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Chapter 8 - Lecture notes 8
Statistical Analysis For Economics And Finance (Baruch College CUNY)
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8.
1 Mul
t
i
pl
eChoi
c
e
1)I
nnonl
i
ne
armode
l
s
,t
hee
xpe
c
t
e
dc
hangei
nt
hede
pe
nde
ntvar
i
abl
ef
orac
hangei
noneoft
he
e
xpl
ana
t
or
yvar
i
abl
e
si
sgi
ve
nby
A)△Y=f
(
X1+X1,X2,
.
.
.Xk
)
.
B)△Y=f
(
X1+△X1,X2+△X2,
.
.
.
,Xk
+△Xk
)
-f
(
X1,X2,
.
.
.
Xk
)
.
C)△Y=f
(
X1+△X1,X2,
.
.
.
,Xk
)
-f
(
X1,X2,
.
.
.
Xk
)
.
D)△Y=f
(
X1+X1,X2,
.
.
.
,Xk
)
-f
(
X1,X2,
.
.
.
Xk
)
.
Ans
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2)Thei
nt
e
r
pr
e
t
a
t
i
onoft
hes
l
opec
oe
ffic
i
e
nti
nt
hemode
lYi=β
n(
Xi
)+uii
sasf
ol
l
ows
:
0+β
1l
A)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
haβ
hangei
nY.
1% c
B)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
hac
hangei
nYof0.
01β
1.
C)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
haβ
1
0
0
%
c
h
a
ngei
nY.
1
D)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
haβ
hangei
nY.
1c
Ans
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3)Thei
nt
e
r
pr
e
t
a
t
i
onoft
hes
l
opec
oe
ffic
i
e
nti
nt
hemode
ll
n(
Yi
)=β
sasf
ol
l
ows
:
0+β
1Xi+uii
A)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
haβ
hangei
nY.
1% c
B)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
ha100β
hangei
nY.
1% c
C)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
hac
hangei
nYof0.
01β
.
1
D)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
haβ
c
h
a
n
g
e
i
n
Y.
1
Ans
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4)Thei
nt
e
r
pr
e
t
a
t
i
onoft
hes
l
opec
oe
ffic
i
e
nti
nt
hemode
ll
n(
Yi
)=β
n(
Xi
)
+uii
sasf
ol
l
ows
:
0+β
1l
A)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
haβ
%
c
h
a
n
g
e
i
n
Y
.
1
B)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
haβ
hangei
nY.
1c
C)ac
hangei
nXbyoneuni
ti
sas
s
oc
i
a
t
e
dwi
t
ha100β
%
c
h
angei
nY.
1
D)a1% c
hangei
nXi
sas
s
oc
i
a
t
e
dwi
t
hac
hangei
nYof0.
01β
1.
Ans
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1
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5)I
nt
hec
as
eofr
e
gr
e
s
s
i
onwi
t
hi
nt
e
r
a
c
t
i
ons
,t
hec
oe
ffic
i
e
ntofabi
nar
yvar
i
abl
es
houl
dbei
nt
e
r
pr
e
t
e
das
f
ol
l
ows
:
A)t
he
r
ear
er
e
al
l
ypr
obl
e
msi
ni
nt
e
r
pr
e
t
i
ngt
he
s
e
,s
i
nc
et
hel
n(
0)i
snotde
fine
d.
B)f
ort
hec
as
eofi
nt
e
r
a
c
t
e
dr
e
gr
e
s
s
or
s
,t
hebi
nar
yvar
i
abl
ec
oe
ffic
i
e
ntr
e
pr
e
s
e
nt
st
hevar
i
ousi
nt
e
r
c
e
pt
sf
or
t
hec
as
ewhe
nt
hebi
nar
yvar
i
abl
ee
qual
sone
.
C)fir
s
ts
e
tal
le
xpl
ana
t
or
yvar
i
abl
e
st
oone
,wi
t
ht
hee
x
c
e
pt
i
onoft
hebi
nar
yvar
i
abl
e
s
.The
nal
l
ow f
ore
ac
h
oft
hebi
nar
yvar
i
abl
e
st
ot
a
keont
heval
ueofones
e
que
nt
i
al
l
y
.Ther
e
s
ul
t
i
ngpr
e
di
c
t
e
dval
uei
ndi
c
a
t
e
st
he
e
ffe
c
toft
hebi
nar
yvar
i
abl
e
.
D)fir
s
tc
omput
et
hee
xpe
c
t
e
dval
ue
sofYf
ore
ac
hpos
s
i
bl
ec
as
ede
s
c
r
i
be
dbyt
hes
e
tofbi
nar
yvar
i
abl
e
s
.
Ne
xtc
ompar
et
he
s
ee
xpe
c
t
e
dval
ue
s
.Eac
hc
oe
ffic
i
e
ntc
ant
he
nbee
xpr
e
s
s
e
de
i
t
he
rasane
xpe
c
t
e
dval
ueor
ast
hedi
ffe
r
e
nc
ebe
t
we
e
nt
woormor
ee
xpe
c
t
e
dval
ue
s
.
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6)Thef
ol
l
owi
ngi
nt
e
r
ac
t
i
onsbe
t
we
e
nbi
nar
yandc
ont
i
nuousvar
i
abl
e
sar
epos
s
i
bl
e
,wi
t
ht
hee
x
c
e
pt
i
onof
A)Yi=β
Xi×Di
)+ui
.
0+β
1Xi+β
2Di+β
3(
B)Yi=β
Xi×Di
)+ui
.
0+β
1Xi+β
2(
C)Yi=(
β
+
D
)
+
β
X
+
u
.
0 i 1i i
D)Yi=β
.
0+β
1Xi+β
2Di+ui
Ans
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7)Ane
xampl
eoft
hei
nt
e
r
ac
t
i
ont
e
r
m be
t
we
e
nt
woi
nde
pe
nde
nt
,c
ont
i
nuousvar
i
abl
e
si
s
A)Yi=β
+
β
X
+
β
D
+
β
(
X
×
D
)
+
u
.
0 1i 2i 3 i i i
B)Yi=β
.
0+β
1X1i+β
2X2i+ui
C)Yi=β
D1i×D2i
)+ui
.
0+β
1D1i+β
2D2i+β
3(
D)Yi=β
+
β
X
+
β
X
+
β
(
X
×
X
)
+
u
.
0 1 1i 2 2i 3 1i 2i i
Ans
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8)I
nc
l
udi
ngani
nt
e
r
ac
t
i
ont
e
r
m be
t
we
e
nt
woi
nde
pe
nde
ntvar
i
abl
e
s
,X1andX2,al
l
owsf
ort
hef
ol
l
owi
ng
e
x
c
e
pt
:
A)t
hei
nt
e
r
a
c
t
i
ont
e
r
ml
e
t
st
hee
ffe
c
tonYofac
hangei
nX1de
pe
ndont
heval
ueofX2.
B)t
hei
nt
e
r
a
c
t
i
ont
e
r
mc
oe
ffic
i
e
nti
st
hee
ffe
c
tofauni
ti
nc
r
e
as
ei
nX1andX2aboveandbe
yondt
hes
um
oft
hei
ndi
vi
duale
ffe
c
t
sofauni
ti
nc
r
e
as
ei
nt
het
wovar
i
abl
e
sal
one
.
C)t
hei
nt
e
r
a
c
t
i
ont
e
r
mc
oe
ffic
i
e
nti
st
hee
ffe
c
tofauni
ti
nc
r
e
as
ei
n
.
D)t
hei
nt
e
r
a
c
t
i
ont
e
r
ml
e
t
st
hee
ffe
c
tonYofac
hangei
nX2de
pe
ndont
heval
ueofX1.
Ans
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9)Anonl
i
ne
arf
unc
t
i
on
A)make
sl
i
t
t
l
es
e
ns
e
,be
c
aus
evar
i
abl
e
si
nt
her
e
a
lwor
l
dar
er
e
l
a
t
e
dl
i
ne
ar
l
y
.
B)c
anbeade
qua
t
e
l
yde
s
c
r
i
be
dbyas
t
r
ai
ghtl
i
nebe
t
we
e
nt
hede
pe
nde
ntvar
i
abl
eandoneoft
he
e
xpl
ana
t
or
yvar
i
abl
e
s
.
C)i
sac
onc
e
ptt
ha
tonl
yappl
i
e
st
ot
hec
as
eofas
i
ngl
eort
woe
xpl
ana
t
or
yvar
i
abl
e
ss
i
nc
eyouc
annotdr
a
w
al
i
nei
nf
ourdi
me
ns
i
ons
.
D)i
saf
unc
t
i
onwi
t
has
l
opet
ha
ti
snotc
ons
t
ant
.
Ans
we
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i
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ar
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i
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nc
.
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10)Ane
xampl
eofaquadr
a
t
i
cr
e
gr
e
s
s
i
onmode
li
s
2
A)Yi=β
+
β
X
+
β
Y
+
u
.
0 1
2
i
B)Yi=β
+
β
l
n
(
X
)
+
u
.
0 1
i
2
C)Yi=β
+
β
X
+
β
X
.
0 1
2 +ui
D)
=β
.
0+β
1X+ui
Ans
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2,t
=607.
3+3.
85I
nc
o
me–0.
0423I
nc
o
me
hef
ol
l
owi
ng
11)(
Re
qui
r
e
sCal
c
ul
us
)I
nt
hee
qua
t
i
on
i
nc
omel
e
ve
lr
e
s
ul
t
si
nt
hemaxi
mum t
e
s
ts
c
or
e
A)607.
3.
B)91.
02.
C)45.
50.
D)c
annotbede
t
e
r
mi
ne
dwi
t
houtapl
otoft
heda
t
a
.
Ans
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12)Tode
c
i
dewhe
t
he
rYi=β
n(
Yi
)=β
st
heda
t
abe
t
t
e
r
,youc
annotc
ons
ul
t
0+β
1X+uiorl
0+β
1X+uifit
2
t
her
e
gr
e
s
s
i
onR be
c
aus
e
A)l
n(
Y)ma
ybene
ga
t
i
vef
or0<Y<1.
B)t
heTSSar
enotme
as
ur
e
di
nt
hes
ameuni
t
sbe
t
we
e
nt
het
womode
l
s
.
C)t
hes
l
openol
onge
ri
ndi
c
a
t
e
st
hee
ffe
c
tofauni
tc
hangeofXonYi
nt
hel
ogl
i
ne
armode
l
.
2
D)t
her
e
gr
e
s
s
i
onR c
anbegr
e
a
t
e
rt
hanonei
nt
hes
e
c
ondmode
l
.
Ans
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13)Youha
vee
s
t
i
ma
t
e
dt
hef
ol
l
owi
nge
qua
t
i
on:
2,
=607.
3+3.
85I
nc
o
me–0.
0423I
nc
o
me
whe
r
eTe
s
t
Sc
o
r
ei
st
hea
ve
r
a
geoft
her
e
adi
ngandma
t
hs
c
or
e
sont
heSt
anf
or
d9s
t
andar
di
ze
dt
e
s
t
t
h
admi
ni
s
t
e
r
e
dt
o5 gr
ades
t
ude
nt
si
n420Cal
i
f
or
ni
as
c
hooldi
s
t
r
i
c
t
si
n1998and1999.I
nc
o
mei
st
he
a
ve
r
a
geannualpe
rc
api
t
ai
nc
omei
nt
hes
c
hooldi
s
t
r
i
c
t
,me
as
ur
e
di
nt
hous
andsof1998dol
l
ar
s
.The
e
qua
t
i
on
A)s
ugge
s
t
sapos
i
t
i
ver
e
l
a
t
i
ons
hi
pbe
t
we
e
nt
e
s
ts
c
or
e
sandi
nc
omef
ormos
toft
hes
ampl
e
.
B)i
spos
i
t
i
veunt
i
laval
ueofI
nc
o
meof610.
81.
C)doe
snotmakemuc
hs
e
ns
es
i
nc
et
hes
quar
eofi
nc
omei
se
nt
e
r
e
d.
D)s
ugge
s
t
sapos
i
t
i
ver
e
l
a
t
i
ons
hi
pbe
t
we
e
nt
e
s
ts
c
or
e
sandi
nc
omef
oral
loft
hes
ampl
e
.
Ans
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:A
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.
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14)Apol
ynomi
alr
e
gr
e
s
s
i
onmode
li
ss
pe
c
i
fie
das
:
A)Yi=β
∙
∙+β
X +ui
.
0+β
1Xi+β
2X +∙
r
B)Yi=β
∙
∙+β Xi+ui
.
0+β
1Xi+β Xi+∙
C)Yi=β
∙
∙+β
Y +ui
.
0+β
1Xi+β
2Y +∙
r
D)Yi=β
X1i×X2i
)+ui
.
0+β
1X1i+β
2X2+β
3(
Ans
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15)Fort
hepol
ynomi
alr
e
gr
e
s
s
i
onmode
l
,
A)youne
e
dne
we
s
t
i
ma
t
i
ont
e
c
hni
que
ss
i
nc
et
heOLSas
s
umpt
i
onsdonotappl
yanyl
onge
r
.
B)t
het
e
c
hni
que
sf
ore
s
t
i
ma
t
i
onandi
nf
e
r
e
nc
ede
ve
l
ope
df
ormul
t
i
pl
er
e
gr
e
s
s
i
onc
anbeappl
i
e
d.
C)youc
ans
t
i
l
lus
eOLSe
s
t
i
ma
t
i
ont
e
c
hni
que
s
,butt
het
s
t
a
t
i
s
t
i
c
sdonotha
veanas
ympt
ot
i
cnor
mal
di
s
t
r
i
but
i
on.
D)t
hec
r
i
t
i
c
alval
ue
sf
r
om t
henor
maldi
s
t
r
i
but
i
onha
vet
obec
hange
dt
o1.
962,1.
963,e
t
c
.
Ans
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16)Tot
e
s
twhe
t
he
rornott
hepopul
a
t
i
onr
e
gr
e
s
s
i
onf
unc
t
i
oni
sl
i
ne
arr
a
t
he
rt
hanapol
ynomi
alofor
de
rr
,
2
A)c
he
c
kwhe
t
he
rt
her
e
gr
e
s
s
i
onR f
ort
hepol
ynomi
alr
e
gr
e
s
s
i
oni
shi
ghe
rt
hant
ha
toft
hel
i
ne
ar
r
e
gr
e
s
s
i
on.
