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1 |
+
|
2 |
+
\documentclass[MSNbibl,nameyear,dvips]{arxstspdf}
|
3 |
+
\usepackage{tikz}
|
4 |
+
\usepackage{graphicx}
|
5 |
+
\usepackage{flushend}
|
6 |
+
\usepackage{stfloats}
|
7 |
+
|
8 |
+
|
9 |
+
\volume{29}
|
10 |
+
\issue{3}
|
11 |
+
\pubyear{2014}
|
12 |
+
\firstpage{363}
|
13 |
+
\lastpage{366}
|
14 |
+
\doi{10.1214/14-STS485} \referstodoi{10.1214/14-STS480}\docsubty{FLA}
|
15 |
+
|
16 |
+
\makeatletter
|
17 |
+
|
18 |
+
|
19 |
+
\newcommand{\lleft}{\left}
|
20 |
+
\newcommand{\rright}{\right}
|
21 |
+
|
22 |
+
\newcommand{\ind}{\mbox{$\,\perp\kern-5.5pt \perp\,$}}
|
23 |
+
|
24 |
+
|
25 |
+
\renewcommand{\citep}[1]{(\citeauthor{#1}, \citeyear{#1})}
|
26 |
+
\newcommand{\citepp}[1]{\citeauthor{#1}, \citeyear{#1}}
|
27 |
+
|
28 |
+
\makeatother
|
29 |
+
|
30 |
+
\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 |
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\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 |
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\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 |
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\bauthor{\bsnm{Haavelmo},~\bfnm{T.}\binits{T.}}
|
530 |
+
(\byear{1958}).
|
531 |
+
\btitle{Hva kan statiske likevektsmodeller fortelle oss?}
|
532 |
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\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 |
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\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 |
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\bauthor{\bsnm{Iwasaki},~\bfnm{Yumi}\binits{Y.}} \AND
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554 |
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\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 |
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\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 |
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\bauthor{\bsnm{Lauritzen},~\bfnm{Steffen~L.}\binits{S.~L.}} \AND
|
582 |
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\bauthor{\bsnm{Richardson},~\bfnm{Thomas~S.}\binits{T.~S.}}
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583 |
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(\byear{2002}).
|
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\btitle{Chain graph models and their causal interpretations}.
|
585 |
+
\bjournal{J. R. Stat. Soc. Ser. B Stat. Methodol.}
|
586 |
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\bvolume{64}
|
587 |
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\bpages{321--361}.
|
588 |
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\bid{doi={10.1111/1467-9868.00340}, issn={1369-7412}, mr={1924296}}
|
589 |
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\bptnote{check related}\end{barticle}
|
590 |
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\bptok{imsref}\endbibitem
|
591 |
+
|
592 |
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\bibitem[\protect\citeauthoryear{{LeRoy}}{2006}]{leroy2006}
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\begin{btechreport}[author]
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\bauthor{\bsnm{{LeRoy}},~\bfnm{Stephen~F.}\binits{S.~F.}}
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595 |
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(\byear{2006}).
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\btitle{Causality in economics}.
|
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\btype{Technical report, Univ. California, Santa Barbara}.
|
598 |
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\end{btechreport}
|
599 |
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\bptok{imsref}\endbibitem
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\bibitem[\protect\citeauthoryear{Manski}{1990}]{manski1990}
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\begin{barticle}[author]
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\bauthor{\bsnm{Manski},~\bfnm{C.~F.}\binits{C.~F.}}
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(\byear{1990}).
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\btitle{Non-parametric bounds on treatment effects}.
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606 |
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\bjournal{American Economic Review}
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607 |
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\bvolume{80}
|
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\bpages{351--374}.
|
609 |
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\end{barticle}
|
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\bptok{imsref}\endbibitem
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611 |
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|
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\bibitem[\protect\citeauthoryear{Morgan}{1991}]{morganstamping1991}
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\begin{bincollection}[author]
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\bauthor{\bsnm{Morgan},~\bfnm{M.~S.}\binits{M.~S.}}
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(\byear{1991}).
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+
\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
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619 |
+
\beditor{\bfnm{Mark}\binits{M.}~\bsnm{Blaug}}, eds.)
|
620 |
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\bpages{237--272}.
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621 |
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\bpublisher{Edward Elgar},
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622 |
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\blocation{Cheltenham}.
|
623 |
+
\end{bincollection}
|
624 |
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\bptok{imsref}\endbibitem
|
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|
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\bibitem[\protect\citeauthoryear{Pearl}{2000}]{pearl2000}
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\begin{bbook}[mr]
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\bauthor{\bsnm{Pearl},~\bfnm{Judea}\binits{J.}}
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629 |
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(\byear{2000}).
|
630 |
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\btitle{Causality: Models, Reasoning, and Inference}.
|
631 |
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\bpublisher{Cambridge Univ. Press},
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632 |
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\blocation{Cambridge}.
