A test of strategic interaction in monetary policy.

Beginning with an example provided by Kydland and Prescott [29], a large literature has evolved concerning strategic interaction between the public and a monetary authority in the conduct of monetary policy. We undertake a direct test of a fundamental assumption of the strategic monetary policy literature. Monetary policy games suppose that the monetary authority and the public are aware of the strategic interdependence between them. That is, these models suppose that the monetary authority chooses an inflation rate that is a function of the public's expected inflation rate and that the public is sophisticated enough to realize that the inflation rate chosen by the monetary authority is a function of its beliefs. Below we extend Cukierman and Meltzer's [18] model of ambiguity and credibility in monetary policy to nest other models. We then use a novel data set that permits a test of the assumption of strategic awareness for agents in the U.S. We find support for our extension of the Cukierman and Meltzer model.

The new data brought to bear is the Federal Reserve's forecast of inflation conditional on the monetary policy it chose. Throughout the sample period, the Federal Open Market Committee (FOMC) met from eight to twelve times per year to consider various short term monetary policies. Data on the forecasted paths for the inflation rate, growth rate in output, and unemployment rate associated with the different monetary policies under consideration is available for some of their meetings; it seems reasonable to suppose that the chosen monetary policy satisfies revealed preference in the forecasts of these variables that it and the rejected monetary policies imply. Therefore, in referring to the data, we use the terms "inflation rate forecast by the Federal Reserve" and "inflation rate chosen by the Federal Reserve" interchangeably. Since theories of time consistent monetary policy have always been cast in the latter terminology, the data are especially well-suited to a direct test of the theory.

Section I briefly reviews the theoretical literature on monetary policy games and empirical literature relevant to the present paper.(1) The econometric tests are generated by a version of the monetary policy game advanced by Cukierman and Meltzer [18] which is explicated in section II. Section III presents and discusses empirical results. Section IV concludes with suggestions for further research.

I. Monetary Policy Games: Theory and Evidence

Burro and Gordon [7] provide a formal model of Kydland and Prescott's example and show the difference between the optimal policy, given by a monetary rule, and the time consistent or discretionary policy. Their model explains the persistence of positive inflation rates as an equilibrium phenomenon in the context of a model embodying the natural rate hypothesis but, because it is a single period model, does not explore how the monetary authority's concern for its reputation can affect policy. Burro and Gordon [6] examine the effects of reputation in infinitely repeated play of their one-shot game. They show that if the creation of surprise inflation by the monetary authority would be followed by a loss of credibility with the public and consequently higher expected inflation in the future, then the monetary authority's desire to avoid the future high inflation caused by a current inflation surprise will cause it to choose a lower inflation rate than is implied by the single period model.(2) They also discuss the implications of including asymmetric information via a taste parameter of the monetary authority that follows a white noise process that is unobserved by the public. Because the public forms expectations using the mean of the taste parameter, from period to period the economy moves along an expectational Phillips curve as the monetary authority's tastes between inflation and unemployment change.

Another set of models with asymmetric information has wage setters contract a nominal wage in advance of the observation of a real or nominal shock to the economy. An early example is Rogoff's [30] analysis of stabilization policy, which among other results, concludes that the selection of a Federal Reserve chair with a strong aversion to inflation can mitigate the inflationary bias of discretionary policy. Another example is Canzoneri's [13] model, in which information is private as well as asymmetric.(3) In his model, the Fed possesses private information on money demand. The public cannot distinguish between an increase in the money supply in response to an increase in money demand from an increase in the money supply intended to come as a surprise and lower unemployment. Canzoneri's model predicts inflation shifts between high and low regimes that mimic the inflation experience of the U.S. better than a single stable regime.(4)

A second approach in the literature on dynamic monetary policy games posits private information with respect to different types of monetary authorities to focus on reputational considerations explicitly. In contrast to the models discussed above, these games have a known, finite horizon that represents the regime of a monetary authority; the preferences of the monetary authority are assumed constant. Backus and Driffill [3; 4] provide the earliest example of this literature.(5) Their finite-horizon model can explain why a disinflationary policy would gain credibility only slowly as the public updates its beliefs in response to unexpectedly low inflation after a change in the monetary regime.

Cukierman and Meltzer [18] use an infinite horizon model to combine aspects of the approaches outlined above and incorporate several additional features of the monetary policy milieu that can be verified with casual empiricism. Rather than treating the monetary authority's objective function as a social welfare function, Cukierman and Meltzer argue that the objective function of the Fed reflects constantly evolving preferences over inflation and unemployment of the changing membership of the FOMC and political pressures sporadically applied to the Fed by Congress and the President. Formally, they represent the changing objective function with a stochastic taste parameter that follows an AR(1) process. Because the Fed's past behavior provides probabilistic information to the public about its current objectives and thus its current and future behavior, the Fed's optimization problem will have an intertemporal linkage. Thus, in the Cukierman and Meltzer model as the Fed chooses the current money supply it must be aware that its current choice will affect the public's future perceptions of what the Fed is seeking to accomplish. Their approach sheds light on what is meant by a change in the regime of the monetary authority. As Cukierman [17, 191-94] observes, if the distribution of the stochastic taste parameter has thin tails, large changes in the money supply regime will occur infrequently in response to infrequent, large draws of the taste parameter. These in turn might reasonably be expected to follow changes in the occupancy of the White House, the Fed chair, or other sources of change in political pressure.(6) The behavior of the public in the Cukierman and Meltzer model rationalizes "Fed watching," the existence of which is verified by casual empiricism. Fed watching is complicated by the fact that the instrument of monetary policy (e.g., M1, M2, the monetary base) is subject to only imperfect control by the Fed. Hence the public cannot precisely infer the Fed's objectives by monitoring the money supply growth rate. Cukierman and Meltzer explicitly define the Fed's credibility at a given point in time as the difference between the most recent money supply growth rate the Fed chose and the money supply growth rate the public expected the Fed to choose.(7) Because the public cannot disentangle a temporary monetary control error from a change in the Fed's tastes, after a large change in the Fed's stochastic taste parameter generates a large absolute value for credibility, credibility will adjust slowly over time. Although their analysis is not couched in terms of unemployment (or some other measure of real activity) and inflation, their model clearly has a rationale for a short run trade-off between real activity and inflation based on the value of credibility.

