ASSESSMENTS OF THE RATE of unemployment that is consistent with holding inflation stable (at a low rate) represent an integral part of the monetary policy framework in most industrial countries. Whether the current rate of unemployment lies above or below the rate that is consistent with inflation stability is a key input into the monetary policy decisionmaking process. Unfortunately, the rate of unemployment that is consistent with inflation stability is not directly observable; policymakers can estimate it only by using other, more observable pieces of information about the state of the economy.

For the United States, estimates of Phillips curves and the nonaccelerating inflation rate of unemployment (NAIRU) were regularly published in the Brookings Papers on Economic Activity during the 1970s and early 1980s by Robert Gordon (for example, 1970, 1975, and 1977). After a period of inactivity in the 1980s when Phillips curves were generally assumed to have broken down, a number of recent papers (Gordon, 1994; and Fuhrer, 1995) have sought to reestimate the Phillips curve and derive new estimates of the natural rate of unemployment. The general approach has been to regress inflation on a measure of expected or lagged inflation, unemployment gaps, and dummy variables to control for various supply shocks, such as the oil shocks and the Nixon price controls.

Despite the variety of techniques used, the common feature of these recent estimates is their use of a linear Phillips curve. A separate strand of the literature has presented evidence supporting the concept of a nonlinear Phillips curve (Macklem, forthcoming; Clark, Laxton, and Rose, 1996; Turner, 1995; and Laxton, Meredith, and Rose, 1994). A convex nonlinear Phillips curve implies that a given fall in the unemployment rate below the NAIRU causes a larger rise in inflation than does a rise of the same magnitude produce a fall in inflation.

The literature on the relationship between inflation and unemployment frequently refers to both the NAIRU and the natural rate of unemployment. With linear specifications of the Phillips curve, the two terms are often used interchangeably. More generally, when allowing for the possibility of a nonlinear Phillips curve and stochastic variability in demand and supply, it is useful to distinguish the NAIRU at a point in time from the expected value over time of the unemployment rate that would be consistent with nonaccelerating inflation, given the stochastic distribution of shocks. We define the former as the deterministic NAIRU because this is the unemployment rate that is consistent with nonaccelerating inflation in the absence of shocks. In the remainder of this paper, we will refer to the deterministic NAIRU as either “D-NAIRU” or “u*.” AS shown below, the true stochastic NAIRU must be greater than u* when there is convexity in the Phillips curve. Following Friedman (1968), this paper uses the term “natural rate of unemployment” to refer to the NAIRU because this is the only sustainable average level of unemployment that would be consistent with nonaccelerating inflation. The nonlinearity in the model implies that policymakers who are less successful in stabilizing the business cycle will induce a larger natural rate of unemployment in their economies (De Long and Summers, 1988; and Laxton, Rose, and Tetlow, 1993b).

Laxton, Rose, and Tetlow (1993a) demonstrate that previous tests for nonlinearity have been severely compromised because researchers have tended to rely upon ad hoc prefiltering techniques that are grossly inconsistent with the key implications of a nonlinear Phillips curve. In this paper, we derive estimates of u* and the natural rate of unemployment for three countries—Canada, the United Kingdom, and the United States—by extracting information from a nonlinear model of the Phillips curve. As in Kuttner (1992 and 1994), we use a Kalman filter and a maximum likelihood procedure to simultaneously estimate the parameters of the model along with model-consistent estimates of u*. We also estimate a linear model of the Phillips curve using the Kalman filter. We show that the nonlinear model fits the data better than the linear model for all three of these countries when plausible restrictions are imposed on the variance of the natural rate.

The first section describes the linear and the nonlinear models that we estimate and describes the Kalman filtering technique that we use in the estimation. The derivation of the inflation expectations series, which is a key component of our model, is described in Section II. Section III presents the results, while Sections IV and V discuss the uncertainty and sensitivity of the results. Section VI concludes.

I. Models and Estimation Technique

This section describes the models and the procedure that we use to estimate the NAIRU. The standard linear expectations-augmented Phillips curve has the following functional form:

where π is inflation, u is the unemployment rate, and u* is the D-NAIRU. The inflation expectations π^e are a linear combination of a backwardand forward-looking component (see the discussion in Section II). The backward-looking component could also reflect inertia in the inflation process. For example, an overlapping-contracts model, such as Fischer (1977), could motivate such a result:

where A(L) and B(L) are polynomial lag operators.

Equations such as (1) have been estimated by Gordon in the 1970s and early 1980s (Gordon, 1970, 1975, and 1977). To proxy inflation expectations, Gordon used lags of the inflation rate of up to two years and also controlled for a number of supply-side influences, such as price controls, relative price changes, and real exchange rate changes. Furthermore, Gordon imposed the constraint that the coefficients on lagged inflation sum to one. Evidence in favor of this restriction is still sometimes interpreted as support for the long-run natural rate hypothesis despite Sargent’s (1971) explanation that this restriction has nothing to do with the long-run natural rate hypothesis. The restriction is inappropriate because it implies that agents always forecast inflation as if it contained a unit root. In the case of the United States, studies either have assumed that the natural rate of unemployment has been constant over the sample period or have estimated some small shifts to control for changes in the composition of the labor force.

More recently, Gordon (1994), Tootell (1994), and Fuhrer (1995) have estimated similar models, which also assume that the natural rate is constant over the sample period. This assumption is supported by both within-sample and out-of sample tests that fail to reject the hypothesis of a constant natural rate. While this assumption may be difficult to reject in the United States, it is likely to be rejected in countries where the natural rate has clearly moved over time (such as the United Kingdom and Canada).

