# American Institute of Mathematical Sciences

July  2020, 7(3): 175-184. doi: 10.3934/jdg.2020012

## Emerging patterns in inflation expectations with multiple agents

 1 Research Group in Economic Dynamics, Universidad de la República, Montevideo, Uruguay 2 IIESS (UNS - Conicet) and Departamento de Economía, Universidad Nacional del Sur, San Andrés 800, Bahía Blanca 8000, Argentina

* Corresponding author: ealvarez@ccee.edu.uy

Received  February 2020 Revised  April 2020 Published  July 2020

Macroeconomic theory and central Banks' policy recommendations have analyzed for decades the link between the expected value of future inflation and its subsequent realization. Agents' inflation expectations have thus become a fundamental input of the economic policy: they allow to know if economic agents are synchronized with the policies and allow the Central Banks to anticipate the market trends. In this paper, we found evidence for the case of Uruguay of a discrepancy between the distribution of agents' inflation expectations and the distribution expected by traditional models. A first consequence is an increase in uncertainty in the estimates; problems related to its asymptotic distribution and the assumptions that arise from this aggregate distribution are analyzed. Another consequence is related to the existence of a structure in the data and the notion of equilibrium in the model. It is concluded that a discussion regarding the nature of the economic phenomenon is essential for the correct specification of the model studied.

