Dynamic Factor Models Cointegration and Error Correction Mechanisms

Transcription

1 Dynamic Factor Models Cointegration and Error Correction Mechanisms Matteo Barigozzi Marco Lippi Matteo Luciani LSE EIEF ECARES Conference in memory of Carlo Giannini Pavia 25 Marzo 2014

2 This talk Statement of the Problem Outline Non-Stationary Dynamic Factor Models Macroeconomic data Impulse-response functions Matteo Barigozzi Factor Models and Cointegration 2/32

3 This talk Statement of the Problem Outline Non-Stationary Dynamic Factor Models Macroeconomic data Impulse-response functions 1 Representation results 2 Information criterion 3 Estimation of non-stationary factors Stationary Dynamic Factor Models (DFM) Forni, Hallin, Lippi & Reichlin (2000); Stock & Watson (2005); Forni, Giannone, Lippi & Reichlin (2009) Cointegration, Error Correction Mechanisms (ECM) Engle & Granger (1987); Johansen (1988, 1991); Stock & Watson (1988) Singular stochastic processes Anderson & Deistler (2008a,b) Matteo Barigozzi Factor Models and Cointegration 2/32

4 Literature Statement of the Problem Introduction Factor models have become increasingly popular Nowadays commonly used by policy institutions They have proven successful 1 Forecasting Stock & Watson (2002); Forni, Hallin, Lippi & Reichlin (2005); Giannone, Reichlin & Small (2008); D Agostino & Giannone (2012) 2 Structural analysis Giannone, Reichlin & Sala (2005); Forni, Giannone, Lippi & Reichlin (2009); Forni & Gambetti (2010); Barigozzi, Conti & Luciani (2013); Luciani (2013) Matteo Barigozzi Factor Models and Cointegration 3/32

5 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated Matteo Barigozzi Factor Models and Cointegration 4/32

6 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Generalized DFM Forni, Hallin, Lippi & Reichlin (2000) x t n 1 = χ t n 1 + ξ t n 1 = B(L) χ t n 1 n q u t q 1 Matteo Barigozzi Factor Models and Cointegration 4/32

7 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 = Λ n r F t r 1 Matteo Barigozzi Factor Models and Cointegration 4/32

8 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 Matteo Barigozzi Factor Models and Cointegration 4/32

9 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 q < r Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛC(L) Matteo Barigozzi Factor Models and Cointegration 4/32

10 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 When q < r = Λ n r F t r 1 = C(L) r q u t q 1 q < r Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛC(L) Matteo Barigozzi Factor Models and Cointegration 4/32

11 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 When q < r generically exists q < r Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛC(L) Matteo Barigozzi Factor Models and Cointegration 4/32

12 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 q < r Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛC(L) When q < r generically exists a finite autoregressive representation Anderson & Deistler (2008a,b) Example Matteo Barigozzi Factor Models and Cointegration 4/32

13 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) D(L) r r x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(0) r q u t q 1 q < r Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛC(L) When q < r generically exists a finite autoregressive representation Anderson & Deistler (2008a,b) Example Matteo Barigozzi Factor Models and Cointegration 4/32

14 Statement of the Problem Introduction Stationary Dynamic Factor Models Fluctuations in the economy, x t I(0), are due to: 1 a few structural shocks, ut, orthogonal white noise 2 several idiosyncratic shocks, ξt, possibly weakly correlated 3 Restricted Generalized DFM Forni, Giannone, Lippi & Reichlin (2009) D(L) r r x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(0) r q u t q 1 q < r When q < r generically exists a finite autoregressive representation Anderson & Deistler (2008a,b) Giannone et al., 2005, Amengual & Watson, 2007, Forni & Gambetti, 2010, and Luciani, 2013 for the US, Barigozzi et al., 2013 for the Euro Area Φ(L) = ΛD(L) 1 C(0) Example Matteo Barigozzi Factor Models and Cointegration 4/32

15 Statement of the Problem Limitations of I(0) Setting Permanent and Transitory Shocks So far, DFM in a stationary setting Exceptions: Bai (2004); Bai & Ng (2004); Peña & Poncela (2004) Other exceptions: Eickmeier (2009); Forni, Sala & Gambetti (2013) Matteo Barigozzi Factor Models and Cointegration 5/32

