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Metu department of economics instructor dr ozan

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METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 7 * 1 1 1 ˆ 1 ˆ ( ) ˆ ( ) t se with T-k-1 degrees of freedom where 1 ˆ = least squares estimate of 1 * 1 = hypothesized value of 1 ( * 0 1 1 : H ) 1 ˆ ( ) se = estimated standard error of 1 ˆ . 2 T= sample size k= total number of estimated parameters excluding intercept term 3 . Note that, under the null hypothesis the true value of 1 is * 1 , but under the alternative hypothesis it is less than or greater than * 1 . Hence we can say that the null hypothesis is a simple hypothesis, whereas the alternative hypothesis is composite: it is what is known as a two-sided hypothesis . Very often such a two-sided alternative hypothesis reflects the fact that we do not have a strong a priori or theoretical expectation about the direction in which the alternative hypothesis should move from the null hypothesis. 2 That is, the square root of the estimated variance of 1 ˆ which is given by 2 ˆ t x . Note that there is no " hat " attached to se because se is already defined as an estimate. 3 k =1 in our model, since 2 degrees of freedom have been used up in estimating the two parameters, 0 ˆ and 1 ˆ .
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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 8 a. Steps of Two-Sided Individual Test The steps for carrying out tests on an individual coefficient are as follows. Step 1. 0 0 : i H versus 0 : i A H Step 2. Construct the same t -statistic ˆ ˆ ˆ ( ) i i i t se , where ˆ i is the estimate and ˆ ( ) i se is its standard error. Under 0 H , it has a t - distribution with T-k-1 degrees 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. Step 3. Look up in the t -table the entry corresponding to T-k-1 degrees of freedom and find the critical point * 1 ( /2) T k t   such that the area to the right of it is one-half the level of significance ( ) . Step 4. Reject the null hypothesis if * ˆ 1 ( /2) i T k t t   . Using p -value approach, reject 0 H if the p -value is less than the level of significance.
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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 9 b. Individual Significance Test Since most regression hypotheses test whether a particular regression coefficient is significantly different from zero, * 1 is typically zero, and the most used form of the t -statistic becomes ˆ ˆ ( 0) ˆ ( ) i i i t se with T-k-1 degrees of freedom, i =0,1. which simplifies to ˆ ˆ ˆ ( ) i i i t se To decide whether to reject or not to reject a null hypothesis based on a calculated t -value, we use a critical t-value.
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