1le solution

1LE Solution

A. The demand for chicken in the United States, 1960-1982

1. 50 points. Derive specifications 1-5. In case assumptions are violated, remedy it if possible.

Specification 1: 1 2 2 3 3ln ln lnt t t tY X X u

Model 1: OLS, using observations 1960-1982 (T = 23) Dependent variable: l_Y coefficient std. error t-ratio p-value --------------------------------------------------------- const 2.03282 0.116183 17.50 1.36e-013 *** l_X2 0.451528 0.0246948 18.28 5.94e-014 *** l_X3 -0.372212 0.0634660 -5.865 9.76e-06 *** Mean dependent var 3.663887 S.D. dependent var 0.187659 Sum squared resid 0.015437 S.E. of regression 0.027783 R-squared 0.980074 Adjusted R-squared 0.978082 F(2, 20) 491.8681 P-value(F) 9.87e-18 Log-likelihood 51.38868 Akaike criterion -96.77735 Schwarz criterion -93.37087 Hannan-Quinn -95.92063 rho 0.049676 Durbin-Watson 1.875601 Log-likelihood for Y = -32.8807 RESET test for specification (squares and cubes) Test statistic: F = 1.354917, with p-value = P(F(2,18) > 1.35492) = 0.283 RESET test for specification (squares only) Test statistic: F = 2.551013, with p-value = P(F(1,19) > 2.55101) = 0.127 RESET test for specification (cubes only) Test statistic: F = 2.522484, with p-value = P(F(1,19) > 2.52248) = 0.129 Fail to reject Ho. Correctly Specified. Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_X2 5.649 l_X3 5.649

VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 1824.2298 Determinant = 31.511333 Reciprocal condition number = 2.0458966e-005 No Multicollinearity. White's test for heteroskedasticity OLS, using observations 1960-1982 (T = 23) Dependent variable: uhat^2 coefficient std. error t-ratio p-value ------------------------------------------------------- const -0.137588 0.132843 -1.036 0.3148 l_X2 -0.0190030 0.0255201 -0.7446 0.4667 l_X3 0.104310 0.102456 1.018 0.3229 sq_l_X2 -0.00412872 0.00294915 -1.400 0.1795 X2_X3 0.0194065 0.0152199 1.275 0.2194 sq_l_X3 -0.0304051 0.0245355 -1.239 0.2321 Unadjusted R-squared = 0.173226 Test statistic: TR^2 = 3.984192, with p-value = P(Chi-square(5) > 3.984192) = 0.551694 Fail to reject Ho. Homoskedastic. rho 0.049676 Durbin-Watson 1.875601 5% critical values for Durbin-Watson statistic, n = 23, k = 2 dL = 1.1682 dU = 1.5435 Durbin-Watson 1.875601 > dU = 1.5435 Fail to Reject Ho. Proceed with Breusch-Godfrey Lag 1: Alternative statistic: TR^2 = 0.065260, with p-value = P(Chi-square(1) > 0.0652602) = 0.798 Lag 2: Alternative statistic: TR^2 = 4.513975, with p-value = P(Chi-square(2) > 4.51398) = 0.105

Lag 3: Alternative statistic: TR^2 = 4.516275, with p-value = P(Chi-square(3) > 4.51628) = 0.211 Lag 4: Alternative statistic: TR^2 = 5.121125, with p-value = P(Chi-square(4) > 5.12112) = 0.275 Lag 5: Alternative statistic: TR^2 = 5.330327, with p-value = P(Chi-square(5) > 5.33033) = 0.377 Lag 6: Alternative statistic: TR^2 = 5.332384, with p-value = P(Chi-square(6) > 5.33238) = 0.502 Lag 7: Alternative statistic: TR^2 = 5.336762, with p-value = P(Chi-square(7) > 5.33676) = 0.619 Lag 8: Alternative statistic: TR^2 = 5.345686, with p-value = P(Chi-square(8) > 5.34569) = 0.72 Lag 9: Alternative statistic: TR^2 = 7.914648, with p-value = P(Chi-square(9) > 7.91465) = 0.543 Lag 10: Alternative statistic: TR^2 = 10.618184, with p-value = P(Chi-square(10) > 10.6182) = 0.388 Lag 11: Alternative statistic: TR^2 = 14.772122, with p-value = P(Chi-square(11) > 14.7721) = 0.193 Fail to Reject Ho. No autocorrelation. Jarque-Bera test = 1.67566, with p-value 0.432649 Fail to Reject Ho. Errors are normally distributed. Summary:

