Regression with Partially Observed Ranks on a Covariate: Distribution-Guided Scores for Ranks

0.26 MB·

by  Yuneung Kim

Regression with Partially Observed Ranks on a
Covariate:
Distribution-Guided Scores for Ranks
7
1 Yuneung Kim, Johan Lim, Young-Geun Choi
0
2 and

n Sujung Choi, and Do Hwan Park
∗
a
J

4

]
E Abstract
M
This work is motivated by a hand-collected data set from one of the largest Internet
. portals in Korea. This data set records the top 30 most frequently discussed stocks
t
a on its on-line message board. The frequencies are considered to measure the attention
t
s paid by investors to individual stocks. The empirical goal of the data analysis is
[
to investigate the effect of this attention on trading behavior. For this purpose, we

1
regress the (next day) returns and the (partially) observed ranks of frequencies. In
v
7 the regression, the ranks are transformed into scores, for which purpose the identity
9 or linear scores are commonly used. In this paper, we propose a new class of scores (a
0
scorefunction)thatisbasedonthemomentsoforderstatistics ofapre-decidedrandom
1
0 variable. The new score function, denoted by D-rank, is shown to be asymptotically
. optimal to maximize the correlation between the response and score, when the pre-
1
0 decided random variable and true covariate are in the same location-scale family. In
7
addition, the least-squares estimator using the D-rank consistently estimates the true
1
: correlation between the response and the covariate, and asymptotically approaches
v
the normal distribution. We additionally propose a procedure for diagnosing a given
i
X score function (equivalently, the pre-decided random variable Z) and selecting one that
r is better suited to the data. We numerically demonstrate the advantage of using a
a
correctly specified score function over that of the identity scores (or other misspecified
scores) in estimating the correlation coefficient. Finally, we apply our proposal to test
the effects of investors’ attention on their returns using the motivating data set.
Keywords: Concomitant variable; investors’ attention; linear regression; moments of
order statistics; optimal scaling; partially observed ranks
∗
Yuneung Kim, Johan Lim and Young-Geun Choi are with the Department of Statistics, Seoul National
University, Seoul, 151-747, Korea. Sujung Choi is with the School of Business Administration, Soongsil
University,Seoul,156-743,Korea. DohwanParkis with DepartmentofMathematics andStatistics, Univer-
sity of Maryland at Baltimore County, Baltimore, MD, 21250, USA. All correspondence are to Johan Lim
(E-mail: [email protected])
1
1 Introduction
This paper ismotivated by a hand-collected data set fromDaum.net, the 2ndlargest Internet
portal in Korea. The Daum.net portal offers an on-line stock message board where investors
can freely discuss specific stocks in which they might be interested. This portal also reports
a ranked list of the top 30 stocks that are most frequently discussed by users on a daily basis.
Thedatasetwascollectedbytheauthorsduringthe537tradingdaysfromOctober4th,2010,
to November 23rd, 2012. Along with therank data, we also collected financial data regarding
individual companies from FnGuide (http://www.fnguide.com). These additional data
include stock-day trading volumes classified in terms of different types of investors, stock
prices, stock returns, and so on.
The purpose of analyzing the collected data is to investigate the shifts in stock returns
caused by variations in investor attention. In finance, researchers are often interested in
determining the motivations that drive buying and selling decisions in stock markets. It is
commonlyassumed thatinvestors efficiently processrelevant informationinatimelymanner,
but in reality, it is nearly impossible to be efficient because of information overload. In
particular, individual investors are often less sophisticated than are institutional investors
and have a limited ability to process all relevant information. For this reason, individual
investors may pay attention only to a limited amount of information, perhaps that which is
relativelyeasytoaccess. Thephenomenonoflimitedattentionisawell-documentedcognitive
bias in the psychological literature (Kahneman, 1973; Camerer, 2003). This phenomenon
affects the information-processing capacities of investors and thus may affect asset prices on
the financial market. To empirically prove the effect of investor attention on stock returns,
we regress the observed stock returns with respect to the partially observed ranks.
