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+
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+ \documentclass[MSNbibl,nameyear,dvips]{arxstspdf}
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+ \usepackage{tikz}
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+ \usepackage{graphicx}
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+ \usepackage{flushend}
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+ \usepackage{stfloats}
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+
8
+
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+ \volume{29}
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+ \issue{3}
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+ \pubyear{2014}
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+ \firstpage{363}
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+ \lastpage{366}
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+ \doi{10.1214/14-STS485} \referstodoi{10.1214/14-STS480}\docsubty{FLA}
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+
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+ \makeatletter
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+
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+
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+ \newcommand{\lleft}{\left}
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+ \newcommand{\rright}{\right}
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+
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+ \newcommand{\ind}{\mbox{$\,\perp\kern-5.5pt \perp\,$}}
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+
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+
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+ \renewcommand{\citep}[1]{(\citeauthor{#1}, \citeyear{#1})}
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+ \newcommand{\citepp}[1]{\citeauthor{#1}, \citeyear{#1}}
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+
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+ \makeatother
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+
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+ \begin{document}
31
+ \begin{frontmatter}
32
+
33
+ \vspace*{12pt}\title{ACE Bounds; SEMs with Equilibrium Conditions}
34
+ \runtitle{ACE Bounds; SEMS with Equilibrium Conditions}
35
+ \begin{aug}
36
+ \author[a]{\fnms{Thomas S.} \snm{Richardson}\ead[label=e1]{thomasr@uw.edu}}
37
+ \and
38
+ \author[b]{\fnms{James M.} \snm{Robins}\ead[label=e2]{robins@hsph.harvard.edu}}
39
+
40
+ \runauthor{T. S. Richardson and J. M. Robins}
41
+
42
+ \affiliation{University of Washington and Harvard School of Public Health}
43
+
44
+ \address[a]{Thomas S. Richardson is Professor and Chair, Department of Statistics, University of Washington,
45
+ Box 354322, Seattle, Washington 98195, USA \printead{e1}.}
46
+
47
+ \address[b]{James M. Robins is Mitchell L. and Robin LaFoley Dong Professor,
48
+ Department of Epidemiology, Harvard School of
49
+ Public Health,
50
+ 677 Huntington Avenue, Boston, Massachusetts 02115, USA \printead{e2}.}
51
+ \end{aug}
52
+
53
+
54
+ \end{frontmatter}
55
+
56
+ We congratulate the author on an enlightening account of
57
+ the instrumental variable approach from the viewpoint of Econometrics.
58
+ We first make some comments regarding the bounds on the ACE under the
59
+ nonparametric IV model,
60
+ and then discuss potential outcomes in the market equilibrium model.
61
+
62
+ \section{ACE Bounds Under the IV Model}
63
+ We consider the model in which $X$ and $Y$ are binary, taking values in
64
+ $\{0,1\}$, while
65
+ $Z$ takes $K$ states $\{1,\ldots,K\}$. We use the notation $X(z_i)$ to indicate
66
+ $X(z = i)$, similarly $Y(x_j)$ for $Y(x = j)$. We consider four
67
+ different sets of assumptions:
68
+ \begin{longlist}[(iii)]
69
+ \item[(i)] $Z \ind Y({x}_0),Y({x}_1), X({z}_1),\ldots,X({z}_{K})$;
70
+ \item[(ii)] $Z \ind Y({x}_0),Y({x}_1)$;
71
+ \item[(iii)] for $i \in\{1,\ldots,K\}$, $j \in\{0,1\}$, $Z \ind
72
+ X(z_i),\linebreak[4] Y(x_j)$;
73
+ \item[(iv)] there exists a $U$ such that $U \ind Z$ and for $j \in\{
74
+ 0,1\}$, $Y(x_j) \ind X,Z \mid U$.
75
+ \end{longlist}
76
+ Condition (i) is joint independence of $Z$ and all potential outcomes
77
+ for $Y$ and $X$. (ii) does not assume independence (or existence)
78
+ of counterfactuals for $X$.
79
+ (iii)~is a subset of the independences in (i), none of which involve
80
+ potential outcomes from different worlds.\footnote{In other words,
81
+ they do not involve both $Y(x_0)$ and $Y(x_1)$, nor $X(z_i)$ and
82
+ $X(z_j$) for $i\neq j$.}
83
+ The counterfactual independencies (i), (ii), (iii) arise most naturally
84
+ in the context
85
+ where the instrument is randomized, as depicted by the DAG in Figure~\ref{figswig}(a).
86
+ Assumption (iii) may be read (via d-separation) from the Single-World
87
+ Intervention Graph (SWIG)\footnote{See \citet{richardsonrobins2013}
88
+ for details.}
89
+ $\mathcal{G}_1(z,x)$, depicted in Figure~\ref{figswig}(b), which
90
+ represents the factorization of $P(Z,X(z),Y(x),U)$, implied by the IV model.
91
+
92
+ Lastly (iv) consists of only three independence statements, but does
93
+ assume the existence of an unobserved variable $U$ that
94
+ is sufficient to control for confounding between $X$ and $Y$. No
95
+ assumption is made concerning the existence of counterfactuals
96
+ $X(z)$; confounding variables ($U^*$) between $Z$ and $X$ are permitted
97
+ (so long as $U^* \ind U$). The DAG $\mathcal{G}_2$ and
98
+ corresponding SWIG $\mathcal{G}_2(x)$ are shown in Figure~\ref{figswig}(c), (d). In \citet{richardsonrobins2014}, we prove the following.
