Searching for
Statistical Causal Models:
Theory and Practice
Richard Scheines
Carnegie Mellon University
1
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
rm1 * rr1 = rm2 * rr2
m1
m2
r1
r2
Early Progenitors
Sewall Wright (1920s,1930s)
Graphical Model
 Causal & Statistical
Interpretation
The Method of Path Coefficients (1934).
Annals of Mathematical Statistics, 5, 161-215.
Social Sciences: 1940s  1970s
• Factor Analysis
• Structural Equation Models
• Instrumental Variable
Estimators
• Simultaneous Equation Models
Economics:
Sociology, Psychometrics, etc.
•
Cowles Commission
•
Franklin Fisher
•
Hubert Blalock
•
Art Goldberger
•
Herb Costner
•
Clive Granger
•
Otis Dudley Duncan
•
Herb Simon
•
David Heise
•
Haavelmo
•
David Kenny
•
R. Strotz
•
Ken Bollen
•
H. Wold
Population level Counterfactuals
1970s & 1980s:Graphical Models &
Independence Structures
• S. Lauritzen
• P. Dawid
• J. Darroch
• D. Cox
• T. Speed
• J. Robins
• H. Kiiveri
• J. Whittaker
• N. Wermuth
• D. Hausman
• D. Papineau
• 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
Graphical
Models
Intervention &
Manipulation
Counterfactuals
Constraints
(Independence)
Modern Non-Parametric Theory of
Statistical Causal Models
Causal Bayes Nets
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)
10
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.
11
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
Chicken Pox
Exposure
Rash
Infection
Rash
12
Causal Graphs
Not Cause Complete
Exposure
Omitted Causes
Infection
Symptoms
Common Cause Complete
Omitted
Common Causes
Exposure
Infection
Symptoms
13
Interventions & Causal Graphs
Model an intervention by adding an “intervention” variable outside the
original system as a direct cause of its target.
Pre-intervention graph
Education
Income
Taxes
Income
Taxes
Income
Taxes
Intervene on Income
“Soft” Intervention
Education
I
“Hard” Intervention
Education
I
14
Structural Equation Models
Education
Causal Graph
Income
Statistical
Causal Model
Longevity
1. Structural Equations
2. Statistical Constraints
15
Structural Equation Models
Education
Causal Graph
Income
Longevity
• 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
16
Structural Equation Models
Causal Graph
Education
Equations:
Education := ed
Income := Educationincome
Longevity
Longevity := EducationLongevity
Income
SEM Graph
Education
(path diagram)
1
Income
Income
2
Longevity
Statistical Constraints:
(ed, Income,Income ) ~N(0,2)
2 diagonal
- no variance is zero
Longevity
17
Two Routes to the Causal Markov Condition
• Assumption 1: Weak Causal Markov Assumption
V1,V2 causally disconnected  V1 _||_ V2
• Assumption 2a:
• Assumption 2b: Determinism, e.g.,
Causal Markov Axiom
Structural Equations
For each Vi  V, Vi := f(parents(Vi))
18
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)
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}
19
Modularity of Intervention/Manipulation
Causal
Graph
Education
Income
Manipulated
Causal
Graph
Longevity
Education
Income
M1
Longevity
Structural Equations:
Education := ed
Longevity := f (Education)Longevity
Income := f (Education)income
Manipulated Structural Equations:
Education := ed
Longevity := f (Education)Longevity
Income := f3 (M1)
Manipulation --> Causal Markov
• 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] LSEPitt Conference: Confirmation, Induction and Science
(London, 8 - 10 March, 2007)
