ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected]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 ut, the OLS estimators 0ˆand 1ˆare normally distributed with means and variances given therein. 0ˆ2202,ttXNTx(5) 1ˆ212,tNx(6) Hence, we can use the normal distribution to make probabilistic statements about 1provided the true population variance2is known. If 2is known, an important property of a normally distributed variable with mean μand variance 2is 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: [email protected]Lecture Notes 5 between μ ± σis about 68 percent, that between the limits μ ± 2σis about 95 percent, and that between μ ± 3σis about 99.7 percent. But 2is rarely known, and in practice it is determined by the unbiased estimator 2ˆ. If we replace by ˆwhere 2212TttuT, (5) and (6) may be written as ˆˆ()iiitsewhere i=0,1. where the 1ˆ()senow refers to the estimated standard error1. It can be shown that the tvariable thus defined follows the tdistribution with T−k−1df (degree of freedom), where Tis total number of observations, kis the number of slope terms and 1is for the intercept term in the regression. Hence, for simple regression model, we can write 000ˆ2121ˆ()ˆTttTtttXTx, and 1It can be denoted by ˆˆiinstead of ˆ()isefor 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: [email protected]Lecture Notes 6 111ˆ2ˆ()ˆttx, with T-K-1degrees of freedom, where 2212TttuT. 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 1is equal to a certain value *1. Formally we wish to test the null hypothesis *011:Hagainst the alternative hypothesis *11:AHWe substitute *11into the formula, and the tstatistic becomes