A. Background

The analytical frameworks that have been developed for addressing monetary policy issues in an open economy traditionally exhibit the monetary policy transmission mechanism depicted in Figure 1. The authorities control a short-term nominal interest rate (rs) with the objective of influencing the rates of inflation (π) and unemployment (u). As shown by the arrows, changes in the policy instrument are transmitted to the policy target variables through several channels. Adjustments in the nominal interest rate can trigger movements in the nominal exchange rate (s), which are transmitted fairly directly to tradable goods prices and inflation and indirectly to unemployment through their effects on the real exchange rate (z) and the gap (y) between actual and potential domestic output. Changes in the nominal interest rate also affect the real interest rate (rs -π^e), both directly and through the response of inflation expectations (π^e); changes in the real interest rate in turn influence unemployment through their effects on aggregate demand and the domestic output gap; and changes in the output gap and unemployment rate influence the inflation rate through channels summarized by the Phillips curve. In addition, an important two-way feedback mechanism is superimposed on the transmission process, with inflation expectations responding (inter alia) to the history of inflation, and with inflation influenced in turn by changes in inflation expectations.

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The Monetary Policy Transmission Mechanism

Citation: IMF Working Papers 2001, 007; 10.5089/9781451842418.001.A001

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Figure 1 does not show any feedback from the policy target variables to the policy instrument. The tasks of identifying and implementing a mechanism for reacting to economic developments in a manner that is conducive to macroeconomic stability is the responsibility of the monetary authorities. In particular, the role of monetary policy is to adjust the policy instrument variable (in this case the nominal interest rate) in reaction to observed and anticipated changes in unemployment, inflation, and other macroeconomic variables, taking account of the behavioral relationships among these variables.

In reality, the operation of monetary policy is greatly complicated by two types of uncertainties: imperfect information about the magnitudes of the various transmission effects shown in the diagram, and difficulties in identifying the effects on macroeconomic variables of various types of economic shocks. In principle, there can be exogenous shocks that directly affect the exchange rate, the observed inflation rate, or the expected inflation rate; and there can be exogenous shifts in the output gap associated with shocks to either aggregate demand or potential aggregate supply.

The operation of monetary policy is also complicated by the fact that policy credibility is imperfect and can vary with the effectiveness of the monetary authorities in achieving desirable outcomes for policy target variables. The endogenous behavior of policy credibility and its role in the monetary policy transmission mechanism has not yet been adequately incorporated into the models that have been used to analyze monetary policy issues.

To motivate our interest in exploring the relevance of endogenous policy credibility for the design of monetary policy strategies, Figure 2 plots quarterly data on selected economic indicators for Australia and the United States. The top two panels show recorded inflation rates and survey measures of inflation expectations. In the United States, inflation has been generally subdued and trendless since the early 1980s, after declining sharply during 1980-82 in the context of a relatively tight monetary policy. In Australia, inflation declined more gradually from the levels experienced during the 1970s and has been generally subdued and trendless only since the early 1990s. Survey measures of Australian inflation expectations have remained persistently above recorded inflation rates during the past decade, while survey measures of U.S. inflation expectations have tracked recorded inflation rates fairly closely. Moreover, with similar recorded inflation rates in Australia and the United States during the 1990s, long-term Australian government bonds have required an interest premium relative to the yield on U.S. government bonds, as shown in the third panel; this is consistent with the expected inflation differential. And finally, despite experiencing similar and fairly stable rates of inflation during the 1990s, the two countries have significantly different unemployment rates, as indicated in the bottom panel.

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Selected Economic Indicators for Australia and the United States

Citation: IMF Working Papers 2001, 007; 10.5089/9781451842418.001.A001

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What can explain why these variables have behaved differently in Australia and the United States? One plausible explanation is that the relatively slow decline and small-sample bias of inflation expectations in Australia, compared with the rapid decline and relative unbiasedness of inflation expectations in the United States, is a reflection of imperfect policy credibility that differs across countries. Imperfect policy credibility may not be the entire explanation, but it is a leading candidate that warrants explicit consideration in modeling the monetary policy transmission mechanism.

