Mind the gap! Stylized facts and structural models. Fabio Canova, - PowerPoint PPT Presentation

Mind the gap! Stylized facts and structural models. Fabio Canova, Norwegian Business School and CEPR Filippo Ferroni, Chicago Fed June 2019

Introduction � Common in macroeconomics to compare dynamics induced by distur- bances using SVAR and DSGE; see e.g. Gali (1999); Christiano et al. (2005); Iacoviello (2005), Basu and Bundik (2017), etc. � Non-invertibility/truncation problems: Ravenna (2007), Fernandez et al. (2007), Giacomini (2013), Plagborg-Moller (2018), Pagan and Robinson (2018), Chahrour and Jurado (2018). � Typically, if VAR has q shocks, use a theory with q or less disturbances. Does the DGP only has q disturbances? Which ones? � Here DGP has q disturbances; empirical model q 1 < q variables. Iden- ti�ed shocks and identi�ed dynamics become mongrels with little eco- nomic interpretation.

� Cross sectional deformation : - Identi�ed shocks need not combine "types" of structural disturbances. - Appropriate theoretical restrictions may be insu�cient. Di�cult to match e.g, identi�ed technology shocks to TFP disturbances . � Time deformation : - Identi�ed shocks are, in general, linear combinations of current and past structural disturbances. Perceived internal transmission stronger than in the DGP .

Punchlines � VARs can not be too small: di�cult to make sense of identi�ed shocks. � If VARs can not be su�ciently large, compare data VARs with the theory reduced to the same VAR observables. Some structural disturbances may not be obtained from a given VAR . � (Corollary) VARs used to derive dynamic facts might change depending on the DGP and the disturbances of interest. To identify monetary policy disturbances may need VARs with di�erent variables if the DGP has �nancial disturbances or not .

� Deformation vs. invertibility. - Problems distinct. - Long lags do not help to reduce cross sectional deformation. � Early literature: Lutkepohl (1984), Hansen and Sargent (1991), Marcet (1991), Braun and Mittnik (1991), Faust and Leeper (1998), Forni and Lippi (1999). � Related literature: Canova and Sahneh (2018), Wolf (2018).

Intuition � Growth model with log preferences, full depreciation, iid shocks to TFP ( Z t ), investment ( V t ), preferences ( B t ). Solution: ��V t Z t K � K t +1 = (1) t (1 � �� ) B t Z t K � C t = (2) t Z t K � Y t = (3) t � System invertible if 0 � � < 1. � Recursive system. All three shocks identi�able if VAR has three variables.

� System 1: (log K t +1 ; log Y t ) log K t +1 = log( �� ) + � log k t + u 1 t (4) log Y t = � log k t + u 2 t (5) u 1 t = log V t + log Z t , u 2 t = log Z t . Cannot recover B t ! cross sec- tional deformation. System maintain recursivity: identi�cation works for log V t ; log Z t . � System 2: (log K t +1 ; log C t ) log K t +1 = log( �� ) + � log K t + u 1 t (6) log C t = log(1 � �� ) + � log K t + u 2 t (7) u 1 t = log V t + log Z t , u 2 t = log B t + log Z t . u t mix demand and supply disturbances. Cross sectional deformation. Recursivity lost; identi�cation does not work.

� System 3: (log C t ; log Y t ) log C t = � log( �� ) + � log C t � 1 + u 1 t (8) log Y t = � log( �� ) + � log Y t � 1 + u 2 t (9) - u 1 t = log B t � � log B t � 1 + log Z t + � log V t � 1 . - u 2 t = log Z t + � log V t � 1 . - Time and cross sectional deformation. - Impossible to go from u jt ; j = 1 ; 2 to demand and supply disturbances. - Dynamics to identi�ed u jt shocks more persistent than dynamics to log B t ; log V t ; log Z t .

Relationship structural disturbances/empirical innovations � (Log-) linear DGP: x t = A ( � ) x t � 1 + B ( � ) e t (10) y t = C ( � ) x t � 1 + D ( � ) e t (11) x t is k � 1 vector of endogenous and exogenous states, e t � (0 ; �), � diagonal, is q � 1 vector of disturbances, y t is m � 1 vector of endogenous controls. A ( � ) is k � k , B ( � ) is k � q , C ( � ) is m � k , D ( � ) is m � q , � structural parameters. � Observables z it = S i [ x t ; y t ] 0 , S i is q i � q matrix.

Case 1: Empirical system eliminates some controls � S 1 = [ I; S 12 ] � Innovations u 1 t generated by u 1 t = z 1 t � E [ z 1 t j � 1 t � 1 ] � z 1 t � ~ F 1 z 1 t � 1 (12) Proposition 1 i) u 1 t = � 1 ( � ) e t , where � 1 ( � ) is q i � q . ii) A su�cient condition for the identi�cation of e j it is that the k-th row ! B ( � ) of G 1 ( � ) � has at most one non-zero element in the j-th S 12 D ( � ) position. � Related to Faust and Leeper (1998).