99
+
100
+
101
+ \begin{thm} Under any of the assumptions \textup{(i)}, \textup{(ii)}, \textup{(iii)}, \textup{(iv)}, the set of possible joint distributions
102
+ $P(Y(x_0), Y(x_1))$ are characterized by
103
+ the $8K$ inequalities:\vspace*{-2pt}
104
+ \begin{eqnarray}
105
+ \label{eqmarg}&&P\bigl(Y(x_i) = y\bigr) \nonumber\\[-1pt]
106
+ &&\quad \leq P(Y = y, X = i | Z = z)\\[-1pt]
107
+ &&\qquad {}+ P(X = 1-i |
108
+ Z = z),\nonumber
109
+ \\[4pt]
110
+ \label{eqjoint}&&P\bigl(Y(x_0) = y, Y(x_1) = \tilde{y}\bigr) \nonumber\\[-1pt]
111
+ &&\quad \leq P(Y
112
+ = y, X = 0 | Z = z)\\[-1pt]
113
+ &&\qquad {} + P(Y = \tilde{y}, X = 1 | Z = z).\nonumber
114
+ \end{eqnarray}
115
+ \end{thm}
116
+
117
+ Thus a distribution $P(X,Y | Z)$ is compatible with the stated
118
+ assumptions if and only if there exists a distribution
119
+ $P(Y(x_0), Y(x_1))$ satisfying (\ref{eqmarg}) and (\ref{eqjoint}).
120
+
121
+ \begin{thm} Under any of the assumptions \textup{(i)}, \textup{(ii)}, \textup{(iii)}, \textup{(iv)}
122
+ for all $i,j \in\{0,1\}$, $P(Y(x_i) = j) \leq
123
+ g(i,j)$, where\vspace*{-2pt}
124
+ {\fontsize{10.9}{12.9}\selectfont{\begin{eqnarray*}
125
+ g(i,j) &\equiv&\min \Bigl\{ \min_{z} \bigl[\vphantom{\hat{P}} P(X
126
+ = i, Y = j | Z = z)\\
127
+ &&\hphantom{\min \Bigl\{\min_{z} \bigl[}{} + P(X = 1-i | Z = z) \bigr],
128
+ \\
129
+ &&\hphantom{\min \Bigl\{} \min_{z, \tilde{z}: z \neq\tilde{z}} \bigl[ P(X = i, Y = j | Z
130
+ = z) \\
131
+ &&\hphantom{\min \Bigl\{\min_{z, \tilde{z}: z \neq\tilde{z}} \bigl[}{}+ P(X = 1-i, Y = 0 | Z = z)
132
+ \nonumber
133
+ \\
134
+ &&\hphantom{\min \Bigl\{\min_{z, \tilde{z}: z \neq\tilde{z}} \bigl[}{} + P(X = i, Y = j | Z =
135
+ \tilde{z}) \\
136
+ &&\hphantom{\min \Bigl\{\min_{z, \tilde{z}: z \neq\tilde{z}} \bigl[}{}+ P(X = 1-i, Y = 1 | Z = \tilde {z}) \bigr] \Bigr\}.
137
+ \nonumber
138
+ \end{eqnarray*}}}
139
+ Furthermore, $P(Y(x_0))$ and $P(Y(x_1))$ are variation independent.
140
+ Consequently,
141
+ \begin{eqnarray*}
142
+ 1-g(1,0)-g(0,1) &\leq& \operatorname{ACE}(X \rightarrow Y)\\
143
+ & \leq& g(0,0)+g(1,1)-1.
144
+ \end{eqnarray*}
145
+ These bounds are sharp.
146
+ \end{thm}
147
+
148
+
149
+ \begin{figure}
150
+
151
+ \includegraphics{485f01.eps}
152
+
153
+ \caption{\textup{(a)} IV model with no confounding between $Z$ and $X$; \textup{(b)}
154
+ SWIG representing $P(Z, X(z),Y(x),U)$;
155
+ \textup{(c)} IV model with confounding between $Z$ and $X$; \textup{(d)} SWIG
156
+ representing $P(Z, X,Y(x),U,U^*)$.}\label{figswig}
157
+ \end{figure}
158
+
159
+
160
+ Note that to evaluate $g(i,j)$ requires finding a minimum over $K^2$
161
+ expressions. In the case
162
+ where $K=2$, these bounds reduce to those given by \citet
163
+ {BalkPearboun1997}, who assume (i).\footnote{\citet{dawid2003} working
164
+ in a non-counterfactual framework also established the bounds for $K=2$
165
+ under the DAG in Figure~\ref{figswig}(a); however,
166
+ his proof also applies to Figure~\ref{figswig}(c). \citet
167
+ {robinsgreenland1996} observed that the Balke--Pearl bounds
168
+ were also sharp under (ii).} \citet{robins1989} and \citet{manski1990} derived what are called
169
+ the ``natural bounds'' on the ACE under the weaker assumption that $Z
170
+ \ind Y({x}_0)$ and $Z \ind Y({x}_1)$.
171
+ As noted by Imbens, without further assumptions these bounds are not
172
+ sharp. However, the natural bounds are sharp under (i) or (iii), if, in
173
+ addition, we assume there are no Defiers (an assumption that has
174
+ testable implications). \citet{chengsmall2006} considered bounds on
175
+ the ACE when $K=3$ under additional assumptions.
176
+
177
+
178
+ \section{Market Equilibrium and BiCausal Models}
179
+ Imbens' clear description of the market equilibrium model is
180
+ particularly informative.