B. The Basic Model, Part I

For purposes of this paper, it is convenient to use the small open-economy model developed by Callen and Laxton (1998), which was estimated using quarterly data for Australia.¹⁹ Table 1 summarizes the assumed behavior of domestic demand, exports, imports, the unemployment rate, the nominal exchange rate, and the rate of inflation. Table 2 describes the treatment of inflation expectations and policy credibility. The variables in the equations are measured quarterly.

Table 1.

A Small Model of the Inflation-Unemployment Process, Part 1

Table 1.

A Small Model of the Inflation-Unemployment Process, Part 1

Domestic demand:
$\begin{array}{ccc}{d}_t=-0.33r_{t-2}+8.40a_t+0.95d_{t-1}+{\epsilon }_t^d& & \left(1\right)\end{array}$
Exports:
$\begin{array}{ccc}{x}_t=0.06z_t+0.48x_{t-1}+{\epsilon }_t^x& & \left(2\right)\end{array}$
Imports:
$\begin{array}{ccc}{m}_t=0.33x_t+1.35d_t-0.09z_t+0.41m_{t-1}+{\epsilon }_t^m& & \left(3\right)\end{array}$
Labor market tightness:
$\begin{array}{ccc}\left({\overline{u}}_t-u_t\right)=0.20y_t+0.77({\overline{u}}_{t-1}-u_{t-1})+{\epsilon }_t^{\left(\overline{u}-u\right)}& & \left(4\right)\end{array}$
Exchange rate:
$\begin{array}{ccc}{S}_t{\left[1+r{s}_t-r{s}_t^{US}\right]}^{.25}=0.38s_{t+1}^{mc}+\left(1-0.38\right)\lfloor s_{t-1}{\left(1+E_t\pi 4_{t+4}-E_t\pi 4_{t+4}^{US}\right)}^{.5}\rfloor +{\epsilon }_t^s& & \left(5\right)\end{array}$
Phillips curve:
$\begin{array}{ccc}{\pi }_t={\pi }_t^c+2.14\frac{\left[u_t^{*}-u_t\right]}{\left[u_t-{\Phi }_t\right]}+{\epsilon }_t^{\pi }& & \left(6\right)\end{array}$
	where
$\begin{array}{ccc}{\pi }_t^c=0.73{\overline{\pi }}_t^e+\left(1-0.73-0.04-0.02\right){\pi }_{t-1}+0.04{\pi }_t^m+0.02{\pi }_{t-1}^m& & \left(7\right)\end{array}$
	and
$\begin{array}{ccc}{\overline{\pi }}_t^e=0.25[E_{t-1}\pi 4_{t+3}+E_{t-2}\pi 4_{t+2}+E_{t-3}\pi 4_{t+1}+E_{t-4}\pi 4_t]& & \left(8\right)\end{array}$

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Table 1.

A Small Model of the Inflation-Unemployment Process, Part 1

Domestic demand:
$\begin{array}{ccc}{d}_t=-0.33r_{t-2}+8.40a_t+0.95d_{t-1}+{\epsilon }_t^d& & \left(1\right)\end{array}$
Exports:
$\begin{array}{ccc}{x}_t=0.06z_t+0.48x_{t-1}+{\epsilon }_t^x& & \left(2\right)\end{array}$
Imports:
$\begin{array}{ccc}{m}_t=0.33x_t+1.35d_t-0.09z_t+0.41m_{t-1}+{\epsilon }_t^m& & \left(3\right)\end{array}$
Labor market tightness:
$\begin{array}{ccc}\left({\overline{u}}_t-u_t\right)=0.20y_t+0.77({\overline{u}}_{t-1}-u_{t-1})+{\epsilon }_t^{\left(\overline{u}-u\right)}& & \left(4\right)\end{array}$
Exchange rate:
$\begin{array}{ccc}{S}_t{\left[1+r{s}_t-r{s}_t^{US}\right]}^{.25}=0.38s_{t+1}^{mc}+\left(1-0.38\right)\lfloor s_{t-1}{\left(1+E_t\pi 4_{t+4}-E_t\pi 4_{t+4}^{US}\right)}^{.5}\rfloor +{\epsilon }_t^s& & \left(5\right)\end{array}$
Phillips curve:
$\begin{array}{ccc}{\pi }_t={\pi }_t^c+2.14\frac{\left[u_t^{*}-u_t\right]}{\left[u_t-{\Phi }_t\right]}+{\epsilon }_t^{\pi }& & \left(6\right)\end{array}$
	where
$\begin{array}{ccc}{\pi }_t^c=0.73{\overline{\pi }}_t^e+\left(1-0.73-0.04-0.02\right){\pi }_{t-1}+0.04{\pi }_t^m+0.02{\pi }_{t-1}^m& & \left(7\right)\end{array}$
	and
$\begin{array}{ccc}{\overline{\pi }}_t^e=0.25[E_{t-1}\pi 4_{t+3}+E_{t-2}\pi 4_{t+2}+E_{t-3}\pi 4_{t+1}+E_{t-4}\pi 4_t]& & \left(8\right)\end{array}$

