demand system and liquidity constraints: simple
Post on 22-Oct-2021
8 Views
Preview:
TRANSCRIPT
DPRIETI Discussion Paper Series 19-E-044
Demand System and Liquidity Constraints:Simple Methodology for Measuring Liquidity Constraint
INOSE, JunyaMitsubishi Research Institute, Inc.
The Research Institute of Economy, Trade and Industryhttps://www.rieti.go.jp/en/
1
RIETI Discussion Paper Series 19-E-044
June 2019
Demand System and Liquidity Constraints: Simple Methodology for Measuring Liquidity Constrainti
Junya INOSE
Mitsubishi Research Institute, Inc.
Abstract
This paper proposed a new methodology, which introduces a stochastic property in the field of
demand analysis, for measuring the skewness of the consumption bundle distribution. We used the
questionnaire data from the National Survey of Family Income and Expenditure, Japan, from 1989 to
2004. We regard two categories of variables such as the total expenditure of each household and the
expenditure share of 10 commodity groups (e.g., food expenditure share, medical expenditure share,
etc.) as random variables, and estimated the probability density function using the “Skew Normal
Distribution.” As a result, distortion in the same direction was detected in all of the 10 commodity
groups. We also calculated that the measured skewness mutually relates to the major proxy of the
liquidity constraint. As a proxy of households under liquidity constraint, here we employ the
measurement of the fraction of Hand-to-Mouth (HtM) households. This paper was motivated by the
need to provide a simple methodology for measuring the liquidity constraint even in countries with
poor statistical literacy. Although several proxies have been proposed to measure the liquidity
constraint, the methodology becomes complex for more rigorous measuring attempts. In contrast, our
methodology only requires micro data on total consumption and a separate total for each commodity
group.
Keywords: Demand System, AIDS, QUAIDS, Skew Normal Distribution.
JEL classification number: C65; D12; E21.
