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Missing Values and the Dimensionality of Expected Returns

Combining many cross-sectional stock return predictors, as in machine learning, often requires imputing missing values. We compare imputation using the expectation-maximization algorithm with simple ad-hoc methods. Surprisingly, expectation-maximization and ad-hoc methods lead to similar results. This similarity happens because predictors are largely independent: Correlations cluster near zero and more than 10 principal components are required to span 50% of total variance. Independence implies observed predictors are uninformative about missing predictors, making ad-hoc methods valid. In an out-of-sample principal components (PC) regression test, 50 PCs are required to capture equal-weighted long-short expected returns (30 PCs value-weighted), regardless of the imputation method.


Introduction
Academic publications have documented more than 200 signals that predict the cross-section of stock returns (Green, Hand and Zhang (2013)). Buried in this literature is the problem of missing values.
Missing values are a relatively minor issue in the original predictability studies. These studies examine at most a handful of predictors, so the standard practice of dropping missing values still leads to a convincing demonstration. This paper shows that missing values are a major problem when combining a large number of cross-sectional predictors, such as in machine learning.
This problem originates from the fact that missingness does not occur in orderly blocks, but is scattered throughout the data in complex ways. As a result, when examining 100 predictors, the vast majority of stocks are missing at least one signal, and dropping stocks with missing values drops at least 90% of the data.
So even though imputing missing values may seem dangerous, when machine learning from 100 predictors or more, the researcher has no choice but to impute.
We find that simply imputing with cross-sectional averages (as in Kozak, Nagel and Santosh (2020); Gu, Kelly and Xiu (2020)) does a surprisingly good job of capturing expected returns. We compare this simple mean imputation with maximum likelihood via the expectation-maximization (EM) algorithm-perhaps the most well-studied method for handling missing values (Efron and Hastie (2016); Little and Rubin (2019)) and the preferred method in psychology (Schafer and Graham (2002); Graham (2009)). Both methods lead to similar mean returns and Sharpe ratios in portfolios sorted on fitted expected returns (a la Lewellen (2015)). This invariance comes from the fact that cross-sectional predictors are largely uncorrelated (Green, Hand and Zhang (2013); McLean and Pontiff (2016) ;Chen and Zimmermann (2022)). Almost all cross-sectional correlations lie between -0.25 and +0.25, and the first 10 principal components span only 50% of total variance. As a result, observed signals provide little information about the missing signals, and one might as well simply impute missing values with constants.
More precisely, our EM imputation boils down to a set of standardized regression coefficients, and these coefficients imply that a one standard deviation change in an observed signal implies less than a 0.1 standard deviation change in the missing signal in the vast majority of cases.
High dimensionality is seen not only in the predictive characteristics, but in their relationship with future returns. We demonstrate this high dimensionality in out-of-sample principal components regression tests. Each month, we sort stocks into long-short portfolios based on a regression of future returns on the principal components (PCs) of 125 predictors. In equal-weighted portfolios, these tests require 50 PCs to capture the potential expected returns: using 50 PCs leads to mean returns of 40% per year, while using "only" 20 PCs yields 25% per year. Value-weighted portfolios lead to qualitatively similar results, though the dimensionality and expected returns are both roughly halved: about 30 PCs are required to capture the 20% annualized mean return found in value-weighted portfolios between 1995 and 2020.
This high dimensionality depends little on whether we use simple mean imputation or the EM algorithm. EM significantly boosts performance if 20 or fewer PCs are used. But if more than 40 PCs are used, both methods lead to similar expected returns and Sharpe ratios. Given that machine learning algorithms generally take advantage of all dimensions of predictability, these results suggest that it matters little whether one uses simple mean imputation or a more sophisticated method.
We also provide a detailed description of the origins of missing values in the Chen and Zimmermann (2022) dataset. The most common driver of missingness is the requirement of having many years of historical data on the stock. Longrun reversal is observed for 62% of stocks, and 5-year sales growth is observed for 40%. These observation rates contrast with size and B/M, which only require recent data, and are observed for about 98% and 70% of stocks, respectively. Using more specialized data can drop the observed share below 40% (e.g. R&D data or analyst forecasts), but low observed shares are also driven by a variety of specialized filters, such as dropping stocks with no payouts or stocks in particular size quantiles. Thus, missingness is often driven by the desire to simplify analyses and avoid the modeling of interactions and non-monotonicity. Modeling these effects would require digging into the code for the hundreds of individual predictors, an issue we leave for future research.
In addition to our empirical results, we review the theory around EM imputation as it relates to cross-sectional asset pricing, including the confusing "missing at random" (MAR) terminology. Though Rubin (1976) described MAR as "rather intuitive," statisticians are still trying to clarify misinterpretations 50 years later (Little (2021)). We explain how the best motivation for EM imputation is that it is maximum likelihood assuming "ignorability" (Rubin (1976)) of the missing data mechanism. Assuming ignorability is arguably necessary for large scale crosssectional predictor data, as the missingness patterns we document are likely too complex to be modeled convincingly.
We post all of our code at https://github.com/jack-mccoy/missing_data. The EM imputed data can be found at https://sites.google.com/site/chenandrewy/.

