Free Essay

Financial Statistics

In: Business and Management

Submitted By imahustla
Words 3762
Pages 16
Ett exempel p˚ samband mellan variabler a c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Vilka data har vi?

Antag tv˚ variabler X och Y som antar v¨rdena a a x1 , x2 , . . . , xn , samt y1 , y2 , . . . , yn .
Om vi ritar dem i ett spridningsdiagram s˚ markerar vi de ordnade a paren
(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ) i figuren. Vi uppfattar dessa par som ett stickprov.

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Korrelation

Vi definierar stickprovskorrelationskoefficienten som r= Kovariansen mellan X och Y
.
(Standardavvikelsen f¨r X ) (Standardavvikelsen f¨r Y ) o o

I t¨ljaren har vi den s˚ kallade stickprovskovariansen mellan X och a a
Y . I n¨mnaren produkten av stickprovsstandardavvikelserna f¨r X a o och Y .

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Korrelation
¨
Overs¨tter vi denna definition i formler har vi a stickprovskorrelationskoefficienten som n i =1 (xi

1 n −1

r=
1
n−1

n i =1 (xi

− x ) (yi − y )

− x )2

1 n−1 n i =1 (yi

− y )2

Det ovanst˚ende st¨mmer v¨l med definitionen f¨r tv˚ a a a oa stokastiska variabler X och Y , vilken ¨r a Corr (X , Y ) =

Cov (X , Y )
Var (X )

Var (Y )

.

F¨r materialet ovan har vi o r = 0.44151.

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a .

Stickprov fr˚n en bivariat normalf¨rdelning a o
Vi kan t¨nka oss att a (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ) a a a ¨r observationer fr˚n en tv˚dimensionell stokastisk variabel (X , Y ).
L˚t oss anta att (X , Y ) f¨ljer en bivariat normalf¨rdelning. Det ¨r a o o a tv˚ normalf¨rdelningar i kors p˚ samma g˚ng. En s˚dan f¨rdelning a o a a a o
2
2 beror p˚ fem parametrar µX , µY , σX , σY , samt ρ. Eftersom vi har a tv˚ stokastiska variabler kan vi tala om korrelationen mellan dem. a Den ges av parametern ρ = Corr (X , Y ).
D˚ kan vi testa a H0 : ρ = 0 mot H1 : ρ = 0.
Under H0 f¨ljer o √ r n−2 t=√ 1 − r2 en t -f¨rdelning med n − 2 frihetsgrader. o c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Korrelationsanalys av v˚rt stickprov a F¨rkasta H0 , om o √

r n−2 r n−2

< −tn−2,α/2 eller √
> tn−2,α/2 .
1 − r2
1 − r2
Naturligtvis ¨r tn−2,α/2 α/2 kvantilen i en t -f¨rdelning med n − 2 a o frihetsgrader. D˚ n = 60 och r = 0.44151 blir a √

3.36244 r n−2
(0.44151) 58

=
= 3.75.
=
0.8972563
1 − r2
1 − (0.44151)2
Kvantilen t58,0.025 saknas bak i boken s˚ vi anv¨nder a a t60,0.025 = 2.000. Vi f¨rkastar nollhypotesen (med besked). o c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Enkel linj¨r regression a Man kan t¨nka sig m˚nga typer av samband, men vi studerar ett a a linj¨rt samband. a Yi = β0 + β1 xi + εi ,

i = 1, 2, . . . , n.

Den skattade linjen yi = b0 + b1 xi + ei ,

i = 1, 2, . . . , n.

Det predicerade v¨rdet ges av a yi = b0 + b1 xi ,
ˆ

i = 1, 2, . . . , n.

och residualerna av ei = yi − yi ,
ˆ

i = 1, 2, . . . , n.

Lite senare redovisar vi antaganden f¨r feltermerna εi . o c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a (1)

Enkel linj¨r regression a V˚r modell a Yi = β0 + β1 xi + εi ,

i = 1, 2, . . . , n

inneh˚ller tv˚ parametrar β0 och β1 . D¨rtill har vi ¨ven variansen a a a a f¨r feltermerna o Var (εi ) = σ 2 .
Vi ¨nskar skatta de tre parametrarna o β0 , β1 , σ 2 .
De skattade parametrarna betecknas
2
b0 , b1 , σ 2 = Se .
ˆ

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Enkel linj¨r regression a De tv˚ f¨rsta β0 och β1 skattar vi med minstakvadratmetoden. ao Feltermernas varians skattar vi p˚ annat s¨tt. a a
Minstakvadratmetoden g˚r ut p˚ att minimera summan av a a kvadraterna p˚ residualerna av a ei = yi − yi ,
ˆ

i = 1, 2, . . . , n.

