing ssems of difference equaions We are going o use a sandard framework for suding dnamic ssems. I is comprised of wo groups of equaions Equaions describing how observables relae o saes. Each of hese akes he form of a linear relaionship, so ha we have a marix ssem of he form (if we absrac from consan and rend erms) 4
Y S S S j = π j + π j +... + π jn n Y Y π π π n S Y π π π n S = =...... Ym πm πm πmn Sn 5 And he saes evolve as S = M S +... M S + G e +... G e j j, jm m, j jn n S S M M M, S S M M M m, = =...... S m Mm Mm Mmm Sm, S m + Ge SGZ [ more Macro 008: Lecure on 5A his par laer] 6
General framework Can be wrien in simple manner Y =Π S S = MS + Ge Is eas o sud and undersand d Can be used for man of differen problems Highlighs imporan consideraions 7 Solving his ssem recursivel Recursive subsiuion from iniial S, given pah of e s S = MS + Ge 0 S = MS + Ge = M[ MS + Ge ] + Ge 0 = + + M S0 MGe Ge 8
coninuing S = M S0 + M Ge + M Ge +... + MGe + Ge [ check : S = M S0 + MGe+ Ge] 9 So, where we are a depends on The iniial sae S 0 The series of e s ha ake place along he pah beween and How imporan hese e s are for S (G) Various powers of M [ver similar o firs order difference equaion, excep in vecors ] 0
Samuelson s s model (wihou g): sae evoluion = α( + β) αβ S = α( + β) αβ = = 0 MS Samuelson s s model (wihou g): oher variables 0 Y = c = 0 α =ΠS i α i = c = α
Samuelson s model (wih g) Assume g = g + e Implies S = g 3 S Samuelson s model (wih g): saes = g α ( + β ) αβ = 0 0 0 + e 0 0 g = MS + Ge 4
Samuelson s s model (wih g): oher variables Y g = S = c g i g 0 0 0 0 = c 0 α 0 g i α SGZ Macro 008: Lecure 5A 5. Analzing he ssem Qualiaive dnamics of ssem (as in Samuelson s regions) are governed b M. Characerisic roos of M are he soluions o he polnomial Iz-M =0 0, whereiisan ideni marix wih same dimension as M and X means deerminan of X. 6
Samuelson s model (wihou g) S = α ( + β) αβ = = 0 MS 7 Finding roos in Samuelson s model Calculaing Iz-M, we find he same resul as in Samuelson s foonoe #: roos are soluion o quadraic equaion 0 ( + α) β αβ 0 = 0 z 0 z ( + α) β αβ = z = z α( + β) z+ αβ 8
This is a quadraic equaion, so There are wo roos There can eiher be wo real roos or wo complex roos The roos can be of varie of sizes in erms of absolue value: Less han one (sable); Equal o one (uni, borderline); Greaer han one (unsable) Higher order polnomials also have roos wih las wo properies (real or complex, small or large) 9 Complex roos Will no be a special focus of our discussion, alhough he were a ke par of Samuelson s analsis (one definiion of business ccles ) Lead o oscillaor naure of dnamic responses (as in Samuelson s able ) Can eiher be sable (damped) or unsable (explosive) 0
3. Sochasic Ssems such as vecor auoregressions Suppose ha we rea he e in he sae ssem above as a series of unpredicable random variables, so ha our expecaion (forecas) of e +j given informaion a is zero. We wrie his as E e +j =0 Then, i is eas o produce a forecas (compue a condiional expecaion) And his forecas akes a simple form In words, he forecas depends on where we sar (iniial S) and he dnamic forces capured b M. ES j + j = M S + E M + j { Ge+... + MGe+ j + Ge+ j} j = M S
Forecasing oher variables We know ha Y is relaed o S b a simple expression, so we ge forecass ver easil Y =ΠS + j + j E Y =Π ES =ΠM S j + j + j 3 Srucure above absracs from consan erms (normal values) bu hese are eas o add, using eiher of wo mahemaicall equivalen approaches APPROACH : "previous ssem is deviaions from normal values" Y Y =Π( S S) S S = M( S S) + Ge APPROACH : "add consan erms o ever equaion, hen work ou normal values" Y A S S B MS = +Π = + + Ge S = ( I M) B; Y= A+ΠS 4
Puing he ssem o work Empirical models (like VARs) Linear raional expecaions models Relaed heoreical device Soluion of LRE model occurs in sae space form 5 Appendix. Analical framework for dnamic linear ssem. Eigenvecor-Eigenvalue decomposiion of M 3. Link o naure of soluion o second order difference equaion (eg Samuelson model) 6
. Analical quesion Recursive soluion akes form S = M S0 + M Ge + M Ge +... + MGe + Ge Wha is behavior of M raised o he j h power? 7 Ke properies from linear algebra An marix M which has disinc (no repeaed) roos can be decomposed ino M=PμP -, where P is a marix of eigenvecors (characerisic vecors) and μ is a diagonal marix wih he eigenvalues on he diagonal and P - is he inverse of P so ha PP - =I. Furher, M =(PμP - )(PμP - ) = (Pμ P - ) and more generall M j = (Pμ j P - ). Since μ is diagonal, μ j is a diagonal marix wih he eigenvalues raised o he jh power on he diagonal. 8
Samuelson model w/o g ( ) m m M α + β αβ = = 0 0 9 P Eigenvecors and Eigenvalues μ μ μ - = P = μ μ μ μ 0 μ = 0 μ μ = [ m+ m + 4 m] μ = [ m m + 4 m] 30
Powers of M j j M = ( Pμ P ) j + j + j + j + μ μ μμ + μμ = j j j j μ μ μ μ μμ + μμ = [ 0 ] j + j M S μ μ μ μ + μ μ = + j+ j+ j+ j+ μ μ μ μ 3 Soluion Hence, he soluion of he difference equaion is of he form described in he main lecure, = θ μ + θ μ θ = ( μ μ μ )/( μ μ ) 0 θ = ( μ + μ μ )/( μ μ ) 0 3