Related Literature
In contemporaneous work, Freyberger, Höppner, Neuhierl and Weber (2021) (FHNW), Bryzgalova, Lerner, Lettau and Pelger (2022), and Beckmeyer and Wiedemann (2022) also study missing values in large-scale crosssectional predictor data. Each paper focuses on a different imputation algorithm, though all avoid modeling the missingness process and effectively assume Rubin ignorability. FHNW use moment conditions designed for cross-sectional predictors, Bryzgalova et al. (2022) use Xiong and Pelger (Forthcoming)'s latent factor structure framework, and Beckmeyer and Wiedemann use a masked language model. Our paper uses the EM algorithm, which is an important benchmark given its long history in the study of missing values (Hartley (1958);Orchard and Woodbury (1972); Rubin (1976) ;Dempster, Laird and Rubin (1977), etc). We also differ by detailing the origins of missingness in predictor data.
But perhaps most important, we are distinct in providing intuition for why simple mean imputation seems to perform well in machine learning style applications. The strong performance of mean imputation is seen in both our PCR portfolios and in FHNW's regularized non-linear strategies. In contrast, Bryzgalova et al. (2022) and Beckmeyer and Wiedemann (2022) do not examine whether their imputation methods can lead to higher Sharpe ratios when combining many predictors.
Our results provide a formal foundation for the ubiquitous use of mean imputation in machine learning papers. Almost every paper that attempts to combine predictors imputes with cross-sectional means or medians, including Gu, Kelly and Xiu (2020) ;Freyberger, Neuhierl and Weber (2020); Kozak, Nagel and Santosh (2020). 1 This result is important because the algorithms in these papers are often complex, so a simple imputation algorithm is critical for transparency.
We add to the various papers that find high dimensionality in the cross-1 Other machine learning style papers that examine imputing with cross-sectional averages section of expected returns. Several studies find that correlations cluster around zero (Green, Hand and Zhang (2013); McLean and Pontiff (2016); Chen and Zimmermann (2022)). These low correlations show up as many dimensions of expected returns using Fama-Macbeth regressions (Green, Hand, and Zhang 2014; or PCA on single predictor long-short returns (Bessembinder, Burt and Hrdlicka (2021)). A limitation of these studies is that they either impute missing values with cross-sectional means or drop missing values, methods that are generally not recommended by the statistics literature (e.g. Little (1992)). We verify that using recommended missing data handling has little effect, providing additional support to the surprising finding that that there are dozens of dimensions of predictability in the cross-section of returns.
These findings complement Lopez-Lira and Roussanov (2020), who document large Sharpe ratios in strategies that hedge the principal components of individual stock returns. Taken together, these studies suggest that cross-sectional predictability typically does not arise from exposure to strong common factors, but from exposure to many distinct sources of information. This "zoo" of information takes time to be impounded into stock prices (Engelberg, McLean and Pontiff (2018); Bowles, Reed, Ringgenberg and Thornock (2020)), leading to a zoo of stock return anomalies.

Missing Values in Cross-Sectional Predictors
We describe the magnitude and origins of missingness among published cross-sectional predictors. We also explain why imputation is required for large scale machine learning studies.

Cross-Sectional Predictors Data
Our predictors data begins with the March 2022 release of the CZ dataset.
This dataset contains 207 cross-sectional predictors published in academic journals.
We drop the 35 discrete predictors because the missing values in this subset are closely related to the interpretation of these predictors by CZ. Many of these predictors draw on event studies (e.g. exchange switch), which CZ interpret as a binary firm characteristic (following McLean and Pontiff (2016)). Others are based on double-sorts (e.g. analyst recommendation and short interest). In both cases, CZ use missing values for any stocks that belong in neither the long or short portfolio, though there are other ways to interpret these studies.
We drop an additional 15 predictors because they have months with fewer than 2 observations in the 1985-2022 sample. This screen ensures the predictor data is relatively well-behaved and avoids the use of special techniques to handle predictor-months with no data.
This section describes missingness for the remaining 157 predictors. When we examine EM imputation (Sections 3 and 4), we focus on the most-observed 125 predictors to keep the algorithm tractable. Table 1 lists the 157 predictors, ranked by the share of stocks with observed signals in 1985, a year that is fairly representative of the full sample. We explain why these predictors have their corresponding missingness in what follows.