Citation: Emiliano Alvarez, Silvia London. Emerging patterns in inflation expectations with multiple agents. Journal of Dynamics and Games, 2020, 7 (3) : 175-184. doi: 10.3934/jdg.2020012
##### References:
 [1] T. Assenza, P. Heemeijer, C. H. Hommes and D. Massaro, Individual expectations and aggregate macro behavior, Tinbergen Institute Discussion Paper 13-016/II, (2013), 59 pp. doi: 10.2139/ssrn.2200424. [2] J. M. Berk, The Preparation of Monetary Policy: Essays on a Multi-Model Approach, Financial and Monetary Policy Studies, 35, Kluwer Academic Publishers, Boston, 2001. [3] G. E. P. Box, Science and statistics, J. Amer. Statist. Assoc., 71 (1976), 791-799.  doi: 10.1080/01621459.1976.10480949. [4] W. A. Branch, Sticky information and model uncertainty in survey data on inflation expectations, Journal of Economic Dynamics and Control, 31 (2007), 245-276.  doi: 10.1016/j.jedc.2005.11.002. [5] W. A. Brock and S. N. Durlauf, Discrete choice with social interactions, Rev. Econom. Stud., 68 (2001), 235-260.  doi: 10.1111/1467-937X.00168. [6] W. A. Brock and C. H. Hommes, A rational route to randomness, Econometrica, 65 (1997), 1059-1095.  doi: 10.2307/2171879. [7] W. A. Brock and C. H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, J. Econom. Dynam. Control, 22 (1998), 1235-1274.  doi: 10.1016/S0165-1889(98)00011-6. [8] P. Cagan, The monetary dynamics of hyperinflation, in Studies in the Quantity Theory of Money, University of Chicago Press, Chicago and London, 1956, 25–117. [9] J. L. Cardoso and N. Palma, The science of things generally?, in Lionel Robbins's Essay on the Nature and Significance of Economic Science–75th Anniversary Conference Proceedings, Londres, London School of Economics, STICERD, 2009,387–402. [10] J. A. Carlson, Are price expectations normally distributed?, Journal of the American Statistical Association, 70 (1975), 749-754.  doi: 10.1080/01621459.1975.10480299. [11] Y. G. Chen, Inflation, inflation expectations, and the phillips curve, Technical Report, CBO Working Paper 2019-07, (2019), 58 pp. [12] A. Clauset, C. R. Shalizi and M. E. Newman, Power-law distributions in empirical data, SIAM Review, 51 (2009), 661-703.  doi: 10.1137/070710111. [13] O. Coibion and Y. Gorodnichenko, Is the phillips curve alive and well after all? Inflation expectations and the missing disinflation, American Economic Journal: Macroeconomics, 7 (2015), 197-232.  doi: 10.3386/w19598. [14] O. Coibion, Y. Gorodnichenko and R. Kamdar, The formation of expectations, inflation and the phillips curve, Journal of Economic Literature, 56 (2018), 1447-1491.  doi: 10.3386/w23304. [15] A. Cornea-Madeira, C. Hommes and D. Massaro, Behavioral heterogeneity in US inflation dynamics, Journal of Business & Economic Statistics, 37 (2019), 288-300. [16] R. Curtin, Inflation expectations and empirical tests: Theoretical models and empirical tests, in Inflation Expectations, Routledge, 2009, 52–79. [17] L. Dräger and M. J. Lamla, Updating inflation expectations: Evidence from micro-data, Economics Letters, 117 (2012), 807-810. [18] G. W. Evans, C. H. Hommes, B. McGough and I. Salle, Are long-horizon expectations (de-) stabilizing? Theory and experiments, Bank of Canada Staff Working Paper, (2019). [19] E. F. Fama, The behavior of stock-market prices, The Journal of Business, 38 (1965), (1965), 34–105. doi: 10.1086/294743. [20] A. M. Fasolo and M. S. Portugal, Imperfect rationality and inflationary inertia: A new estimation of the Phillips Curve for Brazil, Estudos Econômicos (São Paulo), 34 (2004), 725–776. [21] E. Fehr and K. M. Schmidt, The economics of fairness, reciprocity and altruism–experimental evidence and new theories, Economics, 20 (2005), 51. doi: 