Keywords: Inflation objective, learning
Abstract:
The degree to which inflation expectations are anchored at long horizons is important for many issues in macroeconomics and finance. There has been little research examining observable measures of longrun inflation expectations. We investigate the evolution of survey measures of longrun inflation expectations in the United States. Our analysis emphasizes the role of a timevarying inflation objective of monetary policymakers. This focus makes monetary policy actions a key determinant of longrun inflation expectations. Our results have important implications for work on inflation dynamics, monetary policy rules, the costs of disinflation, and the term structure of interest rates.
JEL Codes: E3, E4
1. Introduction
The degree to which inflation expectations are tied down at long horizons, or anchored, is central to many of the key issues in macroeconomics, monetary policy, and finance. An incomplete list of topics influenced by the extent of anchoring in longrun inflation expectations includes
Before turning to our conceptual framework and results, we will note a few important strands of recent research that highlight the role of fluctuations in longrun inflation expectations.
With regard to inflation dynamics, the degree of anchoring of inflation expectations is central in most empirical and theoretical applications  as inflation is a function of inflation expectations in most treatments. On the empirical front, Roberts (2006) and Hooker (2002) have suggested that better anchoring of inflation expectations in the United States has made inflation less sensitive to variations in aggregate demand and to supply shocks (such as sharp increases in energy prices), respectively.. Also, Stock and Watson (2007) and Cogley and Sargent (2007) have suggested that inflation in the United States has become more stable, in the sense of a smaller variance of permanent shocks to inflation; this type of reducedform evidence points to changes in the stability of longrun inflation expectations (either because longrun inflation expectations are an important determinant of inflation, or vice versa). On the theoretical front, standard models of price adjustment, such as those underlying the NewKeynesian Phillips Curve, imply an important role for the longrun or steadystate inflation rate. Empirical investigations of these models show the importance of timevariation in longrun inflation for the ability of such models to capture inflation dynamics (e.g., Cogley and Sbordone (2006) and the subsample analysis of Kiley (2007a)). However, this line of research has not examined observable measures of the public's expectations for inflation at long horizons.
The degree to which public perceptions of the monetary authority's longrun inflation objective fluctuates also has important implications for monetary policy. For example, Ball (1995), Bomfim at al (1997), Erceg and Levin (2003), Roberts (2007), and Kiley (2007b) show that the cost of disinflation in many small and largescale macroeconomic models hinges on the degree to which longrun inflation expectations move in response to policy actions. The interaction between policy actions and the evolution of the public's perceptions of policy objectives plays an important role in many other recent investigations of the interaction between learning and monetary policy (e.g., Bullard and Mitra (2002), Orphanides and Williams (2005), and Gaspar, Smets, and Vestin, (2006)). While Erceg and Levin (2003) and Orphanides and Williams (2005) make some reference to inflation expectations at long horizons, none of this research examines the evolution of survey measures of longrun inflation expectations at more than a superficial level.
Finally, a number of studies have emphasized the potential importance of shifts in longrun inflation expectations for the behavior of the term structure of interest rates. For example, Gürkaynak, Sack, and Swanson (2005) found that longrun inflation expectations, as measured by the difference in yields between nominal and inflationindexed bonds, move in response to news about the economy, rather than remaining unaffected as would be implied by some simple models. More generally, models of the term structure of interest rates fit the data better under the assumption that both inflation expectations and beliefs about the central bank's reaction function are drifting (Kozicki and Tinsley (2001), Rudebusch and Wu (2003); and Cogley (2005)). Most of this research has also ignored available survey measures of longrun inflation expectations. Kozicki and Tinsley (2001) is an important exception. They include an analysis of a survey measure of longrun inflation expectations.^{3} But their work assumes that longrun inflation expectations are a function of an adaptivelearning algorithm, implying that these expectations are solely a function of past inflation. We take a different tack, and consider a model in which monetary policy actions are the key determinant of longrun inflation expectations. Given that inflation at long horizons is (in most models) solely a function of monetary policy choices, our approach seems likely to provide important insights. Our empirical work will compare our approach with an adaptivelearning specification.
