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\begin{document}

\title{A RATIONAL EXPECTATIONS FRAMEWORK FOR SHORT RUN POLICY ANALYSIS}
\author{Christopher A. Sims}
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\thanks{This paper is a revision of one prepared for presentation at the
``New Approaches to Monetary Economics" conference, May 23-24
1985, at the IC$^{2}$ Institute of the University of Texas at
Austin.  The research for the paper was supported in part by NSF
grant SES 8309329.}
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\section{Introduction}

There is increasing recognition that Lucas's [1976] critique of
econometric policy evaluation, at least under its usual
interpretation, is logically flawed.  The point has been made
forcefully recently by Sargent [1984] and by Cooley, Leroy and
Rahman [1984] (henceforth referred to as CLR), as well as in my
own paper [1982].  The problem is that if the parameters of the
policy ``rule" are subject to change, as they must be if it makes
sense to evaluate changes in them, then the public must
recognize this fact and have a probability distribution over the
parameters of the rule.  But then these ``parameters" are
themselves ``policy variables", taking on a time series of values
drawn from some probability law. Predicting how the economy will
behave if we set the parameters of the rule at some value and
keep them there is logically equivalent to predicting the
behavior of the economy conditional on a certain path of a
policy variable.  Yet this is just the kind of exercise which
Lucas claimed to be meaningless.

It is also evident that the methods of policy evaluation which Lucas
criticized are still in wide use nine years after the appearance of his
paper.  During discussions of monetary and fiscal policy, statistical
models prepared by the Congressional Budget Office, the Federal Reserve
Board, numerous other agencies, and by private entities are used to
prepare predictions of the likely future path of the economy conditional
on various possible paths for policy variables. These conditional
projections influence policy-makers' views of the likely consequences of
the choices they must make.  Though Lucas suggested an alternative
paradigm for policy analysis, it is still little used.

Nonetheless, we are not quite to the point where well-trained young
macroeconomists collaborate in preparing and interpreting conditional
projections of the effects of alternative paths for policy variables
without feeling queasy.  Sargent, while clearly explaining the problems
with the rational expectations paradigm for policy analysis and finding
no way around them, claims we must ignore them if we are to avoid the
conclusion that policy recommendations of any kind are meaningless. CLR
show that projections of the future path of the economy conditional on
paths of policy variables are not meaningless, even in the presence of
shifts in policy rule.  But they never explicitly address the central
question of whether such conditional projections could ever be used in
the process of policy choice without invalidating them.  My own paper
[1982] asserts that policy choice based on conditional projection from
models which do not identify the parameters of tastes and technology can
be logically coherent, but it does not support the assertion with formal
modeling and many remain unconvinced.

The remainder of this paper attempts to make that assertion more
convincing by analyzing more closely how properly constructed and
interpreted conditional projections sidestep the Lucas critique and by
presenting examples of model economies in which policy makers steadily
make good use of conditional projections from ``loosely identified"
statistical models.

\section{Finding a Coherent Interpretation of the Lucas Critique}

Lucas formulates his critique most baldly and succinctly at the beginning
of section 6 of his paper, where he writes
    \begin{quote}
    ...there are compelling empirical and theoretical reasons
    for believing that a structure of the form
    \end{quote}

\begin{equation}\label{eq:naive}
  y_{t+1} = F(y_{t},x_{t},\beta,\eta_{t})
\end{equation}

    \begin{quote}
    ($F$ known, $\beta$ fixed, $x_{t}$ ``arbitrary") will not be of use for
    forecasting and policy evaluation in actual economies.
    \end{quote}
He goes on to observe that for short-term forecasting the problem can
avoided, indeed has been avoided in practice, by allowing the
``parameters" $\beta$ to drift in time.  But for policy evaluation he argues
that a completely different approach is required, in which models are
given the form
\begin{gather}\label{eq:rule}
  x_{t} = G(y_{t},\gamma,\varepsilon_{t})\\
\label{eq:response}
  y_{t+1} = F(y_{t},x_{t},\beta(\gamma),\eta_{t}) ,
\end{gather}
where G and F are known, $\gamma$ is a fixed parameter vector, and $\varepsilon_{t}$ and $\eta_{t}$
are vectors of disturbances.  The econometric problem is now that of
estimating the function $\beta(\gamma)$, not a fixed vector of real numbers $\beta$, and
policy evaluation is performed by considering the effects of alternative
settings for $\gamma$, not by comparing choices of paths for $x$.

These assertions make no sense if taken at face value.  The most widely
used method for allowing for parameter drift in forecasting models takes
the ``parameters" $\beta$ to be a time series evolving according to
\begin{equation}\label{eq:tv}
 \beta(t) = H(\beta(t-1),\alpha,v(t)) ,
\end{equation}
where $v$ is a vector of random disturbances, $H$ is a known function, and $\alpha$
is a fixed vector of parameters.  By recursively substituting lagged
versions of \eqref{eq:tv} into itself, we can obtain
\begin{equation}\label{eq:tvback}
\beta(t) = H_{1}(\alpha,V(t),\beta(0)),
\end{equation}
where $V(t) = \{v(1),...,v(t)\}$.  If the influence of $\beta(0)$ on $H_{1}$ is
non-negligible, we can merge it with the parameter vector $\alpha$, to form the
new parameter vector $\alpha^{*}=(\alpha,\beta(0))$, so that
\begin{equation}\label{eq:newbeta}
  \beta(t) = H^{*}(\alpha^{*},V(t)) .
\end{equation}

