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rather we …nd a range more useful  this leads us to report con…dence
intervals
Using the normality assumption we get
β
Pr b1
b
β1 tn
tn 2 ,0 .025 se 2 ,0 .025 se b
β1 β1 b + tn
β1 2 ,0 .025 se b
β1 = 0.95 b = Lower bound of the Con…dence interval
β1 b + tn 2,0.025 se b = Upper bound of the Con…dence interval
β1
β1
0.95 = Con…dence level
tn 2,0.025 = Critical value of twosided test Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 42 / 50 Interpretation of the con…dence interval
The bounds of the interval are random
In repeated samples, the interval that is constructed in the above way
will cover the population regression coe¢ cient in 95% of the cases Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 43 / 50 Con…dence intervals for typical con…dence levels
Pr b1
β tn 2 ,0 .005 se tn 2 ,0 .025 se Pr b1
β tn 2 ,0 .050 se β
Pr b1 b
β1 b
β1
b
β1 β1
β1
β1 b + tn
β1 2 ,0 .005 se b + tn
β1 2 ,0 .050 se b + tn
β1 2 ,0 .025 se b
β1 = 0.99 b
β1 = 0.90 b
β1 = 0.95 Rules of Thumb:
tn 2 ,0 .005 tn 2 ,0 .025 tn 2 ,0 .050 Lecture 4 (econometrics and simple linear regression) = 2.576
= 1.96
= 1.645 EMET2007/6007 13 th March 2013 44 / 50 Relationship between con…dence intervals and hypotheses tests
a 2 interval ) reject H0 : β1 = a
/
in favour of H1 : β1 6= a Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 45 / 50 Example: Determinants of CEO salary \
salary i = 1174 + 0.0155salesi + bi
ε
(0.0089 ) 2 n = 209 R = 0.0144 df = 207 tn
0.0155 = 1.96
1.96 (0.0089) = ( 0.002, 0.033)
2 ,0 .025 The e¤ect of sales on salary is imprecisely estimated as the interval is
very wide. It is not even statistically signi…cant because zero lies in
the interval. Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 46 / 50 Example: Determinants of CEO salary (cont.) \
salary i =
2 898.93 + 262.9 log (salesi ) + bi
ε
(92.36 ) n = 209 R = 0.0377 df = 207 tn
262.9 2 ,0 .025 = 1.96 1.96 (92.36) = (80.82, 444.98) The e¤ect of log (sales ) on salary is relatively less precisely estimated
as the interval is narrow.
However,...
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This note was uploaded on 06/15/2013 for the course EMET 2007 taught by Professor Strachan during the Two '13 term at Australian National University.
 Two '13
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