Quantifying the Beauty Contest: Density Inflation-Forecasts of Professional Japanese Forecasters

Transcription

1 ESRI Discussion Paper Series No.309 Quantifying the Beauty Contest: Density Inflation-Forecasts of Professional Japanese Forecasters Yosuke Takeda May 2014 Economic and Social Research Institute Cabinet Office Tokyo, Japan The views expressed in ESRI Discussion Papers are those of the authors and not those of the Economic and Social Research Institute, the Cabinet Office, or the Government of Japan. (Contact us:

2 Quantifying the Beauty Contest: Density In ation-forecasts of Professional Japanese Forecasters Yosuke Takeda y Abstract The paper aims at quantifying the higher-order expectations that Keynes (1936) compared to the beauty contest, applying a measure of relative entropy to the Japanese ESP Forecast Survey data during the de- ationary period. We conclude that during the de ationary period from June 2009 to April 2010 and from April 2010 to February 2011, professional Japanese forecasters faced the Keynesian beauty contest, in which average expectations dominate agents expectations. Introduction Financial markets play a role in aggregating information acquired and revealed by investors in asset price equilibrium. Information aggregation leads to investors free ride on costly information acquisition of other investors. The impossibility of informational e ciency a la Grossman and Stiglitz (1980) parallels arguments on the higher-order expectations, where it is common knowledge that a macroeconomic structure depends upon the average expectations of agents. As Keynes (1936) wrote of the beauty contest, agents "anticipate what average opinion expects the average opinion to be." The third degree and likewise fourth, fth and higher degrees of expectations do matter, and they have macroeconomic outcomes. This paper aims at quantifying higher-order expectations, and applies a measure of relative entropy to the Japanese ESP Forecast Survey data during the de ationary period. We conclude that during the de ationary period from June 2009 to April 2010 and from April 2010 to February 2011, professional Japanese forecasters took a look at the Keynesian beauty contest, in which average expectations dominate agents expectations. I am grateful to the ESRI and JCER for providing the ESP Forecast data and editorial assistance, to Prof. Takao Komine and members of a study group on the ESP Forecast Survey for giving us some opportunities to discuss the usefulness of the survey, and to seminar participants at Kobe University. y Faculty of Economics, Sophia University, Japan. y-takeda@sophia.ac.jp 1

3 The structure of the paper is as follows: Section 1 quotes descriptions on the beauty contest from Keynes (1936), in relation to the following literature. To quantify the beauty contest, we explain a measure of relative entropy in Section 2. We describe the ESP Forecast Survey data in Section 3, mentioning both dissimilarities and similarities between what Keynes s beauty contest supposes and what the professional forecasters face in the survey response. In Section 4 we estimate the relative entropy measures, as well as absolute forecast errors. Signi cance of the correlation coe cients between both measures is tested. Finally, in Section 5, we make concluding remarks and a statement on further works. 1 The Keynes Beauty Contest John Maynard Keynes(1936) said in Chapter 12 The State of Long-Term Expectation, Thus certain classes of investment are governed by the average expectation of those who deal on the Stock Exchange as revealed in the price of shares, rather than by the genuine expectations of the professional entrepreneur. How then are these highly signi cant daily, even hourly, revaluations of existing investments carried out in practice? (italics inserted by author) and Or, to change the metaphor slightly, professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself nds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fth and higher degrees. (italics inserted by author) The Keynesian beauty contest became not only a parable about how the nancial markets behave in practice, but also a starting point for investigating a behavioral hypothesis on expectations itself in theory, in particular the higherorder expectations (Phelps, 1983; more recently, Woodford, 2003). Phelps 2