181
+ We also strongly endorse the author's contention that the RHS of
182
+ systems of structural equations
183
+ should be interpreted as describing potential outcomes for the
184
+ LHS.\footnote{
185
+ \citet{pearl2000}, \citet{laucausal}, \citet{lauritzen02} argue that these are not
186
+ really ``equations''
187
+ but are better viewed as ``assignments'' in computer languages, for example, $ y
188
+ \leftarrow x +1$;
189
+ see also \citet{strotzwoldrecursive1960}, page 420.}
190
+
191
+ However, we note that this position has important implications both for
192
+ interpretation and inference.
193
+ Furthermore, it does not
194
+ seem to be universally accepted within Economics. \citet{leroy2006}
195
+ states that ``economic models use the equality symbol with its usual
196
+ mathematical meaning,
197
+ not with the meaning of the assignment operator'';
198
+ an approach that is clearly incompatible with an interpretation in
199
+ terms of potential outcomes. For example,
200
+ it becomes permissible to renormalize structural equations to change
201
+ which variable is on the LHS.
202
+
203
+ It has also been argued that statistical analyses of such models should be
204
+ invariant to the normalization; see \citet{hillier1990},
205
+ \citet{basmanncausal1963}.\hskip.2pt\footnote{For example, \citet{greene2003}, page 401, states (in the
206
+ context of the IV model):
207
+ ``one significant virtue of [the Limited Information Maximum Likelihood
208
+ Estimator]
209
+ is its invariance to normalization of the equations.''}
210
+ Contrary to Imbens' remark,\footnote{Footnote 8, page 331.} this
211
+ alternative view does not appear to be motivated by
212
+ considerations of measurement error. \citet{leroy2006} makes clear
213
+ that he does not believe
214
+ that structural equations describe potential outcomes for endogenous
215
+ variables and does not discuss issues relating to measurement.\footnote
216
+ {For example,
217
+ \citet{leroy2006}, page 23, states that ``The assumption that it makes
218
+ sense to delete one or more of the structural equations
219
+ and replace the value of the internal variable so determined by a
220
+ constant without altering the other equations [\ldots] is virtually never satisfied
221
+ in economic models since each external variable typically affects
222
+ equilibrium values of more than one internal variable.''
223
+ He goes on to assert ``In fact, it is difficult to think of nontrivial
224
+ models in any area of research in which the [\ldots] assumption is satisfied.''}
225
+ Rather, this appears to be a fundamental difference in interpretation.
226
+
227
+ The market equilibrium model specifies potential outcomes for
228
+ $Q^d_t(p)$, $Q^s_t(p)$:
229
+ \begin{eqnarray}
230
+ \label{eqqd}Q^d_t(p)&=& \alpha^d + \beta^d p
231
+ + \varepsilon_t^d,
232
+ \\
233
+ \label{eqqs}Q^s_t(p)&=& \alpha^s + \beta^s p
234
+ + \varepsilon_t^s,
235
+ \end{eqnarray}
236
+ and imposes the equilibrium condition:\footnote{To simplify notation,
237
+ throughout we work directly in terms of $\log$ price and $\log$ quantity.}
238
+ \begin{eqnarray}
239
+ Q^d_t(p) = Q^s_t(p).\label{eqequ}
240
+ \end{eqnarray}
241
+ \citet{strotzwoldrecursive1960} described such systems as \textit
242
+ {bicausal}.
243
+ It should be observed that the model does not
244
+ specify potential outcomes for price ($P_t(q_s,q_d)$), nor does it view
245
+ price as externally determined (i.e., exogenous).
246
+ Instead price is determined implicitly as a consequence of the
247
+ equilibrium condition. In this regard, the
248
+ model might be regarded as incomplete: Indeed \citet
249
+ {haavelmowhat1958} is quite critical of this model for failing to
250
+ offer any \emph{explanation} as to
251
+ how the equilibrium price is determined. The model also falls outside
252
+ the scope of non-parametric structural equation models (NPSEM) (see, e.g., \cite{pearl2000}), which require one equation for each
253
+ endogenous variable;\footnote{Indeed \citet{leroy2006} argues against the interpretation of
254
+ structural equations in terms of potential outcomes on the grounds that
255
+ this interpretation, as advanced by Pearl, requires a one-to-one
256
+ mapping between equations and endogenous variables that he argues, does not
257
+ make sense for the market equilibrium model.}
258
+ likewise the model
259
+ defies standard graphical representation, though see Figure~\ref{figone}(a).
260
+ \begin{figure}
261
+
262
+ \includegraphics{485f02.eps}
263
+
264
+ \caption{\textup{(a)} Attempt to depict the bicausal model; \textup{(b)} a schematic
265
+ showing the deterministic system (\protect\ref{eqcon})--(\protect\ref{eqmer});
266
+ the edge
267
+ \protect\tikz\protect\path(0ex,0ex) edge[->] node[above=0pt, black]
268
+ {$\scriptscriptstyle I$} (3ex,0ex);
269
+ denotes that $P$ is the integral of $\Delta P$; see Iwasaki and Simon (\citeyear
270
+ {iwasaki1994}).}\label{figone}
271
+ \end{figure}
272
+
273
+ A related question concerns whether there exist dynamic acyclic (i.e.,
274
+ recursive)
275
+ systems of structural equations
276
+ that lead to the equilibrium distribution corresponding either to a
277
+ cyclic system
278
+ of structural equations or a bicausal system.\footnote{Analysis of this question was stimulated by a heated debate that arose between
279
+ Wold, who advocated a recursive, regression-based approach to demand
280
+ analysis, and Haavelmo
281
+ and the Cowles Commission who advocated simultaneous equations. See
282
+ \citet{haavelmostatistical1943}, \citet{woldbentzelstatistical1946},
283
+ \citet{woldjureendemand1953}, \citet{bentzelhansenrecursiveness1954},
284
+ \citet{strotzwoldrecursive1960}, \citet{basmanncausal1963}; historical overviews
285
+ are given by \citet{morganstamping1991},
286
+ \citet{epsteinhistory1987}.} \citet{fishcorr} provides just such a ``correspondence principle''
287
+ under which the distribution implied by a cyclic linear SEM is obtained
288
+ as a time average of a deterministic
289
+ set of first order difference equations reaching a static equilibrium
290
+ subject to stochastic boundary conditions.