The RIETI Discussion Papers Series aims at widely disseminating research results in the form of professional
papers, with the goal of stimulating lively discussion. The views expressed in the papers are solely those of
the author(s), and neither represent those of the organization(s) to which the author(s) belong(s) nor the
Research Institute of Economy, Trade and Industry.
i This study is conducted as a part of the Project “Heterogeneity across Agents and Economic Growth” undertaken at Research Institute of Economy, Trade and Industry (RIETI)
♥#$♦#♦♥
♥②%% ♦ ♦♥%♠- ♠♥ % ♦♥ ♥ -♦♥③ % -3 %% 3♦ ♠-♦ ♥ ♠-♦ ♦♥♦♠%
♥ ♦ 3 ♠♦%3 ♠♦% ♦- % ♦ ♥% ♥ ♥ -% %- 3 ♥ ♦
①♣♥3- %- ♦- ♦♠♠♦3% % ♥3♦♥ ♦ ♥♦♠ ♠♥ ♥②%% ♦ ♥% %
♦♥-♥ ♥ 3 ①♣♥3♦♥ ♦ ♦- -♥% 3♥ ♦%♦% ♥ -♦%% %3♦♥ %3%
♣- 3♦- ♥ -♦%% %3♦♥ %3% % ♦♥%3♥3 -♦%% ♦%♦% ♥ ♥ % % 3 ♦♠♦♥3②
♦ ♠♥ ♥3♦♥ ♣②% ♥♦ ♣-3 ♥ 3 ♦3- ♥ 3 ♥ ♣ -%3-3♦♥ %3 ♣②% ♣-3
- 3 ♥ ♣ -%3-3♦♥ -♣-%♥3% 3 -3♦♥
∑
i pigi (x,p) = x 3 3♦3 ①♣♥3- x
♣-% pi , p ♥ ♠♥ ♥3♦♥ gi (x,p) ♦- 3 3-3♦♥ 3 %3% %3 3% ♦%♦♥ ♦♦%♥ ♥3♦♥ ♦-♠% ♦ ♣-%% ♦- 3 %-% 3 -%♣3 3♦ 3 3♦3 ♦♥%♠♣3♦♥
%♠♣3♦♥ ♥% 3♦ %3-3 3 ①♣-%%♦♥ ♦ 3 ♠♥ ♦ ♦♠♠♦3② i qi 3♦
qi = fi (x)
- x % 3 3♦3 ①♣♥3- % -3♦♥%♣ % ♦♠♠♦♥② --- 3♦ % ♥ ♥ -
①♣♦-3♦♥ ♦ 3 ♥3♦♥ ♦-♠ ♦ ♥ -% -3 ♥ 3-3- J-% ♥ ♦3-
①♠♥ 3 %♣-♦-3② ♦ 3②♣% ♦ ♥3♦♥ ♦-♠ ♦ ♦-3♠ %♠♦-3♠
♥ ♦ -♣-♦ ♥ ♦♥ ♠♦% % %♦♠ ♠ ♦ %♣-♦-3② ♦- %♦♠ ♦♠♠♦3% ♥
♦- ♣-3 ♦ 3 ①♣♥3- -♥ ❲♦-♥ ♥ %- %3♠3 3 %♠♦-3♠
-3♦♥ ♦ 3 %-% ♥ 3 ♦-3♠ ♦ 3♦3 ①♣♥3- %
ωi = βi + γilog x+ ui
- βi ♥ γi - ♣-♠3-% 3♦ %3♠3 ωi % 3 3 %- ♦ ♦♠♠♦3② i ♥ ui % ♥
--♦- 3-♠ % -3♦♥ ♥ 3% ①3♥%♦♥ % ♥ %3 -S♥3② % 3 ♥ ♣ -%3-3♦♥ %
%3% ♥
∑
i βi = 1 ♥∑
i γi = 0 ♠♦%3 ♠♥ ②%3♠ 3♦♥ ♥ - ♥ ❯ -3 ♠♦%3 ♠♥ ②%3♠ ♥% 3 ♠♣♦②% 3%
①♣-%%♦♥ % %- ♦♥ 3% ♥3♦♥ ♦-♠ 3 33- 3♥ %♦♠ 3-♥3% ♥ 3♦♥ 3♦
3% ①♣-%%♦♥ %♦♠ ♠♣♦② J ♦- J ♥ 3% ♥3♦♥ ♦-♠ ♥ ♦- % ♦
3- ①3 -3♦♥ ♦- -♣-%♥33 ♥3 ♣-♦♣-3% % ♦- ①♠♣ 3- 3
♦♣♠♥3 ♦ 3 %- ♠♣- %3% - 33 -3- 3-♠% ♥ ♥♦♠ - -S-
♦- %♦♠ 3 ♥♦3 ①♣♥3- %- S3♦♥% % ♦- ①♠♣ 3♥%♦♥ 3 ♥
%♠♥ ♥ 3% ♠♣- -%3% % 3♦ 3 ♥ %% ♦ ♠♥ %②%3♠ ❯
♥ ❯ ♥-② 3-3 3♠%-% ♥②%% ♥ 3-♦- βi ♥ γi ♥ %♦ 3 ♦♥3 ♦S-3 3-♠ ♥ % ♦ ❯ ♦♠ ♥3♦♥% ♦ 3 ♣- 3♦- p
♥♦3- -S♥3② % ♥3♦♥ ♦-♠ ♦ ♥ -% % 3 ♦ ♦-3♠ ♠♥② %
♦ 3 ♠♣- ♦♥♥♥ %%♠ 3 ♦ ♦-3♠ ♦-♠ %
log qi = βi + eilog x+∑
k
eiklog pk + ui
3 3♦3 ①♣♥3- %33② ei = ∂log qi/∂log x ♥ ♣- %33② eij = ∂log qi/∂log pj ♥ %②
