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From fundamental signals to stock volatility: A machine learning approach
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This study investigates whether the direction of U.S. implied volatility, VIX index, can be forecast. Multiple forecasts are generated based on standard econometric models, but, more importantly, on several machine learning techniques. Their statistical significance is assessed by a plethora of performance evaluation measures, while real-time investment strategies are devised to appraise the investment implications of the underlying modeling approaches. The main conclusion of the analysis is that the implementation of machine learning techniques in implied volatility forecasting can be more effective compared to mainstream econometric models and model selection techniques, as they are superior both in a statistical and an economic evaluation setting.
We perform a comparative analysis of machine learning methods for the canonical problem of empirical asset pricing: measuring asset risk premiums. We demonstrate large economic gains to investors using machine learning forecasts, in some cases doubling the performance of leading regression-based strategies from the literature. We identify the best-performing methods (trees and neural networks) and trace their predictive gains to allowing nonlinear predictor interactions missed by other methods. All methods agree on the same set of dominant predictive signals, a set that includes variations on momentum, liquidity, and volatility.
Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.
We examine the predictive power of machine learning (ML) models for forecasting realised volatility (RV) using different data sets tested in the literature, viz., variables from the HAR models, limit order book (LOB), and news sentiments. With 3.7 million ML models trained and robustness checked, the high dimensional ML models (viz. 147 data features for up to 21 lags) significantly outperformed HAR on 90% of the out-of-sample period when the actual volatility is not extreme. SHAP values reveal that mid prices, mean bids, and mean asks are the most useful predictive variables. The ML models also better than HAR at capturing forecast dynamics as they evolve through time.
We propose a model selection method to systematically evaluate the contribution to asset pricing of any new factor, above and beyond what a high‐dimensional set of existing factors explains. Our methodology accounts for model selection mistakes that produce a bias due to omitted variables, unlike standard approaches that assume perfect variable selection. We apply our procedure to a set of factors recently discovered in the literature. While most of these new factors are shown to be redundant relative to the existing factors, a few have statistically significant explanatory power beyond the hundreds of factors proposed in the past.
We use deep neural networks to estimate an asset pricing model for individual stock returns that takes advantage of the vast amount of conditioning information, while keeping a fully flexible form and accounting for time-variation. The key innovations are to use the fundamental no-arbitrage condition as criterion function, to construct the most informative test assets with an adversarial approach and to extract the states of the economy from many macroeconomic time series. Our asset pricing model outperforms out-of-sample all benchmark approaches in terms of Sharpe ratio, explained variation and pricing errors and identifies the key factors that drive asset prices.
We construct size and value factors in China. The size factor excludes the smallest 30% of firms, which are companies valued significantly as potential shells in reverse mergers that circumvent tight IPO constraints. The value factor is based on the earnings-price ratio, which subsumes the book-to-market ratio in capturing all Chinese value effects. Our three-factor model strongly dominates a model formed by just replicating the Fama and French (1993) procedure in China. Unlike that model, which leaves a 17% annual alpha on the earnings-price factor, our model explains most reported Chinese anomalies, including profitability and volatility anomalies.
This paper presents a comprehensive study on predicting the cross section of Chinese stock market returns with a large panel of 75 individual firm characteristics. We use not only the traditional Fama-MacBeth regression, but also the “big-data” econometric methods: principal component analysis (PCA), partial least squares (PLS), and forecast combination to extract information from all the 75 firm characteristics. These characteristics are important return predictors, with statistical and economic significance. Furthermore, firm characteristics that are related to trading frictions, momentum, and profitability are the most effective predictors of future stock returns in the Chinese stock market. Keywords: Partial least squares, Machine learning, Firm characteristics, Chinese stock market, Return predictability
We examine whether fundamental measures of volatility are incremental to market-based measures of volatility in (i) predicting bankruptcies (out of sample), (ii) explaining cross-sectional variation in credit spreads, and (iii) explaining future credit excess returns. Our fundamental measures of volatility include (i) historical volatility in profitability, margins, turnover, operating income growth, and sales growth; (ii) dispersion in analyst forecasts of future earnings; and (iii) quantile regression forecasts of the interquartile range of the distribution of profitability. We find robust evidence that these fundamental measures of volatility improve out-of-sample forecasts of bankruptcy and help explain cross-sectional variation in credit spreads. This suggests that an analysis of credit risk can be enhanced with a detailed analysis of fundamental information. As a test case of the benefit of volatility forecasting, we document an improved ability to forecast future credit excess returns, particularly when using fundamental measures of volatility.