16 Statement of the Problem Limitations of I(0) Setting Permanent and Transitory Shocks So far, DFM in a stationary setting Exceptions: Bai (2004); Bai & Ng (2004); Peña & Poncela (2004) Other exceptions: Eickmeier (2009); Forni, Sala & Gambetti (2013) When x t and F t are stationary all common shocks have a permanent effect Is this plausible? Matteo Barigozzi Factor Models and Cointegration 5/32

17 Statement of the Problem Limitations of I(0) Setting Permanent and Transitory Shocks So far, DFM in a stationary setting Exceptions: Bai (2004); Bai & Ng (2004); Peña & Poncela (2004) Other exceptions: Eickmeier (2009); Forni, Sala & Gambetti (2013) When x t and F t are stationary all common shocks have a permanent effect Is this plausible? NO in Macroeconomic theory Matteo Barigozzi Factor Models and Cointegration 5/32

18 Statement of the Problem Limitations of I(0) Setting Permanent and Transitory Shocks So far, DFM in a stationary setting Exceptions: Bai (2004); Bai & Ng (2004); Peña & Poncela (2004) Other exceptions: Eickmeier (2009); Forni, Sala & Gambetti (2013) When x t and F t are stationary all common shocks have a permanent effect Is this plausible? NO in Macroeconomic theory technology shocks monetary policy shocks Matteo Barigozzi Factor Models and Cointegration 5/32

19 Statement of the Problem Limitations of I(0) Setting Permanent and Transitory Shocks So far, DFM in a stationary setting Exceptions: Bai (2004); Bai & Ng (2004); Peña & Poncela (2004) Other exceptions: Eickmeier (2009); Forni, Sala & Gambetti (2013) When x t and F t are stationary all common shocks have a permanent effect Is this plausible? NO in Macroeconomic theory technology shocks monetary policy shocks When x t is non stationary but ξ t is stationary there are no idiosyncratic trends strong implications for cointegration of x t Matteo Barigozzi Factor Models and Cointegration 5/32

20 Statement of the Problem Limitations of I(0) Setting Non-Stationary Dynamic Factor Models Let x t, F t, ξ t I(1) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 Matteo Barigozzi Factor Models and Cointegration 6/32

21 Statement of the Problem Limitations of I(0) Setting Non-Stationary Dynamic Factor Models Let x t, F t, ξ t I(1) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 1 If all q shocks have permanent effects, the genericity argument is valid Matteo Barigozzi Factor Models and Cointegration 6/32

22 Statement of the Problem Limitations of I(0) Setting Non-Stationary Dynamic Factor Models Let x t, F t, ξ t I(1) x t = χ t + ξ t n 1 n 1 n 1 χ t n 1 F t r 1 = Λ n r F t r 1 = C(L) r q u t q 1 1 If all q shocks have permanent effects, the genericity argument is valid 2 If there are only q d common trends, the genericity argument is problematic: rk(c(1)) = q d Matteo Barigozzi Factor Models and Cointegration 6/32

23 Statement of the Problem The research questions The research questions x t = χ t + ξ t χ t = Λ F t F t = C(L)u t Matteo Barigozzi Factor Models and Cointegration 7/32

24 Statement of the Problem The research questions The research questions x t = χ t + ξ t χ t = Λ F t F t = C(L)u t 1 What is the correct autoregressive representation for F t? 2 How many common trends? 3 How to estimate the model? Matteo Barigozzi Factor Models and Cointegration 7/32

25 Outline Representation Theory Representation Theory Matteo Barigozzi Factor Models and Cointegration 8/32

26 Representation Theory Main assumptions Definitions and assumptions F t r 1 = C(L) r q u t q 1 F t is a rational reduced-rank family Matteo Barigozzi Factor Models and Cointegration 9/32

27 Representation Theory Main assumptions Definitions and assumptions F t r 1 = C(L) r q u t q 1 F t is a rational reduced-rank family with cointegration rank c C(L) = ζ η + (1 L)D(L) r c c q Matteo Barigozzi Factor Models and Cointegration 9/32

28 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t Matteo Barigozzi Factor Models and Cointegration 10/32

29 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t Matteo Barigozzi Factor Models and Cointegration 10/32

30 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices Matteo Barigozzi Factor Models and Cointegration 10/32

31 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices β is r c with c = r q + d Matteo Barigozzi Factor Models and Cointegration 10/32

