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首页主成分分析与正则化估计:低秩因子模型研究
主成分分析与正则化估计:低秩因子模型研究
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更新于2024-07-15
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本文档"Principal Components and Regularized Estimation of Factor Models"深入探讨了在大型面板数据中广泛应用的主成分分析(PCA)方法以及正则化技术在因子模型估计中的作用。作者Jushan Bai和Serena Ng在2017年11月的研究中指出,传统的PCA通过迭代最小二乘回归法估计共同因素,但将其替换为ridge回归时,会使得共同成分的奇异值收缩,有可能降低其秩。 在机器学习领域,这种处理方式被用来恢复低秩矩阵。研究者从估计最小秩近似因子模型的角度审视这一过程,发现约束下的因子估计存在偏差,但在均方误差意义上可能更为有效。秩选择的重要性促使他们提出了一个数据依赖的惩罚,用于确定因子的数量。这种新的准则在名义上包含弱因子或测量噪声较大的情况下更为保守,避免了过度拟合。 此外,论文还扩展了该框架,考虑了对载荷(因子与观测变量之间的关系)的先验线性约束。这增加了模型的灵活性,允许根据领域知识对模型参数进行结构化控制。通过这些调整,研究者旨在提供一种更稳健且适应不同场景的因子模型估计方法,尤其是在面对复杂性较高的实际问题时。 总结来说,这篇论文的核心贡献在于揭示了PCA和正则化如何影响因子模型的估计精度和模型选择,特别是在处理噪声、弱因子和模型复杂性方面。它不仅提供了理论上的分析,还为实际应用中的模型选择策略提供了实用指导。对于从事统计学、经济学、金融学或机器学习等领域的人来说,理解和掌握这些方法至关重要,因为它们在处理大量面板数据时能显著提升分析效率和结果可靠性。
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If we write the SVD of X as X = U
˘
DV
′
, the singular values in
˘
D
r
are of O
p
(
√
NT ) magnitude.
To facilitate asymptotic analysis, we consider the scaled data
Z =
X
√
NT
, svd(Z) = UDV
′
, D =
˘
D
√
NT
.
Note that the left and right singular vectors of Z are the same as th ose for X. However, while the
first r singular values of X (i.e.
˘
D
r
) diverge and the remaining N − r are boun ded as N, T → ∞,
the r singular values of Z (i.e. D
r
) are bounded and the r emaining N − r singular values tend to
zero.
The model for the scaled data Z is
Z = F
∗
Λ
∗
′
+ e
∗
(5)
where F
∗
=
F
0
√
T
, Λ
∗
=
Λ
0
√
N
, and e
∗
=
e
√
NT
. Based on th e svd of Z = UDV
′
, we defin e the PC
estimates as:
b
F
z
= U
r
D
1/2
r
(6a)
b
Λ
z
= V
r
D
1/2
r
. (6b)
Notably, the PC and the APC estimates are equivalent up to a scale transformation. In particu-
lar,
b
F
z
=
e
F
D
1/2
r
√
T
and
b
Λ
z
=
e
Λ
D
−1/2
r
√
N
. One can constru ct the APC factor estimates directly from a svd
of XX
′
and rescale the eigenvalues, or one can construct the PC factor estimates from a svd of the
rescaled data Z.
4
While (
b
F
z
,
b
Λ
z
) emerges from the optimization problem of min
F,Λ
kZ −F Λ
′
k
2
F
,
b
F
z
is an estimate for F
∗
=
F
0
√
T
, not for F
0
. Similarly,
b
Λ
z
is an estimate for Λ
∗
=
Λ
0
√
N
. For estimation
of F
0
and Λ
0
, we define
b
F =
√
T U
r
D
1/2
r
(7a)
b
Λ =
√
N V
r
D
1/2
r
. (7b)
That is to say,
b
F =
√
T
b
F
z
and
b
Λ =
√
N
b
Λ
z
. It follow s that
b
F
′
b
F
T
=
b
Λ
′
b
Λ
N
= D
r
,
b
F =
e
F D
1/2
r
,
b
Λ =
e
ΛD
−1/2
r
In contrast, APC uses the normalization
F
′
F
T
= I
r
, and Λ
′
Λ is diagonal. The unit length normal-
ization makes it inconvenient to impose restrictions on the APC estimates of F , a limitation that
is important for the analysis to follow.
