Conversely, growing evidence of the presence of non-linear dynamics in financial time- series, together with the failure of the linear present value model to explain stock prices dynamics suggests a non-linear approach to the price-dividend relationship. Furthermore, an increasing discrepancy between stock and fundamental prices evident in the late 1990s casts a final doubt that the stock prices follow linear stationary perfectly cointegrated behaviour implied by the present value model. Campbell and
Shiller (2001) report an unusually bearish behaviour within the US stock market in 1998 which resulted in a shift in stock prices from the fundamental values and historical averages. However, Lewellen (2004) reports finding strong evidence of stock returns predictability even during the period of the unusual price dynamics. Bohl and Siklos (2004) propose a more plausible approach by taking an assumption that the present value model is valid as a long-run framework for the US stock prices, and recognising the presence of asymmetries in the short-run. As pointed by Bohl and Siklos (2004), there are a number of possible reasons for mixed empirical evidence of the long-term validity of the present value model including the presence of non-linearities, structural breaks and outliers. It is indeed possible to integrate crashes and non-fundamental stock price behaviour that occurred during the 1990s by not including the transversality condition of the standard present value model. Numerous studies were carried out in order to explain stock price behaviour as a function of dividends. An increasing number of researchers conclude that the prices and dividends are in fact cointegrated, however the mean reversion processes is characterised in a non-linear fashion. McMillan (2007), for instance, observed that some researchers argued that the deviation from the fundamentals in the 1990s was a result of an extended bubble that eventually burst, and were concentrating on determining a technique which would allow to capture this type of stock price behaviour. Non-linearities in the present value model are usually explained by the presence of non-fundamental components. In addition, Psaradakis et al. (2004) identified the presence of a time-varying discount factor and the presence of bubbles as possible explanations for short-term deviations of prices from the fundamental values and long-term price-dividend relationship. The most promising theoretical justifications of such dynamics include the presence of speculative bubbles
(Blanchard and Watson, 1982; West, 1988; Evans, 1991); noise traders’ models (Kirman, 1991, 1993; Shleifer, 2000) and the theory on booms and slumps in economic activity (Phelps, 1994; Phelps and Zoega, 2001).
Besides the debate on the presence of stock returns predictability, there is an ongoing discussion concerning the predictability in short- and long-horizons. Rapach et al. (2005) investigated the presence of long-horizon predictability in real stock prices using a predictive regression model with price-dividend and price-earnings ratios as fundamental valuation ratios following previous research that seemed to detect predictability in the long-horizons, but not in the short-horizons (e.g. Campbell and Shiller, 1998). Possible explanation of such pattern of stock price predictability could be attributed to presence of non-linearity. The argument was based on the work by Berkowitz and Giorgianni (1996) who addressed long-horizon predictability of nominal exchange rates using monetary fundamentals as valuation ratios. The researchers argued that a linear framework does not provide sufficient justification for the stock predictability as it implies that the stock predictability is for all horizons or for no horizons. This contradicts with numerous findings of long-horizon predictability and the lack of such in the short-horizons. Using their approach Rapach et al. (2005) adopted the methodology in order to implement the Monte Carlo simulations to the long-horizon stock price predictability. While results from a linear predictive regression demonstrated the ability of both price-dividend and price-earnings ratios to predict stock price in the long- but not short-horizons, the parsimonious exponential smooth transition autoregressive (ESTAR) model proved not only to fit the data sufficiently well, but also to allow for non-linear mean reversion, thus providing plausible explanation for the long-horizon predictability pattern. As a result, Rapach et al. (2005) agree that a non-
linear framework provides a sufficient explanation for the pattern of stock price predictability for at least the dividend-price ratio. Moreover, Kilian (1999) argues that the observed pattern of long-horizon predictability together with the absence of predictability in the short-horizon can be interpreted as indirect evidence of the presence of non-linearity in the data generating process.
Kanas (2005) employed a non-linear cointegration approach to confirm the presence of non-linearities in the stock price and dividend relationship and thus validated the present value model in non-linear fashion. Bali et al. (2008) also found evidence of stock returns predictability by employing a non-linear test of mean reversion. Hartmann et al. (2008) find evidence of predictability of stock returns using macroeconomic variables and incorporating structural breaks by assessing publicly available and easily accessible information on economic and financial crises.