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We estimate a dynamic asset pricing model characterized by heterogeneous boundedly rational agents. The fundamental value of the risky asset is publicly available to all agents, but they have different beliefs about the persistence of deviations of stock prices from the fundamental benchmark. An evolutionary selection mechanism based on relative past profits governs the dynamics of the fractions and switching of agents between different beliefs or forecasting strategies. A strategy attracts more agents if it performed relatively well in the recent past compared to other strategies. We estimate the model to annual US stock price data from 1871 until 2003. The estimation results support the existence of two expectation regimes, and a bootstrap F-test rejects linearity in favor of our nonlinear two-type heterogeneous agent model. One regime can be characterized as a fundamentalists regime, because agents believe in mean-reversion of stock prices toward the benchmark fundamental value. The second regime can be characterized as a chartist, trend following regime because agents expect the deviations from the fundamental to trend. The fractions of agents using the fundamentalists and trend following forecasting rules show substantial time variation and switching between predictors. The model offers an explanation for the recent stock prices run-up. Before the 1990s the trend following regime was active only occasionally. However, in the late 1990s the trend following regime persisted and reenforced an extraordinary deviation of stock prices from the fundamentals. Recently, the activation of the mean-reversion regime has contributed to drive stock prices back closer to their fundamental valuation.

Historical evidence indicates large fluctuations of stock prices compared to indicators of fundamental value. For example, the price to earnings ratio of the S&P500 was around 5 at the beginning of the 1920s, but more than 25 about nine years later to fall back to about 5 again by 1933. In 1995 the price/earnings ratio of the S&P500 was close to 20, went up to more than 40 at the beginning of 2000 and then quickly declined again to about 20 by the end of 2003. Why do prices fluctuate so much compared to economic fundamentals? This question has been strongly debated in financial economics. At the beginning of the 1980s, Shiller (1981) and LeRoy and Porter (1981) claimed that the stock market exhibits excess volatility, that is, stock price fluctuations are significantly larger than movements in underlying economic fundamentals. The debate evolved in two directions. On the one hand, supporters of rational expectations and market efficiency proposed modifications and extensions of the standard theory. In contrast, another part of the literature focused on providing further empirical evidence against the efficiency of stock prices and behavioral models to explain these phenomena. The debate has recently been revived by the extraordinary surge of stock prices in the late 1990s. The internet sector was the main driving force behind the unprecedented increase in market valuations. Ofek and Richardson, 2002 and Ofek and Richardson, 2003 estimated that in 1999 the average price-earnings ratio for internet stocks was more than 600. A recent overview of rational explanations based on economic fundamentals for the increase in stock prices in the late 1990s is e.g. given by Heaton and Lucas (1999). They offer three reasons for the decrease of the equity premium, i.e. the difference between expected returns on the market portfolio of risky stocks and riskless bonds. A first reason is the observed increase of households’ participation in the stock market. This implies spreading of equity risk among a larger population, which could explain a decrease of the risk premium required by investors. Secondly, there is evidence that investors hold more diversified portfolios compared to the past. In the 1970s a large majority of investors concentrated their equity holdings on one or two stocks. More recently households have invested a large proportion of their wealth in mutual funds achieving a much better diversification of risk. Both facts justify a decrease of the required risk premium by investors. Although the wider participation seems unlikely to play an important role in the surge of stock prices in the 1990s, the increased portfolio diversification could at least partly account for the decrease in the equity premium and the unprecedented increase in market valuations. A third, fundamental explanation for the surge of the stock market that has been proposed is a shift in corporate practice from paying dividends to repurchasing shares as an alternative measure to distribute cash to shareholders. In this case dividends do not measure appropriately the profitability of the asset and such a shift in corporate practice explains, at least partly, an increase in price-earnings or price-dividend ratios or equivalently a decrease of the risk premium. Further evidence on this issue is provided by Fama and French (2001). Some recent papers attempt a quantitative evaluation of the decrease in the equity premium. Fama and French (2002) argue that, based on average dividend growth, the real risk premium has significantly decreased from 4.17% in the period 1872–1950 to 2.5% after 1950. Jagannathan et al. (2000) go even further and, comparing the equity yield to a long-term bond yield, reach the conclusion that the risk premium from 1970 onwards is approximately 0.7%. That is, investors require almost the same return to invest in stocks and in 20 years government bonds. The explanations above indicate structural, fundamental reasons for a long-horizon tendency of the risk premium to decrease, or equivalently for an increase of the valuation of the aggregate stock market. However, to quantify the decrease in the equity premium is difficult and the estimates provided earlier are questionable. Although fundamental reasons may partly explain an increase of