Eftersom yi = b0 + b1 xi kan vi skriva
ˆ
n

n

i =1

n

(yi − yi )2 =
ˆ

ei2 = i =1

(yi − b0 − b1 xi )2 . i =1

Detta ¨r en funktion av b0 och b1 som vi kallar H (b0 , b1 ). a Summan av kvadraterna p˚ residualerna betecknas ¨ven med SSE. a a

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Minstakvadratskattning n n

(yi − b0 − b1 xi )2 .

ei2 =

SSE = H (b0 , b1 ) = i =1

∂ SSE
H (b0 , b1 )
=
=
∂ b0
∂ b0

i =1

n

2 (yi − b0 − b1 xi ) (−1) = 0. i =1

Multiplicera med −(1/2) p˚ b¨gge sidor. Anv¨nder vi aa a r¨knereglerna f¨r summor har vi a o n n

yi − nb0 − b1 i =1

xi = 0. i =1

Nu flyttar vi ¨ver b0 och b1 med tillh¨rande multiplikativa o o konstanter i v¨nsterledet: a n

b0 n + b1

n

xi = i =1

c J¨rgen S¨ve-S¨derbergh o a o yi . i =1

Finansiell statistik, v˚rterminen 2011 a (2)

P˚ v¨g mot b0 och b1 aa ∂ SSE
H (b0 , b1 )
=
=
∂ b1
∂ b1

n

2 (yi − b0 − b1 xi ) (−xi ) = 0. i =1

Multiplicera ekvationen med 1/2 och multiplicera ihop resten. n −xi yi + b0 xi + b1 xi2 = 0. i =1

Anv¨nd r¨knereglerna som ovan: a a n −

n

x i y i + b0 i =1

n

xi2 = 0.

xi + b1 i =1

c J¨rgen S¨ve-S¨derbergh o a o i =1

Finansiell statistik, v˚rterminen 2011 a (3)

Normalekvationerna
Vi har allts˚ ett ekvationssystem att l¨sa fr˚n (2) och (3) a o a n

nb0 +

n

xi

b1 =

i =1 n n

xi

xi2

b0 +

i =1

yi i =1 n i =1

b1 =

xi yi i =1

Detta ¨r tv˚ ekvationer i tv˚ obekanta: b0 och b1 . aa a
Ekvationssystemet kallas normalekvationerna.
Dividerar vi den ¨vre ekvationen med n har vi o b0 + xb1 = y

c J¨rgen S¨ve-S¨derbergh o a o ⇔

b0 = y − xb1 .

Finansiell statistik, v˚rterminen 2011 a Normalekvationerna
Substituera uttrycket f¨r b0 in i den nedre ekvationen o n

n

xi

n

xi2

(y − xb1 ) +

b1 =

i =1

i =1

xi yi i =1

och l¨s f¨r b1 . Skriv allt som enskilda termer oo n

n

xi − xb1

y i =1

n

xi + b1 i =1

n

xi2 i =1

=

xi yi i =1

Bryt ut b1 i v¨nsterledet och flytta ¨ver resten i h¨gerledet: a o o n

b1

−x

n

i =1

n

xi2

xi + i =1

c J¨rgen S¨ve-S¨derbergh o a o n

xi yi − y

= i =1

xi i =1

Finansiell statistik, v˚rterminen 2011 a (4)

Normalekvationerna
Uttrycket inom parantesen i (4) skriver vi n xi2 − x i =1

n n n

n

xi2 − n(x )2 .

xi = i =1

i =1

1

Nu p˚st˚r jag att aa n

n

(xi − x )2 = i =1

xi2 − n(x )2 . i =1

Dessutom att h¨gerledet i (4) ¨r lika med: o a n n

(xi − x ) (yi − y ) = i =1

xi yi − nx y . i =1

Dessa likheter bevisar ni p˚ ¨vning 1. ao c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Normalekvationerna
Allts˚ kan vi skriva (4) som a n

n

(xi − x )2

b1 i =1

eller b1 =

(xi − x ) (yi − y ) ,

= i =1

n i =1 (xi − x ) (yi − n 2 i =1 (xi − x )

y)

.

Multiplicerar vi med (1/n)/(1/n) p˚ b¨gge sidor ser vi att vi kan aa t¨nka p˚ b1 som: a a b1 =

Stickprovskovariansen mellan X och Y
.
Stickprovsvariansen f¨r X o c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a ANOVA-tabl˚n a Vi unders¨ker nu noggrannheten hos den skattade o regressionsekvationen. Vi kan skriva yi − yi = yi − y − (ˆi − y ).
ˆ
y
Kvadrera b¨gge sidor och summera ¨ver samtliga observationer a o n n

y
{(yi − y ) − (ˆi − y )}2

(yi − yi )2 =
ˆ
i =1

i =1 n (yi − y )2 + (ˆi − y )2 − 2(yi − y )(ˆi − y ) y y

= i =1 n n

(yi − y )2 +

= i =1

c J¨rgen S¨ve-S¨derbergh o a o n

(ˆi − y )2 − 2 y i =1

(yi − y )(ˆi − y ) y i =1

Finansiell statistik, v˚rterminen 2011 a ANOVA-tabl˚n a Vi f¨rs¨ker g¨ra oss av med den tredje termen. Om vi substituerar oo o in uttrycket f¨r b0 = y − xb1 i (1), d v s yi = b0 + b1 xi o ˆ yi = (y − xb1 ) + b1 xi = y + b1 (xi − x )
ˆ

i = 1, 2, . . . , n.