Staiger, Stock, and Watson (1996) employ a variety of techniques to derive estimates of the natural rate based on Gordon’s approach and on univariate analysis. Their results highlight the uncertainty associated with the estimates. They obtain estimates of the natural rate by assuming, alternatively, that it is constant over the sample period, a constant with occasional shifts, an unobserved random walk, and a specific function of labor market variables. Under one of these assumptions, for example, the degree of uncertainty associated with a point estimate of 6.2 percent in 1990 is a 95 percent confidence interval of 5.1-7.7 percent. In addition, the point estimates vary quite substantially across the different techniques.

Kuttner (1992 and 1994) adopts a strategy that is the closest to the one adopted in this paper. He estimates a model of the Phillips curve allowing for time variation in the level of potential output. He employs a Kalman filter to extract an estimate of the level of potential output, where potential output is assumed to be a random walk with positive drift. However, as in the other models, Kuttner assumes that the Phillips curve has a linear specification.

The key difference between this paper and the previous literature is that we estimate u* in the context of a nonlinear Phillips curve. The nonlinear Phillips curve used here is assumed to have a simple structure of the form where u is the observed unemployment rate and u* is the time-varying unemployment rate at which inflation is constant.¹

A nonlinear Phillips curve may be motivated by the traditional concept of an upward-sloping aggregate supply curve. As the unemployment rate falls below the NAIRU, bottlenecks start to develop that cause further increases in demand to have even larger inflationary consequences. Once the unemployment rate reaches some lower bound, inflation will increase at an almost infinite rate.

As mentioned above, a distinction can be made in the nonlinear model between u* and the natural rate that cannot be made in the linear model. If one defines the natural rate as the expected value of unemployment in the stochastic steady state, the convexity of the nonlinear Phillips curve implies that the natural rate of unemployment will lie above u* by a constant a that embodies the degree of convexity and the nature of the stochastic shocks. Meanwhile, in the traditional Phillips curve model, the linearity ensures that u* and the natural rate are the same.

This distinction is most easily demonstrated if we assume that the Phillips curve takes a slightly different functional form:

The D-NAIRU is given by u*: when the rate of unemployment is equal to u*, inflation is equal to inflation expectations and is neither rising nor falling. If we assume that the error term is normally distributed with zero mean, the average rate of unemployment is given by u* + γ var(u_t)/2. That is, the stochastic steady state rate of unemployment, which we interpret as the natural rate (in the sense of Friedman, 1968), is greater than u*. Furthermore, policies that reduce the variance of unemployment will reduce the natural rate of unemployment. That is, a policymaker who is effective in stabilizing the business cycle will reduce the gap between u* and the natural rate.

Figure 1 is a useful device for illustrating the implications that such asymmetry has for stabilization policy. The nonlinear curve in the figure, the “Phillips curve,” depicts the short-run relationship between inflation adjusted for inflation expectations, π - π^e, and the unemployment rate u. The key assumption underlying the Phillips curve is that the slope of the curve, or the trade-off between unemployment and inflation, worsens as unemployment falls significantly below u*. For illustrative purposes, we assume that u* is equal to 5 percent. Figure 1 illustrates this point by incorporating the incontrovertible assumption that, even in the short run, there is some minimum level of unemployment that cannot be obtained through expansionary policies to manage aggregate demand. For purely illustrative purposes, it is assumed that this minimum level of unemployment is equal to 1 percent of the labor force.

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The Phillips Curve, the D-NAIRU, and the Natural Rate of Unemployment

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Note: α is the difference between the average historical rate of unemployment and the D-NAIRU.

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Because excess demand conditions are more inflationary than excess supply conditions are disinflationary, allowing the economy to enter the region of excess demand implies that the economy will have to operate longer in the region of excess supply to prevent inflation from drifting upward over time. Thus, if disturbances to the economy cause the unemployment rate to vary over time, the natural rate of unemployment—the average unemployment rate that is consistent with stable inflation—will be higher than the u* that enters the Phillips curve because unemployment will have to spend more time above u* in order to offset the greater inflationary tendencies that will be associated with periods when it falls below u*. For illustrative purposes, it is assumed in Figure 1 that the unemployment rate varies between 4 percent and 8 percent, and that the natural rate of unemployment is 6 percent, or 1 percentage point above u*.

This asymmetry in the unemployment-inflation process has important policy implications. Stabilization policy that is not successful in reducing the variability in the business cycle can have undesirable consequences, not only for the variance of unemployment but also for the natural rate of unemployment. Figure 2 illustrates this point by considering the alternative case in which unsuccessful stabilization policy allows the unemployment rate to vary over a wider range than in the top panel, which periodically subjects the economy to serious overheating. In this case, the natural rate (shown as 7 percent) will be even higher than in the top panel because it will take larger excess supply or recessionary states to offset the greater inflationary impetus caused by periodically subjecting the economy to serious overheating.

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Implications of Greater Unemployment Variability for the Natural Rate of Unemployment

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Note: αis the difference between the average historical rate of unemployment and the D-NAIRU.

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The concept of the NAIRU used in this paper is the one generally used in discussions of the “natural rate.” The NAIRU will be affected by the operation of the labor market and embodies the “actual structural characteristics of the labor and commodity markets, including market imperfections” (Friedman, 1968, p. 8). Labor market policy will thus affect the level of the NAIRU through time. However, the distinction that we are focusing on here is that macroeconomic policy will have a further impact on the natural rate through its effect on the variability of the macroeconomy.