%3♠3 -3② -♦♠ ♠♣- %3% %♠♣ 3♦♥ % 3♦
log ωi = βi + (ei − 1) log x+ (eii + 1) log pi +∑
k 6=i
eiklog pk + ui,
♥ %♣② ♥ 3 % ♦ -♦%% %3♦♥ %3% 3 -3♦♥ % - 3♦
log ωi = β′i + γ′
ilog x+ ui.
"#② # "♦ ♥♦# ## # ❯ ♥ ♥② ♦#5 ♥#♦♥ ♦5♠" "5 ♦
" ♦♥"5 " ♥ 5#55② 5"# ♦5 "♦♥ ♦55 ♣♣5♦①♠#♦♥ #♦ ♥② ♠♥ "②"#♠ ♥ #5♦5
♥② ♦" ♦ # ♠♥ "②"#♠ ♦" ♥♦# 5#② "# 5"#5#♦♥" ♦♥ # "♣ ♦ # ♥5# ##②
♥#♦♥ ""♠ # ♠♥ "②"#♠ ❯ # ♦"# ♥#♦♥ " 5C5 #♦ "#"②
# ♦♦♥ ♦♥#♦♥" # ♥♦ ♦#5 "5 5"#5#♦♥" " 5C5 ♦♥ ♥ # ♥♣# ♣5"
♦♠♦♥♦" ♦ 5 ♥ # ♥♣# ♣5" ♥ "♦♥ ♦55 5♥#5♥#
♥ #♦♥ #♦ # ♠♦5 #5#5 ♦♥ # ♠♥ "②"#♠ ♥♦#5 ♣♣5♦ #♦ ♥②③ ♠♥
"②"#♠ ② "♥ ♥ ♠#♠# #♥♦♦② " ♦5 ①♠♣ ❱L"C③5♦ #
♦"# ♦ # ♠♣5 "#" "♠♣② "#♠# # ♣5♦♣♦" ♥ 5" ♦♥ #♠ "5" # ♦
①♣♥#5 ♦#② ♥ ♣5" # ♦5♥5② "# "C5" ♦5 ♠①♠♠ ♦♦ ♠#♦ # "##♥
# ♦5#♠ ♦ ♠♥ ♦ # ♦♠♠♦#② i ♦5 #" # "5 " ①♣♥ 5 ♦5
♠♦"# ♦ ♠♣5 ♥②"" ♦" ♥♦# #5# ♦# # ♦ # # "5 ♥ #♦# ①♣♥#5 "
"#♦"# 5 ♥ #" ♣♣5 ♦" # "#♦"# ♣5♦♣5#② ♦ # # # "5 ♥ #♦#
①♣♥#5 ♥ "5 #" 5#♦♥ ② ♥②③♥ #" ♦♥# ♣5♦#② ♥"#② ♥#♦♥
5♠♥5 ♦ #" ♣♣5 ♣5♦" " ♦♦" #♦♥ "#" ♣ ♦5 "♥ ♠♦ #♦♥
♣5"♥#" # "#♠#♦♥ ♦ ♠♦" ♥ #♦♥ ♦♥"
♦$♦ *② / ♥ ♦
%
❲ " ♦"♦ # "5 ♥ #♦# ♦♥"♠♣#♦♥ # 5♦♠ # "♦♥ )② ♦ ♠② ♥
♦♠ ♥ ①♣♥") 5#5 ♣♥ 5♦♠ #♦ ♥ 5② ②5" " # "
♦#♥ 5♦♠ # #♦♥ ##"# ♥#5 ♣♥ ♥ # "5② " ♦♥# ② # ♥"#5②
♦ ♥#5♥ 5" ♥ ♦♠♠♥#♦♥" #♦ " ♦♥# ♦5 ②5" # ♠5♦ # ♦5
"5② " ♥♦# ♥# ♥ # ♣5 ♥♦5♠#♦♥ " ♥♦# ♠ ♥ # # "# 5♦5
♦♦" #♦ ♦♥# 5♦"" "#♦♥ "#" ♥ "# ♦ # #♠ "5" ♥②"" ♥♠5 ♦ ♦"♦"
♦ " ♥ ♥ ♥ ♥ ♥ ♦"♦ # ♦5 ♠♦5
♣5"♦♥" ♦♥② ♠♦5 ♦♥"♠♣#♦♥ # ♦5 ♦"♦ " ♦# ①♣♥#5 ♦♦ ①
♣♥#5 ♥" ①♣♥#5 #5#② ①♣♥#5 5♥#5 ①♣♥#5 ❲5♥
①♣♥#5 ①♣♥#5 ♦♠♠♥#♦♥ ①♣♥#5 #♦♥ ①♣♥#5
♥#5#♥♠♥# ①♣♥#5 ♥ #5 ①♣♥#5 5 ♠ ""# ♦ ♦"♦" ♦"
♦♥"♠♣#♦♥ ♦5 ①♣♥#5 #♦ " ♣♦"# 5"♦♥ #♦ ♠♣♦② #" 5"#5#♦♥ "
## ♥ ♥②③ ♦ ♦5#♠ ♠♦ # # "5 ♦ # #5# ♦♠♠♦#② 5♦♣
#♦ ♣♦"# #5 "##♥ #" 5"#5#♦♥ # ♥♠5 ♦ ♦"♦" ♦ 5 #♦ ♥
♥ ♥ ♥ ♥
♦♥%♦* ♥ ♥ *
♦5 ♠♦ ♦♥ #♦ # # ♠♦ ①♣♥#♦♥ # " ♥#5♦ # ♦♥#♦5 ♦ "#♦"#
5 # "5 ♥ #♦# ①♣♥#5 5 # " ♣♥ 5♣5"♥#" # "♠
♦5#♠ " # ♦♥# ♣5♦#② ♥"#② "#5#♦♥ ♦ # ♦5#♠ ♦ #♦# ♦♥"♠♣#♦♥
log x ❳ ①" ♥ ♦♦ ①♣♥#5 "5 ωFood ❨ ①" ♥ # 5 5# " ♣♥
5♣5"♥#" # ♦ ♦5#♠ " #" # ♦5#♠ ♦ ♦♦ ①♣♥#5 log ωFood ❨
①" ♥ ♦# "" ♥ ♦"5 ♠♦♥♦#♦♥ 5"♥ ♦♦ ①♣♥#5 "5 # 5"♣# #♦ #
#♦# ♦♥"♠♣#♦♥
"♦ # ♠♣♣♥ ♦ # ♦♥# ♣5♦#② ♥"#② "#5#♦♥" 5" "5 ♥♥" 5"#
♥ ♦5♠♦"# # ♥"#② "#5#♦♥ "♠" #♦ ## ② # ""♥ ♦5 "♦♠ ♦#5 5#