An investment factor, long in low-investment stocks and short in high-investment stocks, helps explain the new issues puzzle.
Adding the investment factor into standard factor regressions reduces the SEO underperformance by about 75%, the IPO underperformance
by 80%, the underperformance following convertible debt offerings by 50%, and Daniel and Titman's (2006) composite issuance effect by 40%. The reason is that issuers invest more than nonissuers, and the investment factor earns
a significantly positive average return of 0.57% per month.
We propose and evaluate explicit tests of the null hypothesis of no difference in the accuracy of two competing forecasts. In contrast to previously developed tests, a wide variety of accuracy measures can be used (in particular, the loss function need not be quadratic and need not even be symmetric), and forecast errors can be non-Gaussian, nonzero mean, serially correlated, and contemporaneously correlated. Asymptotic and exact finite-sample tests are proposed, evaluated, and illustrated.
We use deep neural networks to estimate an asset pricing model for individual stock returns that takes advantage of the vast amount of conditioning information, keeps a fully flexible form, and accounts for time variation. The key innovations are to use the fundamental no-arbitrage condition as criterion function to construct the most informative test assets with an adversarial approach and to extract the states of the economy from many macroeconomic time series. Our asset pricing model outperforms out-of-sample all benchmark approaches in terms of Sharpe ratio, explained variation, and pricing errors and identifies the key factors that drive asset prices.
This paper was accepted by Agostino Capponi, finance.
Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4695 .
We inspect how accurate machine learning (ML) is at forecasting realized variance of the Dow Jones Industrial Average index constituents. We compare several ML algorithms, including regularization, regression trees, and neural networks, to multiple heterogeneous autoregressive (HAR) models. ML is implemented with minimal hyperparameter tuning. In spite of this, ML is competitive and beats the HAR lineage, even when the only predictors are the daily, weekly, and monthly lags of realized variance. The forecast gains are more pronounced at longer horizons. We attribute this to higher persistence in the ML models, which helps to approximate the long memory of realized variance. ML also excels at locating incremental information about future volatility from additional predictors. Lastly, we propose an ML measure of variable importance based on accumulated local effects. This shows that while there is agreement about the most important predictors, there is disagreement on their ranking, helping to reconcile our results.
We survey recent methodological contributions in asset pricing using factor models and machine learning. We organize these results based on their primary objectives: estimating expected returns, factors, risk exposures, risk premia, and the stochastic discount factor as well as model comparison and alpha testing. We also discuss a variety of asymptotic schemes for inference. Our survey is a guide for financial economists interested in harnessing modern tools with rigor, robustness, and power to make new asset pricing discoveries, and it highlights directions for future research and methodological advances.
Expected final online publication date for the Annual Review of Financial Economics, Volume 13 is November 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
With machine learning tools, we document that firm fundamentals have explanatory power on the shape of the option implied volatility (IV) curve that is both economically and statistically significant. We also find that, after accounting for fundamentals, the associated IV process can generate overreaction in the long-term IV with respect to change in the short-term IV, and can allow a positive profit from at-the-money straddle writing, explaining puzzling patterns in the literature. We also provide a simple model linking the IV to firm fundamentals, which permits realistic IV curves and is consistent with the empirical findings.
The Chinese stock market incurs huge illiquidity costs. Liquidity has different aspects but literature rarely measures it from an aggregate perspective. To capture liquidity along various dimensions and more consistently, we propose an aggregate liquidity premium with a partial least squares approach by aggregating information on 12 liquidity-related firm characteristics in the Chinese stock market. The aggregate liquidity predictor generates significantly higher expected stock returns than individual characteristics, remains robust to different information aggregation methods, and reflects the multidimensional feature of liquidity. Behavioral mispricing theory helps explain the aggregate liquidity premium because illiquidity limits arbitrage to correct mispricing. As liquidity is important and traditional factor models limitedly explain it, we develop a new-factor model to capture the aggregate liquidity premium. The aggregate liquidity premium in China indicates that policymakers should improve market liquidity.