291
+ The correspondence assumes that the equilibration time is very fast
292
+ relative to the interval between observations
293
+ so the time averaged variables are in deterministic equilibrium.
294
+ Fisher also derived conditions on the coefficient matrices of a cyclic
295
+ SEM that are required in order for the system
296
+ to reach equilibrium; in fact he further required that each subset of
297
+ structural equations also have this property.
298
+
299
+ However, Fisher's correspondence presumes a normalization under which
300
+ each variable is associated with
301
+ a single equation (as in an NPSEM), and hence would not apply to a
302
+ bicausal system. \citet{richphd}, Chapter~2,
303
+ described a system of finite difference equations that gives rise to
304
+ the bicausal system~(\ref{eqqd})--(\ref{eqequ}):
305
+ \begin{eqnarray}
306
+ \label{eqcon}\mbox{Consumers:}&&\hspace*{4pt} Q^d_{t+(k+1)\delta}(p_{t+k\delta})\nonumber \\[-8pt]\\[-8pt]
307
+ &&\hspace*{4pt}\quad =
308
+ \alpha^d+ \beta^d p_{t+k\delta} + \varepsilon_{t}^d,\nonumber
309
+ \\
310
+ \label{eqsup}\mbox{Suppliers:}&&\hspace*{4pt} Q^s_{t+(k+1)\delta}(p_{t+k\delta}) \nonumber\\[-8pt]\\[-8pt]
311
+ &&\hspace*{4pt}\quad =
312
+ \alpha^s+ \beta^s p_{t+k\delta} + \varepsilon_{t}^s,\nonumber
313
+ \\
314
+ \label{eqmer}\mbox{Merchants:}&&\hspace*{4pt} P_{t+(k+1)\delta}\bigl(q^d_{t+k\delta},
315
+ q^s_{t+k\delta},p_{t+k\delta}\bigr) \nonumber\\[-8pt]\\[-8pt]
316
+ &&\hspace*{4pt}\quad = p_{t+k\delta} +
317
+ \lambda \bigl(q^d_{t+k\delta} - q^s_{t+k\delta}
318
+ \bigr),\nonumber
319
+ \end{eqnarray}
320
+ for $k=\{0,\ldots, \delta^{-1}-1\}$. Note that the disturbances
321
+ $(\varepsilon_{t}^d, \varepsilon_{t}^s)$ represent boundary
322
+ conditions and hence
323
+ remain fixed during the interval $[t,t+1)$.
324
+ As in Fisher's correspondence, the observed variables correspond to
325
+ limiting time-averages over a unit
326
+ interval:
327
+ \begin{eqnarray*}
328
+ \overline{Q}^d_t &=& \lim_{\delta\rightarrow0} \delta
329
+ \sum_{k=0}^{\delta^{-1}-1} {Q}^d_{t+k\delta},\quad
330
+ \overline{Q}^s_t = \lim_{\delta\rightarrow0} \delta
331
+ \sum_{k=0}^{\delta^{-1}-1} {Q}^s_{t+k\delta},\\
332
+ \overline{P}_t &=& \lim_{\delta\rightarrow0} \delta\sum
333
+ _{k=0}^{\delta^{-1}-1} {P}_{t+k\delta}.
334
+ \end{eqnarray*}
335
+ Under suitable conditions on the coefficients, $(\overline{Q}^d_t,
336
+ \overline{Q}^s_t,\allowbreak \overline{P}_t)$ obey
337
+ equations (\ref{eqqd})--(\ref{eqequ}). Note that Merchants'
338
+ equation (\ref{eqmer}) which includes $P$, leads to the equilibrium
339
+ condition (\ref{eqequ}) that does not.\footnote{In causal terms,
340
+ this model is similar to one presented in \citet{wold1959}. Wold
341
+ viewed his model as a formalization of Cournot's theories.}
342
+ It might be objected to the proposed model that there is no disturbance
343
+ term in equation (\ref{eqmer}).
344
+ The explanation for this is that the disturbance terms in the
345
+ nonrecursive model correspond to constant factors in the deterministic
346
+ evolution.
347
+ The equation for price gives the change in price during a small
348
+ interval (length $\delta$) to the discrepancy between supply and demand.
349
+ Adding a disturbance term would say that throughout the observation
350
+ period (length $1$) the Merchants' reaction to change in price was off
351
+ by a constant factor, so that even if quantities supplied and demanded
352
+ were identical, the Merchants would change the price. Thus, if we add
353
+ an error $\varepsilon^p_t$ the model will not, in general, arrive at
354
+ equilibrium
355
+ within the unit interval.\footnote{Having said this, the equations
356
+ (\ref{eqcon}) and (\ref{eqsup}) still imply that producers and
357
+ consumers make systematic errors in computing prices over a time-scale
358
+ of length $\delta$.}
359
+
360
+ \citet{iwasaki1994} represent equilibrating mechanisms via ``causal
361
+ influence diagrams'' in which the derivatives of variables are included.