4 (1983) showed that, instead of the rational expectations hypothesis, the higherorder expectations lead to non-neutrality of money in a model with a macroeconomic structure, depending upon the agents average expectations. The concept of average expectations in the nancial markets, as Keynes mentioned in the beauty contest, also focuses on the role of equilibrium asset prices as information aggregator in nancial markets (Grossman and Stiglitz, 1980; more recently, Morris and Shin, 2002). As Grossman and Stiglitz (1980) suggested, equilibrium asset prices in the nancial markets expose themselves to a signal for other investors demand forecasts. In an informational e ciency in which an equilibrium asset price re ects full information acquired by investors, many investors would have taken the free ride on costly information acquisition of other investors, leading to impossible informational e ciency. 2 Quantifying the Beauty Contest We quantify the beauty contest, where it is assumed in this paper that agents form individual density forecasts of concerned variables, making a guess as near as possible to the average expectations. The density forecast evaluation is considered to be intrinsically linked to the forecast user s loss function (Diebold, Gunther and Tay, 1998). As a measure of how near individual density forecasts are to the average expectations, we adopt relative entropy (Kullback-Leibler distance) measure de ned as D(pjjq) = X p(x) log p(x) q(x) x2 for two discrete density functions p(x) and q(x). The measure represents ine - ciency when an agent presumes a distribution q(x) instead of a true distribution p(x): If and only if p = q, then D(pjjq) = 0. Otherwise, the value should be non-negative. Note that 0 log 0 0 = 0; 0 log 0 q = 0;and p log p 0 = 1. For forecasting variable x of in ation rate, we calculate the relative entropy measure when each forecaster s individual density is q(x) and the total density average is p(x) that would generate the consensus forecast of the variable x. 3 Survey Data of the Professional Forecasters 3.1 The ESP Forecast Survey We use data from the ESP Forecast Survey, conducted by the Association for Economic Planning from 2004 and taken over by the Japan Center for Economic Research from April, The monthly survey aims to obtain consensus on perspectives of the Japanese economy, such as real GDP growth, Consumer Price Index (CPI), and unemployment rate. The respondents on average more than 3

5 40 forecasters deserve to be called professional, since they make some forecasts on the Japanese economy as part of their occupation. The respondent s real names are publicly available, identi ed with their a liations and business titles. The a liations, as well as some think tanks retained by nancial intermediaries or industrial companies, cover trading companies, domestic and foreign security companies, insurance companies, and investment advisory companies. Each forecaster is required to submit the questionnaire response during the rst week, and the compiled survey results are published around 10th of each month online ( The summary is available for free, but the full text is accessible only to dues-paying members. All the information is published in Japanese only, as of today. The questionnaire items are twofold: forecasts for scal years up to 2-3 years ahead; and quarterly forecasts up to 6-10 quarters ahead. The scal year forecasts list a variety of items: nominal and real GDP growth, GDP components, CPI, overall unemployment rate, and so forth, in addition to some nancial market indicators such as the dollar/ yen exchange rate and the Nikkei Stock Average, while the quarterly forecasts pick up a subset of the scal year items. Concerning real GDP growth and CPI among the scal year forecast items, as well as the point forecasts, since June 2008 the survey has also required the forecasters to answer the density forecasts. The density forecasts designate bins of either a step size 0:5 from 6 to 6 or 0:25 from 2 to 5 in percentage, for the annual growth rates of the real GDP or the CPI, respectively. The forecasters distribute each subjective probability into the bins, amounting to 100 percent in total. The forecast years cover the current year, the following year, and the year after that. Note that the coverage of the survey is the annual CPI rate of a scal year for all items less fresh food. Since 2005, once a year, the survey has announced its excellent forecasters, ve persons whose scal year forecasts were relatively superior to those of the others. Professional fame appears to play a role, since some of the prestigious forecasters have frequently appeared in the excellence list. Through 2009 to 2011, four of the ve excellent forecasters have appeared on the list at least twice since Those two forecasters won three times in 2009 (one for 2006, 2008, and 2009; the other for 2006, 2007, and 2009) and in 2010 (one for 2005, 2009, and 2010; the other for 2008, 2009, 2010). Each person won four times in 2010 (for 2005, 2006, 2007, and 2010) and in 2011 (for 2008, 2009, 2010, and 2011), and one forecaster appeared ve times during (the exceptions were in 2008 and 2009). 3.2 Dissimilarity We discuss some dissimilarities and similarities between what the Keynes beauty contest supposes and what the professional forecasters face in responding to the ESP Forecast Survey. Arguments on the beauty contest emphasize a role of equilibrium asset prices played in the information aggregation, as Grossman and Stiglitz (1980) suggested. Absent from privately acquired information on the fundamentals asso- 4

