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Optimal Portfolio Using Factor Graphical Lasso. (2023). Seregina, Ekaterina ; Lee, Tae-Hwy.
In: Working Papers.
RePEc:ucr:wpaper:202302.

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  1. . (A.2) As shown by Friedman et al. (2008), (A.2) can be viewed as a LASSO regression, where the LASSO estimates are functions of the inner products of Wε,11 and b σε,12. Hence, (3.3) is equivalent to p coupled LASSO problems. Once we obtain b β, we can estimate the entries Θε using the formula for partitioned inverses. The procedure to obtain sparse Θε is summarized in Algorithm 1.
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  2. /p. (B.17) For the ease of representation, denote y = ad − b2. Then, using similar technique as in Callot et al. (2019) we get |(b κ1 − κ1)| ≤ y b d − d + y b b − b + |b y − y||d − b| b yy = OP ϱT dT K3/2 sT = oP (1), where the rate trivially follows from Lemma 11. Similarly, we get |(b κ2 − κ2)| = OP ϱT dT K3/2 sT = oP (1).
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  3. = oP (1), where the first inequality was shown in Callot et al. (2019) (see their expression A.50), and the rate follows from Lemmas 11 and 10. We now proceed with the MWC weight formulation. First, let us simplify the weight expression as follows: wMWC = κ1(Θιp/p) + κ2(Θm/p), where κ1 = d − b ad − b2 κ2 = a − b ad − b2 . Let b wMWC = b κ1( b Θιp/p)+b κ2( b Θ b m/p), where b κ1 and b κ2 are the estimators of κ1 and κ2 respectively.
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  4. Ait-Sahalia, Y. and Xiu, D. (2017). Using principal component analysis to estimate a high dimensional factor model with high-frequency data. Journal of Econometrics, 201(2):384–399. 3 Awoye, O. A. (2016). Markowitz Minimum Variance Portfolio Optimization Using New Machine Learning Methods. PhD thesis, University College London. 2 Bai, J. (2003). Inferential theory for factor models of large dimensions. Econometrica, 71(1):135– 171. 9, 17, 18 Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models.

  5. Algorithm 1 Graphical Lasso Friedman et al. (2008) 1: Initialize Wε = b Σε + λI. The diagonal of Wε remains the same in what follows. 2: Repeat for j = 1, . . . , p, 1, . . . , p, . . . until convergence: Partition Wε into part 1: all but the j-th row and column, and part 2: the j-th row and column. Solve the score equations using the cyclical coordinate descent: Wε,11β − b σε,12 + λ Sign(β) = 0. This gives a (p − 1) × 1 vector solution b β. Update b wε,12 = Wε,11 b β. 3: In the final cycle (for i = 1, . . . , p) solve for 1 b θ22 = wε,22 − b β′ b wε,12 and b θ12 = −b θ22 b β.
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  6. December 31, 2019 (240 obs). Markowitz Risk-Constrained Markowitz Weight-Constrained Global Minimum-Variance Return Risk SR Turnover Return Risk SR Turnover Return Risk SR Turnover Without TC EW 2.19E-04 1.98E-02 0.0111 -2.19E-04 1.98E-02 0.0111 -2.19E-04 1.98E-02 0.0111 -Index 2.15E-04 1.16E-02 0.0185 -2.15E-04 1.16E-02 0.0185 -2.15E-04 1.16E-02 0.0185 -FGL 8.86E-04 2.90E-02 0.0305 (0.0450) -3.51E-04 7.07E-03 0.0496 (0.0020) -3.51E-04 6.98E-03 0.0503 (0.0025) -
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  7. Econometrica, 70(1):191–221. 9, 71, 72 Bai, J. and Ng, S. (2006). Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions. Econometrica, 74(4):1133–1150. 17 Bailey, N., Kapetanios, G., and Pesaran, M. H. (2021). Measurement of factor strength: Theory and practice. Journal of Applied Econometrics, 36(5):587–613. 30 Ban, G.-Y., El Karoui, N., and Lim, A. E. (2018). Machine learning and portfolio optimization.