http://philsciarchive.pitt.edu/archive/00003188/01/Detection_of_Unfaithfu
lness_and_Robust_Causal_Inference.pdf
21
SCM Search
Statistical Data  Causal Structure
Data
Equivalence Class of
Causal Graphs
X1
X1
X1
X2
X2
X2
X3
Statistical
Inference
Causal Markov Axiom
(D-separation)
X3
X3
Discovery Algorithm
Independence
Relations
X1
X3 | X2
Background Knowledge
- X2 before X3
22
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.
Tax Rate
b
a
c
Economy
Faithfulness: a ≠ -bc
Tax Revenues
23
Equivalence Classes
Equivalence:
• Independence (M1 ╞ X _||_ Y | Z  M2 ╞ X _||_ Y | Z)
• Distribution (q1 q2 M1(q1) = M2(q2))
• Independence (d-separation equivalence)
• DAGs : Patterns
• PAGs : Latent variable models
• Intervention Equivalence Classes
• Measurement Model Equivalence Classes
• Linear Non-Gaussian Model Equivalence Classes
24
Patterns
X1
X2
Pattern
X4
X3
Represents
X1
X1
X2
X3
X4
25
X2
X3
X4
Patterns: What the Edges Mean
X2
X1 and X2 are not adjacent in any
member of the equivalence class
X1
X2
X1  X2 (X1 is a cause of X2) in
every member of the equivalence
class.
X1
X2
X1  X2 in some members of the
equivalence class, and X2  X1 in
others.
X1
26
PAGs: Partial Ancestral Graphs
X2
X1
PAG
X3
Represents
X2
X1
X2
X1
T1
X3
X3
etc.
X1
X2
X2
X1
T1
T1
X3
27
X3
T2
PAGs: Partial Ancestral Graphs
What PAG edges mean.
X1
X2
X1 and X2 are not adjacent
X1
X2
X2 is not an ancestor of X1
X1
X2
No set d-separates X2 and X1
X1
X2
X1 is a cause of X2
X1
X2
There is a latent common
cause of X1 and X2
28
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
29
Tetrad Demo
http://www.phil.cmu.edu/projects/tetrad_download/
30
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 = 72
PO
degree of political exclusivity
CV
lack of civil liberties
EN
energy consumption per capita (economic development)
FI
level of foreign investment
31
Case Study 1: Foreign Investment
Correlations
fi
en
cv
po
-.175
-.480
0.868
fi
en
0.330
-.391
-.430
32
Case Study 1: Foreign Investment
Regression Results
po = .227*fi
SE
t
(.058)
3.941
- .176*en + .880*cv
(.059)
-2.99
(.060)
14.6
Interpretation: increases in foreign investment
increases political exclusion
33
Case Study 1: Foreign Investment
Alternatives
En
FI
CV
En
FI
CV
En
.31
-.23
FI
CV
.217
.88
-.176
PO
-.48
PO
Regression
Tetrad - FCI
.86
PO
Fit: df=2, 2=0.12,
p-value = .94
No model with testable constraint (df > 0) in which FI
has a positive effect on PO
Case Study 2: Welfare Reform
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)
Aurora Jackson, Richard Scheines
35
Case Study 2: Welfare Reform
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
36
Case Study 2: Welfare Reform
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
37
Case Study 2: Welfare Reform
Background Knowledge
Tier 1:
• Employment Status
Tier 2:
Over 22 million path
models consistent with
these constraints
• Depression
• Self-efficacy
• Mother/Father Relationship
• Father/Child Contact
• Mother’s Parenting/HOME
Tier 3:
• Negative Behaviors
• Cognitive Development
38
Case Study 2: Welfare Reform
Employment
Status
(Time 1)
-.472*
.215**
Conceptual
Model
Mother’s depressive
-.184*
symptoms
(Time 1)
Mother/Father
Relationship
(Time 1)
.407** Father/Child
Contact
(Time 1)
-.166*
.150*
Mother’s
self-efficacy
(Time 1)
Mother’s Parenting/
Home Environment
(Time 1)
.166*
-.291**
2 = 22.3, df = 20, p = .32
Negative
Behaviors
(Time 2)
Cognitive
Development
(Time 2)
Mother/Father
2