� Cases 2-3: The empirical system eliminates/repackages states � S 2 = [ S 21 ; S 22 ]; S 3 = [ S 31 ; 0]. � Innovations u it ; i = 2 ; 3 generated by u it = z it � E [ z it j � it � 1 ] � z it � ~ (13) F i z it � 1 Proposition 2 i) u it = � i ( �; L ) e t , � i is q i � q , each L , i=2,3. ii) u it = i ( �; L ) u 1 t ; i = 2 ; 3.

Dynamics Proposition 3 ! A ( � ) i) If a shock can be identi�ed from u 1 t and if ~ F 1 = , structural S 12 C ( � ) dynamics in the empirical system proportional to those of the DGP. ii) With u it ; i = 2 ; 3 responses to identi�ed shocks distorted at all horizons. � Braun and Mittnik (1991): expression for response biases in VARs.

An example 1 h = 1 � h g t +1 + 1 � h g t + r t � � t +1 (14) � t � t +1 � � � h � t = � t +1 � + k p 1 � h g t + (1 + � n ) n t + k p ( � t � � t ) (15) o t = � t + (1 � � ) n t (16) � � r t = � r r t � 1 + (1 � � r ) � y g t + � p � t + " t (17) g t = a t + o t � o t � 1 (18) = � z � t � 1 + " zt (19) � t a t = � a a t � 1 + " at (20) � t = � � � t � 1 + " �t (21) � t = � � � t � 1 + " �t (22) � t = " mpt (23)

� Minimal state vector x t � 1 = [ o t � 1 ; r t � 1 ; � t � 1 ; a t � 1 ; � t � 1 ; � t � 1 ] 0 (6 � 1) � Control vector y t = [ g t ; o t ; � t ; n t ; r t ] 0 (5 � 1). � Shock vector e t = [ " z t ; " a t ; " � t ; " � t ; " mp t ] 0 (5 � 1) � Set: � = 0 : 33; � = 0 : 99; � n = 1 : 5; h = 0 : 9; k p = 0 : 05; � y = 0 : 1; � p = 1 : 5; � r = 0 : 8; � z = 0 : 5; � a = 0 : 2; � � = 0 : 5; � � = 0 : 0. � How would the shocks/dynamics of an empirical system with q 1 � 4 compare with the shocks/dynamics of the original model?

� System with z t = ( o t ; � t ; n t ; r t ). 1 h � t = � t +1 � 1 � h ( a t +1 + o t +1 � o t ) + 1 � h ( a t + o t � o t � 1 ) + r t � � t +1 (24) � � h � t = � t +1 � + k p 1 � h ( a t + o t � o t � 1 ) + (1 + � n ) n t + k p ( � t � � t ) (25) = � t + (1 � � ) n t (26) o t � � = � r r t � 1 + (1 � � r ) � y ( a t + o t � o t � 1 ) + � p � t + " mpt (27) r t � State vector: x t � 1 = [ o t � 1 ; r t � 1 ; � t � 1 ; a t � 1 ; � t � 1 ; � t � 1 ] 0 . � Law of motion of the states (A, B matrices) unaltered. � Cross sectional deformation, no time deformation distortions .

� System with z t = ( o t ; � t ; n t ). 1 � h ( a t +1 + o t +1 � o t ) + ( h + � r 1 ((1 + � r ) � � r L ) � t = � t +1 � 1 � h + (1 � � r ) � y ) ( a t + o t � o t � 1 ) ( h� r � 1 � h ) ( a t � 1 + o t � 1 � o t � 2 ) + ( � r + (1 � � r ) � p ) � t + " mpt � � t +1 (28) � � h � t = � t +1 � + k p 1 � h ( a t + o t � o t � 1 ) + (1 + � n ) n t + k p ( � t � � t ) (29) o t = � t + (1 � � ) n t (30) x t � 1 = [ o t � 1 ; o t � 2 ; � t � 1 ; a t � 1 ; � t � 1 ; � t � 1 ] 0 . � State vector: ^ � Law of motion of the states (A, B matrices) altered. � Cross-sectional and time deformation distortions .

� System with z t = ( � t ; n t ; r t ). 1 � t = � t +1 � 1 � h ( a t +1 + � t +1 � � t + (1 � � ) ( n t +1 � n t )) h + 1 � h ( a t + � t � � t � 1 + (1 � � ) ( n t � n t � 1 )) + r t � � t +1 (31) � � h � t = � t +1 � + k p 1 � h ( a t + � t � � t � 1 + (1 � � ) ( n t � n t � 1 )) + (1 + � n ) n t + k p ( � t � � t ) (32) � � r t = � r t � 1 + (1 � � ) � y ( a t + � t � � t � 1 + (1 � � ) ( n t � n t � 1 )) + � p � t + " mpt (33) x t � 1 = [ n t � 1 ; r t � 1 ; � t � 1 ; a t � 1 ; � t � 1 ; � t � 1 ] 0 . � State vector: ^ � Law of motion of the states unchanged (given production function n t � 1 proxies for o t � 1 ). � Cross-sectional deformation, limited time deformation distortions .

Cross correlation function: z t = ( o t ; � t ; n t ; r t )

Cross correlation function: z t = ( o t ; � t ; n t )

Cross correlation function: z t = ( � t ; n t ; r t )