The next section discusses our conceptual framework. Section 3 presents the basic empirical results. Section 4 compares our approach to an adaptive learning approach, examines the implications of our results for work on inertia in monetary policy rules, and briefly discusses the link between our findings and work on the cost of disinflation. The final section summarizes our findings and suggests areas for further investigation.
2. The conceptual framework
Suppose that monetary policymakers pursued a policy with a timevarying inflation target consistent with a generalized Taylorstyle nominal interest rate rule. Specifically, the nominal interest rate (i) is a linear function of its own lags, the longrun real interest rate (r*, assumed to be constant), inflation (p), the gap between inflation and a timevarying goal for inflation (pp*), a measure of the output gap or economic activity (y), and a residual (e):
Equation 1
Our focus will be on possible variation in the inflation goal. Following much previous work (e.g., Bomfim et al (1997), Erceg and Levin (2003), Kozicki and Tinsley (2001), and Cogley and Sbordone (2006)), we assume that the longrun inflation goal shifts over time in a randomwalk manner, i.e.,
Equation 2
Of course, the randomwalk specification for the longrun inflation goal is not meant to suggest that monetary policy has allowed for "random" variation in the longrun inflation objective. Rather, this specification is a convenient mathematical representation of the notion that the longrun objective for inflation may have shifted over time in response to changes in economist's and policymaker's understanding of the nature of inflation fluctuations and the costs associated, in the long run, with different levels of inflation. Ireland (2007) provides a reducedform model in which some macroeconomic fundamentals drive time variation in the longrun inflation goal.
With this basic framework, we can derive the efficient method that private agents would use to infer the longrun inflation goal. If agents form their inference so as to minimize the quadratic difference between the true longrun goal for inflation and their estimate, or if the shocks to the policy rule (e(t) in equation 1) and inflation goal (v(t) in equation 2) are governed by a Normal distribution, then the estimate of the inflation goal derived from observations on nominal interest rates, inflation, and the output gap via the Kalman filter is the optimal one.
The solution to this signal extraction problem is standard (e.g., Muth (1961) and Sargent (1987)). We focus on the steadystate, or longrun, equation governing the evolution of the public's beliefs. The implied evolution of the perceived inflation goal Ep(t) is governed by
Equation 3
If the econometrician has available data on the public's perception of the longrun goal for inflation, Equation 3 is estimable directly. However, more efficient coefficient estimates could be obtained by estimating the updating equation (equation 3) with the policy rule (equation 1). But equation 1 is not directly estimable, as it involves the true inflation goal  i.e., any available data on the public's perception should not be inserted into the policy rule. This difficulty can be overcome by first differencing equation 1, yielding
Equation 4
3. Empirical results
3.1 Data
We present empirical results using two measures of the public's expectations of inflation at long horizons. The first measure is from surveys of professional forecasters. These surveys measure expectations of the average rate of change in the Consumer Price Index (CPI) over the next ten years. The summary measure we construct splices together observations from surveys by the private firms DrexelBurnhamLambert and Barclays de Zoete Wedd covering the period from September 1978 to September 1991 with the observations from the Survey of Professional Forecasters covering the period after September 1991. We will call this measure the professional forecasters' measure.
The second measure is from the ReutersMichigan Survey Research Center Survey of Consumers. This series measures the rate of price change expected over the next fivetoten years; we use the median of the responses collected in the survey as our measure of the central tendency of household's longrun expectations. (Of course, there is some dispersion in expectations in both surveys; we do not exploit this dispersion, and view future investigations of the crosssectional variation in the survey measures as an interesting topic for research, as in Mankiw, Reis, and Wolfers (2003)). We will call this measure the household measure.
Both measures of longrun inflation expectations contain a number of missing observations, as the survey questions were only posed at irregular intervals early in our sample period. We linearly interpolate between missing observations. Figure 1 presents the raw and interpolated series for the professional forecasters' measure in the upper panel (with the raw series denoted by the dots and the interpolated series by the line) and the raw and interpolated series for the household measure in the lower panel (with the raw series denoted by the dots and the interpolated series by the line); the period is 1978Q1 to 2007Q3. The raw and interpolated data look very similar; in particular, inflation expectations over the longerrun rose early in the sample, then began a long downward trend beginning around 1981, and appear to have stabilized in the late 1990s. Our empirical analysis will consider both the interpolated and raw series.