Previous work has tried to estimate the determinants of the NAIRU directly with mixed success. For example, Lilien (1982) examines the effect of sectoral changes on the structural unemployment rate. Blanchard and Katz (1996) provide a recent summary of this research. While acknowledging that changes in labor market policy and institutional factors will affect the NAIRU, we do not directly identify the impact of these factors on the NAIRU in our model.

The nonlinear functional form that we employ in our empirical work allows for a hyperbolic shape that imposes an asymptote at zero. In economic terms, this implies that, as the unemployment rate approaches zero, inflation increases at a higher and higher rate. Allowing the asymptote to be at zero is perhaps being overly conservative. It is likely that the economy will run up against insurmountable supply constraints before the unemployment rate reaches zero. Nevertheless, a zero unemployment rate provides an uncontroversial lower limit for the asymptote.

u* is allowed to be time varying in both the linear and the nonlinear models. To estimate the Phillips curve, we need estimates of u*, which is not directly observable. The Kalman filter allows us to estimate the model while simultaneously providing a time-series estimate of the NAIRU. The Kalman filter estimates models of the general form:²

The parameter vector α is time varying in a manner determined by the transition equation (6). In our models, we allow u* to be time varying and, more particularly, a random walk. As mentioned above, u* will be affected by structural changes in the labor market, including those induced by labor market policy. However, for estimation purposes, we have not directly modeled the effects of such changes.

As Kuttner (1994) points out, estimating the natural rate using a Kalman filter has three advantages. First, it allows us to use the information present in the difference between inflation and inflation expectations, rather than relying solely on the univariate properties of the unemployment rate. Second, it allows for smooth, continuous updating of the estimate in real time as new data become available. This will be a particular advantage in the context of policymaking decisions. Third, we can derive estimates of the uncertainty about the natural rate, as in Staiger, Stock, and Watson (1996).

Operationally, we estimate the linear model with the Kalman filter, allowing the constant term γu* in equation (1) to be time varying. We assume that the matrices H and T are identity matrices, and the matrix Q is constructed so that only the constant term is time varying. In the nonlinear model, the coefficient on the inverse of the unemployment rate is time varying. In both models, the time series for u* is obtained by dividing the time varying parameter obtained from the Kalman filter by the (time-invariant) estimated coefficient γ.

Application of the Kalman filter generates two time series of u*. The first comes from the recursive estimation of the model, which uses data that are available only up to the current period (referred to by Kuttner (1992) as "one-sided" estimates). The second (smoothed or “two-sided”) series uses data from the whole sample to estimate a time series for u* and the parameters of the model to maximize the likelihood function. The one-sided estimates allow an assessment of how the model performs in “real time.”

The information that we use to identify movements in u* is the difference between inflation and inflation expectations. We proxy the backwardlooking component of inflation expectations by the first lag of inflation. The estimates of the forward-looking component of inflation in this paper are derived from information present in long-term bond rates. The technique used in this latter step is described in the next section.

II. Data

All data are quarterly data for Canada, the United Kingdom, and the United States; the data are drawn from the Analytical Data Bank of the Organization for Economic Cooperation and Development (OECD). The unemployment and consumer price series are based on official seasonally adjusted estimates for each country. Our approach to measuring inflation expectations is based on the simple idea that long-term interest rates may embody valuable information about policy credibility—for example, see Goodfriend (1993) and McCallum (1996). We use information from bond markets to develop a measure for long-term inflation expectations, ${\pi }_t^{LTE},$, which, in turn, is used to help identify the short-term measure of inflation expectations. ${\pi }_t^e,$ that enters the Phillips curve.

This approach assumes that the inflation expectations of wage and price setters are related to the inflation premiums that participants in the bond market demand to hold long-term, fixed-income securities. Specifically, it is assumed that short-term inflation expectations are a weighted average of long-term inflation expectations and actual inflation in the previous period, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1},$, where δ represents the weight attached to information from the bond market.

The measure of long-term inflation expectations for each country is constructed by subtracting an alternative measure of the equilibrium world real interest rate from a measure of long-term interest rates. After constructing these proxies for long-term inflation expectations, we then test to see whether these proxies provide significant explanatory power for identifying movements in the Phillips curve.

The estimates of the equilibrium world real interest rate that we employ are based on recent empirical work that suggests that the equilibrium real interest rate has been gradually rising over time in response to the large buildup in world government debt—for example, see Ford and Laxton (1995), Helbling and Wescott (1995), and Tanzi and Fanizza (1995).

The basic approach that we follow to derive estimates of the world equilibrium real interest rate is quite simple. First, in order to provide a reasonable benchmark for the equilibrium world real interest rate at the end of the sample, we use data on indexed bonds. Second, after obtaining this benchmark for the end of the sample, we rely on estimates of the effects of government debt on real interest rates to construct a time series that extends back to the early 1970s.

Our preferred set of estimates is based on the empirical work by Tanzi and Fanizza (1995), which suggests that a 1 percentage point increase in the gross government debt-to-GDP ratio increases the world real interest rate by about 7 basis points. Based on a benchmark estimate of 4.5 percent for the real interest rate at the end of the sample, the upward trend in gross public d bt since the late 1970s would suggest that the equilibrium real interest rate has gradually shifted up from an average value of 2.4 percent in the 1970s to 4.5 percent by the first quarter of 1994.³ Other estimates were also constructed using alternative values of 3.5 percent and 5.5 percent for the end-of-sample benchmark.

