Causal Search in Time Series Models

Transcription

1 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Causal Search in Time Series Models Alessio Moneta Institute of Economics Scuola Superiore Sant Anna, Pisa 27 June 2016 Causality in Economics 1/66

2 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References 1. Graphs and equations 2. Structural Equation Models 3. From SEM to VAR 4. Structural VAR Models Causality in Economics 2/66

3 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References 1. Graphs and Equations Causality in Economics 3/66

4 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Graphs and equations There is a one-to-one correspondence between a system of structural equations and a graphical model. Example: X V W Z System of equations: V = a VX X + ε V Z = a ZV V + ε Z W = a WV V + a WX X + ε W where each ɛ j is an independent r.v. Causality in Economics 4/66

5 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Graphs and equations Graphs including error terms: ε X ε V ε Z X V W Complete system of equations: X = ε X V = a VX X + ε V Z = a ZV V + ε Z W = a WV V + a WX X + ε W Z ε W Causality in Economics 5/66

6 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Graphs and equations System of equations (previous example) in matrix notation: Y = X V Z W E = ε X ε V ε Z ε W A = a VX a ZV 1 0 a WX a WY 0 1 AY = E (1) where Σ E = E(EE ) is a diagonal matrix. Causality in Economics 6/66

7 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References 2. Structural Equation Models Causality in Economics 7/66

8 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Structural equations and causality Cowles Commission Approach: dominant approach in macro-econometrics 1940s-1960s Haavelmo s The Probability Approach in Econometrics (1944) Economic theory dictates the causal structure If the structure is adequate: error terms conform to standard probabilistic properties (independence and normality) Measuring the strengths of causal linkages. Causality in Economics 8/66

9 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Example: (borrowed from Hoover 2006: 8ff.) m = αy + ε m (2) y = βm + ε y, (3) where m money, y GDP (both in logs). Statistical properties of ε m and ε y tell whether the model is goods specified. can we get α, β, ε m, ε y from the data? No. Problem of identification m and y are both endogenous variables ε m m y ε y Causality in Economics 9/66

10 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Example: (borrowed from Hoover 2006: 8ff.) m = αy + ε m (2) y = βm + ε y, (3) where m money, y GDP (both in logs). Statistical properties of ε m and ε y tell whether the model is goods specified. can we get α, β, ε m, ε y from the data? No. Problem of identification m and y are both endogenous variables ε m m y ε y Causality in Economics 9/66

11 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Example: (borrowed from Hoover 2006: 8ff.) m = αy + ε m (2) y = βm + ε y, (3) where m money, y GDP (both in logs). Statistical properties of ε m and ε y tell whether the model is goods specified. can we get α, β, ε m, ε y from the data? No. Problem of identification m and y are both endogenous variables ε m m y ε y Causality in Economics 9/66

12 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Problem of identification solved introducing two exogenous variables: m = αy + δr + ε m (4) y = βm + γp + ε y, (5) where r interest rate, p price level (both in logs). can we get α, β, δ, γ ε m, ε y from the data? Yes. Model identified under the assumption that p is not a direct cause of m and r is not a direct cause of y Omitting the error terms: r m y p Causality in Economics 10/66

13 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Problem of identification solved introducing two exogenous variables: m = αy + δr + ε m (4) y = βm + γp + ε y, (5) where r interest rate, p price level (both in logs). can we get α, β, δ, γ ε m, ε y from the data? Yes. Model identified under the assumption that p is not a direct cause of m and r is not a direct cause of y Omitting the error terms: r m y p Causality in Economics 10/66

14 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Example of structural model Problem of identification solved introducing two exogenous variables: m = αy + δr + ε m (4) y = βm + γp + ε y, (5) where r interest rate, p price level (both in logs). can we get α, β, δ, γ ε m, ε y from the data? Yes. Model identified under the assumption that p is not a direct cause of m and r is not a direct cause of y Omitting the error terms: r m y p Causality in Economics 10/66

