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1. • Description: Tobin’s Q—see Freyberger et al. (2020) for more details. • 𝑥𝑑 𝑖,𝑘,𝑡 : total assets (AT) divided by shares outstanding (SHROUT). • 𝑥𝑎 𝑖,𝑘,𝑡 : total assets (AT) minus cash and short-term investments (CEQ), minus deferred taxes (TXDB) scaled by total assets (AT). rel_to_high_price • Description: the price to 52 week high stock price ratio. • 𝑥𝑑 𝑖,𝑘,𝑡 : 52 week high stock price. • 𝑥𝑎 𝑖,𝑘,𝑡 = 0. roc • Description: economic rents over cash ratio. • 𝑥𝑑 𝑖,𝑘,𝑡 : ratio of Short-Term Investments (CHE) to shares outstanding (SHROUT). • 𝑥𝑎 𝑖,𝑘,𝑡 = 0: ratio of long-term debt (DLTT) minus total assets to Cash and Short-Term Investments (CHE).
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12. Choosing the number of nonzero weights effectively chooses the 𝜆1 parameter. I follow Bryzgalova et al. (2020) by using LARS regression with the number of nonzero values of 𝑏 as a hyperparameter.
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17. 𝐹𝑡−1. (A.1) A.1.2 Kozak et al. (2020) - KNS The objective of Kozak et al. (2020) is to create an SDF. The KNS method first performs a standard principle components analysis (PCA) dimension reduction on the factor 𝐹𝑡−1 generated by the linear function 𝑓 (𝑍𝑡) = ˘ 𝑍𝑡. The number of principle components to use is a hyperparameter of the model. This produces a smaller set of factors, which are then reduced to a single portfolio with the vector 𝑏.
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35. Huberman, Gur, and Shmuel Kandel, 1987, Mean-variance spanning, The Journal of Finance 42, 873 – 888.

36. I employ an alternative method based on Bryzgalova et al. (2020). I stack 𝑋𝜏 =             𝑍0 𝑍1 . . . 𝑍𝜏−1             , 𝑌𝜏 =             𝑟1 𝑟2 . . . 𝑟𝜏             (A.9) and fit a random forest of regression trees with feature matrix 𝑋 to predict the target vector 𝑌. The number of trees in the forest, the maximum depth of each tree, and the number of features to consider at each tree are all hyperparameters of this model.
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37. I follow Berk and Green (2004), who have a simple fund flow relationship similar to: 𝜃𝑡 = 𝜙 1 11 𝑡−1 Õ 𝜏=𝑡−12 𝑟𝜏 | {z } fund flow term + 𝑟𝑡−1 |{z} wealth effect ! 𝜃𝑡−1 (B.61) where 𝜙 is an exogenous fund flow parameter based on the previous year’s return excluding the previous month, and 𝑟𝜏 are the statistical arbitrageur excess returns. The wealth effect captures the simple dynamics that if the statistical arbitrageur funds earn more, they have more money to manage. In other words, 𝜃𝑡+1𝐴𝑡+1 − 𝜃𝑡𝐴𝑡 is not a typical fund flow measure, but rather a change in AUM. Thus, this wealth effect should be added as shown, and without it the assumption would be that extra returns are withdrawn by investors.
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38. Γ0 𝛽𝑍0 𝑡𝑍𝑡Γ𝛽 −1 . (A.6) I estimate 𝑏 for the Kelly et al. (2019) model using the Kozak et al. (2020) method described above. The number of factors is a hyperparameter.
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39. Kelly, Bryan, Seth Pruitt, and Yinan Su, 2019, Characteristics are covariances: A unified model of risk and return, Journal of Financial Economics 134, 501 – 524.

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46. Now that 𝑓 is in hand, 𝑏 can be calculated according to Brandt et al. (2009), Kozak et al.
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47. Pástor, Ľuboš, 2000, Portfolio selection and asset pricing models, Journal of Finance 55, 179– 223.
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52. Silverman, B. W., 1988, Density estimation for statistics and data analysis, Chapman & Hall, London 1–176.
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53. Similar to the model of Kelly et al. (2019), this model requires the number of factors 𝑀, as well as 𝜆0 and 𝜆2 as hyperparameters. The standard hyperparameters associated with neural networks, including the number of layers and nodes, are hyperparameters in this model. Table A.3 displays various amounts of hidden layers and nodes that I use during cross validation. For example, (25, 5) signifies two hidden layers with the first hidden layer containing 25 nodes and the second containing 5 nodes. The value () signifies zero hidden layers. As described in Gu et al. (2021), I also use an 𝐿1 penalty for the weights in the neural network, which adds another hyperparameter. 1There are also penalties in the objective function and other details typical of neural network backpropagation.
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54. Some of the descriptions come directly from Freyberger et al. (2020), and I follow them by putting CRSP and Compustat variables in parentheses. I follow their lagged timing convention for when balance sheet variables are used.
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55. Stein, Jeremy C., 2009, Presidential address: Sophisticated investors and market efficiency, The Journal of Finance 64, 1517–1548.

56. The 15 endogenous variables—along with the 47 exogenous variable descriptions— are described in detail in Freyberger et al. (2020). I describe the construction and components of these variables below using this notation, although I do not describe in detail the construction of every variable because it is described so well in Freyberger et al. (2020).
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57. The fund flow term comes from Berk and Green (2004), where flows are a function of the previous year’s returns. Like Berk and Green (2004), there is no intercept in this equation. Note that 𝜃𝑡+1 is not a function of
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58. van Binsbergen, Jules H., and Christian C. Opp, 2019, Real anomalies, Journal of Finance 74, 1659 – 1706.
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59. Vuolteenaho, Tuomo, 2002, What drives firm-level stock returns?, Journal of Finance 57, 233–264.

60. While Brandt et al. (2009) discuss a variety of objective functions to estimate 𝑏, I focus on their model that maximizes the empirical Sharpe ratio. The maximal empirical Sharpe ratio estimate of 𝑏 of these authors can be written as: 𝑏 = argmin ˜ 𝑏 ( 𝐹𝑡−1 − Ω̄𝑡−1 ˜ 𝑏)0 Ω̄−1 𝑡−1( 𝐹𝑡−1 − Ω̄𝑡−1 ˜ 𝑏) which implies 𝑏 = Ω̄−1 𝑡−1
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