Tests Results Decision

ln Y = 2.03 + 0.45 ln X2 - 0.37 ln X3

R2 0.98

Adjusted R2 0.98

F Test 9.87E-18 Model is valid

T Test

const 1.36e-013 *** Significant

l_X2 5.94e-014 *** Significant


RAMSEY'S RESET All p-values > 0.05

Correctly specified

Multicollinearity Test No multicollinearity

l_X2 5.649 l_X3 5.649

Heteroskedasticity Test 0.551694 Homoskedastic

Autocorrelation Test

DW > dU

All p-values > 0.05

No autocorrelation

Normality Test 0.432649 Normally Distributed Errors

Specification 2: 1 2 2 3 3 4 4ln ln ln lnt t t t tY X X X u

Model 3: OLS, using observations 1960-1982 (T = 23) Dependent variable: l_Y coefficient std. error t-ratio p-value --------------------------------------------------------- const 2.12550 0.137882 15.42 3.40e-012 *** l_X2 0.405924 0.0447915 9.063 2.51e-08 *** l_X3 -0.438825 0.0833324 -5.266 4.40e-05 *** l_X4 0.106656 0.0878383 1.214 0.2395 Mean dependent var 3.663887 S.D. dependent var 0.187659 Sum squared resid 0.014326 S.E. of regression 0.027459 R-squared 0.981509 Adjusted R-squared 0.978590 F(3, 19) 336.1808 P-value(F) 1.23e-16 Log-likelihood 52.24812 Akaike criterion -96.49623 Schwarz criterion -91.95426 Hannan-Quinn -95.35394 rho 0.106797 Durbin-Watson 1.778678 Log-likelihood for Y = -32.0213 Excluding the constant, p-value was highest for variable 10 (l_X4) RESET test for specification (squares and cubes) Test statistic: F = 0.735589, with p-value = P(F(2,17) > 0.735589) = 0.494 RESET test for specification (squares only) Test statistic: F = 1.305520, with p-value = P(F(1,18) > 1.30552) = 0.268 RESET test for specification (cubes only) Test statistic: F = 1.281104, with p-value = P(F(1,18) > 1.2811) = 0.273 Fail to reject Ho. Correctly Specified.

Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_X2 19.024 l_X3 9.969 l_X4 32.489 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 2520.6704 Determinant = 3.0793715 Reciprocal condition number = 8.0446316e-006 X2 and X4 are collinear. White's test for heteroskedasticity OLS, using observations 1960-1982 (T = 23) Dependent variable: uhat^2 coefficient std. error t-ratio p-value -------------------------------------------------------- const -0.0881889 0.143060 -0.6164 0.5482 l_X2 -0.0135747 0.0502221 -0.2703 0.7912 l_X3 0.0199224 0.142884 0.1394 0.8912 l_X4 0.0414183 0.123826 0.3345 0.7433 sq_l_X2 -0.00756895 0.00952144 -0.7949 0.4409 X2_X3 0.0265261 0.0499633 0.5309 0.6044 X2_X4 0.00312212 0.0345594 0.09034 0.9294 sq_l_X3 -0.0103037 0.0380486 -0.2708 0.7908 X3_X4 -0.0260893 0.101573 -0.2569 0.8013 sq_l_X4 0.00411950 0.0524370 0.07856 0.9386 Unadjusted R-squared = 0.359382 Test statistic: TR^2 = 8.265797, with p-value = P(Chi-square(9) > 8.265797) = 0.507598 Fail to reject Ho. Homoskedastic. rho 0.106797 Durbin-Watson 1.778678 5% critical values for Durbin-Watson statistic, n = 23, k = 3