Det kan vi skriva som yi − y = b1 (xi − x )
ˆ

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a (5)

ANOVA-tabl˚n a Substituera n y
(yi − y )(ˆi − y )

−2 i =1 n = −2

(yi − y )b1 (xi − x ) i =1 n = −2b1

(yi − y ) (xi − x ) i =1 n n i =1 (xi n i =1 (xi

(yi − y ) (xi − x )

= −2b1 i =1

− x )2
− x )2

1

= −2b1

n i =1 (yi − y ) (xi − n 2 i =1 (xi − x )

x)

n

(xi − x )2 i =1

b1 c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a ANOVA-tabl˚n a n

=

2
−2b1

(xi − x )2 i =1

n
2
b1 (xi − x )2

= −2 i =1 n (b1 (xi − x ))2

= −2 i =1 n (ˆi − y )2 y = −2 i =1

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a ANOVA-tabl˚n a n

n

n

(yi − yi )2 =
ˆ

n

(yi − y )2 +

i =1

i =1

(ˆi − y )2 −2 y i =1

(yi − y )(ˆi − y ) y i =1
−2

n
2
y i =1 (ˆi −y )

n

n

(yi − y )2 −

=

(ˆi − y )2 y i =1

i =1

Vi m¨blerar om, s˚ o a n n

i =1
Total variation i y

n

(yi − yi )2
ˆ

(yi − y )2 = i =1

i =1

Of¨rklarad variation i y o c J¨rgen S¨ve-S¨derbergh o a o (ˆi − y )2 y +

Av regressionen f¨rklarad variation i y o Finansiell statistik, v˚rterminen 2011 a .

ANOVA-tabl˚n a Slutligen kan vi st¨lla upp tabl˚n: a a
Source
Regression

Sum of Squares n 2 y i =1 (ˆi − y )

About regression
About mean

n i =1 (yi n i =1 (yi

− yi )2
ˆ
− y )2

df
1
n−2 n−1 Mean Square
MSR
2
Se =

n
ˆ2
i =1 (yi −yi )

n−2

F¨rkortningen df betyder ‘’degrees of freedom” (frihetsgrader). o Source
Regression
Minitab har andra namn p˚ variationsk¨llorna: a a
Residual Error
Total
Source
Model
SAS anv¨nder: a Error
Corrected Total c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Antaganden f¨r feltermerna εi o E (εi ) = 0 (st¨rningsantagandet); modellen ¨r i genomsnitt o a korrekt. Var (εi ) = E ε2 = σ 2 (homoskedasticitetsantagandet); i feltermerna har samma ¨ndliga varians. a De stokastiska variablerna εi ¨r oberoende. (feltermerna a korrelerar ej med varandra).
Cov (εi , εj ) = E (εi εj ) − E (εi ) E (εj ) = E (εi εj ) = 0. εi ∼ N (0, σ 2 ); feltermerna εi ¨r normalf¨rdelade. a o

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Egenskaper hos punktskattningarna
Vi vet fr˚n ovan att skattningen av β1 ges av formeln a b1 =

n i =1 (xi − x ) (yi − n 2 i =1 (xi − x )

y)

.

Punktskattningen ¨r en stokastisk variabel som har samma a utseende: n (xi − x ) Yi − Y
.
b1 = i =1 n
2
i =1 (xi − x )
Vi har bytt ut de observerade v¨rdena yi mot stokastiska variabler a Yi . Eftersom
(xi − x ) Yi − Y

= (xi − x ) Yi − (xi − x ) Y

s˚ kan vi skriva t¨ljaren som a a n n

(xi − x ) Yi − Y

(xi − x ) Yi −

= i =1

i =1 c J¨rgen S¨ve-S¨derbergh o a o n

(xi − x ) Y i =1

Finansiell statistik, v˚rterminen 2011 a Egenskaper hos punktskattningarna
Eftersom Y ¨r en konstant (den beror inte p˚ i ), s˚ kan vi skriva a a a om den sista termen till n n

(xi − x ) Y

(xi − x )

=Y i =1

i =1

0

Det sista p˚st˚endet skall ni visa p˚ ¨vning 1. aa ao
D¨rf¨r kan vi skriva uttrycket f¨r b1 som ao o b1 =

n i =1 (xi − x ) Yi n 2 i =1 (xi − x )

n

(xi − x )

=

d¨r a ai =

i =1

n i =1 (xi

(xi − n i =1 (xi

− x )2

x)
− x )2

n

Yi =

ai Yi , i =1

.

Fr˚n ovan vet vi att a E (Yi ) = E (β0 + β1 xi + εi ) = β0 + β1 xi + E (εi ) c J¨rgen S¨ve-S¨derbergh o a o 0

Finansiell statistik, v˚rterminen 2011 a Punktskattningen b1 ¨r v¨ntev¨rdesriktig aa a n E (b1 ) = E

ai Yi i =1

n

=

ai E (Yi ) i =1 n =

ai (β0 + β1 xi ) i =1 n = β0

n

ai + β1 i =1

ai xi i =1

= β1 , eftersom n

n

ai = 0 i =1

ai xi = 1.

