Mediating Between Causes and Probabilities: the Use of Graphical Models in Econometrics

1. Alessio Moneta Max Planck Institute of Economics, Jena, and Sant’Anna School of Advanced Studies, Pisa https://mail.sssup.it/~amoneta 16 June 2006 Causality and Probability in the Sciences University of Kent Mediating Between Causes and Probabilities: the Use of Graphical Models in Econometrics

2. Outline 1. Causal inference in macro-econometrics 2. Graphical models 3. Graphical models and structural Vector Autoregressions

3. Causal inference in macro-econometrics • Macro-econometric model: Structural form: A0 Yt + A1 Yt-1 + … + Am Yt-m + B0 Xt + B1 Xt-1 + … + Bn Xt-n = εt Yt : vector of endogenous variables Xt : vector of exogenous variables Reduced form: Yt = P1 Yt-1 + … + Pm Yt-m + Q0 Xt + … + Qn Xt-n + ut, where Pi = -A0-1 Ai, Qi = -A0-1 Bi, and ut = A0-1 εt

4. Causal inference in macro-econometrics • Problem of identification • Underdetermination of theory by data • Formalization of the problem of identification by Haavelmo (1944)

5. Deductivist approaches • Cowles Commission approach: a priori restrictions dictated by “Keynesian macroeconomics” • Lucas Critique (1976): the causal relations identified by the Cowles Commission are not stable (invariant under intervention) • Rational expectations econometrics • Calibration approach (Kydland – Prescott 1982) • Problems with deductivist approaches

6. Inductivist approaches • Sims’s (1980) Vector Autoregressions: Yt = P1 Yt-1 + … + Pm Yt-m + ut • Let us the data speak, “without pretending to have too much a priori theory” • Analysis of the effects of shocks on key variables (impulse response functions) • Structural VAR: A0 Yt = A1 Yt-1 + … + Am Yt-m + εt where Pi = -A0-1 Ai, and ut = A0-1 εt • Problem of identification once again.

7. Inductivist approaches • Granger Causality (1969, 1980). xt causes yt iff: P(yt \yt-1,yt-2 ,… , xt-1,xt-2 ,…,Ω) ≠P(yt \yt-1,yt-2 ,… ,Ω) • Probabilistic conception of causality: xt causes yt if xt renders yt more likely. • Similarities between Suppes’s (1970) and Granger’s account • Shortcomings of probabilistic accounts of causality

8. Graphical causal models • A graphical causal model (Spirtes et al. 2000) is a graph whose nodes are random variables with a joint probability distribution subject to some restrictions. These restrictions concern the type of connections between causal relations and conditional independence relations. • The simplest graphical causal model is the causal DAG. A causal DAG G is a directed acyclic graph whose nodes are random variables with a joint probability distribution P subject to the following condition: • Markov Condition: each variable is independent of its graphical non-descendants conditional on its graphical parents.

9. Causal DAG • Faithfulness condition: each independence condition is entailed by the Markov condition. • Stability

10. Beyond DAGs • Feedbacks (Richardson and Spirtes 1999); • Latent variables; • Non-linearity; • Graphical models for time series.

11. Graphical causal models • Logic of scientific discovery? • Causal Markov Condition and Faithfulness condition are general a priori assumptions • Using output of search algorithm + background knowledge to test single causal hypotheses. • Synthetic approach (Williamson 2003).

12. Graphical models for Structural VAR • Recovering structural analysis in VAR models (Swanson and Granger 1997, Bessler and Lee 2002, Demiralp and Hoover 2003, Moneta 2003). • Procedure to identify the causal structure of VAR models using graphical models. • This procedure uses a graphical algorithm (modified version of the PC algorithm of Spirtes-Glymour-Scheines 2000) to infer the contemporaneous causal structure starting from the analysis of the partial correlations among VAR residuals.

13. Empirical example • VAR model in which Y = (C, I, M, Y, R, ΔP)’ quarterly US data 1947:2 – 1994:1 • Output of the search algorithm: • The output of the algorithm consists of 24 DAGs • Testing the output of the algorithms: 8 DAGs are excluded • Incorporating background knowledge • Sensitivity analysis

14. Conclusions • Mediating between deductivist and inductivist approaches • Causal Markov Condition and Faithfulness Condition as working assumptions • Importance of background knowledge and deductive side of causal inference • Giving background knowledge an explicit causal language • Possibility of testing background knowledge • Under-determination problem and sensitivity analysis