Substituting \eqref{eq:newbeta} into \eqref{eq:naive} produces
\begin{equation}\label{eq:seven}
y_{t+1} = F(y_{t},x_{t},H^{*}(\alpha^{*},V(t)),\eta_{t}) =
                               F^{*}(y_{t},x_{t},\alpha^{*},\eta^{*}(t)) ,
\end{equation}
where $\eta^{*}(t)=(V(t),\eta(t))$.  Now obviously $F^{*}$ in \eqref{eq:seven} has exactly the
same form as $F$ in \eqref{eq:naive}.  If a ``time-varying parameter" model like that
described by $F^{*}$ can in fact provide good forecasts, which Lucas claims
it can, then it would seem he cannot consistently claim also that a
structure of the form \eqref{eq:naive} ``will not be of use for forecasting... in
actual economies." Of course, if $F$ in \eqref{eq:naive} is linear, or in some other
sense ``simple", $F^{*}$ in \eqref{eq:seven} will generally not be.  But Lucas says only
that F is known, not that it is linear, and he undoubtedly meant to be
saying something stronger than that we need nonlinear models.

There is a similar problem with the claim that evaluating policy by using
\eqref{eq:rule} and \eqref{eq:response} to gauge the consequences of changing $\gamma$
is different from
using \eqref{eq:naive} to gauge the consequences of various choices of time path for
$x$.  The ``fixed" parameter $\gamma$ is itself necessarily a time series with at
least two possible values (the value before we change policy and the
value after).  Putting the appropriate subscript on $\gamma$ and substituting
\eqref{eq:rule} into \eqref{eq:response} gives us
\begin{equation}\label{eq:eight}
  y_{t+1} = F(y_{t},G(y_{t},\gamma_{t},\varepsilon_{t}),\beta(\gamma_{t}),\eta_{t})
                = F^{**}(y_{t},\gamma_{t},\eta^{**}_{t}) ,
\end{equation}
where $\eta^{**}_{t} = (\eta_{t},\varepsilon_{t})$ .  Now we have transformed
\eqref{eq:rule} and \eqref{eq:response} into
\eqref{eq:eight}, wherein $F^{**}$ has exactly the functional form of $F$, with $\gamma$ playing
the role of $x$.  There are no unknown parameters in \eqref{eq:eight} at all, but of
course in practice there would be unknown parameters in the functional
form of the $\beta(\gamma_{t})$ function, to take the place of the original fixed $\beta$
parameters.

Again, if $F$ were linear, $F^{**}$ would probably not be, but surely most
economists interpret Lucas's argument as asserting more than that we
will need nonlinear models to do good policy analysis.

It is also true that Lucas wants us to take $\gamma$
as ``fixed".  Of course if
we contemplate changing $\gamma$ it cannot really be fixed, but the spirit of
the argument is that we ought to consider once-and-for-all changes in $\gamma$,
i.e. "paths" of $\gamma_{t}$ which are constant up to some date $T$ and constant
thereafter, with a discontinuity at $T$.  Further according to this
interpretation, we ought to concentrate on predicting the long run
effects of the change, after $\gamma$ has been at its new level long enough for
behavior to have completely adjusted.  From the point of view of equation
\eqref{eq:eight}, this is the suggestion that we should limit ourselves to comparative
statics based on the long run properties of the model.  But this
recommendation, if we admit that $F^{**}$ is the same form as displayed by
standard econometric models, is not at all revolutionary.  And it is a
dubious recommendation -- in practice econometric models are probably
less reliable in their long run properties than in their short run
properties.

Some readers undoubtedly have given Lucas's critique the interpretation
outlined above.  Sargent's paper is one example.  Under this
interpretation, equation \eqref{eq:naive} is interpreted as general enough to
correspond to any statistical model relating endogenous variables $y$ to
lagged endogenous variables, policy variables $x$, some parameters $\beta$, and
random disturbances $\eta$.  The claim is that any use of such a model to make
policy will have to change the model, making the model invalid.  This
leads to skepticism toward all use of data in forming policy, since there
is no way to do so without using a model in the general form \eqref{eq:naive}.
Concentration on choice of the parameters of a policy rule is of no help,
since there is no logical distinction between such ``parameters" and
"policy variables".  Thus on this interpretation, if one takes the
rational expectations critique of standard econometric policy evaluation
seriously it applies with equal force to the rational expectations
program for ``correct" econometric evaluation of policy.

As we will see in the next section through an example, the argument in
this form is simply incorrect.  It is possible for an optimizing,
benevolent, immortal policy maker to make policy every period forever by
choosing among conditional projections from a statistical model while the
model remains accurate.

However, this version of the argument distorts the original.  When Lucas
writes down \eqref{eq:naive} in section 6 of his paper, he introduces it as a
``structure" of the form \eqref{eq:naive}.  By this he probably means to imply that $F$
is not some general statistical model, but the kind of entity he
described in more detail when using \eqref{eq:naive} earlier in the paper.  There he
requires that ``the function $F$ and the parameter vector $\beta$ are derived from
decision rules (demand and supply functions) of agents in the economy,
and these decisions are, theoretically, optimal given the situation in
which each agent is placed."  Furthermore, in the examples Lucas
considers there is in every case one or more functions contained in the
model which represent or are directly affected by agents'
expectation-formation rules.