In addition, the data from the Survey of Professional Forecasters and the Survey of Consumers refer to inflation expectations over some range of future years, including expectations for the nearterm. As a result, these survey measures depend on both near and longterm expectations. In order to focus on longterm expectations, we assume that inflation expectations converge to longrun expectations after three years (a reasonable horizon in models of inflation dynamics like the NewKeynesian Phillips curve, where inflation persistence is not very high  see, for example, Cogley and Sbordone (2006)). With the survey measures (roughly) referring to inflation expectations over the next 10 years (40 quarters), our assumption about convergence to longrun expectations implies that the survey measure of inflation (p is related to nearterm expectations of inflation (p) and longrun expected inflation (p by
Equation 5
Our model assigns monetary policy actions the central role in guiding longrun expectations; in particular, the updating of private sector beliefs is determined by the influence of movements in the policydetermined nominal interest rate relative to movements in inflation and economic activity, where the latter is captured by the output gap. The nominal interest rate is measured by the effective nominal federal funds rate (at an annual rate); inflation is measured by the percent change in the Consumer Price Index over the past four quarters (where the emphasis on a fourquarter change follows Taylor (1993) and most other empirical work); and the output gap is given by the percent deviation of Gross Domestic Product (GDP, in chainweighted 2000 dollars) from trend GDP as measured in the Federal Reserve Board's FRB/US model.
Figure 2 presents the data for CPI inflation, the nominal federal funds rate, and the output gap since 1978. The lowfrequency movements in inflation and the nominal interest rate are readily apparent: inflation and nominal interest rates rose early in the sample and have trended down since the early 1980s. The data for the output gap clearly show the recessions in the early 1980s, early 1990s, and early 2000s.
Our final set of statistics before turning to our analysis examines the basic moments of the data for CPI inflation and our two survey measures of inflation expectations in table 1. We report the mean, median, and standard deviation of each series (where we include interpolated observations for our survey measures). The means and medians for inflation and expected inflation at longer horizons are very similar, while actual inflation is much more volatile than expected inflation at longer horizons (as measured by the standard deviation). Each is these facts is broadly consistent with the ideas underlying our model of the determination of longrun inflation expectations, with relies on updating via the Kalman filter and hence will tend to imply that longrun expectations have the same central tendency as actual inflation but are less volatile.
3.2 Basic results
We now turn to our formal analysis of the determinants of longrun expectations. Our initial results rely simply on estimating the updating equation for longhorizon expectations  equation 3; we defer joint estimation of the updating equation and monetary policy rule for later. For our baseline empirical results, we restrict attention to a simple Taylorrule specification without lags of nominal interest rates. In this specification, the nominal interest rate rule depends only on the longrun real interest rate (r*), inflation, the inflation gap, and the output gap. As in Taylor (1993), we fix the longrun real interest rate (r*) at 2 percent. With these restrictions and the replacement of the public's beliefs about the longrun goal for inflation (Ep*) with the data for the longrun expectation derived from equation 5 (p in equation 3, the estimated equation is
Equation 6
We present results for four versions of equation 6 in table 2. The first two (in columns 1 and 2) use the longhorizon measure of expectations (p from equation 5) based on the professional forecasters' measure. The first specification jointly estimates the updating coefficient ( and the coefficients from the policy rule ( and . The second specification sets the coefficients in the policy rule at the values suggested by Taylor (1993), i.e., 0.5 on both the inflation deviation from target ( and the output gap (; these values are both consistent with a long literature and provide a robustness check on the results. The last two specifications reported in columns 3 and 4 of table 2 are the same as those in columns 1 and 2, with the longhorizon measure of expectations based on the household measure replacing the professional forecasters' measure. In all cases, the equation is estimated by the generalized method of moments. The instruments used include the current and lagged values for the unadjusted survey measure of longhorizon inflation expectations (p(t) and p(t1)), the lagged fourquarter change in the overall CPI (p(t1)), the lagged fourquarter change in the CPI excluding food and energy (p(t1)), the lagged nominal federal funds rate (r(t1)), and the lagged output gap (y(t1)). Standard errors are corrected for heteroskedasticity. The sample extends from 1979Q1 to 2004Q2 (where the last three years of data drop out because of the adjustments to the survey measure induced by equation 5).
