Econometrics-I-8

Prob(reject null|mean=0 = 0.05 prob(reject

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Unformatted text preview: Prob(Reject null|mean=0) = 0.05 Prob(Reject null|mean=.5)=0.07902 Prob(Reject null|mean=1)=0.170066. Increases as the (alternative) mean rises. B E T A .0 8 4 .1 6 8 .2 5 1 .3 3 5 .4 1 9 .0 0 0-3-2-1 1 2 3 4 5-4 N 2 N 0 N 1 Variable &#152;&#152;&#152;™™™ ™ 23/50 Part 8: Hypothesis Testing Test Statistic For the fit measures, use a normalized measure of the loss of fit: ( 29 ( 29 ( 29 ( 29 r r r r r r r 0 since and 0 since ≥ ≥ ′ ′ ′ ′- ′ ′ ≥ ≥ ′ 2 2 u r 2 2 u r 2 u 2 2 u u u yy yy u u u u u u R -R / J F[J,n-K] = R R 1-R / (n-K) Often useful R =1- R =1- S S Insert these in F and it becomes / J F[J,n-K] = / (n-K) e e e e e e e e e e e e e e &#152;&#152;&#152;™™™ ™ 24/50 Part 8: Hypothesis Testing An important relationship between t and F 2 Chi Squared[J]/ J F Chi squared[n K]/ (n K) where the two chi-squared variables are independent. If J = 1, i.e., testing a single restriction, Chi Squared[1]/1 F Chi squared[n K]/ (n K) (N[0,1]) Chi s- =---- =--- =- { } 2 2 quared[n K]/ (n K) N[0,1] = t[1] Chi squared[n K]/ (n K)-- = --- For a single restriction, F[1,n-K] is the square of the t ratio. &#152;&#152;&#152;™™™ ™ 25/50 Part 8: Hypothesis Testing Application Time series regression, LogG = 1 + 2logY + 3logPG + 4logPNC + 5logPUC + 6logPPT + 7logPN + 8logPD + 9logPS + Period = 1960 - 1995. Note that all coefficients in the model are elasticities. &#152;&#152;&#152;™™™ ™ 26/50 Part 8: Hypothesis Testing Full Model---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=LG Mean = 5.39299 Standard deviation = .24878 Number of observs. = 36 Model size Parameters = 9 Degrees of freedom = 27 Residuals Sum of squares = .00855 <******* Standard error of e = .01780 <******* Fit R-squared = .99605 <******* Adjusted R-squared = .99488 <*******--------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X--------+------------------------------------------------------------- Constant| -6.95326*** 1.29811 -5.356 .0000 LY| 1.35721*** .14562 9.320 .0000 9.11093 LPG| -.50579*** .06200 -8.158 .0000 .67409 LPNC| -.01654 .19957 -.083 .9346 .44320 LPUC| -.12354* .06568 -1.881 .0708 .66361 LPPT| .11571 .07859 1.472 .1525 .77208 LPN| 1.10125*** .26840 4.103 .0003 .60539 LPD| .92018*** .27018 3.406 .0021 .43343 LPS| -1.09213*** .30812 -3.544 .0015 .68105--------+------------------------------------------------------------- &#152;&#152;&#152;&#152;™™ ™ 27/50 Part 8: Hypothesis Testing Test About One Parameter Is the price of public transportation really relevant? H0 : 6 = 0. Confidence interval: b6 t(.95,27) Standard error = .11571 2.052(.07859) = .11571 .16127 = (-.045557 ,.27698) Contains 0.0. Do not reject hypothesis Regression fit if drop? Without LPPT, R-squared= .99573 Compare R2, was .99605, F(1,27) = [(.99605 - .99573)/1]/[(1-.99605)/(36-9)] = 2.187 = 1.4722 (with some rounding difference) &#152;&#152;&#152;&#152;™™ ™ 28/50 Part 8: Hypothesis Testing...
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Prob(Reject null|mean=0 = 0.05 Prob(Reject...

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