# Regression/Lösungen

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### Alter und Händlerverkaufspreis

Gegeben: ${\displaystyle s_{xy}=-5,4\qquad s_{y}^{2}=4\qquad R_{yx}^{2}=0,81}$
Es ist ${\displaystyle r_{yx}=s_{yx}/s_{x}s_{y}}$. Daraus folgt: ${\displaystyle s_{x}=s_{yx}/(r_{yx}s_{y})}$
Ferner ist: ${\displaystyle r_{yx}=-0,9}$ (${\displaystyle r_{yx}}$ und die Kovarianz haben das gleiche Vorzeichen); ${\displaystyle s_{y}=2}$
${\displaystyle s_{x}=s_{yx}/r_{yx}s_{y}=-5,4/(-0,9\cdot 2)=3}$

### Arbeitslosenquoten

${\displaystyle \sum _{t=0}^{3}t=6\quad \sum _{t=0}^{3}x_{t}=45,2\quad \sum _{t=0}^{3}tx_{t}=71,5;\quad \sum _{t=0}^{3}t^{2}=14}$ {\displaystyle {\begin{aligned}b&=&{\frac {(T+1)\sum tx_{t}-\sum x_{t}\sum t}{(T+1)\sum t^{2}-(\sum t)^{2}}}\\&=&{\frac {4\cdot 71,5-45,2\cdot 6}{4\cdot 14-6^{2}}}={\frac {286-271,2}{56-36}}={\frac {14,8}{20}}=0,74\\a&=&{\frac {\sum x_{t}}{T+1}}-b{\frac {\sum t}{T+1}}={\frac {45,2}{4}}-0,74\cdot {\frac {6}{4}}=11,3-1,11=10,19\\{\hat {y}}_{i}&=&10,19+0,74\cdot x_{i}\\{\hat {y}}_{4}&=&10,19+0,74\cdot x_{4}=10,19+0,74\cdot 4=13,15\\\end{aligned}}}

### Gesamtkosten und Produktionsmenge

${\displaystyle {\widehat {y}}_{i}=17,253+4,039x_{i}}$

### Gewinn eines Unternehmens

${\displaystyle {\hat {y}}_{i}=a+bx_{i}}$ {\displaystyle {\begin{aligned}a&=&{\frac {\sum y_{i}\sum x_{i}^{2}-\sum x_{i}\sum x_{i}y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}\\&=&{\frac {0-55\cdot 99}{10\cdot 385-55^{2}}}=-6,6\\\\b&=&{\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}\\&=&{\frac {10\cdot 99-0}{10\cdot 385-55^{2}}}=1,2\\\end{aligned}}} ${\displaystyle {\hat {y}}_{i}=-6,6+1,2x_{i}}$

### Hypothekenzinssatz

 ${\displaystyle x_{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle x_{i}-{\bar {x}}}$ ${\displaystyle (x_{i}-{\bar {x}})^{2}}$ ${\displaystyle y_{i}-{\bar {y}}}$ ${\displaystyle (y_{i}-{\bar {y}})^{2}}$ ${\displaystyle (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}$ 1 6 3000 -1 1.0 500 250000.0 -500 2 5 3200 -2 4.0 700 490000.0 -1400.0 3 7 2500 0 0.0 0 0.0 0.0 4 7 2300 0 0.0 -200 40000.0 -0.0 5 8 2000 1 1.0 -500 250000.0 -500 6 9 2000 2 4.0 -500 250000.0 -1000.0 Summe 42 15000 0 10.0 0 1280000.0 -3400 Mittel 7 2500 0 1.7 0 213333.3 -556.7
• ${\displaystyle \displaystyle r={\frac {-3400}{\sqrt {10\cdot 1280000}}}=-0.9503}$
• ${\displaystyle \displaystyle b_{1}={\frac {-3400}{10}}=-340}$, ${\displaystyle \displaystyle b_{0}=2500-(-340)\cdot 7=4880}$

${\displaystyle \displaystyle {\widehat {y}}=4880-340\cdot x}$

• ${\displaystyle \displaystyle R^{2}=r^{2}=-0.9503^{2}=0.9031}$
• ${\displaystyle \displaystyle 4880-340\cdot 4=3520}$ Mio EUR, ${\displaystyle \displaystyle 4880-340\cdot 7,5=2330}$ Mio EUR

### Immobiliensachverständiger

Objekt ${\displaystyle i}$ Alter ${\displaystyle x_{i}}$ Preis ${\displaystyle y_{i}}$ ${\displaystyle x_{i}y_{i}}$ ${\displaystyle x_{i}^{2}}$
1 15 190 2850 225
2 12 210 2520 144
3 3 400 1200 9
4 17 125 2125 289
5 5 300 1500 25
6 8 197 1576 64
${\displaystyle \sum }$ 60 1422 11771 756

