MASTER’S THESIS Financial Econometrics
Heterogeneous Agents and Time-Varying Beliefs about Cyclical and
Non-Cyclical Stock
Thesis Supervisor: Prof. dr. Cees G.H. Diks Second Marker: Prof. dr. Cars H. Hommes
Anouk Phielix – 5922119 July 7, 2014
1 INTRODUCTION ... 2
2 A HETEROGENEOUS AGENTS MODEL APPLIED TO THE STOCK MARKET ... 5
2.1 THE MODEL: OUTLINE AND ASSUMPTIONS ... 5
2.2 HETEROGENEOUS BELIEFS AND PERFORMANCE-DEPENDENT SWITCHING ... 8
3 DATA AND ESTIMATION OF THE FUNDAMENTAL VALUE ... 11
3.1 QUARTERLY AND MONTHLY DATA FOR SIX STOCKS ... 12
3.2 THE FUNDAMENTAL VALUE ... 12
4 EMPIRICAL RESULTS ... 16
4.1 ESTIMATION RESULTS: SOME GENERAL FINDINGS... 17
4.1.1 Model estimates for quarterly versus monthly data ... 19
4.1.2 Model estimates for cyclical versus non-cyclical stocks ... 20
4.2 ESTIMATION OF THE TIME-VARYING FRACTIONS ... 21
5 IN-SAMPLE FORECASTING ... 23
5.1 THE BHM MODEL VERSUS AN AUTOREGRESSIVE MODEL ... 23
5.2 THE DYNAMICS OF IN-SAMPLE FORECASTING ... 26
6 CONCLUSION AND DISCUSSION ... 31
1
Introduction
Stock prices are commonly associated with expectations of future economic events (Fama, 1981; Fischer and Merton, 1984; Estrella and Mishkin, 1998; Aylward and Glen, 2000). This is why it is important to obtain insights into the dynamics of stock prices and their drivers. Many believe that stock prices are influenced by underlying economic fundamentals (Cutler, Poterba and Summers, 1988; Chan, Hamao and Lakonishok, 1991; Heaton and Lucas, 2000). For example, when revenues are high and there is no indication of revenues falling any time soon, the stock price will generally incline due to the attractiveness of this company’s stock. However, several studies of historical stock price movements show that “new information” concerning the fundamental value only partially explains these movements (Shiller, 1981; Le Roy and Porter, 1981). In other words, stock prices exhibit excess volatility, which cannot be attributed to movements in the underlying fundamentals. Moreover, it is broadly acknowledged that stock price movements in cyclical markets differ significantly from stock price movements in non-cyclical sectors. However, is this difference mainly caused by the fundamentals or is there a behavioral explanation?
Much research has been done concerning economic fluctuations. Generally speaking, there are two views concerning the dynamics of business cycles and the sources of economic movements (Hommes, 2013). The first view, introduced by Tinbergen (1930), Frisch (1933), and Slutzky (1937), was that of a stable economic system where random processes can form cyclical patterns. These random processes are described as shocks to economic fundamentals such as earnings or dividends. However, this linear and stable view of economic fluctuations was criticized in the 1940s and 1950s because the explanation given for economic fluctuations was not economic but rather based on external, non-economic forces. Alternatively, Kaldor (1940), Goodwin (1951), and Hicks (1950) proposed nonlinear, endogenous business cycle models in which the economy may be unstable even in the absence of external shocks. However, these models were also criticized for multiple reasons (Hommes, 2013)1. One of the main shortcomings was that in these endogenous models agents behaved irrationally not learning from experience to anticipate future cycle movements. This prompted the rational expectations revolution in the 1960s and 1970s lead be Muth (1961) and popularized by Lucas (1972). Muth (1961) stated that the economy does not waste
information, and that expectations of agents depend on the structure of the entire system. Moreover, Lucas (1972) assumed that exchanges take place at the market clearing price and that agents use the correct conditional distribution in forming expectations, which together imply that the market is efficient. Another well-known advocate of the rational agent approach is Milton Friedman (1953) who claims that non-rational agents will not survive evolutionary competition and will therefore be driven out of the market (Hommes, 2013).
In the late eighties and nineties a number of new developments initialized a paradigm shift from the rational expectations hypothesis to bounded rationality and heterogeneous agent models (HAMs). The first to propose bounded rationality of agents was Simon (1957) who pointed out that people are only partially rational, limited by their cognitive capabilities and the finite amount of time to make a decision. He proposed to replace the idea of utility maximization with a more realistic view of behavior involving satisficing and heuristics (Selten, 1990). This idea was later on popularized by Tversky and Kahneman (1974), who use laboratory experiments to show that individuals often do not behave rationally but instead rely on a limited number of heuristics, which may lead to significant biases. Other contributors to the paradigm shift were Milgrom and Stockey (1982), who argue that if all agents are rational and it is common knowledge among them that the equilibrium trade is feasible and individually rational, agents will never agree to a non-null trade if they begin at their Pareto optimal allocation. Moreover, it was claimed that stock prices exhibit excess volatility, which is difficult to explain by a representative, rational agent model (Shiller, 1981; Le Roy and Porter, 1981). The absence of fundamental economic news during periods of dramatic stock price movements (e.g. late 1987) confirms the idea of excess volatility and hence contradicts the standard view of attributing price fluctuations to changes in the fundamental (Cutler et al., 1988).
One of the first to propose a HAM for the stock market was Zeeman (1974). Even though the model is highly stylized and lacks any micro foundations, it contains a number of important behavioral elements that are also used in current heterogeneous agent modeling (Hommes, 2013). The most essential behavioral element of the model is the introduction of two types of traders i.e. fundamentalists and chartists (trend-followers). Fundamentalists know the “true” value of the stock and believe prices will revert back to that value whereas chartists believe prices to follow trends. In the late eighties and early nineties, Frankel and Froot (1986, 1990) studied the large movements of the US dollar exchange rate in the eighties and use survey data to investigate whether expectations of investors influenced these movements. They associate the longer-term expectations with fundamental expectations, the
shorter-term forecasts with chartist’s expectations, and they stress the importance of chartists since their behavior can exacerbate the movements of asset prices thus amplifying speculative bubbles and crashes. Moreover they find that the weight of the different forecasting strategies shifts over time leading to the introduction of time-varying beliefs. In 1997 and 1998 Brock and Hommes investigate the dynamics of a simple present discounted value asset pricing model with heterogeneous beliefs where agents can choose from a finite set of forecasting strategies and may revise their beliefs in each period in a bounded rationally way, according to a performance measure such as past realized profits. They find that heterogeneity of beliefs can lead to persistent deviations from the fundamental price and chaotic price fluctuations whenever agents are highly sensitive to the performance measure of the strategies. This HAM has been estimated, using annual S&P 500 stock market data, by Boswijk, Hommes and Manzan (2007). Their results show statistically significant behavioral heterogeneity and time variation in the average sentiment of traders. Moreover their model (from now on referred to as the BHM model) suggests that the dot com bubble in the late nineties, triggered by higher expected cash flows of the internet sector, was amplified by trend-following investors motivated by short run profitability. Recently, a standard housing model based on the Brock and Hommes (1997, 1998) HAM was introduced linking housing rental levels to fundamental buying prices using quarterly data for seven different countries (Bolt, Demertzis, Diks, Hommes and van der Leij, 2014). As for the BHM model, heterogeneity in expectations with temporary endogenous switching was found and temporary house bubbles were identified amplified by trend extrapolation and crashes reinforced by fundamentalists. The aim of this thesis is to obtain insights into the expectations feedback system of the stock market. Specifically, expectations about cyclical stock and non-cyclical stock are compared and, since Boswijk et al. used annual stock market data, I investigate whether the model holds for higher frequency data, i.e. quarterly and monthly data.