362
+ Under this scheme, model (\ref{eqcon})--(\ref{eqmer}) is
363
+ represented by the graph in Figure~\ref{figone}(b).
364
+ This example serves to show that time averages of (deterministic)
365
+ equilibrating systems need not have a structural equation for each variable.
366
+ See also \citep{2001.dash.esqaru} for related work.
367
+
368
+
369
+ \section*{Acknowledgments}
370
+ This work was supported by the US National Institutes of Health Grant
371
+ R01 AI032475; Richardson was also supported by the US National Science
372
+ Foundation Grant CNS-0855230.
373
+
374
+
375
+ \begin{thebibliography}{28}
376
+
377
+
378
+ \bibitem[\protect\citeauthoryear{Balke and Pearl}{1997}]{BalkPearboun1997}
379
+ \begin{barticle}[author]
380
+ \bauthor{\bsnm{Balke},~\bfnm{Alexander}\binits{A.}} \AND
381
+ \bauthor{\bsnm{Pearl},~\bfnm{Judea}\binits{J.}}
382
+ (\byear{1997}).
383
+ \btitle{Bounds on treatment effects from studies with imperfect compliance}.
384
+ \bjournal{J. Amer. Statist. Assoc.}
385
+ \bvolume{92}
386
+ \bpages{1171--1176}.
387
+ \end{barticle}
388
+ \bptok{imsref}\endbibitem
389
+
390
+ \bibitem[\protect\citeauthoryear{Basmann}{1963}]{basmanncausal1963}
391
+ \begin{barticle}[author]
392
+ \bauthor{\bsnm{Basmann},~\bfnm{R.~L.}\binits{R.~L.}}
393
+ (\byear{1963}).
394
+ \btitle{The causal interpretation of non-triangular systems of economic relations (with discussion)}.
395
+ \bjournal{Econometrica}
396
+ \bvolume{31}
397
+ \bpages{439--453}.
398
+ \end{barticle}
399
+ \bptok{imsref}\endbibitem
400
+
401
+ \bibitem[\protect\citeauthoryear{Bentzel and Hansen}{1954}]{bentzelhansenrecursiveness1954}
402
+ \begin{barticle}[author]
403
+ \bauthor{\bsnm{Bentzel},~\bfnm{R.}\binits{R.}} \AND
404
+ \bauthor{\bsnm{Hansen},~\bfnm{B.}\binits{B.}}
405
+ (\byear{1954}).
406
+ \btitle{On recursiveness and interdependency in economic models}.
407
+ \bjournal{Rev. Econom. Stud.}
408
+ \bvolume{22}
409
+ \bpages{153--168}.
410
+ \end{barticle}
411
+ \bptok{imsref}\endbibitem
412
+
413
+ \bibitem[\protect\citeauthoryear{Bentzel and Wold}{1946}]{woldbentzelstatistical1946}
414
+ \begin{barticle}[mr]
415
+ \bauthor{\bsnm{Bentzel},~\bfnm{R.}\binits{R.}} \AND
416
+ \bauthor{\bsnm{Wold},~\bfnm{H.}\binits{H.}}
417
+ (\byear{1946}).
418
+ \btitle{On statistical demand analysis from the viewpoint of simultaneous equations}.
419
+ \bjournal{Skand. Aktuarietidskr.}
420
+ \bvolume{29}
421
+ \bpages{95--114}.
422
+ \bid{mr={0017907}}
423
+ \end{barticle}
424
+ \bptok{imsref}\endbibitem
425
+
426
+ \bibitem[\protect\citeauthoryear{Cheng and Small}{2006}]{chengsmall2006}
427
+ \begin{barticle}[mr]
428
+ \bauthor{\bsnm{Cheng},~\bfnm{Jing}\binits{J.}} \AND
429
+ \bauthor{\bsnm{Small},~\bfnm{Dylan~S.}\binits{D.~S.}}
430
+ (\byear{2006}).
431
+ \btitle{Bounds on causal effects in three-arm trials with non-compliance}.
432
+ \bjournal{J. R. Stat. Soc. Ser. B Stat. Methodol.}
433
+ \bvolume{68}
434
+ \bpages{815--836}.
435
+ \bid{doi={10.1111/j.1467-9868.2006.00568.x}, issn={1369-7412}, mr={2301296}}
436
+ \end{barticle}
437
+ \bptok{imsref}\endbibitem
438
+
439
+
440
+ \bibitem[\protect\citeauthoryear{Dash and Druzdzel}{2001}]{2001.dash.esqaru}
441
+ \begin{binproceedings}[author]
442
+ \bauthor{\bsnm{Dash}, \bfnm{Denver}\binits{D.}} \AND
443
+ \bauthor{\bsnm{Druzdzel}, \bfnm{Marek J.}\binits{M. J.}}
444
+ (\byear{2001}).
445
+ \btitle{Caveats for causal reasoning with equilibrium models}.
446
+ In \bbooktitle{Proceedings of the Sixth European Conference on
447
+ Symbolic and Quantitative Approaches\vadjust{\goodbreak} to Reasoning with Uncertainty (ECSQARU), Toulouse, France}
448
+ (\beditor{\bfnm{Salem}\binits{S.} \bsnm{Benferhat}} \AND
449
+ \beditor{\bfnm{Philippe}\binits{P.} \bsnm{Besnard}}, eds.).