B)c
ompar
et
heTSSf
r
om bot
hr
e
gr
e
s
s
i
ons
.
C)l
ooka
tt
hepa
t
t
e
r
noft
hec
oe
ffic
i
e
nt
s
:i
ft
he
yc
hangef
r
om pos
i
t
i
vet
one
ga
t
i
vet
opos
i
t
i
ve
,e
t
c
.
,t
he
nt
he
pol
ynomi
alr
e
gr
e
s
s
i
ons
houl
dbeus
e
d.
et
het
e
s
tof(
r
1)r
e
s
t
r
i
c
t
i
onsus
i
ngt
heFs
t
a
t
i
s
t
i
c
.
D)us
Ans
we
r
:D
17)Thebe
s
twa
yt
oi
nt
e
r
pr
e
tpol
ynomi
alr
e
gr
e
s
s
i
onsi
st
o
A)t
akeade
r
i
va
t
i
veofYwi
t
hr
e
s
pe
c
tt
ot
her
e
l
e
vantX.
B)pl
ott
hee
s
t
i
ma
t
e
dr
e
gr
e
s
s
i
onf
unc
t
i
onandt
oc
al
c
ul
a
t
et
hee
s
t
i
ma
t
e
de
ffe
c
tonYas
s
oc
i
a
t
e
dwi
t
ha
c
hangei
nXf
oroneormor
eval
ue
sofX.
C)l
ooka
tt
het
s
t
a
t
i
s
t
i
c
sf
ort
her
e
l
e
vantc
oe
ffic
i
e
nt
s
.
D)anal
yzet
hes
t
andar
de
r
r
orofe
s
t
i
ma
t
e
de
ffe
c
t
.
Ans
we
r
:B
18)Thee
xpone
nt
i
alf
unc
t
i
on
A)i
st
hei
nve
r
s
eoft
hena
t
ur
all
ogar
i
t
hm f
unc
t
i
on.
B)doe
snotpl
a
yani
mpor
t
antr
ol
ei
nmode
l
i
ngnonl
i
ne
arr
e
gr
e
s
s
i
onf
unc
t
i
onsi
ne
c
onome
t
r
i
c
s
.
x
C)c
anbewr
i
t
t
e
nase
xp(
e)
.
x,whe
D)i
se
r
eei
s3.
1415….
Ans
we
r
:A
4
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i
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ar
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a
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i
on,I
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.
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19)Thef
ol
l
owi
ngar
epr
ope
r
t
i
e
soft
hel
ogar
i
t
hm f
unc
t
i
onwi
t
ht
hee
x
c
e
pt
i
onof
A)l
n(
1/x)=l
n(
x)
.
B)l
n(
a+x)=l
n(
a
)+l
n(
x)
.
C)l
n(
a
x)=l
n(
a
)+l
n(
x)
.
D)l
n(
xa
)al
n(
x)
.
Ans
we
r
:B
20)Thebi
nar
yvar
i
abl
ei
nt
e
r
a
c
t
i
onr
e
gr
e
s
s
i
on
A)c
anonl
ybeappl
i
e
dwhe
nt
he
r
ear
et
wobi
nar
yvar
i
abl
e
s
,butnott
hr
e
eormor
e
.
B)i
st
hes
ameast
e
s
t
i
ngf
ordi
ffe
r
e
nc
e
si
nme
ans
.
C)c
annotbeus
e
dwi
t
hl
ogar
i
t
hmi
cr
e
gr
e
s
s
i
onf
unc
t
i
onsbe
c
aus
el
n(
0)i
snotde
fine
d.
D)al
l
owst
hee
ffe
c
tofc
hangi
ngoneoft
hebi
nar
yi
nde
pe
nde
ntvar
i
abl
e
st
ode
pe
ndont
heval
ueoft
he
ot
he
rbi
nar
yvar
i
abl
e
.
Ans
we
r
:D
21)I
nt
her
e
gr
e
s
s
i
onmode
lYi=β
Xi×Di
)+ui
,whe
r
eXi
sac
ont
i
nuousvar
i
abl
eand
0+β
1Xi+β
2Di+β
3(
Di
sabi
nar
yvar
i
abl
e
,β
3
A)i
ndi
c
a
t
e
st
hes
l
opeoft
her
e
gr
e
s
s
i
onwhe
nD=1.
B)hasas
t
andar
de
r
r
ort
ha
ti
snotnor
mal
l
ydi
s
t
r
i
but
e
de
ve
ni
nl
ar
ges
ampl
e
ss
i
nc
eDi
snotanor
mal
l
y
di
s
t
r
i
but
e
dvar
i
abl
e
.
C)i
ndi
c
a
t
e
st
hedi
ffe
r
e
nc
ei
nt
hes
l
ope
soft
het
wor
e
gr
e
s
s
i
ons
.
D)hasnome
ani
ngs
i
nc
e(
Xi×Di
)=0whe
nDi=0.
Ans
we
r
:C
22)I
nt
her
e
gr
e
s
s
i
onmode
lYi=β
Xi×Di
)+ui
,whe
r
eXi
sac
ont
i
nuousvar
i
abl
eand
0+β
1Xi+β
2Di+β
3(
Di
sabi
nar
yvar
i
abl
e
,β
2
A)i
st
hedi
ffe
r
e
nc
ei
nme
ansi
nYbe
t
we
e
nt
het
woc
a
t
e
gor
i
e
s
.
B)i
ndi
c
a
t
e
st
hedi
ffe
r
e
nc
ei
nt
hei
nt
e
r
c
e
pt
soft
het
wor
e
gr
e
s
s
i
ons
.
C)i
sus
ual
l
ypos
i
t
i
ve
.
D)i
ndi
c
a
t
e
st
hedi
ffe
r
e
nc
ei
nt
hes
l
ope
soft
het
wor
e
gr
e
s
s
i
ons
.
Ans
we
r
:B
23)I
nt
her
e
gr
e
s
s
i
onmode
lYi=β
Xi×Di
)+ui
,whe
r
eXi
sac
ont
i
nuousvar
i
abl
eand
0+β
1Xi+β
2Di+β
3(
Di
sabi
nar
yvar
i
abl
e
,t
ot
e
s
tt
ha
tt
het
wor
e
gr
e
s
s
i
onsar
ei
de
nt
i
c
al
,youmus
tus
et
he
A)t
s
t
a
t
i
s
t
i
cs
e
par
a
t
e
l
yf
orβ
2=0,β
2=0.
B)Fs
t
a
t
i
s
t
i
cf
ort
hej
oi
nthypot
he
s
i
st
ha
tβ
β
0=0,
1=0.
C)t
s
t
a
t
i
s
t
i
cs
e
par
a
t
e
l
yf
orβ
3=0.
D)Fs
t
a
t
i
s
t
i
cf
ort
hej
oi
nthypot
he
s
i
st
ha
tβ
2=0,β
3=0.
Ans
we
r
:D
5
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i
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ar
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a
t
i
on,I
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.
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24)I
nt
hemode
lYi=β
X1×X2)+ui
,t
hee
xpe
c
t
e
de
ffe
c
t
0+β
1X1+β
2X2+β
3(
i
s
A)β
1+β
3X2.
B)β
.
1
C)β
1+β
3.
D)β
+
β
1 3X1.
Ans
we
r
:A
25)I
nt
hel
ogl
ogmode
l
,t
hes
l
opec
oe
ffic
i
e
nti
ndi
c
a
t
e
s
A)t
hee
ffe
c
tt
ha
tauni
tc
hangei
nXhasonY.
B)t
hee
l
as
t
i
c
i
t
yofYwi
t
hr
e
s
pe
c
tt
oX.
C)ΔY/ΔX.
D)
× .
Ans
we
r
:B
26)I
nt
hemode
ll
n(
Yi
)=β
,t
hee
l
as
t
i
c
i
t
yofE(
Y|X)wi
t
hr
e
s
pe
c
tt
oXi
s
0+β
1Xi+ui
A)β
1X
B)β
1
C)
D)Cannotbec
al
c
ul
a
t
e
dbe
c
aus
et
hef
unc
t
i
oni
snonl
i
ne
ar
Ans
we
r
:A
27)As
s
umet
ha
tyouhade
s
t
i
ma
t
e
dt
hef
ol
l
owi
ngquadr
a
t
i
cr
e
gr
e
s
s
i
onmode
l
2
=607.
3+3.
85I
nc
o
me-0.
0423I
nc
o
me.I
fi
nc
omei
nc
r
e
as
e
df
r
om 10t
o11(
$10,
000t
o$11
,
000)
,t
he
n
t
hepr
e
di
c
t
e
de
ffe
c
tont
e
s
t
s
c
or
e
swoul
dbe
A)3.
85
B)3.
850.
0423
C)Cannotbec
al
c
ul
a
t
e
dbe
c
aus
et
hef
unc
t
i
oni
snonl
i
ne
ar
D)2.
96
Ans
we
r
:D
28)Cons
i
de
rt
hepol
ynomi
alr
e
gr
e
s
s
i
onmode
lofde
gr
e
eYi=β
0+β
1Xi+β
2
+.
.
.
+β
r
+ui
.Ac
c
or
di
ng
t
ot
henul
lhypot
he
s
i
st
ha
tt
her
e
gr
e
s
s
i
oni
sl
i
ne
arandt
heal
t
e
r
na
t
i
vet
ha
ti
sapol
ynomi
alofde
gr
e
er
c
or
r
e
s
pondst
o
A)H0:β
.β
r=0vs
r≠ 0
B)H0:β
.β
r=0vs
1≠ 0
C)H0:β
=
0
,
.
.
.
,
β
.H1:al
lβ
≠ 0,j
=3,.
.
.
,r
3
r=0,vs
j
D)H0:β
.
.
,β
.
H1:a
tl
e
as
toneβ
≠ 0,j
=2,.
.
.
,r
2=0,β
3=0.
r=0,vs
j
Ans
we
r
:D
6
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i
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ar
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a
t
i
on,I
nc
.
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29)Cons
i
de
rt
hef
ol
l
owi
ngl
e
as
ts
quar
e
ss
pe
c
i
fic
a
t
i
onbe
t
we
e
nt
e
s
t
s
c
or
e
sandt
hes
t
ude
nt
t
e
ac
he
rr
a
t
i
o:
=557.
8+36.
42l
n(
I
nc
o
me
)
.Ac
c
or
di
ngt
ot
hi
se
qua
t
i
on,a1% i
nc
r
e
as
ei
nc
omei
sas
s
oc
i
a
t
e
dwi
t
h
ani
nc
r
e
as
ei
nt
e
s
ts
c
or
e
sof
A)0.
36poi
nt
s
B)36.
42poi
nt
s
C)557.
8poi
nt
s
D)c
annotbede
t
e
r
mi
ne
df
r
om t
hei
nf
or
ma
t
i
ongi
ve
nhe
r
e
Ans
we
r
:A
30)Cons
i
de
rt
hepopul
a
t
i
onr
e
gr
e
s
s
i
onofl
oge
ar
ni
ngs[
Yi
,whe
r
eYi=l
n(
Ea
r
ni
ng
s
)
]a
gai
ns
tt
wobi
nar
y
i
t
hpe
var
i
abl
e
s
:whe
t
he
rawor
ke
ri
smar
r
i
e
d(
D1i
,whe
r
eD1i
=1i
ft
hei
r
s
oni
smar
r
i
e
d)andt
hewor
ke
r
'
s
t
hpe
ge
nde
r(
D2i
,whe
r
eD2i
=1i
ft
hei
r
s
oni
sf
e
mal
e
)
,andt
hepr
oduc
toft
het
wobi
nar
yvar
i
abl
e
s
Yi=β0+β
D
+
β
D
+
β
(
D
×
D
)
+
u
.
T
h
e
i
n
t
e
r
a
c
t
i
o
n
t
e
r
m
1 1i 2 2i 3 1i 2i i
A)al
l
owst
hepopul
a
t
i
one
ffe
c
tonl
oge
ar
ni
ngsofbe
i
ngmar
r
i
e
dt
ode
pe
ndonge
nde
r
B)doe
snotmakes
e
ns
es
i
nc
ei
tc
oul
dbez
e
r
of
ormar
r
i
e
dmal
e
s
C)i
ndi
c
a
t
e
st
hee
ffe
c
tofbe
i
ngmar
r
i
e
donl
oge
ar
ni
ngs
D)c
annotbee
s
t
i
ma
t
e
dwi
t
houtt
hepr
e
s
e
nc
eofac
ont
i
nuousvar
i
abl
e
Ans
we
r
:A
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i
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ar
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.
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8.
2 Es
s
a
ysandLonge
rQue
s
t
i
ons
1)Fe
mal
e
s
,i
ti
ss
ai
d,make70c
e
nt
st
ot
hedol
l
ari
nt
heUni
t
e
dSt
a
t
e
s
.Toi
nve
s
t
i
ga
t
et
hi
sphe
nome
non,you
c
ol
l
e
c
tda
t
aonwe
e
kl
ye
ar
ni
ngsf
r
om 1,
744i
ndi
vi
dual
s
,850f
e
mal
e
sand894mal
e
s
.Ne
xt
,youc
al
c
ul
a
t
e
t
he
i
ra
ve
r
a
gewe
e
kl
ye
ar
ni
ngsandfindt
ha
tt
hef
e
ma
l
e
si
nyours
ampl
ee
ar
ne
d$346.
98,whi
l
et
hemal
e
s
made$517.
70.
(
a)Cal
c
ul
a
t
et
hef
e
mal
ee
ar
ni
ngsi
npe
r
c
e
ntoft
hemal
ee
ar
ni
ngs
.How woul
dyout
e
s
twhe
t
he
rornott
hi
s
di
ffe
r
e
nc
ei
ss
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
?Gi
vet
woappr
oac
he
s
.
(
b)Ape
e
rs
ugge
s
t
st
ha
tt
hi
si
sc
ons
i
s
t
e
ntwi
t
ht
hei
de
at
ha
tt
he
r
ei
sdi
s
c
r
i
mi
na
t
i
ona
gai
ns
tf
e
mal
e
si
nt
he
l
abormar
ke
t
.Wha
ti
syourr
e
s
pons
e
?