ωi = f (x)
fo (x; Ω) d
d× d Ω Φ(
αTx
)
d
f (x; Ω, α) = 2fo (x; Ω)Φ(
αTx
)
α
α
Z ∼ SN (0, 1, α)
Z =
X0 if U < αX0
−X0 otherwise
X0 U N(0, 1)
X0 U
Z
X0
U U
α > 0
♥(( ♥ * ♦* ①♣♥*5
"♦ ♥ " ♦ + α < 0 ♦♣♣♦"+ ♠♥"♠ ♦0" ♦ ♥0"+♥ +" ♠♥"♠ ♥++② +
♦ ++0 +♦ ""♠ Z " +♦+ ♦♥"♠♣+♦♥ ♥ U " +♦+ ♥♦♠ + ♦"♦ "
♦♥"♠♣+♦♥ ♠♥ ♥ ♥+ X0 < 0 + " ♥♥+♥ ♥0" ♥ + +♦+ ♥♦♠ U > 0+ ♦"♦" + ♦♥"+0♥+ ♦♠ ♦♦" ♥ + ♦"♦ ♥ ♦0 "+ ①♣♥+0
Z = −X0 " ♦♥ " + ♥♥+♥ ♥0" ♥ + +♦+ ♥♦♠ ①" + ♠+ αX0 ♥ +
♦+0 ♥ + ♦"♦ " "+0♦♥ ♦♥"♠♣+♦♥ ♠♥ ♥ ♥+ X0 > 0 + " ♥♥+♥0" ♥ + +♦+ ♥♦♠ U < 0 + ♦"♦ " ♠♦♥② ♦♥"0♥ + + ♦♥"+0♥+ ♥+ ①♣♥+0 ♦♠" ♥0" Z = −X0 " ♦♥ " + ♥♥+♥ 0" ♥ + +♦+ ♥♦♠
①" + ♠+ "♠ " ♦0
"#♠#♦♥ ♥ "#"
♦♥&♦( ♦ & ,(♦&② ♥3&② ♥&♦♥3
0"+ ♥ ♦0♠♦"+ + " ♥+0♦ + ♦♥+♦0 ♦ + ♦♥+ ♣0♦+② ♥"+② ♥+♦♥ ♦ +
"0 ♥ +♦+ ①♣♥+0 0 "♦" + ♦ ♦0+♠ ♣♦+ ♦ + ♦♥+♦0 ♦ + ♦♥+
E ♦ + +♦+ ♦♥"♠♣+♦♥ ♥ ♦♠♠♦+② 0♦♣ ①♣♥+0 "0 ♦♦ ①♣♥+0
♥" ①♣♥+0 +0+② ①♣♥+0 0♥+0 ①♣♥+0 ❲0♥ ①♣♥+0
①♣♥+0 ♦♠♠♥+♦♥ ①♣♥+0 +♦♥ ①♣♥+0 ♥+0+♥♠♥+
①♣♥+0 ♥ +0 ①♣♥+0 ♥ ♥② ♥ + ♦♦ ①♣♥+0 "0 ♥
+0+② ①♣♥+0 "0 "♦" 0 ♥+ +0♥ + 0"♣+ +♦ + ♥0" ♥ + +♦+
♦♥"♠♣+♦♥ ①♣♥+0 "♦ + "♦ ♥♦+ ++ + "♥"" ♥ + ♥" ①♣♥+0
"0 ♦♠" 0② 0 " " ♠♥② " + ♠♣♦②" ♠♣+ 0♥+ " + ♥"
①♣♥+0 ♦0 ♦" ♦♥0" " ♥♦+ ①+② X +♦ + ♠0+ ♦ + ♦" 0♥+
♥ " ♦ + 0♥+0 ❲0♥ ♦♠♠♥+♦♥ ♥ ♥+0+♥♠♥+ ①♣♥+0 "0
+ "+0+♦♥ "♣0 0② ♥ ♠♦"+ ♥♦ ♦00+♦♥ +♥ ♦♠♠♦+② 0♦♣ "0 ♥
+♦+ ♦♥"♠♣+♦♥ ++ + ♥ ♥ ♦♥+0"+ + ♦♥+♦0 ♦ + +♦♥ ①♣♥+0 "0
♥ +♦+ ♦♥"♠♣+♦♥ "♦" "♦♠ ♥+0"+♥ +0" ♥ + ♠ ♥♦♠ 0♦♥ ♦ ♦ + +♦+
♦♥"♠♣+♦♥ ∼ 12.5 ♦0 +" 0♦♥ "♦♠ ♦"♦" ①+" 0+② ♦ +♦♥ ①♣♥+0
"0 ♦♠♣0 +♦ + 0 ♥ + ♦+0 ♥ ♥ + +♦+ ♦♥"♠♣+♦♥ ♥0"" +
+♦♥ ①♣♥+0 "0 ♥♦+② ♥0"" " +0 ♠② "♦ ♥ ♥ ♦0 + ♥+0"+♥ +♦♣
"♦+ ♥♦♠ ♠♦+② "♦ ♥ ♦" ♦♥ + ♦♥+♦0 ♦ +0 ①♣♥+0 "0 ♥
+ +♦+ ♦♥"♠♣+♦♥ ♥ + ♣♦"+ ♦00+♦♥ +0 ①♣♥+0 " + ♦♥②
♦♠♠♦+② 0♦♣ ①+" ♣♦"+ ♦00+♦♥ + 0"♣+ +♦ + +♦+ ♦♥"♠♣+♦♥ 0 +
+0 ①♣♥+0 ♥ ""♦0" " ♥♦♥"♥" +②♣ ♣0♠♠ ♥ 0♦" ♦♥+0+♦♥
$ ♦♥-♦$ ♦ ♦$-♠ ♦ ♦♥3♠♣-♦♥ ❳ ①3 ♥ ♦♠♠♦-② $♦♣ ①♣♥-$
$ ❨ ①3 ♦
$%♠%♦♥ %♦$ ♥ $%$
#$♠$♦♥ ♦ $ + $# 0$$♥ ♥ 0 ♥ 0#0 ♠♥② ♥♦0♠$♦♥ ♦ ♥ ♦0
♥②## ② ♣0♦♥ ♣0♠$0 #0♣$♦♥ ♦ $ ♥#$② ♥$♦♥ ♥ #♠♦0$♠ ♥ ♦
♦0$♠ ♠♦ ♣0♠$0 >$♦♥ $♦ $$ # # #0 ♥ ♥ ♥ $
#$♦#$ ##♠♣$♦♥# #♥ ♣ ♥ ♦♣ ② ③③♥ 0 $ # #♣0 #0
♥# $♦ ①♣♥ ♦ $ #$♠$♦♥ ♦0# ♦#0 ♥② ♦ $ 0 ♥ 0 $ ♠$
0#♦♥ $♦ ##♠ $$ $0 # #♦♠ ♠♦#$ ♣0♦ ♣♦♥$ ♥ $ ♦ ♦ 0$♥ ♦♠♠♦$② 0♦♣