Extant empirical papers that are relevant to strategic monetary policy issues focus on the public's reaction to announced changes in monetary policy rather than time consistency issues. One strategy common to several papers is the "prediction error" technique [10; 14; 15; 33]. Analysts using this method estimate a Phillips curve relationship and/or an interest rate term-structure equation from a sample drawn immediately before the announced change and then investigate the prediction errors of the model after the announced change. In the case of Phillips curve estimation, if the policy change is credible then private-sector inflation expectations change and inflation prediction errors after the change should be negative; that is, for a given level of unemployment, inflation should be over-predicted. For term-structure equations, Blanchard [10] argues that if the policy change is credible in financial and labor markets and if prices and wages are not predetermined, then since agents will have lower real balances in the short run but will anticipate less inflation in the future, long term and short term rates will move in opposite directions and the term structure equation will over-predict the long rates after a policy change. Blanchard (for the U.S. following the Fed's announced procedural changes in 1979), Christensen [14; 15], and Slow [33] all find that announced policy changes gain credibility only gradually with the passage of time. In particular, Blanchard's Phillips curve estimates imply that it took approximately nine quarters for the 1979 policy change to gain credibility.

A second approach taken to test the credibility hypothesis is to define an empirical measure of credibility and to attempt to show that the empirical measure is related to the success of disinflationary policies. Baxter [8], Christensen [16], Weber [35], and Agenor and Taylor [2] adopt this approach. All authors find that their measures of credibility are related to the success of disinflationary policies.

These studies are consistent with theoretical models of time consistent monetary policy by suggesting that the public's inflation expectations are not malleable enough for mere policy announcements to permit monetary authorities to exploit expectations, yet none of these studies directly test for agents' strategic awareness. We are not aware of any attempts to test for strategic interaction directly, which is the key goal of this paper.

II. A Model of Strategic Interaction

In this section, we derive a model of strategic interaction between the Fed and the public that is the basis of the empirical analysis in section III. The model is very similar to Cukierman and Meltzer's [18] model of ambiguity and credibility, in which the Fed's tastes vary over time in a manner that the public can only imperfectly observe. Because the Fed's tastes are serially correlated, the past behavior of the Fed provides information about its future behavior and justifies the public's expenditure of resources to monitor the Fed. Additionally, the Fed is aware that its current decisions will affect not only the present period's outcomes but also the public's beliefs about the Fed's tastes in the next period. Thus the public's inflation expectations are endogenous in the Fed's problem of choosing a money supply growth rate. To focus on the subtle interaction that takes place between the Fed and the public, Cukierman and Meltzer express the Fed's optimization problem and strategic interactions entirely in terms of the money supply growth rate. We recast the issues they consider in the more traditional framework of time consistent monetary policy to introduce variables that empirical analysts observe.(8) This clarifies the relationship of the Cukierman-Meltzer model to other models in the literature on time consistent monetary policy.

Suppose that the price level and output are determined by the quantity theory and a Lucas supply curve,(9)

[P.sub.t] = [M.sub.t] - [Beta][Y.sub.t] + [Theta][D.sub.t] + [v.sub.t]

[Mathematical Expression Omitted]

where [P.sub.t] is the natural log of the price level at time t, [Y.sub.t] is the natural log of real output at time t, [Y.sup.n] is the natural log of the natural level of real output, [Mathematical Expression Omitted] is the natural log of the expected value of the price level at time t based on information available to the public at time t, which does not include the price level, [S.sub.t] is a shock to real output, and [D.sub.t] is a shock to money demand. The errors [v.sub.t] and [[Eta].sub.t] are the logs of a velocity shock and an additional real shock; we assume they are mutually independent, i.i.d. normal variables with mean zero and variances [Mathematical Expression Omitted] and [Mathematical Expression Omitted] and are unobservable to the public and the Federal Reserve at time t. Initially, we suppose [Mathematical Expression Omitted]; later, we will allow for asymmetric information between the Fed and public. Additionally, we suppose that [Mathematical Expression Omitted] at each time period t + j for j [greater than or equal to] 1.(10)

For the data discussed in the next section, it is convenient to work in changes in variables rather than levels. After taking first differences in equations (1), solve for the inflation rate and the growth rate in output to obtain

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Observe that differencing [Mathematical Expression Omitted] yields [Mathematical Expression Omitted]. We have defined [Mathematical Expression Omitted]; then [Mathematical Expression Omitted], where [Mathematical Expression Omitted]. Note that at time t, [[Xi].sub.t - 1] is observable to the Fed and the public. Equations (2) indicate how the rate of change in output and the inflation rate evolve in the economy.

Suppose that the Federal Reserve minimizes the expected present value of a discounted quadratic loss function, defined here over inflation and the growth rate in output. Thus the Federal Reserve's problem is given by

[Mathematical Expression Omitted]

subject to

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [Delta] is a discount factor, [Chi] is the weight the Federal Reserve puts on its growth target relative to the weight it puts on its inflation target, [[Pi].sup.*] is an inflation target, and ([k.sub.t] + [Delta][Y.sup.*]) is the target rate of growth in real output at time t. We allow k and thus the Federal Reserve's growth objective to vary over time, which permits the Federal Reserve to pursue different growth rates in real output at different times in response to various pressures.