10.1016/S1574-0714(06)01008-6. [22] S. Frache and R. Lluberas, New information and inflation expectations among firms, BIS Working Papers, 781, (2019), 42 pp. [23] J. E. Gagnon, Long memory in inflation expectations: Evidence from international financial markets, Board of Governors of the Federal Reserve System, 538 (1996). [24] C. S. Gillespie, Fitting heavy tailed distributions: The poweRlaw package, Journal of Statistical Software, 64 (2015), 1-16. [25] A. P. Kirman, Individual and aggregate behaviour: Of ants and men, Complexity and the Economy: Implications for Economic Policy, (2005), 33–53. doi: 10.1016/j.jmateco.2004.08.001. [26] A. Leijonhufvud, La naturaleza de una economía, Investigación Económica, 70 (2011), 15-36.  doi: 10.22201/fe.01851667p.2011.277.37315. [27] P. Lévy, Calcul des Probabilités, PCMI collection, Gauthier-Villars, 1925. [28] S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics & Games, 6 (2019), 149-158. [29] B. Mandelbrot, The variation of certain speculative prices, The Journal of Business, 36 (1963), 394-419. [30] N. G. Mankiw and R. Reis, Sticky information versus sticky prices: A proposal to replace the new keynesian phillips curve, The Quarterly Journal of Economics, 117 (2002), 1295-1328.  doi: 10.3386/w8290. [31] R. D. McKelvey and T. R. Palfrey, Quantal response equilibria for normal form games, Games and Economic Behavior, 10 (1995), 6-38.  doi: 10.1006/game.1995.1023. [32] J. F. Muth, Rational expectations and the theory of price movements, Econometrica: Journal of the Econometric Society, (1961), 315–335. doi: 10.2307/1909635. [33] M. E. Newman, Power laws, pareto distributions and zipf's law, Contemporary Physics, 46 (2005), 323-351.  doi: 10.1080/00107510500052444. [34] J. P. Nolan, Fitting data and assessing goodness-of-fit with stable distributions, Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics, Washington DC, 1999. [35] D. Phan, M. B. Gordon, and J.-P. Nadal, Social interactions in economic theory: An insight from statistical mechanics, in Cognitive Economics, Springer, 2004,335–358. [36] S. Rinke, M. Busch and C. Leschinski, Long memory, breaks, and trends: On the sources of persistence in inflation rates, Hannover Economic Papers (HEP), (2017). [37] J. Royuela-del Val, et al., Libstable: Fast, parallel, and high-precision computation of $\alpha$-stable distributions in r, c/c++, and matlab, Journal of Statistical Software, 78 (2017). [38] S. S. Shapiro and M. B. Wilk, An analysis of variance test for normality (complete samples), Biometrika, 52 (1965), 591-611.  doi: 10.1093/biomet/52.3-4.591. [39] C. A. Sims, Implications of rational inattention, Journal of monetary Economics, 50 (2003), 665-690.  doi: 10.1016/S0304-3932(03)00029-1. [40] J. Smith and M. McAleer, Alternative procedures for converting qualitative response data to quantitative expectations: An application to australian manufacturing, Journal of Applied Econometrics, 10 (1995), 165-185.  doi: 10.1002/jae.3950100206. [41] L. E. Svensson, Inflation forecast targeting: Implementing and monitoring inflation targets, European Economic Review, 41 (1997), 1111-1146.  doi: 10.3386/w5797. [42] J. B. Taylor, Discretion versus policy rules in practice, in Carnegie-Rochester Conference Series on Public Policy, 39, Elsevier, 1993,195–214. doi: 10.1016/0167-2231(93)90009-L. [43] H. Theil, Economic Forecasts and Policy, North-Holland, 1958. [44] F. Tohmé, C. Dabús and S. London, Processes of evolutionary self-organization in high inflation experiences, in New Tools of Economic Dynamics, Springer, 2005,357–371. [45] M. Woodford, Imperfect common knowledge and the effects of monetary policy, Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, (2003), 25. doi: 10.3386/w8673.