(
c
)Your
e
c
al
lf
r
om yourt
e
xt
bookt
ha
taddi
t
i
onalye
ar
sofe
xpe
r
i
e
nc
ear
es
uppos
e
dt
or
e
s
ul
ti
nhi
ghe
r
e
ar
ni
ngs
.Your
e
as
ont
ha
tt
hi
si
sbe
c
aus
ee
xpe
r
i
e
nc
ei
sr
e
l
a
t
e
dt
o"
ont
hej
obt
r
ai
ni
ng.
"Onef
r
e
que
nt
l
yus
e
d
me
as
ur
ef
or(
pot
e
nt
i
al
)e
xpe
r
i
e
nc
ei
s"
Age
Educ
a
t
i
on6.
"Expl
ai
nt
heunde
r
l
yi
ngr
a
t
i
onal
e
.As
s
umi
ng,
he
r
oi
c
al
l
y
,t
ha
te
duc
a
t
i
oni
sc
ons
t
antac
r
os
st
he1,
744i
ndi
vi
dual
s
,youc
ons
i
de
rr
e
gr
e
s
s
i
nge
ar
ni
ngsona
ge
andabi
nar
yvar
i
abl
ef
orge
nde
r
.Youe
s
t
i
ma
t
et
wos
pe
c
i
fic
a
t
i
onsi
ni
t
i
al
l
y:
=323.
70+5.
15×Ag
e–169.
78×Fe
ma
l
e
,R2=0.
13,SER=274.
75
(
21.
18) (
0.
55)
(
13.
06)
=5.
44+0.
015×Ag
e–0.
421×Fe
ma
l
e
,R2=0.
17,SER=0.
75
(
0.
08)(
0.
002)
(
0.
036)
whe
r
eEa
r
nar
ewe
e
kl
ye
ar
ni
ngsi
ndol
l
ar
s
,Ag
ei
sme
as
ur
e
di
nye
ar
s
,andFe
ma
l
ei
sabi
nar
yvar
i
abl
e
,
whi
c
ht
ake
sont
heval
ueofonei
ft
hei
ndi
vi
duali
saf
e
mal
eandi
sze
r
oot
he
r
wi
s
e
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nt
e
r
pr
e
te
ac
h
r
e
gr
e
s
s
i
onc
ar
e
f
ul
l
y
.Foragi
ve
na
ge
,how muc
hl
e
s
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9
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2)Ane
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10
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.
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Ans
we
r
:
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a)Aonepe
r
c
e
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r
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nve
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t
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Ans
we
r
:
(
a)Thee
r
r
orvar
i
anc
eandt
hevar
i
anc
eoft
hede
pe
nde
ntvar
i
abl
ear
er
e
l
a
t
e
d.Gi
ve
nt
ha
tt
hede
pe
nde
nt
var
i
abl
e(
e
ar
ni
ngs
)i
snotnor
mal
l
ydi
s
t
r
i
but
e
d,i
ti
sdi
ffic
ul
tt
opos
t
ul
a
t
et
ha
tt
hee
r
r
orvar
i
anc
ei
s
nor
mal
l
ydi
s
t
r
i
but
e
d.Us
i
ngl
ogar
i
t
hmsr
e
s
ul
t
si
nadi
s
t
r
i
but
i
ont
ha
ti
sc
l
os
e
rt
oanor
mal
.I
naddi
t
i
on,
t
he
r
es
e
e
mst
obeabe
t
t
e
rfitf
ort
hel
ogl
i
ne
ars
pe
c
i
fic
a
t
i
on,andt
hec
oe
ffic
i
e
nt
sc
anbei
nt
e
r
pr
e
t
e
das
pe
r
c
e
nt
a
gec
hange
s
.
(
b)The
r
es
e
e
mst
obeapa
t
t
e
r
ni
nt
her
e
s
i
dual
swhe
ns
or
t
e
dbya
ge
.Thi
ss
ugge
s
t
sami
s
s
pe
c
i
fie
d
f
unc
t
i
onalf
or
m.
(
c
)Ac
c
or
di
ngt
ot
hes
pe
c
i
fic
a
t
i
on,e
ar
ni
ngsi
nc
r
e
as
ewi
t
ha
geunt
i
lt
hei
ndi
vi
duali
s39ye
ar
sol
d.I
ti
sonl
y
f
r
om a
ge40onwar
dst
ha
tt
her
e
gr
e
s
s
i
onpr
e
di
c
t
sane
ga
t
i
ver
e
l
a
t
i
ons
hi
pbe
t
we
e
ne
ar
ni
ngsanda
ge
.
Ac
c
or
di
ngt
ot
hee
s
t
i
ma
t
e
s
,a30ye
a
r
ol
dwoul
dha
vel
n(
e
a
r
ni
ng
s
)of5.
77,whi
l
et
hepr
e
di
c
t
e
dval
uef
ora
50ye
ar
ol
dwoul
dbe5.
97.Thedi
ffe
r
e
nc
ebe
t
we
e
nt
het
woi
sappr
oxi
ma
t
e
l
y20pe
r
c
e
nt
.
(
d)Thec
r
i
t
i
c
alval
uef
r
om t
heFt
abl
ei
s4.
61a
tt
he1% l
e
ve
l
.He
nc
et
henul
lhypot
he
s
i
si
sr
e
j
e
c
t
e
d.
(
e
)I
ns
t
e
adoft
hei
nve
r
t
e
dVs
hapef
ort
heabover
e
gr
e
s
s
i
on,ani
nve
r
t
e
dUs
hapewoul
dmos
tl
i
ke
l
y
pr
oduc
eabe
t
t
e
rfit
.Thi
sc
anbege
ne
r
a
t
e
dt
hr
ought
heus
eofapol
ynomi
alr
e
gr
e
s
s
i
onmode
lofde
gr
e
e2.
16
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i
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ar
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a
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5)Spor
t
se
c
onomi
c
st
ypi
c
al
l
yl
ooksa
twi
nni
ngpe
r
c
e
nt
a
ge
sofs
por
t
st
e
amsasoneofvar
i
ousout
put
s
,and
e
s
t
i
ma
t
e
spr
oduc
t
i
onf
unc
t
i
onsbyana
l
yzi
ngt
her
e
l
a
t
i
ons
hi
pbe
t
we
e
nt
hewi
nni
ngpe
r
c
e
nt
a
geandi
nput
s
.
I
nMa
j
orLe
a
gueBas
e
bal
l(
MLB)
,t
hede
t
e
r
mi
nant
sofwi
nni
ngar
equal
i
t
ypi
t
c
hi
ngandba
t
t
i
ng.Al
l30
MLBt
e
amsf
ort
he1999s
e
as
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t
c
hi
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i
t
yi
sappr
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ma
t
e
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Te
am Ear
ne
dRunAve
r
a
ge
"(
ERA)
,
andhi
t
t
i
ngqual
i
t
yby"
OnBas
ePl
usSl
uggi
ngPe
r
c
e
nt
a
ge
"(
OPS)
.
Summar
yoft
heDi
s
t
r
i
but
i
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nni
ngPer
c
ent
age
,OnBas
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usSl
uggi
ngPer
c
ent
age
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andTe
am Ea
r
nedRunAver
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orMLBi
n1999
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d
Per
c
ent
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l
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t
i
on
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53
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72 4.
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4.
91 5.
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720 0.
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ng 0.
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0.
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0.
40 0.
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48
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60
Pe
r
c
e
nt
a
ge
Yourr
e
gr
e
s
s
i
onout
puti
s
:
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099×t
e
a
me
r
a+1.
490×o
ps
,R2=0.
92,SER=0.
02.
(
0.
08)(
0.
008)
(
0.
126)
(
a)I
nt
e
r
pr
e
tt
her
e
gr
e
s
s
i
on.Ar
et
her
e
s
ul
t
ss
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
antandi
mpor
t
ant
?
(
b)The
r
ear
et
wol
e
a
gue
si
nMLB,t
heAme
r
i
c
anLe
a
gue(
AL)andt
heNa
t
i
onalLe
a
gue(
NL)
.Onema
j
or
di
ffe
r
e
nc
ei
st
ha
tt
hepi
t
c
he
ri
nt
heALdoe
snotha
vet
oba
t
.I
ns
t
e
adt
he
r
ei
sa"
de
s
i
gna
t
e
dhi
t
t
e
r
"i
nt
he
hi
t
t
i
ngl
i
ne
up.Youar
ec
onc
e
r
ne
dt
ha
t
,asar
e
s
ul
t
,t
he
r
ei
sadi
ffe
r
e
nte
ffe
c
tofpi
t
c
hi
ngandhi
t
t
i
ngi
nt
he
ALf
r
om t
heNL.Tot
e
s
tt
hi
shypot
he
s
i
s
,
youal
l
ow t
heALr
e
gr
e
s
s
i
ont
oha
veadi
ffe
r
e
nti
nt
e
r
c
e
ptand
di
ffe
r
e
nts
l
ope
sf
r
om t
heNLr
e
gr
e
s
s
i
on.Yout
he
r
e
f
or
ec
r
e
a
t
eabi
nar
yvar
i
abl
ef
ort
heAme
r
i
c
anLe
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gue
(
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s
t
i
ma
t
et
hef
ol
l
owi
ngs
pe
c
i
fic
a
t
i
on:
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10×DAL–0.
100×t
e
a
me
r
a+0.
008×(
DAL×t
e
a
me
r
a
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12)(
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008)
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o
ps–0.
187*
(
DAL×o
ps
),R2=0.
92,
SER=0.
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(
0.
163) (
0.
160)
Wha
ti
st
her
e
gr
e
s
s
i
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r
c
e
nt
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xt
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et
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a
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i
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r
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we
e
nt
het
wol
e
a
gue
s
,wha
twoul
dt
her
e
s
ul
t
si
mpl
yi
fal
lc
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e
nt
si
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vi
ng DAL
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r
es
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
?
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i
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ar
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i
on,I
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.
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(
c
)Your
e
me
mbe
rt
ha
ts
e
que
nt
i
al
l
yt
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s
t
i
ngt
hes
i
gni
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anc
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l
opec
oe
ffic
i
e
nt
si
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hes
ameast
e
s
t
i
ng
f
ort
he
i
rs
i
gni
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anc
es
i
mul
t
ane
ous
l
y
.He
nc
eyouas
kyourr
e
gr
e
s
s
i
onpac
ka
get
oc
al
c
ul
a
t
et
heFs
t
a
t
i
s
t
i
c
t
ha
tal
lt
hr
e
ec
oe
ffic
i
e
nt
si
nvol
vi
ngt
hebi
nar
yvar
i
abl
ef
ort
heALar
ez
e
r
o.Yourr
e
gr
e
s
s
i
onpac
ka
gegi
ve
sa
val
ueof0.
35.Looki
nga
tt
hec
r
i
t
i
c
alval
uef
r
om youFt
abl
e
,c
anyour
e
j
e
c
tt
henul
lhypot
he
s
i
sa
tt
he1%
l
e
ve
l
?Shoul
dyouwor
r
yaboutt
hes
mal
ls
ampl
es
i
ze
?
Ans
we
r
:
(
a)Lowe
r
i
ngt
het
e
am ERAbyoner
e
s
ul
t
si
nawi
nni
ngpe
r
c
e
nt
a
gei
nc
r
e
as
eofr
oughl
yt
e
npe
r
c
e
nt
.
I
nc
r
e
as
i
ngt
heOPSby0.
1ge
ne
r
a
t
e
sahi
ghe
rwi
nni
ngpe
r
c
e
nt
a
geofappr
oxi
ma
t
e
l
y15pe
r
c
e
nt
.The
r
e
gr
e
s
s
i
one
xpl
ai
ns92pe
r
c
e
ntoft
hevar
i
a
t
i
oni
nwi
nni
ngpe
r
c
e
nt
a
ge
s
.Bot
hs
l
opec
oe
ffic
i
e
nt
sar
e
s
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
,andgi
ve
nt
hes
mal
ldi
ffe
r
e
nc
e
si
nwi
nni
ngpe
r
c
e
nt
a
ge
,t
he
yar
eal
s
oi
mpor
t
ant
.
(
b)NL:
=–0.
29–0.
100×t
e
a
me
r
a+1.
622×o
ps
.
AL:
=–0.
19–0.
092×t
e
a
me
r
a+1.
435×o
ps
.
Thet
s
t
a
t
i
s
t
i
c
sf
oral
lvar
i
abl
e
si
nvol
vi
ngDALar
e
,i
nor
de
rofappe
ar
anc
ei
nt
heabover
e
gr
e
s
s
i
on,0.
42,
0.
44,and–1.
17.Noneoft
hec
oe
ffic
i
e
nt
si
ss
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
anti
ndi
vi
dual
l
y
.I
ft
he
s
ewe
r
es
t
a
t
i
s
t
i
c
al
l
y
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i
gni
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ant
,
t
he
nt
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c
a
t
et
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tt
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ffic
i
e
nt
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t
we
e
nt
het
wol
e
a
gue
s
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nc
ei
twoul
d
s
ugge
s
tt
ha
tt
hei
nt
r
oduc
t
i
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s
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t
e
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e
l
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t
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)Thec
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i
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i
c
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t
a
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i
s
t
i
ci
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tt
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e
ve
l
,andhe
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e
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e
c
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henul
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s
i
s
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e
ec
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e
nt
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e
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ve
r
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t
a
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i
s
t
i
ci
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e
al
l
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s
t
r
i
but
e
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and,asar
e
s
ul
t
,i
nf
e
r
e
nc
ei
spr
obl
e
ma
t
i
che
r
e
.
18
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ar
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on,I
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.
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6)The
r
ehasbe
e
nmuc
hde
ba
t
eaboutt
hei
mpac
tofmi
ni
mum wa
ge
sone
mpl
oyme
ntandune
mpl
oyme
nt
.