P = .32 GFI = .97 AGFI = .95
Mother’s
depressive
*
=
p
<
.05
**
=
p
<
.01
(20) = 22.3
Employment
-.129* Relationship
symptoms
(Time 1)
.407** Father/Child
Status
(Time
1)
Contact
(Time 1)
(Time 1)
.215**
Tetrad
Equivalence
Class
-.456*
Mother’s
self-efficacy
(Time 1)
2 = 18.87, df = 19, p = .46
.162*
Mother’s Parenting/
Home Environment
(Time 1)
.184*
-.291**
Negative
Behaviors
(Time 2)
.166*
Cognitive
Development
(Time 2)
39
Case Study 2: Welfare Reform
Employment
Status
(Time 1)
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
Mother’s depressive
-.184*
symptoms
(Time 1)
-.472*
.215**
Mother/Father
Relationship
(Time 1)
.407** Father/Child
Contact
(Time 1)
-.166*
.150*
Mother’s
self-efficacy
(Time 1)
Mother’s Parenting/
Home Environment
(Time 1)
.166*
-.291**
Conceptual Model
Negative
Behaviors
(Time 2)
Cognitive
Development
(Time 2)
Mother/Father
2

P = .32 GFI = .97 AGFI = .95
Mother’s
depressive
*
=
p
<
.05
**
=
p
<
.01
(20) = 22.3
Employment
-.129* Relationship
symptoms
(Time 1)
.407** Father/Child
Status
(Time
1)
Contact
(Time 1)
(Time 1)
Points of Disagreement:
•
Depression key cause vs. only
an effect
.215**
-.456*
Mother’s
self-efficacy
(Time 1)
.162*
Mother’s Parenting/
Home Environment
(Time 1)
.184*
-.291**
Tetrad
Negative
Behaviors
(Time 2)
.166*
Cognitive
Development
(Time 2)
40
Case Study 3: Online Courseware
Online Course in Causal & Statistical Reasoning
41
Case Study 3: Online Courseware
Variables
 Pre-test (%)
 Print-outs (% modules printed)
 Quiz Scores (avg. %)
 Voluntary Exercises (% completed)
 Final Exam (%)
 9 other variables
42
Case Study 3: Online Courseware
Printing and Voluntary Comprehension Checks: 2002 --> 2003
2002
2003
-.41**
voluntary
questions
print
.302*
.75**
pre
quiz
print
-.16
voluntary
questions
-.08
pre
.353*
.41*
.323*
.25*
final
final
43
Case Study 4: Charitable Giving
Variables
 Tangibility/Concreteness (Exp manipulation)
 Imaginability (likert 1-7)
 Impact (avg. of 2 likerts)
 Sympathy (likert)
 Donation ($)
44
Case Study 4: Charitable Giving
Theoretical Model
Impact
Tangibility
Imaginability
Donation
Sympathy
study 1 (N= 94) df = 5, 2 = 52.0, p= 0.0000
study 2 (N= 115) df = 5, 2 = 62.6, p= 0.0000
45
Case Study 4: Charitable Giving
Impact
Tangibility
Imaginability
study 1: df = 5, 2 = 5.88, p= 0.32
Donation
study 2: df = 5, 2 = 8.23, p= 0.14
GES
Outputs
Sympathy
Impact
Tangibility
Imaginability
study 1: df = 5, 2 = 3.99, p= 0.55
Donation
study 2: df = 5, 2 = 7.48, p= 0.18
Sympathy
46
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
47
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
48
Other Solutions
• Independent Components, etc
• Background Theory
• Scales
49
Other Solutions:
Background Theory
Specified Model
St1
12
St2
1.2
.
T1
Cognitive
Function
Home
?
St21
12
C2
.
.
T20
Lead
C1
T2
.
.
C20
Key Causal Question
Thus, key statistical question: Lead _||_ Cog | Home ?
50
Other Solutions:
Background Theory
True Model
St1
12
St2
1.2
.
T1
Cognitive
Function
Home
St21
12
T2
.
.
T20
Lead
C1
C2
.
.
“Impurities”
C20
F
Lead _||_ Cog | Home ?
Yes, but statistical inference will say otherwise.
51
Purify
Specified Model
F1
x1
x2
x3
F3
F2
x4
y1
y2
y3
y4
z1
z2
z3
z4
52
Purify
True Model
F1
x1
x2
x3
F3
F2
x4
y1
y2
y3
y4
z1
z2
z3
z4
F
53
Purify
True Model
F1
x1
x2
x3
F3
F2
x4
y1
y2
y3
y4
z1
z2
z3
z4
F
54
Purify
True Model
F1
x1
x2
x3
F3
F2
x4
y1
y2
y3
y4
z1
z2
z3
z4
F
55
Purify
True Model
F1
x1
x2
x3
F3
F2
x4
y1
y2
y3
y4
z1
z2
z3
z4
F
56
Purify
Purified Model
F1
x1
x2
x3
F3
F2
y1
y2
y3
y4
z1
z3
z4
F
57
Other Solutions: Scales
Scale = sum(measures of a latent)
St1
12
Homescale
St2
1.2
.
Homescale = i=1 to 21 (Sti)
Home
St21
12
58
Other Solutions: Scales
True Model
Pseudo-Random Sample: N = 2,000
59
Scales vs. Latent variable Models
True Model
Regression:
Cognition on