Table 2.

A Small Model of the Inflation-Unemployment Process: Part II, Inflation Expectations and Policy Credibility

Table 2.

A Small Model of the Inflation-Unemployment Process: Part II, Inflation Expectations and Policy Credibility

Inflation expectations 1 year ahead:
$\begin{array}{ccc}{E}_t{\pi 4}_{t+4}=0.13\pi 4_t^{mc}+\left(1-0.13\right)E_{t-1}\pi 4_{t+3}+b_t+{\epsilon }^{{\pi }^e}& & \left(9\right)\end{array}$
Low inflation forecasting rule:
$\begin{array}{ccc}\pi 4_t^L=2.5\left(1-0.34\right)+0.34\pi 4_{t-1}+{\epsilon }_t^L& & \left(10\right)\end{array}$
High inflation forecasting rule:
$\begin{array}{ccc}\pi 4_t^H=8.4\left(1-0.81\right)+0.81\pi 4_{t-1}+{\epsilon }_t^H& & \left(11\right)\end{array}$
Expectations bias attributable to imperfect policy credibility:
$\begin{array}{ccc}{b}_t=0.04\left[c_t{E}_t\left\{\pi 4_{t+4}^L\right|\pi 4_t\}+\left(1-c_t\right)E_t\{\pi 4_{t+4}^H\left|\pi 4_t\right\}-{\pi }^{TAR}\right]& & \left(12\right)\end{array}$
Policy credibility
$\begin{array}{ccc}{c}_t=0.83c_{t-1}+0.17{\psi }_{t-1}+{\epsilon }_t^c& & \left(\mathrm{13a}\right)\end{array}$
	where
$\begin{array}{ccc}{\Psi }_t=\frac{{\left[{\epsilon }_t^H\right]}^2}{{\left[{\epsilon }_t^H\right]}^2+{\left[{\epsilon }_t^L\right]}^2}& & \left(\mathrm{13b}\right)\end{array}$

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Table 2.

A Small Model of the Inflation-Unemployment Process: Part II, Inflation Expectations and Policy Credibility

Inflation expectations 1 year ahead:
$\begin{array}{ccc}{E}_t{\pi 4}_{t+4}=0.13\pi 4_t^{mc}+\left(1-0.13\right)E_{t-1}\pi 4_{t+3}+b_t+{\epsilon }^{{\pi }^e}& & \left(9\right)\end{array}$
Low inflation forecasting rule:
$\begin{array}{ccc}\pi 4_t^L=2.5\left(1-0.34\right)+0.34\pi 4_{t-1}+{\epsilon }_t^L& & \left(10\right)\end{array}$
High inflation forecasting rule:
$\begin{array}{ccc}\pi 4_t^H=8.4\left(1-0.81\right)+0.81\pi 4_{t-1}+{\epsilon }_t^H& & \left(11\right)\end{array}$
Expectations bias attributable to imperfect policy credibility:
$\begin{array}{ccc}{b}_t=0.04\left[c_t{E}_t\left\{\pi 4_{t+4}^L\right|\pi 4_t\}+\left(1-c_t\right)E_t\{\pi 4_{t+4}^H\left|\pi 4_t\right\}-{\pi }^{TAR}\right]& & \left(12\right)\end{array}$
Policy credibility
$\begin{array}{ccc}{c}_t=0.83c_{t-1}+0.17{\psi }_{t-1}+{\epsilon }_t^c& & \left(\mathrm{13a}\right)\end{array}$
	where
$\begin{array}{ccc}{\Psi }_t=\frac{{\left[{\epsilon }_t^H\right]}^2}{{\left[{\epsilon }_t^H\right]}^2+{\left[{\epsilon }_t^L\right]}^2}& & \left(\mathrm{13b}\right)\end{array}$