The results are very similar for each survey measure. Consider first the cases where the updating and policy rule coefficients are each estimated (columns 1 and 3). In both cases, the updating coefficient ( is around 0.015 and the policy rule coefficients ( and are large (e.g., uniformly exceeding 1). In addition, the standard errors are large; as a result, none of the coefficients differ from zero in a statistically significant manner, although all have the sign suggested by the model in section 2.
Columns 2 and 4 set the policy rule coefficients at their Taylor (1993) values. As a result, the updating coefficient ( is estimated to be slightly larger  at 0.020 for the professional forecasters' measure and 0.016 for the household measure  and the standard errors are smaller, implying that the coefficients are significant at the 1 percent or 5 percent level, respectively. We will defer discussion of the economic implications of the magnitude of the updating coefficient for section 4. For now, we note that the results from estimating the updating rule (equation 6) provide some support for the basic framework, in that the coefficients have the signs consistent with
Equation 7
Table 3 presents these results; column 1 reports the estimates based on the professional forecasters' measure of expectations, while column 2 reports the results based on the household measure. The updating coefficient ( is estimated at 0.020 and 0.026 for the professional forecasters and household data, respectively, and each is statistically significant at better than the 1 percent level. The coefficient on the output gap ( is large and highly statistically significant for the systems based on both survey measures; however, the coefficient on inflation ( is imprecisely estimated in each case. Overall, the results strengthen those based on the updating equation alone, in that the significance of deviations from the policy rule in influencing longrun inflation expectations  i.e., the size and statistical significance of the updating coefficient  is greater under the system approach.
3.3. Robustness checks
We now turn to some robustness checks. In particular, our analysis so far has been based on the interpolated data and the measure of long horizon inflation expectations that adjusts for realized CPI inflation over the next twelve quarters, p.
We first consider the robustness of the results to measuring longrun expectations by the unadjusted survey measure p (while preserving the interpolations). The results are reported in table 4. Columns 1 and 2 report results from joint estimation of the updating equation and the policy rule (the system of equations 7 and 8) with the unadjusted survey measure. The results are essentially the same as those using the adjusted survey measure for longhorizon inflation expectations: the updating coefficient is highly significant as is the policyrule coefficient on the output gap , while the coefficient on inflation is imprecisely estimated.
Our final robustness check drops the interpolations to the survey measures of expectations (and the adjustment for the next twelve quarters of inflation). Dropping interpolations greatly reduces the sample sizes, as the updating equation is dynamic and hence relies on consecutive nonmissing observations. Table 5 reports the results based on system estimation of the updating and policy rule equations (the system of equations 7 and 8). For the professional forecasters' measure, dropping interpolations has little effect on the estimated coefficients (which remain at values similar to those reported previously). However, the results based on the household measure are no longer supportive of updating of expectations based on monetary policy actions; in particular, the updating coefficient is not significantly different from zero and has the wrong sign. This result is not too surprising, as the sample for the household measure with interpolations includes no observations prior to 1986:Q2 and much of the variation in the data occurs during that earlier period.
To summarize the results, our estimates of the relationship between changes in longrun inflation expectations and deviations of monetary policy from a perceived rule provide some support for the idea that longrun inflation expectations depend importantly on monetary policy actions and the evolution of actual inflation and real economic activity. In particular, the coefficients from estimates of an updating equation for longrun inflation expectations based on our Kalmanfiltering model imply that
4. Implications of our results
4.1 Comparison to adaptive learning