{\displaystyle {\begin{aligned}b_{1}&=&{\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}={\frac {6\cdot 11771-60\cdot 1422}{6\cdot 756-60^{2}}}={\frac {-14694}{936}}=-15,699\\b_{0}&=&{\overline {y}}-b_{1}{\overline {x}}={\frac {1422}{6}}-(-15,699){\frac {60}{6}}=393,99\\{\hat {y}}_{i}&=&b_{0}+b_{1}x_{i}=393,99-15,699\cdot 1=378,291\end{aligned}}}

### Konsumausgaben

• ${\displaystyle {\widehat {y}}_{i}}$ = 211,82 + 0,813${\displaystyle x_{i}}$
• 2488,22 EUR Konsumausgaben

### Konsumausgaben und verfügbares Einkommen

${\displaystyle {\widehat {y}}_{i}=1,94+0,78x_{i}}$

### Kosten und Output

${\displaystyle X={\mbox{Output}},\;Y={\mbox{Kosten}}}$
Gegeben: ${\displaystyle s_{y}^{2}=1801,6;\;s_{yx}=67,2}$
Gesucht: ${\displaystyle b_{1}=s_{yx}/s_{x}^{2}}$
${\displaystyle s_{x}^{2}=\sum x_{i}^{2}/n-{\overline {x}}^{2}=1468/10-12^{2}=146,8-144=2,8}$
${\displaystyle b_{1}=s_{yx}/s_{x}^{2}=67,2/2,8=24}$

### Kunstdünger

• ja
• ${\displaystyle {\widehat {y}}_{i}}$ = 19,93 + 5,0526${\displaystyle x_{i}}$
• 75,5086 dt
• ${\displaystyle B}$ = 0,9753

### Ökonomische Variablen

{\displaystyle {\begin{aligned}b_{1}&=&{\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}={\frac {10\cdot 304-40\cdot 70}{10\cdot 180-40\cdot 40}}={\frac {240}{200}}=1,2\\b_{0}&=&{\frac {\sum y_{i}\sum x_{i}^{2}-\sum x_{i}\sum x_{i}y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}={\frac {70\cdot 180-40\cdot 304}{10\cdot 180-40\cdot 40}}={\frac {440}{200}}=2,2\\b_{0}&=&{\overline {y}}-b_{1}{\overline {x}}=7-1,2\cdot 4=2,2\end{aligned}}}

${\displaystyle x_{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle x_{i}^{2}}$ ${\displaystyle y_{i}^{2}}$ ${\displaystyle x_{i}\cdot y_{i}}$
1 40 12 1 600 144,0 480
2 40 12 1 600 144,0 480
3 40 15 1 600 225,0 600
4 60 12 3 600 144,0 720
5 80 10 6 400 100,0 800
6 80 10 6 400 100,0 800
7 90 9 8 100 81,0 810
8 90 10 8 100 100,0 900
9 90 10 8 100 100,0 900
10 90 10 8 100 100,0 900
Summe 700 110 53 600 1 238,0 7 390
Mittel 70 11 5 360 123,8 739

${\displaystyle \displaystyle b_{1}={\frac {10\cdot 7390-700\cdot 110}{10\cdot 53600-700}}^{2}=-0.0674}$,
${\displaystyle \displaystyle b_{0}={\frac {110}{10}}-{\frac {700}{10}}\cdot -0.0674=15.7174}$,

### Querschnittsanalyse von 11 Unternehmen

• ${\displaystyle \displaystyle \sum _{i=1}^{11}y_{i}=191}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}y_{i}^{2}=5183.6491}$
• ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}=1671.9}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}^{2}=259297.25}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}y_{i}=29829.7}$,

${\displaystyle \displaystyle b_{1}^{(1)}={\frac {11\cdot 29829.7-1671.9\cdot 191}{11\cdot 259297.25-1671.9^{2}}}=0.1542}$,
${\displaystyle \displaystyle b_{0}^{(1)}={\frac {191}{11}}-{\frac {1671.9}{11\cdot 0.15422270096145}}=-6.0768}$,
${\displaystyle \displaystyle R_{y1}^{2}={\frac {(11\cdot 29829.7-1671.9\cdot 191)^{2}}{(11\cdot 259297.25-1671.9^{2})\cdot (11\cdot 3446.92-191^{2})}}=0.9450}$,
${\displaystyle \displaystyle {\hat {y}}_{1}=-6.0768+0.1542x_{1}}$

• ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}=1197.4}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}^{2}=132804.82}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}y_{i}=21344.04}$,