The conceptual underpinnings of the equations in Table 1 are relatively familiar; here we review them briefly and refer readers to Callen and Laxton (1998) for additional discussion and econometric details. The top three equations focus on components of the national income accounts, where domestic demand represents the aggregate domestic expenditures of consumers, investors, and governmental units. The dependent variables refer to detrended measures of domestic demand, exports, and imports.²⁰ The estimated equations suggest that aggregate (detrended) domestic demand (d) exhibits a high degree of persistence (a weight of nearly unity on the lagged dependent variable); it also depends negatively on the (two-quarter lagged) level of the real interest rate (r) and positively on a relevant measure of national wealth (a)—namely, the stock of net claims on the rest of the world (as a percent of GDP). Exports (x) exhibit a substantial degree of persistence (a coefficient of 0.48 on the lagged dependent variable) and also depend importantly on the real exchange rate (z, an increase in which represents a real depreciation). Imports (m) likewise exhibit substantial persistence and depend in a conventional manner on domestic demand, exports, and the real exchange rate. Together, the equations for detrended domestic demand, exports, and imports describe the behavior of aggregate output relative to trend, which is used as a measure of the output gap (y, defined as actual output minus potential output). An important property of these equations is that there are significant lags between changes in real monetary conditions—that is, the real interest rate and the real exchange rate—and the output gap.

The fourth equation in Table 1 describes the behavior of a labor market tightness measure—defined as the amount by which the unemployment rate (u) falls below the NAIRU $\left(\overline{u}\right)$.²¹ The estimated relationship shows a high degree of persistence along with positive dependence on the output gap. The coefficient on the output gap is significantly less than one, which is consistent with casual empiricism: firms hoard labor over the business cycle and, because of hiring and firing costs, adjust labor inputs slowly in response to changes in the demand for their products.

In the exchange rate equation, the left-hand side variable represents the one-period forward exchange rate, which, under covered interest parity, equals the prevailing spot exchange rate appropriately adjusted for the interest rate differential.²² The right-hand side of the equation is a standard backward- and forward-looking components model of the expected future spot exchange rate $(S_{t+1}^e)$, where $S_{t+1}^{mc}$ represents the model-consistent solution and the backward-looking component is simply the lagged spot rate adjusted for the expected inflation differential.²³ The measures of inflation expectations are based on surveys of the rates of inflation expected in Australia and the United States over the year ahead. We use the notation π4 to refer to inflation over four quarters and π to refer to annualized rates of inflation over one quarter; E_tπ4_t+1 is the public’s expectation in quarter t of the rate of inflation over the year through quarter t+4. In estimating equation (5), Callen and Laxton (1998) found a large weight on the backward-looking component, consistent with other evidence that high proportions of the short-run behavior of exchange rates (and other asset prices) cannot be explained by fundamentals, but rather are conditioned by the recent behavior of exchange rates (or other asset prices).²⁴

