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\begin{document}
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\title[Drift and Breaks]{Drift and Breaks in Monetary Policy}%
\author{Christopher A. Sims}%
\address{Department of Economics, Yale University}%
\email{sims@econ.yale.edu}%

\thanks{\copyright 1999 by Christopher A. Sims.  This material may be reproduced for
educational and research purposes so long as the copies are not sold, even to recover costs,
the document is not altered, and this copyright notice is included in the copies.}%
%\subjclass{}%
\keywords{change point, Bayesian methods, vector autoregression,
   identification, time-varying parameters, Lucas critique}%

\date{\today}%
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\begin{abstract}
US monetary policy behavior since 1948 is modeled as nonlinear, changing over
time according to a hidden Markov chain specification.  Though the estimated
Markov chain model implies large shifts in the form of the policy reaction
function, its improvement in fit over a simple linear model comes almost
entirely from its allowance for persistent heteroscedasticity.  A linear model
that allows such heteroscedasticity fits almost as well.  The shifts in policy
regime that are uncovered are not unidirectional---they oscillate, with a given
state seldom persisting more than a few years.  The paper discusses how these
results mesh with attempts to interpret this period through the Lucas critique
and natural rate models.
\end{abstract}
\maketitle
% -------------------------------
\section{Introduction}
   The centerpiece of this paper is a stylized model of the behavior of US
   monetary policy authorities over 1948-97.  It is therefore a contribution to
   the ``reaction function" literature.  Unlike most of that literature,
   however, it considers skeptically the notion that this period can be broken
   into disjoint periods of distinctly different monetary policy behavior.  It
   insists on modeling possible deviation from a simple linear reaction
   function with an explicit stochastic model of the deviations. It does so
   using the hidden Markov chain approach that has recently been applied by
   \citeasnoun{ChibCalcPost} and \citeasnoun{KimNelson} and earlier
   \citeasnoun{HamiltonSwitch}.

   The paper's substantive econometric conclusions are
   \begin{itemize}
   \item there is strong evidence of deviation from a linear, Gaussian policy
   reaction function;
   \item there are variations over time in the size of disturbances to the
   policy rule that are more important to model fit than the variation in the
   coefficients of the reaction function, with the two types of variation
   occurring together;
   \item because of the model's logarithmic form, all of the three estimated
   policy regimes involves strong enough eventual reaction of the interest rate to the
   price level to guarantee existence and uniqueness of equilibrium under a
   passive\footnote{This is the terminology of \citeasnoun{LeeperActPass}.  A
   very similar classification of fiscal policies by \citeasnoun{WoodEMU}
   labels this type of policy ``Ricardian".} fiscal policy.  In the neighborhood
   of historically observed inflation rates, though, the linear approximation
   to only one of the three rules reacts strongly enough to guarantee a unique
   price level.
   \item the variation in policy is not in the form of unidirectional flow from
   one regime to another, as would be expected if policy changes reflected
   steady improvement based on learning; it is in the form of repeated
   oscillations among types of behavior, with ``regimes" generally lasting well
   under a decade;
   \end{itemize}

   The results suggest that there has by and large been continuity in American
   monetary policy, albeit with alternation between periods of erratic,
   aggressive reaction to the state of the economy and periods of more
   predictable and less aggressive response.  The aggressive type of policy seems
   to be more associated with periods of flat or declining interest rates than
   with rising interest rates.  Not only is there no evidence of
   the kind of one-time change in policy rule, accompanied by changes in the
   cyclical behavior of real variables, that is the workhorse of
   theoretical rational expectations policy analysis, there is little evidence
   that monetary policy makers at the beginning of the period had any tendency
   to be less ``tight".\footnote{The data, seen through this model, thus fail to
   confirm either of the competing stories about how American inflation ended
   that \citeasnoun{TJSConquest} discusses.}

   That the nonlinearities in policy behavior we have found do not fit the
   categories of standard rational expectations theory does not mean they are
   uninteresting.  It should be possible to use hidden-state policy models like
   this to probe for nonlinear impacts of monetary policy on the economy.
   The predictability of monetary policy could in principle be important.  Do
   the strong variations in predictability found here have effects on the
   behavior of the economy?  This seems an important topic for further
   research, and this paper includes discussion of how it could be pursued.