15 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

16 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

17 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

18 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

19 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

20 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

21 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

22 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

23 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reduced form model After substituting eq. (4) into eq. (5) and simplifying: m = y = αγ 1 αβ p + δ 1 αβ r + α 1 αβ ε m αβ ε y (6) γ 1 αβ p + βδ 1 αβ r + β 1 αβ ε m αβ ε y (7) This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (8) y = c p + d r + u y (9) From the OLS estimates â, ˆb, ĉ, ˆd, û m, û y, using the respective correspondences between eq. (6) (7) and (8) (9) it is possible to recover α, β, γ, δ, ε m, ε y, i.e. parameter estimates of the structural model. Causality in Economics 11/66

24 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Instrumental variables Notice that the structural-form coefficients (α, β in the ex.) could be equivalently obtained by instrumental variables estimation. Following the previous example: p: instrumental variable for eq. (2) (m = αy + ε m ) r: instrumental variable for eq. (3) (y = βm + ε y ) Two-stage least squares estimation for eq. (2): 1 OLS regression of y on p: obtain ŷ 2 OLS regression of m on ŷ: obtain ˆα IV Causality in Economics 12/66

25 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Instrumental variables Notice that the structural-form coefficients (α, β in the ex.) could be equivalently obtained by instrumental variables estimation. Following the previous example: p: instrumental variable for eq. (2) (m = αy + ε m ) r: instrumental variable for eq. (3) (y = βm + ε y ) Two-stage least squares estimation for eq. (2): 1 OLS regression of y on p: obtain ŷ 2 OLS regression of m on ŷ: obtain ˆα IV Causality in Economics 12/66

26 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Instrumental variables Notice that the structural-form coefficients (α, β in the ex.) could be equivalently obtained by instrumental variables estimation. Following the previous example: p: instrumental variable for eq. (2) (m = αy + ε m ) r: instrumental variable for eq. (3) (y = βm + ε y ) Two-stage least squares estimation for eq. (2): 1 OLS regression of y on p: obtain ŷ 2 OLS regression of m on ŷ: obtain ˆα IV Causality in Economics 12/66

27 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Instrumental variables Notice that the structural-form coefficients (α, β in the ex.) could be equivalently obtained by instrumental variables estimation. Following the previous example: p: instrumental variable for eq. (2) (m = αy + ε m ) r: instrumental variable for eq. (3) (y = βm + ε y ) Two-stage least squares estimation for eq. (2): 1 OLS regression of y on p: obtain ŷ 2 OLS regression of m on ŷ: obtain ˆα IV Causality in Economics 12/66

28 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Instrumental variables Notice that the structural-form coefficients (α, β in the ex.) could be equivalently obtained by instrumental variables estimation. Following the previous example: p: instrumental variable for eq. (2) (m = αy + ε m ) r: instrumental variable for eq. (3) (y = βm + ε y ) Two-stage least squares estimation for eq. (2): 1 OLS regression of y on p: obtain ŷ 2 OLS regression of m on ŷ: obtain ˆα IV Causality in Economics 12/66

29 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Summary on Structural Equation Modeling SEM: provide quantitative assessment of cause-effect relationships. Interpretation of causality: counterfactual manipulability Probabilistic methods are used to measure causality and partially to test it, but not for the sake of causal discovery. Dependence on a priori economic theory: necessity of identifying restrictions. Empiricist query: where does the economic theory come from? Causality in Economics 13/66

30 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality 3. From SEM to VAR Causality in Economics 14/66

31 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality The crisis of the Cowles Commission approach Up to the 1970s: consensus on the Cowles Commission approach Important economic events in the 1970s: Oil crisis (1973), stagflation. Increasing scepticism towards so-called Keynesian Macroeconomics. Two major critiques: Lucas Critique (1976) on economic policy evaluation structural equations are ineffective for policy evaluation: they are unstable under intervention Sims (1980) article Macroeconomics and Reality restrictions used in the Cowles Commission approach are incredible, i.e. empirically not validated Causality in Economics 15/66