Whi
l
emos
toft
hef
oc
ushasbe
e
nont
hee
mpl
oyme
nt
t
opopul
a
t
i
onr
a
t
i
ooft
e
e
na
ge
r
s
,youde
c
i
det
oc
he
c
k
i
fa
ggr
e
ga
t
es
t
a
t
eune
mpl
oyme
ntr
a
t
e
sha
vebe
e
na
ffe
c
t
e
d.Youri
de
ai
st
os
e
ei
fs
t
a
t
eune
mpl
oyme
ntr
a
t
e
s
f
ort
he48c
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a
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oyme
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a
t
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ort
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ames
t
a
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e
si
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ft
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spr
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mpac
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a
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ar
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t
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e
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e
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i
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e
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nt
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l
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put
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=3.
19+0.
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,R2=0.
21,SER=1.
031
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0.
56) (
0.
07)
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a)Ske
t
c
ht
her
e
gr
e
s
s
i
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i
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i
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ot
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r
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r
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t
ht
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i
ne
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i
abl
eDhi
I
mpa
c
tbyal
l
owi
ngt
hes
l
opeandi
nt
e
r
c
e
ptt
odi
ffe
r
,r
e
s
ul
t
si
nt
he
f
ol
l
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ngfit
t
e
dl
i
ne
:
=4.
02+0.
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(
0.
66) (
0.
09)
–3.
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I
mpa
c
t+0.
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Dh
i
I
mpa
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t
×
(
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89)
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t
i
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nvol
vi
ngt
hehi
ghi
mpac
tmi
ni
mum wa
ge
var
i
abl
ear
eze
r
o,i
s42.
16.Canyour
e
j
e
c
tt
henul
lhypot
he
s
i
st
ha
tbot
hc
oe
ffic
i
e
nt
sar
ez
e
r
o?Ske
t
c
ht
het
wo
r
e
gr
e
s
s
i
onl
i
ne
st
oge
t
he
rwi
t
ht
he450l
i
neandi
nt
e
r
pr
e
tt
her
e
s
ul
t
sa
gai
n.
(
c
)Toc
he
c
kt
her
obus
t
ne
s
soft
he
s
er
e
s
ul
t
s
,your
e
pe
a
tt
hee
x
e
r
c
i
s
eus
i
ngane
w bi
nar
yvar
i
abl
ef
ort
hes
oc
al
l
e
dmi
ni
ngs
t
a
t
e(
Dmi
ni
ng)
,i
.
e
.
,t
hee
l
e
ve
ns
t
a
t
e
st
ha
tha
vea
tl
e
as
tt
hr
e
epe
r
c
e
ntoft
he
i
rt
ot
als
t
a
t
e
e
ar
ni
ngsde
r
i
ve
df
r
om oi
l
,gase
xt
r
ac
t
i
on,andc
oalmi
ni
ng,i
nt
he1980s
.Thi
sr
e
s
ul
t
si
nt
hef
ol
l
owi
ng
out
put
:
=4.
04+0.
15×
(
0.
65) (
0.
09)
–2.
92×Dmi
ni
ng+0.
37×(
Dmi
ni
ng×
(
0.
90)
)
,
(
0.
10)
R2=0.
31,SER=0.
997
How c
onfide
ntar
eyout
ha
tt
hepr
e
vi
ous
l
yf
ounde
ffe
c
ti
sduet
omi
ni
mum wa
ge
s
?
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Ans
we
r
:
(
a)Ani
nc
r
e
as
ei
nt
he1985une
mpl
oyme
ntr
a
t
er
e
s
ul
t
si
nani
nc
r
e
as
ei
nt
heune
mpl
oyme
ntr
a
t
ei
n1995of
0.
27pe
r
c
e
nt
.Putdi
ffe
r
e
nt
l
y
,i
fones
t
a
t
ehadaonepe
r
c
e
nthi
ghe
rune
mpl
oyme
ntr
a
t
ei
n1985t
hananot
he
r
s
t
a
t
e
,t
he
nt
hi
sdi
ffe
r
e
nc
ewoul
ds
hr
i
nk,ona
ve
r
a
ge
,t
o0.
27pe
r
c
e
nti
n1995.21pe
r
c
e
ntoft
hevar
i
a
t
i
oni
n
1995s
t
a
t
eune
mpl
oyme
ntr
a
t
e
si
se
xpl
ai
ne
dbyt
her
e
gr
e
s
s
i
on.I
ft
hefit
t
e
dl
i
nec
oi
nc
i
de
dwi
t
ht
he450l
i
ne
,
t
he
nt
heune
mpl
oyme
ntr
a
t
e
si
n1995woul
dr
e
mai
nunc
hange
dwhe
nc
ompar
e
dt
o1985.Thee
s
t
i
ma
t
e
d
r
e
gr
e
s
s
i
oni
mpl
i
e
s
,unr
e
al
i
s
t
i
c
al
l
y
,me
anr
e
ve
r
s
i
oni
nt
heune
mpl
oyme
ntr
a
t
e
s
.
(
b)Thec
r
i
t
i
c
alval
uef
ort
heFs
t
a
t
i
s
t
i
ci
s4.
61a
tt
he1% l
e
ve
landhe
nc
et
henul
lhypot
he
s
i
st
ha
tbot
h
c
oe
ffic
i
e
nt
sar
ez
e
r
oi
nt
hepopul
a
t
i
oni
sr
e
j
e
c
t
e
d.(
Thes
ampl
es
i
zei
ss
mal
l
,howe
ve
r
,s
ot
hedi
s
t
r
i
but
i
onof
t
het
e
s
ts
t
a
t
i
s
t
i
ci
snotr
e
al
l
yknown.
)Thei
nt
e
r
c
e
ptf
ort
hehi
ghi
mpac
ts
t
a
t
e
si
ss
mal
l
e
randt
hes
l
opei
s
s
t
e
e
pe
r
.Thi
ss
ugge
s
t
st
ha
tf
orhi
ghi
mpac
ts
t
a
t
e
st
he
r
ei
sl
e
s
sofame
anr
e
ve
r
s
i
one
ffe
c
tpr
e
s
e
nt
:i
fhi
ghi
mpac
ts
t
a
t
e
shadhi
gh1985une
mpl
oyme
ntr
a
t
e
s
,t
he
nt
he
yar
ee
xpe
c
t
e
dt
oha
vehi
ghe
rune
mpl
oyme
nt
r
a
t
e
si
n1995whe
nc
ompar
e
dt
oal
owi
mpac
ts
t
a
t
e
.Hi
ghandl
ow une
mpl
oyme
ntr
a
t
e
sar
et
he
r
e
bymor
e
pe
r
s
i
s
t
e
ntf
orhi
ghi
mpac
ts
t
a
t
e
s
.
20
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ar
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(
c
)Ther
e
s
ul
t
she
r
ear
es
i
mi
l
art
ot
hos
ei
n(
b)i
nt
ha
tt
her
e
gr
e
s
s
i
onf
ort
hemi
ni
ngs
t
a
t
e
si
ss
t
e
e
pe
rt
hant
he
onef
ort
heot
he
rs
t
a
t
e
s
.Pe
r
hapsomi
t
t
e
dvar
i
abl
e
spl
a
yar
ol
ehe
r
e
,s
uc
hasr
e
l
a
t
i
ve(
oi
l
)pr
i
c
es
hoc
kst
ha
t
affe
c
ts
omes
t
a
t
e
smor
et
hanot
he
r
s
.Oi
lpr
i
c
e
sf
e
l
lc
ons
i
de
r
abl
yove
rt
het
i
mepe
r
i
odandi
ti
spos
s
i
bl
et
ha
t
t
hehi
ghi
mpac
tbi
nar
yvar
i
abl
ec
oe
ffic
i
e
ntpi
c
ksupt
hee
ffe
c
tofomi
t
t
e
dvar
i
abl
e
s
.I
nc
l
udi
ngmor
e
e
xpl
ana
t
or
yvar
i
abl
e
swoul
dbede
s
i
r
abl
e
.
21
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ar
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lOMoARcPSD|27111031
7)Labore
c
onomi
s
t
sha
vee
xt
e
ns
i
ve
l
yr
e
s
e
ar
c
he
dt
hede
t
e
r
mi
nant
sofe
ar
ni
ngs
.I
nve
s
t
me
nti
nhuman
c
api
t
al
,me
as
ur
e
di
nye
ar
sofe
duc
a
t
i
on,andont
hej
obt
r
ai
ni
ngar
es
omeoft
hemos
ti
mpor
t
ant
e
xpl
ana
t
or
yvar
i
abl
e
si
nt
hi
sr
e
s
e
ar
c
h.Youde
c
i
det
oappl
ye
ar
ni
ngsf
unc
t
i
onst
ot
hefie
l
dofs
por
t
s
e
c
onomi
c
sbyfindi
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hede
t
e
r
mi
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sf
orbas
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bal
lpi
t
c
he
rs
al
ar
i
e
s
.Youc
ol
l
e
c
tda
t
aon455pi
t
c
he
r
sf
or
t
he1998bas
e
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ls
e
as
onande
s
t
i
ma
t
et
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ol
l
owi
nge
qua
t
i
onus
i
ngOLSandhe
t
e
r
os
ke
das
t
i
c
i
t
yr
obus
t
s
t
andar
de
r
r
or
s
:
=12.
45+0.
052×Ye
a
r
s+0.
00089×I
nni
ng
s+0.
0032×Sa
v
e
s
(
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(
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0085×ERA,R2=0.
45,SER=0.
874
(
0.
0168)
whe
r
eEa
r
ni
sannuals
al
ar
yi
ndol
l
ar
s
,Ye
a
r
si
snumbe
rofye
ar
si
nt
hema
j
orl
e
a
gue
s
,I
nni
ng
si
snumbe
rof
i
nni
ngspi
t
c
he
ddur
i
ngt
hec
ar
e
e
rbe
f
or
et
he1998s
e
as
on,Sa
v
e
si
snumbe
rofs
a
ve
sdur
i
ngt
hec
ar
e
e
r
be
f
or
et
he1998s
e
as
on,andERAi
st
hee
ar
ne
dr
una
ve
r
a
gebe
f
or
et
he1998s
e
as
on.
(
a)Wha
thappe
nst
oe
ar
ni
ngswhe
nt
hepi
t
c
he
rs
t
a
ysi
nt
hel
e
a
guef
oroneaddi
t
i
onalye
ar
?Compar
et
he
s
al
ar
i
e
soft
wor
e
l
i
e
ve
r
s
,onewi
t
h10mor
es
a
ve
st
hant
heot
he
r
.Wha
te
ffe
c
tdoe
spi
t
c
hi
ng100mor
e
i
nni
ngsha
veont
hes
al
ar
yoft
hepi
t
c
he
r
?Wha
te
ffe
c
tdoe
sr
e
duc
i
nghi
sERA by1.
5?Dot
hes
i
gns
c
or
r
e
s
pondt
oyoure
xpe
c
t
a
t
i
ons
?Expl
ai
n.
(
b)Ar
et
hei
ndi
vi
dualc
oe
ffic
i
e
nt
ss
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
?I
ndi
c
a
t
et
hel
e
ve
lofs
i
gni
fic
anc
eyouus
e
dand
t
het
ypeofal
t
e
r
na
t
i
vehypot
he
s
i
syouc
ons
i
de
r
e
d.
(
c
)Al
t
houghyouar
equi
t
ei
mpr
e
s
s
e
dwi
t
ht
hefitoft
her
e
gr
e
s
s
i
on,s
ome
ones
ugge
s
t
st
ha
tyous
houl
d
i
nc
l
udet
hes
quar
eofye
ar
sandi
nni
ngsasaddi
t
i
onale
xpl
ana
t
or
yvar
i
abl
e
s
.Yourr
e
s
ul
t
sc
hangeas
f
ol
l
ows
:
=12.
15+0.
160×Ye
a
r
s+0.
00268×I
nni
ng
s+0.
0063×Sa
v
e
s
(
0.
05) (
0.
039)
(
0.
00030)
(
0.
0010)
2-0.
2
-0.
0584×ERA–0.
0165×Ye
ar
s
00000045×I
nni
ng
s
(
0.
0165)
(
0.
0026)
(
0.
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R2=0.
69,SER=0.
666
Wha
ti
she
rr
e
as
oni
ng?Ar
et
hec
oe
ffic
i
e
nt
soft
hequadr
a
t
i
ct
e
r
mss
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
?Ar
et
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y
me
ani
ngf
ul
?
(
d)Cal
c
ul
a
t
et
hee
ffe
c
tofmovi
ngf
r
om t
wot
ot
hr
e
eye
ar
s
,asoppos
e
dt
of
r
om 12t
o13ye
ar
s
.
(
e
)Youal
s
ode
c
i
det
ot
e
s
tt
hes
pe
c
i
fic
a
t
i
onf
ors
t
abi
l
i
t
yac
r
os
sl
e
a
gue
s(
Na
t
i
onalLe
a
gueandAme
r
i
c
a
n
Le
a
gue
)byi
nc
l
udi
ngadummyvar
i
abl
ef
ort
heNa
t
i
onalLe
a
gueandal
l
owi
ngt
hei
nt
e
r
c
e
ptandal
ls
l
ope
s
t
odi
ffe
r
.Ther
e
s
ul
t
i
ngFs
t
a
t
i
s
t
i
cf
orr
e
s
t
r
i
c
t
i
ngal
lc
oe
ffic
i
e
nt
st
ha
ti
nvol
vet
heNa
t
i
onalLe
a
guedummy
var
i
abl
et
oze
r
o,i
s0.
40.Compar
et
hi
st
ot
her
e
l
e
vantc
r
i
t
i
c
alval
uef
r
om t
het
abl
eandde
c
i
dewhe
t
he
ror
nott
he
s
eaddi
t
i
onalvar
i
abl
e
ss
houl
dbei
nc
l
ude
d.
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Ans
we
r
:
(
a)Fors
t
a
yi
nganaddi
t
i
onalye
ari
nt
hel
e
a
gue
,t
hepi
t
c
he
rr
e
c
e
i
ve
sa5.
2pe
r
c
e
nti
nc
r
e
as
ei
ne
ar
ni
ngs
.On
a
ve
r
a
ge
,t
her
e
l
i
e
ve
rwi
t
h10mor
es
a
ve
se
ndsupwi
t
h3.
2pe
r
c
e
nthi
ghe
re
ar
ni
ngs
.Pi
t
c
hi
ng100addi
t
i
onal
i
nni
ngsr
e
s
ul
t
si
n8.