The outline of this thesis is as follows. Chapter 2 describes the heterogeneous agents model with performance-dependent switching. In Chapter 3 a description of the data is given and the calculation of the fundamental stock prices and price-to-earnings (PE) ratios is depicted. Chapter 4 presents the estimation results and investigates the dynamics the time-varying beliefs of agents. In Chapter 5 the proposed heterogeneous agents model is compared to a simple autoregressive model and in-sample forecasting is performed. Finally, Chapter 6 concludes.
2
A heterogeneous agents model applied to the stock market
In this chapter I reflect on the heterogeneous asset pricing model with adaptive beliefs introduced by Brock and Hommes (1997, 1998) and reformulate the model in terms of price to cash flow following Boswijk et al. (2007). Agents are assumed to be boundedly rational as they revise their beliefs based on a certain ‘fitness measure’. In other words, agents have heterogeneous beliefs concerning future payoffs and are allowed to switch from one forecasting rule to another, based on the performance of the forecasting rule in the previous period. A behavioral bias is incorporated into the model capturing a common sentiment of the agents regarding the deviation from the fundamental value.
2.1 The model: outline and assumptions
The main assumption of the model presented in this thesis is that there are two assets available, a risky and a risk-free asset. The risk-free asset is in perfect elastic supply and has a constant return r. The risky asset is in zero net supply and denotes the uncertain cash flow generated by the risky asset each period of time. denotes the price of the risky asset at time and the wealth of an agent is described as
, (1)
where
(2)
denotes the excess return of the risky asset and denotes the number of shares of the risky asset purchased at time t. Let [ ] and [ ] denote the expectations of agent h about the conditional mean and the conditional variance of the excess return. The assumption is made that all agents are myopic mean variance maximizers. Consequently, solving this maximization problem for the demand for shares gives
{ [ ] [ ]}
→ (3)
[ ]
[ ]. (4)
In Appendix A.1 the full derivation of the demand for shares is given. Following Brock and Hommes (1997, 1998) and Boswijk et al. (2007), it is assumed that agents are homogeneous with respect to the expectation about the conditional variance and the risk aversion parameter, i.e. [ ] and . This implies that the only source of heterogeneity incorporated in this model is the beliefs about the future payoffs of the risky
asset. Since a zero net supply of the risky asset is assumed, the market clearing equation equals
∑ [ ]
. (5)
As a consequence of the assumption of zero net supply of the risky asset, the risk aversion parameter a and the conditional variance do not play a role in the equilibrium pricing equation which is given by
∑ [ ]. (6)
Notice that according to Eq. (6) the price of the risky asset at time t is given by the discounted, weighted average of the agents’ expectations about the payoffs in the next period. Moreover, Eq. (6) evidently shows that when the fraction of optimistic agents is relatively large, the equilibrium price will be high and contrarily, when the fraction of pessimistic agents is rather large the equilibrium price will be driven to a lower level. Upon rewriting the equilibrium pricing equation (6) as
∑ [ ], (7)
it becomes clear that, in equilibrium, the required rate of return for agents to hold the risky asset equals the discount rate r. During the estimation of the fundamental value in Chapter 3, the discount rate r will be set equal to the risk free interest rate plus the required risk premium on stocks RP. Note that, since the risk aversion parameter does not influence the pricing equation, the risk premium RP is theoretically equal to zero. To allow for a nonzero risk premium, the model should be altered by letting go of the assumption of zero net supply.
Following Boswijk et al. (2007) and Bolt et al. (2014), it is assumed that cash flows follow a geometric Brownian motion with drift i.e.
, (8) where both and are commonly known. Accordingly
(9)
where and which implies that [
] . As stated before, the only heterogeneous factor of this model is the beliefs agents have about the future payoff of the risky asset, which is determined by the current price and next period’s price and cash flow . Since future cash flows are exogenously given by the stochastic process described above and because agents can simply compute the constant growth rate g by running a regression on past realized cash flows, it may be assumed that agents are able to determine the correct beliefs and thus agree on the distribution of next
period’s cash flow. Moreover, the current price of the risky asset is commonly known, which means that the only endogenously determined factor left in the model is the beliefs agents have about the future price of the risky asset . Since prices are affected by the expectations of agents and because a heterogeneous world is assumed, it seems unlikely that agents agree on future prices thus the assumption of heterogeneous beliefs about next period’s price seem realistic.
As the cash flow is exogenously given and known to all agents, a reformulation2 of the pricing equation (6) in terms of price-to-cash flow (PY) ratio seems beneficial hence
{ ∑ [ ]} . (10)
Concerning the fundamental price , the special case is considered where all agents have (the same) rational expectations. If so, the equilibrium pricing equation (6) simplifies to
[ ]. (11)
Consequently, under the assumptions of rational expectations and a constant growth rate of cash flows (dividends), the well-known Gordon growth formula states that the stock price should equal the sum of future cash flows discounted back to their present value. Section 3.2 elaborates on this subject. The fundamental price of the risky asset is therefore given by
, (12)
which holds as long as the expected return of the stock market is larger than . Rewriting (12) in terms of the PY ratio gives
, (13)
which henceforth will be referred to as the fundamental PY-ratio. If all agents were to be rational the pricing equation (10) would become
{ [ ]}. (14)
Define as the deviation of the PY-ratio from its fundamental. Consequently, Eq. (14) may be rewritten as
[ ] (15)
resulting from a rather straightforward computation given in Appendix A.3. Assuming heterogeneity in expectations the pricing equation can ultimately be expressed as
∑ [ ]. (16)
2.2 Heterogeneous beliefs and performance-dependent switching
This section specifies how the different types of agents construct their beliefs and describes how the selection of beliefs is made over time. Following Boswijk et al. (2007) and Bolt et al. (2014) the economy is characterized by two types of traders and both types of traders predict next period’s deviation to be a linear extrapolation of the previous deviation. However, a behavioral bias was added to the model representing a common, persistent feeling agents have about the deviation from the fundamental:
[ ]
[ ] (17)
As Eq. (17) shows, the two types of agents use a different extrapolation parameter when predicting future deviations, characterising the heterogeneity in the model. From now on, when , agents will be referred to as fundamentalists since they believe that the PY-ratio will converge to its fundamental after a while. Contrarily, when , agents will be referred to as trend followers since these agents believe the deviation of the PY-ratio from the fundamental follows a trend i.e. will diverge from the fundamental. An exception is made when the behavioral bias is found to be significantly different from 0. In that case, agents do not believe the fundamental to be true and therefore will never be an equilibrium hence no agents believe the PY-ratio to converge to the fundamental.