32 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality The crisis of the Cowles Commission approach Reactions to the criticisms: Change the theory maintaining the structural, theory-driven approach cfr. New Classical Macroeconomics (Lucas, Sargent), and rational expectations models even more extreme theory-driven approach: calibration Adopt a more data-driven approach: Time-series econometric models, more intensive use of statistical methods Granger Causality (1969) Vector Autoregressive Models (VAR) (Sims 1980) Causality in Economics 16/66

33 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Granger causality Consider two time series X t and Y t. The idea of Granger s (1969, 1980) causality: Y causes X if Y t helps to predict X t+h general definition of G-causality (Granger 1980: 330): Y t is said to cause X t+1 if P(X t+1 A Ω t ) = P(X t+1 A Ω t \Y t ), for a set A and for a set Ω t of relevant information available at time t (Ω t \Y t := relevant information except Y t ). Merely probabilistic notion of causality Importance to appraise usefulness of G-causality but differences with structural causality. Causality in Economics 17/66

34 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Testing G-non-causality Granger non-causality tests: Regression of X t+1 on Ω t (including X t, Y t and their lagged values) + same regression excluding Y t (and its lagged values). Check error variances of the two regressions (with and without Y t ); and/or Test whether the Y t coefficients in the regression of X t+1 on Ω t are zero (t test). Causality in Economics 18/66

35 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Example Example (Hoover 2001: 151ff) Suppose the unobserved DGP / structural model is: Y t = θx t + β 11 Y t 1 + β 12 X t 1 + ɛ 1t (10) X t = γy t + β 21 Y t 1 + β 22 X t 1 + ɛ 2t (11) reduced form model: GNC tests: Y t = Π 11 Y t 1 + Π 12 X t 1 + ν 1t (12) X t = Π 21 Y t 1 + Π 22 X t 1 + ν 2t, (13) where the coefficients in (12) and (13) are functions of the coefficients in (10) and (11) if Π 12 = 0, then X does not G-causes Y if Π 21 = 0, then Y does not G-causes X. Causality in Economics 19/66

36 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Example Example (Hoover 2001: 151ff) Suppose the unobserved DGP / structural model is: Y t = θx t + β 11 Y t 1 + β 12 X t 1 + ɛ 1t (10) X t = γy t + β 21 Y t 1 + β 22 X t 1 + ɛ 2t (11) reduced form model: Y t = Π 11 Y t 1 + Π 12 X t 1 + ν 1t (12) X t = Π 21 Y t 1 + Π 22 X t 1 + ν 2t, (13) Suppose Y causes structurally X, and X does not causes structurally Y. That is: θ = 0, β 12 = 0. Since Π 12 = β 12 + θβ 22 1 θγ Π 21 = γβ 11 + β 21 1 θγ then Π 12 = 0, i.e. X does not G-causes Y, and Π 21 = 0, i.e. Y G-causes X Does structural causality always imply G-causality and vice versa? In general, no Causality in Economics 20/66

37 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Granger causality and VARs Sims (1980) critique: incredible restrictions in structural model VAR model y t m t p t r t = µ + A 1 y t 1 m t 1 p t 1 r t A p y t p m t p p t p r t p + u t GNC: p t m t is tested checking whether the (2, 3) element of the matrices A 1,..., A p are zero. Note that the matrices A 1,..., A p are of dimension (4 4). (14) Causality in Economics 21/66

38 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality Causal Search in Time Series Models 1. Graphs and equations 2. Structural Equation Models 3. From SEM to VAR 4. Structural VAR Models Causality in Economics 22/66

39 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges 1970s Crisis References Granger causality 4. Structural Vector Autoregressive Models and their identification by means of causal search models Causality in Economics 23/66

40 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Identification of SVAR Problem of identification: from the estimate of the VAR y t = A 1 y t A p y t p + u t (15) we want to recover the SVAR: Γ 0 y t = Γ 1 y t Γ p y t p + ε t, (16) y t = By t + Γ 1 y t Γ p y t p + ε t, (17) more parameters in SVAR than in VAR. Notice that: u t = Γ 1 0 ε t = Bu t + ε t (18) Our strategy: from the statistical properties of u t we recover B. Knowing B (and all the parameters in 15) we recover then: Γ 0 = I B Γ 1 = Γ 0 A 1... Γ p = Γ 0 A p Causality in Economics 24/66

