Dynamic models for largedimensional vecor sysems A. Principal componens analysis Suppose we have a large number of variables observed a dae Goal: can we summarize mos of he feaures of he daa using jus a few indicaors? Yields on U.S. Treasury securiies (3 monhs o 10 years) 15.0 12.5 GS3M GS6M GS1 GS2 GS5 GS10 10.0 7.5 5.0 2.5 0.0 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012 n 1 vecor of saionary observaions i T 1 T y i (mean of variable i ii T 1 T y i i 2 y i 1/2 ii y i i y 1,..., y n T 1 T (sample correlaion marix) 1
Goal is o find a scalar and n 1 vecor h so as o minimize T h h Noe: h and are no unique (h h for h qh, q 1 ) bu h is unique. One normalizaion: h h 1. h Scalar explains as much of variaion of as possible. Soluion is called he "firs principal componen" of (deermined up o arbirary scale facor). Elemens of vecor h are called "facor loadings". min h, 1,..., T T h h Concenrae objecive funcion: (1) for any h, find bes 1,..., T (2) subsiue h ino objecive and min wih respec o h 2
(1) for fixed h: min 1,..., T T h h min h h OLS regression of on h h h h 1 h h h I n hh h 1 h (2) minimize over h: min T h h h h h T y I n hh h 1 h max T y hh h 1 h h subjec o h h 1 T h y h h T h Th h max h h h subjec o h h 1 3
Consider eigenvalues of x i ix i for i 1,..,n diag 1,.., n X x 1 x n X X I n X X X X max h h subjec o h h 1 h Le h Xh where X X I n and X X max h h subjec o h h 1 h max h X Xh subjec o h h 1 h h X Xh h h h 1 2 1 h n 2 n max h 2 1 1 h 2 n n h s.. h 1 2 h n 2 1 Soluion: h 1 1 h 2 h 3 h n 0 h x 1 4
Conclusion: he facor loadings are given bhe eigenvecor of associaed wih larges eigenvalue. The firs principal componen is given by h, he produc of his eigenvecor wih de-meaned daa vecor. Example: ineres raes for monh for U.S. Treasury securiies wih mauriies 3m, 6m, 1y, 2y, 5y, 10y n 6 1982:M1-2013:M5 T 1 T Eigenvecor of associaed wih larges eigenvalue: (0.3999, 0.4153, 0.4244, 0.4344, 0.4061, 0.3659) 1/ 6 0.4082 Conclusion: firs principal componen is esseniallhe average of he 6 yields. 5
15.0 12.5 GS10 GS5 GS2 GS1 GS6M GS3M 10.0 7.5 5.0 2.5 0.0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 25 20 15 10 5 0-5 -10-15 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Fied value for yield i: y i i h i 15.0 12.5 GS3M GS3M_PRED GS10 GS10Y_PRED 10.0 7.5 5.0 2.5 0.0-2.5 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 2012 6
Could also ask: suppose I could use 2 variables o summarize he 6 yields. Choose 21 vecor for 1,...,T and n 2 marix H o minimize T H H. Again no unique: Q nonsingular 2 2 marix H HQ Q 1 H H. Normalize H H I 2. min H H y I n HH H 1 H max T y HH H 1 H H T raceh H 1 H y H TH H 1 race H H 7
Soluion: H is in he linear space spanned bhe eigenvecors of associaed wih he wo larges eigenvalues. Second principal componen refers o h 2 for h 2 he eigenvecor of associaed wih he second larges eigenvalue. Noe second PC is orhogonal o he firs: T h 1 y h 2 Th 1 h 2 T 2h 1 h 2 0 Ineres raes: eigenvalues of Ω Eigenvalue Percen 1 56.0309 0.980548 2 1.0423 0.998789 3 0.0519 0.999697 4 0.013 0.999924 5 3.05E-03 0.999978 6 1.28E-03 1 8
GS10 GS5 GS2 GS1 GS6M GS3M Facor loadings associaed wih firs hree principal componens 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 0 12 24 36 48 60 72 84 96 108 120 level (facor 1) slope (facor 2) curvaure (facor 3) ineres raes 15.0 10.0 5.0 0.0 25 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 PC1 15 5-5 -15 3 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 PC2 1-1 -3 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 PC3 1.0 0.6 0.2-0.2-0.6 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Dynamic models for largedimensional vecor sysems A. Principal componens analysis B. Dynamic facor models 9
observed variables unobserved facors u H nr rr D u nn u v D diagd 1, d 2,...,d n H nr Assumpion 1: n 1 H H n u Q H rr wih rank(q H r. Means facors maer for more han jus finie subse of and are differen from each oher (columns of H no oo similar). Assumpion 2: maximum eigenvalue of Eu u is c for all n. Means u does no have is own facor srucure. (e.g., for Eu u 2 11 hen 1 is eigenvecor wih eigenvalue 2 1 1 2 n 10
Suppose hese assumpions held and here was ann r marix W such ha: (i) n 1 W W I r (ii) n 1 W H rr (iii) rank( r For yields example and r 1, W 1, 1,..., 1 H nr u n 1 W n 1 rn rn n 1 W u rn W H nr n 1 W u p 0 (e.g., n 1 n i1 u i p 0 Sock and Wason (JASA, 2002) showed ha under relaed assumpions, he firs r principal componens of provide a consisen esimae of for some nonsingular r r marix. 11