Regression on a (partially observed) rank covariate has not previously been extensively
studied in the literature. A procedure that is commonly used in practice to address rank
2
covariates is to (i) regroup the ranks into only a few groups (if the number of ranks is high)
and (ii) treat the regrouped ranks as an ordinal categorical variable. Ordered categorical
variables frequently arise in various applications and have been studied extensively in the
literature. Score-based analysis is most commonly used for this purpose; see H´ajek (1968),
Hora and Conover (1984), Kimeldorf et al. (1992), Zheng (2008), Gertheiss (2014) and the
references therein. Thus, this typical two-step procedure for addressing a rank covariate is
equivalent to defining a score function for the ranks. However, as in the case of ordinal
categorical variables, such a score-based approach suffers from an inherent drawback related
to the choice of the score function; different choices of scores may lead to conflicting con-
clusions in the analysis (Graubard and Korn, 1987; Ivanova and Berger, 2001; Senn, 2007).
The recommendation for selecting the score function according to the literature is (i) to
choose meaningful scores for the ordinal categorical variable based on domain knowledge of
the data, (ii) to use equally spaced scores if scientifically plausible scores are not available
(see Graubard and Korn (1987)), and (iii) to find a optimal scaling transformed scores that
maximize the correlation with the responses while preserving the assumed characteristics
of the ordinal values(Linting et al., 2007; Costantini et al., 2010; de Leeuw and Mair, 2009;
Mair, and de Leeuw, 2010; Jacoby, 2016).
In this paper, we seek to provide an efficient tool for approach (i) described above, for
the case in which some qualitative knowledge is available regarding the ranks or the ranking
variable (the variable that is ranked). More specifically, we propose a new set of score
functions, denoted by D-rank, and study their use in linear regression. The proposed score
function is based on the moments of order statistics (MOS) of a pre-decided random variable
Z. This score function has several interesting properties related with the regression model,
if the pre-decided random variable is correctly specified as listed below. Here, the correct
specification implies it is within the same location-scale family with the true (unobserved)
covariate X. First, the D-rank is asymptotically optimal in the sense that it maximizes
3
the correlation between the response and score if the distribution of the D-rank is correctly
specified. Second, the least-squares estimator using the D-rank consistently estimates the
true correlation between the response and the covariate and asymptotically approaches the
normal distribution. Finally, the residuals of the fitted regression allow us to diagnose the
given score function (equivalently, the pre-decided random variable X) and to provide a tool
for selecting a score function that is better suited to the data.
The remainder of this paper is organized as follows. In Section 2, we study the properties
of the proposed D-rank. In this section, we show that the proposed D-rank is asymptotically
optimal to maximize the correlation between the response and score. In addition, We also
demonstrate the asymptotic equivalence between the proposed score function and the quan-
tilefunction; thequantilefunctionmayprovideabetterillustrationofthequalitativefeatures
of the score function. In Section 3, we apply the score function to estimate the regression co-
efficient of the linear model or, more precisely, to estimate the correlation coefficient between
the response and the scoring variable X. We prove that the least-squares estimator using
the D-rank consistently estimates the correlation coefficient and is asymptotically normally
distributed. In addition, we discuss the procedure for selecting an appropriate score func-
tion using the residuals. In Section 4, we numerically demonstrate that using the correctly
specified score function significantly reduces the mean square error on the estimation of the
correlation coefficient. In Section 5, we analyze the motivating data set to investigate the
existence of the attention effect. Finally, in Section 6, we briefly summarize the paper and
discuss the application of the proposed scores to regression using other auxiliary covariates.
2 Distribution-Guided Scores for Ranks (D-rank)
Weconsider asimple regression modelinwhich onlypartialranksofa covariateareobserved.
Specifically, suppose that Y ,X , i = 1,2,...,n is the complete set of observations,
i i
where Y is the variable o(cid:8)f(cid:0)primar(cid:1)y interest and X(cid:9) is the covariate related to Y . For
i i i
4
example, in our rank data from Daum.net, for i = 1,2,...,n, Y is a relevant outcome such
i
as earning rate or trading volume, X is the “unobserved” investors’ attention on the ith
i
company measured by the frequency of on-line discussions, and R is the “observed” rank of
i
X among X ,X ,...,X . We make certain assumptions regarding the distributions of X
i 1 2 n
and Y. We assume that the linear model of the relationship between X and Y is
i i
X µ
i X
Y = µ +ρσ − +ǫ , (1)
i Y Y i
σ
X
where the ǫ s are IID values from a distribution of mean 0 and variance σ2. The objective
i ǫ
of this paper is the estimation and inference of ρ = corr Y,X (or the regression coefficient
between Y and X) based on the observed data (Y ,R )(cid:0),i =(cid:1)1,2,...,n . To do it, we aim
i i
to define a good score function S(r) for the obse(cid:8)rved rank r, and consid(cid:9)er the regression of
Y on S(r), where Y is the response Y for R = r.
