# An Econometric Learning Approach to Approximate Expectations in Empirical Macro Models.

JEL Classification C53 * C80 * E1This note discusses a learning-based approach that can be used to approximate agents' expectations about future values of inflation, gross domestic product (GDP), and other economic variables. This approach is particularly useful in cases where survey data on expectations are not easily available.

In theoretical macroeconomic models, agents' expectations are often assumed to be "rational". If this assumption is too strong, then the expectations formation process can be modeled to include some form of "bounded rationality". In empirical models, however, researchers typically do not want to impose much structure on the way agents form expectations. For this reason, expectations are often approximated either from survey data or from generated variables (computed from the observable data).

Survey data on expectations have generally been found to improve the fit of economic models, suggesting that they might better approximate the way agents actually form expectations (Roberts 1995; Coibion et al. 2017). However, survey data are unfortunately not always available. Consider, for instance, the case of inflation expectations, which is by far the most common expectation measure included in macro models. In the U.S., the Survey of Professional Forecasters (SPF) provides data about agents' forecasts of the Consumer Price Index (CPI) inflation from the early 1980s, but core inflation forecasts are only available since 2007. In Europe, the European Central Bank (ECB) Survey of Professional Forecasters records forecasts of the Harmonised Index of Consumer Prices inflation since 1999, but no data about expected core inflation are collected. Canada also faces data difficulties. The Survey of Forecasters conducted by The Conference Board of Canada is only available since 2012, and a new survey managed by the Bank of Canada only started in 2016.

When survey data are not available, researchers can approximate expectations using econometric-based measures normally built upon the method proposed by McCallum (1976). This approach uses an empirical equation and past data to predict agents' expectations. These predicted values are then included as a proxy for expectations in the models to be estimated. I propose extending this approach by adding parameter updating ("econometric learning"). The empirical equations used to predict expectations usually entail regressing the measure of interest (e.g., core inflation) on a number of variables believed to contain information about behavior (such as the output gap, unemployment rate, inflation rate, interest rates, and oil prices). The estimation is typically performed once using all available data. I suggest modifying this step and re-estimating the parameters of these empirical equations in each period using only data up to that period. In practice, this procedure can be easily implemented using parameter-updating formulas, which are usually based on a least squares or constant gain learning algorithm. Evans and Honkapohja (2001) provide further details and the specific updating formulas.

The extension that I propose has several interesting features. First, it delivers measures of expectations that are fully consistent with the set of information available to agents when expectations are formed. Indeed, in each period, the econometric equations used to approximate expectations only employ observable data, and estimated parameters obtained from the data available in that period. Second, the addition of parameter updating helps to avoid possible multicollinearity problems that might arise when econometric equations are used to generate the model variables (in this case, proxies of expectations). Third, this approach is very flexible as it can be used to approximate expectations for any variable and for different horizons.

I assessed the performance of this approach by examining the correlation between computed expectations and the available survey data. I focused on one-quarter-ahead inflation expectations for the U.S. using either a recursive least squares (RLS) or a constant gain (CG) formula to obtain the expectation values. In all exercises, survey data are the mean responses of the SPF participants. However, the correlations are very similar if computed using the median of the individual responses instead. For core CPI and core personal consumption expenditures price index (PCEPI) inflation, the correlations between the survey data and the empirical measure of expectations are 0.64 and 0.8, respectively, under RLS updating, and 0.62 and 0.76, respectively, using a CG formula. For both measures, the predicted expectation values are always within the range of values forecasted by the individual survey participants, with the exception of 2009 Ql for core CPI and 2009 Q2 for core PCEPI. With respect to headline CPI inflation, the correlation between the SPF data and the computed expectation values is 0.69 under RLS and 0.68 under CG. For this variable, I additionally used the proposed approach to predict expectations at longer horizons, again obtaining positive results. As an example, for the one-year ahead inflation rate, the correlation between computed expectations and the SPF data is 0.72, under both RLS and CG updating. Overall, these exercises confirm that the econometric learning approach that I propose is a sound, flexible, and easily available way to approximate agents' expectations that researchers can exploit when survey data are not available.

Francesca Rondina (1)

[??] Francesca Rondina

frondina@uottawa.ca

(1) Department of Economics, University of Ottawa, 120 University Private, Ottawa, ON KIN 6N5, Canada

Published online: 22 November 2017

[c] International Atlantic Economic Society 2017

https://doi.org/10.1007/s11294-017-9662-8

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Title Annotation: | RESEARCH NOTE |
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Author: | Rondina, Francesca |

Publication: | International Advances in Economic Research |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Nov 1, 2017 |

Words: | 871 |

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