Expression 23: left brace, "t" less than or equal to negative 5 : "f" left parenthesis, "x" , right parenthesis , "t" greater than 5 : "f" left parenthesis, "x" , right parenthesis , right bracet≤−5:fx,t>5:fx
23
Unknown true model: left parenthesis, 3 , 1 left brace, "t" less than or equal to negative 5 , right brace , right parenthesis3,1t≤−5
Label:
equals=
left parenthesis, 3 , 1 , right parenthesis3,1
24
Real world: left parenthesis, 3 , 1 left brace, negative 4 less than "t" less than negative 3 , right brace , right parenthesis3,1−4<t<−3
Label:
undefined
25
Goal: find the line that generated the data: left parenthesis, 3 , 1 left brace, negative 3 less than "t" less than negative 2 , right brace , right parenthesis3,1−3<t<−2
Label:
undefined
26
Model estimated by OLS: left parenthesis, 3 , 1.2 left brace, 0 less than "t" less than 2 , right brace , right parenthesis3,1.20<t<2
Label:
undefined
27
Predicted Y: left parenthesis, 3 , 1 left brace, 2 less than or equal to "t" less than 5 , right brace , right parenthesis3,12≤t<5
Label:
undefined
28
Expression 29: left brace, "t" less than or equal to 0 : "A" , "B" , right bracet≤0:A,B
add slider:
29
OLS estimation of the parameters
30
Expression 31: "b" Subscript, "h" , Baseline equals round left parenthesis, mean left parenthesis, "f" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis plus "u" , right parenthesis negative "a" Subscript, "h" , Baseline times mean left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis , 3 , right parenthesisbh=roundmeanfx1+u−ah·meanx1,3
equals=
1.1 3 21.132
31
Expression 32: "a" Subscript, "h" , Baseline equals round left parenthesis, StartFraction, total left parenthesis, left parenthesis, left parenthesis, "x" Subscript, 1 , Baseline minus mean left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis , right parenthesis times left parenthesis, "f" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis plus "u" minus mean left parenthesis, "f" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis plus "u" , right parenthesis , right parenthesis , right parenthesis , right parenthesis Over total left parenthesis, left parenthesis, "x" Subscript, 1 , Baseline minus mean left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis , right parenthesis squared , right parenthesis , EndFraction , 3 , right parenthesisah=roundtotalx1−meanx1·fx1+u−meanfx1+utotalx1−meanx12,3
equals=
0.3 7 90.379
32
Expression 33: "f" Subscript, "h" , Baseline left parenthesis, "x" , right parenthesis equals "a" Subscript, "h" , Baseline times "x" positive "b" Subscript, "h" , Baselinefhx=ah·x+bh
33
Expression 34: "a" Subscript, "t" , Baseline equals left brace, "t" less than 0 : round left parenthesis, StartFraction, "h" left parenthesis, 2 , right parenthesis minus "h" left parenthesis, 1 , right parenthesis Over 2 minus 1 , EndFraction , 2 , right parenthesis , "a" Subscript, "h" , Baseline , right braceat=t<0:roundh2−h12−1,2,ah
equals=
9.2 49.24
34
Expression 35: "b" Subscript, "t" , Baseline equals left brace, "t" less than 0 : round left parenthesis, "h" left parenthesis, 0 , right parenthesis , 2 , right parenthesis , "b" Subscript, "h" , Baseline , right bracebt=t<0:roundh0,2,bh
equals=
negative 15.0 4−15.04
35
`\hat{a}=${a_{t}}`, `\hat{b}=${b_{t}}` left parenthesis, 3 , 1 left brace, negative 2 less than "t" less than 2 , right brace , right parenthesis3,1−2<t<2
Label:
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36
Sum of squared residuals: left parenthesis, 3 , 0.7 5 left brace, negative 2 less than "t" less than 2 , right brace , right parenthesis3,0.75−2<t<2
Label:
undefined
37
`\sum_{k=1}^{32} \left(y_k - (\hat{a}x_k + \hat{b})\right)^2 = ${s}` left parenthesis, 3 , 0.3 5 left brace, negative 2 less than "t" less than 2 , right brace , right parenthesis3,0.35−2<t<2
Label:
undefined
38
Expression 39: "s" equals round left parenthesis, total left parenthesis, left parenthesis, left parenthesis, "f" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis plus "u" , right parenthesis minus "a" Subscript, "t" , Baseline times "x" Subscript, 1 , Baseline minus "b" Subscript, "t" , Baseline , right parenthesis squared , right parenthesis , 2 , right parenthesiss=roundtotalfx1+u−at·x1−bt2,2
