Expected expectations : a heterogeneous model for inflation expectations of professional forecasters

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Expected expectations

A heterogeneous model for inflation expectations of professional forecasters

A.A. Zwartsenberg∗† Faculty Economics and Business

Econometrics thesis

Supervisor: Prof. dr. C.H. Hommes December 23, 2016

Abstract

Heuristics and a Heuristics Switching Model that have proven successful in explaining both individual and aggregate behavior of forecasting behavior by participants in a labo-ratory setting, are fitted to survey data of actual professional forecasters. When adding and modifying certain heuristics and adapting the parameters of the Heuristics Switching Model this heterogeneous model was better able to describe expectations formations than those homogeneous rules.

∗_10808574

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Hierbij verklaar ik, Arthur Zwartsenberg, dat ik deze scriptie zelf geschreven heb en dat ik de volledige verantwoordelijkheid op me neem voor de inhoud ervan. Ik bevestig dat de tekst en het werk dat in deze scriptie gepresenteerd wordt origineel is en dat ik geen gebruik heb gemaakt van andere bronnen dan die welke in de tekst en in de referenties worden genoemd. De Faculteit Economie en Bedrijfskunde is alleen verantwoordelijk voor de begeleiding tot het inleveren van de scriptie, niet voor de inhoud.

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1 Introduction

In an economy decisions of today are founded upon expectations about tomorrow. For example, an investor purchases a stock if he believes that the stock markets will rise. Expectations are therefore the basis of behavior. The realized prize of a stock depends on the number of people that want to buy or sell. According to Anufriev and Hommes (2012), market conditions are therefore the result of the aggregation of human behavior, and thus human expectations. Since market conditions are based on expectations and expectations based on market conditions the market is a feedback system. Gaining a better understanding of how expectations are formed can help us gain better insight into economic dynamics and better understand how economic crises develop.

Anufriev and Hommes (2012) state that academic consensus had been that individuals behave and form expectations rationally. In this rational world expectations of the homo

economicus on average correspond to market realizations and there is no place for irrational

behavior, according to Friedman (1953). However, since research by Simon (1957) it is com-monly accepted that perfect rationality is not a realistic view of the world. According to Simon, a realistic description of the environment of a decision-maker involves a large quantity of information and variables that are too numerous to handle (Simon, 1983). He claims that humans make sense of this web of complexity by making a simplified model of reality and making decisions based on variables in this model. They are only rational within the bounds of this model, or boundedly rational. Experiments have confirmed that decisions by individu-als in situations of uncertainty are not in line with rational behavior. In situations where an outcome was uncertain, especially when the problems posed were more intricate and less trans-parent, individuals made use of simple rules of thumb, called heuristics, to make predictions about the outcome (Tversky & Kahneman, 1974). They found that these intuitive guesses were prone to systematic errors and bias. Concerning boundedly rational agents, Sargent (1993) propagates a theory of evolutionary learning. He states that boundedly rational agents

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and adapt their behavior over time. More recently, models have been developed that include both evolutionary selection and heterogeneous expectations, the property that individuals have diﬀerent expectations (Hommes, 2011).

Most theories in macroeconomics have the assumption that there is no disagreement among agents. Since everyone has the same information set and the same information-processing abil-ities, they all end up with the same expectations (Mankiw, Reis, & Wolfers, 2003). However, there is considerable evidence that there is heterogeneity in forecasting in survey data. For example, Mankiw, Reis and Wolfers (2003) note that there is evidence for heterogeneity in in-flation expectations from forecasts by the Michigan Survey of Consumers, and they show that the data obtained is inconsistent with either rational or adaptive expectations. Furthermore, Capistrán and Timmermann (2009) found that heterogeneity of inflation expectations by pro-fessional forecasters was not constant over time and depended upon the level and variance of

current inflation. There has been a surge in literature that considers heterogeneous behavior.1

Based on these results, several learning-to-forecast experiments were conducted by re-searchers in diﬀerent sessions. In these experiments participants had to forecast the price of a risky asset, which was itself influenced by the forecasts, for multiple consecutive periods (Hommes, Sonnemans, Tuinstra, & Van de Velden, 2005; Hommes, Sonnemans, Tuinstra, & Van de Velden, 2008). Based on these results Anufriev and Hommes (2012) developed a

be-havioral model of learning with four heuristics, that was able to describe both individual and

aggregate behavior in those experiments. The goal of this paper is to determine whether their behavioral model and the heuristics that proved successful in the laboratory, is also successful in predicting real-life forecasts made by professional financial analysts. The paper is organized as follows. Section 2 provides a theoretical background. Section 3 consists of the data and methodology. It is followed by section 4, which provides the results and an analysis. Finally, section 5 concludes.