Following Brock and Hommes (1997) and based on significant evidence found by Shiller (2000) of changing attitudes of investors through time, this model assumes time variation of investor sentiment. Agents are boundedly rational and switch between the two types of strategies based on the relative performance of these strategies. At time , the performance is measured by the realized profits at the end of period , given by
[ ]
, (18)
where is the realized excess return as given in Eq. (2) and is the demand of type for shares of the risky asset at time as given in Eq. (4). Since I reformulated the pricing model in terms of the PY-ratio instead of the stock price itself, it seems convenient to express the realized profit in terms of . In order to do so, two simplifying assumptions are made. The first assumption is the approximation of the stochastic earnings by its expectation at i.e. , which allows me to rewrite Eq. (2) as
( )
This means that under this assumption, the realized profit does not depend on the stochastic growth rate of earnings but rather on the average growth rate . The second simplifying assumption to be made involves the conditional variance of the excess return . Remember that the assumption of an equal conditional variance of return for all types has already been made, i.e. [ ] . Following Boswijk et al. (2007), it is additionally assumed that this homogeneous belief about the conditional variance is equal to fundamentalists’ beliefs about the conditional variance. Therefore, the conditional variance about the return may be rewritten as
[ ] [ ] [ ] [ ]
[ ] where [ ]. Note that, in the fourth line, is replaced by , which holds by Eq. (9). Moreover, is known at time and thus involves no randomness. With the use of Eq. (19) and Eq. (20), the performance measure can now be rewritten in terms of the PY-ratio :
[ ] [ ] [ ] Furthermore, the realized profit can be expressed in terms of , with and , as
( [ ] ) Considering Eq. (22), an intuitive explanation of the performance measure can be given. Notice that, in order to realize a positive profit at time , and ( [ ] ) must have the same sign i.e. both terms must be positive or negative. This means that for example, if was expected to be larger than at time , and at time it indeed turned out that this was the case, the realized profit would be positive.
Following Bolt et al. (2014), I define the latter product of Eq. (22) as the realized utility at time for investor type :
( [ ] ). (23) The time-varying fractions are assumed to evolve according to a discrete choice model with multinomial logit probabilities:
( [ ] [ ]) ( ) where parameter represents the intensity of choice. Otherwise stated, represents the sensitivity of agents to differences in performance. A high value of represents a situation in which agents switch quickly to last period’s best performing stategy whereas a low value of corresponds to a situation in which agents are hesitant to switch to other strategies except if the difference in performance is rather large (Boswijk et al., 2007). The fact that the realized utility is adopted instead of the realized profit as a performance measure has the advantage that the term does not need to be estimated while the intuitive explanation given above still holds. However, since the term has now been absorbed by , it does have the disadvantage that is not comparable between the different stocks. However, since Bolt et al. (2014) found that the model’s forecast accuracy is not very sensitive to the exact value of , and other parameters can to a certain extent compensate for changes in , loosing the possibility of comparing -values does not seem to be a large disadvantage.
In order to clearly illustrate the equilibrium of the model and its local stability condition, the full dynamical system of the heterogeneous asset pricing model is expressed as
From Eq. (25) it follows that the fundamental equilibrium is given by , and . Moreover, by linearizing the dynamics around this equilibrium, the stability condition is given by
| | | | . (26)
Before estimating the full heterogeneous agents model the fundamental value of the PY-ratio is estimated which will be described in the next chapter. Subsequently, assuming white noise disturbances on the pricing equation leading to
, (27)
the parameters of Eq. (27) are estimated using nonlinear least squares (NLS) since the fractions depend nonlinearly on the model parameters.
Regarding the PY-ratio, earnings instead of dividends are used as cash flow. The reason for this is that over the years dividends have frequently dropped to zero, which, when using dividends as cash flow, causes the PY-ratio to be undefined. In the exceptional case of zero earnings at time , is replaced with the average of earnings in the previous and the next period . In other words, all undefined PE ratios caused by zero earnings are replaced by the actual price divided by an average of the previous and next period’s earnings. However, if this “PE-average” is found to be larger than 100 or if at any point in time the PE ratio is larger than 100 caused by very low earnings, the PE ratio is replaced with 100 hence getting rid of all outliers.
3
Data and estimation of the fundamental value
In this chapter a description is given of the data used in this thesis. Moreover it is shown (graphically) how real stock prices compare to calculated fundamental prices and how the PE ratio’s compares to the fundamental ratio . In the second section the estimation of the fundamental PE ratio is described and the values used for this estimation per specific stock are discussed. Finally, based on these values and the volatility of the stock price deviation from the fundamental i.e. the excess volatility, I discuss my expectations.
3.1 Quarterly and monthly data for six stocks
The data used in this thesis consist of quarterly and monthly stock prices and PE ratios. All stock prices and PE ratios were extracted from DataStream. For Coca-Cola, McDonald’s, Johnson & Johnson, Lowe’s, and Johnson Controls data were available from 01-02-1973 to 12-02-2013. For Gap, data were available from 06-02-1976 to 12-12-2013. All stocks trade on either the NYSE or the NASDAQ, and their prices are quoted in US dollars. The real stock prices were derived by deflating nominal prices with the quarterly and monthly Consumer Price Index (CPI), available from the OECD "Main Economic Indicators - complete database". The CPI index for all items in the United States is used since the industries of the stock vary strongly. The CPI is indexed using 2010 as base year.
This thesis discusses three cyclical stocks and three defensive stocks. A definition of these two classifications of stock is given by Doeswijk (2008, p. 177): “Cyclical stocks are
highly sensitive to the economic cycle, i.e. they perform relatively well when expectations about the economy improve and relatively bad during periods with worsening economic expectations. The reverse applies to defensive stocks, whose growth is relatively stable and is not strongly affected by fluctuations in the economic cycle.” Since Coca-Cola and
McDonald’s are both in the food and beverages industry, thus sell products of which demand continues regardless of the economy, their stocks are typical examples of defensive stocks. Furthermore, Johnson & Johnson has such a diversified product portfolio, operating in the segments consumer, healthcare, and medical devices, that it outperforms many companies in these sectors, especially when the economy is weak. Therefore, Johnson & Johnson stock is categorized as defensive stock. Lowe’s is an American chain of retail home improvement and appliance stores, a sector that is very sensitive to economic fluctuations. Gap sells apparel, accessories, and personal care products, which are all commonly seen as luxury products, hence demand declines in times of economic slowdown. Finally, Johnson Controls is a global diversified technology and industrial business company. The company offers services through its three business segments: building efficiency, automotive experience, and power solutions. As these three segments are closely related to the automobile industry, Johnson Controls stock is cyclical.
