1

Richard ScheinesCarnegie Mellon University

Searching for Statistical Causal Models:

Theory and Practice

Goals

1) Policy, Law, and Science: How can we use data to answer

a) subjunctive questions (effects of future policy interventions), or

b) counterfactual questions (what would have happened had things been

done differently (law)?

c) scientific questions (what mechanisms run the world)

2) Rumsfeld Problem: Do we know what we don’t know: Can we tell when

there is or is not enough information in the data to answer causal questions?

Early Progenitors

Charles Spearman (1904)

Statistical Constraints Causal Structure

g

m1 m2 r1 r2

rm1 * rr1 = rm2 * rr2

Early Progenitors

The Method of Path Coefficients (1934).

Annals of Mathematical Statistics, 5, 161-215.

Sewall Wright (1920s,1930s)

Graphical Model Causal & Statistical

Interpretation

Social Sciences: 1940s 1970s

Economics:

• Cowles Commission

• Franklin Fisher

• Art Goldberger

• Clive Granger

• Herb Simon

• Haavelmo

• R. Strotz

• H. Wold

Sociology, Psychometrics, etc.

• Hubert Blalock

• Herb Costner

• Otis Dudley Duncan

• David Heise

• David Kenny

• Ken Bollen

• Factor Analysis

• Instrumental VariableEstimators

• Structural Equation Models

• Simultaneous Equation Models

Population level Counterfactuals

1970s & 1980s:Graphical Models & Independence Structures

• S. Lauritzen

• J. Darroch

• T. Speed

• H. Kiiveri

• N. Wermuth

• D. Hausman

• D. Papineau

• P. Dawid

• D. Cox

• J. Robins

• J. Whittaker

• Judea Pearl 1988

1988 1993:Axioms, Intervention, and

Latent Variable Model Search

• P. Spirtes, C. Glymour, and R. Scheines

• Causal Markov Axiom

• Full model of interventions, both surgical and non-surgical

• Equivalence classes for latent variable models, with search

Modern Non-Parametric Theory of Statistical Causal Models

Intervention & Manipulation

Graphical Models

Counterfactuals

Constraints(Independence)

Causal Bayes Nets

10

Semantics of SCMs Choice 1: Take direct causation as primitive, axiomatize

Causal systems over V

Probabilistic Independence Relations in P(V)

Choice 2: Define direct causation in terms of

intervention, i.e., (hypothetical) treatment)

11

Choice 1: Causal Markov Axiom

If G is a causal graph, and P a probability distribution

over the variables in G, then in <G,P> satisfy the

Markov Axiom iff:

every variable V is independent of its non-effects,

conditional on its immediate causes.

12

Causal Graphs

Causal Graph G = {V,E} Each edge X Y represents a direct causal claim:

X is a direct cause of Y relative to V

Exposure Rash

Exposure Infection Rash

Chicken Pox

13

Causal Graphs

Not Cause Complete

Common Cause Complete

Exposure Infection Symptoms

Omitted Causes

Exposure Infection Symptoms

Omitted

Common Causes

14

Interventions & Causal GraphsModel an intervention by adding an “intervention” variable outside the

original system as a direct cause of its target.

Education Income Taxes Pre-intervention graph

Intervene on Income

“Soft” Intervention Education Income Taxes

I

“Hard” Intervention Education Income Taxes

I

15

Structural Equation Models

1. Structural Equations2. Statistical Constraints

Education

LongevityIncome

Statistical Causal Model

Causal Graph

16

Structural Equation Models

• Structural Equations: One Assignment Equation for each variable V V := f(parents(V), errorV)

for SEM (linear regression) f is a linear function

• Statistical Constraints: Joint Distribution over the Error terms

Education

LongevityIncome

Causal Graph

17

Structural Equation Models

Equations: Education := ed

Income :=Educationincome

Longevity :=EducationLongevity

Statistical Constraints: (ed, Income, Income ) ~N(0, 2) 2diagonal - no variance is zero

Education

LongevityIncome

Causal Graph

Education

Income Longevity

1 2

LongevityIncome

SEM Graph

(path diagram)

18

Two Routes to the Causal Markov Condition

• Assumption 1: Weak Causal Markov Assumption

V1,V2 causally disconnected V1 _||_ V2

• Assumption 2a:

Causal Markov Axiom

• Assumption 2b: Determinism, e.g.,

Structural Equations

For each Vi V, Vi := f(parents(Vi))

