#### 4.1. Comparison with the Observable Data on Expectations

Next, I study the extent to which the estimated path of

${\pi}_{t}^{e}$ is comparable to the available inflation forecasts data series described in

Section 3.

Figure 4 and

Figure 5 report the median, and 16th and 84th percentiles, of the retained draws of

${\pi}_{t}^{e},$ together with the Michigan, Cleveland, and SPF measures of inflation expectations. For the SPF, I report the inflation forecasts for the next quarter (denoted as

$t+1$) and for the next year (denoted as

y).

Figure 4 shows all the available data, while

Figure 5 focuses on the sub-sample

$2007:I-2017:III$, which is the period under analysis in

Coibion and Gorodnichenko (

2015).

Table 2 and

Table 3 report information about the correlations between the estimated path of

${\pi}_{t}^{e}$ and the observable measures of inflation expectations. For each measure, one correlation value is computed with each of the 1000 retained draws of

${\pi}^{e,T}$; the tables report the median, 16th and 84th percentiles of these values.

Table 2 uses all the available data, while

Table 3 splits the sample in two periods: before

$2007:I$ (excluded) and after

$2007:I$ (included). Again, the analysis on the split sample is done to compare the results with those of

Coibion and Gorodnichenko (

2015). In this table, the data from the SPF includes one additional variable, inflation forecasts for the current period, for both headline and core CPI.

11 Figure 4 and

Figure 5 show that, in most periods, the estimated path of

${\pi}_{t}^{e}$ mimics the behaviour of several of the available measures of inflation forecasts. These results are confirmed by the correlations reported in

Table 2 and

Table 3. Most notably, the forecasts of the University of Michigan Survey of Consumers are found to be very highly correlated with the estimated expectation, particularly in the part of the sample before 2007 (the median correlation is 0.9128 for this period). In addition,

Figure 5 shows that the forecasts of the Michigan Survey still behave quite similarly to the estimated

${\pi}_{t}^{e}$ in the years from 2007 to 2012 (result that is consistent with the findings of

Coibion and Gorodnichenko 2015), but the relationship seems to become weaker after 2012. In general,

Table 3 suggests that for several of the measures the correlation with the estimated

${\pi}_{t}^{e}$ decreases after 2007, with the exception of the expectations computed by the Federal Reserve Bank of Cleveland and the SPF forecasts of headline CPI inflation for the current period.

The findings of this paper contribute to the discussion about the “missing disinflation” puzzle, which is the focus of the analysis of

Coibion and Gorodnichenko (

2015). This puzzle refers to the absence of disinflation during the recent Great Recession, when the “slack” in the economy was very high so, according to standard models of the Phillips curve, the inflation rate should have been much lower. A model of the Phillips Curve as the one described by (

1) will rationalize the variation in

${\pi}_{t}$ as due to changes in the measure of slack

${x}_{t}$, to realizations of the shocks

${e}_{t}$, and to the behaviour of expectations. The specific framework employed in this paper allocates this variation among

${x}_{t}$,

${e}_{t}$, and

${\pi}_{t}^{e}$, by estimating the parameters of the models jointly with the entire history

${\pi}^{e,T}$. This implies that, by construction, any missing disinflation (or excess inflation) arising from the estimated model will be accompanied by a path

${\pi}^{e,T}$ that can rationalize it. The exercise of comparing the observable measures of expectations to the estimated history

${\pi}^{e,T}$ can thus also provide information on the extent to which each of these measures can explain the missing disinflation. As shown in

Figure 5, in the years from 2007 to 2012 the inflation forecasts of the Michigan Survey appear to more closely track the behaviour of the estimated expectations relative to the Cleveland or SPF forecasts. This result suggests that, indeed, the expectations of households recorded in the Michigan Survey can explain the missing disinflation puzzle, consistently with what argued by

Coibion and Gorodnichenko (

2015) and

Binder (

2015).

To provide a more thorough analysis of the similarities between the estimated

${\pi}^{e,T}$ and the observable inflation forecasts, I computed a few other statistics in addition to the correlations reported in

Table 2 and

Table 3. These statistics are the average, variance, and first order autocorrelation of the expected inflation series. The results obtained using all the available data are shown in

Table 4, while

Table 5 and

Table 6 focus on 3 different sub-periods: the period before the Great Moderation (1978–1984), the Great Moderation (1985–2006), the Great Recession and post-Great Recession period (2007–2017).

12 For simplicity of the analysis, in this exercise I only used the median of the retained draws of

${\pi}^{e,T}$.

Table 4 and

Table 5 show that the estimated expectations and the observable inflation forecasts imply very similar average expected inflation rates, although the differences are slightly larger for the period 2007–2017 (see

Table 5, third column). In terms of variances,

Table 4 suggests that the estimated series

${\pi}^{e,T}$ is generally more volatile than the observable measures of expectations. However, the results reported in

Table 6 indicate that there are some differences across measures and across periods. For instance, the expectations recorded in the Michigan Survey exhibit a volatility that is quite high, and almost comparable to that of

${\pi}^{e,T}$, in the sub-sample 1978–1984 but not in later years. On the other hand, the SPF forecasts of headline CPI inflation for the current period are as volatile as the estimated expectations across the entire sample period. The variance of all the other observable measures of expectations is lower, particularly in the years from 2007 to 2017. Finally,

Table 4 and

Table 7 report the first order autocorrelations of the series. The Michigan Survey data and the estimated expectations exhibit very similar autocorrelations; this is true across the entire sample period under analysis. With respect to the other measures, many of them (specifically, the SPF inflation forecasts for the next period or the next year and the Cleveland inflation expectations) seem to be more autocorrelated than the estimated

${\pi}^{e,T}$, especially during the sub-period 1985–2006.

Theoretical models of the NKPC are typically clear about what type of expectations should be included in equations like (

1). For instance, in the standard New Keynesian model presented in

Gali (

2008), expectations are assumed to be formed at time

t for the horizon

$t+1$. As these models are often interpreted in terms of quarterly data, the horizon for which they assume expectations to be formed is in fact very short. The framework that I employ in this paper, however, is built to simply reverse-engineer a path of the variable

${\pi}_{t}^{e}$, that evolves according to (

2), and that captures the fraction of the variation of

${\pi}_{t}$ that cannot be explained by

${x}_{t}$ or attributed to the shock

${e}_{t}.$ This implies that the estimated path of

${\pi}_{t}^{e}$ could, in practice, reflect different types of expectations. Overall, the results presented so far seem to indicate that the short term measures of expectations employed in the analysis resemble the estimated path of model-consistent expectations in several directions. However, longer term expectations could in principle perform equally well.

In order to address this question, I considered one additional observable measure of expectations: the Michigan Survey “median forecast of price changes over the next 5 to 10 years”. I examined the correlation with the estimated

${\pi}^{e,T}$, in addition to the variance and the autocorrelation of the series, and I compared these statistics to those obtained in the first part of the analysis for the Michigan Survey “median forecast of price changes over the next 12 months”. As the forecasts over the next 5 to 10 years are only available starting from 1990, I computed all the statistics for the full sample 1990–2017, and for the two sub-samples 1990–2006 and 2007–2017. The results of this exercise are reported in

Table 8. The table shows that for almost all the calculated statistics, the estimated path of expectations is more similar to the forecasts of inflation over the next 12 months than to the forecasts over the next 5 to 10 years. This result seems to indicate that the estimated history

${\pi}^{e,T}$ more closely mirrors expectations for the short horizon, in line with the theoretical assumptions of the NKPC model. However, the differences emerging from

Table 8 are not very large, and do not allow to make strong statements in this respect.

The path of

${\pi}^{e,T}$ estimated in this paper could capture different types of expectations not only with respect to the horizon of reference, but also in other dimensions. For instance, these expectations could refer to alternative measures of inflation, or they could reflect the forecasts of some groups of the population more than others. In terms of inflation measures, the SPF data used in the paper include both headline and core CPI inflation forecasts. This choice was an attempt to account for the fact that agents could base their expectations over different measures of inflation. Unfortunately, the core inflation forecasts are only available for a very short period of time, so the results of this analysis are only indicative. With respect to the expectations of different demographic groups, the recent work of

Binder (

2015) has shown that their impact on the dynamics of inflation might not be uniform.

13 To explore this direction in more depth, I performed one last exercise in which I studied the correlations between the estimated expectations and the Michigan Survey “median forecast of price changes over the next 12 months” for different demographic groups. In this exercise, I used all the available disaggregated data, for all the available years. The results are reported in

Table 9; the first row reproduces the results obtained from the aggregate data, for comparison. The table shows that the correlation between the estimated expectations and the survey data is roughly the same across regions, and is not affected by the gender of the respondents. The forecasts of respondents over 55 years of age or in the bottom 33% income group are less correlated to the estimated

${\pi}^{e,T}$ and, in general, the correlations increase with the income and education of the respondents. These results are consistent with those reported by

Binder (

2015), although, overall, the differences among demographic groups emerging from

Table 9 do not seem to be very large.

As previously mentioned, I performed several robustness checks using different values of the parameters in the prior distributions (more specifically, the means and variances of the normal distributions and the scale parameters of the Inverse Wishart distributions). I also experimented with different burn in periods and sampling lengths in the MCMC estimation procedure, as detailed in the

Appendix A. Finally, I repeated the estimation using core or headline PCE instead of CPI to compute

${\pi}_{t}$. The main results of the paper were materially unaffected by these changes.