If k is i.i.d. and the Fed takes the value of the public's expected inflation as exogenous, then there are no intertemporal linkages in the problem in (3). Hence, the optimization problem in (3) can be solved as a series of one period optimization problems; at time t + j, Nature draws a [k.sub.t + j], and given [k.sub.t + j] and [Mathematical Expression Omitted], the Fed chooses the money supply growth rate that maximizes the single period loss function for period t + j. In each period t + j, the public uses the expected value of k and the reduced form for inflation that appears as a constraint in the optimization problem in (3) to determine [Mathematical Expression Omitted]. This model is quite similar to a model analyzed by Barro and Gordon [6] and, if the mean of k is greater than -[Delta][Y.sup.*], generates the standard result that there is an inflationary bias in the economy that is ultimately due to the Fed's desire to produce something other than the natural rate of output, employment, or, here, output growth. It is also quite straightforward to show that in this framework, a simple bivariate regression of the inflation rate the Fed chooses on the inflation rate expected by the public should have a zero intercept and a slope of one.

Following Cukierman and Meltzer, assume k is a serially correlated random variable,

[k.sub.t] = [Rho][k.sub.t - 1] + [[Phi].sub.t], 0 [less than] [Rho] [less than] 1, (4)

where [Phi] is an i.i.d. random variable independent of [Eta], v, and [Xi], with mean 0 and variance [Mathematical Expression Omitted]. Cukierman and Meltzer incorporate two additional readily verified facts in their model. First, information on many variables that are typically taken as the Fed's instruments (e.g., M1, M2, the monetary base) are routinely reported in the financial press with short lags. Second, the Fed cannot exercise perfect control over any of these variables. Given that recent realizations of the Fed's instruments are reported in the press, if [k.sub.t] persisted long enough and there were no control error, the public would be able to infer [k.sub.t] from its information set and the optimization problem given in equation (3). In the presence of control errors, the public confronts an inference problem in determining [k.sub.t]. We follow Cukierman and Meltzer in supposing that the money supply growth rate is only imperfectly under the Fed's control; specifically, suppose

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the Fed's choice of the money supply growth rate and [Psi] is an i.i.d. random variable distributed independently of [Eta], v, and [Xi] that follows a normal distribution with mean zero and variance [Mathematical Expression Omitted]. Further, assume [Mathematical Expression Omitted].

We formalize the public's problem as choosing the expected rate of inflation to minimize the mean squared error between expected inflation and actual inflation. That is, in each period t, each member of the public solves(11)

[Mathematical Expression Omitted].

In deriving the public's expected inflation, we diverge from the framework considered by Cukierman and Meltzer and adopt an approach taken in VanHoose and Waller [34] that permits us to draw more general conclusions. In the context of a contracting model, they consider optimal wage indexing and the social welfare that emerges from varying the information about demand and supply shocks available to the Fed and the public. For instance, it is possible that the public has forecasts of shocks that the Fed will observe before choosing the money supply growth rate. The advantages of the VanHoose-Waller approach for the present model are twofold. First, the error term in the regressions performed in the next section is generated from the resulting framework. Thus, examining the effects of granting various information sets to the Fed and the public permits an exploration of the contents of the error term of the econometric model in the next section. Second, as Kreps [28, 371-2] observes, it is not the actual timing of moves that matters in the solution to a game, but rather, what players know when they move. Thus, having the public know forecasts of the demand and supply shocks while the Fed knows the realizations of demand and supply shocks sheds light on how the solution to the problem would differ had a contracting model been used.(12) Hence, suppose that in addition to the parameters of the model, information sets of the Fed and public are given by

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted].

An information asymmetry inherent in the present model (as well as Cukierman and Meltzer's) is the Fed's knowledge of [k.sub.t]; given [k.sub.t] and [Mathematical Expression Omitted], the Fed will also be able to calculate the levels and differences of all past demand and supply shocks. It is also assumed that the Fed always knows the public's expectations concerning current shocks, but the converse need not be true. Below, we investigate three scenarios: the traditional contracting framework in which [Mathematical Expression Omitted], the traditional perfect information framework in which [Mathematical Expression Omitted], and a framework in which neither the Fed nor the public observes shocks, but the Fed and the public have different forecasts (i.e., [Mathematical Expression Omitted]). Further, we assume that under all scenarios, the information sets are common knowledge and that it is common knowledge that [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. The latter simply says that if the public has to guess the Fed's expectation of a demand or supply shock, it is common knowledge that the public's guess is its own expectation of the shock. Finally, assume that all expectations of future shocks are unbiased estimates of future shocks and that time series of differences between future shocks and expectations of future shocks are white noise processes.

Because the error term in the regression in the next section depends on the money growth rate prediction error made by the public, it is convenient to treat the public's money supply growth rate prediction error as the Fed's instrument rather than the money supply growth rate. Toward that purpose, rewrite the public's price prediction error from the previous period as

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the public's error in predicting the Fed's money supply growth choice in period t - 1, [[Zeta].sub.t - 1] [equivalent to] -[Beta][[Eta].sub.t - 1] + [v.sub.t - 1], and [Mathematical Expression Omitted]. Equation (8) decomposes the previous period's price prediction error [[Xi].sub.t - 1] into what we take to be the Fed's instrument at time t - 1, [e.sub.t - 1], the random variable [[Zeta].sub.t - 1], which by assumption satisfies [Mathematical Expression Omitted], and the random variable [[Omega].sub.t - 1], a part of the previous period's price prediction error which the public either observes at time period t (if either [Mathematical Expression Omitted], [Mathematical Expression Omitted], or [Mathematical Expression Omitted], [Mathematical Expression Omitted]), or is not in the model as a source of error (i.e., [[Omega].sub.t - 1] [equivalent to] 0, for all t if [Mathematical Expression Omitted], [Mathematical Expression Omitted]).

After substituting equation (8) in for [[Xi].sub.t - 1], taking expectations of the reduced form for inflation given in equations (2) and solving for [Mathematical Expression Omitted] yields

[Mathematical Expression Omitted].

Assume that the public believes that the Fed's money supply growth rate choice is linear in variables about which the public has information

[Mathematical Expression Omitted].