3.2 The fundamental value
In order to estimate the behavioral model described in Eq. (25), the fundamental model parameters must first be estimated. The fundamental price is constructed by using the static
Gordon growth model (Gordon, 1963). This is a dividend discount model assuming a constant growth rate of dividends in perpetuity and a constant cost of equity capital for the concerning company. However, because (growth) stock may sometimes not pay any dividend, a more general version of the static Gordon model is used. In this version the assumption of dividend irrelevance, stated by Miller and Modigliani (1958), is made and the stock’s dividend is replaced by the stock’s earnings per share
, , (27)
where indicates the fundamental price of the asset, stands for the earnings per share at time , and and are averages over time. It follows that, in the model described in the previous chapter, is the fundamental PE ratio.
The average growth rate is directly measurable from the data, however, to estimate the same method is employed as in Fama and French (2002) and Boswijk et al. (2007). They state that the risk premium can be estimated by
, (28)
where denotes the average dividend yield and the risk free interest rate. Again, the dividend yield is replaced by the earnings yield so the estimation of the risk premium becomes
, (29)
where stands for earnings. Because equals the risk free rate plus the risk premium it follows that
, (30)
which makes it unnecessary to estimate or find a measure of the risk free interest rate . Fig. 1 presents the stock prices with their corresponding fundamental values (left panel) and the PE ratios with their corresponding fundamental value (right panel).
Figure 1: Real stock prices and estimated fundamental real stock prices (left-hand side) and PE ratios and year p ri ce 1980 1990 2000 2010 5 1 0 2 0 5 0 1 0 0
Real stock price Fund. real stock pr ice
MCDONALDS year P E 1980 1990 2000 2010 1 0 2 0 5
0 PE RatioFund. Ratio
MCDONALDS year p ri ce 1980 1990 2000 2010 2 5 1 0 2 0 5
0 Real stock price
Fund. real stock pr ice
COCA COLA year P E 1980 1990 2000 2010 1 0 2 0 5 0 1 0 0 PE Ratio Fund. Ratio COCA COLA year p ri ce 1980 1990 2000 2010 5 1 0 2 0 5 0 1 0 0
Real stock price Fund. real stock pr ice
JOHNSON & JOHNSON
year P E 1980 1990 2000 2010 1 0 2 0 3 0 5
0 PE RatioFund. Ratio
JOHNSON & JOHNSON
year p ri ce 1980 1990 2000 2010 0 .5 2 .0 5 .0 2 0
.0 Real stock priceFund. real stock pr ice
LOWE'S year P E 1980 1990 2000 2010 1 0 2 0 5 0 1 0 0 PE Ratio Fund. Ratio LOWE'S year p ri ce 1980 1990 2000 2010 0 .1 0 .5 2 .0 1 0 .0 5 0 .0
Real stock price Fund. real stock pr ice
GAP year P E 1980 1990 2000 2010 5 1 0 2 0 5 0 PE Ratio Fund. Ratio GAP year p ri ce 1980 1990 2000 2010 1 2 5 1 0 5 0
Real stock price Fund. real stock pr ice
JOHNSON CONTROLS year P E 1980 1990 2000 2010 5 1 0 2 0 5 0 1 0 0 PE Ratio Fund. Ratio JOHNSON CONTROLS
The right-hand side graphs in Fig. 1 show that the PE ratio ranges of the three cyclical stocks are wider than the PE ratio ranges of the defensive stocks. This agrees with the statement that cyclical stocks are more volatile than defensive stocks. Note however that the volatility of Johnson & Johnson’s and Lowe’s PE ratio is greatly influenced by extremely low earnings in the early seventies. For all three defensive stocks the PE ratio never falls far below ten whereas GAP’s and Johnson Controls’ PE ratio falls below five in the late eighties. The price of Coca-Cola stock follows a rather steep upward trend in the nineties, suggesting a large fraction of trend-following agents. In contrast, Johnson Controls’ PE ratio lies very close to its fundamental value for a long period of time suggesting the absence of trend-followers. Finally, for all six stocks the deviations from the fundamental are larger upwards than downwards i.e. the highest PE ratio is much further from the fundamental than the lowest PE ratio. This is caused by the fact that earnings per share, relatively speaking, easily drop to zero in times of economic distress whereas stock prices do not, causing the PE ratio to increase dramatically.
Tables 1 and 2 summarize all values discussed in this section for each stock, the former for quarterly data and the latter for monthly data. All these values are used in the estimation of the fundamental. The average earnings yield represents the average of over time, where equals the earnings at time multiplied by four in the case of quarterly data and twelve in the case of monthly data and denotes the last observed stock price. The average growth rate stands for the average annual growth rate of earnings. It is shown that the yield and the annual growth rate differ slightly when comparing quarterly data to monthly data, which seems plausible since monthly data provides one with more information, and thus more movement, than quarterly data. The discount rate , the gross rate of return , and the fundamental PE ratio , are merely calculations based on the found values of and .
For both quarterly and monthly data, the average growth rate of earnings is much higher for the three cyclical stocks than for the defensive stocks. As a consequence, the estimated discount rate is relatively high for cyclical stock. The fundamental PE ratios of the three defensive stocks lie near each other whereas the fundamentals of the cyclical stocks differ significantly; Lowe’s fundamental PE ratio is almost twice as high as Johnson Controls’ fundamental PE ratio. However, this is most likely caused by the fact that earnings were practically zero in the early seventies and nineties, causing Lowe’s PE ratio to increase severely. Finally, when comparing the monthly values with the quarterly values, the values
(especially ) concerning the three cyclical stocks differ more than the values of the three defensive stocks, which could cause relatively large estimation and forecasting differences for the cyclical stocks. A logical explanation for the larger difference between quarterly and monthly values is the relatively volatile stock price movements in between months for cyclical stock.
McDonald's 0.0637 0.1223 0.1860 1.0567 17.6231
Coca-Cola 0.0536 0.0854 0.1390 1.0494 20.2602
Johnson & Johnson 0.0551 0.0973 0.1524 1.0502 19.9298
Lowe's 0.0571 0.1747 0.2318 1.0486 20.5672
GAP 0.0836 0.2086 0.2921 1.0692 14.4588
Johnson Controls 0.0927 0.1720 0.2647 1.0791 12.6446
Table 1: Quarterly data: Values used for the fundamental process. In order to calculate , earnings are
annualized by simply multiplying by 4. The growth rate is annualized by dividing earnings at time by
earnings at time and afterwards subtracting 1.