We have pursued a very focused empirical approach that emphasizes the idea that the public's assessment of the longrun inflation goal of the monetary authority will be influenced by the perceived discrepancy between policy actions and those consistent with the public's previous perception of the longrun inflation goal. Some earlier work has taken a different approach, and suggested that inflation expectations mainly depend on past realizations of inflation. For example, Kozicki and Tinsley (2001) assume that the data on the professional forecasters' measure of longhorizon inflation expectations depends on past inflation and call this assumption an adaptivelearning algorithm. Levin, Natalucci, and Piger (2004) examine the correlation between longhorizon inflation expectations and inflation, but do not consider the role of other factors in influencing longrun inflation expectations.
Our approach and the adaptive learning approach share the feature that private agents' assessment of the longrun goal for inflation is not fixed and depends on incoming data because the inflation goal is not directly observable. The central difference between the adaptive learning approach and our approach is that our emphasis on monetary policy actions gives a role for nominal interest rates and the output gap in the updating of longhorizon inflation expectations. To see the empirical relevance of our approach, we estimate by least squares the following reducedform regressions for the change in both the professional forecasters and household survey measures of longhorizon inflation expectations:
Equation 9
4.2 Estimation of monetary policy rules
We have already noted that the behavior of inflation expectations at long horizons has important implications for a number of important issues in macroeconomics, monetary policy, and finance. We will discuss in detail one area where the model and empirical analysis can provide new insights into an unresolved controversy: the degree of smoothing of monetary policy actions since 1979.
Our empirical results in section 3 ignored a role for lagged nominal interest rates in the policy rule. A more general nominal interest rate rule that allows for smoothing takes the form shown in equation 1, i.e.,
It should be clear that our model of policymaker behavior implies serial correlation in the errors of policy rules that ignore timevariation in the inflation goal, because the inflation goal is a persistent omitted variable from such equations according to our model. In fact, our model implies that such specifications ignore a unitroot component in the policy rule, indicating that specifications with smoothing but without a timevarying inflation goal are quite likely to estimate large coefficients on the lagged dependent variable. This occurs because, according to our model, the policy rules estimated by many researchers omit an important variable by ignoring timevariation in the inflation target. Rudebusch (2002, 2006) has previously suggested that findings of policy smoothing probably reflect an omitted variable from the policy rule.
To examine a separate role for smoothing and a timevarying inflation target, we estimate the following system of equations:
Equation 10
Table 7 reports the results for the professional forecasters and household measures (in columns 1 and 2, respectively). In both cases, there is no support for interest rate smoothing, as the coefficient a is estimated to lie very close to or below zero. The inclusion of the additional variable also lowers the size and significance of the updating coefficient ; given that there is no support for smoothing, the lower size and significance of the updating coefficient may simply reflect the added noise induced into the system by inclusion of the lagged nominal interest rate. We readily admit that this test is a bit weak  it is extremely difficult to differentiate between a lag of the interest rate and a moving average error in a specification like equation 11.
Nonetheless, we interpret these findings as consistent with the idea that empirical evidence for a high degree of interest rate smoothing may stem from omitted variable bias as suggested by Rudebusch. Timevariation in the longrun goal for inflation, as suggested by the variation in longhorizon inflation expectations, seems a likely source of this omitted variable bias. This view is consistent with the variation in longhorizon inflation expectations in figure 1.
4.3 The cost of disinflation
In an economy in which the longrun inflation goal is varying over time, the speed with which the public's perception of the inflation goal is updated is a key determinant of the cost of disinflation, or the sacrifice ratio (Bomfim et al (1997), Erceg and Levin (2003), Kiley (2007b) and Roberts (2007)). Our empirical work captures this speed in the coefficient , and we estimated at a value near 0.02, give or take a bit.
Let's define the sacrifice ratio as the sum of the output gap over a period of disinflation divided by the change in inflation. Previous work has suggested a wide range of values for this measure  from about 2 to 4 over the last forty years in the United States (see Bomfim et al (1997)). In order to go from our estimate of to a value for the sacrifice ratio requires a complete generalequilibrium model, and such an exercise is well beyond the scope of this paper. But Roberts (2007) considers this question in a fairly standard NewKeynesian general equilibrium model (that is not too dissimilar to that used in Erceg and Levin (2003)). He reports that a value for near 0.02 is consistent with a sacrifice ratio near 2 over a fiveyear horizon.