\section{Some Descriptive Statistics}
   Discussion of the recent history of US monetary policy by the economic
   punditry sometimes assumes that it began the postwar era in the thrall of
   Keynesianism, and was thus not vigilant about controlling inflation.  The
   result of this policy is then taken to have been the rise of inflation in
   the 60's and 70's, which was cut short, it is thought, by the advent of a
   new monetary policy that recognized the need to control inflation and the absence
   of any permanent tradeoff between inflation and unemployment.

   Even a quick examination of the facts shows that this story is hard to see
   in the data.  We see in Figure 1 a plot of a short interest rate against the
   consumer price index (CPI) inflation rate for this period. Clearly nominal
   interest rates have tracked inflation rates closely throughout the period.
   For the inflation rate to be determinate and stable, monetary policy must
   eventually increase interest rates at least in proportion to sustained
   increases in the inflation rate.\footnote{This assumes passive fiscal
   policy.}  From Figure 1 we see that this criterion appears to have been met
   throughout the period.  However, in the 70's, as energy price shocks created
   sudden inflation, the real rate was allowed to stay at approximately zero
   over a prolonged period.  In hindsight, knowing that this inflation burst
   was not a temporary burst like the Korean war inflation of the early 50's or
   the temporary ``cost push" inflation of the '57-58 recession, it can be
   argued that pushing the interest rate up above the inflation rate would have
   been advisable.  But note that even the sharp rise in the interest rate up
   to its peak at the beginning of 1980 merely tracked the behavior of CPI
   inflation.  It was only after inflation peaked and began to decline that the
   interest rate, which followed downward with a delay, opened up a gap
   implying a real rate at the level (or slightly above the level) that had
   prevailed in the 60's.

   One can see from the graph that the response of interest rates to inflation
   was probably slower, though perhaps not much less strong in total, in
   the 60's, that the level of interest rates relative to inflation was
   maintained higher then, and that interest rates followed a smoother time
   path month to month in the 60's than in the 70's and 80's.  But it remains
   to be seen whether these differences across periods are sharp enough to give
   nonlinear models a clear edge in fit.

\begin{figure}
  \centering
  \includegraphics[width=.8\textwidth]{cpicprate}
  \caption{CPI Inflation and the 6-month Commercial Paper Rate}\label{fig:cpicprate}
\end{figure}

\section{The Model}
   Because of this paper's exploratory nature, the model of policy behavior has
   been kept drastically simple.  We model monetary policy as setting a short
   term interest rate (the 6 month commercial paper rate) as a function of past
   values of the interest rate and of a commodity price index.  The aim was to
   use a very manageable model in which interest rates tend to rise in response
   to inflationary pressure and in which upward disturbances in the interest
   rate lead to downward movements in prices.  A bivariate system in interest
   rates and commodity prices has just these properties, even though it of
   course ignores the likely dependence of monetary policy on a number of other
   variables.\footnote{It is standard to use the Federal Funds rate in reaction
   function research.  The commercial paper rate has the advantage that it will
   allow extension of the study to the interwar years, though nothing is done
   in this draft with interwar data.}  We use monthly data from 1948:1-1997:10.

   The model is described by the following equations:
   \begin{gather}
      \alpha(L)r_t=c(S_t)+b(S_t)\beta(L)p_{t}+\sigma(S_t)\eps_t \label{eq:reaction}\\
      P[S_t=i\mathbin{|}S_{t-1}=j]=p_{ij}\,,\:i=1,\dots,3\,,\:j=1,\dots,3\:,
   \end{gather}
   where $\eps_t$ is i.i.d. $N(0,1)$ conditional on the entire $\set{S_t}$
   sequence and on $\set{r_s,p_s, s<t}$.  We normalize the initial coefficients
   of the lag polynomials, $\alpha_0$ and $\beta_1$, to 1 (setting $\beta_0=0$) and
   make them 6th order.  In this equation $r$ is the natural log of the
   6-month commercial paper rate, expressed as a percent at annual rates, and
   $p$ is the log of a commodity price index.