Where Does Missingness Come From?
[ Table 1 about here] The most-observed predictors are those that use only the previous month of CRSP data. These predictors include short-term reversal (streversal), idiosyncratic volatility (idiorisk), and size, and are observed for at least 98% of stocks.
Requiring a longer history of CRSP data leads to a non-trivial share of missing values. 12-month momentum is observed for 85% of stocks while medium-run reversal, which uses the past 18 months of CRSP data, is observed for 80%. A similar amount of CRSP data is required for coskewness, beta, and two zero trade predictors (zerotrade and zerotradealt12).
Requiring accounting variables drops the observed share into the 60% to 75% range. Predictors that fall in this range include the popular B/M, profitability (roe, gp), and investment (assetgrowth) predictors. Many other accountingbased predictors fall in this range, including employee growth (hire), leverage, taxes, external financing (xfin), and accruals.
Rates of observability below 50% are driven by a wide variety of issues. Most of these predictors require a long history of accounting or CRSP data. The intangible value predictors (intancfp, intanep, intansp) require 5 years of past cash flow, earnings, and sales, respectively, and are thus only observed for about 40% of stocks. Long histories of accounting data are also required for past 5years sales growth (meanrankrevgrowth), Titman, Wei and Xie (2004)'s investment predictor, and composite debt issuance. Return seasonality in years 11-15 (momseason11yrplus, momoffseasonoff11yrplus) requires 15 years of past returns, leading to a similar observed share. But many of the less-observable predictors are missing values simply because the data items they require are often missing. Predictors that used analyst forecast data are observed for at most 44% of stocks in 1985, as the forecasted EPS (feps) is only observed for this share. Other analyst-based predictors have fewer observations, including analyst revision, analyst value, and earnings streak. Data items are also often missing for less-common accounting variables, like excluded expenses (exclexp), advertising expenditure (adexp), and inventory growth (invgrowth).
Still others are missing values due to a diverse range of specialized filters.
Asset tangibility (tang) applies only to manufacturing firms. Payout yield (payoutyield) drops stocks with non-positive payout. Organizational capital keeps only stocks with fiscal-year ends in December. In CZ's interpretation, turnover volatility (std_turn) drops stocks in the top two size quintiles, and operating profitability (operprof ) drops stocks in the bottom size tercile. The original Piotroski F-score (PS) study only examines stocks in the highest BM quintile.
As seen in these details, missingness is sometimes used as a substitute for modeling non-monotonicity and interaction effects. Modeling these interactions is highly non-trivial. It requires going into the code behind the individual predictors, commenting out the missingness assignments, and carefully introducing new interaction variables if necessary. We leave this work for future research.
At the very bottom of  [ Table 2 about here] Though many cross-sectional predictability studies begin in 1963, at least 5% of predictors have no observations until 1980. Several of these predictors draw on analyst forecast data, which is not generally observed until 1985 (using IBES).

Summarizing Missingness for Individual Predictors
Missingness reaches a steady state in 1985, as the order statistics are largely unchanged after this date. In this steady state, the typical predictor is observed for 60% of stocks, and 75% of predictors are observed for at least 40% of stocks.
While this distribution suggests that missing values are a relatively modest problem, we will see that combining large sets of predictors compounds the issue.

Summarizing Missingness for Sets of Predictors
When combining predictors, one typically needs to have a numerical value for an entire set of predictors in each stock-month. This requirement compounds the magnitudes in Panel (a) of Table 2, leading to a dramatically larger missingness problem.
This problem is illustrated in Figure 1, which shows the "missingness map" for the most observed 125 predictors in June 1985. Roughly half of the map is non-shaded, indicating that roughly half of the data is observed. This does not mean, however, that half of the stocks can be used in a machine learning algorithm without imputation. Only stocks with no missing values can be used, corresponding to columns in the missingness map that are entirely non-shaded.
Almost no columns satisfy this requirement, and thus one must impute data in order to apply machine learning in a meaningful way.
[ Figure 1 about here] Table 2  Returning to Figure 1, the complex missingness patterns illustrate why an algorithm like EM is required. There are dozens of distinct shapes in Figure 1, suggesting that dozens of equations are required to model the missingness mechanism. It is unlikely that an analyst could convincingly present such a complicated model. The only alternative is to assume Rubin ignorability, as in EM imputation.

EM Imputation
The missingness patterns imply that one must impute data when combining more than 100 predictors. These patterns are complex and require a general algorithm that handles arbitrary patterns. EM is one such general algorithm.
In short, our EM imputation assumes that each cross-section of signals is drawn from a multivariate normal distribution, and then uses maximum likelihood to estimate the expected value of the missing signals. This section details our implementation of EM (Section 3.1-3.3) and clarifies common misconceptions about the "missing at random" assumption (Section 3.4).