¨
(Ovning 1 (igen!))

i =1 c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Variansen f¨r punktskattningen b1 o D˚ a Var(Yi ) = Var (β0 + β1 xi + εi ) = 0 + 0 + Var (εi ) = σ 2 har vi att n Var(b1 ) = Var

ai Yi i =1

n

=

Var (ai Yi )

(Yi oberoende)

i =1 n ai2 Var (Yi )

= i =1 n = i =1 c J¨rgen S¨ve-S¨derbergh o a o 2

(xi − x ) n i =1 (xi

2

− x)

σ2.

Finansiell statistik, v˚rterminen 2011 a Variansen f¨r punktskattningen b1 o n i =1 (xi

Var(b1 ) =

− x )2

n i =1 (xi

=

2

− x)

σ2 n i =1 (xi

− x )2

2

σ2, ⇔

.

2
2
Betecknas ¨ven med σb1 i NCT. Eftersom Se skattar σ 2 a v¨ntev¨rdesriktigt s˚ kan vi skatta variansen f¨r punktskattningen a a a o med 2
Se
2 sb1 = n
.
(xi − x )2 i =1

Denna formel ger allts˚ medelfelet f¨r b1 , det som ben¨mnes a o a standard error i datorutskrifterna. c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Skattningen av interceptet β0
Skattningen av β0 ges av b0 = y − b1 x och som stokastisk variabel av b0 = Y − b1 x
E (b0 ) = E Y − b1 x
= E Y − E (b1 x )
= E Y − xE (b1 )
= E Y − x β1

Vi m˚ste ber¨kna E Y for att komma vidare. a a
¨
c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Punktskattningen b0 ¨r v¨ntev¨rdesriktig aa a
F¨rst bildar vi Y av v˚r modell: o a
Yi = β0 + β1 xi + εi ,

i = 1, 2, . . . , n.

Summera ¨ver samtliga observationer o n

n

Yi

=

i =1

(β0 + β1 xi + εi ) i =1 n =

n

β0 + i =1

β1 xi + i =1 n = nβ0 + β1

εi i =1

n

xi + i =1

c J¨rgen S¨ve-S¨derbergh o a o n

εi i =1

Finansiell statistik, v˚rterminen 2011 a Punktskattningen b0 ¨r v¨ntev¨rdesriktig aa a
Dividera med n: n i =1 Yi

n vilket vi skriver som

=

nβ0 β1
+
n

n i =1 xi

n

+

n i =1 εi

n

,

Y = β0 + β1 x + ε.
D˚ ε1 , ε2 , . . . , εn samtliga har E (εi ) = 0, s˚ blir a a
E (ε) = E

ε 1 + ε2 + · · · + εn n 1
E (ε1 + ε2 + · · · + εn ) n 1
=
[E (ε1 ) + E (ε2 ) + · · · + E (εn )] n 1
=
[0 + 0 + · · · + 0] n = 0.
=

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Punktskattningen b0 ¨r v¨ntev¨rdesriktig aa a

Allts˚ blir a EY

= E (β0 + β1 x + ε)
= E (β0 + β1 x ) + E (ε)
= β0 + β1 x

Detta f¨rser oss med resultatet o E (b0 ) = E Y − x β1
= β0 + β1 x − x β1
= β0 .

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Variansen f¨r punktskattningen b0 o Variansen ges av (minns att x ¨r ett fixt tal) a Var(b0 ) = Var Y − b1 x
= Var Y + (−b1 x )
= Var Y + Var (−b1 x ) + 2Cov Y , −b1 x b1 stokastisk

Y och b1 stokastiska

2

= Var Y + x Var (b1 ) − 2x Cov Y , b1

P˚ ¨vning 1 bevisar vi att ao Cov Y , b1 = 0.
D˚ vi k¨nner Var (b1 ) beh¨ver vi endast ber¨kna Var Y . a a o a c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Variansen f¨r punktskattningen b0 o D˚ ε1 , ε2 , . . . , εn ¨r oberoende, s˚ ¨r Y1 , Y2 , . . . , Yn oberoende, d¨r a a aa a
Yi = β0 + β1 xi + εi
Var Y

= Var

och

Var (Yi ) = σ 2 ,

i = 1, 2, . . . , n.

Y1 + Y2 + · · · + Yn n 1 n 2

=

1 n 2

=

1 n 2

=

1 n 2

=
=

σ2 n Var (Y1 + Y2 + · · · + Yn )
[Var (Y1 ) + Var (Y2 ) + · · · + Var (Yn )] σ2 + σ2 + · · · + σ2 nσ 2
(Ett generellt resultat. Se NCT s 270.)

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Variansen f¨r punktskattningen b0 o Allts˚ blir a Var(b0 ) = Var Y + x 2 Var (b1 ) − 2x Cov Y , b1
=

σ2
+ x2 n = σ2

1
+
n

σ2 n i =1 (xi

− x )2

x2 n i =1 (xi

− x )2

.

2
Eftersom Se skattar σ 2 v¨ntev¨rdesriktigt s˚ kan vi skatta a a a variansen f¨r punktskattningen med o 2
2
sb0 = Se

1
+
n

x2 n i =1 (xi

− x )2

.