stock prices, the dramatic movements e.g. in the 1990s are hard to interpret as an adjustment of stock prices toward a new fundamental value. Another strand of recent literature has provided empirical evidence on market inefficiencies and proposed a behavioral explanation. Hirshleifer (2001) and Barberis and Thaler (2003) contain extensive surveys of behavioral finance and empirical results both for the cross-section of returns and for the aggregate stock market. Much attention has been paid to the continuation of short-term returns and their reversal in the long-run. This was documented both for the cross-section of returns by de Bondt and Thaler (1985), and Jegadeesh and Titman (1993) and for the aggregate market by Cutler et al. (1991). At short run horizons of 6–12 months, past winners outperform past losers, whereas at longer horizons of e.g. 3–5 years, past losers outperform past winners. A behavioral explanation of this phenomenon is that at horizons from 3 months to a year, investors underreact to news about fundamentals of a company or the economy. They slowly adjust their valuations to incorporate the news and create positive serial correlation in returns. However, in the adjustment process they drive prices too far from what is warranted by the fundamental news. This shows up in returns as negative correlation at longer horizons. Several behavioral models have been developed to explain the empirical evidence. Barberis et al. (1998), henceforth BSV, assume that agents are affected by psychological biases in forming expectations about future cash flows. BSV consider a model with a representative risk-neutral investor in which the true earnings process is a random walk, but investors believe that earnings are generated by one of two regimes, a mean-reverting regime and a trend regime. When confronted with positive fundamental news investors are too conservative in extrapolating the appropriate implication for the immediate asset valuation. However, they overreact to a stream of positive fundamental news because they interpret it as representative of a new regime of higher growth. The model is able to replicate the empirical observation of continuation and reversal of stock returns. Another behavioral model that aims at explaining the same stylized facts is Daniel et al. (1998), henceforth DHS. Their model stresses the importance of biases in the interpretation of private information. DHS assume that investors are overconfident and overestimate the precision of the private signal they receive about the asset payoff. The overconfidence increases if the private signal is confirmed by public information, but decreases slowly if the private signal contrasts with public information. The model of BSV assumes that all information is public and that investors misinterpret fundamental news. In contrast, DHS emphasize overconfidence concerning private information compared to what is warranted by the public signal. These models aim to explain the continuation and reversal in the cross-section of returns. However, as suggested by Barberis and Thaler (2003), both models are also suitable to explain the aggregate market dynamics. In this paper we consider an asset pricing model with behavioral heterogeneity and estimate the model using yearly S&P 500 data from 1871 to 2003. The model is a reformulation, in terms of price-to-cash flow ratios, of the asset pricing model with heterogeneous beliefs introduced by Brock and Hommes, 1997 and Brock and Hommes, 1998. Agents are boundedly rational and have heterogeneous beliefs about future payoffs of a risky asset. Beliefs about future cash flows are homogeneous and correct, but agents disagree on the speed the asset price will mean-revert back towards its fundamental value. A key feature of the model is the endogenous, evolutionary selection of beliefs or expectation rules based upon their relative past performance, as proposed by Brock and Hommes (1997). The estimation of our model on yearly S&P 500 data suggests that behavioral heterogeneity is significant and that there are two different regimes, a ‘mean-reversion’ regime and a ‘trend following’ regime. To each regime, there corresponds a different (class of) investor types: fundamentalists and trend followers. These two investor types coexist and their fractions show considerable fluctuations over time. The mean-reversion regime corresponds to the situation when the market is dominated by fundamentalists, who recognize a mispricing of the asset and expect the stock price to move back towards its fundamental value. The other trend following regime represents a situation when the market is dominated by trend followers, expecting continuation of say good news in the (near) future and expect positive stock returns. Before the 1990s, the trend regime is activated only occasionally and never persisted for more than two consecutive years. However, in the late 1990s the fraction of investors believing in a trend increased close to one and persisted for a number of years. The prediction of an explosive growth of the stock market by trend followers was confirmed by annual returns of more than 20% for four consecutive years. These high realized yearly returns convinced many investors to also adopt the trend following belief thus reenforcing an unprecedented deviation of stock prices from their fundamental value. The outline of the paper is as follows. Section 2 discusses some closely related literature. Section 3 describes the asset pricing model with heterogeneous beliefs and endogenous switching, while Section 4 presents the estimation results. Section 5 discusses empirical implications of our model, in particular the impulse response to a permanent positive shock to the fundamental and a simulation based prediction of how likely or unlikely high valuation ratios are in the future. Finally, Section 6 concludes and an Appendix contains the details of a bootstrap F-test for linearity and discusses the robustness of our estimation with respect to time variation in the fundamental value.

