Regression/Lösungen

Alter und Händlerverkaufspreis

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

Arbeitslosenquoten

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

Gesamtkosten und Produktionsmenge

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

Gewinn eines Unternehmens

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

Hypothekenzinssatz

	$x_{i}$	$y_{i}$	$x_{i}-{\bar {x}}$	$(x_{i}-{\bar {x}})^{2}$	$y_{i}-{\bar {y}}$	$(y_{i}-{\bar {y}})^{2}$	$(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})$
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

• $\displaystyle r={\frac {-3400}{\sqrt {10\cdot 1280000}}}=-0.9503$
• $\displaystyle b_{1}={\frac {-3400}{10}}=-340$, $\displaystyle b_{0}=2500-(-340)\cdot 7=4880$
  

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

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

Immobiliensachverständiger

Objekt $i$	Alter $x_{i}$	Preis $y_{i}$	$x_{i}y_{i}$	$x_{i}^{2}$
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
$\sum$	60	1422	11771	756

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

Konsumausgaben

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

Konsumausgaben und verfügbares Einkommen

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

Kosten und Output

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

Kunstdünger

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

Ökonomische Variablen

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

Quadratmetermiete

	$x_{i}$	$y_{i}$	$x_{i}^{2}$	$y_{i}^{2}$	$x_{i}\cdot y_{i}$
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

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

Querschnittsanalyse von 11 Unternehmen

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

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

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

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

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

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

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

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

Umsatz und Werbeetat

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

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

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

Zusätzliche statistische Einheit

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