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2 Theoretical background

2.1 Learning-to-forecast

In the learning-to-forecast experiments by Hommes, Sonnemans, Tuinstra and Van de Velden (2005; 2008) participants were told that the realized price was determined by the aggregation of the forecasts of all participants, and that there was positive feedback between their forecasts and the price. After each forecast, participants were informed about the realized price and were asked to give another forecast with this information. The reward given to the participants was inversely proportional to their forecasting errors, so as to stimulate participants to fore-cast accurately. Under rational expectations the realized price level would fluctuate around the fundamental level with a small amplitude. However, their analysis of price behavior re-vealed three qualitatively diﬀerent patterns. These were convergence to equilibrium, persistent oscillations and oscillations with decreasing amplitude. Furthermore, Hommes, Sonnemans, Tuinstra and Van de Velden (2005; 2008) noticed that in every session participants at some point unintentionally coordinated on their forecasting behavior. What is remarkable is that this coordination was achieved with neither coordination between participants nor knowledge of the past forecasts of other participants.

Therefore, the empirical results obtained from the learning-to-forecast experiments by Hommes, Sonnemans, Tuinstra and Van de Velden (2005; 2008) were crucial in gaining insight into how expectations are formed. Participants had not made forecasts based upon rational expectations, but in some cases appeared to be moving toward the rational expectations forecasting rule. Also, three diﬀerent price patterns were observed: monotonic convergence, persistent oscillations and dampening oscillations.

Based on these observations, Anufriev and Hommes (2012, p. 48) derived several

styl-ized facts that applied to individual and aggregated expectations formations. For individual

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• Participants (partially) base their expectations on previous observations.

For aggregated expectations they conclude that:

• In every experiment subjects unconsciously coordinate on which heuristics to use. The

heuristics that participants coordinate on diﬀers between sessions.

• Coordination is far from perfect and some degree of heterogeneity remains.

A model that is appropriate in explaining heterogeneous behavior needs to be able to reproduce these stylized facts, both individually and aggregately.

2.2 The Model

2.2.1 Forecasting

Brock and Hommes (1997) developed a heterogeneous expectations model in which agents switch to forecasting strategies that have performed better. The model is based on the concept of behavioral heterogeneity. This means that participants can choose each heuristic and switch between heuristics if they decide to do so. The decision whether to switch or not is based on how well the heuristic has performed at forecasting the price so far. Anufriev and Hommes (2012) found evidence for this kind of behavior in the learning-to-forecast experiments by Hommes, Sonnemans, Tuinstra and Van de Velden (2005; 2008). Furthermore, they could explain individual heterogeneity in forecasting strategies in the same session, using a model that was an extension of the model of Brock and Hommes (1997). In this model they propose four heuristics that participants can choose from.

Adaptive heuristic. Participants use a heuristic that takes the realisation of the inflation

in the previous time period and the current expectation of the inflation into consideration:

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with w the weighting factor that determines the impact of the previous price

(and 0≤ w ≤ 1). Anufriev and Hommes (2012) used the value of 0.65 for w. For w = 1 the

adaptive heuristic corresponds to naive expectations. In the naive expectations case agents

believe that the next realized inflation rate is equal to the previous realized inflation: π_t+1e =

πt−1.

Trend-following heuristic. The authors estimated another heuristic that extrapolates the

previous change in the inflation rate from the last observed inflation rate.

πe_t+1= πt−1+ γ(πt−1− πt−2) (2)

with γ the strength of the extrapolation. Hommes, Sonnemans, Tuinstra and Van de Velden (2005) estimated the value of γ for each session. The lowest and highest estimated values of γ were 0.4 and 1.3. These values correspond to the weak and strong trend-following rule in this paper, respectively.