The lesson of rational expectations is that when we use a model in whose
functional form is embedded agents' expectational rules, we are likely to
make errors even in forecasting if we insist that those expectational
rules are fixed through time, and that we will make even more serious
errors in policy evaluation if we pretend that those rules will remain
fixed despite changes in policy which make them clearly suboptimal.  The
difference between \eqref{eq:naive} and the system \eqref{eq:rule}-\eqref{eq:response}
as frameworks for policy
analysis is not the superficial one that in the latter we think of
ourselves as choosing a ``fixed parameter" $\gamma$ while in the former we are
choosing ``arbitrary" values of policy ``variables" $x$.  The difference is
that in \eqref{eq:naive} the parameters $\beta$ in fact depend on the hidden policy variable
$\gamma$.  If we try to use \eqref{eq:naive} to guess the effects of various $x$
 paths which
are in fact accompanied by changes in $\beta$, we will make errors.  The
advantage of \eqref{eq:rule}-\eqref{eq:response}, or the equivalent model
\eqref{eq:eight}, is that either of them
takes proper account of the effect of $\gamma$ on $\beta$.

Thus we should not necessarily expect a different mathematical form or a
different probabilistic treatment of policy variables for models which
take proper account of rational expectations.  It is even in principle
possible that such models could have $F^{**}$ functions which turned out to
be a set of linear stochastic difference equations, with the
corresponding mistaken $F$ function being of complicated nonlinear form.

Most important, the problems of identification for rational expectations
models are not fundamentally different from those for what used to be
standard models. In the happy circumstance where the historically
observed data contains exogenous variation in policy along the lines we
are currently contemplating, we can estimate the effects of our policy
choices by reduced form modeling.  We can estimate regressions of current
data on current and past policy variables and correctly use the results
to project the likely effects of our policy choices.  To do this we need
not separate the effects of policy occurring directly from those occurring
indirectly through modifications in expectation-formation rules of the
public.  It is exactly this point which is so neatly laid out by CLR.

This is not to deny that identification is a hard problem.
Identification for purposes of policy evaluation is roughly the same hard
problem whether or not we take account of rational expectations.
Historical data on policy variables will usually reflect some systematic
pattern of response of policy to disturbances originating elsewhere in
the economy.  Therefore conditional distributions of other variables
given policy variables do not necessarily correspond to conditional
distributions of those variables given autonomously induced changes in
policy variables.

\section{Optimal Policy Using Conditional Projections}

If policy is made optimally, it is always reacting correctly to all
available information about the state of the economy.  Presumably then it
does not display capricious or arbitrary variation.  This suggests that
it should contain no autonomous randomness, so that policy variables
should be exact functions of past data.  In this case, there would be no
way to separate the effects of policy variables from the effects of the
variables policy depends on, except by use of strong auxiliary
assumptions.  On the other hand, if policy is made sub-optimally, it might
contain a lot of capricious variation; we might then easily obtain
estimates of the effect of this variation.  Yet if policy makers use
our estimates to improve policy, the amount of autonomous variation in
policy will shrink, the probability structure of the economy is likely to
change, and our estimates may quickly become obsolete.

Is there any way that policy could both be chosen optimally, on the basis
of a correct model, and at the same time contain enough autonomous
variation to allow accurate estimation of the effects of deliberately
induced variations in policy?

It is not hard to see that the answer must be yes.  All that is required
is the existence of some source of variation in policy choice which, as
far as the public is concerned, is indistinguishable from an error or a
capricious shift in policy choice.  One obvious possibility arises when
we recognize that macroeconomic policy is in fact set through a political
process, in which groups with varying knowledge and objectives contest to
influence policy.  The public does not know the identity, the objectives
or the relative political strength of these groups with certainty.  Hence
actual policy always contains an unpredictable element from this source.
The public has no way of distinguishing an error by one of the political
groups in choosing its target policy from a random disturbance in policy
from the political process.  Hence members of a such a group can
accurately project the effects of various settings of policy they might
aim for by using historically observed reactions to random shifts in
policy induced by the political process.  The group will itself, if it
behaves optimally according to its own objective function, make its
target policy a deterministic function of data it observes.  But it can
implement that function correctly by, at each date, using a statistical
model to make conditional projections of the effects of alternative
policy variable paths and choosing the projected path it likes best.