450
+ \bseries{Lecture Notes in Artificial Intelligence}
451
+ \bvolume{2143}
452
+ \bpages{192--203}.
453
+ \bpublisher{Springer},
454
+ \blocation{Berlin}.
455
+ \end{binproceedings}
456
+ \bptok{imsref}\endbibitem
457
+
458
+
459
+
460
+
461
+
462
+ \bibitem[\protect\citeauthoryear{Dawid}{2003}]{dawid2003}
463
+ \begin{bincollection}[mr]
464
+ \bauthor{\bsnm{Dawid},~\bfnm{A.~Philip}\binits{A.~P.}}
465
+ (\byear{2003}).
466
+ \btitle{Causal inference using influence diagrams: The problem of partial compliance}.
467
+ In \bbooktitle{Highly Structured Stochastic Systems}
468
+ (\beditor{\bfnm{P. J.}\binits{P. J.} \bsnm{Green}},
469
+ \beditor{\bfnm{N. L.}\binits{N. L.} \bsnm{Hjort}} \AND
470
+ \beditor{\bfnm{S.}\binits{S.} \bsnm{Richardson}}, eds.).
471
+ \bseries{Oxford Statist. Sci. Ser.}
472
+ \bvolume{27}
473
+ \bpages{45--81}.
474
+ \bpublisher{Oxford Univ. Press},
475
+ \blocation{Oxford}.
476
+ \bid{mr={2082406}}
477
+ \bptnote{check related}\end{bincollection}
478
+ \bptok{imsref}\endbibitem
479
+
480
+ \bibitem[\protect\citeauthoryear{Epstein}{1987}]{epsteinhistory1987}
481
+ \begin{bbook}[mr]
482
+ \bauthor{\bsnm{Epstein},~\bfnm{Roy~J.}\binits{R.~J.}}
483
+ (\byear{1987}).
484
+ \btitle{A History of Econometrics}.
485
+ \bseries{Contributions to Economic Analysis}
486
+ \bvolume{165}.
487
+ \bpublisher{North-Holland},
488
+ \blocation{Amsterdam}.
489
+ \bid{mr={0918969}}
490
+ \end{bbook}
491
+ \bptok{imsref}\endbibitem
492
+
493
+ \bibitem[\protect\citeauthoryear{Fisher}{1970}]{fishcorr}
494
+ \begin{barticle}[author]
495
+ \bauthor{\bsnm{Fisher},~\bfnm{F.~M.}\binits{F.~M.}}
496
+ (\byear{1970}).
497
+ \btitle{A correspondence principle for simultaneous equation models}.
498
+ \bjournal{Econometrica}
499
+ \bvolume{38}
500
+ \bpages{73--92}.
501
+ \end{barticle}
502
+ \bptok{imsref}\endbibitem
503
+
504
+ \bibitem[\protect\citeauthoryear{Greene}{2003}]{greene2003}
505
+ \begin{bbook}[author]
506
+ \bauthor{\bsnm{Greene},~\bfnm{William~H.}\binits{W.~H.}}
507
+ (\byear{2003}).
508
+ \btitle{Econometric Analysis},
509
+ \bedition{5th} ed.
510
+ \bpublisher{Prentice Hall},
511
+ \blocation{Upper Saddle River, NJ}.
512
+ \end{bbook}
513
+ \bptok{imsref}\endbibitem
514
+
515
+ \bibitem[\protect\citeauthoryear{Haavelmo}{1943}]{haavelmostatistical1943}
516
+ \begin{barticle}[mr]
517
+ \bauthor{\bsnm{Haavelmo},~\bfnm{Trygve}\binits{T.}}
518
+ (\byear{1943}).
519
+ \btitle{The statistical implications of a system of simultaneous equations}.
520
+ \bjournal{Econometrica}
521
+ \bvolume{11}
522
+ \bpages{1--12}.
523
+ \bid{issn={0012-9682}, mr={0007954}}
524
+ \end{barticle}
525
+ \bptok{imsref}\endbibitem
526
+
527
+ \bibitem[\protect\citeauthoryear{Haavelmo}{1958}]{haavelmowhat1958}
528
+ \begin{barticle}[author]
529
+ \bauthor{\bsnm{Haavelmo},~\bfnm{T.}\binits{T.}}
530
+ (\byear{1958}).
531
+ \btitle{Hva kan statiske likevektsmodeller fortelle oss?}
532
+ \bjournal{National{\o}konomisk Tidsskrift}
533
+ \bvolume{96 (Suppl.)}
534
+ \bpages{138--145}
535
+ \bnote{(in {N}orwegian). English translation published as: {W}hat can static equilibrium models tell us? \textit{Economic Inquiry} \textbf{12} (1974) 27--34}.
536
+ \end{barticle}
537
+ \bptok{imsref}\endbibitem
538
+
539
+ \bibitem[\protect\citeauthoryear{Hillier}{1990}]{hillier1990}
540
+ \begin{barticle}[mr]
541
+ \bauthor{\bsnm{Hillier},~\bfnm{Grant~H.}\binits{G.~H.}}
542
+ (\byear{1990}).
543
+ \btitle{On the normalization of structural equations: Properties of direction estimators}.
544
+ \bjournal{Econometrica}
545
+ \bvolume{58}
546
+ \bpages{1181--1194}.