The Phillips curve described by equation (6) embodies the convex specification analyzed by Debelle and Laxton (1997) and Debelle and Vickery (1997).²⁵ In the spirit of analyzing the robustness of our simple policy rules, we give equal consideration to linear variants of the model, in which the convex term in the unemployment rate is replaced by a linear approximation, as described in Section IV below.²⁶ For expositional convenience, equation (6) consolidates several terms into a composite “core” rate of inflation (π^c), which is spelled out in equation (7). The first two terms on the right hand side of equation (7) describe the rate of inflation expected over the next quarter. Expectations are assumed to influence inflation dynamics through a wage-and-price setting framework in which the standard contract has a four-quarter horizon, with one-fourth of the contracts respecified every quarter. This is reflected in the ${\overline{\pi }}^e$ term, which is defined in equation (8) as the average of the one-year-ahead inflation expectations that economic agents held during the four quarters in which currently-prevailing contracts were written. Expected inflation is also assumed to depend on the lagged inflation rate (the second term on the right hand side of equation (7)), which can be viewed as a summary indicator of the incentives to incur the costs of revising price or wage contracts before their specified expiration dates. The third and fourth terms on the right hand side capture the influence on consumer price inflation of the contemporaneous and one-quarter-lagged rates of change in import prices (π^m).

Note that the coefficients in equation (7) sum to unity, consistent with the long-run natural rate hypothesis. Figure 3 plots the difference between observed inflation and core inflation (vertical axis) against the unemployment rate (horizontal axis). For purposes of the discussion here, we make the simplifying assumption that core inflation and expected inflation coincide, so that the figure can be viewed as an expectations-augmented Phillips curve. Consistent with the specification in equation (6), the short-run Phillips curve is convex with vertical asymptote at u = Ф and horizontal asymptote at π - π^c = - γ.²⁷ The magnitude of u^* corresponds to the unemployment rate at which actual inflation and expected inflation coincide, such that there would be no systematic pressure for inflation to rise or fall in the absence of stochastic shocks. This corresponds to the non-accelerating-inflation rate of unemployment in a deterministic world. We refer to u^* as the DNAIRU.

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The Convex Phillips Curve

Citation: IMF Working Papers 2001, 007; 10.5089/9781451842418.001.A001

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An important point is that the DNAIRU is not a feasible stable equilibrium in a stochastic world with a convex Phillips curve. The average rate of unemployment consistent with non-accelerating-inflation in a stochastic world, denoted by u bar and referred to as the NAIRU, must be greater than the DNAIRU when the Phillips curve is convex. This can be illustrated in Figure 3 by assuming that actual inflation turned out to be uniformly distributed between plus and minus one percentage point of core (or expected) inflation, which would imply an average rate of unemployment of $\overline{u}=0.5\left(u_1+u_2\right)$. It can easily be seen that with a wider distribution of the actual inflation rate around core inflation, the average rate of unemployment would be even greater. The fact that the difference between the NAIRU and DNAIRU—and hence the average rate of unemployment—depends, in a nonlinear world, on the degree to which the authorities succeed in mitigating the variance of inflation has important implications for monetary policy, as discussed below.

For the most part, the simulation experiments performed with the model start with the economy in a hypothetical equilibrium state. Given the model equations, the simulation results do not depend on historical data, which we generate somewhat arbitrarily. The following points may be noted, however, about the historical data used by Callen and Laxton (1998) in estimating the model equations. The data on inflation expectations for Australia reflect median responses taken from the Westpac bank Melbourne Institute Survey, which asks households for their expectations of inflation one year ahead.²⁸ The analogous data for the United States represent median responses from the Michigan consumer survey of one-year ahead changes in the U.S. consumer price index. And the data on the NAIRU and DNAIRU were constructed by Callen and Laxton (1998) in a manner consistent with the structure of the Phillips curve and historical movements in inflation, inflation expectations, unemployment, and import prices.²⁹

C. Inflation Expectations and Policy Credibility

Table 2 presents the equations that describe the behavior of inflation expectations with endogenous policy credibility.³⁰ The top equation focuses on explaining survey measures of inflation expectations. As noted above, a four-quarter average of these survey measures was assumed in specifying the Phillips curve relation described in Table 1, and the simulation experiments described below use forecasts from equation (9) as hypothetical survey measures of the public’s inflation expectations. The first two terms in the equation are intended to represent a weighted average of a forward-looking model-consistent inflation measure and the one-quarter lag of the survey measure of inflation expectations. For purposes of this paper, the former measure is derived from a proxy for the model—namely, as the fitted values of an auxiliary equation that predicts observed inflation over the year ahead using four lagged values each of the unemployment rate, a long-term interest rate, the survey measure of inflation expectations, and the inflation rate. The third term in equation (9) is intended to capture the expectations bias (b) attributable to imperfect policy credibility.

