Econometrics-I-8

# Relationship is f 1 df t2df a linear function of

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Relationship is F [ 1, d.f.] = t2[d.f.] A linear function of coefficients equals a particular value (linear function of coefficients - value)2 F = ---------------------------------------------------- Variance of linear function Note square of distance in numerator Suppose linear function is k wk bk Variance is kl wkwl Cov[bk,bl] This is the Wald statistic. Also the square of the somewhat familiar t statistic. Several linear functions. Use Wald or F. Loss of fit measures may be easier to compute. ™    40/50

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Part 8: Hypothesis Testing Hypothesis Test: Sum of Coefficients = 1? ---------------------------------------------------------------------- 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 --------+------------------------------------------------------------- ™    41/50
Part 8: Hypothesis Testing Hypothesis Test: Sum of Coefficients Do the three aggregate price elasticities sum to zero? H0 :β7 + β8 + β9 = 0 R = [0, 0, 0, 0, 0, 0, 1, 1, 1], q = [0] Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- LPN| 1.10125*** .26840 4.103 .0003 .60539 LPD| .92018*** .27018 3.406 .0021 .43343 LPS| -1.09213*** .30812 -3.544 .0015 .68105 ™    42/50

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Part 8: Hypothesis Testing Wald Test ™    43/50
Part 8: Hypothesis Testing Using the Wald Statistic --> Matrix ; R = [0,1,0,0,0,0,0,0,0 / 0,0,1,0,0,0,0,0,0]\$ --> Matrix ; q = [1/-1]\$ --> Matrix ; list ; m = R*b - q \$ Matrix M has 2 rows and 1 columns. 1 +-------------+ 1| .35721 2| .49421 +-------------+ --> Matrix ; list ; vm = R*varb*R' \$ Matrix VM has 2 rows and 2 columns. 1 2 +-------------+-------------+ 1| .02120 .00291 2| .00291 .00384 +-------------+-------------+ --> Matrix ; list ; w = m'<vm>m \$ Matrix W has 1 rows and 1 columns. 1 +-------------+ 1| 63.55962 +-------------+ Joint hypothesis: b(LY)   =   1 b(LPG)   =  -1 ™    44/50

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Part 8: Hypothesis Testing Application: Cost Function ™    45/50
Part 8: Hypothesis Testing Regression Results ---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=C Mean = 3.07162 Standard deviation = 1.54273 Number of observs. = 158 Model size Parameters = 9 Degrees of freedom = 149 Residuals Sum of squares = 2.56313 Standard error of e = .13116 Fit R-squared = .99314 Adjusted R-squared = .99277 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| 5.22853 3.18024 1.644 .1023 K| -.21055 .21757 -.968 .3348 4.25096 L| -.85353*** .31485 -2.711 .0075 8.97280 F| .18862 .20225 .933 .3525 3.39118 Q| -.96450*** .36798 -2.621 .0097 8.26549 Q2| .05250*** .00459 11.430 .0000 35.7913 QK| .04273 .02758 1.550 .1233 35.1677 QL| .11698*** .03728 3.138 .0021 74.2063 QF| .05950** .02478 2.401 .0176 28.0108 --------+------------------------------------------------------------- Note: ***, **, * = Significance at 1%, 5%, 10% level. ---------------------------------------------------------------------- ™    46/50

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Part 8: Hypothesis Testing Price Homogeneity: Only Price Ratios Matter β2 + β3 + β4 = 1. β7 + β8 + β9 = 0. ---------------------------------------------------------------------- Linearly restricted regression .................... LHS=C Mean = 3.07162 Standard deviation = 1.54273 Number of observs. = 158 Model size Parameters = 7 Degrees of freedom = 151 Residuals Sum of squares = 2.85625 Standard error of e = .13753 Fit R-squared = .99236 Restrictns. F[ 2, 149] (prob) = 8.5(.0003) Not using OLS or no constant. Rsqrd & F may be < 0 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -7.24078*** 1.01411 -7.140 .0000 K| .38943** .16933 2.300 .0228 4.25096 L| .19130 .19530 .979 .3289 8.97280 F| .41927** .20077 2.088 .0385 3.39118 Q| .45889*** .12402 3.700 .0003 8.26549 Q2| .06008*** .00441 13.612 .0000 35.7913 QK| -.02954 .02125 -1.390 .1665 35.1677 QL| -.00462 .02364 -.195 .8454 74.2063 QF| .03416 .02489 1.373 .1720 28.0108 --------+------------------------------------------------------------- ™    47/50
Part 8: Hypothesis Testing Imposing the Restrictions Alternatively, compute the restricted regression by converting to price ratios and Imposing the restrictions directly. This is a regression of log(c/pf) on log(pk/pf), log(pl/pf) etc.

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