32 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices β is r c with c = r q + d Matteo Barigozzi Factor Models and Cointegration 10/32

33 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices β is r c with c = r q + d d : transitory shocks Matteo Barigozzi Factor Models and Cointegration 10/32

34 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices β is r c with c = r q + d d : transitory shocks r q singularity Matteo Barigozzi Factor Models and Cointegration 10/32

35 Representation Theory Main result Granger Representation Th. for Singular Vectors Theorem For generic values of the parameters of C(L): F t = C(L)u t A(L) F t + αβ F t 1 = h + C(0)u t A(L)is an r r finite-degree polynomial matrices β is r c with c = r q + d d : transitory shocks as if r q transitory shocks had a zero loading Matteo Barigozzi Factor Models and Cointegration 10/32

36 Remarks Representation Theory Main result Representation is not unique A(L) F t + αβ F t 1 = h + C(0)u t 1 the number of error terms varies between d and r (q d), 2 the autoregressive polynomial is not unique Matteo Barigozzi Factor Models and Cointegration 11/32

37 Remarks Representation Theory Main result Representation is not unique A(L) F t + αβ F t 1 = h + C(0)u t 1 the number of error terms varies between d and r (q d), 2 the autoregressive polynomial is not unique empirically this is not a problem 1 choose the maximum value for c = r (q + d) 2 choose the lag of A(L) in a prudent way Matteo Barigozzi Factor Models and Cointegration 11/32

38 Outline How many common trends? How many common trends? Matteo Barigozzi Factor Models and Cointegration 12/32

39 Literature How many common trends? Literature 1 In DFM literature Panel tests (PANIC) Bai & Ng (2004) Information criterion based on PCA of levels x t Bai (2004) 2 Classical methods Use cointegration tests on estimated factors Stock & Watson (1988), Phillips & Ouliaris (1988), Johansen (1991) Matteo Barigozzi Factor Models and Cointegration 13/32

40 Literature How many common trends? Literature 1 In DFM literature Panel tests (PANIC) Bai & Ng (2004) Information criterion based on PCA of levels x t Bai (2004) 2 Classical methods Use cointegration tests on estimated factors Stock & Watson (1988), Phillips & Ouliaris (1988), Johansen (1991) 3 Limitations All the DFM procedures assume q = r The criterion available assumes ξ t stationary The estimation error of factors will affects classical methods Matteo Barigozzi Factor Models and Cointegration 13/32

41 How many common trends? A new Information Criterion Determining the number of common trends χ t = ΛC(L)u t imply: rk(σ χ (0)) = q d = τ Σ χ (θ) = ΛC(e iθ )C (e iθ )Λ lim n λ j χ (0) =, j = 1,..., τ [ ( n ) ] τ = argmin log λ j x (0) + kp(n, T ) τ [0,τ max] j=τ+1 Hallin and Liška (2007) Matteo Barigozzi Factor Models and Cointegration 14/32

42 How many common trends? Simulation results A new Information Criterion ξ i I(1) ξ i I(1), I(0) T N q d DGP 1 DGP 2 DGP 1 DGP Percentage of times we estimate τ correctly DGPs Matteo Barigozzi Factor Models and Cointegration 15/32

43 Outline How to estimate the factors? How to estimate the factors? Matteo Barigozzi Factor Models and Cointegration 16/32

44 Literature How to estimate the factors? Literature 1 PCA on the levels x t Bai (2004) consistent only under the assumption ξ t I(0) implies that all variables are cointegrated an assumption that is not credible in macro-panels example Matteo Barigozzi Factor Models and Cointegration 17/32

45 Literature How to estimate the factors? Literature 1 PCA on the levels x t Bai (2004) consistent only under the assumption ξ t I(0) implies that all variables are cointegrated an assumption that is not credible in macro-panels example 2 PCA on first differences x t, and cumulating (PANIC) Bai & Ng (2004) F t = T t=1 F t F T = χ T = 0 by construction The estimates χ t have finite sample problems Matteo Barigozzi Factor Models and Cointegration 17/32

46 How to estimate the factors? Literature Estimation of χ t with PANIC 20 GDP 15 Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Consumption Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Residential Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Blue: x t ; Matteo Barigozzi Factor Models and Cointegration 18/32

47 How to estimate the factors? Literature Estimation of χ t with PANIC 20 GDP 15 Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Consumption Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Residential Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Blue: x t ; Green: χ t Matteo Barigozzi Factor Models and Cointegration 18/32