9pe
r
c
e
nthi
ghe
re
ar
ni
ngs
,andl
owe
r
i
ngt
heERAby1.
5i
nc
r
e
as
e
se
ar
ni
ngsby1.
3
pe
r
c
e
nt
.ERA,i
nni
ngspi
t
c
he
d,andnumbe
rofs
a
ve
sar
eal
lqual
i
t
yofi
nputi
ndi
c
a
t
or
sands
houl
d
t
he
r
e
f
or
eha
vet
hes
i
gnsasi
nt
her
e
gr
e
s
s
i
onabove
.Ye
ar
si
nt
hema
j
orl
e
a
gue
ss
t
andsasapr
oxyf
oront
he
j
obt
r
ai
ni
ngands
houl
dt
he
r
e
f
or
ec
ar
r
yapos
i
t
i
ves
i
gn.
(
b)Gi
ve
nt
ha
tt
he
r
ei
spr
i
ore
xpe
c
t
a
t
i
onont
hes
i
gnoft
hec
oe
ffic
i
e
nt
s
,yous
houl
dc
onduc
taones
i
de
d
hypot
he
s
i
st
e
s
t
.Al
lvar
i
abl
e
swi
t
ht
hee
x
c
e
pt
i
onofERAc
ar
r
ys
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
antc
oe
ffic
i
e
nt
sa
tt
he5%
l
e
ve
l
.
(
c
)Al
l
owi
ngf
ort
hequadr
a
t
i
ct
e
r
mst
oe
nt
e
rr
e
s
ul
t
si
nani
nve
r
t
e
dUs
hapef
ort
her
e
l
a
t
i
ons
hi
pbe
t
we
e
n
t
hel
ogofe
ar
ni
ngs
,andbot
hye
ar
si
nt
hel
e
a
gueandi
nni
ngspi
t
c
he
d.Bot
hc
oe
ffic
i
e
nt
sar
ehi
ghl
y
s
i
gni
fic
antandha
ver
e
s
ul
t
e
dal
s
oi
nas
i
gni
fic
antERAc
oe
ffic
i
e
nt
.
(
d)Ha
vi
ngpl
a
ye
df
ort
woye
ar
sands
t
a
yi
ngf
oronemor
eye
a
ri
nt
hel
e
a
guer
e
s
ul
t
si
nane
ar
ni
ngsi
nc
r
e
as
e
of7.
8pe
r
c
e
nt
,whi
l
es
t
a
yi
ngf
oranaddi
t
i
onalye
a
raf
t
e
r12ye
ar
si
nt
hema
j
or
sr
e
s
ul
t
si
napr
e
di
c
t
e
d
de
c
r
e
as
eof25.
3pe
r
c
e
nt
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nCha
pt
e
r7.Thes
ol
ut
i
oni
st
ous
ei
ns
t
r
ume
nt
al
var
i
abl
et
e
c
hni
que
s
,al
s
odi
s
c
us
s
e
di
nCha
pt
e
r10.
Thel
i
t
e
r
a
t
ur
ei
nt
hi
sar
e
ahasdones
o,andt
he
c
onve
r
ge
nc
er
e
s
ul
tpe
r
s
i
s
t
s
.
29
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11)Pa
ge
s283284i
nyourt
e
xt
bookc
ont
ai
nananal
ys
i
soft
he"
Re
t
ur
nt
oEduc
a
t
i
onandt
heGe
nde
rGap.
"
Col
umn(
4)i
nTabl
e8.
1di
s
pl
a
ysr
e
gr
e
s
s
i
onr
e
s
ul
t
sus
i
ngt
he2009Cur
r
e
ntPopul
a
t
i
onSur
ve
y
.The
e
qua
t
i
onbe
l
ow s
howst
her
e
gr
e
s
s
i
onr
e
s
ul
tf
ort
hes
ames
pe
c
i
fic
a
t
i
on,butus
i
ngt
he2005Cur
r
e
nt
Popul
a
t
i
onSur
ve
y
.I
nt
e
r
pr
e
tt
hema
j
orr
e
s
ul
t
s
.
=1.
215+0.
0899×e
duc-0.
521×DFe
mme
+0.
0180×(
DFe
mme
×e
duc
)
(
0.
018)(
0.
0011)
(
0.
022)
(
0.
0016)
2-0.
+0.
0232×e
xpe
r-0.
000368×e
xpe
r
058×Mi
d
we
s
t-0.
0098×So
ut
h-0.
030×We
s
t
(
0.
0008)
(
0.
000018)
(
0.
006)
(
0.
0078)
(
0.
0030)
Ans
we
r
:Ther
e
t
ur
nt
oe
duc
a
t
i
onf
ormal
e
si
sappr
oxi
ma
t
e
l
y9% andi
t
sc
oe
ffic
i
e
nthasat
s
t
a
t
i
s
t
i
cof11.
25.
Forf
e
mal
e
s
,t
her
e
t
ur
ni
ss
l
i
ght
l
yhi
ghe
r
,appr
oxi
ma
t
e
l
y11%.Si
nc
et
hebi
nar
yvar
i
abl
ef
orf
e
mal
e
si
s
i
nt
e
r
ac
t
e
dwi
t
ht
henumbe
rofye
ar
sofe
duc
a
t
i
on,t
hege
nde
rgapde
pe
ndsont
henumbe
rofye
ar
sof
e
duc
a
t
i
on.Fort
het
ypi
c
alhi
ghs
c
hoolgr
adua
t
e(
12ye
ar
sofe
duc
a
t
i
on)
,t
hege
nde
rgapi
sappr
oxi
ma
t
e
l
y
27%,whi
l
ef
ort
het
ypi
c
alc
ol
l
e
gegr
adua
t
e(
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ar
sofe
duc
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t
i
on)t
hege
nde
rgapnar
r
owst
o19%.The
pot
e
nt
i
ale
xpe
r
i
e
nc
evar
i
abl
ee
nt
e
r
si
nani
nve
r
t
e
dUs
hape
,whi
c
hi
st
obee
xpe
c
t
e
dgi
ve
nt
hes
hapeof
a
ge
e
ar
ni
ngspr
ofil
e
sandt
hef
ac
tt
ha
tpot
e
nt
i
ale
xpe
r
i
e
nc
ede
pe
ndsont
hea
geoft
hei
ndi
vi
dual
.The
r
ei
s
ade
c
l
i
ni
ngmar
gi
nalval
uef
ore
ac
hye
arofpot
e
nt
i
ale
xpe
r
i
e
nc
eunt
i
li
te
ve
nt
ual
l
ybe
c
ome
sne
ga
t
i
ve
.
No
r
t
he
a
s
ti
st
heomi
t
t
e
dr
e
gi
on,andal
lot
he
rr
e
gi
onsha
vel
owe
r(
l
og)e
ar
ni
ngs
,r
angi
ngf
r
om 0.
8% i
nt
he
So
ut
ht
o5.
8% i
nt
heMi
d
we
s
t
.Al
lc
oe
ffic
i
e
nt
sar
es
t
a
t
i
s
t
i
c
al
l
ys
i
gni
fic
ant
.
30
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8.
3 Ma
t
he
ma
t
i
c
alandGr
aphi
c
alPr
obl
e
ms
1)Gi
vea
tl
e
as
tt
hr
e
ee
xampl
e
sf
r
om e
c
onomi
c
swhe
r
eyoue
xpe
c
ts
omenonl
i
ne
ar
i
t
yi
nt
her
e
l
a
t
i
ons
hi
p
be
t
we
e
nvar
i
abl
e
s
.I
nt
e
r
pr
e
tt
hes
l
opei
ne
ac
hc
as
e
.
Ans
we
r
:Ans
we
r
swi
l
lvar
ybys
t
ude
nt
.Typi
c
alans
we
r
si
nvol
vet
heCobbDougl
aspr
oduc
t
i
onf
unc
t
i
on,
t
hePhi
l
l
i
psc
ur
ve
,e
ar
ni
ngsf
unc
t
i
ons
,and(
gi
ve
nt
het
e
xt
bookdi
s
c
us
s
i
on)s
t
ude
ntpe
r
f
or
manc
eand
i
nc
ome
.
2)Sugge
s
tat
r
ans
f
or
ma
t
i
oni
nt
hevar
i
abl
e
st
ha
twi
l
ll
i
ne
ar
i
zet
hede
t
e
r
mi
ni
s
t
i
cpar
toft
hepopul
a
t
i
on
r
e
gr
e
s
s
i
onf
unc
t
i
onsbe
l
ow.Wr
i
t
et
her
e
s
ul
t
i
ngr
e
gr
e
s
s
i
onf
unc
t
i
oni
naf
or
mt
ha
tc
anbee
s
t
i
ma
t
e
dby
us
i
ngOLS.
(
a)Yi=β
0
(
b)Yi=
(
c
)Yi=
(
d)Yi=β
0
Ans
we
r
:(
a)l
n(
Yi
)=l
n(β
n(
X1i
)+ β
n(
X2i
)
0)+ β
1l
2l
(
b) =β
0
(
c
)l
n
+β
1
=β
0+β
1Xi
(
d)l
n(
Yi
)=l
n(β
n(
X1i
)+β
0)+β
1l
2X2i
3)I
ndi
c
a
t
ewhe
t
he
rornotyouc
anl
i
ne
ar
i
zet
her
e
gr
e
s
s
i
onf
unc
t
i
onsbe
l
ow s
ot
ha
tOLSe
s
t
i
ma
t
i
on
me
t
hodsc
anbeappl
i
e
d:
(
a)Yi=
(
b)Yi=
+ui
Ans
we
r
:
(
a)Thef
unc
t
i
onc
anbel
i
ne
ar
i
ze
dbyt
aki
ngl
ogsonbot
hs
i
de
s
.
(
b)Thef
unc
t
i
onc
annotbel
i
ne
ar
i
ze
dduet
ot
headdi
t
i
vee
r
r
ort
e
r
m.
31
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i
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ar
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i
on,I
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.
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4)Choos
ea
tl
e
as
tt
hr
e
edi
ffe
r
e
ntnonl
i
ne
arf
unc
t
i
onalf
or
msofas
i
ngl
ei
nde
pe
nde
ntvar
i
abl
eands
ke
t
c
h
t
her
e
l
a
t
i
ons
hi
pbe
t
we
e
nt
hede
pe
nde
ntandi
nde
pe
nde
ntvar
i
abl
e
.
Ans
we
r
:Ans
we
r
swi
l
lvar
ybys
t
ude
nt
.Mos
tc
ommonl
yus
e
df
or
msar
et
hequadr
a
t
i
cr
e
gr
e
s
s
i
on,t
he
i
nve
r
s
e(
i
nX)r
e
gr
e
s
s
i
on,andt
hel
ogl
ogmode
l
.
32
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ar
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onEduc
a
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.
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5)I
nt
hec
as
eofpe
r
f
e
c
tmul
t
i
c
ol
l
i
ne
ar
i
t
y
,OLSi
sunabl
et
oe
s
t
i
ma
t
et
hes
l
opec
oe
ffic
i
e
nt
soft
hevar
i
abl
e
s
i
nvol
ve
d.As
s
umet
ha
tyouha
vei
nc
l
ude
dbot
hX1andX2ase
xpl
ana
t
or
yvar
i
abl
e
s
,andt
ha
tX2=X ,s
o
t
ha
tt
he
r
ei
sane
xa
c
tr
e
l
a
t
i
ons
hi
pbe
t
we
e
nt
woe
xpl
ana
t
or
yvar
i
abl
e
s
.Doe
st
hi
spos
eapr
obl
e
mf
or
e
s
t
i
ma
t
i
on?
Ans
we
r
:The
r
ei
snopr
obl
e
mf
ore
s
t
i
ma
t
i
on,s
i
nc
et
hes
e
c
onde
xpl
ana
t
or
yvar
i
abl
ei
snotl
i
ne
ar
l
yr
e
l
a
t
e
d
t
ot
hefir
s
t
.
Thi
si
sane
xampl
eofapol
ynomi
alr
e
gr
e
s
s
i
onmode
lofde
gr
e
e2,whi
c
hi
sf
r
e
que
nt
l
y
e
s
t
i
ma
t
e
di
ne
c
onome
t
r
i
c
s
33
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i
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ar
s
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a
t
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on,I
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.
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6)Thefigur
es
howsi
sapl
otandafit
t
e
dl
i
ne
arr
e
gr
e
s
s
i
onl
i
neoft
hea
ge
e
ar
ni
ngspr
ofil
eof1,
744
i
ndi
vi
dual
s
,t
ake
nf
r
om t
heCur
r
e
ntPopul
a
t
i
onSur
ve
y
.
(
a)De
s
c
r
i
bet
hepr
obl
e
msi
npr
e
di
c
t
i
nge
ar
ni
ngsus
i
ngt
hefit
t
e
dl
i
ne
.Wha
twoul
dt
hepa
t
t
e
r
noft
he
r
e
s
i
dual
sl
ookl
i
kef
ort
hea
gec
a
t
e
gor
yunde
r40?
(
b)Wha
tal
t
e
r
na
t
i
vef
unc
t
i
onalf
or
m mi
ghtfitt
heda
t
abe
t
t
e
r
?
(
c
)Wha
tot
he
rvar
i
abl
e
smi
ghtyouwantt
oc
ons
i
de
ri
ns
pe
c
i
f
yi
ngt
hede
t
e
r
mi
nant
sofe
ar
ni
ngs
?
Ans
we
r
:
(
a)The
r
ewoul
dbemanyove
r
pr
e
di
c
t
i
onsf
ort
hi
sa
gec
a
t
e
gor
yunde
r40,andhe
nc
emor
ene
ga
t
i
ve
r
e
s
i
dual
s
.
(
b)I
twoul
dbebe
t
t
e
rt
ofitaquadr
a
t
i
che
r
e
,i
.
e
.
,apol
ynomi
alr
e
gr
e
s
s
i
onmode
l
,whi
c
hwoul
dpr
oduc
ean
i
nve
r
t
e
dUs
hape
.
(
c
)Ans
we
r
swi
l
lvar
ybys
t
ude
nt
s
,bute
duc
a
t
i
on,ge
nde
r
,r
ac
e
,t
e
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ewi
t
hane
mpl
oye
r
,pr
of
e
s
s
i
onal
c
hoi
c
e
,andabi
l
i
t
yar
et
ypi
c
al
l
ypr
e
s
e
nti
nans
we
r
s
.