In addition, because the effects of government debt on real interest rates vary significantly across different empirical studies, we also calculated estimates that were based on both smaller and larger estimates of the effect of debt buildup on the equilibrium world real interest rate. First, to be consistent with the estimates reported by Ford and Laxton (1995) and Helbling and Wescott (1995), we doubled the assumed effect of world government debt on the equilibrium real interest rate. Second, we assumed that government debt had no effect on the equilibrium real interest rate and that the equilibrium real interest rate has thus been constant since the early 1970s. This last assumption is more consistent with earlier empirical work that suggested that government debt has had no discernible effect on real interest ratesfor example, see Evans (1985).

We found that, despite the different assumptions that were used to construct the equilibrium real interest rate, the resultant measures of long-term inflation expectations were highly significant in all the regressions that were considered.⁴ In fact, in not one of the alternative specifications described above could we reject the hypothesis that long-term bond yields provide significant explanatory power for identifying historical shifts in the Phillips curve. This result is encouraging because, in theory, there should be some relationship between long-term inflation expectations of participants in the bond market and the more short-term inflation expectations that influence the decisions of wage and price setters. In this vein, our results tend to support Goodfriend’s (1993) conclusion that most of the high-frequency variations in long-term bond yields are driven by inflation scares rather than by historical movements in the ex ante real rate of interest.

III. Estimation Results

This section presents empirical estimates of Phillips curves and Phillips lines for Canada, the United Kingdom, and the United States. The basic empirical strategy followed in this paper is to derive model-consistent measures of u* under the assumption that the Phillips curve is really a curve, and then to compare the results of this model with an alternative model that assumes that the curve is linear.

As seen at the top of Table 1, the first model that we estimate assumes that the amount of the actual change in inflation relative to expected rates, π - π^e, is related to the proportional difference between u* and the actual unemployment rate, (u* - u)/u. This specification hypothesizes that the relationship between inflation and unemployment is approximately linear and symmetric in the region where unemployment is close to u* but becomes nonlinear as the unemployment rate moves further away from u*. The results of the assumption of a linear structure, in which the (u* - u)/u term is replaced by (u* - u), are reported in Table 2.

^{Table 1.}

Estimates of Phillips Curves with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

^{Table 1.}

Estimates of Phillips Curves with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_t^{//},where$
	π= year-on-year percent change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent),
	δ= estimated weight on inflation proxy from bond market.
	γ= parameter for measure of labor market tightness (in percent).
	u= unemployment rate (in percent).
	u*= D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function,
	σ2 = variance of residuals,
	α= average historical value of u *-u (in percent).
	Ω = maximum absolute value of u* - u (in perceni), and
	τ = maximum absolute quarterly change in u* (in percent).

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^{Table 1.}

Estimates of Phillips Curves with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_t^{//},where$
	π= year-on-year percent change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent),
	δ= estimated weight on inflation proxy from bond market.
	γ= parameter for measure of labor market tightness (in percent).
	u= unemployment rate (in percent).
	u*= D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function,
	σ2 = variance of residuals,
	α= average historical value of u *-u (in percent).
	Ω = maximum absolute value of u* - u (in perceni), and
	τ = maximum absolute quarterly change in u* (in percent).

Estimation period: 1971:Q2 to 1995:Q2.

Estimation period: 1971:Q2 to 1995:Q2.

Country	γ	δ	LLF	σ	α	Ω	τ
Canada	2.37	0.15	-104.03	0.41	0.86	5.10	0.21
	(4.65)	(3.47)
United Kingdom	2.43	0.20	-176.41	1.81	0.57	3.79	0.65
	(3.70)	(4.4S)
United States	3.55	0.08	-104.85	0.36	0.33	4.01	0.15
	(7.32)	(2.45)

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Estimation period: 1971:Q2 to 1995:Q2.

Country	γ	δ	LLF	σ	α	Ω	τ
Canada	2.37	0.15	-104.03	0.41	0.86	5.10	0.21
	(4.65)	(3.47)
United Kingdom	2.43	0.20	-176.41	1.81	0.57	3.79	0.65
	(3.70)	(4.4S)
United States	3.55	0.08	-104.85	0.36	0.33	4.01	0.15
	(7.32)	(2.45)

becomes nonlinear as the unemployment rate moves further away from u*. The results of the assumption of a linear structure, in which the (u* - u)/u term is replaced by (u* - u), are reported in Table 2.

The estimated parameters were obtained using the maximum likelihood Kalman filter routine in TSP. This technique builds up model-consistent estimates of u* under the assumption that we can approximate historical shifts in u* by a random walk.⁵. In both tables, we include the value of the likelihood function (LLF), the estimated parameter values, and their associated standard errors. We also report the maximum absolute gap between u* and the actual unemployment rate u, as well as the maximum absolute quarterly change in u*, to provide some indication of how jumpy the u*’s have to be to explain inflation in these countries.

^{Table 2.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

^{Table 2.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:
	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_t^{//},where$
	π = year-on-year perceni change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent).
	δ = estimated weight on inflation proxy from bond market.
	γ = parameter for measure of labor market tightness (in percent).
	u = unemployment rate (in percent).
	u* = D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function.
	σ2 = variance of residuals.
	α = average historical value of u - u* (in percent).
	Ω = maximum absolute value ofu - u* (in percent), and
	τ = maximum absolute quarterly change in u* (in percent).

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^{Table 2.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:
	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_t^{//},where$
	π = year-on-year perceni change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent).
	δ = estimated weight on inflation proxy from bond market.
	γ = parameter for measure of labor market tightness (in percent).
	u = unemployment rate (in percent).
	u* = D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function.
	σ2 = variance of residuals.
	α = average historical value of u - u* (in percent).
	Ω = maximum absolute value ofu - u* (in percent), and
	τ = maximum absolute quarterly change in u* (in percent).