Denna formel ger allts˚ medelfelet f¨r b0 , det som ben¨mnes a o a standard error i datorutskrifterna. c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Punktskattningarna ¨r normalf¨rdelade stokastiska a o variabler Om vi antar att feltermerna εi ¨r normalf¨rdelade, s˚ ¨r b˚de b0 a o aa a och b1 normalf¨rdelade: o b0 ∼ N

β0 , σ 2

x2

1
+
n

n i =1 (xi

samt b1 ∼ N β1 ,

− x )2

σ2 n i =1 (xi

− x )2

,

.

Vi skattar feltermernas varians σ 2 genom
2
Se

1
=
n−2

c J¨rgen S¨ve-S¨derbergh o a o n

(yi − yi )2 .
ˆ
i =1

Finansiell statistik, v˚rterminen 2011 a Hypotespr¨vning f¨r β1 : t -test o o
Man kan bevisa att t= b1 − β 1
1
n−2

n
ˆ2
i =1 (yi −yi ) n 2 i =1 (xi −x )

∼ t (n − 2).

Vi ¨nskar testa o H0 : β1 = 0 mot H1 : β1 = 0.
D˚ H0 f¨ljer testvariabeln t en t -f¨rdelning med n − 2 frihetsgrader. a o o F¨rkasta H0 , om o t < −tn−2,α/2 eller t > tn−2,α/2 .

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Konfidensintervall f¨r β1 o Ett 100(1 − α)%-igt konfidensintervall f¨r β1 ges av o b1 ± tα/2 (n − 2)

c J¨rgen S¨ve-S¨derbergh o a o 1 n−2 n i =1 (yi

− yi )2
ˆ
. n 2 i =1 (xi − x )

Finansiell statistik, v˚rterminen 2011 a Hypotespr¨vning f¨r β1 : F -test o o

Ett alternativ till att testa
H0 : β1 = 0 mot H1 : β1 = 0 a a o ¨r att anv¨nda F -f¨rdelningen. Man kan bevisa att kvoten mellan medelkvadraterna i ANOVA-tabl˚n f¨ljer en F -f¨rdelning med 1 ao o frihetsgrad i t¨ljaren och n − 2 frihetsgrader i n¨mnaren. a a
F¨rkasta H0 , om o F > Fα (1, n − 2) .
Fα (1, n − 2) ¨r ett tabellv¨rde. a a

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a Exempel
Vi ber¨knar f¨rst de n¨dv¨ndiga kvadratsummorna. a o oa x
1
2
3
4
5
6
21

y
5
0
1
7
3
8
24

x2
1
4
9
16
25
36
91

xy
5
0
3
28
15
48
99

P˚ detta vis erh˚ller vi a a n (xi − x ) (yi − y ) = 99 − i =1

samt

n

(xi − x )2 = 91 − i =1 c J¨rgen S¨ve-S¨derbergh o a o 1
· 21 · 24 = 15,
6

1
35
· 212 = .
6
2

Finansiell statistik, v˚rterminen 2011 a Exempel

D¨rf¨r blir ao b1 =

n i =1 (xi − x ) (yi − n 2 i =1 (xi − x )

y)

=

15
35
2

6
=.
7

Slutligen har vi b0 = y − b1 x =

c J¨rgen S¨ve-S¨derbergh o a o 24 6 21

= 1.
6
76

Finansiell statistik, v˚rterminen 2011 a Exempel

Ett 95%-igt konfidensintervall f¨r β1 ges av o b1 ± tn−2, α sb1 .
2
D˚ n = 6 och α = 0.05, ska vi anv¨nda t4,0.025 = 2.776. Intervallet a a ges allts˚ av a 6
± (2.776)(0.7478)
7
eller −1.219 till 2.933. Vi ¨r allts˚ 95% s¨kra p˚ att β1 befinner a a a a sig i intervallet (−1.219, 2.933).

c J¨rgen S¨ve-S¨derbergh o a o Finansiell statistik, v˚rterminen 2011 a

Similar Documents

Premium Essay

Case Study

...non-parametric inference tests are also described in the case where the data Sample distribution is not compatible with standard parameter distribution. Thirdly, using multiple resampling methods Computer -generated random sample finally introduced the characteristics of the distribution and estimate Statistical inference. The method of multivariate data processing of the following sections involved. method Clinical trials also briefly review process. Finally, the last section of statistical computer software discussion And through the collection of citations to adapt to different levels of expertise, and to guide readers theme. In this article, there are some methods such as descriptive statistic, probablity distruction, possion ditruction and multivariate methods. For the descriptive statistic, they are tabular, graphical and Numerical Methods basic characteristics of the sample It can be described. While these methods may be the same Used to describe the entire group, they more...

Words: 702 - Pages: 3

Premium Essay

Statistics in Business

...Statistics Statistics is used more often than people realize. They are used for many reasons such as to help one make a difficult decision in their personal or professional life. Statistics is also used to help companies promote their merchandise. Have you ever seen a commercial that used numerical information to show viewers that their product is preferred over their competitors’ product? That is just one of the many times one has probably seen statistics used without even realizing it. Statistics is the result of numerical information that is collected, classified, summarized, organized, analyzed, and interpreted. Once the data is collected and compared, one can draw a conclusion otherwise known as the statistic. There are two types of statistical applications in business, descriptive statistics, and inferential statistics. Descriptive statistics uses methods of organizing, summarizing, and presenting data in an informative and convenient form. Inferential statistics uses sample data to make estimates or predictions about a larger set of data (McClave et al. 2011). In business decision-making, statistics is used both internally and externally. Internally, business owners and managers use statistics to help them make important decisions such as wages, merchandise, operating hours, and the future of the company. For example, a company may view statistics between employees earning hourly and salary wages verse employees earning commission wages. The company can use their findings...