#0 ♥ $ ♦ ♦ ♦$ ♦♥#♠♣$♦♥ ♥ ♦#♦# #$0$# 0♦♥ $ ♠♦#$ ♣0♦
♣♦♥$ ♥ #♥♦0♠ #$0$♦♥ ♥ # #♥# ♦ ♥♦$ 0$② ##♠ ♥② ♥0 ♦0 ♦$0
0$♦♥#♣ $♥ $ ♦ ♦ 0$♥ ♦♠♠♦$② 0♦♣ #0 ♥ $ ♦ ♦ ♦$ ♦♥#♠♣$♦♥
$ #$ #$♠$ $ ♠♦#$ ♣0♦ ♣♦♥$ ># $♦ βi ♥ >$♦♥ ♥ ♥ ♦
♦#♦# #$0$# 0♦♥ $ #$♠$ ♠♦#$ ♣0♦ ♣♦♥$
♥ ♣0#♥$# ♣0♠$0 #$♠$♦♥ 0#$ $ #♥ #♥ ♣ ♥ ❲♥
♦♠♣0 $ #$♠$ ♣0♠$0# ♦$ ♥ # ♦ #♠♦ ♥ ♦♦ ♠♦ ♥ $$ $
#♥## ♣0♠$0 α ♦♠# >$ ♥#$ ♥ $ #♠♦ # #♣② ♥ $ #$♠$♦♥ ♦ $$♦♥ ①♣♥$0 #0 $ #$♠$ ♦ α ①# 104 ♥ ①①# ♥ 107 ♥ ②①#
♥ $ ♦♥$0#$ $♦ $ 0#$ ♥ #♠♦ # $ #♥## ♣0♠$0 α # #$ ♥ $
♦♦ # αTotal $♦$ ①♣♥$0# ♦♦0♥$ ♦ α ♦♠# ♣♦#$ ♦0 ♦♠♠♦$# $ αShare ♦♠♠♦$② 0♦♣ ①♣♥$0 #0# ♦♦0♥$ ♦ α ♦♠# ♥$ ♦0 ♦♠♠♦$# ♦0♥ $♦ $ 0 $# #$ ♦ #♥## ♣0♠$0# ♥$# $$ $ #♥##
♦0# #♣② ♥ $ 0♦♥ ♦ ♦ $♦$ ♦♥#♠♣$♦♥ ♥ ♦♠♠♦$② 0♦♣ ①♣♥$0
#0 ♥ $♦♥ $ #♦ #♦ ♥♦$ $$ $ 0 ♦ #♥## 0# ② ♦♠♠♦$② 0♦♣#
0$② #$0♦♥ #♥## ♥ ♦♠♠♦$② 0♦♣ ①♣♥$0 #0 # $$ ♥ ♦♦ ❲0♥
$♦♥ ♥ $0 ①♣♥$0 0♥$0 ♥$0$♥♠♥$ ♥ $0 ①♣♥$0 ①$#
0$② #$0♦♥ #♥## ♥ $♦$ ♦♥#♠♣$♦♥
❲ ♥ ♥♦$ $♦ ♥$0♣0$ $# $0# 00♥ $ #②♠♠$0$② ♦ $ ♦0$♠ ♦ $
$♦$ ♦♥#♠♣$♦♥ ♥ ♥0 $ ♥♦♠ #$0$♦♥ ♦ ♦#♦# ♦②# +0$♦ ♣♦0 ♥
♥♦♠ 0♦♣ # ♥$0 ②# ♦♥ $ ♥ $ ♣0♦$② ♥#$② #$0$♦♥ ♥ 0 ♥
#♦ $# $ #♥## ♣0♠$0 ♦0 $# ♥$0 ♥ ♥♦$ >$② ①♣♥ $ 0♥ ♦
αTotal ② ♦♠♠♦$② 0♦♣# # # 0$$♥ ♥ $ $ #♥## ♥ ♦♦ ❲0♥ $♦♥
♥ $0 ①♣♥$0 0 #$0♦♥ ♦♠♣0 $♦ ♦$0 ♦♠♠♦$② 0♦♣# # # 0♥$0 0♦0
$ ♦ 0#♦♥ $♦ ♦♥#0 $0 #♦ #♦♠ ♥0②♥ ♣0♦♣0$② ♦0 # ♦♠♠♦$②
0♦♣ 0 $ # ♦♥#0 $ ♠♥♥ ♦ $ #$0♦♥ #♥## ♥ ♦♠♠♦$② 0♦♣ ①♣♥$0
#0 ♥ #♥## ♥ $♦$ ♦♥#♠♣$♦♥ $0② $♦ ①♣♥ $# $0 ♥ ♦$0 ♦0# ♦
♦♥#♠♣$♦♥ ♦#♦# ♠② # $♦ #♣♥ ♠ ♠♦♥② $♦ $ ♦♦ ❲0♥ $♦♥ ♥ $0
①♣♥$0 ♦0 # ♦#♦ ♥ ♥♦$ ♥0# $# ①♣♥$0 #0# $♦ #♦♠ 0#♦♥#
♥ ♣♦## 0#♦♥ ♠$ $ ①#$♥ ♦ $ >$② ♦♥#$0♥$ # $ #♠ ♦ ♦♠♠♦$②
0♦♣# #0 # $♦ $0 # ♠$$♦♥ $♦ ♥0# $ ①♣♥$0 #0 ♦ # ♦♠♠♦$#
# # ♦♦ ❲0♥# $ ♥0 $ >$② ♦♥#$0♥$ ①#$♥ ♦ $ ♠$$♦♥ 0$
#$♦0$♦♥ $♦ $ ①♣♥$0 #0 ♦$♦♥ ♥ $# ♠② ♣0♥$ ♦#♦# 0♦♠ ② ♦♣$♠③♥
$0 ♦0
♥$$ ♥ 6%② ♦♥$%9♥%
♥ $ ♥①$ ♥$0#$ ♦# $♦ ♦ #♠0 $ #♥## ♣0♠$0# ♥ $0$♦♥ >$② ♦♥#$0♥$
♠#0# 0 # ♣0♦①② ♦ ♦#♦# ♥0 >$② ♦♥#$0♥$ 0 ♠♣♦② $ ♠#0♠♥$
♦ $ 0$♦♥ ♦ ♥$♦♦$ $ ♦#♦# ♣♥ ♥ ❱♦♥$ ♥ 0 $
♥ ♦#♦ # $ ♦#♦# $# > $ ♥ # ♣♦#$ ♥ ## $♥
♦0 > $♦ ♦ $# 0♥♥# ♣0 ♣②♣0♦ ♦0 ♥$ ♥ $♥ $ #♠ ♦ ♦ $# ♣0
♣②♣0♦ ♥♦♠ ♥ $# ♦00♦♥ ♠$
αΩ
βαTotal αShare βTotal βShare
♦♦ ①♣♥)+ ,+ 4.72× 10−1 2.47
(
0.225 −0.029−0.029 0.014
)
♥, ①♣♥)+ ,+ 3.26× 104 9.26× 104(
0.211 −0.010−0.010 0.015
)