show all references

##### References:
 [1] T. Assenza, P. Heemeijer, C. H. Hommes and D. Massaro, Individual expectations and aggregate macro behavior, Tinbergen Institute Discussion Paper 13-016/II, (2013), 59 pp. doi: 10.2139/ssrn.2200424. [2] J. M. Berk, The Preparation of Monetary Policy: Essays on a Multi-Model Approach, Financial and Monetary Policy Studies, 35, Kluwer Academic Publishers, Boston, 2001. [3] G. E. P. Box, Science and statistics, J. Amer. Statist. Assoc., 71 (1976), 791-799.  doi: 10.1080/01621459.1976.10480949. [4] W. A. Branch, Sticky information and model uncertainty in survey data on inflation expectations, Journal of Economic Dynamics and Control, 31 (2007), 245-276.  doi: 10.1016/j.jedc.2005.11.002. [5] W. A. Brock and S. N. Durlauf, Discrete choice with social interactions, Rev. Econom. Stud., 68 (2001), 235-260.  doi: 10.1111/1467-937X.00168. [6] W. A. Brock and C. H. Hommes, A rational route to randomness, Econometrica, 65 (1997), 1059-1095.  doi: 10.2307/2171879. [7] W. A. Brock and C. H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, J. Econom. Dynam. Control, 22 (1998), 1235-1274.  doi: 10.1016/S0165-1889(98)00011-6. [8] P. Cagan, The monetary dynamics of hyperinflation, in Studies in the Quantity Theory of Money, University of Chicago Press, Chicago and London, 1956, 25–117. [9] J. L. Cardoso and N. Palma, The science of things generally?, in Lionel Robbins's Essay on the Nature and Significance of Economic Science–75th Anniversary Conference Proceedings, Londres, London School of Economics, STICERD, 2009,387–402. [10] J. A. Carlson, Are price expectations normally distributed?, Journal of the American Statistical Association, 70 (1975), 749-754.  doi: 10.1080/01621459.1975.10480299. [11] Y. G. Chen, Inflation, inflation expectations, and the phillips curve, Technical Report, CBO Working Paper 2019-07, (2019), 58 pp. [12] A. Clauset, C. R. Shalizi and M. E. Newman, Power-law distributions in empirical data, SIAM Review, 51 (2009), 661-703.  doi: 10.1137/070710111. [13] O. Coibion and Y. Gorodnichenko, Is the phillips curve alive and well after all? Inflation expectations and the missing disinflation, American Economic Journal: Macroeconomics, 7 (2015), 197-232.  doi: 10.3386/w19598. [14] O. Coibion, Y. Gorodnichenko and R. Kamdar, The formation of expectations, inflation and the phillips curve, Journal of Economic Literature, 56 (2018), 1447-1491.  doi: 10.3386/w23304. [15] A. Cornea-Madeira, C. Hommes and D. Massaro, Behavioral heterogeneity in US inflation dynamics, Journal of Business & Economic Statistics, 37 (2019), 288-300. [16] R. Curtin, Inflation expectations and empirical tests: Theoretical models and empirical tests, in Inflation Expectations, Routledge, 2009, 52–79. [17] L. Dräger and M. J. Lamla, Updating inflation expectations: Evidence from micro-data, Economics Letters, 117 (2012), 807-810. [18] G. W. Evans, C. H. Hommes, B. McGough and I. Salle, Are long-horizon expectations (de-) stabilizing? Theory and experiments, Bank of Canada Staff Working Paper, (2019). [19] E. F. Fama, The behavior of stock-market prices, The Journal of Business, 38 (1965), (1965), 34–105. doi: 10.1086/294743. [20] A. M. Fasolo and M. S. Portugal, Imperfect rationality and inflationary inertia: A new estimation of the Phillips Curve for Brazil, Estudos Econômicos (São Paulo), 34 (2004), 725–776. [21] E. Fehr and K. M. Schmidt, The economics of fairness, reciprocity and altruism–experimental evidence and new theories, Economics, 20 (2005), 51. doi: 10.1016/S1574-0714(06)01008-6. [22] S. Frache and R. Lluberas, New information and inflation expectations among firms, BIS Working Papers, 781, (2019), 42 pp. [23] J. E. Gagnon, Long memory in inflation expectations: Evidence from international financial markets, Board of Governors of the Federal Reserve System, 538 (1996). [24] C. S. Gillespie, Fitting heavy tailed distributions: The poweRlaw package, Journal of Statistical Software, 64 (2015), 1-16. [25] A. P. Kirman, Individual and aggregate behaviour: Of ants and men, Complexity and the Economy: Implications for Economic Policy, (2005), 33–53. doi: 10.1016/j.jmateco.2004.08.001. [26] A. Leijonhufvud, La naturaleza de una economía, Investigación Económica, 70 (2011), 15-36.  doi: 10.22201/fe.01851667p.2011.277.37315. [27] P. Lévy, Calcul des Probabilités, PCMI collection, Gauthier-Villars, 1925. [28] S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics & Games, 6 (2019), 149-158. [29] B. Mandelbrot, The variation of certain speculative prices, The Journal of Business, 36 (1963), 394-419. [30] N. G. Mankiw and R. Reis, Sticky information versus sticky prices: A proposal to replace the new keynesian phillips curve, The Quarterly Journal of Economics, 117 (2002), 1295-1328.  