7)(
Re
qui
r
e
sCal
c
ul
us
)Show t
ha
tf
ort
hel
ogl
ogmode
lt
hes
l
opec
oe
ffic
i
e
nti
st
hee
l
as
t
i
c
i
t
y
.
Ans
we
r
:Cons
i
de
rt
hede
t
e
r
mi
ni
s
t
i
cpar
tY=AXβ
nl
n(
Y)=β
n(
X)
,whe
r
eβ
n(
A)
.Now
1.The
0+β
1l
0=l
1
Y
Y X
Y
Y
=β
.Al
t
e
r
na
t
i
ve
l
yyouc
ande
r
i
vet
hes
amer
e
s
ul
tbyt
aki
ngt
hede
r
i
va
t
i
ve

1=
1
X
X X Y
Y
f
r
om Y=AXβ
1.
8)As
s
umet
ha
tyouhadda
t
af
orac
r
os
s
s
e
c
t
i
onof100hous
e
hol
dswi
t
hda
t
aonc
ons
umpt
i
onand
pe
r
s
onaldi
s
pos
abl
ei
nc
ome
.I
fyoufital
i
ne
arr
e
gr
e
s
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i
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t
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e
gr
e
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s
i
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ons
umpt
i
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s
pos
abl
e
i
nc
ome
,wha
tpr
i
ore
xpe
c
t
a
t
i
onsdoyouha
veaboutt
hes
l
opeandt
hei
nt
e
r
c
e
pt
?Thes
l
opeoft
hi
s
r
e
gr
e
s
s
i
onf
unc
t
i
oni
sc
al
l
e
dt
he"
mar
gi
nalpr
ope
ns
i
t
yt
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ons
ume
.
"I
f
,i
ns
t
e
ad,youfital
ogl
ogmode
l
,
t
he
nwha
ti
st
hei
nt
e
r
pr
e
t
a
t
i
onoft
hes
l
ope
?Doyouha
veanypr
i
ore
xpe
c
t
a
t
i
onabouti
t
ss
i
ze
?
Ans
we
r
:Fort
hel
ogl
ogs
pe
c
i
fic
a
t
i
on,t
hes
l
opei
st
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l
as
t
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c
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t
y
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nc
et
he
r
ear
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he
or
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e
st
ha
tpr
e
di
c
t
ac
ons
t
anta
ve
r
a
gepr
ope
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i
t
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ons
ume
,t
hee
l
as
t
i
c
i
t
ys
houl
de
qualone
.
34
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ar
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9)Thet
e
xt
books
howst
ha
tl
n(
x+Δx)–l
n(
x)≅
.Show t
ha
tt
hi
si
se
qui
val
e
ntt
ot
hef
ol
l
owi
ng
appr
oxi
ma
t
i
on l
n(
1+y)≅ yi
fyi
ss
mal
l
.Youus
et
hi
si
de
at
oe
s
t
i
ma
t
eade
mandf
ormone
yf
unc
t
i
on,
u
whi
c
hi
soft
hef
or
m m =β
×,
×e whe
r
emi
st
hequant
i
t
yof(
r
e
al
)mone
y
,GDPi
st
he
0×
val
ueof(
r
e
al
)Gr
os
sDome
s
t
i
cPr
oduc
t
,andRi
st
henomi
nali
nt
e
r
e
s
tr
a
t
e
.Youc
ol
l
e
c
tt
hequar
t
e
r
l
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t
a
f
r
om t
heFe
de
r
alRe
s
e
r
veBankofSt
.Loui
sda
t
abank(
"
FRED"
)
,whi
c
hl
i
s
t
st
hemone
ys
uppl
yandGDPi
n
bi
l
l
i
onsofdol
l
ar
s
,pr
i
c
e
sasani
nde
x
,andnomi
nali
nt
e
r
e
s
tr
a
t
e
si
npe
r
c
e
nt
a
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nt
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rye
ar
Youge
ne
r
a
t
et
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i
abl
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si
nyourr
e
gr
e
s
s
i
onpr
ogr
am asf
ol
l
ows
:m =(
mone
ys
uppl
y)
/pr
i
c
ei
nde
x
;GDP
=(
Gr
os
sDome
s
t
i
cPr
oduc
t
/Pr
i
c
eI
nde
x)
,andR=nomi
nali
nt
e
r
e
s
tr
a
t
ei
npe
r
c
e
nt
a
gepoi
nt
spe
rannum.
Ne
xtyoupe
r
f
or
mt
hel
ogt
r
ans
f
or
ma
t
i
onsont
her
e
almone
ys
uppl
y
,r
e
alGDP,andon(
1+R)
.Canyouf
or
s
e
eapr
obl
e
mi
nus
i
ngt
hi
st
r
ans
f
or
ma
t
i
on?
Ans
we
r
:l
n(
x+△x)-l
n(
x)=l
n
=l
n
=l
n(
1+y)
,whe
r
ey=
.Le
ty=0.
05,t
he
nl
n(
1+y)=
0.
049≈ 0.
05.Not
et
ha
tt
hi
sappr
oxi
ma
t
i
ondoe
snothol
dwe
l
lf
orl
ar
ge
rf
r
ac
t
i
ons
,s
uc
has0.
60.The
i
nt
e
r
e
s
tr
a
t
ei
sl
i
s
t
e
di
npe
r
c
e
nt
a
gepoi
nt
s
.Ent
e
r
i
ngRas5,r
a
t
he
rt
han0.
05,make
sβ
qualas
e
mi
2note
e
l
as
t
i
c
i
t
y
.
10)Youha
vee
s
t
i
ma
t
e
dane
ar
ni
ngsf
unc
t
i
on,whe
r
eyour
e
gr
e
s
s
e
dt
hel
ogofe
ar
ni
ngsonas
e
tof
c
ont
i
nuouse
xpl
ana
t
or
yvar
i
abl
e
s(
i
nl
e
ve
l
s
)andt
wobi
nar
yvar
i
abl
e
s
,onef
orge
nde
randt
heot
he
rf
or
mar
i
t
als
t
a
t
us
.Oneoft
hee
xpl
ana
t
or
yvar
i
abl
e
si
se
duc
a
t
i
on.
(
a)I
nt
e
r
pr
e
tt
hee
duc
a
t
i
onc
oe
ffic
i
e
nt
.
(
b)Ne
xt
,s
pe
c
i
f
yt
hebi
nar
yvar
i
abl
e
sandane
qua
t
i
on,whe
r
et
hede
f
aul
ti
sas
i
ngl
emal
e
,wi
t
hout
al
l
owi
ngf
ori
nt
e
r
a
c
t
i
onbe
t
we
e
nmar
i
t
als
t
a
t
usandge
nde
r
.I
ndi
c
a
t
et
hec
oe
ffic
i
e
nt
st
ha
tme
as
ur
et
he
e
ffe
c
tofas
i
ngl
emal
e
,s
i
ngl
ef
e
mal
e
,mar
r
i
e
dmal
e
,andmar
r
i
e
df
e
mal
e
.
(
c
)Fi
nal
l
yal
l
ow f
orani
nt
e
r
ac
t
i
onbe
t
we
e
nt
hege
nde
randmar
i
t
als
t
a
t
usbi
nar
yvar
i
abl
e
s
.Re
pe
a
tt
he
e
x
e
r
c
i
s
eofwr
i
t
i
ngdownt
hevar
i
ouse
ffe
c
t
sbas
e
dont
hef
e
mal
e
/mal
eands
i
ngl
e
/mar
r
i
e
ds
t
a
t
us
.Whyi
s
t
hel
a
t
t
e
rappr
oac
hmor
ege
ne
r
alt
hant
hef
or
me
r
?
Ans
we
r
:
(
a)Thec
oe
ffic
i
e
ntone
duc
a
t
i
ongi
ve
syout
her
e
t
ur
nt
oe
duc
a
t
i
on,i
.
e
.
,i
fe
duc
a
t
i
oni
nc
r
e
as
e
dbyoneye
ar
,
t
he
nbyhow manype
r
c
e
ntdoe
ar
ni
ngsi
nc
r
e
as
e
?
(
b)Le
tDGe
nde
re
qualonei
ft
hei
ndi
vi
duali
saf
e
mal
e
,andbez
e
r
oot
he
r
wi
s
e
.DMa
r
r
i
e
dt
ake
sonaval
ue
ofonei
ft
hei
ndi
vi
duali
smar
r
i
e
dandi
sze
r
oot
he
r
wi
s
e
.Ther
e
gr
e
s
s
i
oni
s
= 0+ 1DGe
nde
r+ 2DMa
r
r
i
e
d+.
.
.
Si
ngl
emal
e
:0;s
i
ngl
ef
e
mal
e
:0+ 1;mar
r
i
e
dmal
e
:0+ 2;mar
r
i
e
df
e
mal
e
:0+ 1+ 2.
= 0+ 1DGe
nde
r+ 2DMa
r
r
i
e
d+ 3(
DGe
nde
r×DMa
r
r
i
e
d+.
.
.
Si
ngl
emal
e
:0;s
i
ngl
ef
e
mal
e
:0+ 1;mar
r
i
e
dmal
e
:0+ 2;mar
r
i
e
df
e
mal
e
:0+ 1+ 2+ 3.Thi
s
appr
oac
hi
smor
ege
ne
r
albe
c
aus
ei
tal
l
owst
hee
ffe
c
tofbe
i
ngmar
r
i
e
dandf
e
mal
et
obedi
ffe
r
e
ntf
r
om
be
i
ngmar
r
i
e
dandmal
e
.I
n(
b)
,bot
hf
e
mal
e
sandmal
e
swe
r
ef
a
c
e
dwi
t
hi
de
nt
i
c
ale
ffe
c
t
sf
r
om be
i
ng
mar
r
i
e
d,2.I
n(
c
)
,t
hi
se
ffe
c
tdi
ffe
r
sduet
ot
headdi
t
i
onalc
oe
ffic
i
e
nt 3.
(
c
)
35
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i
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ar
s
onEduc
a
t
i
on,I
nc
.
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11)Youha
vebe
e
nt
ol
dt
ha
tt
hemone
yde
mandf
unc
t
i
oni
nt
heUni
t
e
dSt
a
t
e
shasbe
e
nuns
t
abl
es
i
nc
et
he
l
a
t
e1970.Toi
nve
s
t
i
ga
t
et
hi
spr
obl
e
m,youc
ol
l
e
c
tda
t
aont
her
e
almone
ys
uppl
y(
m=M/P;whe
r
eM i
sM1
andPi
st
heGDPde
fla
t
or
)
,(
r
e
al
)gr
os
sdome
s
t
i
cpr
oduc
t(
GDP)andt
henomi
nali
nt
e
r
e
s
tr
a
t
e(
R)
.Ne
xt
youc
ons
i
de
re
s
t
i
ma
t
i
ngt
hede
mandf
ormone
yus
i
ngt
hef
ol
l
owi
ngal
t
e
r
na
t
i
vef
unc
t
i
onalf
or
ms
:
(
i
)m=β
0+β
1×GDP+β
2xR+u
u
(
i
i
)m =β
x
×e
0×
(
i
i
i
)m=β
0×
x
u
×e
Gi
veani
nt
e
r
pr
e
t
a
t
i
onf
orβ
ne
ac
hc
as
e
.How woul
dyouc
al
c
ul
a
t
et
hei
nc
omee
l
as
t
i
c
i
t
yi
nc
as
e
1andβ
2i
(
i
)
?
Ans
we
r
:I
n(
i
)
,bot
hc
oe
ffic
i
e
nt
ss
how t
hee
ffe
c
tofauni
ti
nc
r
e
as
eoft
her
e
s
pe
c
t
i
vevar
i
abl
e
sont
he
de
mandf
ormone
y
.I
n(
i
i
)
,t
het
woc
oe
ffic
i
e
nt
sar
ee
l
as
t
i
c
i
t
i
e
s
.I
n(
i
i
i
)
,1i
sane
l
as
t
i
c
i
t
y
,whe
r
e
as 2i
sof
t
e
n
r
e
f
e
r
r
e
dt
oasa"
s
e
mi
e
l
as
t
i
c
i
t
y
.
"Thes
pe
c
i
fic
a
t
i
onbe
c
ome
sl
n(
m)=l
n(0)+ 1l
n(
GDP)+ 2R+u,s
i
nc
e
l
n(
1+R)≈ Rf
ors
mal
lR.He
nc
e 2=
,t
ha
ti
s
,i
ti
ndi
c
a
t
e
sbyhow manype
r
c
e
ntt
he(
r
e
al
)mone
y
de
mandwi
l
li
nc
r
e
as
ef
orape
r
c
e
nt
a
gec
hangei
nt
hei
nt
e
r
e
s
tr
a
t
e
.
12)Youha
vec
ol
l
e
c
t
e
dda
t
af
orac
r
os
s
s
e
c
t
i
onofc
ount
r
i
e
si
nt
wot
i
mepe
r
i
ods
,1960and1997,s
a
y
.Your
t
as
ki
st
ofindt
hede
t
e
r
mi
nant
sf
ort
heWe
al
t
hofaNa
t
i
on(
pe
rc
api
t
ai
nc
ome
)andyoube
l
i
e
vet
ha
tt
he
r
e
ar
et
hr
e
ema
j
orde
t
e
r
mi
nant
s
:i
nve
s
t
me
nti
nphys
i
c
alc
api
t
ali
nbot
ht
i
mepe
r
i
ods(
X1,
a
n
d
X
)
,
T
1,
0
i
nve
s
t
me
nti
nhumanc
api
t
alore
duc
a
t
i
on(
X2,
,andpe
rc
api
t
ai
nc
omei
nt
hei
ni
t
i
alpe
r
i
od
TandX2,
0)
(
Y0)
.Your
unt
hef
ol
l
owi
ngr
e
gr
e
s
s
i
on:
l
n(
YT)=β
n(
Y0)+uT
0+β
1X1,
T+β
2X1,
0+β
3X2,
T+β
4X1,
0+l
Oneofyourpe
e
r
ss
ugge
s
t
st
ha
ti
ns
t
e
ad,
yous
houl
dr
unt
hegr
owt
hr
a
t
ei
npe
rc
api
t
ai
nc
omeove
rt
het
wo
pe
r
i
odsont
hec
hangei
nphys
i
c
alandhumanc
api
t
al
.Fort
hos
er
e
s
ul
t
st
obeapar
s
i
moni
ouspr
e
s
e
nt
a
t
i
on
ofyouri
ni
t
i
alr
e
gr
e
s
s
i
on,wha
tt
hr
e
er
e
s
t
r
i
c
t
i
onswoul
dha
vet
ohol
d?How woul
dyout
e
s
tf
ort
he
s
e
?The
s
amepe
r
s
onal
s
opoi
nt
soutt
oyout
ha
tt
hei
nt
e
r
c
e
ptvani
s
he
si
ne
qua
t
i
onswhe
r
et
heda
t
ai
sdi
ffe
r
e
nc
e
d.