[r:n] [r:n] i i
The D-rank, we propose in this paper, is a set of the MOS of pre-decided random variable
Z, which we assume is in the same location-scale family of the true covariate X. To be
specific, suppose that Z ,Z ,...,Z are independent and identically distributed (IID) copies
1 2 n
of the random variable Z and that Z is the corresponding rth-order statistic for r =
(r:n)
1,2,...,n. The D-rank defines the score of the rank r as S (r) = α := E Z for
n (r:n) (r:n)
r = 1,2,...,n. (cid:0) (cid:1)
We first show that the D-rank maximizes the sample correlation between Y and α ,
[r:n] (r:n)
r = 1,2,...,n, in asymptotic, among all increasing functions S (r) : 1,2,...,n R.
n
{ } →
n
Let S (r) and α be the standardized scores (of S (r) and α ) to make S (r) =
n (r:n) n (r:n) r=1 n
n α = 0 and n S2(r) = n α2 = 1. Let S and S be the coPllection of all
r=1 (r:n) r=1 n r=1 (r:n) n n
iPncreasing functions SP(r) and S (rP), respectively.
n n
Theorem 1. Under the linear model (1), if Z is in the location-scale family of X, the D-rank
maximizes the limit of the sample correlation between Y and S (r) among S (r) S :
[r:n] n n n
∈
n
1
lim S (r) Y Y , (2)
n [r:n] n
n→∞ n σY −
· r=1
X (cid:0) (cid:1)
b 5
where σ2 = 1 n Y Y 2.
Y n r=1 [r:n]− n
P (cid:0) (cid:1)
The prboof of Theorem 1 is followed in Appendix.
Theorem 1 shows the asymptotic optimality of the D-rank for the regression in view of
optimal scaling inthe literature. The optimal scaling finds optimally transformed scores that
explain mostly well the assumed statistical model. It arises in various contexts including Gifi
classification of non-linear multivariate analysis(de Leeuw and Mair, 2009), the aspect (cor-
relational and non-cprrelational aspects) of multivariable(Mair, and de Leeuw, 2010), and
non-linear principal component analysis(Linting et al., 2007; Costantini et al., 2010). Here,
we adopt the idea of the optimal scaling in Jacoby (2016), and find the transformation to
maximize the correlation between the response and transformed scores. Theorem 1 above
shows that the D-rank maximizes the correlation in asymptotic, if pre-determined distribu-
tion for the D-rank is correctly specified.
The proposed score is closely related to the quantile of the underlying distribution of
Z. Let F (z) for z R and Q (q) for q [0,1] be the cumulative distribution function
Z Z
∈ ∈
(CDF) and the quantile function (QF), respectively, of Z. In the estimation of F (z) for
Z
Z ,i = 1,2,...,n , the rth-order statistic Z is the (r/n) 100-th percentile point of
i (r:n)
{ } ×
the empirical CDF, and thus, its expected value is approximately equal to Q (r/n). More
Z
specifically, given p = r , q = 1 p , and Q = Q (p ), we can write
r n+1 r − r r Z r
p q 1
α = Q + r r Q(2) +O ,
(r:n) r 2(n+2) r n2
(cid:0) (cid:1)
where Q(2) = f′ (Q ) f (Q ) 3 and f (z) is the probability density function of Z, which
r − Z r { Z r } Z
is differentiable. We r(cid:14)efer the reader to David (2003, Section 4.6) for the details of the
relationship between the MOS and the quantiles.
Consideration of the QF may provide a better understanding of the qualitative features
of the proposed score function. Suppose we expect the score function S (r) is convex in tail
n
(for r [nc] for a constant c close to 1); in other words, S (r+1) S (r) S (r) S (r 1)
n n n n
≥ − ≥ − −
6
for r [nc]. From the equivalence between the MOS and quantiles, it is known that the
≥
convexity of the scores S (r) is approximately equal to that of the quantile function Q (p).
n Z
Furthermore, the convexity of Q (p) for p c implies the following equivalent statements:
Z
≥
(i) F(z) is concave in z, (ii) f′(z) 0 or (iii) logf(z) is decreasing in z, all for z Q ([nc]).
