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Article Excerpt
1. Introduction
This paper studies the theoretical and empirical implications of forecast accuracy uncertainty on stock returns. Our representative investor receives a disperse range of forecasts regarding a firm's future cash flow growth but is uncertain about the accuracy of the information sources issuing these forecasts. The investor optimally combines the forecasts into an aggregate cash flow estimate. To minimize the mean-squared forecast error of this aggregate estimate, the investor assigns more weight to forecasts issued by more accurate information sources. The corresponding aggregate cash flow estimate represents the investor's expectation of future cash flow growth and determines the firm's stock price.
The investor estimates the accuracy of each information source from their past forecast errors.1 As additional cash flow realizations and forecast errors become available, the investor learns about their respective accuracy. Intuitively, an information source's true accuracy represents its unobservable "skill" at forecasting a firm's cash flow. Investors understand the uncertainty inherent in measuring this skill and gradually update their assessment of each information source's accuracy. (2)
Our model features a risk-neutral representative investor, constant fundamental risk, and a constant discount rate. Expected stock returns are driven entirely by innovations in the investor's aggregate cash flow estimate. These innovations are determined by changes in the information weights and the dynamics of individual forecasts. We focus on the role of time-varying information weights, which has not been previously studied, by assuming that the individual cash flow growth forecasts are, on average, constant over short horizons.
Although the investor in our model immediately incorporates newly issued or revised forecasts into her conditional cash flow expectation, the weights assigned to these forecasts are gradually updated. The gradual updating of the information weights generates return predictability. In particular, earnings momentum and price momentum arise from learning about the relative forecast accuracy of the information sources. For example, after a series of unexpected positive cash flow innovations, the estimated accuracy of relatively optimistic information sources tends to improve. Thus, their information weights increase at the expense of pessimistic information sources. As a consequence, the optimistic information sources exert a greater influence on the aggregate cash flow estimate. This shift in the information weights leads to higher expected cash flow growth and a higher stock price, although the individual forecasts remain unchanged (on average).
Our framework offers several empirical predictions. Momentum is expected to be stronger for stocks with greater fluctuations in their information weights. This unique prediction is verified for both price momentum and earnings momentum using analyst earnings forecasts. In addition, we confirm our model's prediction that momentum is stronger for stocks with greater cash flow uncertainty using analyst forecast dispersion as a proxy. A simulation study also confirms that under reasonable parameters, forecast accuracy uncertainty produces momentum profits whose magnitudes are comparable with existing empirical studies.
Several other predictions from our model are consistent with the empirical evidence in Jiang et al. (2005), Daniel and Titman (2006), Jackson and Johnson (2006), and Zhang (2006), although we provide a new interpretation of their findings. For example, we predict stronger momentum in stocks with fewer available forecast errors; including small firms, young firms, and those undergoing significant changes in their cash flow growth. Stronger momentum for stocks with higher return volatility and higher cash flow volatility are also predicted.
This paper provides a middle ground between behavioral and rational perspectives on momentum. Momentum in our model does not originate from time-varying risk or a time-varying risk premium. Instead, our framework is based on a statistical optimization that combines multiple forecasts of uncertain accuracy into an aggregate cash flow estimate and a learning process that induces slow updating in the weights underlying this aggregate cash flow estimate. The gradual updating of these weights is distinct from the slow diffusion of information in Hong and Stein (1999) and Hong et al. (2007b). Unlike agents in rational expectation models, our investor is not concerned with the impact of her learning on prices. Although our investor is not assumed to be influenced by behavioral biases, [section]2.3 demonstrates that certain characteristics of the information weights mimic behavioral biases that have been invoked to explain momentum, including representativeness and conservatism (Barberis et al. 1998) as well as overconfidence (Daniel et al. 1998). We also demonstrate that information sources whose forecasts are positively correlated with more accurate forecasts are marginalized, which leads to the appearance of limited attention (e.g., Hirshleifer and Teoh 2003, Peng and Xiong 2006).
Our paper is related to recent studies on the ability of parameter uncertainty to generate return predictability (e.g., Timmermann 1993, Lewellen and Shanken 2002). In these studies, investors learn about an unknown parameter regarding a firm's cash flow dynamics. In contrast, our investor does not model cash flow dynamics and does not learn about a firm's cash flow growth parameters from realized cash flows. Instead, our investor's reliance on multiple cash flow forecasts with time-varying weights is crucial.
Our framework also differs from Hong et al. (2007a). Their representative investor uses simple univariate models to forecast cash flow when the true cash flow generating process is multivariate. This investor is limited to a subset of available information and permanently alternates between two incorrect forecast procedures. In contrast, our investor conditions on all available forecasts when forming her cash flow expectation. We are the first to examine the uncertainty surrounding the relative accuracy of different cash flow forecasts.
The remainder of this paper is organized as follows. Section 2 introduces the optimal information weights, the learning mechanism regarding forecast accuracy, and the pricing implications of forecast accuracy uncertainty. Section 3 evaluates the implications of changes in the information weights on stock returns, earnings momentum, and price momentum. Section 4 summarizes and concludes the paper.
2. The Model
Following Barberis et al. (1998), our economy consists of a single risky security (stock) and a risk-neutral representative investor with an exogenous constant discount rate 8. All cash flows N, are paid out as dividends. Under the objective probability, cash flow growth [y.sub.[t + 1]] [equivalent to][.sub.[t + 1]] - [N.sub.t] - [N.sub.t] is assumed to be independent over time, with an unknown and time-varying mean [[theta].sub.[t + 1]]
[y.sub.[t + 1]] = [[theta].sub.[t + 1]] + [[epsilon].sub.[t + 1]], (1)
[[theta].sub.[t + 1]] ~ N [[bar.[theta]], [[sigma].sub.[theta].sup.2], (2)
[[epsilon].sub.[t + 1]] ~ N (0, [[sigma].sub.y.sup.2]). (3)
The parameter [bar.[theta]] represents the unconditional average cash flow growth rate, and is set to zero without loss of generality. A nonzero unconditional mean cash flow growth rate adds a constant term to the average stock return but does not affect our conclusions regarding return predictability. (3) The parameter [[sigma].sub.[theta]] captures the uncertainty surrounding expected cash flow growth, whereas [[sigma].sub.y] measures the stock's fundamental risk. With risk-neutrality, fundamental risk does not influence stock prices.
The critical component of price formation is the investor's conditional expectation of future cash flow growth. In our model, realized cash flow growth is uninformative regarding future cash flow growth. (4) Instead, our investor receives multiple forecasts of future cash flow growth, with each forecast issued by a different information source (such as a sell-side analyst). Information sources issue cash flow growth forecasts for the next period. (5) Specifically, on date t, the investor observes the forecast for [[mu].sub.t.sup.i] for [y.sub.t + 1] issued by the jth information source where j = 1,..., J.
The investor forms their conditional expectation of future cash flow growth by optimally combining the available forecasts into a single aggregate estimate that has the lowest mean-squared forecast error. Intuitively the investor assigns more weight to forecasts issued by more accurate information sources. The crucial assumption is that the investor does not know the true forecast accuracy of the information sources but learns about their accuracy.
Although the investor uses the cash flow growth forecasts to form their conditional expectation of cash flow growth, they cannot directly evaluate the usefulness of these forecasts because the conditional mean of cash flow growth is unobservable and time-varying. We assume that their conditional expectation [E.sub.t][y.sub.[t + 1]][[mu].sub.t.sup.j] under the objective probability coincides with the...
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