graphical causal models for time series econometrics: …

Introduction SVAR GMs Application to SVAR Recent Developments

Graphical Causal Models for Time SeriesEconometrics: Some Recent Developments

and Applications

Alessio Moneta

Max Planck Institute of Economics, Jena

10 December 2009

MAX-PLANCK-GESELLSCHAFT

Mini Symposium: Causality and Time Series AnalysisNIPS 2009, Vancouver

Moneta Causal Search in TS Econometrics

Introduction SVAR GMs Application to SVAR Recent Developments Motivations Overview

B Application of methods of causal search to the problem offinding the appropriate causal order for the Structural VectorAutoregressive models (SVAR).

Overview

B VAR and SVAR model

B Causal search methods: graphical models

B Application to the linear/Gaussian setting

B Extensions:

• Nonparametric setting

• Non-Gaussian case: application of a method based on ICA

Overview

B VAR and SVAR model

B Causal search methods: graphical models

B Application to the linear/Gaussian setting

B Extensions:

• Nonparametric setting

• Non-Gaussian case: application of a method based on ICA

Introduction SVAR GMs Application to SVAR Recent Developments VAR Identification Problem

VAR vs. SVAR model

Basic VAR model (reduced-form):

Yt = A1Yt−1 + . . . + ApYt−p + ut. (1)

• Yt: (yt1, . . . , ytk)′;

• Aj (j = 1, . . . , p) are k× k coefficient matrices;

• ut is the vector white noise process;

• E(utu′t) = Σu.

VAR vs. SVAR model

Yt = A1Yt−1 + . . . + ApYt−p + ut. (1)

• Yt: (yt1, . . . , ytk)′;

VAR vs. SVAR model

Yt = A1Yt−1 + . . . + ApYt−p + ut. (1)

• Yt: (yt1, . . . , ytk)′;

VAR vs. SVAR model

Yt = A1Yt−1 + . . . + ApYt−p + ut. (1)

• Yt: (yt1, . . . , ytk)′;

VAR vs. SVAR model

Wold representation (in case of stationarity):

Yt =∞

∑j=0

Φjut−j, (2)

where Φj = ∑ji=1 Φj−iAi

But for any nonsingular k× k matrix P we get:

Yt =∞

∑j=0

ΦjPP−1ut−j =∞

∑j=0

Ψjεt−j, (3)

where εt−j = P−1ut−j and Ψj = ΦjP (j = 0, 1, 2, ...).

VAR vs. SVAR model

Wold representation (in case of stationarity):

Yt =∞

∑j=0

Φjut−j, (2)

where Φj = ∑ji=1 Φj−iAi

But for any nonsingular k× k matrix P we get:

Yt =∞

∑j=0

ΦjPP−1ut−j =∞

∑j=0

Ψjεt−j, (3)

where εt−j = P−1ut−j and Ψj = ΦjP (j = 0, 1, 2, ...).

VAR vs. SVAR model

If we premultiply equation (1) by P−1 we get

P−1Yt = P−1A1Yt−1 + . . . + P−1ApYt−p + P−1ut. (4)

SVAR model (structural form):

Γ0Yt = Γ1Yt−1 + . . . + ΓpYt−p + εt, (5)

where Γ0 = P−1, Γj = P−1Aj (j = 1, . . . , p).

B choice of P (Γ0 ) based on information about the contemporaneouscausal structure.

VAR vs. SVAR model

Γ0Yt = Γ1Yt−1 + . . . + ΓpYt−p + εt, (5)

where Γ0 = P−1, Γj = P−1Aj (j = 1, . . . , p).

VAR vs. SVAR model

Γ0Yt = Γ1Yt−1 + . . . + ΓpYt−p + εt, (5)

where Γ0 = P−1, Γj = P−1Aj (j = 1, . . . , p).

VAR vs. SVAR model

B choice of P (Γ0 ) in the literature:• Choleski decomposition such that: P is lower diagonal and

Ω = E(εtε′t) = Ik.

• a priori (theoretical, institutional) zero-restrictions;

B Our proposal: inferring P (Γ0 ) starting from the estimatedresiduals ut

• conditional independence relations −→ causal relationships(graphical models)

• independent component analysis (in case of non-Gaussianity)

VAR vs. SVAR model

B choice of P (Γ0 ) in the literature:• Choleski decomposition such that: P is lower diagonal and

Ω = E(εtε′t) = Ik.

• a priori (theoretical, institutional) zero-restrictions;

B Our proposal: inferring P (Γ0 ) starting from the estimatedresiduals ut

• conditional independence relations −→ causal relationships(graphical models)

• independent component analysis (in case of non-Gaussianity)

Introduction SVAR GMs Application to SVAR Recent Developments Causal Search Algorithm

Graphical models

Graphs have two functions:

B Representation of causal structures

B Representation of causal independence relations

Graphical models

Representation of causal structures:

B edge: causal influence

• Undirected edges: X — Y (ambiguous causal influence betweenX and Y)

• Directed edges: X −→ Y, X←− Y

• Bi-directed edges: X←→ Y

B DAGs: Directed acyclic graphs (only directed edges).