The Fed's money growth rule is linear, so that the public's expectation is rational; we proceed toward a derivation of the rule and the coefficients [a.sub.1] - [a.sub.7].

Given the information set for the public in equation (7), take expectations in equation (10) to obtain

[Mathematical Expression Omitted].

In Cukierman and Meltzer's model, [Mathematical Expression Omitted] does not include [k.sub.t - 1]; the empirical analysis of the next section follows their assumption. Hence, turn to a brief discussion of the types of information the public can bring to bear in inferring [k.sub.t].

There are at least two sources of information about the Fed's current preferences. First, for the information sets as defined above, the public observes past realizations of the money supply, [Mathematical Expression Omitted]. Given the relationship between [Delta]M and [Delta][M.sup.F] in equation (5), the public can subtract past realizations of observables in equation (9) from past realizations of [Delta]M to construct [Mathematical Expression Omitted], where [x.sub.t - 1 - j] is given by

[Mathematical Expression Omitted].

Because [Mathematical Expression Omitted] is in fact [Mathematical Expression Omitted], the sequence [Mathematical Expression Omitted] carries noisy information about [Mathematical Expression Omitted], and the serial correlation in the latter sequence makes this informative on [k.sub.t]. The approach taken by Cukierman and Meltzer is to assume that [Psi] and [Phi] follow normal distributions, develop the expectation of [k.sub.t] conditional on [Mathematical Expression Omitted], and substitute that expression into their counterpart to equation (11).(13)

Documents published by the Federal Reserve also provide information. Cukierman [17, 185] notes that the minutes of FOMC meetings along with imprecisely framed directives are available with a six week lag. The "Bluebook," or Monetary Aggregates and Money Market Conditions, and the "Greenbook," or Current Economic and Financial Conditions, both published by the Fed, are additional sources of information about the Fed's preferences. The Greenbook provides the Fed's forecasts of the inflation rate, the growth rate in output, and the unemployment rate conditional on the money supply the Fed chose for the next five quarters from the date the money supply was chosen. The Bluebook reveals the money supply growth rate the Fed chose. Although these publications are released only after a five year waiting period, because the sequence [Mathematical Expression Omitted] is serially correlated and these publications (in principle) reveal past values of elements of the sequence, they are informative on the current value in the sequence.(14)

[TABULAR DATA FOR TABLE I OMITTED]

The forgoing suggests we may proceed under the hypothesis that the public does not know [k.sub.t - 1] when it infers [k.sub.t]. Recall that [e.sub.t] was defined as the public's error in predicting the money supply growth rate and write

[Mathematical Expression Omitted].

We substitute equation (11) for [Mathematical Expression Omitted] into equation (13), and then we substitute (13) into the constraints in the programming problem in (3) for [Mathematical Expression Omitted]. Next we substitute the expression for the public's expected money supply growth rate in equation (11) into the public's inflation expectations in equation (9), and we substitute the resulting expression into the constraints in the programming problem in (3). Finally, we substitute the constraints that result from these substitutions into the objective function. This string of substitutions produces the following programming problem for the Fed:

[Mathematical Expression Omitted]

subject to [e.sub.t - 1] given and [limits of] [[Delta].sup.T][E.sub.t][U.sub.[e.sub.t + T]]([e.sub.t + T], [e.sub.t - 1 + T], [z.sub.t + T], [z.sub.t - 1 + T]) as T approaches [infinity] = 0 where U is the Fed's quadratic loss function, [U.sub.[e.sub.t + T]] is the derivative of this function with respect to [e.sub.t + T], e is the public's money growth prediction error, and z are stochastic variables from the Fed's perspective, including both the observable and unobservable demand and supply shocks, the public's forecasts of the observable demand and supply shocks, and the public's expectation of [k.sub.t]. The problem in (14) can be solved using stochastic Euler equations, as discussed by Sargent [32, 333-38]. Armed with the function for [e.sub.t] that solves (14), the coefficients [a.sub.1] - [a.sub.7] are determined by the condition that [Mathematical Expression Omitted].(15) Here we focus on a discussion of results under the assumption that the public has only [k.sub.t - 2] with which to infer [k.sub.t].

Table I expresses the solution for actual inflation as a function of the Fed's choice of inflation and the Fed's choice of inflation as a function of the public's expected inflation under alternative informational environments. As one would anticipate, the Fed's inflation forecast error consists of the Fed's forecast errors for the velocity shock, the supply shock, the growth rates in these shocks if the growth rates are not in the Fed's information set, and the monetary control error. In addition to these sources of forecast error, the public's inflation forecast error contains its error in forecasting the Fed's taste parameter k and, if the Fed and public have different information sets, differences between the public's forecasts of growth rates in shocks from the Fed's forecast of growth rates in shocks. Thus in several respects, the model is consistent with the data and casual empiricism. For example, the model implies that neither the Federal Reserve nor the public predicts inflation perfectly, nor can the public perfectly predict the Federal Reserve's choice of inflation rates. Further, neither the Federal Reserve's chosen inflation rate nor the public's expected inflation rate are constants. Additionally, as in the model of Cukierman and Meltzer, the model is consistent with persistence in actual inflation rates, Fed watching, and persistence in Fed watching.(16)