4
There may be numerical advantages to using PC over APC. The d ocumentation of prcomp in R notes ’the
calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using
eigenvalues on the covariance matrix. This is generally the preferred method for numerical accuracy.
6
![](https://csdnimg.cn/release/download_crawler_static/12821407/bg8.jpg)
To establish the large sample properties of the PC estimates, we need to compare the estimates
with a rotation of the true factors. Given the relation between
e
F and
b
F , it is natural to defi ne the
new rotation m atrix as
b
H
NT
=
e
H
NT
D
1/2
r
.
This leads to the identities
√
N(
b
F
t
−
b
H
′
NT
F
0
t
) =
√
ND
1/2
r
(
e
F
t
−
e
H
NT
′
F
0
t
),
√
T (
b
Λ
i
−
b
H
−1
NT
Λ
0
i
) =
√
T D
−1/2
r
(
e
Λ
i
−
e
H
−1
NT
Λ
0
i
).
From the limiting distributions of
e
F
t
and
e
Λ
i
, we obtain:
Proposition 1 Suppose that the data are generated by (1) and Assu mption A holds and both N
and T go to infinity. Then the PC-estimates
b
F and
b
Λ satisfy:
(i)
√
N(
b
F
t
−
b
H
′
NT
F
0
t
)
d
−→N
0, D
1/2
r
Avar(
e
F
t
) D
1/2
r
≡ N
0, Avar(
b
F
t
)
;
(ii)
√
T (
b
Λ
i
−
b
H
−1
NT
Λ
0
i
)
d
−→N
0, D
−1/2
r
Avar(
e
Λ
i
) D
−1/2
r
≡ N
0, Avar(
b
Λ
i
)
;
(iii) Let A
NT
(
b
C
it
) =
1
T
b
Λ
′
i
Avar(
b
F
t
)
b
Λ
i
+
1
N
b
F
′
t
Avar(
b
Λ
i
)
b
F
t
. Then (A
NT
(
b
C
it
))
−1/2
(
b
C
it
−C
0
it
)
d
−→N(0, 1).
The PC estimate of F
t
and Λ
i
are
√
N and
√
T consistent respectively, just like the APC estimates.
It follows that the estimated common component
b
C
it
is m in(
√
N,
√
T ) consistent. Analogous to
Lemma 1, we can define asymptotically equivalent rotation matrices. Let
b
H
1,NT
= (Λ
0′
Λ
0
)(
b
Λ
′
Λ
0
)
−1
and
b
H
2,NT
= (F
0′
F
0
)
−1
(F
0′
b
F ). Proposition 1 holds with
b
H
NT
replaced by either
b
H
1,NT
or
b
H
2,NT
.
3 Rank and Nuclear-Norm Minimization
In large dimensional factor analysis, it is customary to choose an r ∈ [0, rmax] that provides a good
fit while taking model complexity into account. Bai and Ng (2002) suggests a class of consistent
factor selection criteria, that is, rules that yield br such that prob(br = r) → 1 as N, T → ∞.
While criteria in this class have little difficulty singling out the dominant factors, they tend to
be too liberal. Over-estimating the number of factors is possible when validity of Assumption A
is questionable. The two leading causes are weak loadings, and idiosyncratic errors with large
variances. The first has the implication that the smaller eigenvalues do not increase sufficiently f ast
with N , an issue emphasized in Onatski (2011) and others. The second has the implication that
r+1-th eigenvalue (which should not increase with N) is not well separated from the r-th eigenvalue
(which should increase with N ). This is because outliers can increase the variance in an otherwise
7
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