41 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA MA representation MA representation of the VAR (under stability): y t = (I A 1 L... A p L p )u t = where Φ 0 = I, Φ i = i j=1 A jφ i j, for i = 1, 2,... Since Γ 0 u t = ε t y t = Φ j u t j (19) j=0 Φ j Γ 1 0 Γ 0u t j = Ψ j ε t j (20) j=0 j=0 Ψ j (k k matrices): structural impulse response functions, they describe how structural shocks affect variables. Causality in Economics 25/66

42 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Graphical Models for SVAR Identification Idea: apply graphical causal search to u t : (u 1t,..., u kt ). The causal structure among u 1t,..., u kt will deliver information on the matrix B Notice that B determines causal relationships among y 1t,..., y kt (see equation 17) Causality in Economics 26/66

43 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Graphical Models for SVAR Identification Graphical causal search applied to u t : (u 1t,..., u kt ): Test conditional independence among (u 1t,..., u kt ) In case of Gaussianity: Fisher s z or Wald tests (see Moneta et al. 2011) on zero partial correlations alternative semi-parametric or NP methods in case of non-gaussianity (see Moneta et al. 2011) Apply search algorithm (e.g. PC algorithm, Lecture 2): Build a complete undirected graph among (u 1t,..., u kt ); Recursively eliminate edges using C.I. tests among (u 1t,..., u kt ); Identify unshielded colliders; Identify chains; Avoid cycles. Causality in Economics 27/66

44 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Example Moneta (2008): King-Stock-Plosser-Watson (1991) data set: C I Y = M Y R P Taking into account non-stationarity / cointegration Get the matrix of residuals û t Causality in Economics 28/66

45 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Results from King et al. (1991) data set: R I Y M P C Configurations R I Y and R I C are excluded. Causality in Economics 29/66

46 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Results from King et al. (1991) data set: Example of one member of the set of possible causal graphs: R I Y M P C Causality in Economics 30/66

47 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Impulse response functions 1.5 Responses of Y to Y 2 Responses of Y to M lags lags 1.5 Responses of Y to I 1.5 Responses of Y to C lags lags Causality in Economics 31/66

48 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Impulse response functions 1.5 Responses of Y to DP 1.5 Responses of Y to R lags lags Causality in Economics 32/66

49 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA ICA-based search Assumptions of the method based on Independent Component Analysis: the structural shocks ε t are non-normal; the elements of ε t : ε 1t,..., ε kt are mutually independent; <acyclicity + causal sufficiency> OR <cycles + causal sufficiency> (OR <acyclicity + latent variables>) Γ 0 (= I B) has unit diagonal (B has zeros on the diagonal). (linearity) Note: Faithfulness condition is not needed here. Causality in Economics 33/66

50 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA ICA-based search Search methods: LiNGAM (Shimizu et al. 2006) applied to VAR models (cfr. also Hyvärinen et al and Moneta et al. 2013) and LiNG (Lacerda et al. 2008). 1 Estimate the VAR model y t = A 1 y t A p y t p + u t. Check whether the residuals are non-gaussian. Denote by Û the K T matrix of estimated residuals. 2 Use FastICA (Hyvärinen et al. 2001) to obtain Û = PÊ, where P is K K and Ê is K T, such that the rows of Ê are the independent components of Û. Note: FastICA finds transformations of the data that maximizes (approximate) negentropy. This is equivalent to minimize mutual information. Causality in Economics 34/66

51 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA ICA-based search Note: the order and scaling of the independent components (output of the FastICA algorithm) is left undetermined. Acyclic case LiNGAM: 3 Let Γ 0 = P 1. Find the permutation of rows of Γ 0 which yields a matrix Γ 0 without any zeros on the main diagonal. The permutation is sought which minimizes i 1/ Γ 0ii. 4 Divide each row of Γ 0 by its diagonal element, to obtain a matrix ˆΓ 0 with all ones on the diagonal. 5 Let B = I ˆΓ 0. 6 Find the permutation matrix Z which makes Z BZ T as close as possible to strictly lower triangular. (Minimize the sum of squares of the permuted upper-triangular elements). Set the upper-triangular elements to zero, and permute back to obtain ˆB which now contains the acyclic contemporaneous structure. Causality in Economics 35/66