Estimation period: 1971 :Q2 to 1995:Q2.

Estimation period: 1971 :Q2 to 1995:Q2.

Country	γ	δ	LLF	σ2	α	Ω	τ
Canada	0.09	0.21	-103.49	0.18	—	17.47	7.66
	(0.77)	(3.23)
United Kingdom	0.63	0.33	-173.40	0.80	—	8.99	3.06
	(2.36)	(5.20)
United States	0.55	0.25	-99.73	0.15	—	4.06	1.33
	(0.13)	(0.06)

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Estimation period: 1971 :Q2 to 1995:Q2.

Country	γ	δ	LLF	σ2	α	Ω	τ
Canada	0.09	0.21	-103.49	0.18	—	17.47	7.66
	(0.77)	(3.23)
United Kingdom	0.63	0.33	-173.40	0.80	—	8.99	3.06
	(2.36)	(5.20)
United States	0.55	0.25	-99.73	0.15	—	4.06	1.33
	(0.13)	(0.06)

Phillips Curve with Model-Consistent u^*’s

As can be seen in Table 1, the estimated parameters δ for the forwardlooking proxy for long-term inflation expectations, as well as the estimated parameters γ for the measure of labor market tightness, are statistically significant. The estimates of δ can be interpreted as implying that wage and price setters place a weight of between 8 percent and 20 percent on forwardlooking inflation expectations. A higher value of γ implies not only that a given amount of excess demand has a larger inflationary impact but also that a given amount of excess supply has a larger disinflationary impact.

Table 1 also shows estimates of α,⁶ the difference between the average historical rate of unemployment and u^*(recall Figure 1 and 2).⁷ As emphasized earlier, in economies where policymakers have been less successful in avoiding boom-and-bust cycles (which may depend on the nature of the shocks hitting the economy), we should expect to observe a larger difference between the average rate of unemployment and u^*. The estimates of a shown in Table 1 suggest either that policymakers in the United States have been more successful in stabilizing their business cycle than their counterparts in the other two countries, or that the shocks that have hit the U.S. economy have been smaller. Other things being equal, this would have contributed to a lower natural rate of unemployment (or average rate of unemployment) in the United States.

The value of α for each country is assumed to be constant here. However, in reality it is likely to be time varying. The value depends on how successful past policymakers were in stabilizing the business cycle. If policymakers have become more successful over time at avoiding large boom-and-bust cycles, one should expect that a would shift down over time as the economy moves to a new stochastic steady state. For example, after the recent disinflation episodes in Canada and the United Kingdom and the subsequent relatively stable inflation process, one might expect that the value of α might decline, reflecting the attainment of a new regime with an improved stabilization performance. However, as suggested in Section IV, α might decline very gradually if it takes a long time for the new regime to develop credibility.

The model-consistent estimates of u^* from the Phillips curve are plotted in Figure 3. This methodology for measuring u^* produces estimates that are fairly smooth, even though the random walk assumption in the measurement equation is capable of producing fairly jumpy measures of u^* if this were necessary to explain inflation in these countries. In fact, the estimates suggest that, from the early 1970s to 1995, the maximum quarterly change in u^* in the United States was only 0.15 percentage point. Although the jumps in u^* are somewhat larger for Canada and the United Kingdom, the estimated u^* series are also fairly smooth in these countries. According to these estimates, the increase in u^* from the low levels measured for the 1950s and 1960s started sooner in Canada and the United States than in the United Kingdom.⁸ In particular, based on this methodology, u^* in the United Kingdom, although fairly low at the beginning of the sample (for example, 3.3 percent in 1973 :Q1), shows a much stronger upward trend than the other two countries over this estimation period. Indeed, these estimates indicate that there has been a slight tendency for u^* to decline over the last decade in the United States.

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Unemployment, the D-NAIRU, and the Natural Rate of Unemployment, Based on the Nonlinear Model

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.

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The estimates of the gap between u^* and the unemployment rate are broadly consistent with a classical characterization of business cycles in these economies. For example, the maximum absolute unemployment gap is 5.0 percent in Canada, 3.8 percent in the United Kingdom, and 4.0 percent in the United States. All three of these estimates are excess supply gaps and, as can be seen in Figure 3, occurred in the early 1980s—about the time of the business cycle troughs that were related to the large disinflationary episode. More precisely, the peak excess supply gaps are dated as 1982:4 in Canada and the United States and 1983:2 in the United Kingdom.

The estimates of u^*’s at the end of the sample (1995:2) are 8.8 percent in Canada, 8.1 percent in the United Kingdom, and 6.1 percent in the United States. Comparing these estimates with the actual unemployment rates at the time of 9.5 percent, 8.3 percent, and 5.6 percent, respectively, suggests that there was still some disinflationary pressure in the Canadian economy, the potential for some inflationary pressure in the U.S. economy, and no pressure in the U.K. economy.⁹

Figure 4 demonstrates the fit of the model in terms of the ability of the estimated excess demand or supply term to explain historical movements in inflation. This figure was constructed by comparing the deviation of inflation from expected inflation,π_t—$\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1},$, with our measure of the effect of labor market tightness, γ (u^* - u)/u. The fit of the simple Phillips curve is remarkably good in all three of these countries, considering the highly parsimonious functional form.¹⁰ The figure illustrates the strong inflationary pressure about the time of the two oil shocks in the 1970s and the disinflation in 1981-82. It also illustrates the disinflation in Canada in the early 1990s.