   The transition matrix $P=[p_{ij}]$ for the unobserved discrete Markov process
   $S$ is controlled by three parameters $\pi_1$, $\pi_2$, and $\pi_3$, so that
   it has the form
   \begin{equation}\label{eq:Padj}
      P=\begin{bmatrix}
         \pi_1 & 1-\pi_1 & 0\\ (1-\pi_2)/2 & \pi_2 & (1-\pi_2)/2\\
         0 & 1-\pi_3 & \pi_3
      \end{bmatrix}\:.
   \end{equation}

   The model implies that the policy reaction function varies among three
   distinct forms as $S_t$ varies among its three possible values 1 to 3.  The
   coefficients on lagged $r$ are the same in all three regimes, as is the form
   of the lag distribution on $p$.  However, the scale of the lag distribution
   on $p$ varies across regimes, as does the regression's constant term and its
   disturbance variance.

   While this model is nonlinear and mixes discrete and continuous components
   of random variation, it falls in the class of what \citeasnoun{ChibCalcPost}
   calls hidden Markov chain models.  This means that its likelihood, the pdf
   of the data conditional on parameters (but integrated over $S$) can be
   calculated recursively analytically, and also that draws from its posterior
   pdf can be generated easily.  The model in \citeasnoun{HamiltonSwitch} is
   also in this class, and he exploited its recursive tractability.  A variety
   of models of this class are discussed in \citeasnoun{KimNelson}.

   There is clearly a lot of room for making this model more complicated.  We
   could allow the coefficients of $\alpha$ and $\beta$ to vary with $S$.  We
   could add more right-hand-side variables, with or without $S$-dependence in
   their coefficients.  There is no magic in the assumption of 3 states---we
   could allow more.  And of course a completely general specification for the
   $P$ matrix would have six free parameters instead of three.

   The form of the $P$ matrix is important to interpretation, and we will
   discuss below the effect of varying it.  We also display the effects of
   reducing (but not of increasing) the number of states.  But note that we
   already have here a system with 23 free parameters.  An unconstrained 4
   variable, 6 lag VAR with a constant term has just 26 free parameters per
   equation (counting the disturbance variance), and we are well used to the
   idea that such a system is likely to forecast poorly unless it is estimated
   with a proper prior distribution.  Since this paper is to some extent
   breaking new ground, we maintain the 23-parameter limit for now.
   Complications may well be worth examining, probably with proper priors, in
   the future.

   Details of how the likelihood is formulated and estimated are described in
   the appendix.

\section{Estimates of the Reaction Function Model}
Maximum likelihood estimates of the parameters of the model, with associated
asymptotic approximations to the standard deviations of the distribution of the
true parameter values about the estimates,\footnote{Here we interpret minus the
inverse second derivative matrix of the log likelihood as the covariance matrix
of a Normal approximation to the likelihood function.  Such an approximation
will become accurate in large samples under broad regularity conditions. Unlike
the usual Normal asymptotic approximation to the distribution of the
coefficient estimates about the true value, this approximate characterization
of the shape of the likelihood is valid even in the presence of unit-root
nonstationarity.} are presented in the left half of Table
\ref{table:PrefModel}.  Note that the three parameters that depend on the
state, $c$, $b$ and $\sigma$, all vary monotonically with the state.  So state
1 has low response to inflation and low disturbance variance, state 3 has high
response to inflation and high disturbance variance, and state 2 is
intermediate.  The fact that state 3 implies the strongest response of interest
rates to inflation does not imply that it is strictly ``tighter" than the other
two regimes.  The constant terms decrease as we go from state 1 to state 3.  It
turns out that with inflation constant, state 1 implies a higher interest rate
than state 2 at annual inflation rates below 60\%, state 1 implies a higher
interest rate than state 3 at inflation rates below 16\%, and state 2 implies a
higher inflation rate than state 3 at inflation rates below 14\%.  Thus the
lower-numbered states imply \emph{higher} interest rates that are less
responsive to inflation, except when commodity price inflation rates reach the
levels observed only at the peak of postwar inflation or in brief bursts at other
times in the period.