I
st
ha
tt
r
ue
?
Ans
we
r
:Ther
e
gr
e
s
s
i
onus
i
nggr
owt
hr
a
t
e
si
sasf
ol
l
ows
:
[
l
n(
YT)-l
n(
Y0)
]= β
X1,
+β
X2,
+(β
n(
Y0)+uT
0+β
1(
T-X1,
0)
3(
T-X1,
0)
5-1)l
Fort
hi
st
obeapar
s
i
moni
ouspr
e
s
e
nt
a
t
i
onoft
hei
ni
t
i
alr
e
gr
e
s
s
i
on,t
hef
ol
l
owi
ngt
wor
e
s
t
r
i
c
t
i
onsmus
t
hol
d:β
=
β
,
a
n
d
β
=
β
.
T
h
e
u
s
e
o
f
a
n
F
t
e
s
t
i
s
r
e
q
u
i
r
e
d
h
e
r
e
t
o
t
e
s
t
t
h
e
r
e
s
t
r
i
c
t
i
o
n
s
s
i
mu
l
t
a
n
e
o
u
s
l
y
.
The
1 2
3 4
i
nt
e
r
c
e
pti
ss
t
i
l
lpr
e
s
e
nti
nt
hee
qua
t
i
on,andt
heas
s
e
r
t
i
ont
he
r
e
f
or
ec
annotbet
r
ue
.
36
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i
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ar
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a
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i
on,I
nc
.
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13)Ear
ni
ngsf
unc
t
i
onsa
t
t
e
mptt
opr
e
di
c
tt
hel
ogofe
ar
ni
ngsf
r
om as
e
tofe
xpl
ana
t
or
yvar
i
abl
e
s
,bot
h
bi
nar
yandc
ont
i
nuous
.Youha
veal
l
owe
df
orani
nt
e
r
a
c
t
i
onbe
t
we
e
nt
woc
ont
i
nuousvar
i
abl
e
s
:e
duc
a
t
i
on
andt
e
nur
ewi
t
ht
hec
ur
r
e
nte
mpl
oye
r
.Youre
s
t
i
ma
t
e
de
qua
t
i
oni
soft
hef
ol
l
owi
ngt
ype
:
= 0+ 1×Fe
mme+ 2×Educ+ 3×Te
nur
e+ 4x(
Educ×Te
nur
e
)+∙
∙
∙
whe
r
eFe
mmei
sabi
nar
yvar
i
abl
et
aki
ngont
heval
ueofonef
orf
e
mal
e
sandi
sz
e
r
oot
he
r
wi
s
e
,Educi
st
he
numbe
rofye
ar
sofe
duc
a
t
i
on,andt
e
nur
ei
sc
ont
i
nuousye
ar
sofwor
kwi
t
ht
hec
ur
r
e
nte
mpl
oye
r
.Wha
ti
s
t
hee
ffe
c
tofanaddi
t
i
onalye
a
rofe
duc
a
t
i
onone
ar
ni
ngs(
"
r
e
t
ur
nst
oe
duc
a
t
i
on"
)f
orme
n?Forwome
n?I
f
youal
l
owe
df
ort
her
e
t
ur
nst
oe
duc
a
t
i
ont
odi
ffe
rf
ormal
e
sandf
e
mal
e
s
,how woul
dyour
e
s
pe
c
i
f
yt
he
abovee
qua
t
i
on?Wha
ti
st
hee
ffe
c
tofanaddi
t
i
onalye
aroft
e
nur
ewi
t
hac
ur
r
e
nte
mpl
oye
rone
ar
ni
ngs
?
Ans
we
r
:Forbot
hmal
e
sandf
e
mal
e
s
,t
hee
ffe
c
tofanaddi
t
i
onalye
arofe
duc
a
t
i
oni
s
= 2+ 4xTe
nur
e
,andhe
nc
ede
pe
ndsonc
ont
i
nuousye
ar
sofwor
kwi
t
ht
hec
ur
r
e
nte
mpl
oye
r
.
Toal
l
ow t
hee
ffe
c
tt
obedi
ffe
r
e
ntf
ormal
e
sandf
e
mal
e
s
,ani
nt
e
r
a
c
t
i
onvar
i
abl
ebe
t
we
e
nFe
mmeandEduc
woul
dha
vet
obei
nt
r
oduc
e
d.Ther
e
t
ur
nt
ot
e
nur
ewi
t
hac
ur
r
e
nte
mpl
oye
ri
s
= 3+ 4xEduc
.
14)Manyc
ount
r
i
e
st
ha
te
xpe
r
i
e
nc
ehype
r
i
nfla
t
i
ondonotha
vemar
ke
t
de
t
e
r
mi
ne
di
nt
e
r
e
s
tr
a
t
e
s
.Asa
r
e
s
ul
t
,s
omeaut
hor
sha
ves
ubs
t
i
t
ut
e
df
ut
ur
ei
nfla
t
i
onr
a
t
e
si
nt
omone
yde
mande
qua
t
i
onsoft
hef
ol
l
owi
ng
t
ypeasapr
oxy:
(
mi
sr
e
almone
y
,andPi
st
hec
ons
ume
rpr
i
c
ei
nde
x)
.
I
nc
omei
st
ypi
c
al
l
yomi
t
t
e
ds
i
nc
emove
me
nt
si
ni
tar
edwar
f
e
dbymone
ygr
owt
handt
hei
nfla
t
i
onr
a
t
e
.
Aut
hor
sha
vet
he
ni
nt
e
r
pr
e
t
e
dβ
he"
s
e
mi
e
l
as
t
i
c
i
t
y"oft
hei
nfla
t
i
onr
a
t
e
.Doyous
e
eanypr
obl
e
ms
1ast
wi
t
ht
hi
si
nt
e
r
pr
e
t
a
t
i
on?
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we
r
:Li
ne
ar
i
zi
ngt
heabovee
qua
t
i
onr
e
s
ul
t
si
nl
n(
m)=l
n(
β
n(
1+Δl
nP)+u.Now t
hi
ss
i
mpl
i
fie
s
0)+β
1l
t
ol
n(
m)=l
n(
β
nP+ui
fΔl
nP,t
hei
nfla
t
i
onr
a
t
e
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ss
mal
l
.I
nt
ha
tc
as
e
,β
e
pr
e
s
e
nt
st
hee
ffe
c
tofa
0)+β
1Δl
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pe
r
c
e
nti
nc
r
e
as
ei
nt
hei
nfla
t
i
onr
a
t
eont
hede
mandf
ormone
y
.Howe
ve
r
,i
ft
hei
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t
i
onr
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t
ei
snots
mal
l
,
asi
st
hec
as
ei
nhype
r
i
nfla
t
i
ons
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he
nt
heappr
oxi
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t
i
ondoe
snothol
danyl
onge
r
.
15)Toi
nve
s
t
i
ga
t
ewhe
t
he
rornott
he
r
ei
sdi
s
c
r
i
mi
na
t
i
ona
gai
ns
tas
ubgr
oupofi
ndi
vi
dual
s
,your
e
gr
e
s
s
t
hel
ogofe
ar
ni
ngsonde
t
e
r
mi
ni
ngvar
i
abl
e
s
,s
uc
hase
duc
a
t
i
on,wor
ke
xpe
r
i
e
nc
e
,e
t
c
.
,andabi
nar
y
var
i
abl
ewhi
c
ht
ake
sont
heval
ueofonef
ori
ndi
vi
dual
si
nt
ha
ts
ubgr
oupandi
sze
r
oot
he
r
wi
s
e
.You
c
ons
i
de
rt
wopos
s
i
bl
es
pe
c
i
fic
a
t
i
ons
.Fi
r
s
tyour
unt
wos
e
par
a
t
er
e
gr
e
s
s
i
ons
,onef
ort
heobs
e
r
va
t
i
onst
ha
t
i
nc
l
udet
hes
ubgr
oupandonef
ort
heot
he
r
s
.Se
c
ond,your
unas
i
ngl
er
e
gr
e
s
s
i
on,butal
l
ow f
orabi
nar
y
var
i
abl
et
oappe
ari
nt
her
e
gr
e
s
s
i
on.Yourpr
of
e
s
s
ors
ugge
s
t
st
ha
tt
hes
e
c
onde
qua
t
i
oni
sbe
t
t
e
rf
ort
het
as
k
a
thand,asl
ongasyoual
l
ow f
oras
hi
f
ti
nbot
ht
hei
nt
e
r
c
e
ptandt
hes
l
ope
s
.Expl
ai
nhe
rr
e
as
oni
ng.
Ans
we
r
:Byr
unni
ngt
her
e
gr
e
s
s
i
onove
rt
hee
nt
i
r
es
ampl
epe
r
i
od,youc
ant
e
s
tf
ore
qual
i
t
yofc
oe
ffic
i
e
nt
s
,
oral
t
e
r
na
t
i
ve
l
y
,f
ort
hes
i
gni
fic
anc
eofbi
nar
yvar
i
abl
e
sc
oe
ffic
i
e
nt
s
.Al
s
o,t
hec
ombi
ne
ds
ampl
ehasmor
e
obs
e
r
va
t
i
ons
,andhe
nc
es
mal
l
e
rs
t
andar
de
r
r
or
s
.
37
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i
ght©2011Pe
ar
s
onEduc
a
t
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on,I
nc
.
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16)Be
i
ngac
ompe
t
i
t
i
vef
e
mal
es
wi
mme
r
,youwonde
ri
fwome
nwi
l
le
ve
rbeabl
et
obe
a
tt
het
i
meoft
he
mal
egol
dme
dalwi
nne
r
.Toi
nve
s
t
i
ga
t
et
hi
sque
s
t
i
on,youc
ol
l
e
c
tda
t
af
ort
heOl
ympi
cGame
ss
i
nc
e1910.
Atfir
s
tyouc
ons
i
de
ri
nc
l
udi
ngvar
i
ousdi
s
t
anc
e
s
,abi
nar
yvar
i
abl
ef
orMar
kSpi
t
z
,andanot
he
rbi
nar
y
var
i
abl
ef
ort
hear
r
i
valandpr
e
s
e
nc
eofEas
tGe
r
manf
e
mal
es
wi
mme
r
s
,buti
nt
hee
ndde
c
i
deonas
i
mpl
e
l
i
ne
arr
e
gr
e
s
s
i
on.Yourde
pe
nde
ntvar
i
abl
ei
st
her
a
t
i
ooft
hef
as
t
e
s
twome
n'
st
i
met
ot
hef
as
t
e
s
tme
n'
st
i
me
i
nt
he100m bac
ks
t
r
oke
,andt
hee
xpl
ana
t
or
yvar
i
abl
ei
st
heye
a
roft
heOl
ympi
c
s
.Ther
e
gr
e
s
s
i
onr
e
s
ul
ti
s
asf
ol
l
ows
,
=4.
42–0.
0017×Ol
ympi
c
s
,
whe
r
eTFo
v
e
r
Mi
st
her
e
l
a
t
i
vet
i
meoft
hegol
dme
dalwi
nne
r
,andOl
ymp
i
c
si
st
heye
a
roft
heOl
ympi
c
Game
s
.Wha
ti
syourpr
e
di
c
t
i
onwhe
nf
e
mal
e
swi
l
lc
a
t
c
hupt
ome
ni
nt
hi
sdi
s
c
i
pl
i
ne
?Doe
st
hi
ss
ound
pl
aus
i
bl
e
?Wha
tot
he
rf
unc
t
i
onalf
or
m mi
ghtyouwantt
oc
ons
i
de
r
?
Ans
we
r
:Ac
c
or
di
ngt
ot
heabover
e
gr
e
s
s
i
on,wome
nwi
l
lc
a
t
c
hupi
nt
heye
ar2011.
76or2012.(
Thi
s
happe
nst
obeanOl
ympi
c
sye
ar
.
)Thi
si
snotpl
aus
i
bl
ef
ors
wi
mmi
ng,andabe
t
t
e
rf
unc
t
i
onalf
or
m woul
d
beTFo
v
e
r
M =β
0+β
1
.
38
Copyr
i
ght©2011Pe
ar
s
onEduc
a
t
i
on,I
nc
.
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17)Ske
t
c
hf
ort
hel
ogl
ogmode
lwha
tt
her
e
l
a
t
i
ons
hi
pbe
t
we
e
nYandXl
ooksl
i
kef
orvar
i
ouspar
ame
t
e
r
val
ue
soft
hes
l
ope
,i
.
e
.
,β
1)
.
1>1;0<β
1<1;β
1=(
Ans
we
r
:
39
Copyr
i
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ar
s
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lOMoARcPSD|27111031
18)Show t
ha
tf
ort
hef
ol
l
owi
ngr
e
gr
e
s
s
i
onmode
l
Yt=
whe
r
eti
sat
i
met
r
e
nd,whi
c
ht
ake
sont
heval
ue
s1,2,…,
T,β
e
pr
e
s
e
nt
st
hei
ns
t
ant
ane
ous(
"
c
ont
i
nuous
1r
c
ompoundi
ng"
)gr
owt
hr
a
t
e
.Show how t
hi
sr
a
t
ei
sr
e
l
a
t
e
dt
ot
hepr
opor
t
i
ona
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er
a
t
eofgr
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c
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s
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al
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ul
a
t
e
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r
om t
her
e
l
a
t
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ons
hi
p
t
Yt=Y0×(
1+g)
whe
nt
i
mei
sme
as
ur
e
di
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s
c
r
e
t
ei
nt
e
r
val
s
.
