The predictability of cross-sectional returns in high frequency.
| Autor | Wang, Yifan |
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Introduction
Stock return forecast is important in the financial world, and people started proposing forecasting methods very early. For example, Dow (1920) examined the role of dividend ratios in stock return prediction. Graham et al. (1934) argued that a high valuation ratio could indicate an undervalued stock. However, making forecasts is not an easy task, due to complex and fast- changing markets, and the many noise variables involved. A successful forecast always requires three things: a set of useful predictors, a reasonable statistical model, and a good estimation of the model.
A variety of factors are used to predict stock returns. Many are company characteristics, such as dividends (Ball, 1978), earnings (Campbell and Shiller, 1988), and book value (Kothari and Shanken, 1997). Some are macro- economic signals, such as inflation (Lintner, 1975) and aggregate consumption (Lettau and Ludvigson, 2001). These data are usually from companies' financial statements and government reports, so the prediction frequency is low (often annually or quarterly). Given that factors used in low- frequency predictions are typically not available with a finer time resolution, high-frequency forecasts are more challenging, and evidence of stock return predictability at high frequency is scarce (Herwartz, 2017).
However, with the rapid growth of the Internet and computing technologies, trading frequency has increased to fractions of seconds (Aldridge and Krawciw, 2017). In high frequency, we can approach the stock return forecast problem using different signals and methods. Some use self-lagged returns to construct time-series models such as ARMA or GARCH (Herwartz, 2017), while others make use of micro-structure data to forecast returns (Huang and Stoll, 1994).
Chinco et al. (2019) propose a method to make one-minute-ahead return forecasts. They argue that since many variables are short-lived in highfrequency prediction, it is challenging both to identify the useful predictors and to estimate the model using traditional regression. They propose using LASSO to identify the "unexpected short-lived and sparse" signals. Since LASSO can select variables effectively, they include all cross-sectional returns of NYSE-listed companies (over 2000), and their three lags--more than 6000 variables, in total--as candidate predictors, then train LASSO on a 30- minute rolling basis. The results are amazing: using predictions from LASSO along with predictions from a traditional AR(3) model can increase the outof-sample fit by more than 2%, which is a huge gain. This indicates high predictability and much valuable information in cross-sectional returns.
Inspired mainly by Chinco et al. (2019), in this article we use the cross- sectional returns of a full set of S&P 500 components to make one-minute-ahead return forecasts of stocks on the Dow Jones index. The S&P 500 are the five hundred largest stocks listed in exchanges in the United States, accounting for more than 60% of market capitalization (cap) of all stocks. The Dow Jones index, on the other hand, contains the thirty large companies whose market caps account for more than 15% of the entire market. Unlike Chinco et al. (2019), we focus on large stocks because they are the most liquid ones on the market, and account for a major portion of total trading volume. Large stocks are definitely more challenging to predict. Focusing on large stocks also allows us to examine the source of the predictability presented by those authors further, by comparing our results with theirs.
Beyond LASSO, we also extend the work of Chinco et al. (2019) to non-linear models. As those authors do, we use LASSO to reduce dimensions effectively. Because LASSO is a linear model, and non-linearity is essential in asset pricing (Freyberger et al., 2020), we introduce non-linearity into our predictions. We choose to use random forest (RF) because it is relatively efficient to train, compared to more complicated models such as neural networks. Trees have also already been proven to perform well in forecasting tasks (Gu et al., 2018). We expect that non-linearity can help improve prediction performance.
We train the models on a rolling basis and set the training window at 30 minutes. Then we make rolling one-minute-ahead predictions using the trained models. Similar to Chinco et al. (2019), we construct several linear regression benchmarks with different benchmark predictors. The benchmark predictors are some long-lived traditional factors, such as market return, size factor, and value factor from the Fama-French three-factor model. Then we use machine-learning methods to see whether cross-sectional returns have predictability, by adding all the cross-sectional returns and their lags as candidate predictors. We eliminate the stocks for which we find no corresponding symbol, or that are no longer in the S&P 500 index, ending up with 397 stocks remaining. The lagged returns of these 397 stocks enter the machine-learning models as candidate predictors. It is impossible to estimate such a large model using linear regression, because the number of variables is much greater than the training-sample size. This is one of the benefits of using machine- learning methods: we do not need to pre-determine the predictors. Instead, we put forward a large set of variables and let the model decide. This can be beneficial, since some signals might carry valuable information for only a short period of time, thus being unlikely for a human to capture.
We measure the accuracy of the predictions using out-of-sample [R.sup.2], which is computed from mean-squared-error:
[Please download the PDF to view the mathematical expression]
where [r.sup.t] is the true value at time t, [[??].sup.model.sub.t] is the prediction given by a certain model, and [[bar.r].sub.t--30:t--1] is the sample average of the training sample. However, this is not what Chinco et al. (2019) use. They run the following regression:
[r.sub.n,t] = [[bar.a].sub.n] + [[bar.b].sub.n][[??].sup.model.sub.n,t] + [e.sub.n,t],
where [r.sub.n,t] is the realized return, and [[??].sup.model.sub.n,t] is the prediction given by the model. They report the [R.sup.2] from this regression. This measures how much variation in true values is explained by the predictions. More details about the metric are provided in later sections. For comparison, we also calculate the metric from their paper. We use it to explain some implications of our results.
Our results are surprising. LASSO and random forest do perform better than linear regression, but all models have a negative out-of-sample [R.sup.2], indicating that they perform worse than the historical mean. This result is stable during the whole sample period, and is robust with different training window sizes. So, we cannot say that there is any predictability in terms of out-of-sample [R.sup.2]. However, results from Chinco's metric suggest that the predictions from random forest can help explain some additional variation in the true values, i.e., adding random forest forecasts in addition to auto- regression predictions increases explanatory power in a statistically-significant way.
This article is arranged as follows. We review related literature in Section 2, then introduce the data and prediction methods in Section 3. Section 4 presents empirical results and implications. Section 5 concludes and provides further research ideas.
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Related work
This article is related to a wide range of literature. First of all, it pertains to research on return predictability. Many scholars have proposed useful signals or methods to make predictions and find predictability in real data (Patelis, 1997; Campbell and Yogo, 2006; Ang and Bekaert, 2007). In particular, the well-known Fama-French three-factor model (Fama and Kenneth, 1993) is tested in many contexts. For example, Griffin (2002) finds that country- specific versions of Fama-French factors work better than global versions. Gaunt (2004) finds that size and value factors improve the predictability of the Fama-French model, using Australian stock market data. Suh (2009) tests it using CRSP data and finds that market premium is the most significant factor among the three. Some papers, however, present frustrating results. Welch and Goyal (2008) provide a good summary of widely-known stock return predictors. They test them on 30 years of data, and find that most signals in the literature are unstable, or even spurious. Moreover, most signals cannot beat the historical average, so give negative out-of-sample [R.sup.2]. Later, Campbell and Thompson (2008) propose adding weak restrictions on regressions to try to solve such challenges.
This article also builds on a large body of work on machine learning methods, especially on LASSO and random forest. LASSO was first introduced by Tibshirani (1996). Under the irrepresentable condition, LASSO achieves model selection consistency (Zhao and Yu, 2006). Also, LASSO has nice properties related to out-of-sample risk under certain mild conditions (Chatterjee, 2013). So LASSO is widely used as a method to select features when one believes that the true model is sparse. In this paper, we implicitly bet on sparsity when we use...
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