  8. For more details regarding constructing b ΣSG, b ΣEL1 and b ΣEL2 see Fan et al. (2018), Sections 3 and 4. Proposition 1. For sub-Gaussian distributions, b ΣSG, b ΛSG K and b ΓSG K satisfy (C.1)-(C.3). For elliptical distributions, b ΣEL1, b ΛEL K and b ΓEL K satisfy (C.1)-(C.3). When (C.1)-(C.3) are satisfied, the bounds obtained in Theorems 2-5 continue to hold.
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  9. ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ = 0.9 (λ3/λ4 = 3.1) (λ3/λ4 = 2.7) (λ3/λ4 = 2.6) (λ3/λ4 = 2.2) (λ3/λ4 = 1.5) (λ3/λ4 = 1.1) ∥b wGMV − wGMV∥1 FGL 1.6900 1.8134 1.8577 1.8839 1.9843 2.0692 FClime 1.9073 1.9524 1.9997 1.9490 1.9898 2.0330 FLW 2.0239 2.0945 2.1195 2.1235 2.2473 2.4745 FNLW 2.0316 2.0790 2.1927 2.2503 2.4143 2.4710 POET 18.7934 28.0493 155.8479 32.4197 41.8098 71.5811 Projected POET 7.8696 8.4915 8.8641 10.7522 11.2092 19.0424 b
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  10. ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ = 0.9 (λ3/λ4 = 3.1) (λ3/λ4 = 2.7) (λ3/λ4 = 2.6) (λ3/λ4 = 2.2) (λ3/λ4 = 1.5) (λ3/λ4 = 1.1) ∥b wGMV − wGMV∥1 FGL 2.3198 2.3465 2.5177 2.4504 2.5010 2.7319 FClime 1.9554 1.9359 1.9795 1.9103 1.9813 1.9948 FLW 2.3445 2.3948 2.5328 2.4715 2.5918 3.0515 FNLW 2.2381 2.3009 2.3293 2.5497 2.9039 3.1980 POET 47.6746 82.1873 43.9722 54.1131 157.6963 235.8119 Projected POET 9.6335 7.8669 10.1546 10.6205 12.1795 15.2581 b
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  11. Γ ′ K. Similarly to Fan et al. (2018), we require the following bounds on the componentwise maximums of the estimators: (C.1) b Σ − Σ max = OP ( p log p/T), (C.2) (b ΛK − Λ)Λ−1 max = OP (K p log p/T), (C.3) b ΓK − Γ max = OP (K1/2 p log p/(Tp)). Let b ΣSG be the sample covariance matrix, with b ΛSG K and b ΓSG K constructed with the first K leading empirical eigenvalues and eigenvectors of b ΣSG respectively. Also, let b ΣEL1 = b D b R1 b D, where b R1 is obtained using the Kendall’s tau correlation coefficients and b D is a robust estimator of variances constructed using the Huber loss. Furthermore, let b ΣEL2 = b D b R2 b D, where b R2 is obtained using the spatial Kendall’s tau estimator. Define b ΛEL K to be the matrix of the first K leading empirical eigenvalues of b ΣEL1, and b ΓEL K is the matrix of the first K leading empirical eigenvectors of b ΣEL2.
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  12. ΦGMV − ΦGMV FGL 0.0033 0.0032 0.0034 0.0027 0.0021 0.0023 FClime 0.0012 0.0012 0.0012 0.0011 0.0010 0.0010 FLW 0.0049 0.0052 0.0061 0.0056 0.0049 0.0059 FNLW 0.0055 0.0060 0.0054 0.0052 0.0066 0.0057 POET 0.0070 0.0122 0.0058 0.0063 0.0103 0.0160 Projected POET 0.0021 0.0022 0.0019 0.0019 0.0018 0.0026 ∥b wMWC − wMWC∥1 FGL 2.3766 2.4108 2.7411 2.6094 2.5669 3.4633 FClime 2.0502 2.0279 2.2901 2.1400 2.1028 3.0737 FLW 2.4694 2.5132 2.8902 2.7315 2.7210 4.0248 FNLW 2.7268 2.3060 2.8984 3.5902 2.9232 3.2076 POET 49.8603 34.2024 469.3605 108.1529 74.8016 99.4561 Projected POET 9.0261 7.4028 8.1899 9.4806 11.9642 13.3890 b