In modeling imperfect policy credibility and quantifying the expectations bias term, we start from the notion that agents who have experienced high inflation in the past are likely to attach a time-varying probability to the prospect that the monetary authorities will remain truly committed and capable of delivering low inflation in the future. To develop this notion in a simple way, we adopt the following paradigm. It is assumed that economic agents, in forming their inflation expectations, distinguish between two scenarios. Under one scenario, monetary policy is successful in achieving and maintaining price stability, and the inflation rate either remains at, or gravitates toward, the authorities’ target rate of inflation. Under the second scenario, policy is less successful in maintaining price stability and inflation gravitates toward a higher steady-state rate. In each scenario the expected inflation rate (i.e., the value of the forecasting rule with an expected error of zero) is assumed to reflect a weighted average of the steady-state inflation rate and the most recently observed inflation rate; this implies that the expected inflation rate exhibits persistence, adjusting each quarter in proportion to, but by a smaller amount than, the observed change in the (lagged) inflation rate. For an inflation target of 2.5 percent, the inflation rate expected under the low-inflation scenario can be described by equation (10), where 0.34 is an estimated parameter. Analogously, under the assumption that the steady-state inflation rate for the high-inflation scenario is 8.4 percent, the inflation rate expected under the high-inflation scenario can be described by equation (11).³¹

Equation (12) defines the measure of expectations bias, where the terms in square brackets measure the discrepancy between the inflation target (π^TAR) and a weighted average of the inflation forecasts under the two scenarios. As reflected in equation (13a), the weight on the low-inflation scenario (c) is treated as a time-varying parameter that exhibits a high degree of persistence (as implied by the model estimates), but changes from period to period by an amount that reflects the extent to which inflation outcomes are more consistent with the low inflation scenario than with the high inflation scenario. The term ψ, as defined in equation (13b), provides our measure of the extent to which inflation outcomes are consistent with the low-inflation scenario.³² Note that if inflation outcomes are repeatedly completely consistent with the low-inflation scenario, ε^L = 0 and ψ = 1, so c converges to unity. Likewise, if inflation outcomes are repeatedly completely consistent with the high inflation scenario, ε^H = 0, ψ = 0, and c converges to zero.

The above two-scenario paradigm bears a resemblance to a two-state regime-switching model with time-varying transition probabilities.³³ Note that the inflation target coincides with the steady-state outcome under the low inflation scenario (i.e., the model is estimated and simulated with π^TAR = 2.5); that c_t can be interpreted as the subjective probability attached to the low inflation state; and that in the limiting case of c_t = 1, long-run inflation expectations coincide with the inflation target. Based on the latter two properties, we regard c_t as a measure of the stock of credibility at time t. Although the associated measure of bias is derived from a relatively simple model of learning³⁴ and might not remain reasonable if the monetary authorities shifted their inflation target significantly away from 2.5, it has the attractive features of behaving in a stable manner (not subject to multiple equilibria) and converging to zero as c converges to one. Furthermore, the estimated behavior of c_t and ψ from the mid-1970s through the mid-1990s seems plausible; see Figure 4. The paradigm thus seems to provide a reasonable way of introducing endogenous policy credibility into a formal model for analyzing monetary policy issues.

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Historical Perspectives on Policy Credibility

Citation: IMF Working Papers 2001, 007; 10.5089/9781451842418.001.A001

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A. The Monte Carlo Experiments

The assumptions underlying our Monte Carlo experiments are as follows:

1. The “true model” of macroeconomic behavior consists of the equations in Tables 1 and 2 (with different versions of equations (6) and (9) under different model variants, as described below), together with the monetary policy rule and a process (described in Appendix I) that generates the evolution of the DNAIRU and NAIRU over time.