48 How to estimate the factors? An estimator of non-stationary factors Estimating the space of common factors 1 Assume x t I(1) with no deterministic component Estimate Λ by PCA on x t F t = ( Λ Λ) 1 Λ x t and χ t = Λ F t Matteo Barigozzi Factor Models and Cointegration 19/32

49 How to estimate the factors? An estimator of non-stationary factors Estimating the space of common factors 1 Assume x t I(1) with x it = a i + b i t + λ if t + ξ it Detrend x it and get x it Estimate Λ by PCA on x t F t = ( Λ Λ) 1 Λ x t and χ t = Λ F t Matteo Barigozzi Factor Models and Cointegration 19/32

50 How to estimate the factors? An estimator of non-stationary factors Estimating the space of common factors 1 Assume x t I(1) with x it = a i + b i t + λ if t + ξ it Detrend x it and get x it Estimate Λ by PCA on x t F t = ( Λ Λ) 1 Λ x t and χ t = Λ F t 2 Comparison with PANIC ( χ P it χ it = λ yit y i0 iλ i (y i0 a i + T ) ) b i t Matteo Barigozzi Factor Models and Cointegration 19/32

51 How to estimate the factors? Detrending vs. Demeaning An estimator of non-stationary factors Matteo Barigozzi Factor Models and Cointegration 20/32

52 How to estimate the factors? Estimation of χ t with BLL An estimator of non-stationary factors 5 0 GDP 20 0 Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Consumption Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 40 Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Residential Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Blue: x t ; Matteo Barigozzi Factor Models and Cointegration 21/32

53 How to estimate the factors? Estimation of χ t with BLL An estimator of non-stationary factors 5 0 GDP 20 0 Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Consumption Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 40 Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Residential Investment Jul60 Jan68 Jul75 Jan83 Jul90 Jan98 Jul05 Blue: x t ; Green: χ t Matteo Barigozzi Factor Models and Cointegration 21/32

54 How to estimate the factors? Simulation results An estimator of non-stationary factors ξ I(0) ξ i I(1), I(0) ξ I(1) T N q d Bai BLL Bai BLL Bai BLL Average R i = Tt=1 ( ˆχ it χ it ) 2 Tt=1 χ 2 it relative to PANIC DGPs Matteo Barigozzi Factor Models and Cointegration 22/32

55 Outline How to estimate the model? How to estimate the model? Matteo Barigozzi Factor Models and Cointegration 23/32

56 How to estimate the model? Estimation: Summary Estimating impulse-response functions x t = ΛF t + ξ t A (L)F t = A(L) F t + αβ F t 1 = h + C(0)u t Impulse responses Φ(L) = Λ(Â (L)) 1 Ĉ(0) c = r τ r Bai & Ng (2002); Alessi, Barigozzi & Capasso (2010), q Hallin & Liška (2007); Onatski (2010), τ Barigozzi, Lippi & Luciani (2014) F t Bai & Ng (2004); Barigozzi, Lippi & Luciani (2014) VECM on F t residuals v t PCA on v t Ĉ(0) Matteo Barigozzi Factor Models and Cointegration 24/32

57 Outline Empirical Analysis Empirical Analysis Matteo Barigozzi Factor Models and Cointegration 25/32

58 Empirical Analysis Set-Up Data and number of shocks 1 Panel of US 103 quarterly series from 1960:Q3 to 2012:Q4 2 Number of common factors and shocks r = 7 q = 3 d = 2 τ = 1 1 permanent shock, 2 transitory shocks Matteo Barigozzi Factor Models and Cointegration 26/32

59 Empirical Analysis Set-Up Data and number of shocks 1 Panel of US 103 quarterly series from 1960:Q3 to 2012:Q4 2 Number of common factors and shocks r = 7 q = 3 d = 2 τ = 1 1 permanent shock, 2 transitory shocks 3 Identification Monetary Policy Shock - Sign Restrictions Barigozzi, Conti & Luciani (2013) Technology Shock - Long-run Restrictions Blanchard & Quah (1992) Matteo Barigozzi Factor Models and Cointegration 26/32

60 Empirical Analysis Monetary Policy Shock Results Gross Domestic Product Consumer Price Index Federal Funds Rate 0 Residential Investments Red: IRF w/o ECM; Matteo Barigozzi Factor Models and Cointegration 27/32