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  13. ΦGMV − ΦGMV FGL 8.62E-04 9.22E-04 7.23E-04 7.31E-04 6.83E-04 5.73E-04 FClime 8.40E-04 8.27E-04 8.02E-04 7.87E-04 7.36E-04 6.71E-04 FLW 1.59E-03 1.73E-03 1.57E-03 1.68E-03 1.69E-03 1.54E-03 FNLW 2.24E-03 2.10E-03 1.83E-03 1.88E-03 2.07E-03 1.29E-03 POET 1.11E-03 1.46E-03 3.59E-03 1.27E-03 1.88E-03 2.51E-03 Projected POET 8.97E-04 8.80E-04 6.83E-04 6.79E-04 7.98E-04 6.55E-04 ∥b wMWC − wMWC∥1 FGL 1.9034 2.2843 1.9118 3.2569 2.7055 2.8812 FClime 2.1193 2.4024 2.0540 3.3487 2.7277 2.8593 FLW 2.2573 2.5809 2.1790 3.5728 3.0072 3.3164 FNLW 2.3207 3.3335 3.5518 3.4282 2.6446 4.8827 POET 15.8824 100.1419 56.9827 33.6483 38.8961 103.0434 Projected POET 6.5386 7.2169 7.8583 9.7342 12.1420 17.7368 b
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  14. In the lower part corresponding to the results with transaction costs, ∗∗∗ indicates p-value < 0.01, ∗∗ indicates p-value < 0.05, and ∗ indicates p-value < 0.10. In-sample: January 1, 1980 - December 31, 1995 (180 obs), Out-of-sample: January 1, 1995 - December 31, 2019 (300 obs). Markowitz Risk-Constrained Markowitz Weight-Constrained Return Risk SR Turnover Return Risk SR Turnover Without TC FGL 1.25E-03 4.09E-02 0.0305 (0.0709) - 3.10E-04 7.86E-03 0.0394 (0.0260) -FClime 3.30E-03 1.30E-01 0.0254 (0.0814) - 2.20E-04 9.61E-03 0.0229 (0.036) -
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  15. January 31, 2020 (4536 obs). Supplemental Appendix This Online Supplemental Appendix is structured as follows: Appendix A summarizes Graphical Lasso algorithm, Appendix B contains proofs of the theorems, accompanying lemmas, and an extension of the theorems to elliptical distributions. Appendix C provides additional simulations for Section 5, additional empirical results for Section 6 are located in Appendix D.
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  16. Journal of Financial Economics, 33(1):3–56. 8 Fama, E. F. and French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1):1–22. 9 Fan, J., Fan, Y., and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model. Journal of Econometrics, 147(1):186 – 197. 22 Fan, J., Liao, Y., and Mincheva, M. (2011). High-dimensional covariance matrix estimation in approximate factor models. The Annals of Statistics, 39(6):3320–3356. 3, 17, 20 Fan, J., Liao, Y., and Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. Journal of the Royal Statistical Society: Series B, 75(4):603–680. 3, 4, 5, 9, 16, 18, 19, 23, 71 Fan, J., Liu, H., and Wang, W. (2018). Large covariance estimation through elliptical factor models.

