Linear state-space models

Transcription

1 Linear sae-space models A. Sae-space represenaion of a dynamic sysem Consider following model Sae equaion: F rr r r Observaion equaion: y n A x nkk v r H nr r Observed variables: y,x w n Unobserved variables:,v,w Marices of parameers: F, A, H v w ~ i.i.d. N 0 0, Q 0 0 R Q rr R n n

2 Example : 2 r r j, L j for j 2,3,...,r, 2 L 3 L 2 L, p L p Observaion equaion: y 2 r y L pu ogeher wih sae equaion: L Ly L Conclusion: any ARMA process can be wrien as a sae-space model. 2

3 Example 2: C sae of business cycle i idiosyncraic componen for secor i C, i unobserved y i growh in secor i (observed) C,, 2,..., n F v F C r Observaion equaion: y y 2 2 y n n n 0 0 3

4 Purpose of sae-space represenaion: sae vecor conains all informaion abou sysem dynamics and forecasing. F v y A x H w Ey j,,...,,y,y,...,y,x j,x j,...,x A x j H F j Linear sae-space models A. Sae-space represenaion of a dynamic sysem B. Kalman filer Purpose of Kalman filer: calculae disribuion of condiional on y,y,...,y,x,x,...,x ~ N,P 4

5 F v y A x H w v w ~ i.i.d. N 0 0, Q 0 0 R Begin wih he prior: 0 ~ N 0 0, P prior bes guess as o value of 0 P 0 0 uncerainy abou his guess (much uncerainy large diagonal elemens of P 0 0 ) F 0 v ~ N 0,P 0 0 F 0 0 P 0 FP 0 0 F Q 5

6 Useful resul: suppose ha y x y 2 x N 2, where i and ij may depend on x. Then y 2 y,x Nm,M m 2 2 y M Here y x, 0 x, 0 N 2, P 0 A x H 0 H P 0 H R 2 P 0 H Hence y,x, 0 N,P 0 P 0 HH P 0 H R y A x H 0 P P 0 P 0 HH P 0 H R H P 0 6

7 Idenical calculaions: if ~ N,P, hen ~ N, P P FP F Q P P P HH P H R H P F y A x H P HH P H R Ieraing on hese calculaions for,2,...,t o produce he sequences T P Kalman filer. T and is called he is he expecaion of given observaion of y, y,..., y, x, x,..., x. P E where hese expecaions condiion on he values of F, Q, A, H, R. 7

8 Forecasing: y A x H w Ey j,x j,f, Q, A, H, R A x j H F j MSE for j : Ey y y y H P H R Smoohed inference: migh also wan o form inference abou using all he daa T : T ~ N T,P T To derive formula, consider insead, ~ N, P Same kind of derivaion as for Kalman filer esablishes ha J J P F P P P J FP Generalizaion: wha if P is singular? 8

9 If P is singular, hen some linear combinaions of can be forecas perfecly from, implying inference abou given and hese linear combinaions of is idenical o inference abou given alone. Le be r and le he rank of P be s r. Define he s vecor H for an arbirary s r marix H such ha P H P H has rank s. For example, migh be he firs s elemens of in which case P would be firs s rows and columns of P. Then, has same disribuion as,. Generalizaion of previous resuls for singular P : J J P H F P P P J H FP 9

10 Nex suppose ha, in addiion o, we had also observed y, y 2,...,y T. This would conain no more informaion abou han was provided by and alone:, T ~ N,P for he same,p. And since, E, T J i follows from law of ieraed expecaions ha E T J T which we can calculae by ieraing backwards for T,T 2,... The MSE s of hese smoohed inferences are given by E T T P T where P T can be found by ieraing on P T P J H P T P H J backward saring from T. 0

11 Procedure o calculae smoohed inferences T T. () Perform Kalman filer recursion and save he values of,,p,p T. (2) Calculae J P H F H P H for,2,...,t, where H is an s r marix selecing he nonredundan elemens of. (3) Calculae T J H T for T where T T, T T, and T T are all known from sep ().