Missing Data Imputation Under "Ignorability"
At its core, EM imputation assumes that that the missing value mechanism is "ignorable" in the sense of Rubin (1976). To formalize this idea, suppose that in month t , stock i has a vector of signals X i ,t drawn from a distribution where x i ,t is a vector of signal realizations and θ t is a vector of parameters. This form means that each month of data is modeled separately. It also means that a stock's observed signals are potentially informative about its unobserved signals (high B/M could mean high tail risk) and is similar to the third moment condition in Freyberger et al. (2021). For ease of exposition, we assume independence across i , but this assumption can likely be weakened to allow for weak dependence as in Wooldridge (1994).
EM allows for general missingness patterns, which results in somewhat unusual notation. Let x obs i ,t be a vector that replaces missing values in x i ,t with a non-numeric value * . Similarly, x mi ss Define the random vector counterparts X mi ss i ,t and X obs i ,t similarly. This notation allows us to write down a quasi-likelihood function given only the observed data where d x mi ss i ,t indicates integrating over the support for each of stock i 's missing predictors. The fundamental assumption of EM imputation is that maximizing Equation (2) provides valid inference about θ t .
The subscript "ign" refers to Rubin (1976) ignorability. Equation (2) is not the full likelihood function, as it ignores the process that determines whether, for a particular stock i , a particular element of X i ,t is missing. As shown in Section 2.2, this process is driven by several distinct mechanisms and the resulting missingness patterns are complex. As such, ignorability is likely a necessary assumption.
This optimization can be achieved with the EM algorithm of Dempster, Laird and Rubin (1977) (see also Hartley (1958); Orchard and Woodbury (1972)). EM iterates between the "expectation" step, which computes a kind of expected quasilikelihood assuming some parameter values, and the "maximization" step, which chooses new parameter values by maximizing the expected quasi-likelihood from the expectation step. We provide further details on the EM algorithm in Appendix A.2.
Givenθ t that maximizes Equation (2), we impute missing values with their expected values conditional on observed signalŝ Since the EM algorithm outputs Equation (3) as a byproduct of the expectation step, it serves as a natural method for imputation.

Multivariate Normal Implementation
The EM algorithm can be quite difficult to implement. In principle, each expectation step requires an extremely large number of high-dimensional integrals, and each maximization step requires solving a global optimization problem.
To keep the problem tractable, we assume that signals are multivariate normal after a transformation. That is, we transform each predictor by winsorizing symmetrically at the 1% level and then apply Hawkins and Weisberg (2017)'s extension of the Box and Cox (1964) method for making each predictor approximately normal. Then we standardize predictors to have zero mean and unit variance. Finally, we assume where MVN (0, Σ t ) is a multivariate normal distribution with mean zero and covariance matrix Σ t and x i ,t represents the transformed signals.
While the MVN assumption may appear restrictive, it can be thought of as an approximation of a more general distribution, and simulation studies often find that the MVN assumption works well even if the underlying data is not normal (Little (1988); Azen et al. (1989); Graham and Schafer (1999)). Little (1988) shows that allowing for fat tails leads to the same algorithm but with smaller weights on the extreme observations. As we winsorize our data before imputation, such a fat tail extension should have little effect on our results.
The Box-Cox transformation is not commonly found in asset pricing, but it is just an extension of the log transformation that is commonly applied to size and B/M (Fama and French (1992)). More broadly, cross-sectional predictability studies rarely impose a functional form on the relationship between the predictor signals and expected returns, so one might as well transform the signals to be approximately normal.
Importantly, our implemention of EM avoids look-ahead bias as each month of data is handled independently. Further details on the implementation are found in Appendix A.1 and A.2.

Predictor Selection for EM
To make the algorithm tractable, we limit our EM analysis to the 125 mostobserved predictors in 1985. (Table 1).
This restriction is important because, even with the MVN assumption, the EM algorithm is still computationally intensive due to complex missingness patterns. While the algorithm boils down to a sequence of matrix operations, every missingness pattern seen in Figure 1 implies a different matrix inversion, and every matrix inversion must be done in every iteration of the algorithm. As a result, it can take up to four hours to impute a single month of data for the 125 mostobserved predictors. This algorithm must then be repeated for every month of the data. We make this computation feasible by imputing all 420 months of data in parallel on a cluster. Table 1 shows that the 125 most-observed predictors span most economic and data categories. Importantly, our list uses data from not only CRSP and Compustat, but also IBES. Including IBES data is important, as it may offer a distinct dimension of predictability not seen in CRSP or Compustat. The predictors omitted from our list are largely those that only apply in specialized subsets of firms, like the Piotroski F-score (see Section 2.2).

Discussion of "Missing at Random" and Related Terms
Section 3.1 does not mention the terms "missing at random" (MAR), "missing completely at random" or "missing not at random." We avoid these terms because they easily cause confusion and are not necessary. Nevertheless, since MAR and related terms commonly arise in missing data analysis (Little and Rubin (2019); Bryzgalova et al. (2022)), we briefly describe their meanings here.
To define MAR, let M I SS i ,t be a random vector indicating the missing values for stock i in month t and mi ss i ,t its realization. Additionally, let φ t be a parameter vector that governs the missing value process, and p M I SS i ,t |X t (·) be the probability of missingness given information on all realized signals for stock i .
An intuitive interpretation of MAR is that missingness is purely random: is a potential realized values of the missing signals and π is a constant vector. Indeed, Equation (5) is how Xiong and Pelger (Forthcoming) (and, it seems, Bryzgalova et al. (2022)) define MAR.
To avoid confusion, we note that much of the missing data literature does not call Equation (5) MAR. Graham (2009), Little (2021, and others describe Equation (5) as "missing completely at random." These papers follow Rubin (1976) and define MAR as for all x mi ss i ,t ,x mi ss i ,t , and φ t , is a realization of the missing signals. As seen in Equation (6), Rubin's MAR does not mean that missingness is random in the sense that it is determined by coin flips. "Missing not at random" is, then, when Equation (6) is not satisfied.
What does Rubin's MAR mean? Perhaps the best description is that it is "the weakest general condition under which pure likelihood inference can be based on [Equation (2)] without needing a model for the missingness mechanism" (Little (2021)). This result is important because Equation (6) is a much weaker assumption than Equation (5). Thus, the EM algorithm does not require that missingness is completely random.
Unfortunately, Equation (6) (2021)). Equation (6) is, in general, untestable (Schafer and Graham (2002)). As such, we describe MAR and related terms merely to avoid misconceptions. To us, the fundamental assumption is simply ignorability (Equation (2)), which is likely necessary given the complexity of the missingness in cross-sectional predictors (Figure 1).