The implications of the theory outlined above that are tested in section III can also be derived from Table I. Time series of inflation, the Fed's choice of inflation, and the public's expected inflation are all integrated of order one. Therefore, if the Fed and the public behave strategically, then the Fed's choice of inflation and the public's expected inflation should be cointegrated with cointegrating vector (1, -1). Further, various models of time consistent monetary policy are nested within the process of the deviations of the Fed's choice of inflation and the public's expected inflation from their long run equilibrium. In particular, if the public does not observe the Fed's taste parameter [k.sub.t] and that parameter is serially correlated as in the Cukierman-Meltzer model and the model presented above, then, under any informational environment, the time series of residuals [Mathematical Expression Omitted] is autocorrelated. The exact form of the autocorrelation depends on the public's procedure for inferring [k.sub.t]; assuming the public observes [k.sub.t - 2] yields an MA(1) process. The estimation strategy followed below is to test the null hypothesis that the Fed's choice of inflation and the public's expected inflation are not cointegrated with cointegrating vector of (1, -1). If that null hypothesis can be rejected, then the Fed and the public behave in a manner consistent with theories of time consistent monetary policy. Further, if the null hypothesis of no cointegration can be rejected, and if the time series of residuals [Mathematical Expression Omitted] is autocorrelated, then the Cukierman-Meltzer model and the present model find support. If the time series of residuals [Mathematical Expression Omitted] is not autocorrelated, then some other model of asymmetric information finds support in the data, such as Canzoneri's model of errors by the public in forecasting money demand or Barro and Gordon's dynamic model in which the Fed's taste parameter is an independent and identically distributed random variable.

III. Estimation

The data for estimation comes from a variety of sources. The Federal Reserve's chosen inflation rate comes from the publication Current Economic and Financial Conditions or, Greenbook, described more fully by Karamouzis and Lombra [27]. This source gives the Federal Reserve's quarterly forecasts of the inflation rate, conditional on the monetary policy it chose, for a series of different lengths for different meetings, but at least five quarters starting with the quarter in which the Federal Open Market Committee met. Although the Greenbook series starts in 1968, the Fed started predicting the CPI only with the change in operating procedures after the third quarter of 1979; prior to 1979, the Fed predicted the GNP deflator. Data on the public's expected inflation rate comes from the Michigan Survey Research Center (SRC) survey, described in more detail by Juster and Comment [26]. This survey is taken quarterly, and, if respondents believe prices will increase, asks the question "By about what per cent do you expect prices to go up, on average, over the next twelve months?" To make these two data sources comparable for estimation, we use the SRC response from the quarter the Federal Open Market Committee met, and take the simple mean of the first four quarters of the Federal Reserve's forecast of inflation conditional on the monetary policy it chose as the annual inflation rate it chose.

We also examine the difference in information held by the Fed and the public. Table I indicates that differences in forecasts of demand and supply shocks, along with asymmetric information about the Fed's taste parameter, accounts for the differences between the Fed's choice of inflation and the public's inflation expectations. We use first differences in the log of the velocity of money and the log of the Bureau of Labor Statistics' wholesale price index for crude oil as proxies for first differences in demand shocks and supply shocks, respectively.(17) Clearly there are a variety of prediction methods the Fed or public could use to forecast growth rates in demand and supply shocks thus defined. To represent the difference between the public and the Fed's forecast of the growth rates of shocks, we use the residuals from AR(1) estimation of actual growth rates in the shocks. Among the variety of methods to construct the series of differences between the Fed's and public's estimates of growth rates in shocks, series constructed in this manner have the virtue of a simple, intuitive interpretation: the Fed knows the value of the current growth rate in the shocks, while the public forecasts the growth rates in shocks by supposing the growth rate follows a first order autoregressive process.

[TABULAR DATA FOR TABLE II OMITTED]

Table III. Sample Statistics, 1979Q4-1988Q4

Variable         Mean    Standard Deviation    Minimum    Maximum

[Pi]             4.748         2.538            1.336     11.812
[[Pi].sup.F]     5.900         2.929            2.250     12.075
[[Pi].sup.e]     5.625         1.991            3.600     11.900
[Delta]S        -0.008         0.136           -0.498      0.373
[Delta]D        -0.001         0.013           -0.032      0.025

Base data series used to construct each series are summarized in Table II. The sample on the Fed's choice of inflation starts with a Federal Open Market Committee meeting in the fourth quarter of 1979 and concludes with a meeting in the last quarter of 1988, and thus contains thirty-seven observations that cover a variety of inflation experience. The SRC survey data runs from the first quarter of 1966 to the fourth quarter of 1989, and our inflation data runs from the first quarter of 1948 to the last quarter of 1992.

Sample means, standard deviations, minima, and maxima for fourth quarter of 1979 to the fourth quarter of 1988 for the variables used are given in Table III. Both the Federal Reserve and the public typically over-predict inflation in the sample. Note also the extreme variation of actual inflation; the variance of the inflation rate is greater than its mean. Finally, observe that wholesale domestic crude oil prices in the sample decreased at an average annual rate of 3.2%, while velocity decreased at an average annual rate of 0.4%.

Table IV presents results from Augmented Dickey-Fuller (ADF) tests in which first differences of each series of inflation and inflation expectations is regressed on the lag of the level [TABULAR DATA FOR TABLE IV OMITTED] of the series, and enough lags of first differences to eliminate serial correlation from the error (standard errors underneath parameter estimates). The ADF [Tau] statistic is the t-statistic for the null hypothesis that the estimated coefficient on the lagged level of each series is not different than zero. For a regression with a constant, the critical values for rejecting the null hypothesis of a unit root for a sample size of 25 are -2.62 at the 10% level of significance, -3.00 at the 5% level of significance, and -3.33 at the 2.5% level of significance; for a sample size of 50, the same critical values are -2.60, -2.93, and -3.22, and for a sample size of 100, -2.58, -2.89, and -3.17. Given our [Tau] statistics, we cannot reject the null hypothesis that all three series have unit roots at the 10% level of significance.(18)