12 (4) Evaluae T J H T for T 2 where righ-hand variables are all known from sep (3). Ierae for T 3,T 4,... Linear sae-space models A. Sae-space represenaion of a dynamic sysem B. Kalman filer C. Maximum likelihood esimaion Saring value for 0: F v If eigenvalues of F are all inside uni circle, se 0 0 E 0 0 P 0 0 E 0 0 vecp 0 0 I r 2 F F vecq 2

13 Alernaively, can use any disribuion 0 N 0 0, P 0 0 Frequenis perspecive: his is uncondiional disribuion of observaion 0 Bayesian perspecive: his is prior beliefs abou 0 If diagonal elemens of P 0 0 are large (e.g. P I r, has lile influence on any resuls How o esimae unknown parameers? Le be vecor conaining unknown elemens of F, Q,A, H,R frequenis principle: chose so as o max likelihood funcion () pick an arbirary iniial guess for (2) run hrough he KF for his fixed calculaing P y A x H C H P HR 3

14 (3) since his implies y, x ; ~ N y, C, choose so as o maximize log likelihood: Tn log 2 log C 2 2 T y 2 y C y y by numerical search (given guess j, find a j associaed wih bigger value for T Classical economerician: choose Asympoic sandard errors from so as o maximize log likelihood: Tn N0 2 log2, C T log C 2 C T 2 logpy y y C y y 2 EM algorihm: convenien numerical algorihm for finding value of ha maximizes. Le py; denoe likelihood (join densiy of y,y 2,...,y T logpy; T logpy ; Consider py,; join densiy of y,y 2,...,y T,, 2,.., T if were observed. 4

15 logpy,; Tr log2 T log Q 2 2 race 2 Q T F F Tr log2 T log R racer T y A x H y A x H The EM algorihm is a sequence of values, 2,... such ha given, he value of maximizes log py,; py,; d ) Why does EM algorihm work? FOC will saisfy py,; py,; py,; d 0 5

16 If we had a fixed poin (, py,; py,; py,; d py,; py; 0 d py,; d so a fixed poin is he MLE Furhermore, i can be shown ha ha is, each sep of EM algorihm increases he log likelihood (2) How do we implemen EM algorihm? Suppose ha was observed direcly and we wan o choose o max logpy,; Tr log2 T log Q 2 2 race 2 Q T F F Tr log2 T log R racer T y A x H y A x H 6

17 Consider firs parameers of F. If we observed hese would be found from OLS regression of on : T T F. T T F In sep of he EM algorihm we don observe so don maximize log py,; bu insead max he expecaion of log py, ;. In oher words, we inegrae ou condiioning on daa Y and assuming he previous ieraion s value for.. E Y, P T T T S where T and P T denoe he esimaes coming from he Kalman smooher evaluaed a previous ieraion s value for. 7

18 Similarly E Y, for P, T T T S, T P, T J P T J P F P FOC if observed: T T F. Value of F chosen by sep of EM algorihm: T T S T F T T S, T F T T S T T T S, T. Likewise, if we were esimaing Q wih observed, we would max Tr 2 log2 T 2 log Q T 2 race Q F F Q T T F F. Sep of EM chooses Q T T S T F S, S, F F S T F. 8

19 Analogously, wih observed we would choose A, H o max Tr 2 log2 T 2 log R T 2 racer y A x H y A x H If were observed we would do OLS regression of y on z : z x A H T T z z y z. Sep of EM algorihm hus uses A H T y x T y T T x x T x T T T x T S T 9

20 Linear sae-space models A. Sae-space represenaion of a dynamic sysem B. Kalman filer C. Maximum likelihood esimaion D. Applicaions. Time-varying parameer models and missing observaions Suppose ha F, Q, A,H, R are known funcions of (or more generally, known funcions of x : F v Ev v Q y A x H w Ew w R Then Kalman filer recursion immediaely generalizes o: P F P F Q P P P H H F P H R H P y A x H P H H P H R 20

21 One simple rick for handling missing observaions: if observaion y i is missing for dae, se ih rows of A and H o zero, ake y i 0, se row i, col i of R o and all oher elemens of row i or col i of R o zero. Why i works: suppose for illusraion he firs r elemens of y are missing. A 0 Ã H 0 H R I r 0 0 R Then H 0 H P H 0 P H H P H H P H 2

22 P H H P H R 0 P H I r 0 0 H P H R 0 P HH P H R P H H P H R 0 P HH P H R P H H P H R acs as if firs r elemens of y weren here Linear sae-space models D. Applicaions. Time-varying parameer models and missing observaions 2. Using mixed-frequency daa as hey arrive in real ime 22

23 Pracical problem for economic forecasers: Differen daa are of differen, asynchronous frequencies and are subsequenly revised Example: Inroducing he Euro-Sing: Shor Term Indicaor of Euro Area Growh, Maximo Camacho and Gabriel Perez- Quiros Assumpion: here is an unobserved scalar f represening he monhly growh rae of real economic aciviy. z h 4 vecor of hard indicaors of f z h i z h z h 2 z h 3 z h 4 indusrial producion growh reail sales growh new indusrial orders growh Euro area expor growh k h i h h i f u i 23