Learned anchor and adjustment heuristic. Other participants used a heuristic that

extrapolated the previous price change from a reference point or anchor. This rule is an adaptation of the anchor and adjustment rule, which uses the fundamental price as anchor. (Anufriev & Hommes, 2012). However, in the experiments by Hommes, Sonnemans, Tuinstra and Van de Velden (2005; 2008) participants were unaware of the fundamental price and

substituted as anchor the average inflation of all observations so far: πav_t₋₁.

π_t+1e = 0.5(πt−1+ π_t−1av ) + (πt−1− πt−2) (3) with π_tav₋₁= 1 t t ∑ i=1 πt−i.

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2.2.2 The Heuristics Switching Model

The Heuristics Switching Model (HSM) allows participants to switch between heuristics. In each time period every participant ’chooses’ a heuristic h to forecast the inflation in the next

time period from the available pool of heuristics, with h∈ H. Based on the performance of

the heuristics so far, participants update their choice. The performance of the heuristic is inversely proportional to the forecasting error by that heuristic. This performance measure looks as follows:

Uh,t−1 =−(πt−1− πh,te −1)2+ ηUh,t−2 (4)

A more negative value indicates a worse performance. The parameter η (with 0 ≤ η ≤ 1)

represents the significance participants appropriate to previous forecasting errors by heuristic

h, or memory. Given this performance measure the impact of heuristic h at time t is described

by:

nh,t = δnh,t−1+ (1− δ)

eβUh,t−1

Zt−1

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where δ (with 0 ≤ δ ≤ 1) is the fraction of agents that sticks with the previous strategy

and (1 - δ) the fraction that switches to a diﬀerent rule. β (with β ≥ 0) is the intensity of

choice, meaning how sensitive agents are to performance of the heuristics. If β = 0, agents

are indiﬀerent to forecasting errors made and move to an equal distribution. If the coeﬃcient of β is high, agents switch to better-performing rules more quickly. In the extreme version of

this, with β =∞, all participants who update their heuristics, the fraction 1 − δ, switch to

the best predictor. Zt−1=

H

∑

h=1

eβUh,t−1 _{is a normalization factor, so as to ensure that the sum}

of all heuristics is 1. Naturally,

H

∑

h=1

nh,t = 1

3 Method

The goal of this thesis is to determine whether the Heuristics Switching Model that has been successful in describing the forecasting strategies in the laboratory (Anufriev & Hommes, 2012)

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is also successful in describing the forecasting strategies of professional forecasters.

3.1 Data

The data is from the Survey of Professional Forecasters (SPF), which is a survey administered by the Federal Reserve Bank of Philadelphia. In this survey professional forecasters are asked to make predictions of several key economic indicators of the U.S. economy. Because it is the oldest such survey in existence, choosing this data source gives us most data points. Their quarterly forecasts of the GDP price index (PGDP) from 1968Q4:2015Q2 are used. As many forecasters dropped out and new ones entered and identification numbers were reused, individual forecasts were unreliable. Therefore, only average forecasts were used. The PGDP as forecasted by the professional analysts as well as the actual inflation rate is plotted in Figure 1. As inflation is the relative change in prices, inflation will be calculated as log-detrended relative change of the price level, as done by Corneo-Madeira, Hommes and Massaro (2016).

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -1 -0.5 0 0.5 1 1.5 2 2.5 Inflation Inflation SPF Forecast

Figure 1: Forecasts by the Survey of Professional Forecasters together with the actual infla-tion rate over time.

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3.2 Method and parametrization

The naive rule and each of the four heuristics that have been identified by Anufriev and Hommes (2012) is used to determine how accurate it is in describing the forecasts made by the SPF. The goodness of fit will be determined by calculating the mean squared error (MSE), with the MSE for heuristic h defined as

M SEh = 1 T T ∑ t=1 (πt+1− πeh,t+1)2 (6)

Then the Heuristic Switching Model (HSM) is applied to the data. In this heterogeneous model agents can use each of the four rules at any time. The fit of the HSM to the data is also determined by the MSE. The MSE of the HSM is defined as:

M SEHSM = 1 T T ∑ t=1 (πt+1− πet+1)2 (7)

with πe_t the weighted average of the forecasts of all heuristics. This weighted average is given

by: πe_t+1= H ∑ h=1 nh,tπh,t+1e (8)

The MSE’s of the individual rules and the MSE of the HSM are calculated to determine whether the homogeneous heuristics or the heterogeneous HSM fits the data best. As evi-denced by Anufriev and Hommes (2012), the relative weight of the heuristics changes over time in response to performance of the heuristics. When applying the HSM to the data, the distribution of the weight of the heuristics is determined.

