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# 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: https://form.cao.go.jp/esri/en_opinion-0002.html)
# Quantifying the Beauty Contest: Density Inflation-Forecasts of Professional Japanese Forecasters $ ^{*} $

Yosuke Takeda $ ^{†} $

## 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 deflationary period. We conclude that during the deflationary 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.

## I ntroduction

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 efficiency 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, fifth 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 deflationary period. We conclude that during the deflationary 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.
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. Significance of the correlation coefficients 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 significant 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 finds 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, fifth and higher degrees. (italics inserted by author)

The Keynesian beauty contest became not only a parable about how the financial markets behave in practice, but also a starting point for investigating a behavioral hypothesis on expectations itself in theory, in particular the higher-order expectations (Phelps, 1983; more recently, Woodford, 2003). Phelps
(1983) showed that, instead of the rational expectations hypothesis, the higher-order 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 financial markets, as Keynes mentioned in the beauty contest, also focuses on the role of equilibrium asset prices as information aggregator in financial markets (Grossman and Stiglitz, 1980; more recently, Morris and Shin, 2002). As Grossman and Stiglitz (1980) suggested, equilibrium asset prices in the financial markets expose themselves to a signal for other investors' demand forecasts. In an informational efficiency in which an equilibrium asset price reflects full

information acquired by investors, many investors would have taken the free ride on costly information acquisition of other investors, leading to impossible informational efficiency.

## 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 defined as

$$ D(p||q)=\sum_{x\in\chi}p(x)\log\frac{p(x)}{q(x)} $$

for two discrete density functions $ p(x) $ and $ q(x) $ . The measure represents inefficiency when an agent presumes a distribution $ q(x) $ instead of a true distribution $ p(x) $ . If and only if p = q, then $ D(p||q) = 0 $ . Otherwise, the value should be non-negative. Note that $ 0 \log \frac{0}{0} = 0 $ , $ 0 \log \frac{0}{a} = 0 $ , and $ p \log \frac{p}{0} = \infty $ .

For forecasting variable x of inflation 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, 2012. 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
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 identification 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 2008. 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, deflation 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 fiscal year 2008. Each figure represents the measure for the density forecast of a designated code of the forecaster, from June 2008 to April 2009. 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 fiscal years 2009 to 2012, Tables 2-A and 2-B to 5-A and 5-B indicate similar results. Note that a definition of the relative entropy $ p \log \frac{p}{0} = \infty $ requires us
to fill 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 an 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 inflation 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 affiliation 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, defined as $ |x^{e}-x| $ . Table 6-A and Table 6-B indicate the absolute forecast errors for the fiscal year 2008. Similarly, Tables 7-A and 7-B to 9-A and 9-B present the errors for the fiscal year 2009 to 2011.

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 coefficients either between both measures themselves (Pearson product-moment correlation coefficient) or between the rankings in both measures (Spearman's rank correlation coefficient). 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 significance of the correlation coefficients differs according to the forecasting period. It is evident from the Pearson correlation coefficients that the
relative entropy measure and the absolute forecast error have significantly positive correlations during a period from June 2009 to April 2010 when the forecasters made a prediction of the fiscal year 2010. As for the fiscal year 2011 prediction, both measures are significantly positive-correlated from April 2010 to February 2011. Though the evidence is not so clear in Spearman's rank correlation coefficients, they are indicative of less significant 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 deflation in Figure 2, up to the annual inflation rate -2.4% recorded in August 2009. During the deflationary 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 significance 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. Significantly positive correlation coefficients of the measures with absolute forecast errors suggest that during the deflationary 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 inflation 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 inflation rate. In the current version of the paper, the correspondences are primitively estimated via the correlation coefficients, 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. 863-883.
[2] Grossman, Sanford J., and Joseph E. Stiglitz. "On the Impossibility of Informationally Efficient Markets." The American Economic Review, vol. 70. no. 3, 1980, pp. 393-408.

[3] Keynes, John Maynard. The General Theory of Employment, Interest and Money. London, Macmillan & Co., LTD, 1936.

[4] Morris, Stephen, and Hyun Song Shin. "Social Value of Public Information." The American Economic Review, vol. 92, no. 5, 2002, pp. 1521-1534.

[5] Phelps, Edmund. S. "The Trouble with Rational Expectations and the Problem of Inflation Stabilization." in R. Frydman and E. S. Phelps (eds.), Individual Forecasting and Aggregate Outcomes: 'Rational Expectations' Examined, Cambridge: Cambridge University Press, 1983.

