Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

HIGH-DIMENSIONALITY EFFECTS IN THE MARKOWITZ PROBLEM AND OTHER QUADRATIC PROGRAMS WITH LINEAR CONSTRAINTS: RISK UNDERESTIMATION

Noureddine El Karoui
The Annals of Statistics
Vol. 38, No. 6 (December 2010), pp. 3487-3566
Stable URL: http://www.jstor.org/stable/29765272
Page Count: 80
  • Read Online (Free)
  • Download ($19.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
HIGH-DIMENSIONALITY EFFECTS IN THE MARKOWITZ PROBLEM AND OTHER QUADRATIC PROGRAMS WITH LINEAR CONSTRAINTS: RISK UNDERESTIMATION
Preview not available

Abstract

We first study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high-dimensional setting where p, the number of variables in the problem, is of the same order of magnitude as n, the number of observations used to estimate the parameters. The Markowitz problem in Finance is a subcase of our study. Assuming normality and independence of the observations we relate the efficient frontier computed empirically to the "true" efficient frontier. Our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix. In particular, the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 - p/n. Therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters. We also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities. We investigate the robustness of our Gaussian results by extending the study to certain elliptical models and models where our n observations are correlated (in "time"). We show a lack of robustness of the Gaussian results, but are still able to get results concerning first order properties of the quantities of interest, even in the case of relatively heavy-tailed data (we require two moments). Risk underestimation is still present in the elliptical case and more pronounced than in the Gaussian case. We discuss properties of the nonparametric and parametric bootstrap in this context. We show several results, including the interesting fact that standard applications of the bootstrap generally yield inconsistent estimates of bias. We propose some strategies to correct these problems and practically validate them in some simulations. Throughout this paper, we will assume that p, n and n — p tend to infinity, and p

Page Thumbnails

  • Thumbnail: Page 
3487
    3487
  • Thumbnail: Page 
3488
    3488
  • Thumbnail: Page 
3489
    3489
  • Thumbnail: Page 
3490
    3490
  • Thumbnail: Page 
3491
    3491
  • Thumbnail: Page 
3492
    3492
  • Thumbnail: Page 
3493
    3493
  • Thumbnail: Page 
3494
    3494
  • Thumbnail: Page 
3495
    3495
  • Thumbnail: Page 
3496
    3496
  • Thumbnail: Page 
3497
    3497
  • Thumbnail: Page 
3498
    3498
  • Thumbnail: Page 
3499
    3499
  • Thumbnail: Page 
3500
    3500
  • Thumbnail: Page 
3501
    3501
  • Thumbnail: Page 
3502
    3502
  • Thumbnail: Page 
3503
    3503
  • Thumbnail: Page 
3504
    3504
  • Thumbnail: Page 
3505
    3505
  • Thumbnail: Page 
3506
    3506
  • Thumbnail: Page 
3507
    3507
  • Thumbnail: Page 
3508
    3508
  • Thumbnail: Page 
3509
    3509
  • Thumbnail: Page 
3510
    3510
  • Thumbnail: Page 
3511
    3511
  • Thumbnail: Page 
3512
    3512
  • Thumbnail: Page 
3513
    3513
  • Thumbnail: Page 
3514
    3514
  • Thumbnail: Page 
3515
    3515
  • Thumbnail: Page 
3516
    3516
  • Thumbnail: Page 
3517
    3517
  • Thumbnail: Page 
3518
    3518
  • Thumbnail: Page 
3519
    3519
  • Thumbnail: Page 
3520
    3520
  • Thumbnail: Page 
3521
    3521
  • Thumbnail: Page 
3522
    3522
  • Thumbnail: Page 
3523
    3523
  • Thumbnail: Page 
3524
    3524
  • Thumbnail: Page 
3525
    3525
  • Thumbnail: Page 
3526
    3526
  • Thumbnail: Page 
3527
    3527
  • Thumbnail: Page 
3528
    3528
  • Thumbnail: Page 
3529
    3529
  • Thumbnail: Page 
3530
    3530
  • Thumbnail: Page 
3531
    3531
  • Thumbnail: Page 
3532
    3532
  • Thumbnail: Page 
3533
    3533
  • Thumbnail: Page 
3534
    3534
  • Thumbnail: Page 
3535
    3535
  • Thumbnail: Page 
3536
    3536
  • Thumbnail: Page 
3537
    3537
  • Thumbnail: Page 
3538
    3538
  • Thumbnail: Page 
3539
    3539
  • Thumbnail: Page 
3540
    3540
  • Thumbnail: Page 
3541
    3541
  • Thumbnail: Page 
3542
    3542
  • Thumbnail: Page 
3543
    3543
  • Thumbnail: Page 
3544
    3544
  • Thumbnail: Page 
3545
    3545
  • Thumbnail: Page 
3546
    3546
  • Thumbnail: Page 
3547
    3547
  • Thumbnail: Page 
3548
    3548
  • Thumbnail: Page 
3549
    3549
  • Thumbnail: Page 
3550
    3550
  • Thumbnail: Page 
3551
    3551
  • Thumbnail: Page 
3552
    3552
  • Thumbnail: Page 
3553
    3553
  • Thumbnail: Page 
3554
    3554
  • Thumbnail: Page 
3555
    3555
  • Thumbnail: Page 
3556
    3556
  • Thumbnail: Page 
3557
    3557
  • Thumbnail: Page 
3558
    3558
  • Thumbnail: Page 
3559
    3559
  • Thumbnail: Page 
3560
    3560
  • Thumbnail: Page 
3561
    3561
  • Thumbnail: Page 
3562
    3562
  • Thumbnail: Page 
3563
    3563
  • Thumbnail: Page 
3564
    3564
  • Thumbnail: Page 
3565
    3565
  • Thumbnail: Page 
3566
    3566