ch03 - Chapter 3 Interval Estimation and Hypothesis Testing Principles of Econometrics 4th Edition Chapter 3 Interval Estimation and Hypothesis Testing

# ch03 - Chapter 3 Interval Estimation and Hypothesis Testing...

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Principles of Econometrics, 4t h Edition Page 1 Chapter 3: Interval Estimation and Hypothesis Testing Chapter 3 Interval Estimation and Hypothesis Testing
Principles of Econometrics, 4t h Edition Page 2 Chapter 3: Interval Estimation and Hypothesis Testing 3.1 Interval Estimation 3.2 Hypothesis Tests 3.3 Rejection Regions for Specific Alternatives 3.4 Examples of Hypothesis Tests 3.5 The p -value 3.6 Linear Combinations of Parameters Chapter Contents
Principles of Econometrics, 4t h Edition Page 3 Chapter 3: Interval Estimation and Hypothesis Testing 3.1 Interval Estimation
Principles of Econometrics, 4t h Edition Page 4 Chapter 3: Interval Estimation and Hypothesis Testing There are two types of estimates Point estimates The estimate b 2 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 confidence intervals . We prefer to call them interval estimates because the term ‘‘confidence’’ is widely misunderstood and misused 3.1 Interval Estimation
Principles of Econometrics, 4t h Edition Page 5 Chapter 3: Interval Estimation and Hypothesis Testing The normal distribution of b 2 , the least squares estimator of β 2 , is A standardized normal random variable is obtained from b 2 by subtracting its mean and dividing by its standard deviation: 3.1.1 The t -Distribution ± ² ¸ ¸ ¹ · ¨ ¨ © § ³ ¦ 2 2 2 2 , ~ x x N b i V E ± ² ± ² 1 , 0 ~ 2 2 2 2 N x x b Z i ¦ ³ ³ V E Eq. 3.1 3.1 Interval Estimation
Principles of Econometrics, 4t h Edition Page 6 Chapter 3: Interval Estimation and Hypothesis Testing
Principles of Econometrics, 4t h Edition Page 7 Chapter 3: Interval Estimation and Hypothesis Testing We know that: Substituting: Rearranging: ± ² 95 . 0 96 . 1 2 96 . 1 2 2 2 ¸ ¸ ¸ ¹ · ¨ ¨ ¨ © § d ³ ³ d ³ ¦ x x b P i V E ± ² 95 . 0 96 . 1 96 . 1 d d ³ Z P ± ² ± ² 95 . 0 96 . 1 96 . 1 2 2 2 2 2 2 2 ¸ ¹ · ¨ © § ³ ´ d d ³ ³ ¦ ¦ x x b x x b P i i V E V 3.1 Interval Estimation 3.1.1 The t -Distribution
Principles of Econometrics, 4t h Edition Page 8 Chapter 3: Interval Estimation and Hypothesis Testing The two end -points provide an interval estimator. In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter β 2 . This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term σ 2 . ± ² ¦ ³ r 2 2 2 96 . 1 x x b i V 3.1 Interval Estimation 3.1.1 The t -Distribution