\begin{table}
  \centering
  \begin{tabular}{l|c|c||c|c}
  &\multicolumn{2}{c||}{Full Model}&\multicolumn{2}{c}{Variances Only}\\ \hline
   parameter&estimate&std. error&estimate&std. error\\ \hline
 &    0.0091    &0.0079& &\\
c&    0.0023    &0.0149&0.02204  &   0.094115\\
&   -0.0422    &0.1143& &\\ \hline
 &    0.2869    &0.0596& &\\
b&    0.3559    &0.1559& 0.2476   &   0.84093\\
 &    2.5163    &0.6908& &\\ \hline
&    1.4320    &0.0429 &     1.4424   &   0.16037 \\
&   -0.5925    &0.0703  &   -0.60474    &  0.32806 \\
$\alpha$&   0.2668  &0.0681     &   0.27231   &   0.30502\\
&   -0.1386    &0.0662  &   -0.15331  &   0.084267\\
&    0.0304    &0.0646  &   0.046084   &  0.077247\\
&   -0.0063    &0.0369  &  -0.010583  &   0.061189\\ \hline
&   -0.6170    &0.0101      &      -0.1960  &     3.0638\\
 &  -0.7078    &0.2098       &      -1.3872  &     4.9638\\
$\beta$&    0.8687    &0.2975 &      1.3887  &      2.002\\
&   -0.4225    &0.1388       &     -0.5951  &     5.0373\\
&   -0.1164    &0.1432       &     -0.2165  &     5.2359\\ \hline
&   0.0259 &  0.0015   &     0.0258 &0.0050\\
$\sigma$ &  0.0530   &0.0029 &0.0529&     0.0203\\
&    0.1110 &   0.0127       &0.1221 &     0.0149\\ \hline
&    0.9748   & 0.0093      &0.9764  &0.0183\\
$\pi$   &   0.9578  & 0.0152&  0.9591&0.0248\\
&      0.9093& 0.0336       & 0.9192 & 0.0105\\ \hline
log LH & \multicolumn{2}{c||}{1005.75}&\multicolumn{2}{c}{1000.66}
\end{tabular}
\medskip

  \caption{MLE parameter estimates of preferred model}\label{table:PrefModel}
\parbox{.8\textwidth}{\noindent \small Note: The parameters $c$, $b$, and
$\sigma$ have three values each, listed in order for the states we label 1, 2,
and 3.  The coefficients in $\alpha$ are those on lags 1 through 6 of $r$ on
the right-hand side of the equation.  The coefficients in $\beta$ are the
scaled coefficients on lags 2 through 6 of $p$ on the right-hand side, with the
scaled coefficient on the first lag normalized to one and $b$ providing the
scale factor.  See equation \eqref{eq:reaction} in the text.  The listed
standard errors are based on the square roots of the diagonal elements of minus
the inverse of the second derivative of the log likelihood, as approximated by
the BFGS update process during calculation of the MLE. }
\end{table}

Responses to inflation, as measured by $b$, vary by a factor of 8 between
states 1 and 3, and standard deviations of disturbances vary by a factor of 4.
The usual asymptotic $\chi^2(2)$ statistic for equality is 174 for the variance
parameters and 10.5 for the $b$ parameters.  Notice that though the difference
in size of the parameters is bigger for $b$, the standard errors for $b$ are
relatively bigger.  In fact, by the Schwarz criterion the differences among
states in $b$ are (just barely) not sufficient to favor this model allowing
variation in $b$ across states over a model that restricts $b$ to be constant
across states. The asymptotic arguments that justify the Schwarz criterion may
not apply in a model like this one that implies high likelihood for unit roots
in the dynamics.  However, direct application of the method of
Laplace\footnote{\cite[section 7.4.3]{Schervish}. The method of Laplace is
simply the idea of approximating posteriors, likelihoods, or products of these
objects with functions of the underlying random variable, by second order
Taylor expansions of their logs about their peaks---in other words, by their
usual asymptotic Normal estimators.  The Schwarz criterion follows from this
approach under the assumption that the second derivative of the log likelihood
is growing as $O(T)$, where $T$ is sample size.  In other contexts, where this
assumption is not valid but the asymptotic normal approximation still holds (as
in the presence of unit roots), the role of $\log T$ in the Schwarz criterion
is taken over by the difference in log determinants of the second derivative
matrices for the restricted and unrestricted models.} (from which the Schwarz
criterion is derived) would in this instance even more strongly favor the
simpler model.