Words: 456 - Pages: 2

Premium Essay

Mgt 498

...Statistics in Business QNT/351 Aug 21, 2013 Edward Balian Statistics Investopedia defines statistics as a type of mathematical analysis involving the use of quantified representations, models and summaries for a given set of empirical data or real world observations. Statistical analysis involves the process of collecting and analyzing data and then summarizing the data into a numerical form. ("Investopedia", 2013) Types and Levels Descriptive statistics, inferential statistics, ratio-level data, interval- level data, ordinal-level data, and nominal-level date are some types and levels of statistics. Descriptive statistics utilizes numerical and graphical methods to look for patterns in a data set, to summarize the information revealed in a data set, and to present the information in a convenient form. Inferential statistics utilizes sample data to make estimates, decisions, predictions, or other generalizations about a larger set of data. Business decision making When it comes to the role of statistics in business decision-making it is applied in many ways in terms of consumer preferences or even financial trends.   For example, managers across any type of business formulate problems, they decide on a question relating to the problem and then form a statistical formulation of the question is used to determine answers to all of the above.   An example of a business question may be how many calls are answered...

Words: 399 - Pages: 2

Free Essay

Qnt 351 Week 1 Paper

...Statistics in Business QNT/351 Statistics in business The purpose of this essay is to examine the purpose of statistics in business. Our text, Lind (2011) defines statistics as “The science of collecting, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions” (p.5). Types and levels of statistics There are two major types of statistics, descriptive and inferential. Descriptive statistics is defined by Lind (2011) as “methods of organizing, summarizing, and presenting data in an informative way” (p.6). An example of descriptive statistics would be a high school report showing that it had 300 graduates in 1990 and 450 graduates on 1991. The information that they provided described the amount of graduates that they had for each year. Inferential statistics is defined by Lind (2011) as “the methods used to estimate a property of a population on the basis of a sample” (p.7). If the same high school sent out a report showing the graduate numbers for 1999- the present to estimate the number of graduates that they would have for this school year, those statistics would be inferential because they are used to estimate future outcomes. There are four levels of statistical data: nominal, ordinal, interval and ratio. The nominal level deals with qualitative variables such as colors and blood types that can only be counted and classified. Ordinal data measurement is a variable rating system that ranks data according...

Words: 651 - Pages: 3

Premium Essay

Statistics in Business

...Statistics in Business Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data. It is applicable to a wide variety of academic disciplines, from the natural and social sciences to the humanities, government and business ("Statanalysis Consulting", 2013). Statistics is used in decision making that affects our everyday lives. The study of statistics is divided into two categories and has four levels of measurements. The two types of statistics are descriptive statistics and inferential statistics. Descriptive statistics is the organizing, presenting, and analyzing of data in an informative way. Inferential statistics is the methods used to estimate a property of a population on the basis of a sample. The four levels of measurements in statistics are nominal, ordinal, interval, and ratio. The first scale is nominal. The nominal level of measurement is the lowest level. Nominal data deals with names, categories, or labels. The next level is called the ordinal level of measurement. Data at this level can be ordered, but no differences between the data can be taken that are meaningful. The interval level of measurement deals with data that can be ordered, and in which differences between the data does make sense. The fourth and highest level of measurement is the ratio level. Data at the ratio level possess all of the features of the...

Words: 530 - Pages: 3

Premium Essay

Finer Diner Sales Proposal

...effective business decisions. Calculating time cost, money cost, and return on forecast, all are based on data mining, marketing research, information analysis and findings. Statistics is way to finding the answer. Statistics – The Method of Organizing Data Generally, statistics is a set of disciplines to analyze quantitative information. Statistics entails all aspects of information: comprehending, collecting, communicating, organizing, and interpreting. All of these are the key reference for forecasting consequences or decision making. Thus, it permits us to estimate the extent of our errors. Purchasing a Business It is not an easy task or decision to purchase a business. Before the final decision is made there are many things to consider. To start with, what exactly do you want to achieve? For whatever reason, you must be sure that it is something that you are ready to devote a large amount of time and energy too. Otherwise, you might be trapped into doing something that you loathe. You must ask yourself how far you are ready to commit. How much of your own time, energy, and money are you willing to sacrifice? Finer Diner Sales Proposal The owner of the Finer Diner submitted a proposal to you in hopes of selling the business to you. His asking price is $250,000. Your financial institution advises that your monthly payment to finance that amount would be $1850.00. This is in addition to other business expenses you would incur such as product, payroll...