52 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Example Steps 2-6: Suppose the actual structure is: 1 0 γ α 1 β Equivalently: u 1 u 2 u 3 Causal order: = u 3 u 1 u 2 u 3 = 0 0 γ α 0 β e 1 e 2 e 3 u 1 u 2 u 3 + e 1 e 2 e 3 u 1 u 2 Causality in Economics 36/66

53 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA Suppose from step 2 one finds: 2α 2 2β u u 2 = 3 0 3γ u 3 From step 3: 3 0 3γ 2α 2 2β From step 4-5: u 1 u 2 u 3 = Step 6: u 3 u 1 u 2 = u 1 u 2 u 3 = 0 0 γ α 0 β γ 0 0 β α 0 e 1 e 2 e 3 e 1 e 2 e 3 u 1 u 2 u 3 u 3 u 1 u e 1 e 2 e 3 e 3 e 1 e 2 Causality in Economics 37/66

54 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges GMs for SVAR References ICA ICA-based search Cyclic case LiNG 3 Let Γ 0 = P 1. Test hypothesis of non-zero coefficients by bootstrap sampling. Prune out very statistically insignificant elements of Γ 0. Find all row permutation matrices of Γ 0, which yield Γ 0 that has zeroless diagonal. Note: There might be several candidates of Γ 0, depending how sparse is Γ 0. 4 Divide each row of Γ 0 by its diagonal element, to obtain a matrix ˆΓ 0 with all ones on the diagonal. 5 Let B = I ˆΓ 0. Final step (both for LiNG and LiNGAM): Calculate lagged causal effects ˆΓ i = (I ˆB 0 )Â i, for i = (1,..., p) Causality in Economics 38/66

55 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Empirical application: Chilean micro data Exporting and productivity as part of the growth process: Results from a structural VAR by Tommaso Ciarli (U. of Sussex), Alex Coad (U. of Sussex), and Alessio Moneta (Sant Anna) Focus on the dynamic interaction of export and productivity growth of exporting firms in Chile ( ) Causality in Economics 39/66

56 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Our data Survey of manufacturing plants (Encuesta Nacional Insustrial Manufacturera) collected by Chilean Statistical Institute (INE): plants with more than 10 employees, more than 6 months activity, classified in manufacturing sectors (ISIC 4 digits) exporting in years size, proxied by employment (employ) output, proxied by (deflated) total sales (output) = domestic sales (domsales) + exports (exp) productivity, estimated by total factor productivity (tfp) Causality in Economics 40/66

57 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Our variables 1 gr-domsales:= log domsales t = log domsales t log domsales t 1 2 gr-empl:= log empl t 3 gr-exp:= log exp t 4 gr-tfp:= log tfp t sample size: 4021 observations 1309 plants 5 time periods (not for all firms: the panel is not balanced) 10 manufacturing sectors Causality in Economics 41/66

58 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Causal model search Our strategy: estimate a (pooled-panel) Vector Autoregressive (VAR) model on the basis of the estimated residuals search for the underlying Structural VAR Causality in Economics 42/66

59 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness VAR and SVAR Recall: we can estimate (VAR): y t = A 1 y t A p y t p + u t, we want to recover (SVAR): Γ 0 y t = Γ 1 y t Γ p y t p + ε t or y t = By t + Γ 1 y t Γ p y t p + ε t Causality in Economics 43/66

60 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Results from GCM-SVAR Ouput of PC algorithm applied to VAR-estimated residuals (OLS, 2 lags), partial correlation tests: DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t DS EX L P Causality in Economics 44/66

61 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Results from GCM-SVAR Ouput of PC algorithm applied to VAR-estimated residuals (OLS, 1 lag), both partial correlation test and nonparametric conditional independence test: DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t DS EX L P Causality in Economics 45/66