As a check on the validity of the asymptotics\footnote{We can check the
validity of the asymptotics because we are using Bayesian asymptotics, which
only make assertions about the shape of the likelihood, which can be examined
directly.}, we re-estimate the model with variation across states in the
disturbance variances only, obtaining the results shown in the right half of
Table \ref{table:PrefModel}.  This produces an asymptotic $\chi^2(4)$ statistic
of 10.18, which is at a marginal significance level of .038.  In a sample of
size 592, this is weak evidence in favor of the null, since the Schwarz
criterion would require that the statistic be 25.5 before it favored the larger
model.  Furthermore, note that in the restricted model the parameters $b$ and
$\beta$ are locally very ill-determined.  Direct application of the method of
Laplace would therefore imply even stronger weight toward the more restricted
model, because in effect the restricted model has nearly redundant parameters,
and is therefore effectively smaller than a count of its parameters would suggest.

It might appear from the ``insignificance" of $b$ and $\beta$ estimates in the
restricted model that the evidence for any effect of $p$ on $r$ at all is weak.
This is not true, however.  When the model is restricted to allow no effect of
$p$ on $r$, so that it is just a univariate model of $r$ with switching among
residual variance regimes and constant terms, the log likelihood maximum is
reduced to 983.15.  Eliminating the 6 parameters associated with $p$ therefore
produces a $\chi^2(6)$ statistic of $2\times(1005.75-983.15)=43.2$, which has a
marginal significance of around $10^{-7}$ and favors the model including prices
by the Schwarz criterion.

It is difficult to assess the strength of the estimated response of $r$ to $p$
from these individual coefficients.  In the full model, one minus the sum of
coefficients on lagged $r$ is 0.00812269, with a standard error of .002624.  So
even though the sum is small, implying a root close to one in $\alpha(L)$,
likelihood sharply distinguishes the sum from zero.  The sum of coefficients in
$\beta(L)$ is .0050 with a standard error of .021, so the estimated long run
response of the level of interest rates to the level of commodity prices is not
distinguished from zero, by model fit.  The point estimate
of the long run response of the log of the interest rate to the log of
commodity prices is about $.62b(S_t)$, which ranges from .18 to 1.56.  The
largest of these implies that to invoke an increase in the interest rate from,
say, 5\% to 6\%, the price level would have to increase by 13\%, so the
estimated responses are quite modest.

The cumulative sum of coefficient in $\beta(L)$ is the lag distribution on
monthly inflation rates.  If we take $b$ times the first 5 elements of this
cumulative sum as the lag distribution on inflation (setting to zero the 6th
and later coefficients, which are of course all .0050), sum the result, and
divide by $\alpha(1)$, we have an estimate of the long run response of $\log r$
to monthly inflation, under the assumption of zero long run response to the
price level. Dividing the result by 12 to get responses to annual (rather than
monthly) inflation rates gives us, as long run responses in the three states, 5.09,
6.31 and 44.6.  Because of the logarithmic specification, the interest rate
levels above which these estimates imply more than one-for-one response of $r$
to the commodity price inflation rate are 20\%, 16\%, and 2.2\%, respectively.
It is interesting to note that the middle state implies more than one-for-one
response starting at just the level of interest rates actually reached in the
late 70's and early 80's

Because the logarithmic specification matters for the interpretation of the
model, it is worth noting that the likelihood, corrected for the Jacobian of
the data transformation, is quite a bit lower when the model is estimated in
the levels of the interest rate.

The model was also estimated with symmetric jumping among the states, i.e. with
$p_{ij}=(1-\pi_{i})/2,\,j\ne i$.  This results in a slightly lower log
likelihood of around 1003.  The MLE for this symmetric model implies that only
two of 16 likely points of transition between states in the sample involved
jumps between non-adjacent states, an unlikely result if jump probabilities are
in fact symmetric.  The model was also estimated with the states restricted to form an
irreversible chain, i.e. with $p_{12}=1-\pi_1$, $p_{23}=1-\pi_2$, $p_{ij}=0$
for all other $j\ne i$, and $\pi_3=1$. This specification has one fewer free
parameter, and produces much lower log likelihood (around 960).\footnote{The
log likelihoods cited here are vague, because the exact numerical results were
lost as this draft is being written, leaving no time to reproduce them.}