[6] Woodford, Michael. "Imperfect Common Knowledge and the Effects 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, 2003.
Figure 1 Consumer Price Index: 1995-2012 (annual rate; all items less fresh food)
Table 1-A
Relative Entropy	Code for Professional Forecasters
Fiscal Yea	Timing	34	41	25	16	22	20	24	2	42	37	7	6	45	35	18	5	12	8	46	3	40	26
2008	Jun-08	0.468768	1.273015	0.935218	2.144086	0.535721	0.788262	0.624673	0.57094	2.284271	0.844413	0.617785	1.692779	1.531812	4.052316	0.633653	1.633862	4.197172	1.02554	1.397589	0.743624	1.579105
Jul-08	1.554395	1.122441	1.57852	2.206154	1.574324	0.236002	2.315426	1.184071	3.511602	3.870833	1.82599	1.419885	3.31609	3.860872	0.157327	3.049311	4.009484	1.713399	3.887121	0.602954	2.129338
Aug-08	0.69709	1.775827	4.121849	0.904984	0.506884	0.547592	1.275541	0.644195	1.881829	2.908925	4.506734	2.08562	0.951887	3.153061	0.253296	1.33604	1.5694	1.273838	2.160889	1.869052
Sep-08	0.505379	0.801369	1.072812	0.158223	0.194457	0.070704	0.070526	2.391874	2.052782	1.072812	2.989584	0.744975	2.106746	0.088267	2.402396	0.678073	0.738339	0.22008	2.572087	1.072812
Oct-08	0.32883	0.583952	2.468976	0.807292	0.138251	0.161497	0.421898	0.092475	2.66672	2.454613	2.617782	2.123422	0.646249	0.751988	0.05596	2.387744	0.534031	0.750839	2.292104	0.727179
Nov-08	1.095798	0.368103	1.024325	0.368103	0.015405	1.682301	0.422825	2.406862	2.820175	1.268609	1.097079	0.131362	2.391672	0.081597	1.909865	2.992926	0.312419	6.739851	1.010321
Dec-08	0.318361	0.828427	3.124398	0.877192	0.828427	0.698128	0.847227	0.919492	1.997985	4.238627	3.626056	1.554351	0.278964	3.124398	0.30808	1.783798	0.877192	0.37465	1.997985	1.036447
Jan-09	1.288839	0.460259	7.300015	0.951346	0.423466	1.385306	1.03755	1.709484	1.822367	7.270156	1.06215	0.931192	1.609406	0.462583	1.709484	0.951346	0.424334	0.363429	1.481855	0.93353
Feb-09	0.406957	0.410208	0.560039	0.276803	1.921225	0.489547	0.353986	3.034553	1.417968	7.558926	1.67942	0.376154	1.531666	0.469619	1.531666	1.693755	1.455298	0.258251	0.469619	0.494967
Mar-09	1.535128	0.298402	0.454505	0.236893	0.454505	0.279078	0.231724	2.585512	1.337508	0.203407	0.279078	0.279078	0.279078	0.279078	0.310064	2.585512	1.686108	1.380662	0.379494	1.337508	0.317387
Apr-09	1.370518	0.342512	1.180741	0.515988	0.356399	1.250368	1.058039	0.223607	1.222588	0.23393	1.250368	0.238964	2.038233	1.355925	1.074501	0.238964	1.048725	0.387064	2.038233	0.366589
Ranking
2008	Jun-08	3	20	18	29	5	15	10	8	30	16	9	28	25	34	11	27	35	19	21	13	26
Jul-08	16	10	19	24	17	3	25	12	30	32	22	13	29	31	1	28	36	20	34	7	23
Aug-08	11	23	35	14	6	7	17	10	25	31	36	28	15	33	4	18	21	16	29	24
Sep-08	13	21	25	7	9	2	1	31	29	25	34	20	30	5	32	17	19	11	33	25
Oct-08	13	17	32	24	7	11	14	3	34	31	33	27	19	23	1	29	15	22	28	21
Nov-08	25	12	23	12	1	28	15	31	32	27	26	5	30	3	29	33	10	34	22
Dec-08	12	17	34	21	17	16	20	23	32	37	36	27	9	34	11	30	21	13	32	25
Jan-09	24	11	35	17	7	25	21	29	32	34	22	15	28	12	29	17	8	3	27	16
Feb-09	14	15	24	6	32	22	7	34	26	35	30	12	28	20	28	31	27	5	20	23
Mar-09	30	15	24	6	24	10	5	33	27	2	10	10	10	18	33	31	29	22	27	19
Apr-09	33	12	27	20	13	29	24	2	28	4	29	5	35	32	25	5	22	17	35	16