dL = 1.0778 dU = 1.6597 Durbin-Watson 1.778678 > dU = 1.6597 Fail to Reject Ho. Proceed with Breusch-Godfrey Lag 1: Alternative statistic: TR^2 = 0.335777, with p-value = P(Chi-square(1) > 0.335777) = 0.562 Lag 2: Alternative statistic: TR^2 = 3.530854, with p-value = P(Chi-square(2) > 3.53085) = 0.171 Lag 3: Alternative statistic: TR^2 = 3.687262, with p-value = P(Chi-square(3) > 3.68726) = 0.297 Lag 4: Alternative statistic: TR^2 = 4.786886, with p-value = P(Chi-square(4) > 4.78689) = 0.31 Lag 5: Alternative statistic: TR^2 = 4.786984, with p-value = P(Chi-square(5) > 4.78698) = 0.442 Lag 6: Alternative statistic: TR^2 = 5.025121, with p-value = P(Chi-square(6) > 5.02512) = 0.541 Lag 7: Alternative statistic: TR^2 = 5.356404, with p-value = P(Chi-square(7) > 5.3564) = 0.617 Lag 8: Alternative statistic: TR^2 = 5.382452, with p-value = P(Chi-square(8) > 5.38245) = 0.716 Lag 9: Alternative statistic: TR^2 = 15.014217, with p-value = P(Chi-square(9) > 15.0142) = 0.0905 Lag 10: Alternative statistic: TR^2 = 15.821113, with p-value = P(Chi-square(10) > 15.8211) = 0.105 Lag 11: Alternative statistic: TR^2 = 18.001362, with p-value = P(Chi-square(11) > 18.0014) = 0.0815 Fail to Reject Ho. No autocorrelation. Jarque-Bera test = 1.06914, with p-value 0.58592 Fail to Reject Ho. Errors are normally distributed. Summary:


ln Y = 2.12 + 0.41 ln X2 - 0.44 ln X3 + 0.11 lnX4

R2 0.98

Adjusted R2 0.98


T Test




l_X4 0.2395 Not Significant


Correctly specified

Multicollinearity Test X2 and X4 are collinear l_X2 19.024

l_X3 9.969

l_X4 32.489



DW > dU

All p-values > 0.05

No autocorrelation



Model 4: OLS, using observations 1960-1982 (T = 23) Dependent variable: l_Y coefficient std. error t-ratio p-value --------------------------------------------------------- const 2.03936 0.122320 16.67 8.46e-013 *** l_X2 0.440867 0.0524182 8.411 7.91e-08 *** l_X3 -0.381500 0.0763394 -4.997 8.00e-05 *** l_X5 0.0213660 0.0920071 0.2322 0.8188 Mean dependent var 3.663887 S.D. dependent var 0.187659 Sum squared resid 0.015394 S.E. of regression 0.028464 R-squared 0.980131 Adjusted R-squared 0.976994 F(3, 19) 312.4186 P-value(F) 2.44e-16 Log-likelihood 51.42127 Akaike criterion -94.84254 Schwarz criterion -90.30056 Hannan-Quinn -93.70025 rho 0.051662 Durbin-Watson 1.868772 Log-likelihood for Y = -32.8481