62 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Non-Gaussianity Histogram of X[, i] Histogram of X[, i] Histogram of X[, i] Histogram of X[, i] Density Normal Q Q Plot Normal Q Q Plot X[, i] Normal Q Q Plot X[, i] Sample Quantiles Sample Quantiles Density Density Normal Q Q Plot X[, i] Sample Quantiles Shapiro-Wilk, Shapiro-Francia, Jarque-Bera tests strongly reject the H 0 : normality. Causality in Economics 46/66

63 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Results from VAR-LiNGAM Ouput of LiNGAM applied to VAR-estimated residuals (LAD, 1 lag): DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t DS EX L P LAD (least absolute deviations) preferred to OLS in case of fat tails distributions lag selection: 1 lag according to different criteria (Akaike Information, the Hannan-Quinn or the Schwarz Criterion) Causality in Economics 47/66

64 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Estimation B Γ 1 gr domsales gr empl gr exp gr tfp l gr domsales l gr empl l gr exp l gr tfp gr domsales gr empl gr exp gr tfp Bootstrap standard errors Causality in Economics 48/66

65 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Contemporaneous structure DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t positive effects negative effects DS EX L P Causality in Economics 49/66

66 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Contemporaneous structure L P EX DS Causality in Economics 50/66

67 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Lagged effects L L P 7 P EX EX 7 DS DS (t-1) Lagged effects displayed are only those significant at 0.05 significance level (t) L: gr-empl t P: gr-tfp t EX: gr-exp t DS: gr-domsales t positive effects negative effects Causality in Economics 51/66

68 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Main causal mechanisms Primus motor is employment growth, directly affecting domestic sales and exports Employment growth negative effect on TFP growth downsizing firms better able to improve productivity than investing firms TFP growth positive impacts on growth of domestic sales and of exports firms better off pursuing TFP growth as a prerequisite for sales growth Export growth has a negative impact on growth of domestic sales. international firms focus on one market TFP export? Within the period: TFP export and not vice versa Exporting growth has a small positive impact on subsequent TFP growth (much smaller effect: (s.e ) vs (s.e )) Causality in Economics 52/66

69 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Robustness checks Checking the robustness of the contemporaneous causal structure through a bootstrap procedure, percentage of links found: DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t positive effects negative effects EX 100% 100% 99% L 100% 100% P 47% DS 53% Causality in Economics 53/66

70 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Results from LiNG Results from LiNG algorithm (feedback allowed): DS: gr-domsales t L: gr-empl t EX: gr-exp t P: gr-tfp t positive effects negative effects DS EX L P Causality in Economics 54/66

71 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Lagged effects (LiNG) L L P 7 P EX EX 7 DS DS (t-1) (t) L: gr-empl t P: gr-tfp t EX: gr-exp t DS: gr-domsales t positive effects negative effects Lagged effects displayed are only those significant at 0.05 significance level. Instantaneous effects are not displayed here. Causality in Economics 55/66

72 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges Data Causal References search Robustness Stability of causal orders across bootstrap samples control variables: dummies different measure of productivity sectors size old exporters new exporters so far: stability of the link TFP t export t, absence of the link TFP t export t, and weak evidence for the link export t TFP t+1 Impulse Response Analysis confirms this pattern. Causality in Economics 56/66

73 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References What can we learn from causal-search methods? Gains the emphasis is on the structural model (data generating process), it tries to goes beyond the reduced form model (associational model) on the basis of an adequate characterization of the joint distribution of the observed variables it allows discriminating the possible causal structures the automatic features of these methods helps perform a rigorous robustness analysis Causality in Economics 57/66

74 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Pitfalls possible sensitivity measured variables, controls, rescaling, lags heterogeneity (across individuals and over time) feedback and latent variables drawback of VAR analysis: number of shocks = number of equations Causality in Economics 58/66

75 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Challenges integrating causal search methods with panel data analysis allowing across-individual heterogeneity integration with instrumental variable estimation allowing for common shocks / common factors robust causal search Causality in Economics 59/66

76 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Suggested Readings Causality in Economics 60/66