As the trend-following heuristics and the learned anchor and adjustment heuristic require

πt−1 and πt−2, the realized prices of one and two periods ago respectively, forecasts can only

be obtained from t = 3 onwards for these heuristics2. As the adaptive heuristic requires no

variable that goes back further than t− 1, it can give a forecast for period 3. However, to

2

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compare heuristics the same starting point is used. MSE calculations are made from t = 17, because some homogeneous models that are not discussed in this paper require 16 periods to initialize. That way MSE’s by other models can be more easily compared.

The learned anchor and adjustment heuristic uses the average inflation rate π_tav₋₁ which

includes all previous observed inflation rates to make its prediction. The research question also includes how many previous observations of the inflation should be included in the prediction to optimalize the forecasts, since it may not be realistic for agents to keep track of so many previous inflation rates.

To obtain the performance measure for the first forecasting period the second term of equation (4) is omitted to obtain:

Uh,t =−(πt− πh,te )2 (9)

This is done for all heuristics h = 1, 2, 3, 4. Then, an initial distribution of nh,t for h = 1, ..., h

needs to be obtained. The calculation of nh,tis determined by the forecasting error of heuristic

h and by U_h,t−1, as seen in equation (4). However, at the beginning no forecasts have been

made yet, which means there is no value for Uh,t−1. Therefore, the model is initialized by

taking an initial distribution nh,t = 0.25 for h = 1, 2, 3, 4. Before forecasts are made no

forecast is preferred, therefore the distribution is equal3.

In their article, Anufriev and Hommes (2012) used β = 0.4, δ = 0.9 and η = 0.7 for their benchmark model. These values are used at first. However, these values were used to describe behavior of participants in a laboratory. Therefore, values of β, δ and η that yield a better description of the data with the HSM are chosen.

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Because the first time period for which every heuristic has made a forecasts is t = 4, the first distribution nh,tfor h = 1, 2, 3, 4 is at t = 5

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4 Results

4.1 Benchmark model

An overview of the heuristics as put forth by Anufriev and Hommes (2012) is given in Table 1. When fitting the four heuristics as seen in Table 1 and the naive expectations rule, together

Table 1: The heuristics

Adaptive heuristic (AH) π_t+1e = 0.65πt−1+ 0.35πte

Learned anchor and adjustment heuristic (LAAH) π_t+1e = 0.5(πt−1+ πtav−1) + (πt−1− πt−2)

Weak trend-follow heuristic (WTFH) π_t+1e = πt−1+ 0.4(πt−1− πt−2)

Strong trend-follow heuristic (STFH) π_t+1e = πt−1+ 1.3(πt−1− πt−2)

with the benchmark model as put forth by Anufriev and Hommes (2012) to the SPF forecasts we get the results in Figures 2, 3, 4, 5 and 6. The predictions made by the HSM are seen in Figure 7. 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast Naive 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 2: Forecasts of naive expectations (top panel) together with the prediction errors of the naive rule relative to the SPF forecasts (bottom panel)

From Figure 2 and Figure 3 we can see that the naive rule and the adaptive heuristic do well, especially around the years 1990-2000, where the prediction errors are close to zero.

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast AH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 3: Forecasts of the AH (top panel) together with the prediction errors of the AH relative to the SPF forecasts (bottom panel)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast LAAH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 4: Forecasts of the LAAH (top panel) together with the prediction errors of the LAAH relative to the SPF forecasts (bottom panel)

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast WTFH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 5: Forecasts of the WTFH (top panel) together with the prediction errors of the WTFH relative to the SPF forecasts (bottom panel)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast STFH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 6: Forecasts of the STFH (top panel) together with the prediction errors of the STFH relative to the SPF forecasts (bottom panel)

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) SPF Forecast HSM 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 7: Forecasts of the HSM (top panel) together with the prediction errors of the HSM relative to the SPF forecasts (bottom panel)

However, the naive rule and the AH seem to miss the more extreme observations, especially around 1975, and around 2007 and 2009 to a lesser degree. As seen in Figure 4, the LAAH seems to be giving large forecasting errors from the period 1970-1990, alternating between overestimating and underestimating the inflation expectations. The weak trend-following rule is quite accurate at forecasting from 1990-2000, as seen in Figure 5, just like naive expectations and AH. However, it overestimates the expectations from 2005-2010, with the exception of 2009, when it underestimates the expectations.