19

X is a direct cause of Y relative to S, iff z,x1 x2 P(Y | X set= x1 , Z set= z)

P(Y | X set= x2 , Z set= z)

where Z = S - {X,Y}

Choice 2: Define Direct Causation from Intervention

X is a cause of Y iff x1 x2 P(Y | X set= x1) P(Y | X set= x2)

Structural Equations: Education := ed

Longevity := f (Education) Longevity

Income := f (Education) income

Education

LongevityIncome

Modularity of Intervention/Manipulation

Causal Graph

Manipulated Structural Equations: Education := ed

Longevity := f (Education) Longevity

Income := f3(M1)

ManipulatedCausal Graph

Education

Longevity Income

M1

21

• Manipulation conception of causation and Modularity

--> weak version of CMA

• Zhang, Jiji and Spirtes, Peter (2007) Detection of

Unfaithfulness and Robust Causal Inference. In [2007] LSE-

Pitt Conference: Confirmation, Induction and Science

(London, 8 - 10 March, 2007)

http://philsci-archive.pitt.edu/archive/00003188/01/Detection_of_Unfaithfulness_and_Robust_Causal_Inference.pdf

Manipulation --> Causal Markov

22

SCM SearchStatistical Data Causal Structure

Background Knowledge

- X2 before X3

X3 | X2 X1

Independence Relations

Data

Statistical Inference

X2 X3 X1

Equivalence Class of Causal Graphs

X2 X3 X1

X2 X3 X1

Discovery Algorithm

Causal Markov Axiom (D-separation)

23

Faithfulness

Constraints on a probability distribution P generated by a causal structure G hold for all parameterizations of G.

Revenues = aRate + cEconomy + Rev.

Economy = bRate + Econ.

Faithfulness: a ≠ -bcTax Revenues

Economyc

ba

Tax Rate

24

Equivalence Classes

• Independence (d-separation equivalence)• DAGs : Patterns• PAGs : Latent variable models• Intervention Equivalence Classes

• Measurement Model Equivalence Classes• Linear Non-Gaussian Model Equivalence Classes

Equivalence:• Independence (M1 ╞ X _||_ Y | Z M2 ╞ X _||_ Y | Z)

• Distribution (q1 q2 M1(q1) = M2(q2))

25

Patterns

X2

X4 X3

X1

X2

X4 X3

Represents

Pattern

X1 X2

X4 X3

X1

26

Patterns: What the Edges Mean

X2 X1

X2 X1X1 X2 in some members of theequivalence class, and X2 X1 inothers.

X1 X2 (X1 is a cause of X2) inevery member of the equivalenceclass.

X2 X1 X1 and X2 are not adjacent in anymember of the equivalence class

27

PAGs: Partial Ancestral Graphs

X2

X3

X1

X2

X3

Represents

PAG

X1 X2

X3

X1

X2

X3

T1

X1

X2

X3

X1

etc.

T1

T1 T2

28

PAGs: Partial Ancestral Graphs

X2 X1

X2 X1

X2 X1

X2 There is a latent commoncause of X1 and X2

No set d-separates X2 and X1

X1 is a cause of X2

X2 is not an ancestor of X1

X1

X2 X1 X1 and X2 are not adjacent

What PAG edges mean.

29

Overview of Search Methods

• Constraint Based Searches• TETRAD (SGS, PC, FCI)• Very fast – max N ~ 1,000• Pointwise Consistent

• Scoring Searches• Scores: BIC, AIC, etc.• Search: Hill Climb, Genetic Alg., Simulated Annealing• Difficult to extend to latent variable models• Meek and Chickering Greedy Equivalence Class (GES)• Very slow – max N ~ 30-40

30

Tetrad Demo

http://www.phil.cmu.edu/projects/tetrad_download/

31

Case Study 1: Foreign Investment

Does Foreign Investment in 3rd World Countries cause Repression?

Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.