Excluding the constant, p-value was highest for variable 11 (l_X5) RESET test for specification (squares and cubes) Test statistic: F = 2.722305, with p-value = P(F(2,17) > 2.72231) = 0.0943 RESET test for specification (squares only) Test statistic: F = 4.782213, with p-value = P(F(1,18) > 4.78221) = 0.0422 RESET test for specification (cubes only) Test statistic: F = 4.696225, with p-value = P(F(1,18) > 4.69623) = 0.0439 Reject Ho. Misspecified. Summary:


ln Y = 2.04 + 0.44 ln X2 - 0.38 ln X3 + 0.02 lnX5

R2 0.98

Adjusted R2 0.98


T Test





RAMSEY'S RESET Two p-values < 0.05

Misspecified

Specification 4: 1 2 2 3 3 4 4 5 5ln ln ln ln lnt t t t t tY X X X X u

Model 5: OLS, using observations 1960-1982 (T = 23) Dependent variable: l_Y coefficient std. error t-ratio p-value --------------------------------------------------------- const 2.18979 0.155715 14.06 3.77e-011 *** l_X2 0.342555 0.0832663 4.114 0.0007 *** l_X3 -0.504592 0.110894 -4.550 0.0002 *** l_X4 0.148545 0.0996726 1.490 0.1535 l_X5 0.0911049 0.100716 0.9046 0.3776 Mean dependent var 3.663887 S.D. dependent var 0.187659

Sum squared resid 0.013703 S.E. of regression 0.027591 R-squared 0.982313 Adjusted R-squared 0.978383 F(4, 18) 249.9282 P-value(F) 1.67e-15 Log-likelihood 52.75935 Akaike criterion -95.51870 Schwarz criterion -89.84123 Hannan-Quinn -94.09083 rho 0.082235 Durbin-Watson 1.826069 Log-likelihood for Y = -31.51 Excluding the constant, p-value was highest for variable 11 (l_X5) RESET test for specification (squares and cubes) Test statistic: F = 1.757534, with p-value = P(F(2,16) > 1.75753) = 0.204 RESET test for specification (squares only) Test statistic: F = 2.984631, with p-value = P(F(1,17) > 2.98463) = 0.102 RESET test for specification(cubes only) Test statistic: F = 2.913951, with p-value = P(F(1,17) > 2.91395) = 0.106 Fail to Reject Ho. Correctly Specified. Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_X2 65.115 l_X3 17.486 l_X4 41.433 l_X5 42.307 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 3266.5534 Determinant = 0.23109959 Reciprocal condition number = 3.6880068e-006 All vars are collinear. White's test for heteroskedasticity OLS, using observations 1960-1982 (T = 23)

Dependent variable: uhat^2 coefficient std. error t-ratio p-value -------------------------------------------------------- const 0.0666492 0.132097 0.5045 0.6275 l_X2 0.0903519 0.110755 0.8158 0.4382 l_X3 -0.0957348 0.146949 -0.6515 0.5330 l_X4 -0.0346548 0.140236 -0.2471 0.8110 l_X5 -0.0492160 0.105977 -0.4644 0.6547 sq_l_X2 -0.0287863 0.0327508 -0.8789 0.4051 X2_X3 -0.00502249 0.0992081 -0.05063 0.9609 X2_X4 0.0518875 0.0840126 0.6176 0.5540 X2_X5 0.0182509 0.0690208 0.2644 0.7981 sq_l_X3 0.0149539 0.0571317 0.2617 0.8001 X3_X4 0.0336122 0.131514 0.2556 0.8047 X3_X5 -0.0263042 0.0975837 -0.2696 0.7943 sq_l_X4 -0.0156263 0.0595316 -0.2625 0.7996 X4_X5 -0.0656163 0.0987854 -0.6642 0.5252 sq_l_X5 0.0336460 0.0383519 0.8773 0.4059 Unadjusted R-squared = 0.725482 Test statistic: TR^2 = 16.686094, with p-value = P(Chi-square(14) > 16.686094) = 0.273292 Fail to reject Ho. Homoskedastic. rho 0.082235 Durbin-Watson 1.826069 5% critical values for Durbin-Watson statistic, n = 23, k = 4 dL = 0.9864 dU = 1.7855 Durbin-Watson 1.826069 > dU = 1.7855 Fail to Reject Ho. Proceed with Breusch-Godfrey Lag 1: Alternative statistic: TR^2 = 0.266910, with p-value = P(Chi-square(1) > 0.26691) = 0.605 Lag 2: Alternative statistic: TR^2 = 2.471524, with p-value = P(Chi-square(2) > 2.47152) = 0.291 Lag 3: Alternative statistic: TR^2 = 3.028455, with p-value = P(Chi-square(3) > 3.02845) = 0.387 Lag 4: Alternative statistic: TR^2 = 3.496698,