61 Empirical Analysis Monetary Policy Shock Results Gross Domestic Product Consumer Price Index Federal Funds Rate 0 Residential Investments Red: IRF w/o ECM; Black: IRF with ECM Matteo Barigozzi Factor Models and Cointegration 27/32

62 Empirical Analysis Technology Shock - GDP Results Gross Domestic Product Red: IRF w/o ECM; Matteo Barigozzi Factor Models and Cointegration 28/32

63 Empirical Analysis Technology Shock - GDP Results Gross Domestic Product Red: IRF w/o ECM; Black: IRF with ECM Detrended data Dedola & Neri (2007); Smets & Wouters (2007) Matteo Barigozzi Factor Models and Cointegration 28/32

64 Empirical Analysis Results Technology Shock - Hours Hours worked What happens after a positive technology shock? Macroeconomic theory: increase Real Business Cycle model decrease New Keynesian model Empirical evidence: decrease Galí (1999); Francis & Ramey (2005) increase Christiano, Eichenbaum & Vigfusson (2003); Dedola & Neri (2007) Matteo Barigozzi Factor Models and Cointegration 29/32

65 Empirical Analysis Results Technology Shock - Hours 6 Hours Worked Red: IRF w/o ECM; Black: IRF with ECM Matteo Barigozzi Factor Models and Cointegration 29/32

66 Summary Empirical Analysis Summary Correct AR specification for F t 1 VECM representation IRF consistent with macroeconomic theory 2 VAR representation IRF not necessarily consistent due to cumulation of IRF unrealistically high and persistent Matteo Barigozzi Factor Models and Cointegration 30/32

67 Conclusions Conclusions Non-stationary Dynamic Factor models Representation results Granger Representation Theorem for Singular I(1) Vectors Criterion for number of common trends Estimation of non-stationary common factors Estimation of impulse response functions Monetary policy shocks Technology shocks Matteo Barigozzi Factor Models and Cointegration 31/32

68 Thank You!

69 Data Generating Processes DGP 1: DGP 2: χ it = λ if t F t = Φ(L)F t 1 + Gu t χ it = λ i;0f t + λ i;1f t 1 f t = Φ(L)f t 1 + u t Matteo Barigozzi Factor Models and Cointegration - Appendix

70 Data Generating Processes DGP 1: χ it = λ if t F t = Φ(L)F t 1 + Gu t g 11 0 G = g g 32 0 g 42 (1 φ j L)(1 L)F jt = g j1 u 1t j = 1, 2 permanent (1 φ j L)F jt = g j2 u 2t j = 3, 4 transitory BACK! - Criterion BACK! - Estimation Matteo Barigozzi Factor Models and Cointegration - Appendix

71 Data Generating Processes DGP 2: χ it = λ i;0f t + λ i;1f t 1 f t = Φ(L)f t 1 + u t (1 ρ 1 L)(1 L)f 1t = u 1t permanent (1 ρ 2 L)f 2t = u 2t transitory BACK! - Criterion BACK! - Estimation Matteo Barigozzi Factor Models and Cointegration - Appendix

72 Cointegration and Idiosyncratic Component Suppose x t = ΛF t + ξ t x it βx jt = z t z t = (λ i βλ j )F t + (ξ it βξ jt ) if ξ it, ξ jt I(0), then we can take β = λ i λ j which implies z t = ξ it βξ jt z t I(0) that is x it and x jt are cointegrated BACK! Matteo Barigozzi Factor Models and Cointegration - Appendix

73 Example for the Genericity argument y 1t = u t + au t 1 y 2t = u t + bu t 1 Matteo Barigozzi Factor Models and Cointegration - Appendix

74 Example for the Genericity argument y t = ( ) 1 + al u 1 + bl t Matteo Barigozzi Factor Models and Cointegration - Appendix

75 Example for the Genericity argument y t = y t Ay t 1 = ( ) 1 + al u 1 + bl t ( 1 1) u t, Matteo Barigozzi Factor Models and Cointegration - Appendix

76 Example for the Genericity argument y t = y t Ay t 1 = ( ) 1 + al u 1 + bl t ( 1 1) u t, BACK! A = 1 ( ) ab a 2 b a b 2 ab Matteo Barigozzi Factor Models and Cointegration - Appendix