Graphical models

Representation of conditional independence relations:

B edge: statistical dependence

• X ⊥⊥ Y | ∅ : no edge between X and Y• X ⊥⊥/ Y: X — Y; X −→ Y; X←− Y• X ⊥⊥ Z | Y: X −→ Y −→ Z; X←− Y←− Z; X←− Y −→ Z• X ⊥⊥/ Z | Y: X −→ Y←− Z

Graphical models

Rules of Inference:

B edge: from conditional independence 7→ causal relationships

• DAGs among V1, . . . , Vk

• Causal Markov Condition: conditioned on its parents every node isindependent of its nondescendants; or:conditioned on its direct causes every variable is independent of itsnon-effects

• Faithfulness Condition: every conditional independence relation amongV1, . . . , Vk is entailed by the Causal Markov Condition

Search algorithm:

PC, SGS, modified PC algorithm (Spirtes, Glymour and Scheines2000):

• Input: conditional independence tests

• Output: set of Markov equivalent DAGs

B Start: complete undirected graph among V1, . . . , Vk

B First step: elimination of edges whenever ⊥⊥

B Second step: statistical orientation of edges• search for unshielded colliders: X −→ Y←− Z

B Third step: logical orientation of edges

• X −→ Y — Z 7→ X −→ Y −→ Z• orient edges in order to avoid cycles

Search algorithm:

Introduction SVAR GMs Application to SVAR Recent Developments Graphical Models and SVAR

GMs applied to SVAR

• Cfr. Swanson and Granger (1997), Bessler and Lee (2002), Demiralp andHoover (2003) (among others)

• Estimate reduced form VAR

Yt = A1Yt−1 + . . . + ApYt−p + ut. (6)

• King-Stock-Plosser-Watson (1991) updated data setUS data 1947:2 - 1994:1 (quarterly data)

CIMYR∆P

per capita consumptionper capita investmentmoney M2 / priceper capita private incomenominal interest rateprice inflation

• Taking into account non-stationarity / cointegration

• Get the matrix of residuals Ut

GMs applied to SVAR

• Cfr. Swanson and Granger (1997), Bessler and Lee (2002), Demiralp andHoover (2003) (among others)

• Estimate reduced form VAR

Yt = A1Yt−1 + . . . + ApYt−p + ut. (6)

• King-Stock-Plosser-Watson (1991) updated data setUS data 1947:2 - 1994:1 (quarterly data)

CIMYR∆P

per capita consumptionper capita investmentmoney M2 / priceper capita private incomenominal interest rateprice inflation

• Taking into account non-stationarity / cointegration

• Get the matrix of residuals Ut

GMs applied to SVAR

• Causal graph among u1t, . . . , ukt ≡ causal graph amongy1t, . . . , ykt

• C.I. relations among u1t, . . . , ukt tested via Wald tests onzero-partial correlations

• Gaussianity assumption

Results (Moneta 2008):

R I Y M ∆P

Configurations R −→ I←− Y and R −→ I←− C are excluded.

Possibilities:B sensitivity analysisB bootstrap analysis (cfr. Demiralp, Hoover and Perez 2008)

R I Y M ∆P

Configurations R −→ I←− Y and R −→ I←− C are excluded.

Possibilities:B sensitivity analysisB bootstrap analysis (cfr. Demiralp, Hoover and Perez 2008)

One of the 16 DAGs:

R - I - Y M - ∆P

Impulse Response Analysis

0 3 6 9 12 15 18 21 24 27

Responses of Y to C

0 3 6 9 12 15 18 21 24 27

Responses of Y to I

0 3 6 9 12 15 18 21 24 27

Responses of Y to Y

0 3 6 9 12 15 18 21 24 27

Responses of Y to M

0 3 6 9 12 15 18 21 24 27

Responses of Y to R

0 3 6 9 12 15 18 21 24 27

Responses of Y to DP

Introduction SVAR GMs Application to SVAR Recent Developments Nonparametric causal search Causal search based on ICA Application

Extensions

• Non-parametric approach• Start from non-parametric tests of conditional independence

• Semi-parametric approach• Assumption of linearity and non-Gaussianity

Extensions

• Non-parametric approach• Start from non-parametric tests of conditional independence

• Semi-parametric approach• Assumption of linearity and non-Gaussianity

Nonparametric approach

Chlaß and Moneta (2009):

• X ⊥⊥ Y | Z iff f (X|Y, Z) = f (X|Z)

• Test f (X, Y, Z)f (Z) = f (X, Z)f (Y, Z).