CLR include regime switches in their model, but provide no explanation
for why policy differs under the two regimes.  It should be apparent,
though, that something at least very much like their model could emerge
if their two regimes were generated by two optimizing political
coalitions.\footnote{Recently Roberds [1985] has produced an explicit model which goes
some way toward capturing optimal behavior of stochastically alternating
regimes of policy makers.  Like CLR, he has the regime switch a purely
exogenous random process, but instead of arbitrarily changing policy
rules, regime switches in Roberds' model change policy makers' objective
functions.  His model is not ready to be tacked on to CLR's, however,
because it does not follow CLR in giving the public an inference problem
in trying to determine the current regime.  This means also that, unlike
CLR, Roberds does not make the stochastic regime switches a source of
identifying variation in policy; the public sees regime switches directly
in Roberds' model, so they are an extra observable policy variable rather
than an underlying source of variation in observable policy variables.
Roberds also ignores the interesting problem of strategic interaction
between the two regimes.  Each regime takes account of the fact that the
public is modeling its behavior, but neither attempts to model the
behavior of the other regime or consider the possibility that its own
behavior could affect the behavior of the other regime.  It would be
interesting to see work along the lines Roberds has begun which tied more
directly to the setting CLR dealt with.}

While policy randomness due to political struggles is probably the most
realistic and important source of identifying variation in policy, it
leads to analytically challenging models.  A simpler way to approach the
question of how to design good policy posits a unitary policy authority.
In such a framework, the most plausible source of identifying variation
in policy is noisy information available to the policy authority which is
visible to the econometrician, if at all, only with a delay.  The
optimizing policy maker will use the noisy information, but because it is
noisy his use of it will introduce into policy a ``random error".  The
reaction of the economy to this random error will provide a statistical
basis for determining the reaction of the economy to hypothetical
optimization errors.  To some it may be apparent immediately that this
setup will ``work".  But since I am not aware of a closely similar
construction in the literature, we work out an explicit model here.

The typical agent in this economy chooses the vector $C_{t}$ (consumption) at
$t$.  The government chooses $G_{t}$ at t, government activity per capita.
Utility of the agent is
\begin{equation}\label{eq:nine}
   E \left[ \sum_{t=1}^{\infty} \rho^{t}.5(C_{t}^{\prime} AC_{t} + G_{t}'BG_{t} +
   K_{t}'FK_{t})\right]
\end{equation}
The technology imposes the constraint
\begin{equation}\label{eq:ten}
   K_{t} = HC_{t} + MG_{t} + NK_{t-1} + X_{t}  ,
\end{equation}
where the stochastic process $X$ is exogenous to both private agents and the
government.  Though this framework is very general and looks like the
canonical quadratic-linear control problem, it is unconventional in that
it does not assert that $X_{t}$ is i.i.d., only that its evolution is
unaffected by choices of $G$, $C$, or $K$.

The government and the agents both try to maximize the same objective
function \eqref{eq:nine}.  There is an ``information process" containing three
subvectors, $Q_{t}$, $W_{t}$ and $Z_{t}$.  When private agents choose $C_{t}$, they do
so with knowledge of all values of $Q_{s}$, $W_{s}$ and $Z_{s}$ for $s \leq t$.  The
government must choose $G_{t}$ based on knowledge only of these variables
for $s \leq t-1$, plus an observation on $Z_{t}$.

We assume special relationships among $Q$, $W$, $X$ and $Z$ to generate our
example: $Q$ and
 $W$ are Granger causally prior to $Z$, $X$ is a linear function
of current and past $Q$, and $W$ alone, and $Z_{t}$ has the form of a noisy
measurement of $W_{t}$.  I.e., assuming $Q$, $W$, and $Z$ form a linear process
with an autoregressive representation,
\begin{align}\label{eq:eleven}
      E_{t}\begin{bmatrix}
        Q_{t+1} \\
        W_{t+1}
      \end{bmatrix}
      = a*\begin{bmatrix}
      Q_{t} \\
      W_{t}
      \end{bmatrix} \\
        \label{eq:twelve}
         E[Z_{t}| W_{s},Q_{s},Z_{s-1}, \text{all} s \leq t] = W_{t} \\
        \label{eq:thirteen}
        X_{t} = b*\begin{bmatrix}
        Q_{t} \\
        W_{t}
        \end{bmatrix}\: ,
\end{align}
where $a(s)=b(s)=0$ for $s<0$ ($a$ and $b$ are one-sided) and ``$E_{t}[.]$" stands for
``$E[.|Q_{s},W_{s},Z_{s}, \text{all} s \leq t]$".

The Granger causal priority assumption means that the information the
government obtains at $s \leq t$ is strictly redundant from the point of view of
agents at $s \leq t$.  This means in turn that the solution to this problem can be
found as if there were a single optimizing agent who must choose $G_{t}$ at
a stage where his information set is smaller than it is at the next
stage, when he chooses $C_{t}$.   This would be impossible if, say, private
agents also observed only noisy information on $X_{t}$ at $s \leq t$, but with a
noise different from that facing the government.

Because $K_{t-1}$ enters the constraint \eqref{eq:ten}, the public cares about future
values of
 $G$, leaving a role for the rational expectations hypothesis.

Because the model is quadratic-linear, we can solve it using dynamic
certainty-equivalence.

The main point of the example does not depend on the explicit solution of
the model.  Consider the government's problem at $s \leq t$, which is to choose
$G_{t}$.  Dynamic certainty equivalence tells us that a correct approach to
this problem can begin by forming forecasts of the paths
 $X$ using the data
available to the government at this point, i.e. values of $Q_{s}$, $W_{s}$ for
s<t together with the current observation on $Z_{t}$.  The objective
function \eqref{eq:nine} can then be maximized with respect to the future paths of
 $C$,$G$ and $K$, subject to the constraint \eqref{eq:ten}, replacing future X's by
their expectations.  The $G_{t}$ which emerges from this computation will be
the correct period-$s \leq t$ decision for the government.