24 z h i k h i h h i f u i f f a f a 2 f 2 a 6 f 6 f u h i N0, c h h i u i, c h h i2 u i,2 c h h h i,6 u i,6 i h i N0, 2 hi z h k h h h f u u h C h h u C h h 2 u 2 f, f,..., f 5, u h, u h C h h 6 u 6,..., u h 5 h Also have some sof survey measures inended o reflec year-over-year growh s z s z 2 s z 3 s z 4 s z 5 Belgium overall business indicaor Euro-zone economic senimen German IFO business climae Euro manufacuring purchasing managers index services PMI s z i s u i k s i is s j0 f j u i c s s i u i, c s i2 u i,2 s s c i,6 s u i,6 s i 24

25 q rue monhly growh rae of real GDP in deviaion from mean (no observed) q 3 q f u q u q c q q u c q q 2 u 2 c q q 6 u 6 q Every hree monhs we do observe a second revision of quarerly GDP growh y 2 k 2 q 3 2 q 3 q 2 2 q 3 3 q days earlier a more preliminary firs revision was available y y 2 e 2 20 days before ha he iniial flash esimae of GDP was released y 0 y e 25

26 Model also uses quarerly employmen growh. Poenial observaion vecor: y y 2,z h, z s,, y, y 0. Poenial observaion vecor: y y 2, z h,z s,,y, y 0. In every monh, some of hese (e.g., y 2,, and y 0 ) are reaed as missing observaions On any given day before he end of he monh, a smaller subse is observed. f, f,...,f, u q, u q,...,u q 5,... u h h,...,u 5,u s s,...,u 5,u,...,u 5 Model allows forecas of any variable using all informaion available as of any day 26

27 Real-ime forecass of 2007:Q4 real GDP growh from release of second revision on 2007/07/2 unil 2008/02/3 Linear sae-space models D. Applicaions. Time-varying parameer models and missing observaions 2. Using mixed-frequency daa as hey arrive in real ime 3. Esimaion of dynamic sochasic general equilibrium models Basic approach: () Find a sae-space model ha approximaes soluion o DSGE (2) Esimae parameers of DSGE by maximizing implied likelihood 27

28 Example: max c,k 0 E 0 0 log c s.. c k e z k k,2,... z z,2,... k 0,z 0 given N0, If,, 0,, soluion akes he form c ck, z ; k kk, z ; Problem: The funcions c. and k. canno be found analyically. () Take a fixed numerical value for,,,,. (2) Find values c, c k, c z, k, k k, k z as funcions of for which ck, z ; c c k k kc z z kk, z ; k k k k kk z z. 28

29 (3) If we hink of observed variables (c, k as differing from model analogs (c, k by measuremen or specificaion error, c c c k k k, hen resuling sysem has sae-space represenaion wih sae vecor k, z for k k k (deviaions from mean). sae equaion: k k k k k z z z z observaion equaion: c c c k k c z z c k k k k How do we achieve sep (2)? One approach: perurbaion mehods. Consider a coninuum of economies indexed by and use Taylor s Theorem o find approximaion in neighborhood of 0 (ha is, as economy becomes deerminisic). 29

30 Firs-order condiions: c E k expz c c k e z k k z z (2a) Find seady-sae (soluions for he case 0 : c c c k k k z 0 0 c k c c k k k In his case c and k can be found analyically: k c k k. 30

31 More generally, could be found numerically. (a) Make arbirary iniial guess ( 0 for c 0,k 0. (b) Calculae c k 2 c c k k k 2. (c) Find beer guessc, k unil objecive funcion accepably small. Wha if daa are nonsaionary? One approach: le c denoe some measure of derended consumpion, and assume c c c for c he magniude described by model. Alernaive approach: explicily model rend in z, find ransformaion in model ha induces a saionary magniude c, and apply same ransformaion o daa. (2b) Use Taylor s Theorem o find approx linear coefficiens. (Take case for illusraion) Wrie F.O.C. as E ak, z ;, 0 a k, z ;, ck,z ; kk,z ; expz ckk,z ;,z ; a 2 k, z ;, ck,z ;kk, z ; e z k k 3

32 Firs-order approximaion: Since E ak, z ;, 0 for all k,z ;, i follows ha E a k k,z ;, 0 for a k k,z ;, ak,z;, k likewise E a z k, z ;, E a k,z ;, 0 E c 2 a k,z ;, k kk,z 0,0 c k k2 c k k k c c 2 k k k Since c and k are known from previous sep, seing his o zero gives us an equaion in he unknowns c k and k k where for example c k ck,z ; k k k,z 0,0 a 2 k,z ; k k k,z 0,0 c k k k k This is a second equaion in c k, k k, which ogeher wih he firs can now be solved for c k, k k as a funcion of c and k 32