2. The DNAIRU follows a bounded random walk³⁵ with floor at 4 percent and ceiling at 10 percent. Conditional on not hitting either bound, the DNAIRU changes from one quarter to the next by a random amount drawn from a normal distribution with mean zero and standard deviation corresponding to that of the distribution of estimated DNAIRUs during the historical sample period;³⁶ for cases in which the DNAIRU could not change by the full amount of the random draw without moving below 4 percent or above 10 percent, the DNAIRU moves to its floor or ceiling, respectively.

3. In each period t, the monetary authorities update their estimate of the DNAIRU and set the period-t interest rate based on an information set Ω_t that includes: the complete specification of the true model except for the process that generates the DNAIRU and NAIRU (i.e., complete information about equations (1)-(3) and (5)-(14)); the history of all observable variables (including the survey measures of inflation expectations) through period t-1, as well as the inflation rate for period t; and the probability distributions (but not the realizations) of the shocks for period t and all future periods. This implicitly assumes that the inflation rate is the first relevant period-t statistic that becomes known to the authorities, and that the period-t interest rate is set immediately following the arrival of information about the period-t inflation rate.³⁷ Although the authorities do not know the true processes that generate the DNAIRU and NAIRU, they update their estimates of these parameters each period on the basis of their information about the structure of the model and the history of unemployment and inflation.³⁸

4. The exogenous shocks are drawn from independent normal distributions with zero means and standard deviations that reflect the unexplained variances of the dependent variables during the historical periods over which the equations were estimated.

5. The authorities use a prespecified policy rule, along with the assumption that the realizations of random shocks beyond period t will coincide with their expected values of zero, to determine the interest rate setting for period t and to generate forecasts of the complete future timepaths of all the macroeconomic variables in the model, including interest rates and the DNAIRU.

Given these assumptions, for each candidate policy rule we simulate a hypothetical path of the economy over a horizon of 100 quarters. Starting in period t, the monetary authorities observe data through period t-1, update their forecasts for all variables through the end of a 50-quarter horizon,³⁹ including their forecasts for the period-t values of the variables that enter the policy rule (with perfect foresight or advanced knowledge of the period-t inflation rate), and determine the period-t interest rate setting. After the forecasts are generated and the interest rate is set, the shocks for period t are drawn randomly (but consistently with the prespecified probability distributions) and the period-t values of relevant variables are determined from the true model. The determination of the period-t solution is based on the assumptions that the (expected) future path of the real interest rate coincides with the authorities’ forecast, and that the NAIRU follows the process described by the true model. After the period-t solution is added to the hypothetical history of the economy, the authorities generate updated forecasts (including updated estimates of the NAIRU) and set the interest rate for period t+1. And so forth.

The process of generating 100-quarter simulations is conducted 100 times for the given policy rule, starting each time from the same initial position of the economy. Together the 100 simulations provide 10,000 observations for each variable in the model. The process is then replicated for the other policy rules, using the same 100 sequences of randomly drawn shocks.⁴⁰ For purposes of evaluating the relative attractiveness of the different rules and rule calibrations, we define the loss function

where θ, β, and υ are parameters and $u_t^{*}$ is the DNAIRU (deterministic NAIRU). For β = 0 this corresponds to the specification that has been used in other recent simulation studies of monetary policy rules.⁴¹ More generally, it also allows us, somewhat in the spirit of Barro and Gordon (1983a, 1983b) and Rogoff (1985), to consider cases in which the authorities’ preferences with regard to unemployment are not symmetric around the DNAIRU but center on an unemployment rate below the DNAIRU (i.e., cases with β > 0).⁴² For each of the alternative rules and rule calibrations we compare the simulated sample means and standard deviations of the inflation rate and the unemployment rate and calculate, as a summary statistic, the sum of the 10,000 observations on L_t for the base-case parameter settings (β, θ, v) = (1, 1, 0.5).⁴³

B. The Model Variants

The relative performance of the policy rules were evaluated within each of six variants of the basic model. In the first three variants, the convex Phillips curve (equation (6) in Table 1) was replaced with the linear approximation⁴⁴

For the first of these cases, as well as the first case with a convex Phillips curve, we assume that the public forms its inflation expectations in a backward-looking manner; thus, we replace equation (9) with

Under the second case for each class of Phillips curves, it is assumed that the public forms its inflation expectations in a forward-looking manner but treats policy credibility as exogenous. For these cases the simulations retain equation (9) but set b_t = 0 for all t. The third model variant for each class of Phillips curve is forward-looking with endogenous policy credibility.

