ECON301_Handout_08_1213_02

ˆ ˆ ˆ 095 i i i i p t se t se 0025 0025 ˆ ˆ ˆ

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ˆ ˆ ˆ ( ) ( ) 0.95 i i i i P t se t se 0.025 0.025 ˆ ˆ ˆ ˆ ( ) ( ) 0.95 i i i i i P t se t se  
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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 3 0.025 0.025 ˆ ˆ ˆ ˆ . ( ) . ( ) 0.95 i i i i i P t se t se     and arranging, 0.025 0.025 ˆ ˆ ˆ ˆ . ( ) . ( ) 0.95 i i i i i P t se t se  Thus the 95% confidence interval for i is 0.025 0.025 ˆ ˆ ˆ ˆ . ( ) . ( ) i i i i i t se t se  with T-k-1 degrees of freedom or 0.025 ˆˆ . ( ) i i i t se  with T-k-1 degrees of freedom The meaning of the 95% confidence interval is that there is a 0.95 probability of including the true value of the population parameter in the interval 0.025 . ( ) ii t se  (with T-k-1 degrees of freedom). o However, note that we can not say that the probability is 95% that the specific interval [for example, (0.4215, 0.5987)] contains the true i because in this case the interval will be fixed and it would be no longer random contrary to the case of 0.025 . ( ) t se , where in each case of repeated sampling, 0.025 . ( ) t se make result to different specific intervals. o We can say that, given the confidence coefficient of 95%, in repeated sampling, in 95 out of 100 cases, intervals calculated like 0.025 . ( ) t se will contain the true value i .
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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 4 If we generalize, we can write as follows: 100.(1- )% confidence interval for i is: /2 ˆˆ . ( ) ii t se  with T-k-1 degrees of freedom or, ˆ ˆ ˆ ˆ . ( ) . ( ) 1 i i i i i P t se t se        (1) Notice that the larger the standard error, the larger the width of the confidence interval. 1 Before moving on, we want to discuss a bit further the interpretation of equation (1). Given the values of the regressor, the random variable in the probability statement is ˆ i . Equation (1) says that the probability is 0.95 that the interval ˆ ˆ ( . , . ) tt  will contain i . In other words, assume that we could take a large number
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ˆ ˆ ˆ 095 i i i i P t se t se 0025 0025 ˆ ˆ ˆ ˆ 095...

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