N = 72PO degree of political exclusivityCV lack of civil libertiesEN energy consumption per capita (economic development)FI level of foreign investment

32

Correlations

po fi en fi -.175 en -.480 0.330 cv 0.868 -.391 -.430

Case Study 1: Foreign Investment

33

Regression Results

po = .227*fi - .176*en + .880*cv SE (.058) (.059) (.060)t 3.941 -2.99 14.6

Interpretation: increases in foreign investment increases political exclusion

Case Study 1: Foreign Investment

Alternatives

.217

FI

PO

CV En

Regression

.88 -.176

FI

PO

CV En

Tetrad - FCI

FI

PO

CV En

Fit: df=2, 2=0.12, p-value = .94

.31 -.23

.86 -.48

Case Study 1: Foreign Investment

No model with testable constraint (df > 0) in which FI has a positive effect on PO

35

Aurora Jackson, Richard Scheines

Single Mothers’ Self-Efficacy, Parenting in the Home Environment, and  

Children’s Development in a Two-Wave Study(Social Work Research, 29, 1, 7-20)

Case Study 2: Welfare Reform

36

Sampling Scheme

• Longitudinal Data

o Time 1: 1996-97 (N = 188)

o Time 2: 1998-99 (N = 178)

• Single black mothers in NYC

• Current and former welfare recipients

• With a child who was 3 – 5 at time 1,

and 6 to 8 at time 2

Case Study 2: Welfare Reform

37

Constructs/Scales/Measures

• Employment Status

• Perceived Self-efficacy

• Depressive Symptoms

• Quality of Mother/Father Relationship

• Father/Child Contact

• Quality of Home Environment

• Behavior Problems

• Cognitive Development

Case Study 2: Welfare Reform

38

Background Knowledge

Tier 1: • Employment Status

Tier 2: • Depression• Self-efficacy• Mother/Father Relationship• Father/Child Contact• Mother’s Parenting/HOME

Tier 3:• Negative Behaviors• Cognitive Development

Over 22 million path models consistent with these constraints

Case Study 2: Welfare Reform

39

Employment Status

(Time 1)

Mother’s self-efficacy

(Time 1)

Mother’s depressive symptoms (Time 1)

Mother/Father Relationship

(Time 1) Father/Child Contact (Time 1)

Mother’s Parenting/ Home Environment

(Time 1)

Negative Behaviors (Time 2)

Cognitive Development

(Time 2)

.215** -.456*

-.129* .407**

.162*

-.291** .166*

* = p < .05 ** = p < .01

.184*

2 (19) = 18.87 P = .46 GFI = .97 AGFI = .95

Employment Status

(Time 1)

Mother’s self-efficacy

(Time 1)

Mother’s depressive symptoms (Time 1)

Mother/Father Relationship

(Time 1) Father/Child Contact (Time 1)

Mother’s Parenting/ Home Environment

(Time 1)

Negative Behaviors (Time 2)

Cognitive Development

(Time 2)

.215** -.472*

-.184* .407**

.150*

-.291** .166*

-.166*

2 (20) = 22.3 P = .32 GFI = .97 AGFI = .95

* = p < .05 ** = p < .01

Tetrad Equivalence Class

Conceptual Model

2 = 22.3, df = 20, p = .32

2 = 18.87, df = 19, p = .46

Case Study 2: Welfare Reform

40

Employment Status

(Time 1)

Mother’s self-efficacy

(Time 1)

Mother’s depressive symptoms (Time 1)

Mother/Father Relationship

(Time 1) Father/Child Contact (Time 1)

Mother’s Parenting/ Home Environment

(Time 1)

Negative Behaviors (Time 2)

Cognitive Development

(Time 2)

.215** -.456*

-.129* .407**

.162*

-.291** .166*

* = p < .05 ** = p < .01

.184*

2 (19) = 18.87 P = .46 GFI = .97 AGFI = .95

Employment Status

(Time 1)

Mother’s self-efficacy

(Time 1)

Mother’s depressive symptoms (Time 1)

Mother/Father Relationship

(Time 1) Father/Child Contact (Time 1)

Mother’s Parenting/ Home Environment

(Time 1)

Negative Behaviors (Time 2)

Cognitive Development

(Time 2)

.215** -.472*

-.184* .407**

.150*

-.291** .166*

-.166*

2 (20) = 22.3 P = .32 GFI = .97 AGFI = .95

* = p < .05 ** = p < .01

Tetrad

Conceptual Model

Points of Agreement:• Mother’s Self-Efficacy mediates

the effect of Employment on all other variables.