)+)② ①♣♥)+ ,+ 9.52× 10−1
(
0.255 −0.015−0.015 0.002
)
+♥)+ ①♣♥)+ ,+ 2.91× 10−1 3.21× 101(
0.209 0.0000.000 0.002
)
❲+♥ ①♣♥)+ ,+ −7.07× 10−2 2.28× 101(
0.209 0.0010.001 0.003
)
①♣♥)+ ,+ 4.40× 10−1 3.72× 102(
0.209 −0.001−0.001 0.003
)
♦♠♠♥)♦♥ ①♣♥)+ ,+ 2.81× 10−1
(
0.215 0.0120.012 0.017
)
)♦♥ ①♣♥)+ ,+ 6.44× 104 5.91× 107(
0.225 0.0260.026 0.020
)
♥)+)♥♠♥) ①♣♥)+ ,+ 8.56× 10−2 1.01× 101(
0.209 0.0010.001 0.010
)
)+ ①♣♥)+ ,+ 2.49× 10−2
(
0.238 0.0400.040 0.033
)
,)♠) H+♠)+, α, β ♥ Ω ♥ ♥ ,♠♦+)♠ ,
αΩ
βαTotal αShare βTotal βShare
♦♦ ①♣♥)+ ,+
(
0.346 −0.243−0.243 0.307
)
♥, ①♣♥)+ ,+
(
0.221 −0.493−0.493 14.301
)
)+)② ①♣♥)+ ,+
(
0.364 −0.301−0.301 0.417
)
+♥)+ ①♣♥)+ ,+
(
0.378 −0.208−0.208 0.874
)
❲+♥ ①♣♥)+ ,+
(
0.218 −0.089−0.089 1.590
)
①♣♥)+ ,+
(
0.245 −0.221−0.221 2.062
)
♦♠♠♥)♦♥ ①♣♥)+ ,+
(
0.265 −0.120−0.120 0.580
)
)♦♥ ①♣♥)+ ,+
(
0.213 0.0060.006 3.543
)
♥)+)♥♠♥) ①♣♥)+ ,+
(
0.330 −0.212−0.212 0.824
)
)+ ①♣♥)+ ,+
(
0.214 0.0320.032 1.124
)
,)♠) H+♠)+, α, β ♥ Ω ♥ ♥ ♦♦+)♠ ,
$ +-② ♦♥2-$♥- ♥ ♥22 - $♦♠ <$♠-$2 ♥ ♦♦ ①♣♥-$
$ ♣$2♥-2 - 2-$-♦♥ ♦ ♦2♦2 - $2♣- -♦ - 2♥22 Φ(
αTx
)
♥
♥ + - ♥ ② - ♣②♣$♦ ♥♦♠ 2 ♥ ♦$$2♣♦♥2 -♦
- ♣♣$ ♥ ♦$ ♠- ♦$ - - ♦2♦2 - 2 ♥-$2-♥ -- - ♣ ♦ - 2-$-♦♥
♦- ♦♥ - ♣♣$ ♠- ♦ - - -2 + - ♥ 2 + -♦ - ♦
-2 $♥♥2 ♣$ ♣②♣$♦ ♦$♥ -♦ - 2-$-♦♥ ♥ - $ - ♦- ♦ - ♦♦♥
2--♠♥-2 $ -$ ♥ ♣ ♣ - ♦2♦2 - ♦2♦2 - 2♥22 ♦♠2
♠♦$-② ♥ ♥ ♣ ♣ ♦2♦ - 2♥22 - ♦2♦2 - -$ +
- ♥ ♦$ $♥♥2 $♦♥ 0.5 ♦♠2 ♠♦$-② 2 ♥♥2 $2 -- - 2♥22♥ +-② ♦♥2-$♥- ♠-② $- ♥ - 2♥22 ♣$♠-$ ♠- ♦♦ ♣$♦①② -♦
♠2$ - +-② ♦♥2-$♥-
$♥')♦♥ ♦ ) ♥'' 1$♠)$ ♦$ ♠
♥ ♦♥2$ - 2♥22 ♣$♠-$ 2 ♥ ♠♦♥- ♦ 2-♦$-♦♥ ♣$♥- ♦2♦2 $♦♠
② ♦♣-♠③♥ -$ ♦$ - 2 2♦ ♠♥♥ -♦ ♥②③ ♦ -2 ♣$♠-$2 ♥
♦$ -♠ ♣$2♥-2 - ♥ ♥ 2-♠- ♣$♠-$2 $♦♠ -♦ $2- ♦♥2$
- 2-♠- $2-2 ♦$ αShare αShare ♦ ♦♦ $♥-$ ♥ ♥-$-♥♠♥- ①♣♥-$ 2$
$22 2-② - -♠ αShare ♦ -♦♥ ♥ -$ ①♣♥-$ 2$2 $
♥$2♥ 2 $2- ♥-2 -- - 2-♦$-♦♥ ♥ ♦♦ $♥-$ ♥ ♥-$-♥♠♥- ①♣♥-$
2 $2♥ - -♦♥ ♥ -$ ①♣♥-$ 2 ♥$2♥ ♦♥ ♦♥-$2- 2♣② -
2-♦$-♦♥ ♥ -♦♥ ♥ -$ ①♣♥-$ 2♦ ♠♣2③ ♥ 2♥22 -- - 2-♦$-♦♥
♥ -2 ①♣♥-$ 2$ 2 ♥ ♥$2♥
♦♥ - 2 ♦♥2$ - ②♥♠2 ♥ αTotal ♠$ -♦ - 222♦♥ ♥ - ②♥♠2 ♦ αShare
- 2-♦$-♦♥ ♥ αTotal 2♦ ♥$2$2 ② ♦♠♠♦-② $♦♣ -$♦ - 2$② ♦$ -
♥-- ♥$2-♥♥ ♦ -2 2-♦$-♦♥ 2 ♠ ♠♦$ - -♥ - ♥②22 ♥ αShare ♥ ♥$
