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High-Dimensional Predictive Regression in the Presence Of High-dimensional predictive regression in the presence of cointegration⇤ Bonsoo Koo Heather Anderson Monash University Monash University Myung Hwan Seo† Wenying Yao Seoul National University Deakin University Abstract We propose a Least Absolute Shrinkage and Selection Operator (LASSO) estimator of a predictive regression in which stock returns are conditioned on a large set of lagged co- variates, some of which are highly persistent and potentially cointegrated. We establish the asymptotic properties of the proposed LASSO estimator and validate our theoreti- cal findings using simulation studies. The application of this proposed LASSO approach to forecasting stock returns suggests that a cointegrating relationship among the persis- tent predictors leads to a significant improvement in the prediction of stock returns over various competing forecasting methods with respect to mean squared error. Keywords: Cointegration, High-dimensional predictive regression, LASSO, Return pre- dictability. JEL: C13, C22, G12, G17 ⇤The authors acknowledge financial support from the Australian Research Council Discovery Grant DP150104292. †Corresponding author. His work was supported by Promising-Pioneering ResearcherProgram through Seoul National University(SNU) and from the Ministry of Education of the Republic of Korea and the Na- tional Research Foundation of Korea (NRF-0405-20180026). Author contact details are [email protected], [email protected], [email protected], [email protected] 1 1 Introduction Return predictability has been of perennial interest to financial practitioners and scholars within the economics and finance professions since the early work of Dow (1920). To this end, empirical studies employ linear predictive regression models, as illustrated by Stam- baugh (1999), Binsbergen et al. (2010) and the references therein. Predictive regression allows forecasters to focus on the prediction of asset returns conditioned on a set of one-period lagged predictors. In a typical example of a predictive regression, the dependent variable is the rate of return on stocks, bonds or the spot rate of exchange, and the predictors usually consist of financial ratios and macroeconomic variables. The literature that has investigated the predictability of stock returns is very large, and it has proposed many economically plausible predictive variables and documented their em- pirical forecast performance. Koijen and Nieuwerburgh (2011) and Rapach and Zhou (2013) provide recent surveys of this literature. However, despite the plethora of possible predictors that have been suggested, there is no clear-cut evidence as to whether any of these variables are successful predictors. Indeed, many studies have concluded that there is very little dif- ference between predictions derived from a predictive regression and the historical mean of stock returns. Welch and Goyal (2008) provided a detailed study of stock return predictability based on many predictors suggested in the literature, and concluded that ”a healthy skepticism is appropriate when it comes to predicting the equity premium, at least as of 2006”. Least squares predictive regression has provided the main estimation tool for empirical efforts to forecast stock returns, but this involves several econometric issues. See Phillips (2015) for a summary of these issues, together with relevant econometric theory. Among these issues, we restrict our attention to two specific problems, which we believe could be tackled by using a different estimation approach. Firstly, the martingale difference features of excess returns are not readily compatible with the high persistence in many of the predictors suggested by the empirical literature. It is well-documented that excess returns are stationary and that financial ratios are often highly persistent and potentially non-stationary processes. The regression of a stationary variable on a non-stationary variable provides an example of an unbalanced regression, and such a regression is difficult to reconcile with the assumptions required for conventional estimation.1 This so-called unbalanced regression has the potential 1Although the issues associated with unbalanced regression can be tackled by incorporating an accommo- dating error structure such as (fractionally) integrated noise, this error structure is not readily compatible with making predictions. 2 to induce substantial small sample estimation bias, and it also renders many of the usual types of statistical inference conducted in standard regression contexts invalid. See Campbell and Yogo (2006), Phillips (2014) and Phillips (2015) for more details. Secondly, the literature does not provide guidance on which predictors are to be included or excluded from predictive regressions. Given a large pool of covariates available for return predictability, selecting an appropriate subset of relevant covariates can improve the precision of prediction. However, the choice of the right predictors is extremely difficult in practice, and precision is often sacrificed for parsimony, especially if there are many predictors but the sample size used for estimation is small. Quite commonly a single predictor, or only a handful of predictors are used. Against this backdrop, our paper studies the use of the highly celebrated LASSO (orig- inally proposed by Tibshirani, 1996) to estimate a predictive regression when some of the predictors are persistent and it is possible that some of these persistent predictors are cointe- grated. We are particularly interested in (i) the LASSO’s capability in model selection in this context; and (ii) the properties of LASSO estimation when potentially cointegrated predictors are involved, especially if such properties can circumvent issues associated with unbalanced regression. As will become clear in our empirical study, there is at least one cointegrating relationship between persistent predictors for stock returns, which lends strong support to an estimation approach that allows for cointegration. In a setting in which the target vari- able is stationary and there are many persistent predictors, the fact that the model selection mechanism of the LASSO eliminates variables with small coefficients, plays a role with re- spect to excluding persistent variables that do not contribute to a cointegrating relationship. This leads to ”balance” between the time series properties of the target variable and a linear combination of the regressors, and tractable prediction errors. It is also possible that the in- corporation of cointegration in this context of predicting returns will improve forecasts, as Christoffersen and Diebold (1998) have noted in other settings. Although predictive regression enables us to analyze the explanatory power of each indi- vidual predictor, our primary objective is to improve the overall prediction of stock returns given a set of predictors. To do this, we focus on the out-of-sample forecasting performance of the set of predictors selected via LASSO estimation. In addition, we investigate the large sample properties of the LASSO estimation of the linear predictive regression. We show the consistency of the LASSO estimator of coefficients on both stationary and nonstationary pre- dictors within the linear predictive regression framework. Furthermore, in the presence of 3 cointegration, we derive the limiting distribution of the cointegrating vector under certain regularity conditions, in contrast with the usual LASSO estimation, in which case the limit- ing distribution of the estimator is not readily available. To the best of our knowledge, this paper is the first work that establishes the limiting distribution for the LASSO estimator of a cointegrating vector. The remainder of this paper is organized as follows. We briefly discuss related literature in Section 2. Section 3 introduces predictive regression models, and Section 4 discusses the LASSO selection of relevant predictors within the model framework. In particular, Section 4.1 demonstrates the consistency of the proposed LASSO estimators, and Section 4.2 derives the limiting distribution of the LASSO estimator of a cointegrating vector. Simulation results are discussed in Section 5. Also, a real application of our methodology based on the Welch and Goyal (2008) data set is given in Section 6. Section 7 concludes. All proofs of theorems and lemmas are found in A. 2 Relevant literature Most of the literature on forecasting stock returns has focused on whether a certain variable has predictive power. The present-value relationship between stock prices and their cash flows suggests a wide array of financial ratios and other financial variables as possible pre- dictors. Documented examples include the earnings-price ratio (Campbell and Shiller, 1988), the yield spread (Fama and Schwert, 1977; Campbell, 1987), the book-to-market ratio (Fama and French, 1993; Pontiff and Schall, 1998), the dividend-payout ratio (Lamont, 1998) and the short interest rate (Rapach et al., 2016). Various macroeconomic variables such as the consumption-wealth ratio (Lettau and Ludvigson, 2001) and the investment rate (Cochrane, 1991) have been considered as well. More recently, Neely et al. (2014) have suggested the use of technical indicators. An issue that arises when returns are conditioned on just one predictor, is that prediction can be biased if other relevant predictors are omitted. This issue might partially explain the failure of single predictor models, although the use of several predictors in a standard regression
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