33 E a k,z ;, z c 2 c z k2 c k c 2 a 2k,z ; z c z k z k kk,z 0,0 c k k z c z kk,z 0,0 k z k c seing hese o zero allows us o solve for c z, k z a k,z ;, k k,z 0,0 c c 2 k2 c k c 2 a 2 k,z ; c k k k c c k k c z c k k,z 0,0 Taking expecaions and seing o zero yields c 2 c k2 c k k c 2 c k k c 0 c k 0 which has soluion c k 0 volailiy, risk aversion play no role in firs-order approximaion 33

34 In his example, values for c k, c z, k k, k z could be found analyically. More generally, we will have a quadraic sysem of equaions in unknowns like c k, c z, k k,k z. Common approach o solving: recognize as linear raional-expecaions model and use numerical mehods o find sable soluion (assuming exiss and is unique). In general, beer soluions obained (paricularly for rending daa) if linearize in logs raher han levels. c logc, k logk exp c E expk expz exp c exp c expk e z expk expk c ck,z ; k k k,z ; (3) Once we have sae-space represenaion for observed daa y c, k associaed wih his fixed, we can choose o maximize likelihood. 34

35 (3) since his implies y, x ; ~ N y, C, choose so as o maximize log likelihood: Tn log 2 log C 2 2 T y 2 y C y y by numerical search (given guess j, find a j associaed wih bigger value for T Cauion: he DSGE is ofen simulaneously () overidenified, (2) underidenified, and (3) weakly idenified. () Overidenificaion: The DSGE implies a sae-space model F v y ah w where F, a, H saisfy complicaed nonlinear resricions. A specificaion wih less resricive F, a, H may fi daa much beer. 35

36 Empirical applicaions ypically have much richer dynamics han simple heoreical models. Example: Smes and Wouers, Journal of European Economic Associaion, insananeous uiliy funcion: U a b c C hc c a L h 0 habi persisence a b b b b a b i.i.d. N0, 2 b shock o ineremporal subs a L L L L a shock o inraemporal subs Le C denoe deviaion of logc from is seady-sae value () C h h C h h h c h R E h c â b b E â E C 36

37 capial evoluion: K K Sa I I /I I S. adjusmen coss a I I I a I (2) K K Î (3) Î Î E Î /S" Q Eâ I â I Q value of capial sock (4) Q R r k E Q r k Q r k E r K, r K rae of reurn o capial Q acked on 37

38 oupu from producer of inermediae good of ype j j y a a K j L j fixed cos a a produciviy shock a a a a a a L j aggregae of labor hired from each household L j /w, d w, 0 w w, w w shock o workers marke power wage sickiness: a fracion w of workers are no allowed o change heir wage bu insead have heir wage increase from he previous value by P /P 2 w w degree of indexing 38

39 (5) w Ew w E w w h w w L L c h C hc â L w h w w w w L/ w w labor demand W L j r K,z K j z capial uilizaion (6) L w r K, K parameer based on cos of uilizing capial inermediae goods sold o final goods producer wih marke power of firm j governed by p, p p prices p fracion allowed o adjus p indexing parameer 39

40 (7) p E h p p p r K, w â a p h p p p p p goods marke equilibrium (8) Y K/Y G/Y C K/YÎ G/Yâ G producion funcion hen deermines r K, (9) Y â a K r K, L s.s. coss moneary policy (Taylor Rule) (0) R R r r Y Y Y P r r Y Y p Y Y p Y R inflaion arge Y p oupu level if prices perfecly flexible 40

41 y C,C, R,R, K,K,Î,Î, Q, w, w,l,,, Y,r K, x â b,â I, Q, â L, w, â a, p,â G,, R equaions ()-(0) (along wih lag definiions) can be wrien as AE y By Cx while shocks saisfy x x (noe also E x x ) (2) A he same ime ha DSGE implies refuable overidenifying resricions, iself may be unidenified (Komunjer and Ng, Economerica, 20). Wih Gaussian errors, he observable implicaions of he sae-space represenaion are enirely summarized by y N, for y y,..., y T a T a known funcion of e.g., diagonal blocks of given by H H R vec I r 2 FF vecq 4

42 Model is unidenified a 0 if here exiss a 0 such ha a 0 a 0 Can check his locally by numerically calculaing derivaives wih respec o (3) Finally, oher parameers of may only be weakly idenified 0 bu 0 small for 0 large. Common approach: fix some parameers such as using a priori informaion. Bayesian esimaion wih informaive priors can help wih some of hese numerical problems. 42