Since 0025 ˆ t t we reject the null hypothesis and

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Since, 0 0.025 ˆ t t , we reject the null hypothesis and conclude that the estimate of 0 is statistically significant from zero at 0.05 level of significance. For individual significance test of slope coefficient, we wish to test the hypothesis 0 1 : 0 H Against the alternative hypothesis 1 : 0 A H The t statistic for slope coefficient is as follows: 1 1 ˆ 1 ˆ 0.7 3.3 0.21 ˆ ( ) t se Taking the level of significance as 0.05 ( =0.05), the critical values of t for T-k-1=20-1-1=18 degrees of freedom are: -t 0.025 =-2.101, and
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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 13 t 0.025 =2.101. Hence the acceptance region 1 0.025 0.025 ˆ t t t is 1 ˆ 2.101 2.101 t . Since, 1 0.025 ˆ t t , we reject the null hypothesis and conclude that 1 is statistically significant from zero at 0.05 level of significance. Example 2 Suppose that we have the ten pairs of observations on X (price) and Y (quantity supplied) shown in table below. Obs. Y (Quantity) X (Price) 1 69 9 2 76 12 3 52 6 4 56 10 5 57 9 6 77 10 7 58 7 8 55 8 9 67 12 10 53 6 11 72 11 12 64 8 Using the formulas for intercept and slope terms, we got the following estimated supply function: ˆ 33.75 3.25 t t Y X
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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 14 For the estimation we have used the OLS formulas in mean deviation form: 1 1 2 1 156 ˆ 3.25 48 T t t t T t t x y x and, 0 1 ˆ ˆ 63 (3.25)(9) 33.75 Y X Obs. Y (Quantity) X (Price) y x xy x 2 1 69 9 6 0 0 0 2 76 12 13 3 39 9 3 52 6 -11 -3 33 9 4 56 10 -7 1 -7 1 5 57 9 -6 0 0 0 6 77 10 14 1 14 1 7 58 7 -5 -2 10 4 8 55 8 -8 -1 8 1 9 67 12 4 3 12 9 10 53 6 -10 -3 30 9 11 72 11 9 2 18 4 12 64 8 1 -1 -1 1 Y = 63 X = 9 xy=156 x 2 =48 0 ˆ 33.75 1 ˆ 3.25 Show that the standard errors are: 0 0 2 2 0 ˆ ˆ 2 ˆ ˆ ˆ (or ( )or ) 8.28 t t X s se T x 1 1 2 1 ˆ ˆ 2 1 ˆ ˆ ˆ (or ( )or ) 0.89 t s se x
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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 15 The t values for the two parameter estimates are 0 0 ˆ 0 ˆ 33.75 4.08 8.28 ˆ ( ) t se 1 1 ˆ 1 ˆ 3.25 3.65 0.89 ˆ ( ) t se Since 0 0.025 ˆ 4.08 t t and 1 0.025 ˆ 3.65 t t , both estimates are statistically significant at 0.05 level of significance. ii. One Sided (or One Tailed) t Test The most common use of the one-sided t -test is to determine whether a regression coefficient has the sign or value range predicted by economic theory. Possible one sided hypotheses around zero are: 0 1 1 : 0 : 0 A H H and 0 1 1 : 0 : 0 A H H
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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 16 Possible one-sided hypotheses based on hypothesized values other than zero are: * 0 1 1 * 1 1 : : A H H and * 0 1 1 * 1 1 : : A H H a. Steps of One-Sided Individual Test Step 1. 0 0 : i H versus 0 : i A H Step 2. Construct the t-statistic ˆ ˆ ˆ ( ) i i i t se , where ˆ
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