Phillips Line with Model-Consistent u^*’s

As noted above, the estimation results for the linear model with modelconsistent u^*’s are reported in Table 2. The problem with the linear model is evident in the final two columns of Table 2. The maximum absolute value of the gap takes on values that are excessively large in Canada and the United Kingdom. Furthermore, u^* changes by as much as 7.7 percent and 3 percent in one quarter in Canada and the United Kingdom, respectively.

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Historical Performance of the Nonlinear Model

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a ${\pi }_t-\delta {\pi }_t^{LTE}-(1-\delta ){\pi }_{t-1}\mathrm{.}$^b $\gamma ({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Clearly, such estimates are not economically sensible. This problem is illustrated by Figure 5. The series for u^* are excessively volatile in Canada and the United Kingdom, and volatile to a lesser extent in the United States. The estimates for u^* for Canada, which range from about minus 5 percent to over 20 percent, are not plausible.

The nonlinear and linear models are not nested; however, because each model has the same number of parameters, we can simply compare the values of the likelihood function in Table 1 and 2 to determine which model fits the data best.¹¹ This comparison shows that the linear model has a significantly better fit in all three of these countries. However, unlike the nonlinear model, the fit in the linear model is achieved by allowing extreme volatility in the u^* series for each country, as shown in Figure 5. Indeed, the u^* series generated by the linear model are considerably more volatile than the actual unemployment series.

One advantage of the Kalman filter approach is that it is fairly straightforward to impose prior restrictions on volatility in u^* series. Because the linear model produced such implausible estimates of u^*’s for these countries, we estimated an alternative model in which we imposed some prior restrictions on large jumps in u^*’s. In order to make the results comparable to the results for the nonlinear model, we reestimated the linear model subject to the constraint that the maximum absolute quarterly change in u^* was equal to that obtained in the nonlinear model (that is, 0.21 percent, 0.65 percent, and 0:15 percent for Canada, the United Kingdom, and the United States, respectively). The results are reported in Table 3. When this restriction is imposed, there is a significant deterioration in the fit of the equation. Indeed, in this case a comparison of the values of the likelihood function indicates that the nonlinear model is the preferred model.

IV. Uncertainty and Recursive Estimates of u^*

A fundamental problem that policymakers face in attempting to stabilize the business cycle is the considerable uncertainty in their inferences about current excess demand pressures. Measures of the natural rate of unemployment and, hence, labor market tightness are especially uncertain for countries that have undergone major structural reforms or are regularly subjected to significant supply shocks that require a continuous reallocation of labor resources across sectors. An important lesson from history is that it may be counterproductive to place too large a weight on stabilizing unemployment if there is considerable uncertainty about the level around which it should be stabilized, that is, the underlying natural rate.

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Unemployment and the D-NAIRU with a Linear Phillips Curve

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.

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^{Table 3.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

^{Table 3.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:
	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_I^{//},where$
	π = year-on-year perceni change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent).
	δ = estimated weight on inflation proxy from bond market.
	γ = parameter for measure of labor market tightness (in percent).
	u = unemployment rate (in percent).
	u* = D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function.
	σ2 = variance of residuals.
	α = average historical value of u - u* (in percent).
	Ω = maximum absolute value ofu - u* (in percent), and
	τ = maximum absolute quarterly change in u* (in percent).

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^{Table 3.}

Estimates of Phillips Lines with Model-Consistent D-NAIRUs

(t-statistics in parentheses)

Estimated equation:
	${\pi }_t={\pi }_t^e+\gamma (u^{*}-u)/u+{\in }_I^{//},where$
	π = year-on-year perceni change in the CPI.
	${\pi }_t^e,$ = inflation expectations, $\delta {\pi }_t^{LTE}+(1-\delta ){\pi }_{t-1}$ (in percent).
	${\pi }_t^{LTE},$ = long-term inflation expectation proxy from bond market (in percent).
	δ = estimated weight on inflation proxy from bond market.
	γ = parameter for measure of labor market tightness (in percent).
	u = unemployment rate (in percent).
	u* = D-NAIRU estimates derived from Kalman filter (in percent).
	LLF = value of the likelihood function.
	σ2 = variance of residuals.
	α = average historical value of u - u* (in percent).
	Ω = maximum absolute value ofu - u* (in percent), and
	τ = maximum absolute quarterly change in u* (in percent).

Estimation period: 1971 :Q2 to 1995:Q2.

Estimation period: 1971 :Q2 to 1995:Q2.

Country	γ	δ	LLF	σ2	α	Ω	τ
Canada	0.25	0.14	-104.81	0.41	—	4.76	0.21
	(4.55)	(3.33)
United Kingdom	0.46	0.22	-179.75	1.83	—	5.28	0.65
	(3.56)	(4.52)
United States	0.47	0.08	-106.61	0.38	—	3.68	0.15
	(7.00)	(2.27)

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Estimation period: 1971 :Q2 to 1995:Q2.

Country	γ	δ	LLF	σ2	α	Ω	τ
Canada	0.25	0.14	-104.81	0.41	—	4.76	0.21
	(4.55)	(3.33)
United Kingdom	0.46	0.22	-179.75	1.83	—	5.28	0.65
	(3.56)	(4.52)
United States	0.47	0.08	-106.61	0.38	—	3.68	0.15
	(7.00)	(2.27)

The difficulties invovled in measuring the natural rate of unemployment can contribute to significant policy errors. Indeed, one popular interpretation of overheating in the 1970s is that policymakers significantly underestimated the increases in the NAIRU in their countries. This type of policy error can have deleterious implications for the economy if the monetary authorities take considerable time to reestablish the credibility of their commitments to low inflation. In this sense, the asymmetric model predicts that the seeds of large contractions are sown when the monetary authority defers dealing with rising inflation and allows excess demand conditions to become entrenched in inflation expectations (Clark and Laxton, 1997).