However, suppose historical data on $C$, $G$, $K$, and $W$ are publicly available,
the government has access to an econometrician capable of estimating a
vector autoregression (VAR) for these variables, and current and past
values of these variables constitute an information set equivalent to
current and past $Q$,
 $W$, and $Z$.  In this case the government can simplify
its problem by having the econometrician estimate the vector
autoregression and generate predictions with it conditional on various
possible paths for $G$.

In particular, the government should use its observation on $Z_{t}$ to form
an estimate of $W_{t}$.  It should ask the econometrician to put his vector
autoregression in triangular form, in the order $W$, $G$, then $K$ and $C$ (I.e.,
the regression equations for $C$, $G$, and $K$ should all contain
contemporaneous $G$ and $W$, and the regression equation for $G$ should contain
contemporaneous $W$.)  Then the conditional projections should be formed
with all disturbances in the VAR set to zero, except for those in the G
equation (which are varied to generate various paths for G) and for the
contemporaneous disturbance in the $W$ equation (which is determined by the
government's estimate of $W_{t})$.  If the government evaluates the paths
generated this way using the utility function \eqref{eq:nine}, ignoring uncertainty,
and chooses the value of $G_{t}$ associated with the projected path
maximizing utility, it will make exactly the optimal choice of $G_{t}$.

In proceeding this way the government avoids any need to know the
structure of the constraints \eqref{eq:ten}, trusting that the private sector has
systematically taken care of optimizing relative to these constraints and
will continue to do so.

The reason this procedure works is that a class of paths for $C$, $G$, and $K$
in which only the part of the innovation in $G$ which is orthogonal to $W$
varies is a class in which the path of $X$ remains fixed.  This is true
because the only contemporaneous innovation on which $G_{t}$ depends is that
of $Z_{t}$, which in turn depends only on the $W_{t}$-innovation and $e_{t}$.  So
the part of $G_{t}$ orthogonal to the $W_{t}$-innovation is a function of $e_{t}$
alone, which is independent of $X_{t}$.  But if in this class of paths
 $X$ is being fixed at its projected value given $Z_{t}$, while
 $G$ varies, the paths are all satisfying \eqref{eq:ten} with this fixed
 $X$.  Thus the paths being considered are a subset of those considered in the full
certainty-equivalent solution when $C$, $K$ and $G$ are all varied while \eqref{eq:ten}
is imposed and the path of $X$ held fixed at its predicted value.  The only
question remaining is whether we can be sure that this subset of the
feasible paths actually contains the optimum.  But it does, because
another way to characterize the certainty-equivalent solution is that at
$s \leq t$ it sets the paths of $G$, $C$, $K$ and $W$ equal to their conditional
expectations given information available to the government, i.e. data up
to $s \leq t-1$ together with $Z_{t}$.  But this is just what is obtained from the
VAR when the innovations in $X_{t}$ and $G_{t}$ are set at their conditional
expectations given $Z_{t}$ and past data, while all other innovations in the
triangularized system at $t$ and later are set to zero.

The foregoing verbal argument is complete except for the absence of any
proof that it is possible for current and past $G$, $C$, $K$, and $W$ to be an
information set equivalent to current and past $Q$, $W$, and $Z$.  Since the
foregoing argument is intricate and possibly hard to follow, and
since the general conditions for equivalence of the information sets
appear difficult to set down, we will work out a specific example in some
detail below.  First, though, we display the general form of the first
order conditions:
\begin{align}
\label{eq:fourteen}
     C&:  &AC_{t} &= -H\gamma_{t} &&\\
\label{eq:fifteen}
     G&:  &BG_{t} &= -ME_{zt}[\gamma_{t}] &&\\
\label{eq:sixteen}
        K&:  &FK_{t} &= \gamma_{t} - \rho NE_{t}[\gamma_{t+1}] &&\: ,
\end{align}
where $\gamma_{t}$ is a stochastic Lagrange multiplier vector and ``$E_{zt}[.]$"
means \[``E[.|Z_{s},Q_{s-1},W_{s-1}, \text{ all } s \leq t]"\:.\]

Because this is a quadratic-linear problem, we could solve it in closed
form by assuming some simple form for the autoregressive representation
of the $Q$, $W$, $Z$ process, then deriving the decision rules.  Even for
simple cases, however, the parameters of the decision rules and therefore
of the AR representation of the system are fairly complicated nonlinear
functions of the problem's original parameterization.  To keep the
algebra simpler, we will derive a solution ``backwards" (see my paper
[1984].  We proceed by not using \eqref{eq:thirteen}, instead assuming directly
\begin{equation} \label{eq:seventeen}
        \gamma_{t} = f*\begin{bmatrix}Q_{t}\\W_{t}\\ Z_{t}\end{bmatrix} \: ,
\end{equation}
treating \eqref{eq:ten} now as defining $X_{t}$ after (14)-(16) have been used
to determine $C$, $G$, and $K$ from $Q$, $W$, and $Z$.  We cannot use the most
straightforward backwards solution method, because we cannot choose $f$
arbitrarily and still guarantee that the constraint in \eqref{eq:thirteen} -- that $X_{t}$
not depend on current $Z_{t}$ -- be satisfied.  Nonetheless, it proves to be
easy to adjust our choice of $f$ to impose this constraint.

