Comparing linear and nonlinear forecasts for stock returns
Introduction
It is now well documented that aggregate stock returns cannot be satisfactorily explained by the present value (PV) model. Based on a constant discount rate and rational expectations, the PV model establishes a linear relation between stock prices and dividends, i.e., the fundamental variable. Efforts to incorporate speculative bubbles into the linear PV model also failed to improve upon its empirical performance Blanchard & Watson, 1982, Flood & Garber, 1980.2 To account for these failures, several new theoretical models were introduced, namely the intrinsic bubbles specification (Froot & Obstfeld, 1991), the trigger strategists model (Krugman, 1987) and the fads model (Summers, 1986). These models establish a nonlinear relation between stock prices and dividends, and thus imply a departure from the linear PV model. In contrast to the dismal performance of the linear PV model, these nonlinear models were found to adequately describe aggregate stock prices Froot & Obstfeld, 1991, van Norden & Schaller, 1994. Another strand of theoretical and empirical research has introduced models in which trading volume is an important determinant of stock prices Brock, 1993, Hiemstra & Jones, 1994. These models depart from a linear relation between stock prices and fundamental variables and establish a nonlinear relation between stock returns and trading volume.
The purpose of this article is to compare the forecasting performance of linear and nonlinear models of monthly aggregate stock returns. Our aim is to examine whether forecasts from a nonlinear stock returns model are preferable to forecasts from a linear stock returns model in terms of forecast accuracy as well as forecast encompassing. Although testing for forecast accuracy is a natural way to proceed when two competing forecasts are available, Granger and Newbold (1986) indicated that a more stringent requirement would be that the competing forecasts embody no useful information absent in the preferred forecasts. Clements and Hendry (1993) refer to this situation as forecast encompassing. In this article, we examine whether a nonlinear stock returns model encompasses a competitor linear model, in the sense of being able to explain the forecast errors made by the linear model.
An artificial neural network (ANN) methodology is employed to estimate a nonlinear model for stock returns, and out-of-sample (nonlinear) stock return forecasts are obtained from this model. The ANN methodology is preferred to other nonlinear models because it is nonparametric, and thus appropriate here since we do not want to examine a specific nonlinear functional form between stock prices and fundamentals. The input layer of the ANN contains three input variables, namely lagged trading volume, dividends, and lagged stock index returns. The specification of the lagged trading volume and dividends is postulated by recent work establishing a nonlinear relationship between stock returns and these variables Brock, 1993, Campbell et al., 1993, Froot & Obstfeld, 1991, Hiemstra & Jones, 1994, Krugman, 1987. The lagged stock index return is included due to extensive evidence supporting the inclusion of this variable in a conditional mean specification. Its forecasting competitor is a linear model with the same explanatory variables. We seek to examine whether out-of-sample short-run (one-step) forecasts generated by the ANN model are more accurate than out-of-sample forecasts generated by the linear model, using the testing procedure developed in Diebold and Mariano (1995). Furthermore, we examine whether the nonlinear out-of-sample forecasts encompass the linear out-of-sample forecasts of its linear competitor, following the testing procedure in Clements and Hendry (1998). It should be noted that although the ANN model is expected to have a superior in-sample performance, since it may nest the linear model, there is no guarantee that it will dominate the linear model out-of-sample (Donaldson & Kamstra, 1996a). Evidence of ANN forecasts encompassing linear forecasts might imply that nonlinearities in the relation between stock returns, trading volume, and dividends do matter in forecasting. This would be consistent with the previous work suggesting that the relation between stock returns and these variables is nonlinear, thereby explaining the failure of linear PV models in describing stock market fluctuations.
The remainder of this article is organised as follows. In the second section, we discuss previous studies that explain a nonlinear relationship between stock returns and trading volume, and stock returns and dividends. In the fourth section, we outline the neural network methodology employed in estimating the models. In the fifth section, we discuss the results from the forecast encompassing tests. The final section provides a summary and concludes.
Section snippets
Nonlinearity in the relation between stock returns, trading volume, and dividends
Recently developed theoretical approaches to modelling the behaviour of stock prices introduce nonlinearity in the relation between stock prices and dividends, and stock returns and trading volume. With regard to the stock price–dividend relation, Froot and Obstfeld (1991) introduced the intrinsic bubbles specification in which the bubbles are driven by the dividends. An important property of an intrinsic bubble is that for a given level of dividends the bubble will remain constant over time.
Data and preliminary statistics
This study uses monthly data for aggregate stock returns, trading volume, and dividends for two countries, namely the UK and the US. The stock index for the UK is the Financial Times All Share Index (FT), and the Dow Jones Industrial Average (DJ) for the US. We construct stock index returns that include dividend payments, namely total stock index returns to holding stocks, as these are a relatively large portion of the monthly returns.3
A nonlinear neural network model for stock returns
We employ the technique of ANN estimation to obtain out-of-sample forecasts from a nonlinear model. An ANN model represents an attempt to emulate certain features of the way in which the brain processes information. ANN models have received considerable attention as a useful vehicle for forecasting financial variables (Swanson & White, 1995). An important feature of ANN is that they are nonparametric models and, therefore, are appropriate for our purposes as we do not rely on a specific
Out-of-sample forecasting performance results
This section focuses on the out-of-sample forecasting ability of the ANN and the linear models. It is important to note that we examine exclusively the out-of-sample forecasting ability of the two competing models and, therefore, we do not automatically favour the model with the higher flexibility to fit the data in-sample, i.e., the ANN model. Although ANN is expected to have a superior in-sample performance, since it nests the linear model, there is no guarantee that it will dominate the
Conclusions
The paper compared out-of-sample forecasts of monthly returns from both DJ and FT indices, generated by two competing models, namely a linear model and a nonlinear ANN model. We consider two fundamental variables, namely the trading volume and the dividend, and the lagged returns series as explanatory variables in the linear model and input variables in the ANN model. The comparison of out-of-sample forecasts is carried out on the basis of two approaches: forecast accuracy and forecast
Acknowledgements
We wish to thank two anonymous referees for several constructive comments that helped us improve the current version of the paper. Thanks are also due to John Goddard, Yue Ma, and Laurence Copeland for helpful comments and discussions when preparing this draft. The first author gratefully acknowledges partial financial assistance from the Research Committee, University of Crete. The usual disclaimer applies.
References (25)
- et al.
An artificial neural network–GARCH model for international stock return volatility
Journal of Empirical Finance
(1997) - et al.
How effective are neural networks at forecasting and prediction? A review and evaluation
Journal of Forecasting
(1998) - et al.
Bubbles, rational expectations, and financial markets
- Brock, W. A. (1993). Pathways to randomness in the economy: emergent nonlinearity and chaos in economics and finance....
- et al.
Trading volume and serial correlation in stock returns
Quarterly Journal of Economics
(1993, November) - et al.
The econometrics of financial markets
(1998) - et al.
The dividend–price ratio and the expectations of future dividends and discount factors
Review of Financial Studies
(1988) - et al.
Econometric evaluation of linear macroeconomic models
Review of Economic Studies
(1986) Combining forecasts: a review and annotated bibliography
International Journal of Forecasting
(1989)- et al.
On the limitations of comparing Mean Square Forecast Errors
Journal of Forecasting
(1993)
Forecasting economic time series
Comparing predictive accuracy
Journal of Business and Economic Statistics
Cited by (0)
- 1
Tel.: +30-831-77405.