We add to the emerging literature on empirical asset pricing in the Chinese stock market by building and analyzing a comprehensive set of return prediction factors using various machine learning algorithms. Contrasting previous studies for the US market, liquidity emerges as the most important predictor, leading us to closely examine the impact of transaction costs. The retail investors’ dominating presence positively affects short-term predictability, particularly for small stocks. Another feature that distinguishes the Chinese market from the US market is the high predictability of large stocks and state-owned enterprises over longer horizons. The out-of-sample performance remains economically significant after transaction costs.
Most anomalies fail to hold up to currently acceptable standards for empirical finance. With microcaps mitigated via NYSE breakpoints and value-weighted returns, 65% of the 452 anomalies in our extensive data library, including 96% of the trading frictions category, cannot clear the single test hurdle of the absolute t-value of 1.96. Imposing the higher multiple test hurdle of 2.78 at the 5% significance level raises the failure rate to 82%. Even for replicated anomalies, their economic magnitudes are much smaller than originally reported. In all, capital markets are more efficient than previously recognized.
Received June 12, 2017; editorial decision October 29, 2018 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.
What capital allocation role can China’s stock market play? Counter to perception, stock prices in China have become as informative about future profits as they are in the US. This rise in stock price informativeness has coincided with an increase in investment efficiency among privately owned firms, suggesting the market is aggregating information and providing useful signals to managers. However, price informativeness and investment efficiency for state-owned enterprises fell below that of privately owned firms after the postcrisis stimulus, perhaps reflecting unpredictable subsidies and state-directed investment policy. Finally, evidence from realized returns suggests Chinese firms face a higher cost of equity capital than US firms.
We construct a robust stochastic discount factor (SDF) summarizing the joint explanatory power of a large number of cross-sectional stock return predictors. Our method achieves robust out-of-sample performance in this high-dimensional setting by imposing an economically motivated prior on SDF coefficients that shrinks contributions of low-variance principal components of the candidate characteristics-based factors. We find that characteristics-sparse SDFs formed from a few such factors—e.g., the four- or five-factor models in the recent literature—cannot adequately summarize the cross-section of expected stock returns. However, an SDF formed from a small number of principal components performs well.
We take up Cochrane's (2011) challenge to identify the firm characteristics that provide independent information about average U.S. monthly stock returns by simultaneously including 94 characteristics in Fama-MacBeth regressions that avoid overweighting microcaps and adjust for data-snooping bias. We find that while 12 characteristics are reliably independent determinants in non-microcap stocks from 1980 to 2014 as a whole, return predictability sharply fell in 2003 such that just two characteristics have been independent determinants since then. Outside of microcaps, the hedge returns to exploiting characteristics-based predictability also have been insignificantly different from zero since 2003. (JEL G12, G14)
Managed portfolios that take less risk when volatility is high produce large alphas, increase Sharpe ratios, and produce large utility gains for mean-variance investors. We document this for the market, value, momentum, profitability, return on equity, investment, and betting-against-beta factors, as well as the currency carry trade. Volatility timing increases Sharpe ratios because changes in volatility are not offset by proportional changes in expected returns. Our strategy is contrary to conventional wisdom because it takes relatively less risk in recessions. This rules out typical risk-based explanations and is a challenge to structural models of time-varying expected returns.
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Aggregate stock return volatility is both persistent and countercyclical. This paper tests whether it is possible to improve volatility forecasts at monthly and quarterly horizons by conditioning on additional macroeconomic variables. I find that several variables related to macroeconomic uncertainty, time-varying expected stock returns, and credit conditions Granger cause volatility. It is more difficult to find evidence that forecasts exploiting macroeconomic variables outperform a univariate benchmark out-of-sample. The most successful approaches involve simple combinations of individual forecasts. Predictive power associated with macroeconomic variables appears to concentrate around the onset of recessions.
A five-factor model directed at capturing the size, value, profitability, and investment patterns in average stock returns performs better than the three-factor model of Fama and French (FF, 1993). The five-factor model's main problem is its failure to capture the low average returns on small stocks whose returns behave like those of firms that invest a lot despite low profitability. The model's performance is not sensitive to the way its factors are defined. With the addition of profitability and investment factors, the value factor of the FF three-factor model becomes redundant for describing average returns in the sample we examine.
A volatility model must be able to forecast volatility; this is the central requirement in almost all financial applications. In this paper we outline some stylized facts about volatility that should be incorporated in a model: pronounced persistence and mean-reversion, asymmetry such that the sign of an innovation also affects volatility and the possibility of exogenous or pre-determined variables influencing volatility. We use data on the Dow Jones Industrial Index to illustrate these stylized facts, and the ability of GARCH-type models to capture these features. We conclude with some challenges for future research in this area.