We have proposed a behavioral asset pricing model with endogenous evolutionary switching of investors between different forecasting strategies according to their relative past performances and estimate the model on yearly S&P500 data from 1871–2003. Our estimation results show statistically significant behavioral heterogeneity and substantial time variation in the average sentiment of investors. Investors believe that fundamentals are driving the long term dynamics of stock prices, but they interpret the persistence of the deviation of stock prices from their benchmark fundamentals in a different way. If a recent increase in stock prices is observed, agents tend to extrapolate that the mispricing will increase even further and allocate more capital to the trend following belief. However, in periods of gradual price changes they believe that the deviation is transitory and will revert back to its historical mean. This type of time variability of agents average sentiment is also supported by the survey evidence in Shiller (2000). In particular, our model suggests an evolutionary explanation of the ‘irrational exuberance’ of stock prices in the late nineties. Starting in 1996 the behavior of stock prices was at odds with the evidence that when deviations are large they tend to revert back to their long run mean. From 1996 until 1999 the PD-ratio indicated that the stock market was overvalued and it was likely to correct back to the fundamentals. The PE-ratio gave the same indication, although less clearly and somewhat later in time. Despite the common feeling among investors that stocks were overvalued, the market continued to grow by approximately 30% a year. The estimation of our model shows that a large majority of investors had explosive, trend following beliefs about the persistence of the deviations from the fundamentals. Apparently, investors neglected the role of fundamental news and continued to buy stocks for purely speculative reasons. The extraordinary performance of the trend following strategy convinced most investors to adopt this type of beliefs. The outcome of our model is consistent with the view that fundamentalists with mean-reverting expectations had limited capital to arbitrage the mispricing away and force stock prices back to the fundamental values. Our behavioral model suggests that in the mid-nineties optimistic, boundedly rational investors, motivated by short run profitability, reinforced the rise in stock prices triggered by higher expected cash flows of the internet sector. An important topic for future research is to investigate the robustness of behavioral heterogeneity in financial market data. In particular, we have chosen a very simple fundamental process, the static Gordon growth model with constant growth rate of dividends or earnings and constant discount rate, allowing only for one jump in the estimated risk premium based on dividends in 1950, as in Fama and French (2002). For deviations of this simple benchmark our estimation results show significant behavioral heterogeneity of fundamentalists and trend following trading strategies, both for fundamental valuation based on dividends and earnings. As a first step, we show in the Appendix that our estimation results are fairly robust, by considering deviations from a benchmark fundamental with time variation in the cash flow growth rate. With more time variation in the benchmark fundamental, the estimation results are similar and behavioral heterogeneity is significant. An important topic for future work is to investigate whether similar results can be found at higher frequencies, e.g. for quarterly, monthly weekly or daily stock market data. Let us finally discuss some other recent related work linking nonlinear econometric models to a speculative model of asset prices.3van Norden and Schaller (1999) study a time series switching model with two regimes, an explosive bubble regime and a collapsing bubble regime. The probability of being in the explosive regime depends negatively on the relative absolute deviation of the bubble from the fundamental. Stated differently, the larger the absolute relative deviation from the fundamental, the larger the probability that the (positive or negative) bubble collapses. Brooks and Katsaris (2005) extend this model to three regimes, adding a third dormant bubble regime where the bubble grows at the required rate of return without explosive expectations. Another novel feature of their extension is the observation that an abnormal trading volume is a signal of changing market expectations about the future of a speculative bubble. In their model the probability that the bubble collapses increases when trading volume becomes abnormally high. This could be related to the empirical evidence provided by Kandel and Pearson (1995) who find a significant relation between heterogeneity of analysts expectations and abnormal trading volume. These speculative models are somewhat different in spirit and do not start off, at least not explicitly, from micro foundations with heterogeneous agents and evolutionary selection of behavioral rules according to past recent performance. But these nonlinear switching models can be reconciled with our heterogeneous agent model and the framework of Brock and Hommes, 1997 and Brock and Hommes, 1998, by modifying the evolutionary fitness measure and include absolute relative deviations from the fundamental price and trading volume in the fitness measure for strategy selection. A large absolute deviation from the fundamental would then act as a far from equilibrium stabilizing force, while abnormally high trading volume would act as a trigger event for bubble collapse. A distinctive feature of our approach is that investors’ behavior is driven by short run profits. This may be particularly important for getting bubbles started. After a few positive random shocks to fundamentals, motivated by short run profits trend followers may reinforce the price rise and contribute to the start of a bubble. Another important difference concerns the probability of extreme, long lasting bubbles as in the late nineties. When investors’ expectations are mainly based upon relative deviations from fundamentals a long lasting bubble becomes more and more likely to burst. In contrast, as long as investment strategies are to a large extent driven by short run profit opportunities such bubbles may continue for a long time and cause extreme deviations from fundamentals. Further empirical work on estimating various heterogeneous agent models and determining the main driving forces of large deviations from fundamentals is an important topic for future work.