The finding that the Fed's choice of inflation and the public's inflation expectations have unit mots implies that simple regression of the Fed's choice of inflation on the public's inflation expectations can yield spurious regression results, as discussed by Granger and Newbold [23]. In the presence of variables with unit roots and with prior beliefs generated by our model that (1, -1) is a cointegrating vector for the two series, the correct strategy to test for agents' strategic awareness is to determine if the time series of residuals [Mathematical Expression Omitted] is stationary. To test the null hypothesis that a pair of time series is not cointegrated with cointegrating vector (1, -1), we use the ADF test on the series of simple differences of the two series to determine if that time series of residuals is stationary. Table V presents tests of the null hypotheses that (1, -1) is not a cointegrating vector for each pair of series among actual inflation, Fed's choice of inflation, and public's expected inflation. For the same critical values of the [Tau] statistic as for Table IV, we can reject the null hypothesis that (1, -1) is not a cointegrating vector for the series of Fed's choice of inflation and the public's inflation expectation at a level of significance between 2.5% and 1%. The null hypothesis that (1, -1) is not a cointegrating vector for actual inflation and the public's expected inflation can be rejected at about the 5% level of significance. However, one cannot reject the null hypothesis that (1, -1) is not a cointegrating vector for the series of actual inflation and the Fed's choice of inflation at the 10% level of significance.(19)

The finding that (1, -1) is a cointegrating vector for the Fed's choice of inflation and the public's inflation expectation supports theories of time consistent monetary policy because it implies that the time series of simple differences in the Fed's choice of inflation and the public's expected inflation are stationary series, The finding that (1, -1) is a cointegrating vector for actual inflation and the public's inflation expectations also supports theories of time consistent monetary policy because the public is assumed to forecast inflation rationally. The finding that (1, -1) is not a cointegrating vector for actual inflation and the Fed's choice of inflation is inconsistent with theories of time consistent monetary policy inasmuch as in those models, actual inflation is determined by the Fed's choice of inflation plus a stationary error. However, the latter result may be due to the low power of the test that stems from our small sample size and the low power of ADF tests. Note also that theory implies that the constant in the regression should be zero; if a zero constant is imposed a priori, then one can reject the null hypothesis that (1, -1) is not a cointegrating vector for inflation and the Fed's forecast of inflation at the 5% level of significance.

To discriminate among models of time consistent monetary policy, recall that the Cukierman-Meltzer [TABULAR DATA FOR TABLE V OMITTED] model and the model presented above imply that the time series of residuals [Mathematical Expression Omitted] is serially correlated. The regression of [Mathematical Expression Omitted] on a constant yields a Durbin-Watson statistic of .442, which is evidence of positive serial correlation that supports the Cukierman-Meltzer model and the model presented above (other regression results not reported). If the public observes the Fed's taste parameter with a one period lag as in the model of section II, then the time series [Mathematical Expression Omitted] follows an MA(1) process and is determined by differences between the Fed's and public's forecasts of growth rates in demand and supply shocks. Table VI presents estimates from the regression of [Mathematical Expression Omitted] on the current quarter's forecast error and one quarter lagged forecast error for first differences in velocity and oil price shocks, where the forecast error is derived from separate AR(1) regressions of first differences in the log of velocity and the log of the crude petroleum price difference. We estimate the regression in Table VI supposing that the time series [Mathematical Expression Omitted] follows an MA(1) process by minimizing the sum of squared residuals among OLS regressions conditional on values of the parameter on the lagged error in the interval [-1, 1].(20) Coefficient estimates in Table VI indicate that among the quarterly forecast errors, the [TABULAR DATA FOR TABLE VI OMITTED] only error which has an estimated coefficient significantly different than zero is the current quarter's forecast error for the growth rate in crude petroleum prices. An interpretation consistent with the model of section II is that the Fed and the public share the same information sets with respect to velocity, but not with respect to petroleum prices. Thus the difference between the Fed's choice of inflation and the public's expected inflation is explained in part by their different information with respect to crude petroleum price shocks. Observe that the Durbin-Watson statistic implies that, once the MA(1) error is accounted for, we can reject the null hypothesis of positive serial correlation in the errors at the 1% level. The implication that the errors of the regression in Table VI are not serially correlated provides additional support for the model of section II. Results here are qualified somewhat by our small sample size and the proxies we use for demand and supply shocks.

Because of asymmetric information concerning the value of the serially correlated taste parameter k, the model of section II also implies that if the public's inflation expectations and the Fed's inflation choice were for the next quarter's inflation rate rather than the next year's inflation rate, then the time series [Mathematical Expression Omitted] would not be serially correlated, while the time series [Mathematical Expression Omitted] would be serially correlated. However, Bryan and Gavin [12] observe that the difference in the length of the forecast horizon (annual) and the sampling frequency (quarterly) will induce serial correlation in the public's forecast error for inflation. Similarly, changes in the Fed's serially correlated taste parameter from quarter to quarter will cause it to choose a series of inflation rates that are serially correlated, and thereby induce serial correlation in its forecast of annual inflation. To avoid the problem of induced serial correlation, we use data for which the forecast horizon matches the sampling frequency. For the Fed, we subtracted the forecast made at the last meeting of the FOMC in quarter t for the annualized quarter t + 1 inflation rate from the annualized inflation rate for quarter t + 1. Table VII presents the results of regressing these residuals on a constant and includes tests of the null hypothesis of no serial correlation for lags up to four quarters using the method developed by Breusch [11] and Godfrey [24]. Although the Fed's forecast errors appear to be independent over time, one can reject the null hypothesis that the Fed's one quarter ahead forecast of inflation is an unbiased prediction of actual inflation at the .8% level.

[TABULAR DATA FOR TABLE VII OMITTED]

To investigate whether the public's inflation forecast errors are serially correlated, we split the quarterly observations from SRC survey over the 1966 to 1989 period into four samples such that each sample contains forecasts of annual inflation taken from the same quarter of each year. Table VIII presents the results of regressing the forecast errors of each of these four samples on a constant, and includes tests of the null hypothesis of no serial correlation for lags of up to two years using the method developed by Breusch and Godfrey. As implied by theories of time consistent monetary policy, we would have to accept the null hypothesis that the public's prediction of inflation is unbiased for each of the samples. However, as implied by the theory of section II, in three of the four regressions, we can reject the null hypothesis of no serial correlation at the 10% level or less.