McDonald's 0.0636 0.1220 0.1856 1.0567 17.6350
Coca-Cola 0.0535 0.0855 0.1390 1.0493 20.2758
Johnson & Johnson 0.0552 0.0972 0.1524 1.0503 19.8895
Lowe's 0.0570 0.1957 0.2528 1.0477 20.9717
GAP 0.0825 0.2345 0.3169 1.0668 14.9641
Johnson Controls 0.0912 0.1905 0.2816 1.0766 13.0580
Table 2: Monthly data: Values used for the fundamental process. In order to calculate , earnings are
annualized by simply multiplying by 12. The growth rate is annualized by dividing earnings at time by
earnings at time and afterwards subtracting 1.
4
Empirical results
This chapter discusses the estimation results of the heterogeneous beliefs switching model applied to the 12 datasets described in the previous chapter. First, an explanation is given of why 3 is fixed rather than estimated and some general findings are discussed. Subsequently, the similarities and dissimilarities between the quarterly and monthly model are discussed after which I focus on the differences between the estimation of the model on cyclical and non-cyclical stock. Moreover, the time-varying fractions and AR(1) coefficients of the model are graphically shown and compared.
4.1 Estimation results: some general findings
Due to the fact that utility instead of return is used as the performance measure to determine the fractions of agents, is not comparable between the different stocks. Moreover, Bolt et al. (2014) state that the model’s forecast accuracy is rather insensitive to the exact value of . Because of these two findings, and the fact that the number of the degrees of freedom is rather high since was incorporated in the model, was set to a certain (fixed) value in the estimation procedure for each dataset4. These values were chosen through trial and error, using 1, 10 and 100 as possible -values, making sure the BIC value5
of the model was as low as possible. The optimal is found to be equal to 1 for more than half the datasets and equal to 10 for the others. However, the BIC value only increased slightly (less than one point) when changing the value of confirming what Bolt et al. (2014) found; the model’s sum of squared residuals (SSR) is not very sensitive to the precise value of .
Tables 3 and 4 show the estimation results for respectively the quarterly and the monthly data. Even though the behavioral bias is only found to be significantly different from 0 once, I chose to include in the estimation since an interesting interpretation may be made in the case of the quarterly GAP model. Besides, the estimation results (parameter values, significance and BIC value) only differ slightly when fixing . The estimation results of the model without the behavioral bias ( ) are given in Tables 7 and 8 in Appendix A.4. As mentioned above, the bias is found to be significantly different from 0 in only one case, namely in the quarterly GAP model. This finding indicates that either the fundamental value calculated based on the quarterly data for GAP is not accurate or that both types of investors believe the price of GAP stock to be higher than the fundamental. Stated differently, from the investors’ perspective the calculated fundamental is quite low resulting in a positive behavioral bias towards GAP stock. The fact that is 0.5 larger when calculated based on monthly data (see Tables 1 and 2) and the, albeit insignificant, bias is found to be much smaller in the estimation of monthly model confirms the first indication that the fundamental calculated with the quarterly data is inaccurate.
Another noticeable result is that in most of the cases and are significantly different from each other, the four exceptions being the quarterly estimation for GAP and Johnson Controls and the monthly estimation for Coca-Cola and Johnson & Johnson. A rather
4 Estimating , thus estimating five parameters, often lead to singular convergence, which means that the data do
not support the estimation of five or more parameters. 5
straightforward explanation can be given for the absence of heterogeneity in the quarterly model for Johnson Controls: the PE ratio of Johnson Controls lies very close to its fundamental value for a long period of time, indicating the absence of trend-followers. The fact that was found to be significantly smaller than 1 and was not found to be significantly different form 0 confirms the absence of trend-followers and the absence of heterogeneity. Moreover, the fact that the monthly estimation for Johnson Controls does find heterogeneity in beliefs indicates that traders do follow trends for shorter periods of time i.e. monthly. An additional note may be made concerning , namely that for five out of twelve datasets is found to be smaller than 1. This means that both types of agents have fundamental beliefs; only one type of agent has a stronger reverting belief than the other.
Parameter is found to be significantly larger than 0 (and by restriction smaller than 1) for all but one stock, i.e. Coca-Cola stock. The fact that is found to be larger than 0 in almost all cases indicates that a certain fraction of agents update their beliefs each period, therefore suggesting the presence of time-varying heterogeneity. It must be noted that for a few sets of data approximates 1, which means that barely any switching, or a very slow transition of beliefs, takes place. Finally, assuming a behavioral bias equal to 0, the implied ratio’s and are smaller than 1 for all datasets which suggests that the fundamental equilibrium is locally stable6. Note that this does not hold for the quarterly GAP model because is not the local eigenvalue since is found to be significantly larger than 0.
6The implied ratios and in the model where the bias is fixed to 0
McDonald's Coca-Cola
Johnson&
Johnson Lowe's GAP
Johnson Controls 1 1 1 10 1 1 0.576*** (0.1362) 0.713*** (0.0768) 0.637*** (0.1138) 0.283* (0.1294) 0.219 (0.4933) 0.683** (0.2513) 0.429 * (0.2078) 0.352** (0.1080) 0.296 ˚ (0.1618) 0.643** (0.2020) 0.887 (0.8030) 0.471 (0.5059) 0.176 (0.3322) 0.202 (0.4012) 0.011 (0.3222) 0.348 (0.6412) 1.773* (0.8650) 0.379 (0.3660) 0.6765*** (0.1476) 0.303 (0.2179) 0.580* (0.2311) 0.589*** (0.1491) 0.975*** (0.0169) 0.860*** (0.2524) BIC 931.221 985.140 921.466 1149.397 1037.526 933.105 impl. 1.005 1.065 0.933 0.926 1.106 1.153 1.0139 1.0121 1.0123 1.0119 1.0169 1.0192 impl. 0.780 0.879 0.776 0.605 0.652 0.901
Table 3: Estimated model parameters for the quarterly data. Significance codes: 0.00 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘˚’ 0.10.
McDonald's Coca-Cola
Johnson&
Johnson Lowe's GAP
Johnson Controls 10 1 10 10 1 10 0.837*** (0.0541) 0.968*** (0.0348) 0.901*** (0.0426) 0.697*** (0.0562) 0.733*** (0.0420) 0.417* (0.2088) 0.163* (0.0781) 0.015 (0.0584) 0.075 (0.0649) 0.304** (0.0962) 0.322*** (0.0553) 1.027* (0.4320) 0.024 (0.1079) 0.051 (0.1297) 0.015 (0.1091) 0.075 (0.2581) 0.245 (0.1875) 0.431 (0.2804) 0.815*** (0.1013) 0.544 (2.2169) 0.927*** (0.1045) 0.949*** (0.0348) 0.364** (0.1154) 0.959*** (0.0289) BIC 2238.851 2391.383 2245.580 3103.863 2464.361 3078.186 impl. 0.999 0.983 0.975 1.001 1.054 1.444 1.0046 1.0042 1.0041 1.0039 1.0054 1.0062 impl. 0.914 0.972 0.934 0.845 0.889 0.925
Table 4: Estimated model parameters for the monthly data. Significance codes: 0.00 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘˚’ 0.10.