ABSTRACT Two easily measured variables, size and book-to-market equity, combine to capture the cross-sectional variation in average stock returns associated with market {3, size, leverage, book-to-market equity, and earnings-price ratios. Moreover, when the tests allow for variation in {3 that is unrelated to size, the relation between market {3 and average return is flat, even when {3 is the only explanatory variable. THE ASSET-PRICING MODEL OF Sharpe (1964), Lintner (1965), and Black (1972)
The determination of the level of corporate borrowing and the choice of debt maturity are the two most important concerns in the management of capital structure. In this study, we examine analytically and empirically the impact of debt maturity changes on the expected returns of common stocks. Using the US stock market data and financial statement data, we examine the cross-sectional relation between expected returns and financial leverage ratios. We find reliable evidence that common stock expected returns are positively related to the extent of short-term debt financing. The positive relation is significant even after we control for systematic risk, total debt ratio and firm size. The results suggest that an increase in short-term debt which displaces the same amount of long-term debt increases the expected returns of common stocks, possibly because the substitution transfers risk from long-term debtholders to shareholders.
We revisit the relation between stock market volatility and macroeconomic activity using a new class of component models that distinguish short run from secular movements. We study long historical data series of aggregate stock market volatility, starting in the 19th century, as in Schwert (1989). We formulate models with the long term component driven by inflation and industrial production growth that are at par in terms of pseudo out-of-sample prediction for horizons of one quarter and are at par or out-perform more traditional time series volatility models at longer horizons. Hence, imputing economic fundamentals into volatility models pays off in terms of long horizon forecasting. We also find that at a daily level, inflation and industrial production growth, account for between 10 % and 35 % of one-day ahead volatility prediction. Hence, macroeconomic fundamentals play a significant role even at short horizons. Unfortunately, all the models -purely time series ones as well as those driven by economic variables -feature structural breaks over the entire sample spanning roughly a century and a half of daily data. Consequently, our analysis also focuses on subsamples -pre-WWI, the Great Depression era, and post-WWII (also split to examine the so called Great Moderation). Our main findings remain valid across subsamples.
This paper examines whether a simple accounting-based fundamental analysis strategy, when applied to a broad portfolio of high book-to-market firms, can shift the distribution of returns earned by an investor. I show that the mean return earned by a high book-to-market investor can be increased by at least 7 percent annually through the selection of financially strong high BM firms while the entire distribution of realized returns is shifted to the right. In addition, an investment strategy that buys expected winners and shorts expected losers generates a 23 percent annual return between 1976 and 1996 and the strategy appears to be robust across time and to controls for alternative investment strategies. Within the portfolio of high BM firms, the benefits to financial statement analysis are concentrated in small and medium sized firms, companies with low share turnover and firms with no analyst following, yet this superior performance is not dependent on purchasing firms with low share prices. A positive relationship between the sign of the initial historical information and both future firm performance and subsequent quarterly earnings announcement reactions suggests that the market initially underreacts to the historical information. In particular, 1/6th of the annual return difference between ex ante strong and weak firms is earned over the four three-day periods surrounding these quarterly earnings announcements. Overall, the evidence suggests that the market does not fully incorporate historical financial information into prices in a timely manner.
This study maintains that the value relevance of accounting information cannot be measured solely by reference to Ball-Brown (ERC) analysis. The contribution of accounting information to return volatility is equally crucial. Synthesizing and generalizing results from earlier studies, this paper rigorously develops Ball-Brown (1968) and volatility measures of the information content of earnings from underlying primitives using the accounting return decomposition model of Vuolteenaho (2002). The well-known Ball-Brown metric is obtained by assuming that (i) the time series of (log deflated) earnings is stationary AR(1), (ii) other information shocks are non-existent and (iii) expected future discount rates are inter-temporally constant. This measure is initially extended to include other information shocks and dynamic discount rates that are also stationary AR(1). Generalized Ball-Brown and variance measures of information content are further obtained where the time series of earnings, other information and expected future discount rates follow either stationary ARMA(p,q) processes or a log-linear stationary VAR process. Closed form solutions of information content are also obtained for earnings components (e.g., cash flows and accruals). Applications of the theory are offered that show how this framework improves on extant empirical methodologies and suggests further lines of research.