Summarizing our findings, a result consistent with all theories of time consistent monetary policy is that we can reject the null hypothesis that the public's inflation expectations are not cointegrated with cointegrating vector (1, -1) with both inflation and the Fed's choice of inflation. Evidence of serial correlation in the public's inflation forecast errors, serial independence in the Fed's inflation forecast errors, and serial correlation in the difference of the Fed's and public's forecasts of inflation support the Cukierman-Meltzer model and the model of section II. Two findings are inconsistent with any of the theories of time consistent monetary policy: the Fed's prediction of quarterly inflation from the first quarter of 1980 to the first quarter of 1989 is biased, and we cannot reject the null hypothesis that (1, -1) is not a cointegrating vector for the Fed's choice of annual inflation from the fourth quarter of 1980 to the fourth quarter of 1988 and inflation over the same period. However, the public's forecast error for inflation over the fourth quarter of 1979 to the fourth quarter of 1988 is also biased, which suggests unanticipated demand and supply shocks made inflation difficult to predict unbiasedly over that short sample period.

IV. Conclusion

In commenting on the papers in the empirical literature that tests the credibility hypothesis (discussed in section I), Blackburn and Christensen state "It is tempting to regard the foregoing empirical evidence on credibility as a rather damning critique of the prolific theoretical literature" [9, 40]. While the credibility hypothesis is not supported by the prediction error models they [TABULAR DATA FOR TABLE VIII OMITTED] review, the theoretical literature on time consistent monetary policy, including the present model, does not have the credibility hypothesis as an implication. The support we find for the analysis of Cukierman and Meltzer and the previous empirical tests of the credibility hypothesis are consistent and, we believe, impressive evidence of strategic interaction in monetary policy. Further investigation into strategic interaction in monetary policy is clearly warranted, and several areas suggest themselves.

Direct estimation of the Euler equation from the Fed's optimization problem using one or more of what are taken to be the instruments of monetary policy rather than the Fed's inflation choice as a dependent variable could provide a further test of the theory. Additionally, the implications of the adoption of different procedures by the public to infer the Fed's tastes on the precise form of the autocorrelation in the public's error in predicting the Fed's choice of inflation can be investigated. Also, by introducing the observable variable of the public's expected inflation, the model presented above makes it feasible to empirically test many of the theories discussed by Cukierman [17]. We are not aware of the availability of predictions of inflation by monetary authorities and citizenries of countries other than the U.S. but, to the extent that such data is available, any of these analyses could be replicated for other countries.

Additional theoretical applications of the Cukierman and Meltzer model can also be suggested. Several papers in the time consistent monetary policy literature stem from seminal game-theoretic ideas that first appeared in industrial organization as models of collusion. While we have striven to present the Cukierman and Meltzer model as following the literature that preceded it, the framework they consider more naturally extends to regulatory environments rather than collusive environments. For example, a regulatory agency may have a budget that is partially under its discretion but determined by Congress, and may have to respond to changing political pressures in allocating its resources among uses (e.g., enforcement, research, etc.) that affect firms in the regulated industry.

The authors thank Normand Bernard of the Federal Reserve Bank staff, William Hutchinson, Maureen Lage, Nicholas Noble, and Michael Treglia for making the data available. Additionally, we thank David Bivin, William Even, Antonio Ligeralde, Nicholas Noble, and Prosper Raynold for helpful conversations and an anonymous referee for especially useful comments. Finally, we would like to express special thanks to George Davis for numerous conversations which clarified our thoughts and substantially improved our exposition. Errors of ommission or commission remain those of the authors.

1. Along with an apology for omissions due to space constraints, note that several excellent surveys of the literature are available. Reviews by Rogoff [31], Blackburn and Christensen [9], and especially Cukierman [17] contain thoughtful discussions of results and modeling issues.

2. While the ideas that motivate the dynamic Barro and Gordon model are plausible, a serious technical problem remains. As various Folk Theorems indicate will generally be the case for repeated games, an infinite number of equilibria exist. In the Barro and Gordon [6] model, each equilibrium is generated by varying the length of time over which the monetary authority loses credibility if it creates surprise inflation. In the context of a public that consists of atomistic agents, it is difficult to understand how a particular equilibrium would be selected. An unexplored possibility is that deterrence might occur without coordination among the public, based on some summary statistic of the distribution of the number of periods for which the monetary authority loses credibility among members of the public.

3. By asymmetric information, we mean information on variables that the Fed currently observes which become observable to the public, such as the most recent forecast of GNP; by private information, we mean variables that the public may never observe, such as the Fed's current marginal rate of substitution between inflation and unemployment.

4. Note that Canzoneri's model suffers from the same coordination problem for the public as Barro and Gordon's model with respect to the length of time for the loss of credibility. Additionally, note that in the industrial organization literature on collusion from which these monetary policy games are arbitraged, the non-stochastic version of Barro and Gordon's model corresponds to Friedman's [22] paper on non-cooperative collusion, the more realistic stochastic model supplied by Canzoneri's model corresponds to Green and Porter's [25] model of collusion under imperfect monitoring, and the optimal punishment strategy for the Green and Porter model is given by Abreu, Pearce, and Stachetti [1]. The latter is relevant because Abreu, Pearce and Stachetti show that rather than the punishment episodes of a fixed length implied by the Green and Porter model and in Canzoneri's model, under the optimal punishment strategy, the regimes in the Green and Porter model follow a first order Markov process, which is more plausible than high inflation regimes of fixed length.

5. See Cukierman [17, 309-32] for an excellent discussion of several additional papers in this literature.

6. Ball [5] provides an infinite horizon model similar to Cukierman and Meltzer's in which the policymaker is either "wet" or "hard nosed" as in Backus and Driffill [3]. The type of the policymaker currently in power is unknown to the public, and follows an exogenously given Poisson process. His analysis has implications similar to the Cukierman and Meltzer model.

7. To make credibility an observable, define it as the difference between the inflation rate the Fed chooses and the inflation rate expected by the public. We take this approach in section II to generate the empirical analysis of section III.

