UCLA ECON 103 - Econ-103-Lecture-04 (80 pages)

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Econ-103-Lecture-04



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Econ-103-Lecture-04

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Pages:
80
School:
University of California, Los Angeles
Course:
Econ 103 - Introduction to Econometrics
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Lecture Note 4 Interval Estimation Hypothesis Testing Moshe Buchinsky UCLA Fall 2014 Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 1 80 Topics to be Covered 1 Interval Estimation 2 Hypothesis Tests 3 Rejection Regions for Speci c Alternatives 4 Examples of Hypothesis Tests 5 The p value 6 Linear Combinations of Parameters Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 2 80 Interval Estimation Interval Estimation There are two types of estimates point estimates and interval estimates Point estimates The estimate b2 is a point estimate of the unknown population parameter in the regression model Interval estimates Interval estimation proposes a range of values in which the true parameter is likely to fall Providing a range of values gives a sense of what the parameter value might be and the precision with which we have estimated it Such intervals are often called con dence intervals We prefer to call them interval estimates because the term con dence is widely misunderstood and misused Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 3 80 Interval Estimation The normal distribution of b2 the least squares estimator for 2 is 2 b2 N 2 N 2 i 1 xi x A standardized normal random variable is obtained from b2 by subtracting its mean and dividing by its standard deviation Z r Buchinsky UCLA b2 2 2 N i 1 xi Econ 103 Lecture 4 N 0 1 x 3 1 2 Fall 2014 4 80 Interval Estimation We know that Pr 1 96 Substituting 0 B Pr B 1 96 r 1 96 95 Z b2 2 2 N i 1 xi 1 C 1 96C A 95 x 2 Rearranging Pr b2 1 96 s 2 x 2 N i 1 xi s b2 1 96 Buchinsky UCLA Econ 103 Lecture 4 2 2 N i 1 xi x 2 95 Fall 2014 5 80 Interval Estimation The two end points b2 interval estimator 1 96 r 2 N i 1 xi x 2 provide an In repeated sampling 95 of the intervals constructed this way will contain the true value of the parameter 2 This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term 2 Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 6 80 Interval Estimation b2 creates a random variable t Replacing 2 with t r b2 2 b 2 N i 1 xi b2 2 b2 2 q b Se b2 b b2 Var x 2 t N 3 2 Se b2 has a t distribution with N The ratio t b2 2 b degrees of freedom which we denote as t t N 2 2 2 In general we can say if assumptions SR1 SR6 hold in the simple linear regression model then t Buchinsky UCLA bk k b Se bk t N 2 Econ 103 Lecture 4 for k 1 2 3 3 Fall 2014 7 80 Interval Estimation The t distribution is a bell shaped curve centered at zero It looks like the standard normal distribution except it is more spread out with a larger variance and thicker tails The shape of the t distribution is controlled by a single parameter called the degrees of freedom often abbreviated as df Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 8 80 Interval Estimation Figure 3 1 Critical values from a t distribution Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 9 80 Interval Estimation We can nd a critical value from a t distribution such that Pr t tc Pr t tc 2 where is a probability often taken to be 0 01 or 0 05 The critical value tc for degrees of freedom m is the percentile value t 1 2 m Each shaded tail area contains 2 of the probability so that 1 of the probability is contained in the center portion Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 10 80 Interval Estimation Consequently we can make the probability statement Pr tc t tc 1 1 or Pr tc or Pr bk Buchinsky UCLA tc b Se bk bk k b Se bk k tc 3 4 bk t c b Se bk 1 Econ 103 Lecture 4 Fall 2014 3 5 11 80 Interval Estimation When bk and b Se bk are estimated values numbers based on a given sample of data then bk tc b Se bk is called a 100 1 interval estimate of k Equivalently it is called a 100 1 con dence interval Usually 0 01 or 0 05 so that we obtain a 99 con dence interval or a 95 con dence interval Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 12 80 Interval Estimation The interpretation of con dence intervals requires a great deal of care The properties of the interval estimation procedure are based on the notion of repeated sampling Any one interval estimate based on one sample of data may or may not contain the true parameter k and because k is unknown we will never know whether it does or does not When con dence intervals are discussed remember that our con dence is in the procedure used to construct the interval estimate it is not in any one interval estimate calculated from a sample of data Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 13 80 Interval Estimation For the food expenditure data Pr b2 2 024b Se b2 2 b2 2 024b Se b2 0 95 3 6 The critical value tc 2 024 which is appropriate for 0 05 and 38 degrees of freedom To construct an interval estimate for 2 we use the least squares estimate b2 10 21 and its estimated standard error q p b b b2 4 38 2 09 Se b2 Var Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 14 80 Interval Estimation A 95 con dence interval estimate for 2 b2 Se b2 10 21 tc b 2 024 2 09 5 97 14 45 When the procedure we used is applied to many random samples of data from the same population then 95 of all the interval estimates constructed using this procedure will contain the true parameter Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 15 80 Interval Estimation Is 2 actually in the interval 5 97 14 45 We do not know and we will never know What we do know is that when the procedure we used is applied to many random samples of data from the same population then 95 of all the interval estimates constructed using this procedure will contain the true parameter The interval estimation procedure works 95 of the time What we can say about the interval estimate based on our one sample is that given the reliability of the procedure we would be surprised if 2 is not in the interval 5 97 14 45 Buchinsky UCLA Econ 103 Lecture 4 Fall 2014 16 80 Interval Estimation What is the usefulness of an interval estimate of 2 When reporting regression results we always give a point estimate such as b2 10 21 However the point estimate alone gives no sense of its …


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