Z
≤ ≥
3 Simple Linear Regression
In this section, we consider a simple regression model in which only partial ranks of a
covariate are observed. Specifically, suppose that Y ,X , i = 1,2,...,n is the complete
i i
set of observations from the linear model (1), and R(cid:8)(cid:0)is the(cid:1)rank of X amon(cid:9)g X ,X ,...,X .
i i 1 2 n
The rank R of X is indirectly measured by the frequency of on-line discussions of the ith
i i
company.
In this paper, we consider the case in which the ranks R are partially observed in the
i
sense that we observe only that U = R I R m + m+I R > m rather than R , where
i i i i i
≤
m+ is an arbitrary constant that is greate(cid:0)r than m(cid:1). Finally(cid:0), the obs(cid:1)ervations are
(Y ,U ), i = 1,2,...,n .
i i
(cid:8) (cid:9)
We let Y = Y I R = r for r = 1,2,...,m, and denote the above partially observed data
[r:n] i i
by Y for notati(cid:0)onal sim(cid:1)plicity.
[m]
The objective of this section is to identify a good estimator of ρ = corr Y,X (or the
regression coefficient between Y and X) and to test : ρ = 0 versus : ρ(cid:0) = 0(cid:1)or ρ > 0
0 1
H H 6
using the observed data Y .
[m]
3.1 Least-Squares Estimator
To estimate ρ, we recall assumptions regarding the distributions of X and Y. We assume
that the linear model of the relationship between X and Y is
i i
X µ
i X
Y = µ +ρσ − +ǫ ,
i Y Y i
σ
X
7
where the ǫ s are IID values from a distribution of mean 0 and variance σ2. By ordering on
i ǫ
the X s, we have for r = 1,...,n
i
σ
Y
Y = µ +ρ X µ +ǫ , (3)
[r:n] Y (r:n) X [r:n]
σ −
X
(cid:0) (cid:1)
where ρ = corr Y,X and
(cid:0) (cid:1)
E Y = µ +ρσ α (4)
[r:n] Y Y (r:n)
var(cid:0)Y (cid:1) = σ2 ρ2β +1 ρ2
[r:n] Y (rr:n) −
cov Y (cid:0),Y (cid:1) = ρ2σ(cid:0)2β , r = s (cid:1)
[r:n] [s:n] Y (rs:n) 6
(cid:0) (cid:1)
with
X µ X µ X µ
(r:n) X (r:n) X (s:n) X
α = E − and β = Cov − , −
(r:n) (rs:n)
σ σ σ
(cid:26) X (cid:27) (cid:18) X X (cid:19)
for r,s = 1,2,...,n (David and Galambos, 1974; David, 2003).
We are motivated by the identities (3) and (4) given above and propose the least-squares
estimator
1 [ns] α Y µ
ρ s r=1 (r:n) [r:n]− Y (5)
≡ σY · P [rn=s(cid:8)1] α(2r:n) (cid:9)
(cid:0) (cid:1) b
as an estimator of ρ with s =b m/n, where, µ =P n Y /n and σ2 = n (Y µ )2/n are
b Y i=1 i Y i=1 i− Y
the empirical estimators of the mean and variancPe, respectively, of Y.P
b b b
We claim that, if X is drawn from a location-scale family generated by Z, then the
least-squares estimator ρ s with s = m/n in (5), that is calculated based on the partial
observations Y , is consi(cid:0)st(cid:1)ent and asymptotically normally distributed with an appropriate
[m]
b
scale, as shown in Theorem 2. Suppose that
[ns] [ns] [ns] [ns]
1 1 1
ΨI(s) := α2 σ2 ,ΨII(s) := α α β2 , andΦ (s) := α2 ,
n n (r:n) (r:n) n n (r1:n) (r2:n) (r1,r2:n) n n (r:n)
r=1 r1=1r2=1 r=1
X X X X
where σ2 = σ2 X , and let ΨI (s), ΨII(s) and Φ (s) be the limits of ΨI(s), ΨII(s)
(r:n) (r:n) ∞ ∞ ∞ n n
and Φ (s), respec(cid:0)tively ((cid:1)under the assumption that they exist).
n
8
Theorem 2. Under the assumption that X is drawn from a distribution of a location-scale
family with a finite variance, the distribution of √n ρ(s) ρ converges to the normal dis-
−
tribution of mean 0 and variance ΨI (s)/σ2 +ρ2Ψ(cid:0)II(s) /Φ2(cid:1)(s).