We investigate the joint hypothesis that (1) tax expense contains information about core profitability that is incremental to reported earnings and (2) that information is reflected in stock prices with a delay. We find that seasonally differenced quarterly tax expense, our proxy for tax expense surprise, is related positively to future returns. This anomaly is separate from previously documented pricing anomalies based on financial and tax variables. Additional investigation reveals that tax expense surprise is related positively to changes in future quarterly earnings and tax expense, and both those future changes are related positively to future returns. While the returns to investing in predictable future earnings changes has been documented before, these results suggest that predicting changes in future tax expense also generates incremental future returns.
A fundamental assumption in cost accounting is that the relation between costs and volume is symmetric for volume increases and decreases. In this study, we investigate whether costs are “sticky”—that is, whether costs increase more when activity rises than they decrease when activity falls by an equivalent amount. We find, for 7,629 firms over 20 years, that selling, general, and administrative (SG&A) costs increase on average 0.55% per 1% increase in sales but decrease only 0.35% per 1% decrease in sales. Our analysis compares the traditional model of cost behavior in which costs move proportionately with changes in activity with an alternative model in which sticky costs occur because managers deliberately adjust the resources committed to activities. We test hypotheses about the properties of sticky costs and how the degree of stickiness of SG&A costs varies with firm circumstances.
Recent explanations of aggregate stock market fluctuations suggest that countercyclical stock market volatility is consistent with rational asset evaluations. In this paper, I develop a framework to study the causes of countercyclical stock market volatility. I find that countercyclical risk premia do not imply countercyclical return volatility. Instead, countercyclical stock volatility occurs if risk premia increase more in bad times than they decrease in good times, thereby inducing price–dividend ratios to fluctuate more in bad times than in good. The business cycle asymmetry in the investors’ attitude toward discounting future cash flows plays a novel and critical role in many rational explanations of asset price fluctuations.
Forecast evaluation often compares a parsimonious null model to a larger model that nests the null model. Under the null that the parsimonious model generates the data, the larger model introduces noise into its forecasts by estimating parameters whose population values are zero. We observe that the mean squared prediction error (MSPE) from the parsimonious model is therefore expected to be smaller than that of the larger model. We describe how to adjust MSPEs to account for this noise. We propose applying standard methods [West, K.D., 1996. Asymptotic inference about predictive ability. Econometrica 64, 1067–1084] to test whether the adjusted mean squared error difference is zero. We refer to nonstandard limiting distributions derived in Clark and McCracken [2001. Tests of equal forecast accuracy and encompassing for nested models. Journal of Econometrics 105, 85–110; 2005a. Evaluating direct multistep forecasts. Econometric Reviews 24, 369–404] to argue that use of standard normal critical values will yield actual sizes close to, but a little less than, nominal size. Simulation evidence supports our recommended procedure.
We survey and interview more than 400 executives to determine the factors that drive reported earnings and disclosure decisions. We find that managers would rather take economic actions that could have negative long-term consequences than make within-GAAP accounting choices to manage earnings. A surprising 78% of our sample admits to sacrificing long-term value to smooth earnings. Managers also work to maintain predictability in earnings and financial disclosures. We also find that managers make voluntary disclosures to reduce information risk and boost stock price but at the same time, try to avoid setting disclosure precedents that will be difficult to maintain.
This paper performs a financial statement analysis that combines a large set of financial statement items into one summary measure which indicates the direction of one-year-ahead earnings changes. Positions are taken in stocks on the basis of this measure during the period 1973–1983, which involve canceling long and short positions with zero net investment. The two-year holding-period return to the long and short positions is in the order of 12.5%. After adjustment for ‘size effects’ the return is about 7.0%. These returns cannot be explained by nominated firm risk characteristics.
This article presents a dynamic, rational expectations equilibrium model of asset prices where the drift of fundamentals (dividends)
shifts between two unobservable states at random times. I show that in equilibrium, investors' willingness to hedge against
changes in their own “uncertainty” on the true state makes stock prices overreact to bad news in good times and underreact
to good news in bad times. I then show that this model is better able than conventional models with no regime shifts to explain
features of stock returns, including volatility clustering, “leverage effects,” excess volatility, and time-varying expected
returns.