5. Summary
An understanding of the historical variation in inflation expectations, especially at long horizons, is important for questions in macroeconomics and finance such as the nature of inflation dynamics, the costs of disinflation, and the relationship between yields on longterm bonds and shortterm interest rates. We have provided an empirical examination of the variation in longrun inflation expectations that emphasizes the role of monetary policy actions.
We reach three conclusions regarding the experience in the United States since the late 1970s. First, longrun inflation expectations are not a function of past inflation alone, but depend on monetary policy actions, inflation, and economic activity. Reducedform regressions provide weak evidence for these conclusions; our structural model provides stronger evidence. These results point to straightforward and easily interpretable results: tighter monetary policy lowers longrun inflation expectations, and higher inflation and stronger economic activity raise longrun inflation expectations absent a monetary policy response. Second, previous work that finds large interestrate smoothing in monetary policy reaction functions has largely ignored timevariation in the inflation goal of the monetary authority; when such variation is taken into account, there appears to be little support for interest rate smoothing. And finally, our estimates for the speed with which monetary policy actions may influence longhorizon inflation expectations appears to be consistent with reducedform evidence on the cost of disinflation, or the size of the sacrifice ratio, that has been reported in other studies.
Our research raises a number of interesting questions for further research. Examples include the following:
Ball, Laurence M. (1995) "Disinflation with Imperfect Credibility," Journal of Monetary Economics, February 1995, pp. 523.
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Cogley, Timothy and Argia M. Sbordone (2006) Trend Inflation and Inflation Persistence in the New Keynesian Phillips Curve. Mimeo, University of California, Davis, July.
Cogley, Timothy, and Thomas J. Sargent (2007). "Inflation Gap Persistence in the U.S. (5.7 MB PDF)" University of California, Davis, working paper, January.
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Erceg, Christopher, and Andrew Levin (2003). "Imperfect Credibility and Inflation Persistence," Journal of Monetary Economics, vol. 50 (May), pp. 91544.
Gaspar, Vitor, Frank Smets, and David Vestin (2006). "Adaptive Learning, Persistence, and Optimal Monetary Policy," Journal of the European Economic Association, vol. 4 (AprilMay), pp. 37685.
Gürkaynak, Refet, Brian Sack, and Eric Swanson (2005). "The Sensitivity of Longterm Interest Rates to Economic News: Evidence and Implications for Macroeconomic Models," American Economic Review, vol. 95 (March), pp. 42536.
Hooker, Mark A. (2002) "Are Oil Shocks Inflationary? Asymmetric and Nonlinear Specifications versus Changes in Regime." Journal of Money, Credit, and Banking, May 2002, v. 34, iss. 2, pp. 54061.
Ireland, Peter N. (2007) "Changes in the Federal Reserve's Inflation Target: Causes and Consequences," Journal of Money, Credit, and Banking, December 2007.
Kiley, Michael T. (2007a) "A Quantitative Comparison of StickyPrice and StickyInformation Models of Price Setting," Journal of Money, Credit, and Banking, vol. 39 (February), pp. 101125.
Kiley, Michael T. (2007b) Inflation Dynamics and Monetary Policy: Comparing Small and LargeScale Policy Models. Paper presented at the Federal Reserve System policy Conference, May 16, 2007, St. Louis, MO.
Kozicki, Sharon, and Peter Tinsley (2001). "Shifting Endpoints in the Term Structure of Interest Rates," Journal of Monetary Economics, vol. 47(June), pp. 61352.
Levin, Andrew, Fabio Natalucci, and Jeremy Piger (2004). "The Macroeconomic Effects of Inflation Targeting (406 KB PDF)," Federal Reserve Bank of St. Louis, Review, vol. 86 (July), pp. 5180.
Mankiw, N. Gregory, Ricardo Reis, and Justin Wolfers (2003). "Disagreement about Inflation Expectations," NBER Macroeconomics Annual, pp. 209248.
Orphanides, Athanasios, and John C. Williams (2005). "Inflation Scares and Forecastbased Monetary Policy," Review of Economic Dynamics, vol. 8 (April), pp. 498527.
Orphanides, Athanasios (2007) "Taylor Rules." Federal Reserve Board Finance and Economics Discussion Series 200718.
Roberts, John (1998) "Inflation Expectations and the Transmission of Monetary Policy." Federal Reserve Board Finance and Economics Discussion Series 199843.
Roberts, John (2006). "Monetary Policy and Inflation Dynamics," International Journal of Central Banking, vol. 2 (September), pp. 193230.
Roberts, John (2007) "Learning, Sticky Inflation, and the Sacrifice Ratio." Mimeo, Federal Reserve Board.
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Rudebusch, Glenn (2006) "Monetary Policy Inertia: Fact or Fiction?" International Journal of Central Banking 2 (4), December 2006, 85135.
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CPI inflation  Longhorizon expectation, professional forecasters  Longhorizon expectation, households  