6 ciated with an asset, investors could extract some information on other investors demand from a signal of the equilibrium asset price. In contrast, in a monthly response to the ESP Forecast Survey, the forecasters cannot rely upon equilibrium price, if any, as information aggregator. All that the forecasters can resort to is, as Morris and Shin (2003) assume in a Lucas-island-economy model, to extract fundamental information from both publicly announced signals and privately acquired ones. The signal extraction process involves errors in the signals, which also results in forecasting accuracy depending upon how accurate the public and private signals are relative to each other. 3.3 Similarity We might not treat the ESP Forecast in the same way as Keynes s beauty contest, but it is nonetheless worth mentioning some similarities between them. First of all, there is an incentive for the forecasters to make as accurate a prediction as possible in the ESP Forecast Survey, since the identi cation of the excellent forecasters would provide carrot and stick motivation for the forecasters, who would have a horse race over reputation as a professional forecaster even in the absence of accuracy. Those labeled as excellent forecasters might also have an advantage in the job markets. Second, the consensus forecasts in the ESP Forecast Survey are well known to have good predictive capability. The ESP Forecast Survey aims to obtain the consensus forecasts of the Japanese economy, an indicator close to the average expectations of Keynes (1936) and Phelps (1983). It is probable that such a good predictive capability of the consensus forecasts suggests a role of the average expectations in the ESP Forecast Survey, too. 4 Estimation We use density forecasts of the professional forecasters on the annual CPI rate. The ESP Forecast Survey began to compile the density forecast in June Our data set covers from June 2008 to the present. The period is characteristic of undergoing ups and downs in the CPI rate (Figure 1). Especially from April to August 2009, de ation accelerated up to 2:4 per annum (Figure 2). 4.1 Relative Entropy Measure The relative entropy measures are indicated in Table 1-A and Table 1-B each code for professional forecasters for the scal year Each gure represents the measure for the density forecast of a designated code of the forecaster, from June 2008 to April The relative entropy magnitude is also ranked among all the forecasts for the period, shown below in Table 1-A and Table 1-B. As for the scal years 2009 to 2012, Tables 2-A and 2-B to 5-A and 5-B indicate similar results. Note that a de nition of the relative entropy p log p 0 = 1 requires us 5

7 to ll positive probability values in each bin of the individual density q, even if the probabilities are empty in the bins. We replace the empty bins with a imperceptible probability of 0:01 in percentage. Also note that the relative entropy measure in a period cannot be comparable with one in another period. The ranking of the relative entropy shows high persistence of good predictions nearer to the average expectations, for a small group of the forecasters with the code numbers 2; 5; 3; and 1. On the other hand, a larger group of the forecasters persistently predict the in ation rate poorly distant from the average expectations, with the code numbers 16; 42; 37; 45; 18; 26; and 36. Unfortunately, we have no information for identifying who is designated for each code. My conjecture on the clustered ranking is, since each code is a permanent number designated for each forecaster or the a liation even if there have been changes in person in charge, a group of the forecasters with the small-number codes might have been learning to match the forecasts to the average expectations, from experience in the previous ESP Forecast Survey. We also wonder if the forecasters with the small-number codes might be those who have frequently been named excellent forecasters. 4.2 Accuracy in Forecasting The nearer the density forecasts are to the average expectations, the smaller value the relative entropy measure of the forecasts has. However, it does not suggest the forecast s high accuracy. We calculate an absolute forecast error of point forecast x e on the annual CPI rate x, de ned as jx e xj. Table 6-A and Table 6-B indicate the absolute forecast errors for the scal year Similarly, Tables 7-A and 7-B to 9-A and 9-B present the errors for the scal year 2009 to The absolute forecast errors show that, compared with the relative entropy measure, there are not more noticeable clusters in the accuracy ranking. As it were, the forecasters with codes 7 and 9 have better predictive capability, while those with codes 37; 46; 38; 36; and 50 seem to be inferior to other forecasters. In a comparatively weaker meaning than the average expectations, I would say the salience in the relative accuracy might imply a role of learning experience in accurate forecasts. 4.3 Correlation Finally, we connect the relative entropy measure with the absolute forecast error for the same period, estimating two correlation coe cients either between both measures themselves (Pearson product-moment correlation coe cient) or between the rankings in both measures (Spearman s rank correlation coe cient). The t-statistics for testing the null hypothesis H 0 : no correlation against either one-sided alternative H 1 : positive correlation or two-sided alternative H 1 : non-zero correlation is indicated in Table 10. The signi cance of the correlation coe cients di ers according to the forecasting period. It is evident from the Pearson correlation coe cients that the 6