To make the example really simple, we assume that $C$, $K$ and $G$ all have dimension one.
 $W$ and $Z$ must then be scalar also. (Since $Z$'s innovation affects only
 $G$, we will not be able to recover $Z$ from data on $C$, $K$, $G$,and $W$ unless $G$'s
 dimension at least matches $Z$'s; and $W$ and $Z$ are assumed to be of the same dimension.)  If
 $C$, $K$, $G$ and $W$ are not to be jointly singular, we must have the dimension of
 $C$ and $K$ jointly no greater than the dimension of $Q$, and if we are to be able to recover
 the history of $Q$, $W$, and $Z$ from that of $C$, $K$, $G$, and $W$, we must have
 $C$ and $K$ jointly no smaller in dimension than $Q$.  So $Q$ is 2x1.

We assume $A=B=F=H=M=N=1$.  We take $a$ in \eqref{eq:eleven} to be such that
\begin{equation}\label{eq:eighteen}
    E_{t}\begin{bmatrix} Q_{t+1}\\W_{t+1} \end{bmatrix}
       =  diag(\zeta) \begin{bmatrix} Q_{t}\\W_{t} \end{bmatrix} \: ,
\end{equation}
where $\zeta$ is a vector with distinct elements all less than one in absolute
value.  We use lower case letters to refer to the innovations in the
corresponding upper case stochastic processes.  We define $\phi$ as the vector
of coefficients projecting the innovations in $Q$ and $W$ onto the innovation
in $Z$, i.e.
\begin{equation}\label{eq:nineteen}
     E_{t}\begin{bmatrix}q_{t}\\w_{t}\end{bmatrix}   =  \phi z_{t}  .
\end{equation}

Obviously $\phi$ is determined by the covariance matrix of the innovations in
the information process $Q$, $W$, $Z$.  We choose $f$ in \eqref{eq:seventeen} so that
\begin{equation}\label{eq:twenty}
     -\gamma_{t} = \mathbf{1}^{\prime}Q_{t} + W_{t} + \frac{\varepsilon}{(1-\beta L)}Z_{t}\:.
\end{equation}

It will turn out that we can keep $Z$ from affecting $X$ (maintain the
validity of \eqref{eq:thirteen}) by setting $\varepsilon$ and $\beta$ properly.

Now we can use our simplifying assumptions to rewrite the first order
conditions  \eqref{eq:fourteen}-\eqref{eq:sixteen} as
\begin{gather}\label{eq:twentyone}
     C_{t} = \mathbf{1}^{\prime} Q_{t} + W_{t} + \frac{\varepsilon}{(1-\beta L)}Z_{t}  \\
\label{eq:twentytwo}
     G_{t} = \mathbf{1}^{\prime} diag(\zeta)
                \begin{bmatrix}
                Q_{t-1} \\ W_{t-1}
                \end{bmatrix}
                 + \frac{\varepsilon}{(1-\beta L)}Z_{t} + \mathbf{1}^{\prime}
                 \phi(Z_{t}-\zeta_{3}W_{t-1}) \\
\label{eq:twentythree}
     K_{t} = -\mathbf{1}^{\prime}Q_{t} - W_{t}
        - \frac{\varepsilon}{(1-\beta L)}Z_{t} + \rho N \left(\mathbf{1}^{\prime}diag(\zeta)
        \begin{bmatrix}
        Q_{t} \\ W_{t}
        \end{bmatrix} + \varepsilon\zeta_{3}W^t\right) \: .
\end{gather}

The simplified version of \eqref{eq:ten}, written as solved for $X$, is
\begin{equation}\label{eq:twentyfour}
     X_{t} = K_{t} - NK_{t-1} -C_{t} - G_{t}   .
\end{equation}

It is easy to see that the terms in $Z$ which result when \eqref{eq:twentyone}-\eqref{eq:twentythree} are
substituted into \eqref{eq:twentyfour} are given by
\begin{equation}\label{eq:twentyfive}
 \left[-{3\varepsilon/(1-\beta L)} - \mathbf{1}^{\prime}\phi
           - NL\frac{\varepsilon}{(1-\beta L)}\right] Z_{t}\:.
\end{equation}

This term vanishes if and only if
\begin{equation}\label{eq:twentysix}
     \varepsilon(3+NL) = -\mathbf{1}^{\prime}\phi(1-\beta L)  ,
\end{equation}
which will be true if $\beta=-N/3$ and $\varepsilon=-\mathbf{1}^{\prime}\phi/3$.
We can choose $\beta$ and $\varepsilon$ to satisfy these relations so long as $|N|<3$.

To verify that the relation between $C$, $K$, $G$ on the one hand and $Q$,$Z$ on
the other is invertible, we must check the characteristic roots of the
matrix polynomial in the lag operator which makes up the right hand side
of \eqref{eq:twentyone}-\eqref{eq:twentythree}, i.e. of

\begin{equation}\label{eq:twentyseven}
        \begin{bmatrix}
        1   &        1    &       \varepsilon/(1-\beta L) \\
     \zeta_{1}L &   \zeta_{2}L & \varepsilon /(1-\beta L)+\mathbf{1}^{\prime}\phi \\
        -1+RN\zeta_{1} &  -1+RN\zeta_{2}    &  -\varepsilon / (1-\beta L)
        \\
        \end{bmatrix}
\end{equation}

Some algebra leads to the conclusion that there is just one root to the
determinant of this matrix, and it is less than one in absolute value so
long as $|N|<4$.