- 2-♦$-♦♥ ♥ x ①2 ♦$$2♣♦♥2 -♦ - 2-♦$-♦♥ ♦ - -♦♣ ♦ - ♣$♦-② 2-$-♦♥ ♥-♦♥♦$ - ♦ -♦- ♦♥2♠♣-♦♥ 2 2♠ $♦22 2-♠-♦♥ ♥ - 2 - -♦ 22♠ -
2-♦$-♦♥ ♥ x ①2 -♦ $ ② ♦♠♠♦-② $♦♣ ♥ ♣♦22 ♥$2-♥♥ 2 -- - ♥
♥ αTotal $-2 - ♥ ♥ ♦$$-♦♥ -♥ - -♦- ♦♥2♠♣-♦♥ ♥ - ①♣♥-$ 2$
αTotal αShare
♦♦ ①♣♥/1 21
♥2 ①♣♥/1 21
/1/② ①♣♥/1 21
1♥/1 ①♣♥/1 21
❲1♥ ①♣♥/1 21
①♣♥/1 21
♦♠♠♥/♦♥ ①♣♥/1 21
/♦♥ ①♣♥/1 21
♥/1/♥♠♥/ ①♣♥/1 21
/1 ①♣♥/1 21
2/♠/ I1♠/12 α, β ♥ Ω ♥ ♥ 2♠♦1/♠ 2
♦ ♦♠♠♦/② 1♦♣ 2 21 ♥ / 1 / 1122♦♥ ♥ ♦11/♦♥ ♦♥/
12 ② ♦♠♠♦/② 1♦♣2 1♥ ♥ 1122♦♥ ♦♥/ 12/2 ♥ / 2♦♣ ♦ / ♣/
1/ 22♥ ♥ / 1♥ ♥ /2 2♦♣ ♠② / / 1/♦♥ ♦ / 2/♦1/♦♥
♦♦♥((♦-
♦ 1 ♦♥② 222 / ♦ / 2/♠/ ♣1♠/12 / ♥♦/ 11 ♥② 2♥♥ ♦1
2/♠/♦♥ ♥ /2 2/♦♥ / / ♦♦♥22♦/ ② /♥ ♥ II ♣♦/
1 ♣♦/ 1♣12♥/2 ♥/♥/ ♣♦/ ♥ 2 1♣ /♦♦ /♦ ♣ 2222♥ /1 2/
♦ / ♣2② ♠ 1♦♠ 2♦♠ /♦1/ 2/1/♦♥ ♥ /2 2 / 2 ♥♦1♠ 2/1/♦♥
♣♦/ ♦♠♣12 /♦ ♣1♦/② 2/1/♦♥2 ♥ ♠♦2/ 2 /♦1/ ♥ ♠♣1 2 ②
♣♦//♥ /1 \♥/2 ♥2/ ♦/1 /2 /♦ 2/1/♦♥2 1 2♠1 / ♣♦/ 2
2/1/ ♦♥ / ♥ y = x1 2 ♥♦/1 ♠/♦♦♦② /♦ 2222 / 2♠1/② ♦ 2/1/♦♥ II ♣♦/ II ♣♦/ 1♣12♥/2
I1♦/②I1♦/② ♣♦/ ♥ ♦♠♣12 / /♦ ♠/ 2/1/♦♥ ♥/♦♥2 ♥2/
♦/1 1♥ ♥ ♥ II ♣♦/ 2 ♦/ ♥♦♥ 2 ♣1♦/② ♣♦/2 ♥ / 1♥ 2
/ ♥/♦♥ /②♣ ♦ / ♦♠♣12♦♥ ♣♦/ 22 ♥♦♥♠/ 2/1/♦♥ ♥/♦♥ ♦1 ♥2/②
♥/♦♥ /♦ / /2 \♥/2 II ♣♦/ 22 ♠/ 2/1/♦♥ ♥/♦♥
1 ♥ ♣12♥/2 ♣♦/ ♦ ♦♦ ❲1♥ ♥ ①♣♥/1 21 ♥
2♠♦1/♠ 1 ♥ ♦ ♦1/♠ 1 2 / 2 ♥/12/♥ /♦ ♦2 ♦♥
♦♠♣12♦♥ / /2 12/2 ♦♦ ①♣♥/1 21 ♣112 2♠♦ 2 12 ❲1♥
♥ ①♣♥/1 21 ♣112 ♦♦ 2 2 12/2 2 2♠ ♥ II ♣♦/ 1
♥ ♦1 ♠♦2/ ♦ ♥ II ♣♦/2 1 ♥♦/ \/② 2♥♥/ ♥ /2 2 ♠②
♥ 1/1 ♠♦/♦♥ ♦ / ♠♦
&♠♥-♦♥ -&♠&♦♥
" ♥ ♠'♦♦♦② /♦ ♥ / '♦ /'♠' ♣"♠'"/ ♥ /♥ /'♠'♦♥ ② /♥ ♠'
"' / ♥♦"♠ /'"'♦♥ "/' ♦♥/" ♠♥/♦♥ ♣/ /♣ ♦♥//' ♦ ' '♦' ①
♣♥'" ♥ ♦♠♠♦'② "♦♣/ ①♣♥'" /" ❲ ♠♥' ♦♠♠♦'② "♦♣ ♥/
①♣♥'" /" "♦♠ '♦ ♣"♥' ♠'/♣♥"② ♣"/♥'/ ' /'♠' ♣"♠
'"/ ♦" '/ ♠'"' /'♠'♦♥ ♥ ♦♠♣" '/ "/' ' / /
♦♠♣"♥ αShare ♥ ♥ α ♥ ' /♥// /'② ♥"// ♥ ♠♦/' ♦♠♠♦'②"♦♣/
α βΩ
/'♠' Q"♠'"/ α, β ♥ Ω ♥ ♥ ♠'"' ♦♦"'♠ /♦' ♦' ①♣♥'" ♦♦ ①♣♥'" /" '"'② ①♣♥'" /" "♥'"
①♣♥'" /" ❲"♥ ①♣♥'" /" ①♣♥'" /" ♦♠♠♥'♦♥
①♣♥'" /" '♦♥ ①♣♥'" /" ♥'"'♥♠♥' ①♣♥'" /" ♥
'" ①♣♥'" /"
" ♦♥'♦" ♦ ♦"'♠ ♦ ♦♥/♠♣'♦♥ ❳ ①/ ♥ ♦♦ ①♣♥'" " ❨ ①/
" ' /♦ ♠♣/③ '' ' /'♠'♦♥ ♥ ♠'"' / ♥♦"♠ / ♥♦' /' ♦♠♣"
'♦ ' "' / '② '♦ /'♠' ♣"♠'"/ ♦ ♥ ♥ /♥ ♣
'♦ ' ♠'"' / ♥♦"♠ /'♠'♦♥ / /♠♣ ♥ ♣♦" / /'♠'♦♥ ♠' ♥♦'
♣♣$♦♣$' ♥ )♥) ♦ ♦' )''② ♥ )♥♥ ) ♣$)♥' ♥ $ ' ♦♦♥))