doi: 10.3386/w8290. [31] R. D. McKelvey and T. R. Palfrey, Quantal response equilibria for normal form games, Games and Economic Behavior, 10 (1995), 6-38.  doi: 10.1006/game.1995.1023. [32] J. F. Muth, Rational expectations and the theory of price movements, Econometrica: Journal of the Econometric Society, (1961), 315–335. doi: 10.2307/1909635. [33] M. E. Newman, Power laws, pareto distributions and zipf's law, Contemporary Physics, 46 (2005), 323-351.  doi: 10.1080/00107510500052444. [34] J. P. Nolan, Fitting data and assessing goodness-of-fit with stable distributions, Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics, Washington DC, 1999. [35] D. Phan, M. B. Gordon, and J.-P. Nadal, Social interactions in economic theory: An insight from statistical mechanics, in Cognitive Economics, Springer, 2004,335–358. [36] S. Rinke, M. Busch and C. Leschinski, Long memory, breaks, and trends: On the sources of persistence in inflation rates, Hannover Economic Papers (HEP), (2017). [37] J. Royuela-del Val, et al., Libstable: Fast, parallel, and high-precision computation of $\alpha$-stable distributions in r, c/c++, and matlab, Journal of Statistical Software, 78 (2017). [38] S. S. Shapiro and M. B. Wilk, An analysis of variance test for normality (complete samples), Biometrika, 52 (1965), 591-611.  doi: 10.1093/biomet/52.3-4.591. [39] C. A. Sims, Implications of rational inattention, Journal of monetary Economics, 50 (2003), 665-690.  doi: 10.1016/S0304-3932(03)00029-1. [40] J. Smith and M. McAleer, Alternative procedures for converting qualitative response data to quantitative expectations: An application to australian manufacturing, Journal of Applied Econometrics, 10 (1995), 165-185.  doi: 10.1002/jae.3950100206. [41] L. E. Svensson, Inflation forecast targeting: Implementing and monitoring inflation targets, European Economic Review, 41 (1997), 1111-1146.  doi: 10.3386/w5797. [42] J. B. Taylor, Discretion versus policy rules in practice, in Carnegie-Rochester Conference Series on Public Policy, 39, Elsevier, 1993,195–214. doi: 10.1016/0167-2231(93)90009-L. [43] H. Theil, Economic Forecasts and Policy, North-Holland, 1958. [44] F. Tohmé, C. Dabús and S. London, Processes of evolutionary self-organization in high inflation experiences, in New Tools of Economic Dynamics, Springer, 2005,357–371. [45] M. Woodford, Imperfect common knowledge and the effects of monetary policy, Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, (2003), 25. doi: 10.3386/w8673.
Responses to Firms' Expectations Survey, forecasting 12 months ahead. Source: BCU, INE
Estimation of parameters $\alpha, \beta, \gamma$ and $\delta$. In dotted lines 95 % confidence intervals, calculated using bootstrap
Information obtained from Firms' Expectations Survey. Source: BCU-INE, Uruguay. Firms were asked about their inflation expectations in 18 months ahead until July 2013, and about their inflation expectations in 24 months ahead from July 2013. It is related to the change in the monetary policy horizon of the Central Bank of Uruguay
 Variable Firms expectations Data Monthly Span 2012.06 to 2017.12 Surveyed Business financial Managers Observations 522 firms Inflation Forecast Current year 12 months 18 months 24 months
 Variable Firms expectations Data Monthly Span 2012.06 to 2017.12 Surveyed Business financial Managers Observations 522 firms Inflation Forecast Current year 12 months 18 months 24 months
Proportion of months for which the null hypothesis is rejected. P-values obtained by bootstrap in (a) and by the Shapiro-Wilk test in (b)
 12 months 18 to 24 months (a) PL: p-value $<$ 0.05 0 % 1.5 % (b) Normal: p-value $<$ 0.05 100 % 100 %
 12 months 18 to 24 months (a) PL: p-value $<$ 0.05 0 % 1.5 % (b) Normal: p-value $<$ 0.05 100 % 100 %
Sensitivity analysis of the analyzed variables, based on the calculations made by bootstrap
 Variable Expectation 2.5% 25% 50% 75% 97.5 % (a) $\alpha$ 12 months 6.43 7.74 8.46 9.66 17.38 18-24 months 5.55 6.53 7.49 8.46 10.65 (b) $x_{min}$ 12 months 8 10 10 10 14 18-24 months 8 10 10 11 14.35 (c) p-value 12 months 0.3067 0.6908 0.9483 0.9999 1.0000 18-24 months 0.1123 0.6304 0.9497 0.9997 1.0000
 Variable Expectation 2.5% 25% 50% 75% 97.5 % (a) $\alpha$ 12 months 6.43 7.74 8.46 9.66 17.38 18-24 months 5.55 6.53 7.49 8.46 10.65 (b) $x_{min}$ 12 months 8 10 10 10 14 18-24 months 8 10 10 11 14.35 (c) p-value 12 months 0.3067 0.6908 0.9483 0.9999 1.0000 18-24 months 0.1123 0.6304 0.9497 0.9997 1.0000
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