8 relative entropy measure and the absolute forecast error have signi cantly positive correlations during a period from June 2009 to April 2010 when the forecasters made a prediction of the scal year As for the scal year 2011 prediction, both measures are signi cantly positive-correlated from April 2010 to February Though the evidence is not so clear in Spearman s rank correlation coe cients, they are indicative of less signi cant but qualitatively similar correlations for those periods. The months from June 2009 to April 2010 and from April 2010 to February 2011 are characteristic of accelerating de ation in Figure 2, up to the annual in ation rate 2:4% recorded in August During the de ationary period, it turns out that the Japanese professional forecasters faced a surge in the Keynesian beauty contest. 5 Conclusion We explored Keynes s beauty contest by quantifying the signi cance in the professional forecasters. Using the Japanese ESP Forecast Survey data, the relative entropy measures were estimated for how near the individual density forecasts are to the average expectations. Signi cantly positive correlation coe cients of the measures with absolute forecast errors suggest that during the de ationary period, Japanese professional forecasters were dominated by Keynes s beauty contest, where macroeconomic structure depends on the average expectations over economic agents who, so as to rationally expect the in ation rates, are led to forecast the forecast of others, who in turn link their forecast to other forecasts, as the Keynes beauty contest model of Phelps (1983) indicates. There remain further works for elaborating the paper. Among them, what seems to me the most challenging is to estimate in a semi-parametric method correspondence, both between individual density forecasts and the average expectations, and between the average expectations and a latent true distribution of the in ation rate. In the current version of the paper, the correspondences are primitively estimated via the correlation coe cients, Pearson and Spearman. Another further task is to analyze a learning process of individual forecasters in a Bayesian way. Also in the current paper, not treating learning explicitly is assumed, though the relative entropy and the absolute forecast error suggest roles of learning among the experienced forecasters. References [1] Diebold, Francis X., Todd A. Gunther and Anthony S. Tay. "Evaluating Density Forecasts with Applications to Financial Risk Management." International Economic Review, vol. 39, no. 4, Symposium on Forecasting and Empirical Methods in Macroeconomics and Finance, 1998, pp

9 [2] Grossman, Sanford J., and Joseph E. Stiglitz. "On the Impossibility of Informationally E cient Markets." The American Economic Review, vol. 70. no. 3, 1980, pp [3] Keynes, John Maynard. The General Theory of Employment, Interest and Money. London, Macmilan & Co., LTD, [4] Morris, Stephen, and Hyun Song Shin. "Social Value of Public Information." The American Economic Review, vol. 92, no. 5, 2002, pp [5] Phelps, Edmund. S. "The Trouble with Rational Expectations and the Problem of In ation Stabilization." in R. Frydman and E. S. Phelps (eds.), Individual Forecasting and Aggregate Outcomes: Rational Expectations Examined, Cambridge: Cambridge University Press, [6] Woodford, Michael. "Imperfect Common Knowledge and the E ects of Monetary Policy." in P. Aghion, R. Frydman, J. Stiglitz and M. Woodford (eds.), Knowledge, Information, and Expectations in Macroeconomics: In Honor of Edmund S. Phelps, Princeton, NJ: Princeton University Press,

10 Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 Jan-01 Jul-01 Jan-02 Jul-02 Jan-03 Jul-03 Jan-04 Jul-04 Jan-05 Jul-05 Jan-06 Jul-06 Jan-07 Jul-07 Jan-08 Jul-08 Jan-09 Jul-09 Jan-10 Jul-10 Jan-11 Jul-11 Jan-12 Jul-12 % 3 Figure -3 1 Consumer Price Index: (annual rate; all items less fresh food) 系列 1,

11 Jan-08 Mar-08 May-08 Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 May-09 Jul-09 Sep-09 Nov-09 Jan-10 Mar-10 May-10 Jul-10 Sep-10 Nov-10 Jan-11 Mar-11 May-11 Jul-11 Sep-11 Nov-11 Jan-12 Mar-12 May-12 Jul-12 Sep-12 Nov-12 % 3 Figure 2 Consumer Price Index: Jun Dec (annual rate; all items less fresh food)

12 Table 1-A Relative Entropy Code for Professional Forecasters Fiscal Yea Timing Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

13 Table 1-B Relative Entropy Fiscal Yea Timing Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

14 Table 2-A Relative Entropy Code for Professional Forecasters Fiscal Yea Timing Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

15 Table 2-B Relative Entropy Fiscal Yea Timing Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

16 Table 3-A Relative Entropy Code for Professional Forecasters Fiscal Yea Timing Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

17 Table 3-B Relative Entropy Fiscal Yea Timing Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

18 Table 4-A Relative Entropy Code for Professional Forecasters Fiscal Yea Timing Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr