Behavioral Heterogeneity in U.S. Inflation Dynamics

ABSTRACT

In this article we develop and estimate a behavioral model of inflation dynamics with heterogeneous firms. In our stylized framework there are two groups of price setters, fundamentalists and random walk believers. Fundamentalists are forward-looking in the sense that they believe in a present-value relationship between inflation and real marginal costs, while random walk believers are backward-looking, using the simplest rule of thumb, naive expectations, to forecast inflation. Agents are allowed to switch between these different forecasting strategies conditional on their recent relative forecasting performance. We estimate the switching model using aggregate and survey data. Our results support behavioral heterogeneity and the significance of evolutionary learning mechanism. We show that there is substantial time variation in the weights of forward-looking and backward-looking behavior. Although on average the majority of firms use the simple backward-looking rule, the market has phases in which it is dominated by either the fundamentalists or the random walk believers.

KEY WORDS:

• Evolutionary selection
• Heterogeneous expectations
• Phillips curve.

Acknowledgments

The authors thank Peter Boswijk and João Madeira for comments. This article was presented at the Society for Computational Economics (2012), Asian Econometric Society Meeting (2013), European Econometric/Economic Society Meeting (2013), Bocconi University (2013), New York Federal Reserve Board (2013), Royal Economic Society (2014), and International Economic Association World Congress (2014). Financial support from the British Academy’s PDF/2009/370 and from the EU 7th framework collaborative project “Monetary, Fiscal and Structural Policies with Heterogeneous Agents (POLHIA),” 1grant no. 225408 and from the EU 7th framework collaborative project “Complexity Research Initiative for Systemic InstabilitieS (CRISIS),” grant no. 288501 is gratefully acknowledged.

Notes

1 The assumption of RE in the formulation of intertemporal optimization decisions was also criticized by Hendry and Mizon (2010, 2014) in the presence of unanticipated structural breaks on the grounds that the law of iterated expectations needs not to hold when distributions shift, as integrals are taken over different weighted intervals. Castle et al. (2014) found evidence for such shifts when fitting the hybrid NKPC to U.S. inflation data and demonstrate that a potentially spurious outcome can arise when the NKPC is estimated under the assumption of RE.

2 Arifovic, Bullard, and Kostyshyna (2012) considered a DSGE model in which agents’ beliefs evolve through social learning dynamics and show that the Taylor principle is not necessary for convergence to the minimum state variable solution under social learning.

3 Heterogeneity in individual expectations has also been found in other settings. For example, Frankel and Froot (1990), Allen and Taylor (1990), and Ito (1990) found that financial experts use different forecasting strategies to predict exchange rates, while Hommes et al. (2005), Adam (2007), Pfajfar and Zakelj (2014), and Assenza et al. (2011) found evidence for heterogeneity in learning to forecast laboratory experiments with human subjects.

4 See also Stock and Watson (2007) and Groen, Paap, and Ravazzolo (2013) who acknowledged the possibility of the changing time-series properties of inflation.

5 The model derivation in the presence of a discrete number of belief types, which is a particular case of the general specification derived in this section, is outlined in the online appendix.

6 See the online appendix for a detailed derivation.

7 See the online appendix for details.

8 Notice also that we cannot directly impose a structure on ξ_t since we will make assumptions about how agents forecast inflation but not about how agents forecast prices. From a behavioral point of view, forecasting prices is rather different than forecasting inflation (see, e.g., Tuinstra and Wagener 2007). In fact, while we will make specific assumptions on inflation expectations on the basis of observable statistical or theoretical properties of the inflation process, it is more difficult to model price expectations since in reality agents rarely collect information or read news about prices in levels.

9 We justify the fact that the law of iterated expectations holds at the individual level in the presence of evolutionary switching by appealing to the learning literature which models the selection of forecasting rules as a distinct statistical problem. Thus agents choose a forecasting model and then use that model to solve for their optimal plan in the anticipated utility sense, as in Kreps (1998) and Sargent (1999).

10 Technically, because the discounted sum of real marginal costs starts at k = 1, we measure it using ${(I-\delta {\widehat{A}}_{\left(t\right)})}^{-1}{\widehat{A}}_{\left(t\right)}{Z}_t$ instead of ${(I-\delta {\widehat{A}}_{\left(t\right)})}^{-1}{Z}_t$.

11 In principle the switching metric should depend on the degree of price stickiness, that is, the stickier the prices the longer the horizon the firms should take into account. We leave this issue for future research and, in what follows, we consider the average forecast error over the previous K periods as switching metric.

12 The estimation results are robust to alternative specifications of the fitness measure. We chose relative absolute forecast error for numerical convenience, since it restricts the support of the fitness measure to the interval [ − 1, 0].

13 We repeated our estimation exercise using samples with different starting dates and results are not qualitatively different.

14 From a behavioral point of view it seems a sensible choice to pick K = 4 for quarterly data, meaning that the fitness measure takes an average of the forecast errors over the past year. Experimentation with different values of K shows that our results are robust to the choice of the number of lags in the performance measure.

15 The order of magnitude of β is more difficult to interpret as it is conditional on the functional form of the performance measure U.

16 The series are in deviation from the mean.

17 In fact, in the presence of homogeneous firms we have that ξ_t = 0 and, substituting the fundamental forecast in Equation (4(4) $${\pi }_t=\delta {\bar{E}}_t{\pi }_{t+1}+\gamma m{c}_t+{\xi }_t,$$(4) ), we get π_t = δγ∑^∞_{k = 1}δ^{k − 1}E^f_tmc_{t + k} + γmc_t = γ∑^∞_{k = 0}δ^kE^f_tmc_{t + k} which corresponds to the inflation path implied by the RE closed form solution in eq. (6(6) $${\pi }_t=\gamma \sum _{k=0}^{\infty }{\delta }^k{E}_{t}m{c}_{t+k}.$$(6) ), when the discounted sums of current and future expected marginal costs are estimated in the same way. The difference between the model with only fundamentalists and the standard model under RE is the fact that typically in the latter the matrix of coefficients in the VAR model (9(9) $$Z_t=A{Z}_{t-1}+{ϵ}_t,$$(9) ) is estimated using the full sample, that is, ${\widehat{A}}_{\left(T\right)}$, while consistently with our behavioral model, fundamentalists only use available information in each period implying that the matrix of coefficients A is estimated period by period, that is, ${\widehat{A}}_{\left(t\right)}$. The outcome of the test reported in Table 3 below does not change if we substitute model M₂ with the actual RE closed form solution. Moreover, in Section 4 we estimate our behavioral model by giving the fundamentalists the same “informational advantage” as in standard models with RE, hence using the full sample to estimate the VAR. The qualitative results in Table 3 are not altered when considering a behavioral model in which the fundamentalists have an informational advantage as in RE models.

18 For completeness, we also compared the HSM model to the nested static model M₄ using a likelihood ratio test. We rejected the null of a restricted static model at the 1% level.

19 For the sake of completeness, we also test the joint significance of the alternative models against our benchmark nonlinear switching model. The resulting F-statistic is 4.828, which leads to the acceptance of the null hypothesis of joint insignificance of alternative models M₁, M₂, M₃, and M₄ against HSM (p-value χ²(4) = 0.305).

20 We thank an anonymous referee for this suggestion.

21 We use π_{t − 1} in the construction of the VAR to be consistent with the information set of fundamentalist firms in the model. In fact, as standard in learning models, current values of endogenous variables are not observable at time t because they depend on the heterogeneous beliefs in the economy which are not known to the individual firm.

22 The point forecasts of the UC-SV model are computed as the median of the posterior distribution.

23 Details on the construction of iterated and direct forecasts of the HSM model are reported in the online appendix, which also reports starting values and prior specification of the UC-SV model.

24 We have also considered the prewhitened quadratic spectral estimator of Andrews and Monahan (1992) and obtained similar results.

25 Since the UC-SV model is the benchmark model, the HSM baseline, M₁, M₂, M₃ are non nested under the null of equal forecast accuracy in finite sample.

26 The conclusions are similar when we use iterated point forecasts with a rolling window scheme (for the UC-SV, M₁, M₂, M₃ models) and direct point forecasts, both with recursive and rolling window scheme (for M₁, M₂, M₃ models, see Tables G.1, G.2, and G.3 in the online appendix).

27 Branch (2004) fit a model with similar dynamic selection among three predictors, namely naive, adaptive and VAR expectations and costs associated with the adoption of each predictor, to the Michigan survey data on inflation expectations. The author found support for switching behavior between different rules depending on the relative mean squared errors of the predictors.

28 The “Real Time Data Set for Macroeconomists,” made available by the Federal Reserve Bank of Philadelphia, reports vintages of the major macroeconomic data available at quarterly intervals in real time. However, the dataset does not include the labor share of income, while other variables are not available for the full sample considered in this article.

29 For the sake of completeness, we also compare average SPF expectations data with expectations paths generated by models in which n_{f, t} = 0, 1. The R² (log-likelihood) for the HSM model is 0.83 (894) and for the models with n_{f, t} = 0, 1 they are, respectively, 0.79 (874) and 0.15 (742).