Denoting the matrix polynomial in the lag operator in \eqref{eq:twentyseven} by $\theta(L)$, the
full relation between $C$, $G$, $K$, $W$ and $Q$, $Z$, $W$ is
\begin{equation}\label{eq:twentyeight}
\begin{bmatrix}
C \\ G \\ K \\W
\end{bmatrix} =
\begin{bmatrix}
    \theta(L) &    1  \\
     \zeta_{3}(1-1'\phi)L &     Z  \\
    -1+RN\zeta_{3} & (1+\varepsilon) \\
    0   &    1
\end{bmatrix}
\begin{bmatrix}
Q \\ Z \\ W
\end{bmatrix}\:,
\end{equation}
where the partition of the columns of the matrix on the right-hand side
corresponds to a partition of the $Q$-$Z$-$W$ vector between $Q$,$Z$ and
 $W$.  This means that the relation between innovations in the two processes is
 \begin{equation}\label{eq:twentynine}
 \begin{bmatrix}
 c \\ g\\ k \\ w
 \end{bmatrix} =
 \begin{bmatrix}
 1     &      1   &        \varepsilon   &    1   \\
0   &        0       &    \varepsilon+1'\phi &  0 \\
-1+RN\zeta_{1} & -1+RN\zeta_{2} & -\varepsilon -1+RN\zeta_{3}(1+\varepsilon) \\
0     &      0     &      0  &     1
 \end{bmatrix}
 \begin{bmatrix}
 q_1 \\ q_2 \\ z
 \end{bmatrix}
 \end{equation}
We see from \eqref{eq:twentynine} confirmation of what we have already argued for the
general case:  the innovation in $G$ is a function of the innovation in $Z$
alone.  From \eqref{eq:eighteen} we can observe that in this example as in the general
case $Q$ and $W$ can be expressed entirely in terms of current and lagged
values of their own innovations (a consequence of their not being
Granger-caused by $Z$).  Thus a class of projections of future values of
 $G$,$C$, and $K$ generated by varying only the part of $g$ orthogonal to
 $w$ will leave the corresponding (via the inverse of \eqref{eq:twentyeight}) class of
 projections of
future $Q$, $W$ and $Z$ unchanged in $Q$ and $W$, with only $Z$ varying.  Optimal
choice of $G_{t}$ is described by the second row of \eqref{eq:twentyeight} or
\eqref{eq:twentynine}, and the
jointly optimal certainty-equivalent path for all variables conditional
on the government's information at $s \leq t$ is obtained by setting $z_{t}$ to its
actual value and $q_{t}$,$w_{t}$ according to \eqref{eq:nineteen}, generating future values for
$Q$,W,$Z$ by solving the equations of their VAR with these initial
conditions, and translating these to future values for
 $C$, $G$, $K$, and $W$ via
\eqref{eq:twentyeight}.  But as we can now see explicitly, this whole process can be
described equivalently as setting $w_{t}$ and $g_{t}$, then generating future
values for $C$, $G$, $K$ and $W$ according to the VAR for these variables
themselves.

\section{Remarks}

The example in the preceding section is set in a canonical
quadratic-linear form, with no constants or linear terms in the objective
function and all stochastic terms in the constraints.  Models with
stochastic terms in the objective function, or cross-product and linear
terms there, can be transformed into the canonical form.  Thus even if
the government could deduce the form of some or all of the constraints
on the canonical form from the VAR (which it can't in this example), it
would not necessarily have found the economy's ``technology".

If $Z_{t}$ were an observable variable, there would be an exact linear
relation connecting $Z_{t}$ and past data to current $G_{t}$ which could be
quickly deduced from the data.  Our example depends on thinking of policy
makers as agents who can collect unpublished, informal data and use it to
make a guess at $W_{t}$, but can't characterize their data as a number and,
say, plot it against their previous choices of $G_{t}$.  This is of course
not very realistic, but neither is the idea of a unitary, optimizing
macro policy maker.  If policy makers were truly rational agents, of
course, they could fit and solve econometric models in their heads and
would have no need of econometricians.

The example is rigged so that identification does not even require
instrumental variables estimation -- it uses a Wold causal chain style of
identification.  It could have been rigged so that the identification
problem was a little more interesting.

The Lucas critique raises no problems for the example in the preceding
section, not because expectation-formation does not enter the model (it
is an important part of the model's dynamics), but because the model is
one in which policy is already optimal and persists in being so.  Thus
the process of policy choice does not change the expectations formation
behavior implicit in the model's structure.  The point of the example is
that it is one in which the Lucas statement, quoted above, about the
uselessness of models in the general form of \eqref{eq:naive}, is completely incorrect
if taken at face value.