However, lieraure on erm srucure of ineres raes suggess ha firs 3 PC of ineres raes do no span se of linear combinaions mos useful for forecasing (e.g., Gregory Duffee, "Informaion in (and no in) he erm srucure," Rev Financial Sudies, 2011) Selecing he number of facors r V r H r, r H H nr H, arg min H, 1,..., T subjec o H H I r H H Bai and Ng, Economerica (2002): Choose r o minimize log V r H r, r r nt logminn,t nt 12
Ahn and Horensein, Economerica (2013): T 1 T 1 larges eigenvalue of n smalles eigenvalue of Choose r o be value for which r/ is larges. Dynamic models for largedimensional vecor sysems A. Principal componens analysis B. Dynamic facor models C. Nowcasing wih large jagged-edge realime daase Giannone, Reichlin, and Small, JME, 2008 Suppose we have poenially hundreds of differen monhly indicaors, only some of which are currenly available. Wha is he opimal esimae of wha his quarer s GDP growh will urn ou o be? 13
n 1 vecor of saionary indicaors ha will evenually be available for monh y i y i y i / i. Calculae firs r PC using larges T for which full sample is available y i i h i w i Ew 2 2 i r i Could calculae analyically or wih OLS regressions for i 1,..., n H nr w Ew w R diagr 1 2,..., r n 2 obs eq for sae-space model F rr Ev v Q v sae eq (could esimae by OLS) 14
Conclusion: we could use Kalman filer o obain opimal esimae of facors for las dae T for which we have complee daa, T y T,..., y 1 N T T, P T T, and forecas for T 1: T1 y T,..., y 1 N T1 T, P T1 T. If we had full observaion of y T1, we would updae inference wih T1 T1 T1 T P T1 T HH P T1 T H R 1 T1 T1 y T1 H T1 T If insead for some day T1 we only have some subse of y T1, jus se rows of H and T1 corresponding o missing obs equal o 0: T1 T1 T1 T P T1 T H T1 H T1 P T1 T H T1 R 1 T1 T1 i,t1 T1 i,t1 T1 y i,t1 i h i T1 T if i is observed 0 if i is no observed 15
If our goal is o esimae GDP growh for quarer q, regress i on facor value in hird monh of quarer (q using full se of observed daa: y q q e q q 1, 2,..., Q, Q T ŷ q Tk T3 Tk where for example F 2 T3 T1 T1 T1 16
Sample of news releases during week of Oc 21, 2013 Dae 21-Oc Time (ET) Saisic For Acual Briefing Marke Forecas Expecs Prior Revised From 10:00 AMExising Home Sales Sep 5.29M 5.15M 5.30M 5.39M 5.48M 22-Oc 8:30 AMNonfarm Payrolls Sep 148K 165K 183K 193K 169K 22-Oc 8:30 AMUnemploymen Rae Sep 7.20% 7.30% 7.30% 7.30% - 22-Oc 8:30 AMHourly Earnings Sep 0.10% 0.20% 0.20% 0.30% 0.20% 24-Oc 8:30 AMIniial Claims Oc 350K 330K 341K 362K 358K 25-Oc 8:30 AMDurable Orders Sep 3.70% 4.20% 3.50% 0.20% 0.10% Durable Goods -ex 25-Oc 8:30 AMransporaion Sep -0.10% 0.30% 0.30% -0.40% -0.10% Michigan Senimen - 25-Oc 9:55 AMFinal Oc 73.2 73 74.5 75.2 - Source: hp://biz.yahoo.com/c/e.hml Nowcass of 2013:Q4 U.S. real GDP growh (quarerly rae) Oc 22: 0.81 Oc 24: 0.76 Oc 25: 0.71 Source: www.now-casing.com Nowcass of 2013:Q3 U.S. real GDP growh rae (quarerly rae) Source: www.now-casing.com 17
Forecass of 2014:Q1 U.S. real GDP growh rae (quarerly rae) Source: www.now-casing.com Dynamic models for largedimensional vecor sysems A. Principal componens analysis B. Dynamic facor models C. Nowcasing wih large jagged-edge realime daase D. Facor-Augmened Vecor Auoregressions (FAVAR) Bernanke, Boivin and Eliasz, QJE, 2005 n 1 vecor of observed variables n 120 x m 1 subse of of special ineres or imporance. BBE ake x r (he fed funds rae) or x fed funds rae, indusrial producion, and inflaion, in deviaions from heir means. 18
Facor-Augmened VAR: x m1 11 L rr 21 L mr 12 L rm 22 L mm x m1 1 2 m1 ij L 1 ij L 1 2 ij L 2 p ij L p Could esimae space spanned by bha spanned by, he firs r principal componens of. Quesion: how o idenify moneary policy shock? Noe: since r is included in, each elemen of H is linear funcion of r. Claim: a moneary policy shock does no affec "slow-moving variables" (wages, prices) in he curren monh. 19
y subse of ha is "slow-moving" firs r PC of firs r PC of y (1) regress i i i r e i for i 1,...,r (2) Calculae i i i r (3) Esimae VAR for x,r x Lx (4) Calculae nonorhogonalized impulse-response funcion L I L 1 s x s and Cholesky facorizaion T 1 T P P 20
(5) Effec of moneary policy shock (u M on x x is x s u M sp for p he las column of P (6) Since H nr h r r, effec of moneary policy on any variable is s u M H h r sp Effecs of moneary policy shock 21
Effecs of moneary policy shock Effecs of moneary policy shock Bu how do we updae his approach now ha fed funds rae is suck a zero? Fischer Black, Journal of Finance, 1995: Can hink of laen or shadow shor rae ha is allowed o be negaive Acual shor rae is maximum of his and (say) 0.25 Wu and Xia (UCSD, 2013) develop convenien algorihm o calculae shadow rae 22
Wu and Xia fail o rejec hypohesis ha FAVAR coefficiens since 2009 are same as hose using fed funds for hisorical daa Daa available a hp://econweb.ucsd.edu/~faxia/policyrae.hml 23