4.1.1 Model estimates for quarterly versus monthly data
Regarding the significance of the estimated parameters, it is shown that the model holds quite well for both frequencies, especially for the monthly model where , , and are found to be significant on a level for all stocks except Coca-Cola (and for Johnson & Johnson), thus confirming time-varying heterogeneity in beliefs. One of the most noticeable results is that is, in five out of six cases, larger for the monthly model than for the quarterly model. This result suggests that agents believe the process of reverting to the fundamental takes time, i.e. more reversion may take place when the timespan is larger. Another remarkable result is the fact that is smaller for the monthly model, again in five
out of six cases. This indicates that when forecasting prices a month ahead, agents with different beliefs seem to disagree less than when forecasting prices a quarter ahead. This seems a fairly logical conclusion since the further one looks into the future the more uncertainty there is. Concerning , no consistent differences are found between the monthly and the quarterly model.
Another noteworthy result is the fact that, for five out of six stocks, is larger in the monthly model than in the quarterly model. This indicates that, similar to reverting, agents find that updating their beliefs takes time, i.e. a larger fraction of agents re-evaluate and update their beliefs in three months than in one month. Finally, the implied ratio
is smaller than for all six stocks, indicating that the fundamental equilibrium is more stable in the quarterly model than in the monthly model. 4.1.2 Model estimates for cyclical versus non-cyclical stocks
First of all, let me emphasize that significant heterogeneity and significant time-variance of beliefs was found for both types of stocks. Regarding the values of the estimated parameters, Tables 3 and 4 show that is smaller for cyclical stock than for defensive stock (except for the quarterly Johnson Controls model). This finding indicates that fundamental agents believe cyclical stocks to revert back to their fundamental more quickly than defensive stocks. Furthermore, is larger for cyclical than for defensive stocks (except for the quarterly Lowe’s model), suggesting that trend-followers believe that cyclical stocks follow stronger trends than non-cyclical stocks. These two results match the theory of cyclical stocks being more volatile and following more extreme movements than defensive stocks. Moreover, it corresponds with the fact that is larger for cyclical stock than for defensive stock indicating that agents disagree more about the future price of cyclical stock than that of non-cyclical stock. Additionally, notice that is smaller than 1 for all monthly defensive stock models. This finding suggests that, based on the monthly data, both types of agents do not believe price trends to occur for non-cyclical stock. Considering the estimation of , no apparent differences are found between cyclical and non-cyclical stock. Regarding , no conclusions may be drawn since is found to be significant only once. Note however that for all stocks and both frequencies is larger for cyclical stock than for non-cyclical stock. This suggests that the estimation of the fundamental PE ratio is less accurate when estimating on cyclical stock data.
4.2 Estimation of the time-varying fractions
Fig. 2 shows the deviation of the PE ratio from its fundamental (upper panel), the fractions of agents forming expectations based on the first type of belief ( ) (middle panel), and the AR(1) coefficients (lower panel). The quarterly and monthly models are paired per stock, the left plot based on the quarterly data, the right plot based on the monthly data.
As Fig. 2 shows, the switching dynamics of the models where approaches 1 are rather smooth, i.e. the fractions of agents with a fundamental-reverting belief slowly converge to 0.5. This dynamic is the most obvious for the quarterly GAP model. Contrarily, when is estimated to be relatively low, the dynamics of are very volatile (around 0.5) which is clearly shown by the middle panels of the quarterly Coca-Cola and Lowe’s estimations and the monthly GAP estimation. Furthermore, the graph for the monthly Coca-Cola dataset evidently shows strong fluctuations of when the PE ratio deviates strongly from its fundamental (around 2001/2002). This is most likely the result of the phenomenon that, when the deviation from the fundamental value is relatively large, the utilities of the two beliefs vary more than usual, thus strongly influencing the fraction . More specifically, the McDonald’s and Johnson & Johnson graphs show that in times of an inclining deviation from the fundamental i.e. a price bubble, declines denoting an increasing number of trend-following agents. When subsequently the bubble collapses, simultaneously increases denoting an increase in the number of fundamentalists. This finding confirms the theory proposed and corroborated by respectively Boswijk et al. (2007) and Bolt et al. (2014) that the extraordinary performance of a trend-following belief during a price bubble convinces investors to adopt this belief, even though fundamental news indicates that the stock is highly overvalued.
Taking a look at the initial values of it becomes apparent that whenever the PE ratio of a stock deviates strongly from its fundamental at the beginning of the time series, has an extremely low initial value. This phenomenon has a quite straightforward explanation: the initial value of is defined as , which is equal to the part
of equation (24) that determines how many of the agents who re-evaluate their strategy switch to the fundamental belief. Since, during a period of inclining deviations from the fundamental, a trend-following belief performs relatively well, the fraction of fundamentalists will be very small. I also estimated the model setting the initial value of to 0.5, equal to the equilibrium value. However, this didn’t improve the fit of the model nor did it drastically
change the dynamics of . Moreover, for the models with extreme deviations from the fundamental at the start of the time series, it often led to singular convergence. Finally, the plots show that has rather similar dynamics for the quarterly and monthly models, except that the monthly dynamics are often less extreme. This fits the finding previously discussed, that reverting and switching costs time.
Figure 2 (part A): PE ratio deviations from the fundamental (upper panels), fractions of agents with fundamental beliefs of type 1 (middle panels) and the implied AR(1) coefficients (bottom panels).
− 1 0 0 1 0 2 0 3 0 4 0 5 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .6 5 0 .7 5 0 .8 5 Time A R (1 ) MCDONALDS Quarterly − 1 0 0 1 0 2 0 3 0 4 0 5 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .8 8 0 .9 2 0 .9 6 Time A R (1 ) MCDONALDS Monthly 0 2 0 4 0 6 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .7 0 0 .8 0 0 .9 0 1 .0 0 Time A R (1 ) COCA COLA Quarterly 0 2 0 4 0 6 0 8 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .9 6 6 0 .9 7 0 0 .9 7 4 0 .9 7 8 Time A R (1 ) COCA COLA Monthly − 1 0 0 1 0 2 0 3 0 4 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .6 5 0 .7 0 0 .7 5 0 .8 0 0 .8 5 0 .9 0 Time A R (1 )
JOHNSON & JOHNSON Quarterly − 1 0 0 1 0 2 0 3 0 4 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .9 2 0 .9 3 0 .9 4 0 .9 5 0 .9 6 0 .9 7 Time A R (1 )
JOHNSON & JOHNSON Monthly 0 2 0 4 0 6 0 8 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 Time A R (1 ) LOWE'S Quarterly 0 2 0 4 0 6 0 8 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .8 0 0 .8 5 0 .9 0 0 .9 5 Time A R (1 ) LOWE'S Monthly
Figure 2 (part B): PE ratio deviations from the fundamental (upper panels), fractions of agents with fundamental beliefs of type 1 (middle panels) and the implied AR(1) coefficients (bottom panels).