The only coherent interpretation of the Lucas critique is that it states
that if one uses a model which incorrectly describes the reaction of
expectations formation to policy choice, it will produce incorrect
evaluations of policy.  The implication is not that econometric
evaluation of policy using models fitted to history is impossible, but
that it requires correct specification of the reaction of the economy to
policy.  And the notion strongly associated with the rational
expectations literature, that the only correct way to model such
reactions is to ignore the probabilistic character of policy choice, is
spurious baggage.

Some readers might be convinced by the example, but suppose that it only
shows a minor qualification to the Lucas critique -- that when there are
no changes in the policy rule, policy choice using econometric
conditional projections is at least possibly logically coherent.  When
we contemplate truly important changes in policy, or changes which
may approach permanence, then we are in the realm of
changes in policy rule, and it may be thought that Lucas is still right
that econometric conditional projections cannot be used for analyzing
changes in rule.

But this brings us back to the argument in the first part of the paper.
Changes in rule are changes in policy variables, just as is any other
kind of change in policy.  Very persistent or very large changes in
policy are likely to generate nonlinear effects in the reaction of the
economy to policy.  These nonlinearities -- e.g. that the reaction of the
economy to 6 years of below average inflation cannot be correctly
determined by adding up 12 copies of the reaction of the economy to 2
quarters of below average inflation -- must be modeled accurately if
conditional projections with the model are to be accurate.  But there is
no logical distinction between using an accurate nonlinear model to
project the effects of persistent or large policy changes and using an
accurate linear model to project the effects of smaller, short run policy
changes.  It is true that the nonlinear models appropriate to projecting
the effects of large or persistent policy changes are likely to be
complicated, and that the uncertainty about how the data should be
interpreted is likely therefore to be larger in such cases.  There may be
plausible alternative identifying assumptions available which yield
different conditional projections without conflicting with the data.
Putting the same thing another way, the role of untested subjective
judgement (i.e., ``a priori theory") may be large.  But this is to some
extent true in any econometric policy analysis and raises no new issues
of principle.

There may be some policy issues where the simple rational expectations
policy analysis paradigm -- treating policy as given by a rule with
deterministic parameters, which are to be changed once and for all, with
no one knowing beforehand that the change may occur and no one doubting
afterward that the change is permanent -- is a useful approximate
simplifying assumption.  To the extent that the rational expectations
literature has led us to suppose that all ``real" policy change must fit
into this internally inconsistent mold, it is has led us onto sterile
ground.

\section{Conclusions}

It should be clear that, though the model in the previous section is
specially rigged, what is crucial to the rigging is the information
structure.  Whenever the policy authority makes imperfectly predictable
choices because it observes a noisy information variable, while the
public finds the  government's information variable redundant, we will
reproduce the identifying assumptions justifying the form of econometric
policy evaluation which is valid in this model.

In more complicated models, identification of the structural relation
connecting policy actions to the private economy's reaction is likely to
be less easy than it is here.  Within the context of models which like
that in this paper yield a stationary linear structure while policy
varies, different assumptions about the source of the variation in policy
could make reaction unidentifiable or could lead to a need to identify it
by use of techniques which could be close to those of standard
simultaneous equations theory.  Examples of identification techniques
which work directly with a VAR reduced form appear in Blanchard and
Watson [1984] and Litterman [1984].

The generic version of the point this paper makes is as follows:  we can
ordinarily expect that accurate conditional projections of the effects of
policy choices will be useful to policy makers; we can expect that there
will be autonomous variation in policy in the data, so that statistical
estimation will be valuable in preparing such conditional projections;
and the identifying assumptions required to bring the data to bear in
forming the projections will be much less than a complete behavioral
interpretation of each equation in the probability model, in fact not
even requiring separation of expectational dynamics from other dynamics.

\section*{REFERENCES}

Blanchard, Olivier, and Mark Watson [1984], ``Are Business Cycles
All Alike", NBER discussion paper, presented at March 1984
conference.

Cooley, Thomas F., Stephen F. LeRoy, and Neil Rayman [1984],
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California, Santa Barbara, and University of Missouri, Columbia.

Kydland, F. and E. Prescott [1977], ``Rules Rather than
Discretion: the Inconsistency of Optimal Plans," \emph{Journal of
Political Economy},v.85.

Litterman, R.L. [1984], ``The Costs of Intermediate Targeting,"
Federal Reserve Bank of Minneapolis Discussion Paper.

Lucas, Robert E., Jr. [1976], in \emph{The Phillips Curve and Labor
Markets}, Karl Brunner and Alan Meltzer, ed.'s.  Amsterdam:
North Holland.

Roberds, Will [1985], ``Policy Analysis under Stochastic
Replanning," Federal Reserve Bank of Minneapolis Research
Department Working Paper number 278.

Sargent, T.J. [1984], ``Vector Autoregressions, Expectations, and
Advice," \emph{American Economic Review}, May proceedings issue.

Sims, C.A. [1975], ``Exogeneity and Causal Orderings," in \emph{New
Methods in Business Cycle Research}, Federal Reserve Bank of
Minneapolis.

\underline{\phantom{Sims, C.A.}} [1982], ``Policy Analysis with Econometric Models,"
\emph{Brookings Papers on Economic Activity}.

\underline{\phantom{Sims, C.A.}} [1984], ``Solving Stochastic Equilibrium Models
"Backwards", Univerisity of Minnesota Center for Economic
Research Discussion Paper number 206.
\end{document}