♦' ♥ == ♣♦' ♣$♦) )) )♥♥ '♦♥) ♦ ♠♣$ $♦♠ '
'♦$' ♥$) ' ' ' ♥ ♣♦' ♥ ' ♠ ♥ == ♣♦' ♥ ') '$) ♣♣$
♥ ) ♦ ' )) )♥♥ ♥ ♥$
♠♠%② ♥ +++♦♥
) ♣♣$ ♣$♦♣♦) ♥ ♠'♦♦♦② '♦ ♠)$ ' )♥)) ♦ ' ♦♥)♠♣'♦♥ ♥ )'$'♦♥
' ) )♦ ' '' ' ♠)$ )♥)) ♠'② $') '♦ ' ♠♦$ ♣$♦①② ♦ ' G'②
♦♥)'$♥' G ' ♥ ♦$ $♥♥) ♣$ ♣②♣$♦ ) ♣♣$ ) ♠♦'' ② ' ♥)
'♦ ♣$♦ )♠♣ ♠'♦♦♦② '♦ ♠)$ ' G'② ♦♥)'$♥' ♥ ♥ ♦♥'$② ' ♣♦♦$
)'')' '$② '♦ )$ ♣$♦①) '♦ ♠)$ ' G'② ♦♥)'$♥' $ ♣$♦♣♦) '
♠'♦♦♦② '♥ '♦ ♦♠ ♠♦$ ♦♠♣① '$② '♦ ♠)$ ' ♠♦$ $♦$♦)② ♥ ♦♥'$)' ♦$
♠'♦♦♦② ♦♥② $G$) ♠$♦ ' ♦♥ '♦' ♦♥)♠♣'♦♥ ♥ ') $ ♦♥ ♦$ ♦♠♠♦'②
$♦♣
♥ ' ♦'$ ♥ )$ ))) $ ' ♦$ $'$ )))♦♥) $)' ' ♦$♥ ♦ ' )♥))
)♦ ♦♥)$ $② '$♦ ' )'')' ♥②)) ) ) ))) ♥ ))♠
' ♦$♥ ♦ ' )♥)) '♦ $)' $♦♠ )'♦)' ♥$)$) ♥ ' ♦♥)♠♣'♦♥ ♦ ♦'$
♦♠♠♦'② $♦♣ ♦$ ' ' ♦ ' )♥)) ♥ ' )'$'♦♥ ♦ ' '♦' ♦♥)♠♣'♦♥ ♦$
♥♦♠ ) ♦ ) )'$♦♥ ♥ '♦ ♥'② ' ♣②) )♥♥' $♦ '♦ )'♠' '
)♥)) ♣$♠'$ α♦♥ ' ♠♦'♦♥ ♦ ' ♠♦ $G$ '♦ ♠♣$♦ ' ♦♦♥))♦' )♣ ♦
♥ == ♣♦' $) ② ♦♠♠♦'② $♦♣) ♥ )'♠' ②$) )♦ ♠♦$ ♦♠♣$♥) ♦$ ♥$
♠♦ ♠' $G$ $ $ )♦♠ ♦'$ ))) ♦$ ' ) )'$'♦♥ ) ) ) ②
)'$'♦♥) ) ♣' )'$'♦♥) ' ' ♥$)'♥♥) ♦ )'$'♦♥ ♥
♦) ♦♥)$'♦♥ ♦ ' ♦♥♦♠ ♥'$♣$''♦♥ '♦ ' ''$ ♥$)'♥♥) ♦ ' ♥
②♥♠)
$ ' ♥''♦♥ ♦ ' ♣$♠'$ ♦$ ' ♣$♦①② '♦ ♠)$ ' G'② ♦♥)'$♥' )
♥♦'$ ♠♣♦$'♥' )) ♥ ) ' )♥)) ♣$♠'$ ♦ ♦♦ ①♣♥'$ ) ♣$♦①② '♦
♠)$ ' G'② ♦♥)'$♥' ♦$ ' ♥''♦♥ ♦ ' ♣$♠'$ ) ♣$♦①② ♥
$'$ )))♦♥) ♥ '♦♥ '$♦ ♠♥② )'')' ♥②)) ♥ $♦) ♦♥'$) ♥ ♦ '
♠'♦♦♦② '♦ ♠)$ ' )♥)) ) '♦ ) ' )'♠' ♣$♠'$) ♥ ♠'$' )
)'$'♦♥ ' ') ♠'♦♦♦② ♦) ♥♦' ♦$ ♥ ♦$ ♥②)) '♦ )) )''② ♥ )♥♥
) )))♦♥ )♦ ♦♥'♥ ♥ ♥ ' ' ♠♦ ♠♦'♦♥
#♥&
!♥%♦♥ ♦♠ ♥ !♠ ♣♥♥ ♦♥ ♦♦ ♥ C♦♠ ! ♠②
①♣♥!C C② ♦♥♦♠ ♦C♥
③③♥ ♥ ♣!♥♦ ♦C♠ ♥ ! ♠% ♠C ❯♥C%!②
UC%%
♥% ♥ ♥ C! ♥ C% ♥ ♦♥%♠C ♠♥
♦ ♦♥♦♠% ♥ !!%!% ♣♣
%!② C% ♥C① ♥ C♥ ♥ ♠C♥ C♠
C♥% ♥ %♦! ♥♦♠ ♦!② ♥ ♥ ♣♣
!♦♥ ♥ C ♦♥♦♠% ♥ ♦♥%♠C ♦C ♠C ❯♥C%!②
UC%%
♥ UC♦!♦♥% ♥ ♦♥%♠♣!♦♥%C]!♥%% % _♥C% %♥ !♥
♥%!!! ♥!C♥!♦♥ !!%!` ♣♣
C!C③♥ ♦♠ ♦C♠♥ ♥ C% ♦♥♦♠!C ♣♣
C ❯♥②♠ ♥ ❲♥C ❲!② ♥ !♦ ♦! ♥ ♣♥ ♦♥♦♠
!!C% ♣♣
%♠♥ ❲ ② ♥ U♦ ♦♥♥C CC♦C% ♥ ❱C% %!♠!♦♥ ♦
♦♠ ♥ C% ♦C♥ ♦ ♦♥♦♠!C% ♣♣
♣♥ ♥ ❱♦♥! ♦ ♦ ! ♦♥%♠♣!♦♥ %♣♦♥% !♦ % !♠%
U②♠♥!% ♦♥♦♠!C ♣♣
%C ❱ ♦C♠% ♦ ♥ ♥!♦♥% ♦♥♦♠!C ♣♣
UC% ♥ ♦!C ♥②%% ♦ ♠② !% ♠C ❯♥C%!②
UC%% ♥ !♦♥
❱e%`③C♦ %!♦ ♠ U ♥ ① %!♠!♦♥ ♦
♠♥ ②%!♠% ♦♣ ♣♣C♦ ♦C♥ ♦ ♣♣ ♦♥♦♠!C% ♣♣
❲♦C♥ !!%! % ♦ ♠② ①♣♥!C ♦C♥ ♦ ♠C♥ !!%! %%♦
!♦♥ ♣♣
top related