Predictability Results
Having described the need for EM imputation, we are finally in a place to present our main results. This section compares the performance of EM imputation and simple ad-hoc alternatives for predicting long-short returns.
We begin with a warm-up exercise that studies portfolios sorted on individual predictors (Section 4.1). Section 4.2 presents the main results, which examines long-short portfolios that combine information from 125 predictors.

Single-Predictor Strategies
We start with single-predictor strategies because they are so familiar to asset pricing researchers. It is unlikely that missing data handling has a large effect here, but these results are helpful "descriptive" statistics before the main results.
Our single-predictor strategies go long the 500 stocks with the strongest signal and short the 500 stocks with the weakest signal each month. We do this for two missing data handling methods: (1) EM imputation and (2)  This design allows us to examine the most common method for handling missing values (dropping missing) while simultaneously removing "signal dilution" effects that can occur when using imputed data. Intuitively, the imputed data covers more stocks, and if these additional stocks have more neutral signals, the extreme deciles using imputed data will display less predictability simply because of this additional coverage. Fixing the number of stocks in each leg controls for this issue. For simplicity, we drop predictor-months that have less than 1000 stock-predictor observations. This removes 1% of predictor-months. Table 3 shows the resulting distribution of mean returns and Sharpe ratios.
EM generally improves performance, on average. However, the effects of EM imputation are generally small. In equal-weighted portfolios, EM increases the median annualized return from 5.9% to 7.0% and the median annualized Sharpe Ratio from 0.53 to 0.56. In value-weighted portfolios, the median performance is almost unchanged. EM increases somewhat the dispersion of performance measures, as seen in the slightly larger right and left tails.
[ Table 3 about here] The "EM Improvement" row computes the difference between EM and miss-ing dropped performance at the predictor level. The typical difference is tiny, at around 0.1 percentage point per year for expected returns. The largest difference is the 90th percentile for EM improvement for equal-weighted long short returns, and this difference is a rather moderate 3.6 percentage points per year.

Strategies that Combine 125 Predictors
Here we examine the most compelling setting for studying missing values: strategies that combine information from many predictors. We combine information using principal components regression (PCR), a a simple method for handling correlated predictors and potential overfitting problems. PCR also provides a natural connection with the dimensionality of expected returns.
Our PCR analysis forms "real-time" long-short strategies as follows: For each month t between January 1995 and December 2020, we 1. Find the principal components (PCs) of the predictors using data from the past 120 months.