Due to its large value of γ (1.3) the STFH is extreme at following the trend, over- or underestimating the inflation expectations, as can be seen in Figure 6. From 1990 onwards it seems to consistently overestimate the expectations. Furthermore, it seems to be following its own trend that deviates from the SPF forecasts. In Figure 7 we can see that the HSM matches the predictions quite nicely. Its forecasting seems to be comparable to the forecasting of the AH, being very accurate from the years 1990-2000. In Figure 8 we can see the distribution

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Proportion AH LAAH WTFH STFH

Figure 8: Distribution of the heuristics in the HSM fitted to the SPF data over time

to its accurate forecasting, while the STFH seems to be losing followers, especially in the beginning, due to its extreme behavior. From around 1990 onwards the forecasts move to an equal distribution while around 2008 there is a jump again.

The heuristics seem to be giving accurate forecasts when the inflation is low and stable, during a time of economic prosperity. Around 1975, when the oil crisis hit, and around 2008 when the financial crisis hit, the heuristics and the HSM seem to be making their largest pre-diction errors. Professional forecasters seem to be cautious and modest in their expectations. So while the realized inflation rate and the predicted rates by the heuristics and the HSM go up, the inflation expectations by the forecasters stay behind, yielding larger errors.

To quantitatively compare the the accuracy of the heuristics the MSE’s are summarized in Table 2. What is noteworthy is that despite the HSM being able to switch to the best

Table 2: MSE’s of the diﬀerent heuristics. Lowest MSE shown in bold.

Naive AH LAAH WTFH STFH HSM

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predictor, under these circumstances it seems to be outperformed by two homogeneous rules, the naive rule and the adaptive heuristic. The forecasting of the HSM and its resultant MSE is largely determined by the values of β, δ and η, which in this case were 0.4, 0.9 and 0.7, respectively. For these values the HSM was accurate in describing the behavior of participants in the laboratory setting of Anufriev and Hommes (2012). However, the MSE of the HSM seems to suggest that for these values the HSM is not accurate in describing behavior in the real world. As we can see from Figure 9, the higher the β the higher the intensity of choice of

0 2 4 6 8 10 β 0.06 0.07 0.08 0.09 0.1 0.11 0.12 MSE HSM

Figure 9: MSE of the HSM for diﬀerent values of β

agents. This means that they are more sensitive to forecasting errors made by other heuristics and switch to more accurate heuristics more quickly; this will result in the HSM picking the better-performing rules more often and therefore a lower MSE. It seems to be reaching a value asymptotically, around 0.06. So at a certain point increasing the β will not result in a lower MSE. The β used for the benchmark model is 0.4, so the MSE can be improved. As we can see in Figure 10 the MSE seems to be slowly decreasing as δ increases. This means that the more agents stick with their previous strategy, the more accurate the model is. Around the value of 0.9 the MSE reaches its minimum, and continues to increase from there onwards. As in our benchmark model we have used the value 0.9 not much improvement will be gained

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0 0.2 0.4 0.6 0.8 1 δ 0.108 0.11 0.112 0.114 0.116 0.118 MSE HSM

Figure 10: MSE of the HSM for diﬀerent values of δ

0 0.2 0.4 0.6 0.8 1 η 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 MSE HSM

Figure 11: MSE of the HSM for diﬀerent values of η

their decision-making, the more accurate the model is. Our current value from the benchmark model is 0.7, so the MSE can be improved by varying η. Plots in Figures 9, 10 and 11 confirm that taking diﬀerent values of β, δ and η yield lower MSE’s. So in order to improve the fit of the HSM to the data we take values of β, δ and η so that the MSE of the HSM is minimized. However, individually varying the parameters β, δ and η may not yield the optimal model. Therefore the parameters are varied simultaneously (see Appendix A, Figures 17, 18, 19, 20,

21 and 22). Minimizing for β∈[0, 10], δ∈[0, 1]and η∈[0, 1]yields a minimal MSE (0.0408)

