ECON301_Handout_06_1213_02

N0 1 for 1 ˆ where 2 2 ˆ 2 t t x t x and 1 2 ˆ 2 t

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N(0, 1) for 1 ˆ where 0 2 2 ˆ 2 t t X T x and 1 2 ˆ 2 t x

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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 4 2. Testing Hypotheses In this section we discuss three types of hypothesis testing: (A) testing the statistical significance of individual coefficients, (B) testing several regression coefficients jointly, and (C) testing a linear combination of regression coefficients. A. Testing Individual Coefficients: Student’s t Test As we have shown, with the normality assumption for u t , the OLS estimators 0 ˆ and 1 ˆ are normally distributed with means and variances given therein. 0 ˆ 2 2 0 2 , t t X N T x (5) 1 ˆ 2 1 2 , t N x (6) Hence, we can use the normal distribution to make probabilistic statements about 1 provided the true population variance 2 is known. If 2 is known, an important property of a normally distributed variable with mean μ and variance 2 is that the area under the normal curve:
ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 5 between μ ± σ is about 68 percent, that between the limits μ ± 2σ is about 95 percent, and that between μ ± 3σ is about 99.7 percent. But 2 is rarely known, and in practice it is determined by the unbiased estimator 2 ˆ . If we replace by ˆ where 2 2 1 2 T t t u T , (5) and (6) may be written as ˆ ˆ ( ) i i i t se where i=0,1. where the 1 ˆ ( ) se now refers to the estimated standard error 1 . It can be shown that the t variable thus defined follows the t distribution with T −k−1 df (degree of freedom), where T is total number of observations, k is the number of slope terms and 1 is for the intercept term in the regression. Hence, for simple regression model, we can write 0 0 0 ˆ 2 1 2 1 ˆ ( ) ˆ T t t T t t t X T x , and 1 It can be denoted by ˆ ˆ i instead of ˆ ( ) i se for i =0, 1.

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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 6 1 1 1 ˆ 2 ˆ ( ) ˆ t t x , with T-K-1 degrees of freedom, where 2 2 1 2 T t t u T . With the above transformation formulae we may conduct tests of any hypothesis concerning the true value of the population parameter i . i. Two Sided (or Two Tailed) t Test Suppose we want to test the null hypothesis that the true parameter 1 is equal to a certain value * 1 . Formally we wish to test the null hypothesis * 0 1 1 : H against the alternative hypothesis * 1 1 : A H We substitute * 1 1 into the formula, and the t statistic becomes