8. We emphasize that one can obtain a model similar to Cukierman and Meltzer's from the model presented below. Because output and inflation depend on the money supply growth rate the Fed chooses, the objective function can be written in terms of the money supply growth rate. Further, since the Fed's choice of the money supply growth rate is both how it influences economic outcomes and how the public infers the Fed's tastes, the public's inflation expectations necessarily depend on its expectations concerning the money supply growth rate. The reason for recasting their model in traditional terms is that the available data is for the public's expected inflation, and not for its expected money supply growth rate.

9. The hypotheses tests of the next section are carried out on the inflation rate, the Fed's inflation choice, and the public's inflation expectations. Given that hypotheses are tested on nominal variables, either the Lucas supply curve or a contracting model could be used to generate the model of time consistent monetary policy in the present section. Without taking a stance on which is more realistic, we choose the Lucas supply curve framework because it involves less algebra.

10. Solving the Fed's dynamic programming problem requires taking expectations of future values of the growth rates of these variables. Positing that the expected future levels of the variables are constants implies growth rates of zero. Alternatively, we could suppose that the future growth rates of these variables is expected to be constant by both the Fed and the public. Then discounted sums of these constant growth rates would appear in the Fed's Euler equation, and also as a constant term in the public's inflation expectations. The assumption of constant levels and zero growth rates is made without additional loss of generality in the rest of our conclusions in order to economize on space.

11. As a referee pointed out in response to an earlier version, it is important to treat the public as a set of atomistic agents solving the same problem. Additionally, as Cukierman [17, 34-5] notes, one must be careful in using the Lucas supply curve formulation for time consistent monetary policy that there is a reason in the model for the public to be confused about the money growth rate, or else the Phillips curve will be vertical. In the model presented below, uncertainty over the Fed's tastes explicitly leads to the public's confusion between nominal and real shocks. Ideally, one should start from a representative agent's problem in disentangling real and nominal sources of price movements in developing a model of time consistent monetary policy based on the Lucas supply curve. Such an approach lies outside the scope of the present paper; we take the parameter [Alpha] in equations (1) to be given when (6) is solved.

12. Of course, a feature of the contracting approach is that the lapse of time between when the public chooses its wage and the monetary authority chooses its control provides a ready rationale for the difference in the information sets of the Fed and public.

13. In the present setting, if one adopts their approach, the presence of {[[Psi].sub.t - 1 - j], [Delta][[Eta].sub.t - 1 - j], [Delta][v.sub.t - 1 - j]} in the reduced form equations for inflation and the growth rate in output ensure that past values of all other observable variables are uninformative on [k.sub.t] for the public.

14. In Cukierman and Meltzer's analysis, [Mathematical Expression Omitted] is a linear function of all past values of x. Recognizing that there is a longer lag in the release of the Greenbook and Bluebook than in the reporting of money supply growth rates in the financial press, if we include say, [k.sub.t - 2] in [Mathematical Expression Omitted] and assume that [Phi] is i.i.d. normal, then [Mathematical Expression Omitted] turns out to be a linear function of [k.sub.t - 2] and [Delta][M.sub.t - 1], where the coefficients of the linear function are a convex combination of the variance in the control error and [Rho] times the variance of serial correlation error [Phi]. Yet the fact that we do not observe the public anxiously awaiting the release of the Greenbook and Bluebook suggests that the combination of the length of period over which k is constant and the value of the serial correlation coefficient may be such that these publications are too dated to be of much value in predicting the current value of k.

15. The cumbersome but straightforward mathematics that solves the problem in (14) is available from the authors by request.

16. Persistence in Fed watching would not occur if either (i) k is a constant or (ii) k is not serially correlated. Even if [k.sub.t - 2] were not observed, repeated observation of the money supply growth rate would allow precise enough estimation of the constant k in ease (i) and the mean of the distribution of k in case (ii) so that the expenditure of resources to obtain more precise estimates would eventually fail to satisfy marginal benefit-marginal cost considerations.

17. We used nominal GNP divided by the nominal M2 money supply as a measure of velocity. Cointegration tests provided in Friedman and Kuttner [21] indicate that the M2 velocity has been stationary through the 1980's, while the M1 velocity has not.

18. We followed a hypothesis testing strategy advocated by Enders [20, 227] for quarterly observations; start with twelve lagged first differences, and drop lags 9 through 12 if an F-test indicates that coefficients on these lags are not significantly different than zero and Durbin h statistics indicated that the errors in the regression with 8 lagged first differences were not serially correlated. Even if the regressions were run with eight lags of first differences, the hypothesis tests with or without a constant included indicate that the null hypothesis of a unit root cannot be rejected at the 10% level of significance with one exception. The exception is that the null hypothesis that the Fed's choice of inflation has a unit root can be rejected at the 1% level if one uses 8 lagged first differences and includes a constant. However, the F-test for the regression of the Fed's choice of inflation on twelve lagged first differences, lagged level, and a constant indicates that the joint null hypothesis that the coefficients on the lagged differences 9 through 12 are equal to zero can be rejected at a level of significance of 1%. An additional problem indicated in Table IV is that the regression errors for that regression may be serially correlated. Since we would have only twenty degrees of freedom if we included another four lags, we did not estimate a model with sixteen lagged first differences.

19. The hypothesis testing strategy used to obtain estimates for Table V is the same as described in footnote 18, supra.

20. The estimation technique is provided by Davidson and MacKinnon [19, 355-56]. They show that if one includes an initial observation defined as the error that starts the process (and with the regressand and regressors set equal to 0), the regressors for subsequent periods can be recursively defined in a manner such that minimizing the sum of squared residuals produced by OLS in a grid search over [-1, 1] for the moving average parameter yields consistent estimates of all parameters. The procedure introduces one new observation, the observation that starts the error process, and one new parameter to estimate, the error that starts the error process, and hence has the same number of degrees of freedom.

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