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fraction of agents switch and agents use all memory. 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 4 3 2 1 AH LAAH WTFH STFH

Figure 12: Ranks of the heuristics over time, from 1 (best) to 4 (worst)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Proportion AH LAAH WTFH STFH

Figure 13: Distribution heuristics for the HSM with lowest MSE (0.0408), with parameters

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minimal MSE that is (almost) equal to the MSE of the AH (0.0408). As we have seen from Table 1, the AH is the best forecaster by far, and as we can see from Figure 12, the AH gives the best forecasts most of the time. It appears that the optimal value for the HSM in this case means that all heuristics except the adaptive heuristic are simply unused. Indeed, our suspicion is confirmed when looking at Figure 13.

Under these economic conditions the HSM will not give an accurate forecast because only one rule is best. If one rule consistently outperforms the other heuristics the HSM will move to a distribution where only the best rule is used, functioning essentially as a homogeneous model. The power of the HSM is picking the best among equals, but if there are no equals it is no better than a homogeneous model. In order to better examine the performance of the HSM the heuristics of the HSM are adapted.

4.2 Adapted Heuristics Switching Model

4.2.1 LAA with diﬀerent inflation history

If forecasters use simple rules of thumb, how accurate is it that all previous inflation rates are taken into account for the LAA? It may be the case that agents only have limited capacity to keep track of all previous inflation rates. To refresh, the formula for the LAAH is:

πe_t+1= 0.5(πt−1+ 1 t t ∑ i=1 πt−i) + (πt−1− πt−2) (10)

However, we now vary the previous time period used by the formula. Currently, all previous

realized inflation rates are used in the forecast4. It may not be the case that the inflation of

40 years ago has any eﬀect on the forecast of today. Therefore, we adapt the model so that

the average inflation rate of the past J periods is used5. If the number of previous inflation

rates taken into account J is larger than the amount of observations, all observations are used. 4

Because π_t−1av is dependent on all observations from the past, this means that the inflation rate of 1968Q4 has influence on the forecast for 2015Q2.

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This looks like this: πav_t−1 =              1 j t−1 ∑ i=t−j+1 πt−i, if j ≤ t 1 t t−1 ∑ i=1 πt−i, if j > t

This is done for j = 1, ..., 1876 Plotting the MSE’s for all periods yields the following graph:

20 40 60 80 100 120 140 160 180 J 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 MSE

Figure 14: MSE of the LAAH as a function of the number of previous inflation rates taken into account when calculating the average past inflation rate

From Figure 14, we see that the more previous periods are taken into account the higher the MSE. We see that from around j = 183 the MSE stays constant. This is because the first forecast was made at t = 4 onwards so going back further is not possible. A minimum can be seen at j = 23. However, even the ’best’ value of the LAAH has a MSE (0.1275) that is much higher than the other heuristics. In the adapted HSM this rule is therefore dropped.

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4.2.2 Composition of the HSM

We add the naive expectations rule because it seems to be giving an accurate estimation (MSE of 0.076 as seen from Table 2). Furthermore, the STFH seems to be missing the mark quite often, and while the WTFH is more accurate it is still almost universally inferior to the AH. In order to have a trend-follow rule that is more competitive we vary the coeﬃcient γ as seen in equation (2) so as to obtain a trend-follow heuristic that is more accurate. Varying γ can

be seen in Figure 15. We obtain a minimal MSE for γ = −0.44. Because the coeﬃcient is

-1 0 1 γ 0.05 0.1 0.15 0.2 0.25 MSE Trend-follow heuristic

Figure 15: MSE for the trend-follow rule for diﬀerent values of γ. A minimum can be seen

at γ =−0.44

negative, this is sometimes called a contrarian rule (Kozyra & Lento, 2011). For this value of

γ the MSE is low, so this seems to be giving an accurate description of the data. Adding the

naive rule, dropping the LAAH and merging the weak and the strong trend-following heuristics

into the adapted trend-following heuristic with γ = −0.44 we can obtain an overview of the

heuristics used in the AHSM (Adapted Heuristics Switching Model) in Table 3. Table 3: The heuristics in the AHSM

Naive expectations (NE) π_t+1e = πt−1

Adaptive heuristic (AH) π_t+1e = 0.65πt−1+ 0.35πte

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4.2.3 Diﬀerent values of β, δ and η