• HOME environment mediates the effect of all other factors on outcomes: Cog. Develop and Prob. Behaviors

Points of Disagreement:• Depression key cause vs. only

an effect

Case Study 2: Welfare Reform

41

Online Course in Causal & Statistical Reasoning

Case Study 3: Online Courseware

42

Variables

Pre-test (%)

Print-outs (% modules printed)

Quiz Scores (avg. %)

Voluntary Exercises (% completed)

Final Exam (%)

9 other variables

Case Study 3: Online Courseware

43

Printing and Voluntary Comprehension Checks: 2002 --> 2003

.302*

-.41**

.75**

.353* .323*

pre

print voluntary questions

quiz

final

2002

-.08

-.16

.41* .25*

pre

print voluntary questions

final

2003

Case Study 3: Online Courseware

44

Variables

Tangibility/Concreteness (Exp manipulation)

Imaginability (likert 1-7)

Impact (avg. of 2 likerts)

Sympathy (likert)

Donation ($)

Case Study 4: Charitable Giving

45

Theoretical Model

Case Study 4: Charitable Giving

Imaginability Tangibility

Impact

Sympathy

Donation

study 1 (N= 94) df = 5, 2 = 52.0, p= 0.0000

study 2 (N= 115) df = 5, 2 = 62.6, p= 0.0000

46

GES Outputs

Case Study 4: Charitable Giving

Imaginability Tangibility

Impact

Sympathy

Donation

study 1: df = 5, 2 = 5.88, p= 0.32

Imaginability Tangibility

Impact

Sympathy

Donation

study 2: df = 5, 2 = 8.23, p= 0.14

study 1: df = 5, 2 = 3.99, p= 0.55

study 2: df = 5, 2 = 7.48, p= 0.18

47

The Causal Theory Formation Problem for Latent Variable Models

Given observations on a number of variables, identify the latent variables that underlie these variables and the causal relations among these latent concepts.

Example: Spectral measurements of solar radiation intensities. Variables are intensities at each measured frequency.

Example: Quality of a Child’s Home Environment, Cumulative Exposure to Lead, Cognitive Functioning

48

The Most Common Automatic Solution: Exploratory Factor Analysis

• Chooses “factors” to account linearly for as much of the variance/covariance of the measured variables as possible.

• Great for dimensionality reduction• Factor rotations are arbitrary• Gives no information about the statistical and thus the

causal dependencies among any real underlying factors.

• No general theory of the reliability of the procedure

49

Other Solutions

• Independent Components, etc• Background Theory• Scales

50

Other Solutions: Background Theory

St112

Home

St212

St2112

.

.

T1

Lead

.

.

Cognitive Function

T2

T20

C1 C2 C20 . .

?

Key Causal Question

Thus, key statistical question: Lead _||_ Cog | Home ?

Specified Model

51

St112

Home

St212

St2112

.

.

T1

Lead

.

.

Cognitive Function

T2

T20

C1 C2 C20 . .

F

Lead _||_ Cog | Home ?

Yes, but statistical inference will say otherwise.

Other Solutions: Background Theory

True Model

“Impurities”

52

F1

x1 x2

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

Specified Model

53

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

True Model

54

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

True Model

55

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

True Model

56

F1

x1 x2

F

F2 F3

x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

Purify

True Model

57

F1

x1 x2

F

F2 F3

x3 y1 y2 y3 y4 z1 z3 z4

Purify

Purified Model

58

Scale = sum(measures of a latent)

Other Solutions: Scales

St112

Home

St212

St2112

.

.

Homescale = i=1 to 21 (Sti)

Homescale

59

True Model

Other Solutions: Scales

Pseudo-Random Sample: N = 2,000

60

Scales vs. Latent variable Models

Regression:Cognition on Home, Lead

 Predictor Coef SE Coef T PConstant -0.02291 0.02224 -1.03 0.303Home 1.22565 0.02895 42.33 0.000Lead -0.00575 0.02230 -0.26 0.797 S = 0.9940 R-Sq = 61.1% R-Sq(adj) = 61.0%

Insig.

True Model

61

Scales vs. Latent variable Models

Scales homescale = (x1 + x2 + x3)/3leadscale = (x4 + x5 + x6)/3cogscale = (x7 + x8 + x9)/3

True Model

62

Scales vs. Latent variable Models

Cognition = - 0.0295 + 0.714 homescale - 0.178 Lead  Predictor Coef SE Coef T PConstant -0.02945 0.02516 -1.17 0.242homescal 0.71399 0.02299 31.05 0.000Lead -0.17811 0.02386 -7.46 0.000

Regression:Cognition on

homescale, Lead

Sig.