547
+ \bid{doi={10.2307/2938305}, issn={0012-9682}, mr={1079413}}
548
+ \end{barticle}
549
+ \bptok{imsref}\endbibitem
550
+
551
+ \bibitem[\protect\citeauthoryear{Iwasaki and Simon}{1994}]{iwasaki1994}
552
+ \begin{barticle}[mr]
553
+ \bauthor{\bsnm{Iwasaki},~\bfnm{Yumi}\binits{Y.}} \AND
554
+ \bauthor{\bsnm{Simon},~\bfnm{Herbert~A.}\binits{H.~A.}}
555
+ (\byear{1994}).
556
+ \btitle{Causality and model abstraction}.
557
+ \bjournal{Artificial Intelligence}
558
+ \bvolume{67}
559
+ \bpages{143--194}.
560
+ \bid{doi={10.1016/0004-3702(94)90014-0}, issn={0004-3702}, mr={1281657}}
561
+ \end{barticle}
562
+ \bptok{imsref}\endbibitem
563
+
564
+ \bibitem[\protect\citeauthoryear{Lauritzen}{2001}]{laucausal}
565
+ \begin{bincollection}[mr]
566
+ \bauthor{\bsnm{Lauritzen},~\bfnm{Steffen~L.}\binits{S.~L.}}
567
+ (\byear{2001}).
568
+ \btitle{Causal inference from graphical models}.
569
+ In \bbooktitle{Complex Stochastic Systems ({E}indhoven, 1999)}.
570
+ \bseries{Monogr. Statist. Appl. Probab.}
571
+ \bvolume{87}
572
+ \bpages{63--107}.
573
+ \bpublisher{Chapman \& Hall/CRC},
574
+ \blocation{Boca Raton, FL}.
575
+ \bid{mr={1893411}}
576
+ \end{bincollection}
577
+ \bptok{imsref}\endbibitem
578
+
579
+ \bibitem[\protect\citeauthoryear{Lauritzen and Richardson}{2002}]{lauritzen02}
580
+ \begin{barticle}[mr]
581
+ \bauthor{\bsnm{Lauritzen},~\bfnm{Steffen~L.}\binits{S.~L.}} \AND
582
+ \bauthor{\bsnm{Richardson},~\bfnm{Thomas~S.}\binits{T.~S.}}
583
+ (\byear{2002}).
584
+ \btitle{Chain graph models and their causal interpretations}.
585
+ \bjournal{J. R. Stat. Soc. Ser. B Stat. Methodol.}
586
+ \bvolume{64}
587
+ \bpages{321--361}.
588
+ \bid{doi={10.1111/1467-9868.00340}, issn={1369-7412}, mr={1924296}}
589
+ \bptnote{check related}\end{barticle}
590
+ \bptok{imsref}\endbibitem
591
+
592
+ \bibitem[\protect\citeauthoryear{{LeRoy}}{2006}]{leroy2006}
593
+ \begin{btechreport}[author]
594
+ \bauthor{\bsnm{{LeRoy}},~\bfnm{Stephen~F.}\binits{S.~F.}}
595
+ (\byear{2006}).
596
+ \btitle{Causality in economics}.
597
+ \btype{Technical report, Univ. California, Santa Barbara}.
598
+ \end{btechreport}
599
+ \bptok{imsref}\endbibitem
600
+
601
+ \bibitem[\protect\citeauthoryear{Manski}{1990}]{manski1990}
602
+ \begin{barticle}[author]
603
+ \bauthor{\bsnm{Manski},~\bfnm{C.~F.}\binits{C.~F.}}
604
+ (\byear{1990}).
605
+ \btitle{Non-parametric bounds on treatment effects}.
606
+ \bjournal{American Economic Review}
607
+ \bvolume{80}
608
+ \bpages{351--374}.
609
+ \end{barticle}
610
+ \bptok{imsref}\endbibitem
611
+
612
+ \bibitem[\protect\citeauthoryear{Morgan}{1991}]{morganstamping1991}
613
+ \begin{bincollection}[author]
614
+ \bauthor{\bsnm{Morgan},~\bfnm{M.~S.}\binits{M.~S.}}
615
+ (\byear{1991}).
616
+ \btitle{The stamping out of process analysis in econometrics}.
617
+ In \bbooktitle{Appraising Economic Theories}
618
+ (\beditor{\bfnm{Neil}\binits{N.}~\bparticle{de}~\bsnm{Marchi}} \AND
619
+ \beditor{\bfnm{Mark}\binits{M.}~\bsnm{Blaug}}, eds.)
620
+ \bpages{237--272}.
621
+ \bpublisher{Edward Elgar},
622
+ \blocation{Cheltenham}.
623
+ \end{bincollection}
624
+ \bptok{imsref}\endbibitem
625
+
626
+ \bibitem[\protect\citeauthoryear{Pearl}{2000}]{pearl2000}
627
+ \begin{bbook}[mr]
628
+ \bauthor{\bsnm{Pearl},~\bfnm{Judea}\binits{J.}}
629
+ (\byear{2000}).
630
+ \btitle{Causality: Models, Reasoning, and Inference}.
631
+ \bpublisher{Cambridge Univ. Press},
632
+ \blocation{Cambridge}.
633
+ \bid{mr={1744773}}
634
+ \end{bbook}
635
+ \bptok{imsref}\endbibitem
636
+
637
+ \bibitem[\protect\citeauthoryear{Richardson}{1996}]{richphd}
638
+ \begin{bphdthesis}[author]
639
+ \bauthor{\bsnm{Richardson},~\bfnm{T.~S.}\binits{T.~S.}}
640
+ (\byear{1996}).
641
+ \btitle{Models of feedback: Interpretation and discovery}.
642
+ \btype{Ph.D. thesis, Carnegie-Mellon Univ}.