Mean  4.1  4.2  4.1 
Median  3.4  3.6  3.5 
Std. Dev.  3.2  1.8  1.6 
Note: CPI inflation is measured by the percent change in the Consumer Price Index from the previous quarter, at an annual rate. The longrun expectations refer to the interpolated series presented in figure 1 based on several different surveys, as discussed in the text.
Parameter  (1) Professional Forecasters 
(2) Professional Forecasters 
(3) Household Measure 
(3) Household Measure 

0.017
(0.011) 
***0.020
(0.007) 
0.013
(0.013) 
**0.016
(0.007) 

1.03
(1.52) 
0.5  1.76
(2.55) 
0.5  
2.24
(1.51) 
0.5  3.27
(3.30) 
0.5 
Note: Estimated via GMM. See text for discussion of instruments. Standard errors reported in parentheses (corrected for heteroskedasticity). ***, **, * denotes statistically different from zero at the 1, 5, and 10 percent levels, respectively.
Parameter  (1) Professional Forecasters 
(2) Household Measure 

***0.020
(0.006) 
***0.026
(0.005) 

0.30
(0.38) 
0.22
(0.32) 

***1.22
(0.26) 
***1.59
(0.31) 
Note: Estimated via GMM. See text for discussion of instruments. Standard errors reported in parentheses (corrected for heteroskedasticity and firstorder serial correlation). ***, **, * denotes statistically different from zero at the 1, 5, and 10 percent levels, respectively.
Parameter  (1) Professional Forecasters 
(2) Household Measure 

***0.015
(0.005) 
***0.019
(0.004) 

0.30
(0.45) 
0.20
(0.37) 

***1.00
(0.26) 
***1.44
(0.31) 
Note: Estimated via GMM. See text for discussion of instruments. Standard errors reported in parentheses (corrected for heteroskedasticity and firstorder serial correlation). ***, **, * denotes statistically different from zero at the 1, 5, and 10 percent levels, respectively.
Parameter  (1) Professional Forecasters 
(2) Household Measure 

***0.015
(0.005) 
0.007
(0.010) 

0.02
(0.52) 
0.30
(0.31) 

***0.83
(0.29) 
***0.62
(0.19) 
Note: Estimated via GMM. See text for discussion of instruments. Standard errors reported in parentheses (corrected for heteroskedasticity and firstorder serial correlation). ***, **, * denotes statistically different from zero at the 1, 5, and 10 percent levels, respectively. Dropping
interpolations changes the sample periods. In column 1, the sample for the updating equation is 1980Q3 to 2004Q2, with observations missing in 1991Q3 and 1991Q4; the sample for the rule equation is 1980Q4 to 2004Q2, with observations missing in 1991Q4 and 1992Q1. In column 2, the sample for the
updating equation is 1986Q2 to 2004Q2, with observations missing from 1988Q1 to 1990Q2; the sample for the rule equation is 1986Q3 to 2004Q2, with observations missing from 1988Q2 and 1990Q3.
Parameter  Professional Forecasters  Households 

a  ***0.044 (0.011)  0.015
(0.022) 
a  ***0.014 (0.005)  0.005
(0.009) 
a  0.001
(0.007) 
***0.031
(0.011) 
Parameter  (1) Professional Forecasters 
(2) Household Measure 

a  0.21
(0.23) 
0.02
(0.13) 
0.003
(0.002) 
0.001
(0.003) 

0.27
(0.39) 
0.32
(0.33) 

***1.26
(0.32) 
***1.07
(0.33) 
Note: Estimated via GMM. See text for discussion of instruments. Standard errors reported in parentheses (corrected for heteroskedasticity and firstorder serial correlation). ***, **, * denotes statistically different from zero at the 1, 5, and 10 percent levels, respectively.