Econometrics-I-5

# Part 5 regression algebra and fit ordinary least

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Part 5: Regression Algebra and Fit +----------------------------------------------------+ | Ordinary least squares regression | | LHS=LOGBOX Mean = 16.47993 | | Standard deviation = .9429722 | | Number of observs. = 62 | | Residuals Sum of squares = 25.36721 | | Standard error of e = .6984489 | | Fit R-squared = .5323241 | | Adjusted R-squared = .4513802 | +----------------------------------------------------+ +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |t-ratio |P[|T|>t]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ |Constant| 11.9602*** .91818 13.026 .0000 | |LOGBUDGT| .38159** .18711 2.039 .0465 3.71468| |STARPOWR| .01303 .01315 .991 .3263 18.0316| |SEQUEL | .33147 .28492 1.163 .2500 .14516| |MPRATING| -.21185 .13975 -1.516 .1356 2.96774| |ACTION | -.81404** .30760 -2.646 .0107 .22581| |COMEDY | .04048 .25367 .160 .8738 .32258| |ANIMATED| -.80183* .40776 -1.966 .0546 .09677| |HORROR | .47454 .38629 1.228 .2248 .09677| |PCBUZZ | .39704*** .08575 4.630 .0000 9.19362| +--------+------------------------------------------------------------+ ™    21/33

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Part 5: Regression Algebra and Fit Adjusted R Squared p Adjusted R2 (for degrees of freedom?) = 1 - [(n-1)/(n-K)](1 - R2) p Degrees of freedom” adjustment suggests something about “unbiasedness.” The ratio is not unbiased. p includes a penalty for variables that don’t add much fit. Can fall when a variable is added to the equation. 2 R 2 R ™    22/33
Part 5: Regression Algebra and Fit Adjusted R2 What is being adjusted? The penalty for using up degrees of freedom. = 1 - [ ee /(n – K)]/[ yM 0 y /(n-1)] uses the ratio of two ‘unbiased’ estimators. Is the ratio unbiased? = 1 – [(n-1)/(n-K)(1 – R2)] Will rise when a variable is added to the regression? is higher with z than without z if and only if the t ratio on z is in the regression when it is added is larger than one in absolute value. 2 R 2 R 2 R 2 R ™    23/33

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Part 5: Regression Algebra and Fit Full Regression (Without PD) ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 9 Degrees of freedom = 27 Residuals Sum of squares = 596.68995 Standard error of e = 4.70102 Fit R-squared = .99334 <********** Adjusted R-squared = .99137 <********** Info criter. LogAmemiya Prd. Crt. = 3.31870 <********** Akaike Info. Criter. = 3.30788 <********** Model test F[ 8, 27] (prob) = 503.3(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -8220.38** 3629.309 -2.265 .0317 PG| -26.8313*** 5.76403 -4.655 .0001 2.31661 Y| .02214*** .00711 3.116 .0043 9232.86 PNC| 36.2027 21.54563 1.680 .1044 1.67078 PUC| -6.23235 5.01098 -1.244 .2243 2.34364 PPT| 9.35681 8.94549 1.046 .3048 2.74486 PN| 53.5879* 30.61384 1.750 .0914 2.08511 PS| -65.4897*** 23.58819 -2.776 .0099 2.36898 YEAR| 4.18510** 1.87283 2.235 .0339 1977.50 --------+------------------------------------------------------------- ™    24/33
Part 5: Regression Algebra and Fit PD added to the model. R2 rises, Adj. R2 falls ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 10 Degrees of freedom = 26 Residuals Sum of squares = 594.54206 Standard error of e = 4.78195 Fit R-squared = .99336 Was 0.99334 Adjusted R-squared = .99107 Was 0.99137 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -7916.51** 3822.602 -2.071 .0484 PG| -26.8077*** 5.86376 -4.572 .0001 2.31661 Y| .02231*** .00725 3.077 .0049 9232.86 PNC| 30.0618 29.69543 1.012 .3207 1.67078 PUC| -7.44699 6.45668 -1.153 .2592 2.34364 PPT| 9.05542 9.15246 .989 .3316 2.74486 PD| 11.8023 38.50913 .306 .7617 1.65056 (NOTE LOW t ratio) PN| 47.3306 37.23680 1.271 .2150 2.08511 PS| -60.6202** 28.77798 -2.106 .0450 2.36898 YEAR| 4.02861* 1.97231 2.043 .0514 1977.50 --------+------------------------------------------------------------- ™    25/33

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Part 5: Regression Algebra and Fit Linear Least Squares Subject to Restrictions Restrictions : Theory imposes certain restrictions on parameters. Some common applications Dropping variables from the equation = certain coefficients in b forced to equal 0. (Probably the most common testing situation. “Is a certain variable significant?”) Adding up conditions: Sums of certain coefficients must equal fixed values. Adding up conditions in demand systems. Constant returns to scale in production functions. Equality restrictions: Certain coefficients must equal other coefficients. Using real vs. nominal variables in equations.

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