2.
Regress stock-level returns on lagged first K PCs, either equally weighting stocks or weighting stocks by market value. We examine K ranging from 1 to 75.
3. Go long the top decile of predicted returns and short the bottom decile, equal-or value-weighting to match the choice in step 2. 4. Hold the position for one month.
We apply this procedure using three missing data handling procedures (1) EM imputation, (2) simple imputation with cross-sectional means, and (3) "available case" imputation. As seen in Section 2.4, some form of imputation is required when combining 125 predictors.
Available case imputation is the same as EM imputation (Sections 3.1-3.2), but it computes each entry in the covariance matrix by using the sample covariance from all available predictor-stock observations for corresponding entry (called "pairwise.complete.obs" in R). This alternative method is much more tractable than the EM imputation, which can take weeks to compute without a large cluster. Available case covariances are also examined in Bryzgalova et al. (2022), though they focus on the largest PCs among a set of 45 characteristics. Figure 2 shows the result. Panel (a) shows the mean return as a function K , the number of PCs in the forecasting regression. It shows that the dimensionality of expected returns depends strongly on whether one considers small-cap stocks or focuses on large-caps. Equal-weighting (red) implies a very high dimensionality while value-weighting (blue) implies a relatively lower dimensionality. Nevertheless, all results point to far more dimensions of predictability than is commonly believed among asset pricing researchers.
[ Figure 2 about here] Equal-weighting results in an extremely high 50 dimensions of predictability. That is, adding more PCs to the expected return regression steadily increases mean returns and Sharpe ratios until one uses 50 PCs. These increases in returns are large. Using 5 PCs leads to a 10%-15% annualized mean return, 20 PCs leads to about 22%, but using 50 PCs can lead to 40%-roughly 3 times as much profitability as using 5 PCs. A similar tripling of performance is seen in annualized Sharpe ratios, which increase from around 0.7 to 2.5.
Value-weighting results in roughly 30 dimensions of predictability. While 30 is far smaller than 50 dimensions seen in equal-weighted strategies, it is still 6 times larger than the 5 factors in Fama and French (2015). Available case imputation has mixed results. Under equal-weighting, available case generally produces lower mean returns than simple mean imputation, especially when more than 20 PCs are used. This underperformance is likely because of estimation noise. Available case computes covariances with overlapping observations, and for some predictor pairs the overlap is small, leading to noisy estimates. Noise shows up in higher order PCs and obscures the signal from the better-observed predictor pairs. These results are robust to factor model adjustments. Figure 3 adjusts the long-short mean returns for exposure to the market excess return (CAPM, Panel (a)) and the Fama and French (2015) 5-factor model + momentum (FF6, Panel (b)). These adjustments affect levels. The CAPM alphas are generally larger than the raw mean returns, while the FF6 alphas are generally smaller. But the key patterns are similar: dozens of PCs are required to capture the potential alphas, and simple mean imputation performs surprisingly well.
[ Figure 3 about here] This high dimensionality is consistent with previous papers. Allowing for equal-weighting, Green, Hand and Zhang (2014) and Bessembinder, Burt and Hrdlicka (2021) find that there are dozens of independent contributors to expected returns. In contrast, Green, Hand and Zhang (2017) and Lettau and Pelger (2020) find a much smaller dimensionality in value-weighted analyses. Our results generally find higher dimensionality than any of these studies, perhaps because of our use of firm-level predictors and the CZ dataset. Firm-level data likely provides more information about expected returns, and the CZ dataset is much larger than the Green, Hand and Zhang (2017) and Kozak, Nagel and Santosh (2020) datasets studied in Green, Hand and Zhang (2017) and Lettau and Pelger (2020), respectively.
Our contribution to these papers is to show that high dimensionality is littleaffected by missing value handling. This invariance is important because high dimensionality is strongly counterintuitive. Asset pricing theory implies that a single SDF prices all assets, which suggests that the zoo of predictors originate from a single latent factor. The simplest way to square this idea with the aforementioned papers is to point out a statistical error. Mishandling of missing values is one potential error, as Green, Hand, and Zhang (2014; impute with cross-sectional means, and Bessembinder, Burt and Hrdlicka (2021) build on CZ's strategies, which drop missing values. Neither of these methods are recommended by the statistics or applied literature (Little (1992); Graham (2009); Little and Rubin (2019)). Our analysis shows that state-of-the-art methods have little effect on these conclusions, though it is difficult to know this result a priori.

Imputation and Dimensionality
The predictability results in Figure 2 may be counterintuitive from the perspective of asset pricing theory. This section provides a statistical intuition that links imputation with dimensionality.

Theory and Simulation Demonstration
Our multivariate normal assumption (Equation (4)) implies that the imputation (Equation (3)) boils down to a regression equation where x mi ss i , j ,t is a component of the vector x mi ss i ,t and the imputation slope vector whereΣ obs|i ,obs|i ,t is the submatrix ofΣ t corresponding to the observed predictors for stock i , andΣ j ,obs|i ,t is the subvector ofΣ t corresponding to the covariance between predictor j and the observed predictors.
In essence, our imputation forecasts missing values (LHS of Equation (7)) based on their covariance with observed values (RHS of Equation (8)), just as in OLS. If there was no missing data, Equation (8) (2022)). We then conduct a PCA variance decomposition and evaluate the imputation expression (Equation (7)) assuming that a random 1/3 of predictor pairs are missing. For simplicity, we assume that the actual covariance matrix is known. Results average over 20 simulations. [ Figure 4 about here] Consistent with intuition, low dimensionality implies that a careful imputation is quite different than just imputing with observed means. This difference is seen in the right panel, which plots the distribution of imputation slopes. More precisely, we stack all the imputation slope vectors β i , j ,t i =1,...,N t ; j =1,...,J into one large vector and plot the distribution. This distribution shows that many slopes are meaningfully different than zero. Indeed, the mean absolute slope is 1.7, indicating that EM imputation would be very different than simple mean imputation.
In contrast, Panel (c) describes a high-dimensional setting. Here, the partial correlations are given by a beta distribution with shape parameters both equal to 15. This setting implies that almost all full correlations are less than 0.5 in absolute value, and thus even 10 PCs span less than 50% of total variance. Consistent with intuition, the imputation slopes cluster close to zero, with an average absolute value of 0.14. In this case, EM imputation would be very similar to simple mean imputation.

Panel (b) shows an intermediate setting.
Here, the beta shape parameters are equal to 4, and the resulting charts show dimensionality and imputation slopes in between those of Panels (a) and (c). Thus, the importance of careful imputation is an issue of magnitudes, with higher dimensionality implying less importance, in a generally smooth manner.