True Model

63

Scales vs. Latent variable Models

Modeling Latents

True Model

Specified Model

64

Scales vs. Latent variable Models

(2 = 29.6, df = 24, p = .19)

B5 = .0075, which at t=.23, is correctly insignificant

True Model

Estimated Model

65

Scales vs. Latent variable Models

Mixing Latents and Scales

(2 = 14.57, df = 12, p = .26)

B5 = -.137, which at t=5.2, is incorrectly highly significantP < .001

True Model

66

Build Pure Clusters

Output - provably reliable (pointwise consistent):

Equivalence class of measurement models over a pure subset of measures

L1 L2 L3

m1 m2 m3 m4 m5 m6 m7 m8 m9

Stress Dep Health

m1 m2 m3 m4 m5 m6 m7 m8 m9 m11 m10

m

BPC

True Model

Output

67

Build Pure ClustersQualitative Assumptions1. Two types of nodes: measured (M) and latent (L)

2. M L (measured don’t cause latents)

3. Each m M measures (is a direct effect of) at least one l L

4. No cycles involving M

Quantitative Assumptions:1. Each m M is a linear function of its parents plus noise

2. P(L) has second moments, positive variances, and no deterministic relations

68

Case Study 4: Stress, Depression, and Religion

MSW Students (N = 127) 61 - item survey (Likert Scale)

• Stress: St1 - St21

• Depression: D1 - D20

• Religious Coping: C1 - C20

p = 0.00

St112

Stress

St212

St2112

.

.

Dep112

Coping

.

. Depression

Dep212

Dep2012

C1 C2 C20 . .

+

- +

Specified Model

69

Build Pure Clusters St3

12

Stress

St412 St1612

Dep912

Coping

Depression Dep1312

Dep1912

C9 C12 C15

St1812

St2012

C14

Case Study 4: Stress, Depression, and Religion

70

Assume Stress temporally prior:

MIMbuild to find Latent Structure: St312

Stress

St412 St1612

Dep912

Coping

Depression Dep1312

Dep1912

C9 C12 C15

St1812

St2012

C14

+

+

p = 0.28

Case Study 4: Stress, Depression, and Religion

71

Case Study 5: Test Anxiety

Bartholomew and Knott (1999), Latent variable models and factor analysis

12th Grade Males in British Columbia (N = 335)

20 - item survey (Likert Scale items): X1 - X20:

X2

Emotionality Worry

X8

X9

X10

X15

X16

X18

X3

X4

X5

X6

X7

X14

X17

X20

Exploratory Factor Analysis:

72

Build Pure Clusters:

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Cares About Achieving

Self-Defeating

Case Study 5: Test Anxiety

73

Build Pure Clusters:

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Worries About Achieving

Self-Defeating

X2

Emotionality Worry

X8

X9

X10

X15

X16

X18

X3

X4

X5

X6

X7

X14

X17

X20

p-value = 0.00 p-value = 0.47

Exploratory Factor Analysis:

Case Study 5: Test Anxiety

74

X2

Emotionalty

X8

X9

X10

X11

X16

X18

X3

X5

X7

X14

X6

Worries About Achieving

Self-Defeating

MIMbuild

p = .43

Emotionalty-Scale

Worries About Achieving-Scale

Self-Defeating

Unininformative

Scales: No Independencies or Conditional Independencies

Case Study 5: Test Anxiety

75

Economics

Bessler, Pork Prices

Hoover, multiple

Cryder & Loewenstein,

Charitable Giving

Other Cases

Educational Research

Easterday, Bias & Recall

Laski, Numerical coding

Climate Research

Glymour, Chu, , Teleconnections

Biology

Shipley,

SGS, Spartina Grass

Neuroscience

Glymour & Ramsey, fMRI

Epidemiology

Scheines, Lead & IQ

76

References

Biology

Chu, Tianjaio, Glymour C., Scheines, R., & Spirtes, P, (2002). A Statistical Problem for Inference to Regulatory Structure from Associations of Gene Expression Measurement with Microarrays. Bioinformatics, 19: 1147-1152.

Shipley, B. Exploring hypothesis space: examples from organismal biology. Computation, Causation and Discovery. C. Glymour and G. Cooper. Cambridge, MA, MIT Press.

 Shipley, B. (1995). Structured interspecific determinants of specific leaf area in 34 species of

herbaceous angeosperms. Functional Ecology 9.

General

Spirtes, P., Glymour, C., Scheines, R. (2000). Causation, Prediction, and Search, 2nd Edition, MIT Press.

Pearl, J. (2000). Causation: Models of Reasoning and Inference, Cambridge University Press. 