The post-sample probabilities of state 1 and state 3 (that for state
2 of course is 1 minus the sum of the other two), together with the dependent
variable in the regression, are plotted in Figure \ref{fig:xadjphist}.  Note
that the high-variance, high-response state 3 is estimated as likely only
during the early 80's reserve-targeting period, plus a brief period in the 50's
when the interest rate came down sharply and a similar but even briefer period
in the mid-70's.  The rest of the period shows regular oscillations between
states 1 and 2, except for the 60's, during which there was an unusually long
period of prevalence of state 1.  Futhermore, the advent of state 3 in every
case coincides with the end of a period of rising interest rates.  It seems to
be required to explain the rapid declines in rates that occur at the starts of
recessions and the ends of inflations.  The switch to state 3 at the start of
the 80's occurs in the first month of 1980, just before the interest rate peak
of 16.5\% the next month, which was then only very slightly exceeded in two
later months that year.  Most of the period 1980:1-1982:10 during which this
state is estimated to have persisted is a time of constant or declining
interest rates and declining inflation.  The responsiveness of interest rates
to inflation in this state explains the speed with which interest rates
declined over this period, though it also contributes at the beginning of the
period, when lagged inflation was high, to the high level of interest rates.
\begin{figure}
  \centering
  \includegraphics*[width=.8\textwidth]{xadjphistNude}
  \caption{Post-Sample Probabilities of States}\label{fig:xadjphist}
\end{figure}
\section{Implications of the Estimates}
The case for recognizing time variation in the variance of disturbances in the
reaction function is very strong.  Despite the unrealistically simplified form
of this model, there seems little doubt that this result would carry over to
more realistic specifications.  That there is heteroscedasticity in policy
reaction function disturbances in linear VAR's has has been recognized before.
This paper shows that it is well modeled as persistent.\footnote{The symmetric
jumping form of the model includes as a special case equal probabilities on all
states in all periods, which would imply a simple mixture of Gaussian
disturbance distributions, i.e. fat tails rather than persistent
heteroscedasticity.  The point estimates of that model were very far from this
form.  However, we have not directly checked for a general three-state mixture
model without persistence, which would be what is implied by having all columns
of $P$ constant, but not necessarily at the same level. }

The fact that the states seem to arrange themselves one-dimensionally suggests
that it might make sense to expand to more than three, while keeping the
parameterization from expanding too much by constraining $P$ to take a form
that concentrates along its diagonal.  There would be no computational problem
in such an expansion if it could keep the $P$ matrix dependent on no more than
3 or 4 parameters.  One way to do this would be to choose $P$ so that it can be
thought of as a discrete approximation to a continuous AR process.

One should treat more cautiously substantive conclusions about the nature of
policy based on this model, because they could easily shift with the addition
of more variables or the use of the non-recursive identification schemes that
we know can give better results.\footnote{Much of the identified VAR literature
using postwar data has stuck to recursive identification schemes, probably as a
matter of convenience. \citeasnoun{CapeConf} shows that extending structural
VAR methods to the interwar period seems to require use of non-recursive
identification.}  Nonetheless, this model is no more oversimplified than some
that have been used to spin interesting stories about policy, so it is
interesting to proceed.

The results present a paradox, in that they show that a story that asserts a
sudden, large increase in the strength of reaction to inflation in the early
80's is consistent with the data, yet that the improvement in fit from modeling
this change in the reaction to inflation is quite modest.  My own interim
conclusion is that it is not yet time to toss out linear reaction
function models entirely, so long as they are amended to allow persistent
heteroscedasticity.

The point estimates of the full model, though consistent with a toughening of
anti-inflationary policy in the early 80's, do not imply an irreversible
learning process.  Indeed fit deteriorates badly if policy is not allowed to
revert to previous forms.  The 80's are estimated as quite different, in other
words, but also as quite temporary.  One interpretation of this pattern fits
the pessimistic conclusion of \citeasnoun{TJSConquest}: the experience of the
seventies taught policy-makers the importance of anti-inflationary action, but
the accumulating experience of a stable economy in the late 80's and 90's could
be simply undoing the learning process, preparing the stage for another
inflationary crisis.