Evidence suggests the volatility of stock prices cannot be accounted for by information about future dividends. The authors argue that some of the volatility of stock prices in excess of fundamentals results from fluctuations in the amount of public information over time. Their model assumes that dividends and consumption are constant in the aggregate but that there are good firms and bad firms whose identity may be unknown to the public, as in George Akerlof's (1970) 'lemons' problem. In that case, the collective valuation of the constant dividend stream depends on the degree of informational asymmetry. Copyright 1994 by Oxford University Press.
We model consumption and dividend growth rates as containing (1) a small long-run predictable component, and (2) fluctuating economic uncertainty (consumption volatility). These dynamics, for which we provide empirical support, in conjunction with Epstein and Zin's (1989) preferences, can explain key asset markets phenomena. In our economy, financial markets dislike economic uncertainty and better long-run growth prospects raise equity prices. The model can justify the equity premium, the risk-free rate, and the volatility of the market return, risk-free rate, and the price-dividend ratio. As in the data, dividend yields predict returns and the volatility of returns is time-varying. Copyright 2004 by The American Finance Association.
This paper analyzes the relation of stock volatility with real and nominal macroeconomic volatility, economic activity, financial leverage, and stock trading activity using monthly data from 1857 to 1987. An important fact, previously noted by Robert R. Officer (1973), is that stock return variability was unusually high during the 1929-39 Great Depression. While aggregate leverage is significantly correlated with volatility, it explains a relatively small part of the movements in stock volatility. The amplitude of the fluctuations in aggregate stock volatility is difficult to explain using simple models of stock valuation, especially during the Great Depression. Copyright 1989 by American Finance Association.
Lawrence R. Klein pioneered the work on aggregation, in particular in production functions, in the 1940s. He paved the way for researchers to establish the conditions under which a series of micro production functions can be aggregated so as to yield an aggregate production function. This work is fundamental in order to establish the legitimacy of theoretical (neoclassical) growth models and empirical work in this area (e.g., growth accounting exercises, econometric estimation of aggregate production functions). This is because these models depend on the assumption that the technology of an economy can be represented by an aggregate production function, i.e., that the aggregate production function exists. However, without proper aggregation one cannot interpret the properties an aggregate production function. The aggregation literature showed that the conditions under which micro production functions can be aggregated so as to yield an aggregate production function are so stringent that it is difficult to believe that actual economies can satisfy them. These results question the legitimacy of growth models and their policy implications. Scientifi c work cannot proceed as if production functions existed. For this reason, the profession should pause before continuing to do theoretical and applied work with no sound foundations and dedicate some time to studying other approaches to estimating the impact of economic policies in order to understand what questions can legitimately be posed to the empirical aggregate data. Lawrence R. Klein fue uno de los pioneros del campo de la agregaci�n, en particular en el �rea de las funciones de producci�n, durante la d�cada de los 40. Sus contribuciones ayudaron a defi nir el problema de la agregaci�n para que investigadores posteriores establecieran formalmente las condiciones formales bajo las que funciones de producci�n microecon�micas con propiedades neocl�sicas pudieran ser agregadas con el fi n de generar una funci�n
Post-1970, share issuance exhibits a strong cross-sectional ability to predict stock returns. This predictive ability is more statistically significant than the individual predictive ability of size, book-to-market, or momentum. Our finding is related to research that finds that long-run returns are associated with share repurchase announcements, seasoned equity offerings, and stock mergers, although our results remain strong even after exclusion of the data used in these studies. We estimate the issuance relation pre-1970 and find no statistically significant predictive ability for most holding periods. Copyright 2008 by The American Finance Association.
A volatility model must be able to forecast volatility; this is the central requirement in almost all financial applications. In this paper we outline some stylised facts about volatility that should be incorporated in a model; pronounced persistence and meanreversion, asymmetry such that the sign of an innovation also affects volatility and the possibility of exogenous or pre-determined variables influencing volatility. We use data on the Dow Jones Industrial index to illustrate these stylised facts, and the ability of GARCH-type models to capture these features. We conclude with some challenges for future research in this area. Keywords: volatility modelling, ARCH, GARCH, volatility forecasting. JEL Classification Code : C22 * Please send comments or questions to rengle@stern.nyu.edu. 2 1. INTRODUCTION A volatility model should be able to forecast volatility. Virtually all the financial uses of volatility models entail forecasting aspects of future returns. Typically a volati...