Now that we have three heuristics that seem to be approximately equal, with MSE’s of 0.0760, 0.0408 and 0.0640 for the NE, AH and the ATFH respectively, the HSM might be optimal if we minimize for β, δ and η. Minimizing the MSE by changing the values of β, δ and η yields us a minimal MSE for β = 1.8, δ = 0.94 and η = 1. The MSE’s are shown in Table 4. We

Table 4: MSE’s heuristics AHSM and AHSM. Lowest MSE shown in bold

NE AH ATFH AHSM

0.0760 0.0408 0.0640 0.0510

can see that the MSE of the HSM (0.0510) is higher than the MSE of the best homogeneous predictor, the AH (0.0408). Even for values of β, δ and η that yield an optimal MSE of the HSM, the HSM is worse than the AH. In the case of the HSM with four rules the optimal MSE was equal to the MSE of the best homogeneous predictor.

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Proportion ATFH AH NE

Figure 16: Distribution of the three heuristics used in the optimal AHSM (β = 1.8, δ = 0.94 and η = 1, with MSE 0.0510

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After 1975 it appears to be doing best once again and almost everyone switches to the AH. Due to the relatively weak performance of the AH in the first few years the proportion of the ATFH increased. However, it seems to be the case that the AH was the better predictor after these years. Because the HSM was slow to adapt the MSE turned out to be higher. Furthermore,

because only three heuristics were used, the starting distributions of each heuristic was 1

3,

which means that if a rule was still weak, it took longer to eliminate the rule, yielding a higher MSE.

5 Forecasting Inflation

As we can see from Figure 1, the forecasts of the inflation rate seem to not be completely accurate. Forecasters are too careful apparently, making modest forecasts even when the realized inflation rate turns out higher. The HSM is unable to account for this risk-aversion, which results in large prediction errors when the realized inflation rate turns out high. To determine whether the HSM is more accurate at forecasting the real inflation rate the forecasts are plotted (in Appendix B), and the MSE’s are shown in Table 6: The AH seems to be Table 5: MSE’s of the diﬀerent heuristics for the real inflation rate. Lowest MSE shown in bold.

NE AH LAAH WTFH STFH HSM

0.1035 0.0855 0.2268 0.1303 0.2709 0.1430

doing best. It appears that the ranking of all the heuristics relative to eachother has stayed approximately the same, with the AH doing best and the STFH doing worst. However, while

the HSM was second-best, losing only to the AH when forecasting inflation expectations7, the

WTFH seems to be more accurate than the HSM in forecasting the real inflation rate. These MSE’s were obtained with the benchmark values of β, δ and η, so the MSE minimized by changing their values. The optimal MSE is 0.0855 (β = 10, δ = 0.14 and η = 1). The MSE

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of the optimal HSM is equal to the MSE of the AH, which was also the case when forecasting SPF expectations. The same problem that applied to the expected forecasts seems to apply here. The HSM is transformed into the AHSM by using the same three rules as seen in Table 3. The optimal values of β, δ and η are determined in order to minimize the MSE. The MSE’s of the three rules together with the MSE of the optimal AHSM are summarized in Table 6.

Table 6: MSE’s heuristics of the AHSM and the AHSM itself. Lowest MSE shown in bold

NE AH ATFH AHSM

0.1035 0.0855 0.0993 0.0904

The lowest MSE achieved by the HSM is 0.0904 for β = 1.3, δ = 0.94 and η = 1. The optimal AHSM with three rules is worse than the best homogeneous predictor, which was also the case in forecasting SPF predictions.

6 Conclusions

Whether the forecasting heuristics and the Heuristics Switching Model (HSM) that proved accurate in predicting the expectations of economic variables by participants in a laboratory experiment, were also accurate in describing the inflation expectations of professional forecast-ers, was the goal of this paper.