My own preferred interpretation would treat the policy rule as stable, though
nonlinear.  Policy reacts to commodity price inflation, but it does not react
strongly to temporary fluctuations in it, which are frequent.  It reacts with
increasing strength as the evidence of strong and persistent inflation
accumulates, and is capable of very vigorous action when the threat of
inflation is very clear.  If the public understands this pattern of behavior,
theory implies that the episodes in which strong counter-inflationary action is
needed may be rare.  Indeed it is easy to construct models in which, so long
as the public is convinced of the readiness of the policy authority to react
strongly, the conditions under which the policy authority needs to do so never
arise.

The most interesting next directions for research, it seems to me, are checking
for whether the states that are useful in modeling nonlinearity in policy
behavior also have direct effects on the private sector.  That we should look
for this possibility is the real lesson of the Lucas critique.  Policy changes
need not be well modeled as zero-mean, stationary random shifts in the constant
term of a linear reaction function.  They could instead need to be modeled as
stochastic, more or less persistent, changes in the coefficients or variance
parameters of those relationships.  If they do have this character,
extrapolating the effects of policy changes from linear approximations to the
true policy rule could be quite misleading.\footnote{This way of looking at the
Lucas critique is not new.  See \citeasnoun{CooleyLeroyRaymon}
and\citeasnoun{CASREframePolAnal}.}  To take account of the Lucas critique in
practice, we need to look for nonlinearity in policy behavior, and then to
examine the connections between the sources of variation in policy behavior and
their effects (if any) on private sector behavior.  To do this, we must
identify variation that originates in policy behavior and distinguish it form
variation originating in the private sector, a difficult task.  As the
structural VAR literature has shown, however, it is possible to approach this
task without insisting that the model be formulated entirely in terms of fully
interpretable ``deep" parameters of taste and technology.

It is quite possible that the nonlinearities in policy behavior that this
paper's model points to are unimportant to modeling the reaction of the public
to monetary policy.  That is, the public's behavior might depend on the history of
interest rates and reserves directly and more or less linearly, without regard
to the current state of the policy reaction function.  But rational
expectations has given us a rich set of examples to show that it could be
that the public reacts differently to the same pattern of changes in
interest rates under policy states 1 and 3.  Further exploration, in
multivariate models, to see whether this possibility emerges as practically
important, is clearly desirable.  It is likely to have the additional benefit
that, by bringing more information to bear, it should sharpen inference on the
fundamental question that this paper's results leave open:  In telling interesting
stories about policy shifts with this papers's Markov chain models, are we just
making a man in the moon out of random meteor craters?

\appendix
\section{Calculating the Likelihood}
Following \citeasnoun{HamiltonSwitch}, we treat the initial lagged values of
$r_{1-s}$ and $p_{1-s}$, $s=1,\dots,6$, as independent of $S_1$, treat $S_1$ as
drawn from its steady-state distribution (as implied by $P$), and condition on
the initial lagged values of $r$ and $p$ in forming the likelihood.

We will use $\mathcal{I}_t$ to refer to the observable information set at $t$, i.e.
$set{r_{t-s},\,p_{t-s},\,s\ge 0}$.  The model implies
\begin{equation}\label{eq:condpdf}
  q(r_t\mathbin{|}\mathcal{I}_t, S_t) =
  \frac{1}{\sigma(S_t)}\phi\left(\frac{\alpha(L)r_t-b(S_t)\beta(L)p_t-a(S_t)}{\sigma(S_t)}\right)\:.
\end{equation}
Here as elsewhere we use $q$ as a generic symbol for a pdf when the random
variable and the conditioning set are apparent from $q$'s arguments.  Also, we
use $\phi$ as the symbol for a standard $N(0,1)$ Gaussian pdf. We can form the
joint one-step-ahead density
\begin{equation}\label{eq:onesteprS}
  q(r_t, S_t\mathbin{|}\mathcal{I}_{t-1})=
   q(S_t \mathbin | \mathcal{I}_{t-1})q(r_t\mathbin{|}\mathcal{I}_{t-1},
   S_t)
\end{equation}
and from this the one-step-ahead predictive density for $r_t$ as
\begin{equation}\label{eq:onesteprOnly}
  q(r_t\mathbin | \mathcal I_{t-1})=\sum_{S_t=1}^{3}q(r_t,
  S_t\mathbin{|}\mathcal{I}_{t-1})\:.
\end{equation}
We can form
\begin{equation}\label{eq:Sgivent}
  q(S_t \mathbin | \mathcal I_t)=\frac{q(r_t,S_t \mathbin | \mathcal I_{t-1})}
   {\sum_{S_t}q(r_t,S_t \mathbin | \mathcal  I_{t-1})}\:.
\end{equation}
This last step rests on the assumption that the distribution of $p_t\mathbin
|\mathcal I_{-1},r_t,S_t$ is independent of $S_t$.  In other words, all
influence of policy on current $p$ comes through current $r$, with no direct
perception of and reaction to $S_t$ by the public within the current period.

