20.06.2024 Test Exam#
Example Regression#
How to do a regression by hand:
Given Values:
y = testscore |
89 |
71 |
69 |
58 |
42 |
55 |
|---|---|---|---|---|---|---|
x = time |
6 |
8 |
4 |
6 |
14 |
12 |
\(\bar{ x }\): 25/3
\(\bar{ y }\): 64
\(\sum_{i=1}^n x_i^2\) = 492
\(\sum_{i=1}^n x_i y_i\) = 2974
A: \(b_1\)#
\[\begin{split}
\hat{\beta_1} = \frac
{\widehat{Cov}_{xy}}
{\widehat{Var}_x}
= \frac
{\sum x_i y_i - n \bar{ x } \bar{ y }}
{\sum x_i^2-n \bar{ x }^2}
\\
= \frac{ 2974-6*64*25/3 }{492-6*(25/3)^2} \\
= \frac{ -226 }{226/3} = -3
\end{split}\]
B: \(b_0\)#
\[\begin{split}
\beta_0 =(\bar{y}- \hat{\beta_1} \bar{x})
\\
= (64-(-3)\frac{ 25 }{3}) = 89
\end{split}\]
Formel: \(y = 89-3*x\)
C: \(R^2\)#
calculate fitted values
calculate residuals
1 |
2 |
3 |
4 |
5 |
6 |
|
|---|---|---|---|---|---|---|
\(\hat{ y }\) |
71 |
65 |
77 |
71 |
47 |
53 |
u |
18 |
6 |
-8 |
-13 |
-5 |
2 |
\(\sum u = 0\) !!!
\(\sum u^2 = 622\)
\(\sum y_i^2 = 25876\)
\[\begin{split}
R^2 = 1-\frac{ \sum \hat{ u }^2 }{\sum y_i^2- n*\bar{ y }^2}
\\
=1- \frac{ 622 }{25876-6*64^2} = 0.5215
\end{split}\]
D: \(Std. Error: Regression\)#
\[\begin{split}
\hat{ \sigma }^2 = \frac{ 1 }{n-2} * \sum_{i=1}^n \hat{ u }^2
\\
= 1/4*622 = 155.5
\\
\hat{ \sigma } = \sqrt{ \sigma^2 } = 12.4600
\end{split}\]
E: \(Std. Error: \beta_1\)#
\[\begin{split}
se(\beta_1) = \frac{ \sigma^2 }{\sqrt{SST_x}} \\
SST= \sum (x-\bar{ x })^2
\end{split}\]
1 |
2 |
3 |
4 |
5 |
6 |
|
|---|---|---|---|---|---|---|
\((x-\hat{ x })^2\) |
ToDo#
Normale Regression berechnen
SE der Regression
SE von einem Koeffizienten
R^2 aus R-Ergebnissen berechnen (7.4)
Konfidenzintervall (7.3)
Hypothesen und Test
t-Verteilung
Normalverteilung
partielle Effekte
Marginal Effekte
R-Code anschauen
Percentage / percentage Points