5 In-sample forecasting
This chapter compares a relatively simple autoregressive (AR) model to the modified version of the BHM model introduced in this thesis. First, an AR model applied to each dataset is estimated and the fit of the estimated autoregressive models are compared to the fit of the heterogeneous agents model by means of the residual standard error. Successively, in-sample forecasts of the BHM model and AR(5) models are compared by investigating the dynamics of the density forecasts and their performance based on the squared residuals.
5.1 The BHM model versus an autoregressive model
Table 5 shows the residual standard errors of the BHM model and the AR( ) model. The Akaike Information Criterion (AIC) is used to determine the order of the autoregressive models. Moreover, the Yule-Walker method is used to solve for the parameters and a restriction on the order of the model is set to a maximum order of 5. For both the quarterly and the monthly datasets the heterogeneous agents model outperforms the AR model for all but one stock i.e. the residual standard errors of the BHM models are considerably smaller than those of the AR models for all stocks except Johnson Controls. Another noteworthy outcome is that the standard errors of the residuals are bigger for the cyclical stocks than for the non-cyclical stocks, again the exception being Johnson Controls. This confirms the
− 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .7 5 0 .8 5 0 .9 5 1 .0 5 Time A R (1 ) GAP Monthly − 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .6 0 .7 0 .8 0 .9 1 .0 Time A R (1 ) GAP Quarterly − 1 0 0 1 0 2 0 3 0 4 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .7 5 0 .8 0 0 .8 5 0 .9 0 0 .9 5 1 .0 0 Time A R (1 ) JOHNSON CONTROLS Quarterly 0 2 0 4 0 6 0 8 0 X 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 n1 1980 1990 2000 2010 0 .8 0 .9 1 .0 1 .1 1 .2 1 .3 Time A R (1 ) JOHNSON CONTROLS Monthly
statement that cyclical stock prices show more excess volatility than defensive stock prices. Keep in mind that Johnson Controls has had rather stable movements around the fundamental for a long period of time, which, even though Johnson Controls is listed as a cyclical stock, is a characteristic normally observed for defensive stock. Finally, it is shown that the residual standard errors are smaller for the monthly model than for the quarterly model, indicating that both models (BHM and AR) capture more variation when looking at higher-frequency data.
Fig. 3 shows the plots of the residuals for the quarterly model7; the left-hand side plots show the residuals of the BHM model and the right-hand side plots show the residuals of the fitted AR( ) model. It is clearly observable that the dynamics of the BHM residuals are not very different from those of the AR( ) residuals. Therefore, the rather large differences in the residual standard errors of the two models (shown in Table 5) are most likely caused by a few outliers e.g. see the first and last few data points in the McDonald’s residual plot. Moreover, the plots show that the residuals are largest, for both models, whenever the PE-ratio deviates strongly from its fundamental e.g. around 2000 for Coca-Cola stock and around 2010 for Johnson Controls stock.
Quarterly Monthly
Residual standard error Order Residual standard error Order
BHM AR( ) BHM AR( )
McDonald's 4.083 5.610 4 2.323 3.327 2
Coca-Cola 4.827 5.306 4 2.715 2.915 4
Johnson & Johnson 3.961 5.090 1 2.339 3.030 1
Lowe's 5.971 8.596 2 5.625 6.470 4
GAP 5.344 7.405 5 3.676 3.886 2
Johnson Controls 4.107 4.042 2 5.479 5.141 5
Table 5: Residual standard errors of the fitted AR models of order .
7
Figure 3 (part A): Plots of the residuals. The left-hand side plots show the residuals of the BHM model and the right-hand side plots show the residuals of the fitted AR( ) models.
year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 MCDONALDS − BHM year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 MCDONALDS − AR( 4 ) year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 3 0 COCA COLA − BHM year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 3 0
COCA COLA − AR( 4 )
year re s id u a ls 1980 1990 2000 2010 − 2 0 − 1 0 0 1 0 2 0
JOHNSON & JOHNSON − BHM
year re s id u a ls 1980 1990 2000 2010 − 2 0 − 1 0 0 1 0 2 0
JOHNSON & JOHNSON − AR( 1 )
year re s id u a ls 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 LOWE'S − BHM year re s id u a ls 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 LOWE'S − AR( 2 )
Figure 3 (part B): Plots of the residuals. The left-hand side plots show the residuals of the BHM model and the right-hand side plots show the residuals of the fitted AR( ) models.
5.2 The dynamics of in-sample forecasting
Figs. 4-9 show the 5, 15, 50, 85, and 95% quantiles of the density forecasts. The 5 and 95 quantiles are represented by red dashed lines, the 15 and 85 quantiles are shown as green dashed lines and the median forecast is represented by a blue dashed line. Finally, the deviation from the fundamental is shown as a black solid line. Since the heterogeneous agents model contains five parameters (i.e. , and ), an AR order of 5 was used to guarantee that both models have the same amount of parameters. In-sample forecasting was done for both the quarterly and the monthly model, for all six stocks. Forecasts were made for the last 7.5 years of the sample, hence forecasting 30 periods for the quarterly model and 90 periods for the monthly model. The model parameters were estimated based on the whole data sample. Table 6 shows the mean squared residuals of the in-sample median forecasts for the quarterly and monthly model, comparing the BHM model forecasts with the AR(5) forecasts.
year re s id u a ls 1980 1990 2000 2010 − 3 0 − 1 0 1 0 3 0 GAP − BHM year re s id u a ls 1980 1990 2000 2010 − 3 0 − 1 0 1 0 3 0 GAP − AR( 5 ) year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 3 0 JOHNSON CONTROLS − BHM year re s id u a ls 1980 1990 2000 2010 − 1 0 0 1 0 2 0 3 0
Figure 4: In-sample forecasts for McDonald’s stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). De left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
Figure 5: In-sample forecasts for Coca-Cola stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). The left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
MCDONALDS − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
MCDONALDS − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
MCDONALDS − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
MCDONALDS − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
COCA COLA − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 4 0 0 2 0 6 0
COCA COLA − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 4 0 0 2 0 6 0
COCA COLA − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 4 0 0 2 0 6 0
COCA COLA − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 4 0 0 2 0 6 0
Figure 6: In-sample forecasts for Johnson & Johnson stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). The left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
Figure 7: In-sample forecasts for Lowe’s stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). The left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
JOHNSON & JOHNSON − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
JOHNSON & JOHNSON − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
JOHNSON & JOHNSON − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
JOHNSON & JOHNSON − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 1 0 3 0 5 0
LOWE'S − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
LOWE'S − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
LOWE'S − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
LOWE'S − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
Figure 8: In-sample forecasts for GAP stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). The left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
Figure 9: In-sample forecasts for Johnson Controls stock: estimates of the 5, 15, 50, 85 and 95% quantiles of the density forecasts (dashed lines). The left-hand side panels show the forecasts for the BHM model based on the estimated parameters in Tables 3 and 4, the right-hand side panels show the forecasts based on an AR(5) model.