∞ Y ∞b ∞
(cid:8) (cid:9)
The proof of Theorem 2 is provided in the Appendix.
We conclude this section with two remarks regarding Theorem 2. First, in Theorem 2,
from the tower property of the conditional expectation,
1 1
var √nρ > = 1,
1 n [ns] α2 ≥ var X
r=1 (r:n)
(cid:0) (cid:1)
b (cid:0) (cid:14) (cid:1)P (cid:0) (cid:1)
and when ρ = 0, the asymptotic variance of √nρ is larger than 1, which is the variance of
the least-squares estimator in the case where X is completely observed. Second, it is possible
b
to test the hypothesis : ρ = 0 using the statistic T = √nρ, which has an asymptotically
0
H
normal distribution of mean 0 and variance 1 Φ (s).
∞
b
(cid:14)
3.2 Residual Analysis
As in the classical linear model, the residuals can provide guidance for identifying a better
model and score function. The residuals are defined as e∗ = Y µ /σ ρα for
[r:n] [r:n]− Y Y − (r:n)
r = 1,2,...,[ns]. Statistical properties of the residuals, which a(cid:0)re analogo(cid:1)us to those in the
b
classical linear model, are summarized as follows.
Theorem 3. Under the assumptions of Theorem 2, the following statements are true for the
residuals: (i) E e∗ = 0; (ii)
[r:n]
(cid:0) (cid:1)
1 ΨI(s) 1
var e∗ = ρ2β + 1 ρ2 +α2 n 2
[r:n] (rr:n) − (r:n)nσ2 Φ2(s) − [ns] α2
n o Y n r=1 (r:n)
(cid:0) (cid:1) (cid:0) (cid:1)
[ns] P
ρ2 α α β +α2 (1 ρ2)
× (w:n) (r:n) (rw:n) (r:n) − 
 Xw=1 
(iii) E e∗ α = 0; and(iv) E e∗ Y∗ = 0, where Y∗ = µ  ρα .
[r:n] (r:n) [r:n] [r:n] [r:n] Y − (r:n)
(cid:0) (cid:1) (cid:0) (cid:1)
b b b
9
The proof of Theorem 3 requires only simple algebra and is thus omitted here. The theorem
states that the residuals have mean 0 and finite variance, and also states that they are
uncorrelated with the scores α and the predicted values Y . Thus, the residual plots,
(r:n) [r:n]
which are the plots of (i) r versus e∗ , (ii) α versus e∗ , and (iii) Y versus e∗ ,
[r:n] (r:n) [rb:n] [r:n] [r:n]
have the same interpretations as those of the classical linear model. We plug in µ and σ
b Y Y
with their empirical estimators and use e = Y µ /σ ρα .
[r:n] [r:n] Y Y (r:n)
− − −
The residual sum of squares may be another(cid:0)useful tool for(cid:1)measuring the goodness of fit
b b b
of the proposed model, as in the classical linear model. The residual sum of squares in our
model is defined as
[ns] Y µ 2
[r:n] Y
RSS = − Y
[r:n]
σ −
r=1 (cid:18) Y (cid:19)
X b
and will be used along with the residual plots as a guidebfor selecting a better score function.
b
Finally, the proposed least-squares estimator (5) assumes that theregression line between
α and Y µ has an intercept (at the y axis) of 0. Thus, if the model (or the score
(r:n) [r:n] Y
−
function)i(cid:0)scorrectlys(cid:1)pecified, thentheinterceptestimatedbytheregression(withintercept)
b
should be close to 0, and the estimated intercept therefore serves as a measure for checking
the correctness of the score function. Note that the regression (without intercept) performed
in this paper is based on observations of the top [ns] ranks and assumes that the function
passes through the origin (see Figure 4).
3.3 An Estimator with Unranked Observations
The least-squares estimator presented in Section 3.2 does not fully use the information
contained in Y := Y I(R = r),r > m ; it is used only to estimate µ and σ , not to
[r:n] i i Y Y
estimate ρitse(cid:8)lf. Inthissection, webriefly d(cid:9)emonstratehowρcanbemodifiedtoincorporate
these unranked observations.
b
10