C. The Policy Rules

For each of the six model variants we used our stochastic simulation experiments to evaluate macroeconomic performance under two forms of policy rules: Taylor rules and analogous inflation forecast based (IFB) rules.⁴⁵ Part of the motivation for focusing on these linear forms of policy rules is pragmatic; particularly in the nonlinear variants of our model, the task of deriving the optimal functional form would be horrendous. In addition, simple classes of rules are transparent and relatively appealing to policymakers, Taylor rules have been popular in the literature since the early 1990s, and IFB rules have been shown to deliver reasonable economic performances over a wide range of disturbances.⁴⁶ It may be noted that linear IFB rules, by focusing on the deviation from target of the authorities’ inflation forecast, have the appealing feature (in comparison with Taylor rules) of taking account of any nonlinearities in the macroeconomic model insofar as the inflation forecast reflects the structure of the model.

The two classes of rules were specified in the general forms:

and

where

Here rs_t is the nominal interest rate setting at time t; r_t is the concept of the real interest rate on which aggregate demand depends; E_tπ4_t+4 denotes the public’s expectations at time t of the inflation rate over the year ahead; r^* is a constant corresponding to the equilibrium real interest rate in a deterministic world under the prespecified initial conditions of the economy; π^TAR denotes the target rate of inflation; ${\tilde{E}}_t\{|{\Omega }_t\}$ denotes a model-consistent forecast at time t based on the authorities information set Ω_t, which includes information about the model along with the observed values of the inflation rate through period t and all other economic variables through quarter t-1; π_t and u_t represent the rates of inflation and unemployment; and ${\overline{u}}_t$ is the authorities’ estimate of the NAIRU based on observed data through period t-1 (see Appendix I). The bracketed terms in equations (15) and (16) are relatively traditional components of policy reaction functions discussed in the literature, corresponding to the deviation of inflation (or the authorities’ inflation forecast) from target and the deviation of the unemployment rate from the NAIRU.⁴⁷ Experimentation suggested that specifying the second class of rules in terms of an inflation forecast that looks ahead three quarters was capable of producing reasonable macroeconomic stability.⁴⁸

Note that the policy reaction functions are specified in the form of rules for real interest rate adjustment. Although monetary policy operates by setting the nominal interest rate, in our model (and most others) the extent to which monetary policy adjustment stimulates or restrains aggregate demand depends on the change in the real interest rate. It would thus make no sense to propose that policy be guided by a nominal interest rate rule that could not be explicitly translated into an economically reasonable rule for the real interest rate on which aggregate demand depends.

Note also that in four of the six model variants, the Taylor rule defined by equations (15) and (17) embodies a forward-looking measure of the real interest rate. By contrast, the conventional form of the Taylor rule embodies a backward-looking measure of the real interest rate, as in the two model variants that impose condition (9a). In general, a conventional completely-backward-looking Taylor rule performs worse than the Taylor rule defined by equations (15) and (17).⁴⁹

The simulation results we report below focus first on two specific calibrations of the (α, γ) parameters for each class of rules. This allows us to assess robustness by comparing the performances of specific rules across the six model variants. The first calibration, (α, γ) = (.5, 1), corresponds roughly to the weights originally suggested for the U.S. case by Taylor (1993), after adjusting γ for the fact that Taylor specifies his rule in terms of the output gap rather than the unemployment gap, and that the unemployment gap tends to vary about half as much over business cycles as the output gap. The second parameter setting, (α, γ) = (2, 1), is based on a calibration of the inflation forecast based (IFB) rule that we found to perform relatively well in another study.⁵⁰ We later report, and discuss the limited relevance of, simulation results for the (approximately) optimal Taylor rule calibrations for two of the model variants.