GAP − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0
GAP − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0
GAP − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0
GAP − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0
JOHNSON CONTROLS − BHM forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
JOHNSON CONTROLS − AR forecasts − Quarterly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
JOHNSON CONTROLS − BHM forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
JOHNSON CONTROLS − AR forecasts − Monthly
year X 1980 1990 2000 2010 − 2 0 0 2 0 4 0 6 0 8 0
Quarterly Monthly
BHM AR(5) BHM AR(5)
McDonald's 52.778 48.432 47.830 48.130
Coca-Cola 13.932 27.801 17.863 30.950
Johnson & Johnson 19.252 34.411 21.232 29.509
Lowe's 28.712 46.513 30.979 42.762
GAP 53.169 33.468 19.273 33.868
Johnson Controls 173.233 207.380 5.457E+28 531.086
Table 6: The mean squared residuals of the in-sample forecast for the last 7.5 years.
Based on Figs. 4-9 and Table 6 a few conclusions can be drawn. First of all, it must be noted that in the case of the heterogeneous model, a one-step-ahead model is used to construct forecasts over multiple step forecast horizons. This may lead to sub-optimal forecasts, especially when the forecast horizon in quite large. Nevertheless, for most of the stocks, the mean squared residuals of the BHM in-sample forecasts are smaller than those of the AR(5) model. Secondly, for the BHM model, three out of twelve density forecasts show explosive behavior. This suggests that that the system is globally unstable. However, this conclusion should be interpreted with caution since, as stated before, a one-step-ahead model is used which can cause sub-optimality. Finally, the width of the predictive intervals for the BHM model is, for the non-explosive forecasts, smaller than the width of the AR(5) predictive intervals. This is a consequence of the fact that the width of the predictive density is partially dependent on the standard deviation of the residuals. Table 5 shows that the residual standard deviations of the best fitting AR( ) models are larger than those of the BHM models (except for Johnson Controls stock) thus positively influencing the width of the predictive interval.
It can be observed that the density forecasts of the BHM model and the AR(5) model vary significantly for a few stocks whereas for other stocks the forecasts differ marginally. For example, the forecasts for McDonald’s are quite similar; the widths of both predictive intervals increase initially but after a while (around 2010) the widths remain fairly stable. Moreover, with respect to the median, both models predict the deviation from the fundamental to remain close to 0. There are two reasonable explanations for this occurrence. First of all, the PE ratio at the start of the in-sample forecast is close to the fundamental ratio, which causes both models to produce similar one-step ahead forecasts. Secondly, the extrapolation parameter of the type 2 investor ( ) is very weak for the quarterly model and smaller than 1 for the monthly model, so even if the fraction of type 2 agents were to be rather
large, no (strong) trend would be predicted. Finally, the mean of the squared residuals confirm that the two medians lie very close to each other.
Three obvious examples of heterogeneous forecasts that differ notably from the autoregressive forecasts are the monthly predictions for GAP and both predictions for Johnson Controls. Regarding the BHM forecast for GAP, the 85 and 95% quantiles show (upwards) explosive behavior. An explanation is given by the fact that the model parameter is found to be significantly larger than 0 thus forcing the top quantiles upwards, unable to find equilibrium. Regarding the density forecasts for Johnson Controls stock, two coherent clarifications can be given explaining the explosive behavior. First of all, both the quarterly and the monthly models have a rather large trend-following parameter , which according to Bolt et al. (2014) causes bubbles and crashes in a system. Secondly, the switching parameter is quite high for both models, indicating that only a small fraction ( ) of agents updates their beliefs each period. This means that if, at a certain point, the fraction of trend-following agents is quite large, it will take a long time for that fraction to decrease. Note that, even though the forecasting behavior is explosive, the BHM forecasts do capture the price bubble around 2010, whereas the AR(5) model does not.
6 Conclusion and Discussion
I have proposed a slightly modified version of the Boswijk et al. (2007) heterogeneous agents model in which agents switch between a fundamentalists’ and a trend-following forecasting rule based upon their relative performance. The model is estimated for three cyclical stocks, Lowe’s, Gap and Johnson Controls and three defensive stocks, McDonald’s, Coca-Cola and Johnson & Johnson. Both quarterly and monthly stock market data is used from 1973-2013, resulting in a total of twelve datasets.
It is found that the majority of the datasets support heterogeneity in expectations. However, for four out of six non-cyclical datasets both type of agents have fundamental expectations i.e. neither believe defensive stock to follow trends. Moreover, investors believe cyclical stock prices to revert more quickly to their fundamental than defensive stock prices and they disagree more about the future prices of cyclical stocks than those of defensive stocks, which corresponds with the relatively volatile character of cyclical stocks. Another relevant result is the fact that agents believe reverting costs time i.e. the fundamentalists’ extrapolation parameter is smaller in the quarterly models than in the monthly models.
However, they do not believe that trend-following costs time; some of the trend-following extrapolation parameters are larger for the monthly model than for the quarterly model, others smaller. Finally, agents disagree more about future quarterly stock prices than future monthly stock prices, which reflects the uncertainty agents have when predicting relatively far into the future.
For all stocks, except Coca-Cola, significant switching is found between the two forecasting strategies and since switching takes time, a larger fraction of agents switch in the quarterly models than in the monthly models. Furthermore, price bubbles are amplified by trend-following beliefs and when bubbles burst the fraction of fundamentalists increases reinforcing the rapid fall of stock prices. For all but one dataset, the behavioral bias is not found to be significantly different from 0, indicating that agents believe the calculated fundamental value to be correct. In-sample forecasting is performed and compared to the density forecasts of a linear AR benchmark model. The qualitative predictions show large differences whenever agents have a significant bias or whenever agents believe stock prices to follow strong trends. In these two cases, the density quantiles of the heterogeneous model show explosive behavior whereas the density quantiles of the AR model do not. However, by measures of the mean squared residuals, the BHM model outperforms the AR model in all other cases.
An essential topic for future research is to investigate whether the results of this heterogeneous agents model, i.e. behavioral heterogeneity and time-varying beliefs about cyclical and non-cyclical stock, are robust. This may be investigated by using indices for the cyclical and non-cyclical sector instead of individual stocks. Moreover, a rather simple and static model is used to determine the fundamental PE value, which induces a constant fundamental over time. Allowing for time variation in the fundamental value may improve the results, especially for stocks as GAP where a significant bias was found. Finally, both forecasting rules included only one lag thus incorporating more memory into the agents’ models, especially for high frequency stock market data, seems like an interesting research topic.