77

References

Scheines, R. (2000). Estimating Latent Causal Influences: TETRAD III Variable Selection and Bayesian Parameter Estimation: the effect of Lead on IQ, Handbook of Data Mining, Pat Hayes, editor, Oxford University Press.

Jackson, A., and Scheines, R., (2005). Single Mothers' Self-Efficacy, Parenting in the Home Environment, and Children's Development in a Two-Wave Study, Social Work Research , 29, 1, pp. 7-20.

Timberlake, M. and Williams, K. (1984). Dependence, political exclusion, and government repression: Some cross-national evidence. American Sociological Review 49, 141-146.

 

78

ReferencesEconomics

Akleman, Derya G., David A. Bessler, and Diana M. Burton. (1999). ‘Modeling corn exports and exchange rates with directed graphs and statistical loss functions’, in Clark Glymour and Gregory F. Cooper (eds) Computation, Causation, and Discovery, American Association for Artificial Intelligence, Menlo Park, CA and MIT Press, Cambridge, MA, pp. 497-520.

Awokuse, T. O. (2005) “Export-led Growth and the Japanese Economy: Evidence from VAR and Directed Acyclical Graphs,” Applied Economics Letters 12(14), 849-858.

Bessler, David A. and N. Loper. (2001) “Economic Development: Evidence from Directed Acyclical Graphs” Manchester School 69(4), 457-476.

Bessler, David A. and Seongpyo Lee. (2002). ‘Money and prices: U.S. data 1869-1914 (a study with directed graphs)’, Empirical Economics, Vol. 27, pp. 427-46.

Demiralp, Selva and Kevin D. Hoover. (2003) !Searching for the Causal Structure of a Vector Autoregression," Oxford Bulletin of Economics and Statistics 65(supplement), pp. 745-767.

Haigh, M.S., N.K. Nomikos, and D.A. Bessler (2004) “Integration and Causality in International Freight Markets: Modeling with Error Correction and Directed Acyclical Graphs,” Southern Economic Journal 71(1), 145-162.

Sheffrin, Steven M. and Robert K. Triest. (1998). ‘A new approach to causality and economic growth’, unpublished typescript, University of California, Davis.

79

ReferencesEconomics

Swanson, Norman R. and Clive W.J. Granger. (1997). ‘Impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions’, Journal of the American Statistical Association, Vol. 92, pp. 357-67.

Demiralp, S., Hoover, K., & Perez, S. A Bootstrap Method for Identifying and Evaluating a Structural Vector Autoregression Oxford Bulletin of Economics and Statistics, 2008, 70, (4), 509-533

- Searching for the Causal Structure of a Vector Autoregression Oxford Bulletin of Economics and Statistics, 2003, 65, (s1), 745-767

Kevin D. Hoover, Selva Demiralp, Stephen J. Perez, Empirical Identification of the Vector Autoregression: The Causes and Effects of U.S. M2*, This paper was written to present at the Conference in Honour of David F. Hendry at Oxford University, 2325 August 2007.

Selva Demiralp and Kevin D. Hoover , Searching for the Causal Structure of a Vector Autoregression, OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 65, SUPPLEMENT (2003) 0305-9049

A. Moneta, and P. Spirtes “Graphical Models for the Identification of Causal Structures in Multivariate Time Series Model”, Proceedings of the 2006 Joint Conference on Information Sciences, JCIS 2006, Kaohsiung, Taiwan, ROC, October 8-11,2006, Atlantis Press, 2006.

References

• Causation, Prediction, and Search, 2nd Edition, (2000), by P. Spirtes, C. Glymour, and R. Scheines ( MIT Press)

• Causality: Models, Reasoning, and Inference (2000). By Judea Pearl, Cambridge Univ. Press

• Computation, Causation, & Discovery (1999), edited by C. Glymour and G. Cooper, MIT Press

• Eberhardt, F., and Scheines R., (2007).“Interventions and Causal Inference”, in PSA-2006, Proceedings of the 20th biennial meeting of the Philosophy of Science Association 2006 http://philsci.org/news/PSA06

• Silva, R., Glymour, C., Scheines, R. and Spirtes, P. (2006) “Learning the Structure of Latent Linear Structure Models,” Journal of Machine Learning Research, 7, 191-246.

• TETRAD IV: www.phil.cmu.edu/projects/tetrad • Web Course on Causal and Statistical Reasoning :

www.phil.cmu.edu/projects/csr/• Causality Lab: www.phil.cmu.edu/projects/causality-lab