  17. Management Science, 64(3):1136–1154. 27 Barigozzi, M., Brownlees, C., and Lugosi, G. (2018). Power-law partial correlation network models.
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  18. One of the standard and commonly used approaches is to determine K in a data-driven way (Bai and Ng (2002); Kapetanios (2010)). As an example, in their paper Fan et al. (2013) adopt the approach from Bai and Ng (2002). However, all of the aforementioned papers deal with a fixed number of factors. Therefore, we need to adopt a different criteria since K is allowed to grow in our setup. For this reason, we use the methodology by Li et al. (2017): let bi,K and ft,K denote K × 1 vectors of loadings and factors when K needs to be estimated, and BK is a p × K matrix of stacked bi,K. Define V (K) = min BK ,FK 1 pT p X i=1 T
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  19. Proceeding to the MWC risk exposure, we follow Callot et al. (2019) and introduce the following notation: x = a2 − 2b + d and x̂ = â − 2b̂ + ˆ d to rewrite b ΦMWC = p−1(x̂/ŷ). As shown in Callot et al. (2019), y/x = O(1) (see their equation A.42). Furthermore, by Lemma 11 (b)-(d) |x̂ − x| ≤ |â − a|2 + 2 b̂ − b + ˆ d − d = OP (ϱT dT K3/2 sT ) = oP (1), and by Lemma 11 (f): |ŷ − y| = â ˆ d − b̂2 − (ad − b2 ) = OP (ϱT dT K3/2 sT ) = oP (1). Using the above and the facts that y = O(1) and x = O(1) (which were derived by Callot et al.
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  20. Proof. The proof of Lemma 1 can be found in Fan et al. (2011) (Lemma B.1). Lemma 2. Under Assumption (A.4), maxt≤T PK s=1|E[ε′ sεt]|/p = O(1).
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  21. Proof. The proof of Lemma 2 can be found in Fan et al. (2013) (Lemma A.6). Lemma 3. For b K defined in expression (3.6), Pr b K = K → 1.
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  22. Proof. The proof of Lemma 3 can be found in Li et al. (2017) (Theorem 1 and Corollary 1).
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  23. The Annals of Statistics, 46(4):1383–1414. 3, 4, 9, 17, 23, 59, 60, 65 Friedman, J., Hastie, T., and Tibshirani, R. (2008). Sparse inverse covariance estimation with the Graphical Lasso. Biostatistics, 9(3):432–441. 2, 11, 13, 14, 23, 40, 41 Gabaix, X. (2011). The granular origins of aggregate fluctuations. Econometrica, 79(3):733–772. 10 Goto, S. and Xu, Y. (2015). Improving mean variance optimization through sparse hedging restrictions.
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  24. The choice of the penalty function is similar to Bai and Ng (2002). Throughout the paper we let b K be the solution to (D.2).
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  25. Using the expressions (A.1) in Bai (2003) and (C.2) in Fan et al. (2013), we have the following identity: b ft − Hft = V p −1 1 T T X s=1 b fs E[ε′ sεt] p + 1 T T X s=1 b fsζst + 1 T T X s=1 b fsηst + 1 T T X s=1 b fsξst # , (B.1) where ζst = ε′ sεt/p − E[ε′ sεt] /p, ηst = f′ s Pp i=1 biεit/p and ξst = f′ t Pp i=1 biεis/p. Lemma 4. For all i ≤ b K, (a) (1/T) PT t=1 h (1/T) PT t=1 ˆ fisE[ε′ sεt] /p i2 = OP (T−1), (b) (1/T) PT t=1 h (1/T) PT t=1 ˆ fisζst/p i2 = OP (p−1), (c) (1/T) PT t=1 h (1/T) PT t=1 ˆ fisηst/p i2 = OP (K2/p), (d) (1/T) PT t=1 h (1/T) PT t=1 ˆ fisξst/p i2 = OP (K2/p).
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  26. X t=1 rit − 1 √ K b′ i,Kft,K 2 , (D.1) where the minimum is taken over 1 ≤ K ≤ Kmax, subject to normalization B′ KBK/p = IK. Hence, F̄′ K = √ KR′BK/p. Define b F′ K = F̄′ K(F̄KF̄′ K/T)1/2, which is a rescaled estimator of the factors that is used to determine the number of factors when K grows with the sample size. We then apply the following procedure described in Li et al. (2017) to estimate K: b K = arg min 1≤K≤Kmax ln(V (K, F̂K)) + Kg(p, T), (D.2) where 1 ≤ K ≤ Kmax = o(min{p1/17, T1/16}) and g(p, T) is a penalty function of (p, T) such that (i) Kmax g(p, T) → 0 and (ii) C−1 p,T,Kmax g(p, T) → ∞ with Cp,T,Kmax = OP max h K3 max √ p , K 5/2 max √ T i .
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  34. Knowing factors or factor loadings, or neither? Evaluating estimators of large covariance matrices with noisy and asynchronous data. (2019). Dai, Chaoxing ; Xiu, Dacheng ; Lu, Kun.
    In: Journal of Econometrics.
    RePEc:eee:econom:v:208:y:2019:i:1:p:43-79.