To complete the recursion, then, we form
\begin{equation}\label{eq:Snew}
  q(S_{t+1}\mathbin |\mathcal I_t) = \begin{bmatrix}
   q(S_t=1\mathbin |\mathcal I_t)\\q(S_t=2\mathbin |\mathcal I_t)\\q(S_t=3\mathbin |\mathcal I_t
\end{bmatrix}'\cdot P \:.
\end{equation}
With \eqref{eq:Snew} in hand, we can return to \eqref{eq:condpdf} and
\eqref{eq:onesteprS} to generate the time-$t+1$ predictive pdf, and thus
recursively the entire likelihood.\footnote{Of course in models like this with
predetermined, but possibly not strictly exogenous, right-hand side variables,
what we are forming is not a complete pdf for the observed data given
parameters or even a conditional pdf for it. It is a factor of the complete pdf
that includes all the components that depend on the parameters we are
interested in; so though it is not a pdf for the data, it is still proportional to
the likelihood function.}
To start off the algorithm, we need only have in hand $q(S_0\mathbin |\mathcal
I_0)$, which as we stated at the start is taken to be given by the steady-state
distribution of $S$, extracted as a left eigenvector of $P$ corresponding to an
eigenvalue of 1.

With this recursion complete for the whole sample $t=1,\dots,T$, we can then
execute a backward recursion to evaluate $q(S_t\mathbin | \mathcal I_T)$.  We
can do this because
\begin{equation}\label{eq:qback}
  q(S_{t+1},S_t\mathbin | \mathcal I_T)=\kappa(\mathcal I_T)q(S_{t+1}\mathbin
  |\mathcal I_T)q(S_{t+1}\mathbin | S_t)q(S_t\mathbin | \mathcal I_t)\:,
\end{equation}
where $\kappa(\mathcal I_T)$ is a normalizing constant set so that the expression sums
to one across $S_t,S_{t+1}$. We can therefore find the desired
$q(S_t\mathbin | \mathcal I_T)$ by summing \eqref{eq:qback} over $S_{t+1}$.
That \eqref{eq:qback} holds depends on two assumptions we have made:
\begin{gather}
   q(S_{t+1}\mathbin |S_t,\mathcal I_t)=q(S_{t+1}\mathbin | S_t)\label{Sind}\\
   q(r_{t},p_{t}\mathbin | \mathcal I_{t}, \set{S_v,\,v\le
   t})=q(r_t,p_t\mathbin | \mathcal I_t,\,S_t)\label{mkvI}\:.
\end{gather}

The class of models we consider here does not include the models labeled
``Markov switching models" in most of the literature.  Those models,
following the specification in \citeasnoun{HamiltonSwitch}, take the form
\begin{equation}\label{eq:HamEq}
  \alpha(L)(y_t-\mu_{S_t})=\eps_t\:.
\end{equation}
Even if $S$ in this model is a first-order Markov process, the appearance of
higher than a first order lag term in $\alpha$ will require that the full model
for $y$ and $S$ treat the state vector as including as many lagged values of
$S$ as of $y$. This is not as much of a computational problem as it might
sound.  The point here is not that Hamilton-style Markov switching models are
any worse than the ones that have been fitted in most of this paper, but that
they are different.

In the computations, to avoid the need to handle constraints on the parameters
explicitly, the $\sigma(S_t)$ parameters were logged and the $\pi$ parameters
were given a logit transformation (i.e., $\log(\pi_t/(1-\pi_t))$).  The
reported parameters in Table \ref{table:PrefModel} undo these transformations
for both the parameter values and their estimated standard errors.

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