In order to truly assess the inflationary risks of overheating, it is important to develop measures of uncertainty so that confidence bands can be established around any particular point estimate. As mentioned earlier, an advantage of the Kalman filter is that it allows the calculation of real-time estimates of the model. Each period, the filter uses the new information to revise its estimates of the model’s parameters and the estimate of u^*. This exercise replicates to an extent the process that a policymaker would undertake in using this framework to determine inflationary pressures. One can then assess with the advantage of hindsight whether the recursive estimates give a significantly different picture of the degree of inflationary pressure from the full-sample estimates, which incorporate more information.

In the framework used in this paper, there are two sources of uncertainty: parameter uncertainty and uncertainty about the natural rate. The solid lines in Figure 6a, 7a, and 8a show the recursive or period-by-period estimates of u^* and the model parameters for the nonlinear model. Figure 6b, 7b, and 8b display the same information for the linear model. The dotted lines in the upper panels of Figures 6-8 provide confidence bands of one standard error around our full-sample estimates of u^* (which are shown by the bars). These standard error bands were obtained by imposing the full-sample parameter estimates for γ and δ for the whole sample, thus removing the effect of parameter instability, and then reestimating the model. This estimation produces an end-of-sample standard error on the estimate of u^* that is applied for the whole sample.¹²

The parameter estimates are generally stable and approach their fullsample values relatively quickly. Each period, the filter assigns variation in the left-hand-side variable (in effect, the difference between inflation and inflation expectations) among variation in u^*, variation in the parameters, and the error term. Thus there is likely to be a negative correlation between movements in the recursive estimates of u^* and movements in the recursive parameter estimates.

The recursive estimates of u^* fluctuate generally within one standard error of the full-sample estimate. In particular, there is little difference in the United States between the recursive estimates of u^* and the full-sample estimates. When the recursive estimate lies above the full sample estimate, the policymaker using this framework is overestimating the excess demand in the economy and thus may be running an overly restrictive policy. Such a situation can arise when a negative shock to inflation occurs, as this will tend to pull up the recursive estimate of u^* until more observations can make clear whether the shock is temporary or permanent, and until the curvature of the Phillips curve can be more exactly estimated.

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Recursive Estimates of the D-NAIRU and Parameters for Canada: Nonlinear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Recursive Estimates of the D-NAIRU and Parameters for Canada: Linear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Recursive Estimates of the D-NAIRU and Parameters for United Kingdom: Nonlinear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Recursive Estimates of the D-NAIRU and Parameters for United Kingdom: Linear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Recursive Estimates of the D-NAIRU and Parameters for United States: Nonlinear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Recursive Estimates of the D-NAIRU and Parameters for United States: Linear Phillips Curve Model

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a$({u^{*}}_t-u_t)/u_t\mathrm{.}$

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Nevertheless, the standard error bands are wide in all three countries. Even in the United States, a confidence interval of approximately 66 percent (one standard error) for u^* is 1 percentage point wide. This outcome is consistent with the uncertainty surrounding other estimates of u^*; the estimates of u^* by Staiger, Stock, and Watson (1996), for example, are associated with a 66 percent confidence interval that is 1.3 percentage points wide.¹³

Turning to the linear model estimates, we see that the recursive estimates of u^* are at least as variable as the volatile full-sample estimates. The 66 percent confidence interval for the Canadian u^* is almost 8 percentage points wide. Furthermore, in Canada and the United States, the recursive estimates often lie outside the standard error bands of the full sample estimates, implying that a policymaker using the linear model is more likely to misread the extent of excess demand or supply in the economy.

In the nonlinear model, the standard error bands also suggest that it is considerably easier to measure u^* in the United States than it is in Canada or the United Kingdom. Indeed, perhaps one reason why policymakers in the United States have been more successful in avoiding boom-and-bust cycles is because they have had less difficulty in obtaining reliable measures of the natural rate of unemployment. That being said, the enormous uncertainty in these estimates reinforces the view that it may be desirable for policymakers to exercise caution by setting monetary conditions in a way that guards against the serious overheating that was allowed in the 1970s.

V. Information Content of Long-Term Inflation Expectations

As indicated above, our basic methodology in modeling inflation expectations π^e involves using information from bond markets to develop proxies for long-term inflation expectations π^LTE, and then testing to see whether these proxies are useful for identifying shifts in the Phillips curve. This approach to measuring inflation expectations may have some important advantages over reduced-form, distributed lag models if these measures of long-term inflation expectations embody information about how wage and price setters revise their expectations in response to shocks or changes in policy regimes.

Because there is nothing really fundamental to tie down the distribution of future monetary policies—beyond the reputation of today’s policymakers— it may take a considerable amount of time for agents to become convinced that governments are committed to low inflation. Furthermore, it may be entirely rational for market participants, when confronted with a new regime, to discount recent inflation performance along the transition path under the new regime and to weight heavily long moving averages of past inflation performance until it is evident that policymakers are committed to living with any adverse consequences of low inflation. Laxton, Ricketts, and Rose (1993) show that, where the monetary authorities are gaining credibility along the transition path, one should observe persistent excess supply gaps. This explanation is consistent with the results reported in Figure 4, which suggest that Canada and the United Kingdom have been operating mainly in the region of excess supply since the great disinflationary episode that started in the early 1980s.