Empirical Dimensionality and Imputation Slopes
Figure 5 examines dimensionality in empirical predictor data from several perspectives. The "Observed" lines illustrate dimensionality without carefully accounting for missing data. These lines compute the correlation between two predictors using all pairs of stocks that have data in the given month (sometimes called "available case" or "pairwise complete" correlations). The left panels show that these observed correlations cluster close to zero, consistent with Chen and Zimmermann (2022). This distribution suggests a very large number of PCs are required to describe the data.
[ Figure 5 about here] The right panels verify this intuition. Based on observed correlations, more than 9 PCs are required to capture just half of total variance. This suggests that the high-dimensionality setting in Figure 4 is the empirically relevant setting, though the available case analysis is not generally a recommended solution from the older missing data literature (Little (1992)).
The "EM Algo" lines in Figure 5 examine a recommended solution: correlations estimated from the EM algorithm. Despite the additional rigor, Figure 5 shows that both methods lead to nearly identical distributions. The similarity is reminiscent of Xiong and Pelger (Forthcoming), who find that available case covariance matricies are consistent estimators under some assumptions on the missing data process. This result also makes sense given that the most significant problems with available case analysis seem to occur when the data are highly correlated (Haitovsky (1968)). We should note that similar covariance distributions does not mean similar covariances, and as we saw in Section 4.2 EM and available case do lead to different inferences about long-short strategy performance.
The panels in Figure 5 show these patterns are very stable over time. Regardless of whether one computes the correlations in June 1990, June 2000, or June 2020, almost all correlations like between -0.25 and +0.25, 10 PCs span only 50% of total variance, and both the EM algorithm and observed correlations lead to nearly identical distributions and PCA decompositions.
Though the overall distributions are nearly identical, the EM algorithm does have an effect on correlations at the predictor-pair level. These effects are seen in Figure 6, which plots the distribution of difference between EM correlations and observed correlations for each predictor pair. In other words, while Figure  [ Figure 6 about here] Panel (a) of Figure 6 shows that the EM and observed correlations differ on average by about 0.05 in absolute value. This difference is tiny in absolute terms, but given that correlations are typically close to zero the difference is non-trivial, relatively speaking. This alternative perspective is seen in Panel (b), which shows the substantial changes in correlations scaled by the absolute value of the observed correlation. The meaningfulness of the changes in correlations is also seen in Figure 2, which shows that EM generally outperforms available case for forming long-short strategies.
Given this high dimensionality, the simulations in Section 5.1 imply that imputation slopes will be close to zero. That is, our EM imputation boils down to a simple regression equation (Equation (7)), and the slopes in this equation lie close to zero if the dimensionality of the data is high (Figure 4).
This idea is confirmed in Figure 7, which shows the distribution of imputation slopes in selected months between 1990 and 2010. The slopes are computed by plugging in the EM-estimated covariance matrix into Equation (8), and then stacking all of the slope vectors into one long vector before plotting the distribution.
[ Figure 7 about here] Figure 7 shows that almost all slopes lie between -0.2 and +0.2, regardless of the month one examines. Indeed the mean absolute slope is just 0.065. As our predictor data is standardized to have unit variance, these coefficients imply that a one standard deviation increase in an observed signal implies a 6.5% change to the missing signal-almost no change at all. As a result, simple mean imputation and EM imputation will lead to similar results for examining expected returns.

Conclusion
In the age of machine learning, missing values can no longer be ignored.
Classical studies of missing values find that there is no general solution. Every missing value problem seems to present a new challenge, with the ideal solution tailored to the nature of the missing value problem.
We argue that likelihood-based inference assuming Rubin (1976) ignorability is the ideal missing data method for cross-sectional stock return predictors. The missingness patterns in these data are highly complex, originating from a wide variety of sources. As a result, one has little choice but to ignore the missing data mechanism.
It turns out that likelihood-based inference leads to similar inferences regarding expected returns as simpler ad-hoc methods like imputing with crosssectional means. This invariance happens because cross-sectional predictors are largely independent. As a result, observed signals are largely uninformative about missing signals.
More broadly, our paper adds to the evidence that the cross-section of expected stock returns is surprisingly high-dimensional Zhang (2014, 2017); Chen and Zimmermann (2022); Bessembinder, Burt and Hrdlicka (2021)). An important outstanding issue in this literature is missing values, which could potentially bias these papers toward finding higher dimensionality than is really there. We find that this literature is robust to this concern and provide further support for the need for asset pricing theories that accommodate dozens of dimensions of predictability.

A.1 Extended Box-Cox Transformation
For each month-predictor combination, we symmetrically winsorize 1% of observations and then transform the winsorized data using Hawkins and Weisberg (2017)'s extension of the Box and Cox (1964) method. As we apply this transformation within each month-predictor separately, this avoids look-ahead bias in our portfolio tests (Sections 4.1 and 4.2).
Box and Cox assume the data is normally distributed after a power transform where λ is a parameter. The common practice of taking logs to reduce the skewness of predictor signals (e.g. size and B/M in Fama and French (1992)) is a special case of this transformation (λ=0). Box and Cox recommend estimating λ by maximum likelihood.
A limitation of Box and Cox is that it requires that the untransformed data is always positive. Hawkins and Weisberg (2017) apply the transformation g HW y|γ = 0.5 y + y 2 + γ 2 and then assume g BC g HW x|γ |λ is normally distributed. Maximizing the likelihood then gives the estimated parameter values γ and λ as well as the transformed data.
We choose the Hawkins and Weisberg transformation because it is implemented in the popular "Companion to Applied Regression" R package (Fox and Weisberg (2018)).