The HSM works by switching to the best forecasting rule. A necessary requirement for accurate forecasting therefore requires an approximately equal performance of all heuristics. If all heuristics are doing equally well, the HSM is able to each time pick the best forecasting rule. However, if one heuristic consistently performs best the HSM distribution will move to always pick that forecasting rule, functioning essentially as a homogeneous model. As we have seen, inflation forecasts of the U.S. economy by the Survey of Professional Forecasters was almost universally best described by the adaptive heuristic, as measured by the MSE. In

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Because forecasters made modest forecasts, the heuristics and the HSM were also fitted to real inflation data to determine whether this fit was more accurate. The result was that, once again, the homogeneous adaptive heuristic was best at forecasting. Optimalizing the HSM in this case yielded a MSE that was equal to the MSE of the best homogeneous rule. When the rules incorporated into the HSM were changed, namely the inaccurate heuristics (LAAH, STFH) were dropped, the accurate naive expectations was added and the WTFH was adapted into the more accurate contrarian ATFH, the optimal HSM was worse than the best homogeneous heuristic at describing SPF forecasts. When using this adapted HSM to forecast real inflation data the optimal HSM was once again worse than the best homogeneous rule. Apparently, reducing the number of rules used in the HSM, even though the rules were more accurate by themselves, resulted in a worse fit of the data by the HSM.

Further research might investigate whether using less heuristics was indeed the cause of the MSE that was higher than the most accurate homogeneous rule, as was the case with our AHSM. If this is the case, adding more rules to the HSM might improve its accuracy. Furthermore, determining heuristics that are more accurate than the current heuristics used might aid in improving the fit of the HSM to the behavior of professional forecasters, and whether the HSM functions better with these heuristics. The heuristics and the HSM were able to predict the forecasts quite accurately if there were no economic shocks, judging by the small prediction errors in prosperous years, while there were large prediction errors in 1975, around the oil crisis, and around 2008, the time of the financial crisis. Further research might investigate how the HSM can be adapted so as to ensure that it also forecasts accurately during economic shocks.

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References

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A MSE of the HSM for simultaneously varied β, δ and η

0.04 0 0.06 0 0.2 0.08 MSE 2 0.1 0.4 η 0.12 4 β 0.6 ₆ 0.8 ₈ 1 10

Figure 17: The MSE of the benchmark HSM with four rules for diﬀerent values of β and η

0 1 2 3 4 5 6 7 8 9 10 β 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η 0.05 0.06 0.07 0.08 0.09 0.1 0.11

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0.04 1 0.06 0.8 ₁ 0.08 MSE 0.6 0.8 0.1 η 0.6 δ 0.4 0.12 0.4 0.2 _0.2 0 0

Figure 19: The MSE of the benchmark HSM with four rules for diﬀerent values of δ and η

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η 0.06 0.07 0.08 0.09 0.1 0.11

Figure 20: A top view of the above graph of the MSE of the benchmark HSM with four rules for diﬀerent values of δ and η. Blue indicates a lower MSE while yellow indicates a higher MSE.

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0.06 1 0.08 0.8 ₁₀ MSE 0.6 0.1 8 δ 6 β 0.4 0.12 4 0.2 ₂ 0 0

Figure 21: The MSE of the benchmark HSM with four rules for diﬀerent values of β and δ

0 1 2 3 4 5 6 7 8 9 10 β 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115

Figure 22: A top view of the above graph of the MSE of the benchmark HSM with four rules for diﬀerent values of β and δ. Blue indicates a lower MSE while yellow indicates a higher MSE.

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B Forecasts of the heuristics and the HSM versus real inflation

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation NE 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 23: Forecasts of naive expectations (top panel) together with the prediction errors of the naive rule relative to the realized inflation rate (bottom panel)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation AH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 24: Forecasts of the AH (top panel) together with the prediction errors of the AH relative to the realized inflation rate (bottom panel)

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation LAAH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 25: Forecasts of the LAAH (top panel) together with the prediction errors of the LAAH relative to the realized inflation rate (bottom panel)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation WTFH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 26: Forecasts of the WTFH (top panel) together with the prediction errors of the WTFH relative to the realized inflation rate (bottom panel)

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1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation STFH 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 27: Forecasts of the STFH (top panel) together with the prediction errors of the STFH relative to the realized inflation rate(bottom panel)

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 3 Inflation (%) Inflation HSM 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -2 -1 0 1 2 Inflation (%) Prediction error

Figure 28: Forecasts of the HSM (top panel) together with the prediction errors of the HSM relative to the realized inflation rate(bottom panel)

Expected expectations : a heterogeneous model for inflation expectations of professional forecasters