Ans
we
r
:l
n(
Yt
)=β
nc
eβ
0+β
1×t+uandhe
1=
=
t
.Fr
om Yt=Y0×(
1+g)
,wege
tl
n(
Yt
)=
l
n(
Y0)+l
n(
1+ g)
t=β
,whe
r
eβ
n(
Y0)and β
1+g)≈ gf
ors
mal
lg.He
nc
ei
fgi
ss
mal
l
,t
he
n
0+β
1t
0=l
1=(
r
e
gr
e
s
s
i
ngt
hel
ogofavar
i
abl
eont
i
mege
ne
r
a
t
e
sas
l
opec
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e
ntwhi
c
hi
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oxi
ma
t
e
l
yt
he
pr
opor
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i
ona
t
er
a
t
eofgr
owt
hf
ors
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lgr
owt
hr
a
t
e
s
.
19)Yourt
as
ki
st
oe
s
t
i
ma
t
et
hei
c
ec
r
e
am s
al
e
sf
orac
e
r
t
ai
nc
hai
ni
nNe
w Engl
and.Thec
ompanymake
s
a
vai
l
abl
et
oyouquar
t
e
r
l
yi
c
ec
r
e
am s
al
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Y)andi
nf
or
msyout
ha
tt
hepr
i
c
epe
rgal
l
onhasappr
oxi
ma
t
e
l
y
r
e
mai
ne
dc
ons
t
antove
rt
hes
ampl
epe
r
i
od.Youga
t
he
ri
nf
or
ma
t
i
onona
ve
r
a
gedai
l
yt
e
mpe
r
a
t
ur
e
s(
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dur
i
ngt
he
s
equar
t
e
r
sandr
e
gr
e
s
sYonX,addi
ngs
e
as
onalbi
nar
yvar
i
abl
e
sf
ors
pr
i
ng,s
umme
r
,andf
al
l
.
The
s
evar
i
abl
e
sar
ec
ons
t
r
uc
t
e
dasf
ol
l
ows
:DSpr
i
ngt
ake
sonaval
ueof1dur
i
ngt
hes
pr
i
ngandi
sz
e
r
o
ot
he
r
wi
s
e
,DSumme
rt
ake
sonaval
ueof1dur
i
ngt
hes
umme
r
,e
t
c
.Spe
c
i
f
yt
hr
e
er
e
gr
e
s
s
i
onf
unc
t
i
ons
whe
r
et
hef
ol
l
owi
ngc
ondi
t
i
onshol
d:t
her
e
l
a
t
i
ons
hi
pbe
t
we
e
nYandXi
s(
i
)f
or
c
e
dt
obet
hes
amef
ore
ac
h
quar
t
e
r
;(
i
i
)al
l
owe
dt
oha
vedi
ffe
r
e
nti
nt
e
r
c
e
pt
se
ac
hs
e
as
on;(
i
i
i
)al
l
owe
dt
oha
vevar
yi
ngs
l
ope
sand
i
nt
e
r
c
e
pt
se
ac
hs
e
as
on.Ske
t
c
ht
hedi
ffe
r
e
nc
ebe
t
we
e
n(
i
)and(
i
i
)
.How woul
dyout
e
s
twhi
c
hmode
lfit
st
he
da
t
at
hebe
s
t
?
Ans
we
r
:(
i
)Yi=β
;
0+β
1Xi+ui
(
i
i
)Yi=β
+
β
X
+
β
D
S
p
r
i
n
g
+
β
r+β
l
l+ui
;
0 1i 2
3DSumme
4DFa
(
i
i
i
)Yi=β
i
ng+β
r+β
l
l
0+β
1Xi+β
2DSpr
3DSumme
4DFa
+β
DSpr
i
ng×Xi
)+β
DSumme
r×Xi)+β
DFa
l
l×Xi
)+ui
;
5(
6(
7(
(
i
i
i
)i
st
hemos
tge
ne
r
aloft
hemode
l
s
,t
heot
he
r
sar
ene
s
t
e
d.He
nc
eyouc
anus
et
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e
s
tt
os
e
ei
fc
e
r
t
ai
n
r
e
s
t
r
i
c
t
i
onshol
d.Fore
xampl
e
,(
i
)i
sapar
s
i
moni
ousr
e
pr
e
s
e
nt
a
t
i
onof(
i
i
i
)i
fal
lc
oe
ffic
i
e
nt
si
nvol
vi
ngt
he
s
e
as
onalbi
nar
yvar
i
abl
e
sar
es
i
mul
t
ane
ous
l
ye
qualt
oz
e
r
o.
40
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i
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ar
s
onEduc
a
t
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on,I
nc
.
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20)I
ne
s
t
i
ma
t
i
ngt
heor
i
gi
nalr
e
l
a
t
i
ons
hi
pbe
t
we
e
nmone
ywa
gegr
owt
handt
heune
mpl
oyme
ntr
a
t
e
,
Phi
l
l
i
psus
e
dUni
t
e
dKi
ngdom da
t
af
r
om 1861t
o1913t
ofitac
ur
veoft
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ol
l
owi
ngf
unc
t
i
onalf
or
m
(
whe
r
e
+β
0)=β
1×
u,
×e
i
st
hepe
r
c
e
nt
a
gec
hangei
nmone
ywa
ge
sanduri
st
heune
mpl
oyme
ntr
a
t
e
.Ske
t
c
ht
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f
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t
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ol
edoe
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i
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art
r
ans
f
or
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ont
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tal
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owsyout
oe
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t
i
ma
t
et
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0pl
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t
i
onus
i
ngOLS?I
f
,a
f
t
e
rt
aki
ngl
ogar
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t
hmsonbot
hs
i
de
soft
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qua
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i
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i
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r
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l
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e
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)
,
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wha
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m mi
ghtyour
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nt
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t
ht
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e
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t
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i
devar
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ors
omeoft
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e
r
va
t
i
ons
?
Ans
we
r
:Gi
ve
nt
hes
hapeoft
hePhi
l
l
i
psc
ur
ve
,β
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lbene
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l
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ve
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ue
sofβ
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ras
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+β
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+β
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ur
)+u,whi
c
hc
annotbee
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t
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1)
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or
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i
ngas
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e
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e
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ogr
am s
uc
hasEx
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ott
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eX, i=
,whe
r
e 0=-4.
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e
rval
ue
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nt
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s
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umn
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om 0andt
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e
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nt
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lyour
e
ac
h2.
0.The
ne
nt
e
rt
hel
ogi
s
t
i
cf
unc
t
i
on
f
or
mul
ai
nt
hene
xtc
ol
umn.Fi
nal
l
ypr
oduc
eas
c
a
t
t
e
rpl
ot
,c
onne
c
t
i
ngt
hepr
e
di
c
t
e
dval
ue
swi
t
hal
i
ne
.
Ans
we
r
:
41
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i
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22)Tabl
e8.
1onpa
ge284ofyourt
e
xt
bookdi
s
pl
a
yst
hef
ol
l
owi
nge
s
t
i
ma
t
e
de
ar
ni
ngsf
unc
t
i
oni
nc
ol
umn
(
4)
:
=1.
503+0.
1032×e
d
uc-0.
451×DFe
mme+0.
0143×(
DFe
mme
×e
d
uc
)
(
0.
023)(
0.
0012)
(
0.
024)
(
0.
0017)
2-0.
+0.
0232×e
xpe
r-0.
000368×e
xpe
r
058×Mi
d
we
s
t-0.
0098×So
ut
h-0.
030×We
s
t
(
0.
0012)
(
0.
000023)
(
0.
006)
(
0.
006)
(
0.
007)
n=52.
790, 2=0.
267
Gi
ve
nt
ha
tt
hepot
e
nt
i
ale
xpe
r
i
e
nc
evar
i
abl
e(
e
xpe
r
)i
sde
fine
das(
Age
Educ
a
t
i
on6)findt
hea
gea
twhi
c
h
i
ndi
vi
dual
swi
t
hahi
ghs
c
hoolde
gr
e
e(
12ye
ar
sofe
duc
a
t
i
on)andwi
t
hac
ol
l
e
gede
gr
e
e(
16ye
ar
sof
e
duc
a
t
i
on)ha
vemaxi
mum e
ar
ni
ngs
,hol
di
ngal
lot
he
rf
a
c
t
or
sc
ons
t
ant
.
Ans
we
r
:Theans
we
rc
anbef
ounde
i
t
he
rbyus
i
ngc
a
l
c
ul
usorgr
aphi
c
al
/s
pr
e
ads
he
e
tt
e
c
hni
que
s
.
Maxi
mum e
ar
ni
ngsoc
c
ur
sa
tpot
e
nt
i
ale
xpe
r
i
e
nc
eof31.
5.He
nc
ewi
t
h12ye
ar
sofe
duc
a
t
i
on,t
he
maxi
mum e
ar
ni
ngshappe
na
ta
ge49.
5,whi
l
ef
orape
r
s
onwi
t
h16ye
ar
sofe
duc
a
t
i
ont
he
s
eoc
c
ura
t53.
5
ye
ar
s
.(
Si
nc
et
aki
ngl
ogar
i
t
hmsr
e
s
ul
t
si
namonot
oni
s
t
i
ct
r
ans
f
or
ma
t
i
onoft
heor
i
gi
nalda
t
a
,t
hes
ame
r
e
s
ul
t
shol
df
ort
hel
ogofe
ar
ni
ngsasf
ore
ar
ni
ngs
)
.
23)Cons
i
de
rat
ypi
c
alb
e
t
ac
o
nv
e
r
g
e
nc
er
e
gr
e
s
s
i
onf
unc
t
i
onf
r
om mac
r
oe
c
onomi
c
s
,whe
r
et
hegr
owt
hofa
c
ount
r
y'
spe
rc
api
t
ai
nc
omei
sr
e
gr
e
s
s
e
dont
hei
ni
t
i
all
e
ve
lofpe
rc
api
t
ai
nc
omeandvar
i
ousot
he
r
e
c
onomi
cands
oc
i
oe
c
onomi
cvar
i
abl
e
s
.As
s
umet
ha
tt
wooft
he
s
evar
i
abl
e
sar
et
hea
ve
r
a
genumbe
rof
ye
ar
sofe
duc
a
t
i
oni
nt
hes
pe
c
i
ficc
ount
r
yandabi
nar
yvar
i
abl
ewhi
c
hi
ndi
c
a
t
e
swhe
t
he
rornott
he
c
ount
r
ye
xpe
r
i
e
nc
e
das
i
gni
fic
antnumbe
rofye
ar
sofc
i
vi
lwar
/unr
e
s
t
.Expl
ai
nwhyi
twoul
dmakes
e
ns
e
t
oha
vet
he
s
et
wovar
i
abl
e
se
nt
e
rs
e
par
a
t
e
l
yandal
s
owhyyous
houl
dus
eani
nt
e
r
a
c
t
i
ont
e
r
m.Wha
ts
i
gns
woul
dyoue
xpe
c
tont
het
hr
e
ec
oe
ffic
i
e
nt
s
?
Ans
we
r
:Si
mpl
ee
xt
e
ns
i
onsoft
hes
t
andar
dne
oc
l
as
s
i
c
algr
owt
hmode
ls
ugge
s
tt
ha
tt
henumbe
rofye
ar
sof
e
duc
a
t
i
onha
veapos
i
t
i
vee
ffe
c
tonc
ondi
t
i
onalgr
owt
hi
nt
hewe
al
t
hofana
t
i
on(
pe
rc
api
t
ai
nc
ome
)
.A
c
i
vi
lwarwoul
dha
veane
ga
t
i
vee
ffe
c
tont
hei
nve
s
t
me
nt
/out
putr
a
t
i
o(
s
a
vi
ngsr
a
t
e
)andyouwoul
d
t
he
r
e
f
or
ee
xpe
c
tane
ga
t
i
ves
i
gnont
hec
oe
ffic
i
e
nt
.Howe
ve
r
,i
ti
si
mpor
t
antt
oi
nt
e
r
ac
tt
hevar
i
abl
e
s
be
c
aus
enoma
t
t
e
rhow muc
he
duc
a
t
i
ont
hea
ve
r
a
gepe
r
s
onhas
,t
he
r
ewi
l
lbevi
r
t
ual
l
ynoi
nve
s
t
me
nti
na
c
ount
r
ydur
i
ngac
i
vi
lwar
.He
nc
eyouwoul
de
xpe
c
tane
ga
t
i
ves
i
gn,whi
c
hwoul
di
ndi
c
a
t
et
hee
ffe
c
tt
ha
ta
c
i
vi
lwarhasont
hee
duc
a
t
i
one
ffe
c
t
.
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ar
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i
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24)Cons
i
de
rt
hef
ol
l
owi
ngr
e
gr
e
s
s
i
onoft
e
s
t
s
c
or
e
sonani
nt
e
r
c
e
pt
,abi
nar
yvar
i
abl
et
ha
te
qual
s1i
ft
he
s
t
ude
nt
t
e
ac
he
rr
a
t
i
oi
s20ormor
e(
Hi
STR)andanot
he
rbi
nar
yvar
i
abl
et
ha
te
qual
s1i
ft
hepe
r
c
e
nt
a
geof
Engl
i
s
hl
e
ar
ne
r
si
s10% ormor
e(
Hi
EL)
.
=664/1-1.
9×Hi
STR-18.
2×Hi
EL-3.
5×(
Hi
STR×Hi
EL)
Us
i
ngt
het
wobyt
wot
abl
ebe
l
ow,fil
li
nt
hee
xpe
c
t
e
dt
e
s
t
s
c
or
e
sofas
t
ude
ntwi
t
hvar
i
ousc
ombi
na
t
i
onsof
t
hehi
gh/l
ow s
t
ude
ntt
e
ac
he
rr
a
t
i
oandt
hehi
gh/l
ow pe
r
c
e
ntofEngl
i
s
hl
e
ar
ne
r
s
.
STR<20
STR
EL<10%
EL 10%
Ans
we
r
:
EL<10%
EL 10%
STR<20
664.
1
645.
9
STR
20
662.
2
640.
5
43
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i
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ar
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i
on,I
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.
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20