• Distance measures between kernel density functions:• Euclidean distance (Szekely and Rizzo 2004; Baringhaus and Franz

• Weighted Hellinger distance (Su and White 2008)

• Curse of dimensionality

• Bootstrap procedure

• X ⊥⊥ Y | Z iff f (X|Y, Z) = f (X|Z)

Causal search based on ICA

Moneta, Entner, Hoyer and Coad (2009)

B VAR-LiNGAM algorithm (Hyvarinen, Shimizu and Hoyer 2008)

• Estimate the reduced-form VAR

• Check that the residuals are non-Gaussian

• Use an ICA algorithm to decompose the residuals matrix

• Order the variables (residuals) so as to obtain a matrix P that isclose to lower triangular

• Once the instantaneous effects are identified, identify laggedeffects

• No need of assumptions such as Faithfulness.

Empirical Application

Panel VAR on Firm Growth and R&D Expenditures

Coad-Rao (2007) data set: US firm 1973:2004 (manufacturing sector).

• Employees (growth rate)• Total sales (growth rate)• R&D expenditure (growth rate)• Profits (growth rate)

LAD (least absolute deviation) estimation

Structural coefficients one-lag model:

Empl.gr(t+1)

RnD.gr(t+1)

Sales.gr(t+1)

Opinc.gr(t)

RnD.gr(t)

Sales.gr(t)

Empl.gr(t)

Opinc.gr(t+1)

Solid green arrows: positive causal influence; dashed red arrows: negative influence.Only major effects are displayed.

Structural coefficients two-lags model:

Empl.gr(t+2)

RnD.gr(t+2)

Sales.gr(t+2)

Opinc.gr(t+2)Opinc.gr(t)

RnD.gr(t)

Sales.gr(t)

Empl.gr(t) Empl.gr(t+1)

Sales.gr(t+1)

RnD.gr(t+1)

Opinc.gr(t+1)

Solid green arrows: positive causal influence; dashed red arrows: negative influence.Only major effects are displayed.

Conclusions

B Identification of causal structure in time series /non-experimental settings

B SVAR model: possibility of applying causal search to i.i.d.residuals

B Graphical causal search applied to linear / Gaussian data

B Assumptions (rules of inference): Causal Markov andFaithfulness Condition

B Extensions:

• Nonparametric conditional independence tests. No distributionalassumptions but same rules of inference.

• Causal search based on ICA. Semiparametric approach. Weakerassumptions.

Conclusions

B Extensions:

Conclusions

B Extensions:

Conclusions

B Extensions:

Conclusions

B Extensions:

Conclusions

B Extensions:

Thank you!

References

Baringhaus, L. and C. Franz, 2004, On a new multivariate two-sample test, Journal ofMultivariate Analysis, 88(1), 190-206.

Bessler, D.A., and S. Lee, 2002, Money and prices: US data 1869-1914 (a study withdirected graphs), Empirical Economics, 27, 427-446.

Chlaß, N., and A. Moneta, 2009, Can Graphical Causal Inference Be Extended toNonlinear Settings?, An Assessment of Conditional Independence Tests, in in M.Dorato, M. Redei, M. Suarez (eds.) EPSA Epistemology and Methodology of Science,Springer Verlag.

Demiralp, S., and K.D. Hoover, 2003, Searching for the Causal Structure of a VectorAutoregression, Oxford Bulletin of Economics and Statistics, 65, 745-767.

Demiralp, S., K.D. Hoover and S.J. Perez, 2008, A Bootstrap Method for Identifyingand Evaluating a Structural Vector Autoregression, Oxford Bulletin of Economics andStatistics.

Hyvarinen, A. and S. Shimizu and P. O. Hoyer, 2008, Causal modelling combininginstantaneous and lagged effects: an identifiable model based on non-Gaussianity,Proceedings of the 25th International Conference on Machine Learning, 424-431.

. Moneta Causal Search in TS Econometrics

References

Moneta, A. 2008, Graphical Causal Models and VARs: an Empirical Assessment of theReal Business Cycles Hypothesis, Empirical Economics.

Moneta, A., D. Entner, P. Hoyer, and A. Coad, 2009, Causal Inference by IndependentComponent Analysis with Applications to Micro- and Macroeconomic Data, mimeo.

Szekely, G. J. and M.L. Rizzo, 2004, Testing for Equal Distributions in High Dimension,InterStat, November (5).

Spirtes, P., C. Glymour, and R. Scheines, 2000, Causation, Prediction, and Search, The MITPress.

Su, L. and H. White, 2008, A Nonparametric Hellinger Metric Test for ConditionalIndependence, Econometric Theory, 24, 829-864.

Swanson, N.R., and C.W.J. Granger, 1997, Impulse Response Function Based on aCausal Approach to Residual Orthogonalization in Vector Autoregressions, Journal ofthe American Statistical Association, 92(437), 357-367.

graphical causal models for time series econometrics: …

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