Words: 1152 - Pages: 5

Premium Essay

Chap1 Solution

... Introduction to Statistics LEARNING OBJECTIVES The primary objective of chapter 1 is to introduce you to the world of statistics, enabling you to: 1. Define statistics. 2. Be aware of a wide range of applications of statistics in business. 3. Differentiate between descriptive and inferential statistics. 4. Classify numbers by level of data and understand why doing so is important. CHAPTER OUTLINE 1.1 Statistics in Business Best Way to Market Stress on the Job Financial Decisions How is the Economy Doing? The Impact of Technology at Work 1.2 Basic Statistical Concepts 1.3 Data Measurement Nominal Level Ordinal Level Interval Level Ratio Level Comparison of the Four Levels of Data Statistical Analysis Using the Computer: Excel and MINITAB KEY TERMS census ordinal level data descriptive statistics parameter inferential statistics parametric statistics interval level data population metric data ratio level data nominal level data sample nonmetric data statistic nonparametric statistics statistics STUDY QUESTIONS 1. A science dealing with the collection, analysis, interpretation, and presentation of numerical data is called _______________. 2. One way to subdivide the field of statistics is into the two branches...

Words: 1190 - Pages: 5

Premium Essay

Statistics: Highly Informative

...Statistics: Highly Informative Latosha Greer BUS308: Statistics for Managers Instructor Hayes June 1, 2014 In this essay I am aim to discuss the differences between descriptive statistics and Inferential statistics and the reasons why we use them. I will also discuss hypothesis development and testing, when to select the appropriate statistical test, and how to evaluating statistical results. In this class I learned the difference between descriptive statistics and inferential statistics. We use descriptive statistics to measure and analysis data. There are a number of reasons why we use Descriptive statistics. We use it, because Descriptive statistics numerical summaries measure the central tendency of a data set, it can include graphical summaries that show the spread of the data, and they provide simple summaries about the sample that help interpret and analyze data. First, there are a number of reasons why we use descriptive statistics we use it because descriptive statistics numerical summaries that either measure the central tendency of a data set. In business therefore descriptive statistics helps in making conclusions about various issues and therefore helps in making decision. Description statistics is the first step in analyzing data before making inferences of data, therefore it is important in analyzing any data collected that will help in describing the characteristics of data collected. There are three measurements that we tend to use. One measurement...

Words: 1422 - Pages: 6

Premium Essay

Gfgfg

...the Development of Statistics September 2007 This paper has been prepared by the Development Data Group of the World Bank and the PARIS21 Secretariat at the request of the PARIS21 Steering Committee. It is part of the follow-up to the Third International Roundtable on Managing for Development Results in Hanoi in February 2007 and the meeting on scaling up support for statistics held during the World Bank and IMF Spring Meetings in Washington in April 2007. Scaling-up Support to Statistics Executive summary This paper briefly discusses why scaling up investment in National Statistical Systems is needed and then focuses on how this could be achieved by applying a system-wide approach, drawing lessons from the Sector-Wide Approaches (SWAps) used successfully in areas such as health, education and agriculture. It argues that an effective and efficient National Statistical System is essential for managing for development results. Investments in statistics are needed, just as much as in areas such as financial management and procurement. National Statistical Systems provide the data needed to develop appropriate policies, to target scarce resources, to measure progress, to make effective use of aid and to monitor and evaluate outcomes. At the Third International Roundtable on Managing for Development Results, which was held in Hanoi in February 2007, participating country teams emphasized that to be successful any scaling up of support to statistics must be grounded in...

Words: 7652 - Pages: 31

Premium Essay

Statistics

...be approximately equal to the mean of the population. Furthermore, all of the samples will follow an approximate normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size. This statistical theory is very useful when examining returns for a given stock or index because it simplifies many analysis procedures. An appropriate sample size depends on the data available, but generally speaking, having a sample size of at least 50 observations is sufficient. Due to the relative ease of generating financial data, it is often easy to produce much larger sample sizes. • Null Hypothesis: States the assumption (numerical) to be tested, for Example: The average number of TV sets in U.S. Homes is at least three (H0: μ ≥ 3). 1. Is always about a population parameter, not about a sample statistic. ✓ H0: μ ≥ 3 X H0: [pic] ≥ 3 Always begins with the assumption that the null hypothesis is true, similar to the notion of innocent until proven guilty. Refers to the status quo. Always contains “=”, “≤” or “≥” sign. May or may not be rejected. 1. • The Alternate Hypothesis : Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S. homes is less than 3 ( HA: μ<...

Words: 1168 - Pages: 5

Premium Essay

Purefoods

...easily think of two people you know where the shorter one is heavier than the taller one. Nonetheless, the average weight of people 5'5'' is less than the average weight of people 5'6'', and their average weight is less than that of people 5'7'', etc. Correlation can tell you just how much of the variation in peoples' weights is related to their heights. Although this correlation is fairly obvious your data may contain unsuspected correlations. You may also suspect there are correlations, but don't know which are the strongest. An intelligent correlation analysis can lead to a greater understanding of your data. Techniques in Determining Correlation There are several different correlation techniques. The Survey System's optional Statistics Moduleincludes the most common type, called the Pearson or product-moment correlation. The module also includes a variation on this type called partial correlation. The latter is useful when you want to look at the relationship between two variables while removing the effect of one or two other variables. Like all statistical techniques, correlation is only appropriate for certain kinds of data. Correlation works for quantifiable data in which numbers are meaningful, usually quantities of some sort. It cannot be used for purely categorical data, such as gender, brands purchased, or favorite color. Rating Scales Rating scales are a controversial middle case. The numbers in rating scales have meaning, but that meaning isn't very...