The three panels in Figure 9 provide plots of our measures of inflation π, inflation expectations π^e, and long-term inflation expectations π^LTE. These panels are useful for identifying the role of the long-term inflation proxy in generating the statistical properties of the π — π^e measure reported above. As can be seen in Figure 8, because of the larger estimated weight on the lagged inflation component (δ ≤ .2 for all three countries), the quarterly variation in the short-term inflation expectations that enter the Phillips curve is motivated by information about recent inflation performance π_t-1. However, the persistent deviation between π and π^e over longer periods of time is influenced to a significant extent by the long-term inflation proxy.

According to these estimates, market participants revise their expectations of long-term inflation very slowly in response to observed inflation performance. This interpretation of our results seems to reconcile the findings of our simple Phillips curves with other empirical work in the literature. First, as Gordon and many others have found, it takes fairly long lags on past inflation and a host of other “supply-shock” variables to save the Phillips curve. Obviously, in our highly simplistic model of the inflation process in these countries, the proxy for long-term inflation expectations fulfills this role. Indeed, our empirical results are consistent with some recent evidence that suggests that trends in long-term interest rates have long-term memory components. For example, Gagnon (1996) shows that the Fisher equation holds surprisingly well if long moving averages of past inflation are used to measure long-term inflation expectations.

Because a traditional interpretation of the demand-determined view of the business cycle requires that measures of the business cycle, such as u^* - u, must mimic measures of π - π^e, the trick of reduced-form modelers has been to search distributed lag models in order to find specifications that fit both the story and the data. In Figure 4, we have shown that, if information from the bond market is used to identify inflation expectations, one can obtain a parsimonious nonlinear specification of the unemployment-inflation process in these countries without appealing to implausible estimates of u^*. In other words, our measures of π - π^e, are consistent with both convexity in the Phillips curve and a fairly traditional interpretation of the business cycle, namely, that it is driven principally by demand shocks, with moderate changes in the underlying u^*.

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Consumer Price Inflation and Inflation Expectations

(In percent)

Citation: IMF Staff Papers 1997, 002; 10.5089/9781451947243.024.A004

Source: OECD, Analytical Data Bank.^a π^tb${\pi }_t^e=\delta {\pi }_t-(1-\delta ){\pi }_{t-1}$^c ${\pi }_t^{LTE},$

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Given the problems associated with modeling inflation expectations with fixed-parameter, reduced-form models, we are somewhat surprised that more research has not focused on using information from the bond market to help identify shifts in the Phillips curve. As McCallum (1996) points out, without reliable survey measures of inflation expectations in most countries, information from long-term bond yields is probably the only objective indicator of policy credibility. One possibility why researchers have not utilized this information more in the past may be related to the problems with measuring the ex ante real interest rate. Although our procedure of developing proxies for the ex ante real interest rate is admittedly crude, it is interesting that many of our results are not overly sensitive to alternative assumptions.

For example, one might argue that the benchmark for the end-of-sample estimate of 4.5 percent may be too high for the United States. When we use a lower estimate of the equilibrium real interest rate—3.5 percent—there is only a slight change in our estimate of γ, and α rises from 0.33 percent to 0.48 percent in the United States.¹⁴ Similar results are obtained for the other two countries. As we mentioned above, several alternatives were employed in which we changed the assumed effect of government debt on the equilibrium world real interest rate. In fact, in one polar case we doubled the effect to be more consistent with some recent estimates that suggest that the effects have been large in some countries. In another case, we assumed that the world ex ante real interest rate has been unaffected by world government debt. In addition, we used alternative assumptions about the end-ofsample benchmark ranging from 3.5 percent to 5.5 percent. In all cases, it was impossible to reject the hypothesis that valuable information in the bond market will help identify historical shifts in the Phillips curve.

VI. Conclusions and Policy Implications

Previous tests for convexity in the Phillips curve have been biased because researchers have employed filtering techniques for u^* that have been fundamentally inconsistent with the existence of convexity. A preferred statistical methodology would place both the linear and nonlinear models on an equal statistical footing by estimating model-consistent measures of u^*. This paper proposes a simple method for estimating model-consistent measures of u^* that allows for either convexity or linearity in the unemployment-inflation process. After imposing plausible restrictions on variability in u^*, we find that the nonlinear model fits the data best in Canada, the United Kingdom, and the United States.

The implications of convexity for the macroeconomic policy debate are that policymakers that are unsuccessful in stabilizing the business cycle will induce a higher natural rate of unemployment in their economies. Uncertainty about the true level of u^* reinforces the case for a cautionary strategy of raising interest rates before the economy reaches potential. Thus, rather than trying to fine-tune policy with a view to ensuring that all resources are fully employed at all points in time, it may be optimal in the context of uncertainty to develop a strategy that attempts to avoid large boom-and-bust cycles.

Indeed, in order to avoid the necessity of generating large recessions to reign in inflationary forces and reestablish the credibility of the commitment to low inflation, it may be optimal for policymakers to provide a buffer zone to guard against the possibility of serious overheating. The width of the buffer zone could be positively related both to the degree of uncertainty about the natural rate of unemployment and the degree of asymmetry in the unemployment-inflation process. As noted above, in the presence of an asymmetry in the unemployment-inflation trade-off, such a buffer zone strategy could raise the average level of output and reduce the average level of unemployment over time if it is successful in avoiding boom-and-bust cycles.