Home, Lead
Predictor
Constant
Home
Lead
S = 0.9940
Coef
-0.02291
1.22565
-0.00575
SE Coef
0.02224
0.02895
0.02230
R-Sq = 61.1%
T
-1.03
42.33
-0.26
P
0.303
0.000
0.797
Insig.
R-Sq(adj) = 61.0%
60
Scales vs. Latent variable Models
True Model
Scales
homescale = (x1 + x2 + x3)/3
leadscale = (x4 + x5 + x6)/3
cogscale = (x7 + x8 + x9)/3
61
Scales vs. Latent variable Models
True Model
Regression:
Cognition on
homescale, Lead
Cognition = - 0.0295 + 0.714 homescale - 0.178 Lead
Predictor
Constant
homescal
Lead
Coef
-0.02945
0.71399
-0.17811
SE Coef
0.02516
0.02299
0.02386
T
-1.17
31.05
-7.46
P
0.242
0.000
0.000
Sig.
62
Scales vs. Latent variable Models
True Model
Modeling
Latents
Specified
Model
63
Scales vs. Latent variable Models
True Model
Estimated Model
(2 = 29.6, df = 24, p = .19)
B5 = .0075, which at t=.23, is
correctly insignificant
64
Scales vs. Latent variable Models
True Model
Mixing Latents
and Scales
(2 = 14.57, df = 12, p = .26)
B5 = -.137, which at t=5.2, is
incorrectly highly significant
P < .001
65
Build Pure Clusters
Output - provably reliable (pointwise consistent):
Equivalence class of measurement models over a pure subset of measures
Dep
Stress
True Model
m1 m2 m3
Health
m4 m5 m6 m10
m11
m7 m8 m9
m
BPC
L1
Output
m1 m2 m3
L2
L3
m4 m5 m6 m7 m8 m9
66
Build Pure Clusters
Qualitative Assumptions
1.
Two types of nodes: measured (M) and latent (L)
2.
M
3.
Each m  M measures (is a direct effect of) at least one l  L
4.
No cycles involving M
L (measured don’t cause latents)
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
67
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
St1
12
St2
1.2
.
Specified Model
Stress
Depression
+
-
+
St21
12
C1
C2
.
.
.
.
Dep20
12
Coping
p = 0.00
Dep1
12
Dep2
12
C20
68
Case Study 4: Stress, Depression, and Religion
Build Pure Clusters
St3
12
St4
12
St16
12
St18
12
St20
12
Stress
Depression
Coping
C9
C12
C14
Dep9
12
Dep13
12
Dep19
12
C15
69
Case Study 4: Stress, Depression, and Religion
Assume Stress temporally prior:
MIMbuild to find Latent Structure:
St3
12
St4
12
St16
12
St18
12
St20
12
+
Stress
Depression
+
Coping
C9
C12
C14
C15
Dep9
12
Dep13
12
Dep19
12
p = 0.28
70
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
Exploratory Factor Analysis:
X3
X4
X8
X5
X9
X10
Emotionality
Worry
X6
X7
X15
X14
X16
X17
X18
X20
71
Case Study 5: Test Anxiety
Build Pure Clusters:
X3
X2
X8
Cares About
Achieving
X9
X10
X11
X16
X5
X7
Emotionalty
SelfDefeating
X6
X14
X18
72
Case Study 5: Test Anxiety
Build Pure Clusters:
Exploratory Factor Analysis:
X3
X2
X4
X8
X3
X2
X8
Worries About
Achieving
X5
X5
X9
X9
X10
Emotionality
Worry
X6
X10
X7
Emotionalty
X7
X15
X14
X16
X17
X18
X20
p-value = 0.00
X11
X16
X6
SelfDefeating
X14
X18
p-value = 0.47
73
Case Study 5: Test Anxiety
MIMbuild
X3
X2
X8
Worries About
Achieving
X9
X10
X11
X16
Scales: No Independencies or
Conditional Independencies
X5
X7
Emotionalty
SelfDefeating
X6
X14
Worries About
Achieving-Scale
EmotionaltyScale
SelfDefeating
X18
p = .43
Unininformative
74
Other Cases
Economics
Climate Research

Bessler, Pork Prices


Hoover, multiple

Cryder & Loewenstein,
Charitable Giving
Glymour, Chu, , Teleconnections
Epidemiology

Scheines, Lead & IQ
Biology
Educational Research

Easterday, Bias & Recall

Laski, Numerical coding

Shipley,

SGS, Spartina Grass
Neuroscience

Glymour & Ramsey, fMRI
75
References
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.
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.
76
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.
77
References
Economics
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.
78
References
Economics
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), 509533
- 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.
79
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