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633 |
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\bid{mr={1744773}}
|
634 |
+
\end{bbook}
|
635 |
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\bptok{imsref}\endbibitem
|
636 |
+
|
637 |
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\bibitem[\protect\citeauthoryear{Richardson}{1996}]{richphd}
|
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+
\begin{bphdthesis}[author]
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639 |
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\bauthor{\bsnm{Richardson},~\bfnm{T.~S.}\binits{T.~S.}}
|
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(\byear{1996}).
|
641 |
+
\btitle{Models of feedback: Interpretation and discovery}.
|
642 |
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\btype{Ph.D. thesis, Carnegie-Mellon Univ}.
|
643 |
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\end{bphdthesis}
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\bptok{imsref}\endbibitem
|
645 |
+
|
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\bibitem[\protect\citeauthoryear{Richardson and Robins}{2013}]{richardsonrobins2013}
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\begin{btechreport}[author]
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648 |
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\bauthor{\bsnm{Richardson},~\bfnm{Thomas~S.}\binits{T.~S.}} \AND
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649 |
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\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}}
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650 |
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(\byear{2013}).
|
651 |
+
\btitle{{S}ingle {W}orld {I}ntervention {G}raphs {(SWIGs)}: A unification of the counterfactual and graphical approaches to
|
652 |
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causality}.
|
653 |
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\btype{Technical Report 128, Center for Statistics and the Social Sciences, Univ.
|
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Washington, Seattle, WA}.
|
655 |
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\end{btechreport}
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\bibitem[\protect\citeauthoryear{Richardson and Robins}{2014}]{richardsonrobins2014}
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\begin{bunpublished}[author]
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660 |
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\bauthor{\bsnm{Richardson},~\bfnm{T.~S.}\binits{T.~S.}} \AND
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\bauthor{\bsnm{Robins},~\bfnm{J.~M.}\binits{J.~M.}}
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(\byear{2014}).
|
663 |
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\btitle{Assumptions and bounds in the instrumental variable model}.
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664 |
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\bnote{Preprint}.
|
665 |
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\end{bunpublished}
|
666 |
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\bptok{imsref}\endbibitem
|
667 |
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|
668 |
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\bibitem[\protect\citeauthoryear{Robins}{1989}]{robins1989}
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\begin{bincollection}[author]
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\bauthor{\bsnm{Robins},~\bfnm{J.~M.}\binits{J.~M.}}
|
671 |
+
(\byear{1989}).
|
672 |
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\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 |
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(\beditor{\bfnm{L.}\binits{L.}~\bsnm{Sechrest}},
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675 |
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\beditor{\bfnm{H.}\binits{H.}~\bsnm{Freeman}} \AND
|
676 |
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\beditor{\bfnm{A.}\binits{A.}~\bsnm{Mulley}}, eds.).
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\bpublisher{U.S. Public Health Service},
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\blocation{Washington, DC}.
|
679 |
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\end{bincollection}
|
680 |
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\bptok{imsref}\endbibitem
|
681 |
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|
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\bibitem[\protect\citeauthoryear{Robins and Greenland}{1996}]{robinsgreenland1996}
|
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\begin{barticle}[author]
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\bauthor{\bsnm{Robins},~\bfnm{James~M.}\binits{J.~M.}} \AND
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685 |
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\bauthor{\bsnm{Greenland},~\bfnm{Sander}\binits{S.}}
|
686 |
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(\byear{1996}).
|
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\btitle{Identification of causal effects using instrumental variables: Comment}.
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688 |
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\bjournal{J. Amer. Statist. Assoc.}
|
689 |
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\bvolume{91}
|
690 |
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\bpages{456--458}.
|
691 |
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\bid{issn={01621459}}
|
692 |
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\end{barticle}
|
693 |
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\bptok{imsref}\endbibitem
|
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|
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\bibitem[\protect\citeauthoryear{Strotz and Wold}{1960}]{strotzwoldrecursive1960}
|
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\begin{barticle}[mr]
|
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\bauthor{\bsnm{Strotz},~\bfnm{Robert~H.}\binits{R.~H.}} \AND
|
698 |
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\bauthor{\bsnm{Wold},~\bfnm{H.~O.~A.}\binits{H.~O.~A.}}
|
699 |
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(\byear{1960}).
|
700 |
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\btitle{Recursive vs. nonrecursive systems: An attempt at synthesis}.
|
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\bjournal{Econometrica}
|
702 |
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\bvolume{28}
|
703 |
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\bpages{417--427}.
|
704 |
+
\bid{issn={0012-9682}, mr={0120034}}
|
705 |
+
\end{barticle}
|
706 |
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\bptok{imsref}\endbibitem\vfill\eject
|
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+
|
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\bibitem[\protect\citeauthoryear{Wold}{1959}]{wold1959}
|
709 |
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\begin{bincollection}[mr]
|
710 |
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\bauthor{\bsnm{Wold},~\bfnm{Herman~O.~A.}\binits{H.~O.~A.}}
|
711 |
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(\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 |
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\bpublisher{Almqvist \& Wiksell},
|
716 |
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\blocation{Stockholm}.
|
717 |
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\bid{mr={0109088}}
|
718 |
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\end{bincollection}
|
719 |
+
\bptok{imsref}\endbibitem
|
720 |
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|
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\bibitem[\protect\citeauthoryear{Wold and Jur{\'e}en}{1953}]{woldjureendemand1953}
|
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\begin{bbook}[author]
|
723 |
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\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 |
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\btitle{Demand Analysis}.
|
727 |
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\bpublisher{Wiley},
|
728 |
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\blocation{New York}.
|
729 |
+
\end{bbook}
|
730 |
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\bptok{imsref}\endbibitem\vfill
|
731 |
+
|
732 |
+
|
733 |
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\end{thebibliography}
|
734 |
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\end{document}
|