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  35. Estimating a Large Covariance Matrix in Time-varying Factor Models. (2019). Jung, Jaeheon.
    In: Papers.
    RePEc:arx:papers:1910.11965.

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  36. Large Volatility Matrix Prediction with High-Frequency Data. (2019). Song, Xinyu.
    In: Papers.
    RePEc:arx:papers:1907.01196.

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  37. Time series models for realized covariance matrices based on the matrix-F distribution. (2019). Zhu, Ke ; Li, Wai Keung ; Jiang, Feiyu ; Zhou, Jiayuan.
    In: Papers.
    RePEc:arx:papers:1903.12077.

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  38. Generalized Dynamic Factor Models and Volatilities: Consistency, rates, and prediction intervals. (2019). Hallin, Marc ; Barigozzi, Matteo.
    In: Papers.
    RePEc:arx:papers:1811.10045.

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  39. State-Varying Factor Models of Large Dimensions. (2019). Xiong, Ruoxuan ; Pelger, Markus.
    In: Papers.
    RePEc:arx:papers:1807.02248.

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  40. Machine Learning Macroeconometrics: A Primer. (2018). Korobilis, Dimitris.
    In: Working Paper series.
    RePEc:rim:rimwps:18-30.

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  41. A nonparametric eigenvalue-regularized integrated covariance matrix estimator for asset return data. (2018). Feng, Phoenix ; Lam, Clifford.
    In: LSE Research Online Documents on Economics.
    RePEc:ehl:lserod:88375.

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  42. A nonparametric eigenvalue-regularized integrated covariance matrix estimator for asset return data. (2018). Lam, Clifford ; Feng, Phoenix.
    In: Journal of Econometrics.
    RePEc:eee:econom:v:206:y:2018:i:1:p:226-257.

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  43. Simultaneous multiple change-point and factor analysis for high-dimensional time series. (2018). Barigozzi, Matteo ; Fryzlewicz, Piotr ; Cho, Haeran .
    In: Journal of Econometrics.
    RePEc:eee:econom:v:206:y:2018:i:1:p:187-225.

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  44. Testing against constant factor loading matrix with large panel high-frequency data. (2018). Kong, Xin-Bing ; Liu, Cheng.
    In: Journal of Econometrics.
    RePEc:eee:econom:v:204:y:2018:i:2:p:301-319.

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  45. Adaptive thresholding for large volatility matrix estimation based on high-frequency financial data. (2018). Kim, Donggyu ; Wang, Yazhen ; Li, Cui-Xia ; Kong, Xin-Bing.
    In: Journal of Econometrics.
    RePEc:eee:econom:v:203:y:2018:i:1:p:69-79.

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  46. High-dimensional covariance forecasting based on principal component analysis of high-frequency data. (2018). Jian, Zhi Hong ; Zhu, Zhican ; Deng, Pingjun.
    In: Economic Modelling.
    RePEc:eee:ecmode:v:75:y:2018:i:c:p:422-431.

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  47. Generalized Dynamic Factor Models and Volatilities: Consistency, Rates, and Prediction Intervals. (2018). Hallin, Marc ; Barigozzi, Matteo.
    In: Working Papers ECARES.
    RePEc:eca:wpaper:2013/278905.

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  48. Uniform Inference for Characteristic Effects of Large Continuous-Time Linear Models. (2018). Yang, Xiye ; Liao, Yuan.
    In: Papers.
    RePEc:arx:papers:1711.04392.

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  49. Uniform Inference for Conditional Factor Models with Instrumental and Idiosyncratic Betas. (2017). Yang, Xiye ; Liao, Yuan.
    In: Departmental Working Papers.
    RePEc:rut:rutres:201711.

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