TESTS FOR ROBUST VERSUS LEAST
SQUARES FACTOR MODEL FITS
R. Douglas Martin*
Computational Finance Program Director
Departments of Applied Mathematics and Statistics
University of Washington
[email protected]
R-Finance 2014 May 16, 2014, Chicago
* Joint work with Tatiana Maravina (PhD, Boeing Company) and Kjell Konis, Department of
Applied Mathematics University of Washington.
1
Time Series Factor Models
rt  ft β   t ,
t  1, 2, , T
Robust M-Estimates
 r  f β 
βˆ  argminβ    t t 
 sˆo 
t=1


T
 r  f β 
ft   t t   0

 sˆo 
t=1


T
Use lmRob in package robust
2
Favorite Rho and Psi Functions
Bisquare
Optimal
8
RHO(x)
4
6
2
-5
0
x
5
-5
0
x
5
0
x
5
85 %
90 %
95 %
99 %
PSI(x)
0
-1
-2
-2
-1
PSI(x)
0
1
2
85 %
90 %
95 %
99 %
1
2
85 %
90 %
95 %
99 %
0
0
2
RHO(x)
4
6
8
85 %
90 %
95 %
99 %
-5
0
x
5
-5
Optimal bias robust: Svarc, M., Yohai, V. J., & Zamar, R. H. (2002).
3
Test Statistic T1 (Hausman-type)
H1: Errors have a normal distribution
K1: Errors have a symmetric or skewed non-normal distribution
K2: Joint distribution of asset and factor returns is bias producing


V βˆ M  βˆ LS  VM  VLS  (1  EFFM ) VM
Efficient under H1 (see Hausman, 1978)
Test Statistic T2 (Wald-type)
H2: Errors have a normal distribution or a non-normal distribution
K2: Joint distribution of asset and factor returns is bias producing
−1
2
𝑛 𝜷𝐿𝑆 − 𝜷𝑀𝑀 ) → 𝑵(𝟎, 𝛿𝐿𝑆
𝑪
,𝑀𝑀 𝐟
4
R-Implementation
New functions in package robust (Kjell Konis), to be
submitted to CRAN by Sunday 5/18:
lsRobTest
> args(lsRobTest)
function (object, test = c("T2", "T1"), ...)
Object =
5
an lmRob fitted model object
10 15 20
0
5
^
Robust:
1.8 0.09
^
OLS:
1.5 0.1
-10 -5
MER Returns, %
MER
-10
-5
0
5
10
Market Returns, %
6
> lsRobTest(fit.mm, test="T1")
Test for least squares bias
H0: normal regression error distribution
Individual coefficient tests:
LS Robust
Delta Std. Error
Stat
p-value
x 1.497 1.798 -0.3009
0.009612 -31.31 3.889e-215
> lsRobTest(fit.mm, test="T2")
Test for least squares bias
H0: composite normal/non-normal
regression error distribution
Individual coefficient tests:
LS Robust
Delta Std. Error
Stat
p-value
x 1.497 1.798 -0.3009
0.08383 -3.589 0.0003315
7
DD
-5
-15
-25
DD Returns, %
0
5
^
Robust:
1.2 0.128
^
OLS:
1.19 0.076
20-Oct-1987
-25
-20
-15
-10
-5
0
5
Market Returns, %
T1 p-value = .65
T2 p-values = .82
8
References
Bailer, Maravina and Martin (2011). “Robust betas in asset
management”, Handbook of Quantitative Asset Management, Oxford
University Press.
Maravina and Martin (2014). “A Hausman type test of robust versus
least-squares regression fits”, submitted to SSRN on 5/18/2014.
Maravina and Martin (2014). “A Wald type test of robust versus leastsquares regression fits”, in preparation.
9
Appendix: Test Statistics T1 and T2
T1:
𝑇1𝑖 =
T2:
𝑇2𝑖 =
10


n βˆ M  βˆ LS  N  0, Vdiff
𝛽𝑀𝑀,𝑖 − 𝛽𝐿𝑆,𝑖
1 − 𝐸𝐹𝐹 ⋅ 𝑠𝑒 𝛽𝑀𝑀,𝑖

Vdiff  (1  EFFM )     2  Cf
𝑠𝑒 𝛽𝑀𝑀,𝑖 =
E 2 ( )
E 2 ( )
1
−1
𝜏𝜎1 2 𝐶𝑥,𝑖𝑖
𝑛
−1
2
𝑛 𝜷𝐿𝑆 − 𝜷𝑀𝑀 ) → 𝑵(𝟎, 𝛿𝐿𝑆
𝑪
,𝑀𝑀 𝐟
𝛽𝑀𝑀,𝑖 − 𝛽𝐿𝑆,𝑖
1 2
−1
𝛿𝐿𝑆,𝑀𝑀 𝐶𝑥,𝑖𝑖
𝑛
2
𝛿𝐿𝑆
,𝑀𝑀
=𝐸
𝑢𝐿𝑆 −
𝜎𝜓2
𝐸𝜓2′
𝑢𝑀𝑀
𝜎
𝑢𝑀𝑀
𝜎
2