77 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References On Philosophy of Causation and Economics Hitchcock, C. (2010), Probabilistic Causation, Stanford Encyclopedia of Philosophy. www Hoover, K. (2001) Causality in Macroeconomics, CUP, Chapter 1. Hoover, K. (2001) The Methodology of Empirical Macroeconomics, CUP, Chapter 4. Hoover, K. (2008), Causality in economics and econometrics, in The New Palgrave Dictionary of Economics. www Holland, P. (1986). Statistics and Causal Inference, Journal of the American Statistical Association, 81(396), www. Hoover, K. (2001) Causality in Macroeconomics, CUP, Chapter 2. Reiss, J. (2013), Philosophy of Economics. A Contemporary Introduction, Routledge. Scheines, R. (2005), The similarity of causal inference in experimental and non-experimental studies, Philosophy of Science, 72, pp Williamson, J. (2007), Causality, Handbook of Philosophical Logic, vol 14, Springer. www Woodward, J. (2008), Causation and Manipulation, Stanford Encyclopedia of Philosophy. www Causality in Economics 61/66

78 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References On Causal Inference and Statistics Spirtes, Glymour, Scheines (2000), Causation, Prediction, and Search, MIT Press. Chapters 1 and 2; Sections 3.3; 3.4; 3.5; 3.7; 5.4; 5.5 Pearl (2000), Causality: Models, Reasoning, and Inference, CUP. Section 1.1, 1.2, 1.3, 2.9 Hoover, K. (2001), Causality in Macroeconomics. CUP. pp Spanos, A. (1999), Probability Theory and Statistical Inference. CUP. Section 2.2 and 6.4 Causality in Economics 62/66

79 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Further reading: Bryant, Bessler, Haigh (2009), Disproving Causal Relationships Using Observational Data, OBES. www Cooper, G.F. (1999), An Overview of the Representation and Discovery of Causal Relationships Using Bayesian Networks, in C. Glymour, G.F. Cooper, Computation Causation, and Discovery, MIT Press. Richardson and Spirtes (1999), Automated Discovery of Linear Feedback Models, in Glymour and Cooper (1999), MIT Press, pp Scheines, R. (1997), An Introduction to causal inference. www TETRAD Project: Causality in Economics 63/66

80 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References On model identification Pearl, J. (2000), Causality: Models, Reasoning, and Inference, CUP: pp Hoover, K. (2001), Causality in Macroeconomics. CUP: p. 132 and pp Hoover, K. (2006), Economic Theory and Causal Inference. www Further reading: Moneta, A. (2007), Mediating between causes and probabilities: the use of graphical models in econometrics. www Angrist, J.D. and Krueger, A.B. (2001) Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments, Journal of Economic Perspectives, 15(4). www Causality in Economics 64/66

81 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reading List (SVAR and Graphical Models) Demiralp, S. and K. Hoover (2003), Searching for the Causal Structure of a Vector Autoregression, OBES 65, pp www Lütkepohl, H. (2006), Vector Autoregressive Models, in T.C. Mills and K. Patterson (eds.), Palgrave Handbook of Econmetrics, Vol. 1 Econometric Theory, Palgrave Macmillan, pp Further reading: Moneta, A. (2008) Graphical causal models and VARs: an empirical assessment of the real business cycles hypothesis, Empirical Economics, 35(2), www Moneta, A., N. Chlaß, D. Entner, and P. Hoyer (2011), Causal Search in Structural Vector Autoregressive Models, in Journal of Machine Learning Research: Workshop and Conference Proceedings, 12: www Causality in Economics 65/66

82 Graphs SEM SEM-VAR SVAR Empirical application Gains, pitfalls & challenges References Reference on ICA and SVAR Hyvärinen, A. and E. Oja (2000), Independent component analysis: Algorithms and Applications, Neural Networks, 13(4-5): , www Moneta, A., D. Entner, P. Hoyer, and A. Coad, Causal Inference by Independent Component Analysis: Theory and Applications, Oxford Bulletin of Economics and Statistics www Causality in Economics 66/66