${\displaystyle \displaystyle b_{1}^{(2)}={\frac {11\cdot 21344.04-1197.4\cdot 191}{11\cdot 132804.82}}-1197.4^{2}=0.2245}$,
${\displaystyle \displaystyle b_{0}^{(2)}={\frac {191}{11}}-{\frac {1197.4}{11\cdot 0.2245}}=-7.0749}$,
${\displaystyle \displaystyle R_{y2}^{2}={\frac {(11\cdot 21344.04-1197.4\cdot 191)^{2}}{(11\cdot 132804.82-1197.4^{2})\cdot (11\cdot 3446.92-191^{2})}}=0.9513}$,
${\displaystyle \displaystyle {\hat {y}}21=-7.0749+0.2245x_{2}}$

• ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}=29.2}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}^{2}=88.44}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}y_{i}=519.52}$,

${\displaystyle \displaystyle b_{1}^{(3)}={\frac {11\cdot 519.52-29.2\cdot 191}{11\cdot 88.44}}-29.2^{2}=1.1441}$,
${\displaystyle \displaystyle b_{0}^{(3)}={\frac {191}{11}}-{\frac {29.2}{11\cdot 1.1441}}=14.3266}$,
${\displaystyle \displaystyle R_{y3}^{2}={\frac {(11\cdot 519.52-29.2\cdot 191)^{2}}{(11\cdot 88.44-29.2^{2})\cdot (11\cdot 3446.92-191^{2})}}=0.1096}$,
${\displaystyle \displaystyle {\hat {y}}_{3}=14.3266+1.1441x_{3}}$

• ${\displaystyle r_{y1}={\sqrt {R_{y1}^{2}}}=0.9721}$; ${\displaystyle r_{y2}={\sqrt {R_{y2}^{2}}}=0.9753}$; ${\displaystyle r_{y3}={\sqrt {R_{y3}^{2}}}=0.3311}$;
• ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}=1671.9}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}^{2}=259297.25}$,

${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}=1197.4}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}^{2}=132804.82}$,
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}x_{i2}=185557.02}$,
${\displaystyle \displaystyle r_{12}={\frac {11\cdot 185557.02-1671.9\cdot 1197.4}{\sqrt {(11\cdot 259297.25-1671.9^{2})\cdot (11\cdot 132804.82-1197.4^{2})}}}}$
${\displaystyle \displaystyle r_{12}=0.9973}$
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}=1671.9}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}^{2}=259297.25}$,
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}=29.2}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}^{2}=88.44}$,
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i1}x_{i3}=4478.28}$,
${\displaystyle \displaystyle r_{12}={\frac {11\cdot 4478.28-1671.9\cdot 29.2}{\sqrt {(11\cdot 259297.25-1671.9^{2})\cdot (11\cdot 88.44-29.2^{2})}}}}$
${\displaystyle \displaystyle r_{12}=0.16868}$
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}=1197.4}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}^{2}=132804.82}$,
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}=29.2}$, ${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i3}^{2}=88.44}$,
${\displaystyle \displaystyle \sum _{i=1}^{11}x_{i2}x_{i3}=3203.89}$,
${\displaystyle \displaystyle r_{12}={\frac {11\cdot 3203.89-1197.4\cdot 29.2}{\sqrt {(11\cdot 132804.82-1197.4^{2})\cdot (11\cdot 88.44-29.2^{2})}}}}$
${\displaystyle \displaystyle r_{12}=0.1545}$

### Umsatz und Werbeetat

Schätzung des Parameters ${\displaystyle b_{1}}$ in der linearen Regressionsfunktion ${\displaystyle {\hat {y}}_{i}=b_{0}+b_{1}x_{i}}$.

Filiale 1 2 3 4 5 6 ${\displaystyle \sum }$
${\displaystyle y_{i}:{\mbox{Umsatz (1000 EUR)}}}$ 20 16 18 17 12 13 96
${\displaystyle x_{i}:{\mbox{Werbeetat (100 EUR)}}}$ 29 25 28 26 20 22 150
${\displaystyle x_{i}^{2}}$ 841 625 784 676 400 484 3810
${\displaystyle x_{i}y_{i}}$ 580 400 504 442 240 286 2452

{\displaystyle {\begin{aligned}b_{1}&=&{\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{n\sum x_{i}^{2}-\sum x_{i}\sum x_{i}}}\\&=&{\frac {6\cdot 2452-150\cdot 96}{6\cdot 3810-150^{2}}}={\frac {14712-14400}{22860-22500}}={\frac {312}{360}}=0,866666\\&=&0,867\end{aligned}}}

### Zusätzliche statistische Einheit

Lösung g)
${\displaystyle b_{1}=(10\cdot 304-40\cdot 70)/(10\cdot 180-40^{2})=240/200=1,2}$
${\displaystyle b_{0}=(70\cdot 180-40\cdot 304)/10\cdot 180-40^{2})=440/200=2,2}$