643
+ \end{bphdthesis}
644
+ \bptok{imsref}\endbibitem
645
+
646
+ \bibitem[\protect\citeauthoryear{Richardson and Robins}{2013}]{richardsonrobins2013}
647
+ \begin{btechreport}[author]
648
+ \bauthor{\bsnm{Richardson},~\bfnm{Thomas~S.}\binits{T.~S.}} \AND
649
+ \bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
650
+ (\byear{2013}).
651
+ \btitle{{S}ingle {W}orld {I}ntervention {G}raphs {(SWIGs)}: A unification of the counterfactual and graphical approaches to
652
+ causality}.
653
+ \btype{Technical Report 128, Center for Statistics and the Social Sciences, Univ.
654
+ Washington, Seattle, WA}.
655
+ \end{btechreport}
656
+ \bptok{imsref}\endbibitem
657
+
658
+ \bibitem[\protect\citeauthoryear{Richardson and Robins}{2014}]{richardsonrobins2014}
659
+ \begin{bunpublished}[author]
660
+ \bauthor{\bsnm{Richardson},~\bfnm{T.~S.}\binits{T.~S.}} \AND
661
+ \bauthor{\bsnm{Robins},~\bfnm{J.~M.}\binits{J.~M.}}
662
+ (\byear{2014}).
663
+ \btitle{Assumptions and bounds in the instrumental variable model}.
664
+ \bnote{Preprint}.
665
+ \end{bunpublished}
666
+ \bptok{imsref}\endbibitem
667
+
668
+ \bibitem[\protect\citeauthoryear{Robins}{1989}]{robins1989}
669
+ \begin{bincollection}[author]
670
+ \bauthor{\bsnm{Robins},~\bfnm{J.~M.}\binits{J.~M.}}
671
+ (\byear{1989}).
672
+ \btitle{The analysis of randomized and non-randomized AIDS treatment trials using a new approach to causal inference in longitudinal studies.}
673
+ In \bbooktitle{Health Service Research Methodology: A Focus on {AIDS}}
674
+ (\beditor{\bfnm{L.}\binits{L.}~\bsnm{Sechrest}},
675
+ \beditor{\bfnm{H.}\binits{H.}~\bsnm{Freeman}} \AND
676
+ \beditor{\bfnm{A.}\binits{A.}~\bsnm{Mulley}}, eds.).
677
+ \bpublisher{U.S. Public Health Service},
678
+ \blocation{Washington, DC}.
679
+ \end{bincollection}
680
+ \bptok{imsref}\endbibitem
681
+
682
+ \bibitem[\protect\citeauthoryear{Robins and Greenland}{1996}]{robinsgreenland1996}
683
+ \begin{barticle}[author]
684
+ \bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}} \AND
685
+ \bauthor{\bsnm{Greenland},~\bfnm{Sander}\binits{S.}}
686
+ (\byear{1996}).
687
+ \btitle{Identification of causal effects using instrumental variables: Comment}.
688
+ \bjournal{J. Amer. Statist. Assoc.}
689
+ \bvolume{91}
690
+ \bpages{456--458}.
691
+ \bid{issn={01621459}}
692
+ \end{barticle}
693
+ \bptok{imsref}\endbibitem
694
+
695
+ \bibitem[\protect\citeauthoryear{Strotz and Wold}{1960}]{strotzwoldrecursive1960}
696
+ \begin{barticle}[mr]
697
+ \bauthor{\bsnm{Strotz},~\bfnm{Robert~H.}\binits{R.~H.}} \AND
698
+ \bauthor{\bsnm{Wold},~\bfnm{H.~O.~A.}\binits{H.~O.~A.}}
699
+ (\byear{1960}).
700
+ \btitle{Recursive vs. nonrecursive systems: An attempt at synthesis}.
701
+ \bjournal{Econometrica}
702
+ \bvolume{28}
703
+ \bpages{417--427}.
704
+ \bid{issn={0012-9682}, mr={0120034}}
705
+ \end{barticle}
706
+ \bptok{imsref}\endbibitem\vfill\eject
707
+
708
+ \bibitem[\protect\citeauthoryear{Wold}{1959}]{wold1959}
709
+ \begin{bincollection}[mr]
710
+ \bauthor{\bsnm{Wold},~\bfnm{Herman~O.~A.}\binits{H.~O.~A.}}
711
+ (\byear{1959}).
712
+ \btitle{Ends and means in econometric model building}.
713
+ In \bbooktitle{Probability and Statistics: {T}he {H}arald {C}ram\'er Volume} ({U}. {G}renander, ed.)
714
+ \bpages{355--434}.
715
+ \bpublisher{Almqvist \& Wiksell},
716
+ \blocation{Stockholm}.
717
+ \bid{mr={0109088}}
718
+ \end{bincollection}
719
+ \bptok{imsref}\endbibitem
720
+
721
+ \bibitem[\protect\citeauthoryear{Wold and Jur{\'e}en}{1953}]{woldjureendemand1953}
722
+ \begin{bbook}[author]
723
+ \bauthor{\bsnm{Wold},~\bfnm{H.~O.~A.}\binits{H.~O.~A.}} \AND
724
+ \bauthor{\bsnm{Jur{\'e}en},~\bfnm{L.}\binits{L.}}
725
+ (\byear{1953}).
726
+ \btitle{Demand Analysis}.
727
+ \bpublisher{Wiley},
728
+ \blocation{New York}.
729
+ \end{bbook}
730
+ \bptok{imsref}\endbibitem\vfill
731
+
732
+
733
+ \end{thebibliography}
734
+ \end{document}