Words: 2622 - Pages: 11

Premium Essay

Anova

...the telecommunications sector, the financial sector, and the energy sector, we will be using the .05 significance level. The research question is “Does the financial sector, the telecommunication sector and the energy sector all share the same average net income.” ANOVA or analysis of variance allows team E to compare more than two means simultaneously. In formulating the hypothesis statement using ANOVA, the team has decided on a null hypothesis that mean net income from the three different financial sectors are equal. After developing the null and alternate hypothesis, Team E preformed the five steps of hypothesis testing using MegaStat and Microsoft Excel (Doane & Seward, 2007). Step 1: State the Hypotheses The hypotheses to be tested are Ho: μt =μf =μe H1: Not all the means are equal (at least one mean is different). Step 2: State the Decision Rule There is c = 3 groups and n = 75 observations, so degrees of freedom for the F test are: Numerator: d.f. 1 = c -1 = 3 -1 =2 (between treatments, factor) Denominator: d.f.2 = n –c = 75 -3 =71 (within treatments, error) Step 3: Select a Level of Significance We will use a = .05 for the test. The 5 percent right-tail critical value from Appendix F is F(3,25) = 3.13 Step 4: Perform the Calculations/ Test Statistics Using MegaStat for the calculations, we obtained the results shown in the Excel Spread sheet. Excels default is a = .05. Step 5: Make the decision Since the test statistic F = 7.60 exceeds the critical value...

Words: 464 - Pages: 2

Premium Essay

Statistical Engineering: Principles and Examples

... Lynne s Background (Why me?) Lynne’s Background (Why me?) • 40+ years in industry – Nabisco, then Kraft , – Unilever (Lipton) – Hunt‐Wesson Foods • Academic – AB Math The Colorado AB Math, The Colorado  College – MS, Applied and  Mathematical Statistics,  Rutgers – PhD, Interdisciplinary, Rutgers • Government: NIST Government: NIST • Academia – Cal. St. Fullerton (MBAs) – Rutgers (Experiment Station) ( ) • ASA – Chair P&Q Division – Fellow ‘94 • Consulting – Consumer goods – Pharmaceuticals – R&D, Manufacturing, Quality • ASQ – Chair Statistics Division – Fellow ’86 – Column Quality Progress Column:  Quality Progress Slide 3 Statisticians?? Not sure N t Yes No Don’t care.  This is the only session  that looked remotely interesting. that looked remotely interesting 4 How can you tell? How can you tell? • If you have more than one pen with you If you have more than one pen with you • If you know more than one joke about the  binomial distribution binomial distribution • If your glasses are thicker than mine • If you are too shy to be an accountant • If you talk to your colleagues in SAS y y g 5 1.  Motivation 1 Motivation Statistical Engineering 1. Motivation The State of Statistics as...

Words: 2285 - Pages: 10

Premium Essay

Statistics in Business

...Statistics in Business Oswaldo Acosta-Villasenor University Of Phoenix QNT/351 ON14BSB02 7/14/15 Dr. JYOTIRMAY DEB Statistics in Business The Random House College Dictionary defines statistics as “the science that deals with the collection, classification, analysis, and interpretation of information or data.” Thus, a statistician isn’t just someone who calculates batting averages at baseball games or tabulates the results of a Gallup poll. Professional statisticians are trained in statistical science—that is, they are trained in collecting numerical information in the form of data, evaluating it, and drawing conclusions from it. Furthermore, statisticians determine what information is relevant in a given problem and whether the conclusions drawn from a study are to be trusted. Statistics is the science of data. It involves collecting, classifying, summarizing, organizing, analyzing, and interpreting numerical information. (Sincich, Benson,, & McClave, 2011, p. ). Many companies use statistics to make major business decisions, this can be financial, personnel, acquisitions, marketing and others. Another entity that benefits from statistics are government...

Words: 349 - Pages: 2

Premium Essay

Store 24

...increasing store level employees retention. The extent to which the profitability of the stores was related to “people factors” was considered to be an important factor. This could be captured by estimating of the actual financial impact based on the tenure of the manager and the crew at a store. The site-location factors such as population, number of competitors, pedestrian access, visibility, location and timings were also to be considered as key drivers of profitability. However, a huge amount of variation existed with manager and crew tenure. Also the stores in the sample appeared to be widely geographically disbursed, complicating site-location factors. An opinion was to be formed as to whether increasing wages, implementing a bonus program, instituting new training programs, or developing a career development program would be the best course of action. Analysis (A) REGRESSION OF PROFITABILITY WITH MANAGER & CREW TENURE Inference: Let the regression equation be represented by , where y and x are the variables, a is the y-axis intercept. The R square statistic has a value 0.41 which signifies less than half the points fit the regression line. Now we represent the NULL HYPOTHESIS as H0(b=0) and ALTERNATE HYPOTHESIS as Ha(b≠0). The p-statistic value corresponding to the Mtenure is 0.009 which is within a significance level of 0.05 so the null hypothesis can be accepted and the regression result is significant. However, Ctenure’s p-value 0.809 does not lie within...

Words: 920 - Pages: 4