A.2 EM Algorithm Details
As the ignorable quasi-likelihood (Equation (4)) is in the exponential family, the estimation step (E-step) boils down to finding the conditional sufficient statistics (conditional first and second moments) and the maximization step (Mstep) just sets the new parameters to match the conditional moments from the E-step. In iteration k, these two steps are as follows: 1. E-step: Given parameter guess Σ t , j , find the conditional moments where these expectations can be found using standard MVN equations.
2. M-step: Given S j set the next iteration's parameters to match moments The algorithm ends when the maximum norm Σ t ,k − Σ t ,k+1 ∞ ≤ 10 −4 .
Dempster, Laird and Rubin (1977) show that L i g n Σ t ,k+1 | x obs 1,t , ..., x obs N t ,t ≥ L i g n Σ t ,k | x obs 1,t , ..., x obs N t ,t , so every iteration improves the likelihood. Sometimes the EM algorithm is described as repeatedly updating missing values with their conditional expectation (e.g. Efron and Hastie (2016)). While this description is precise for the first moment due to the linearity of and for the cross terms the variance terms acquire a second order uncertainty adjustment where X mi ss|i ,t is the subvector of X i ,t that is missing, and Σ mi ss|i ,obs|i ,t and Σ mi ss|i ,mi ss|i ,t are defined similarly. The second and third terms in Equation (9) account for the uncertainty inx mi ss|i ,t .

Figure 2: Missing Data Effects on Strategies that Combine 125 Predictors
Each month, we forecast next month's returns using principal components regression on the past 10 years of data. Strategies go long stocks in the top decile of forecasted returns and short the bottom decile. 'EM' imputes missing values with the EM algorithm. 'Simple mean' imputes with cross-sectional means. 'Available Case' uses Equation (4) but uses sample covariances from all available pairs of predictor-stocks with observed data. Predictor data runs 1985-2020. Real-time returns use data from 1995-2020. Interpretation: There are many dimensions of predictability, particularly for small stocks. Simple mean imputation works surprisingly well, especially if a large number of PCs are used.

Figure 3: Alphas on Strategies that Combine 125 Predictors
Each month, we form long-short portfolios based on principal components regression using the past 10 years of data (See Figure 2). Panel (a) shows CAPM alphas and Panel (b) shows Fama and French (2015) 5 factors + momentum alphas. 'EM' imputes missing values with the EM algorithm. 'Simple mean' imputes with cross-sectional means. 'Available Case' uses Equation (4) but uses sample covariances from all available pairs of predictor-stocks with observed data. Real-time returns use data from 1995-2020. Interpretation: Patterns seen in raw expected returns (Figure 2) are also found after factor model adjustments.

Figure 4: Dimensionality and Imputation Simulations
We simulate correlation matricies, compute eigenvalue decompositions, and imputation slopes. Imputation slopes are the response of an imputed missing value to a one standard deviation change in an observed predictor (Equation (7)). Slopes are the elements of the vectorsβ ′ i , j ,t computed with Equation (8), reprinted here:β ′ i , j ,t =Σ j ,obs|i ,tΣ −1 obs|i ,obs|i ,t .
Simulations assume symmetric-beta-distributed partial correlations following Lewandowski et al. (2009). Panel (a) uses a beta parameter of 1.2, Panel (b) uses 4, and Panel (c) uses 15. Interpretation: High dimensionality implies that slopes are close to zero, in which case simple mean imputation is similar to EM imputation. All signals are transformed using an extended Box-Cox algorithm to have approximately normal marginal distributions at the predictor level (Section 3.2). 'Observed' uses all available cases at the predictor-pair level (a.k.a. pairwisecomplete). 'EM Algo' uses the EM-estimated covariance matrix (Σ t from Equation (4)). Interpretation: The correlations cluster near zero, implying that Panel (c) of Figure 4 is the relevant setting, and that simple mean imputation works fairly well.

Figure 6: A Closer Look at EM vs Observed Correlations
Data is from June 2000. We compute pairwise correlations using the EM algorithm and observed available cases, and then take differences (Panel (a)). Panel (b) normalizes the differences by the absolute value of the observed available case correlation. Interpretation: Though the distribution of correlations is visually identical using EM or observed available case (Figure 5), at the predictorpair level the correlations differ, and with the median difference about 30% of the available case correlation. Imputation slopes are the response of an imputed missing value to a one standard deviation change in an observed predctor (Equation (7)). Slopes are the elements of the vectorβ ′ i , j ,t from Equation (8), reprinted here: obs|i ,obs|i ,t .
Interpretation: Consistent with the intuition in Section 5.1, the high dimensionality of signals implies imputation slopes are close to zero, and that simple mean and EM imputation are similar.