with p-value = P(Chi-square(4) > 3.4967) = 0.478 Lag 5: Alternative statistic: TR^2 = 3.570441, with p-value = P(Chi-square(5) > 3.57044) = 0.613 Lag 6: Alternative statistic: TR^2 = 3.815685, with p-value = P(Chi-square(6) > 3.81569) = 0.702 Lag 7: Alternative statistic: TR^2 = 4.165470, with p-value = P(Chi-square(7) > 4.16547) = 0.761 Lag 8: Alternative statistic: TR^2 = 4.171108, with p-value = P(Chi-square(8) > 4.17111) = 0.841 Lag 9: Alternative statistic: TR^2 = 17.471803, with p-value = P(Chi-square(9) > 17.4718) = 0.0418 Lag 10: Alternative statistic: TR^2 = 17.539138, with p-value = P(Chi-square(10) > 17.5391) = 0.0633 Lag 11: Alternative statistic: TR^2 = 17.599731, with p-value = P(Chi-square(11) > 17.5997) = 0.0913 Fail to Reject Ho. No autocorrelation. Jarque-Bera test = 1.07777, with p-value 0.5834 Fail to Reject Ho. Errors are normally distributed. Summary:


ln Y = 2.19 + 0.34 ln X2 - 0.50 ln X3 + 0.15 lnX4 + 0.09 lnX5

R2 0.98

Adjusted R2 0.98


T Test


l_X2 0.0007 *** Significant





Correctly specified

Multicollinearity Test All vars are collinear l_X2 65.115

l_X3 17.486

l_X4 41.433

l_X5 42.307



DW > dU

All p-values > 0.05

No autocorrelation



Model 6: OLS, using observations 1960-1982 (T = 23) Dependent variable: l_Y coefficient std. error t-ratio p-value --------------------------------------------------------- const 2.02986 0.118682 17.10 5.36e-013 *** l_X2 0.481286 0.0681877 7.058 1.02e-06 *** l_X3 -0.350628 0.0793939 -4.416 0.0003 *** l_X6 -0.0610352 0.129960 -0.4696 0.6440 Mean dependent var 3.663887 S.D. dependent var 0.187659 Sum squared resid 0.015260 S.E. of regression 0.028340 R-squared 0.980303 Adjusted R-squared 0.977193 F(3, 19) 315.2063 P-value(F) 2.24e-16 Log-likelihood 51.52141 Akaike criterion -95.04282 Schwarz criterion -90.50084 Hannan-Quinn -93.90052 rho 0.030523 Durbin-Watson 1.910653 Log-likelihood for Y = -32.748 Excluding the constant, p-value was highest for variable 12 (l_X6) RESET test for specification (squares and cubes) Test statistic: F = 1.284077, with p-value = P(F(2,17) > 1.28408) = 0.302 RESET test for specification (squares only) Test statistic: F = 2.398063, with p-value = P(F(1,18) > 2.39806) = 0.139 RESET test for specification (cubes only) Test statistic: F = 2.369678, with p-value = P(F(1,18) > 2.36968) = 0.141 Fail to Reject Ho. Correctly Specified.

Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_X2 41.389 l_X3 8.495 l_X6 57.887 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 2549.3133 Determinant = 1.4984896 Reciprocal condition number = 9.8716999e-006 X2 and X6 are collinear. White's test for heteroskedasticity OLS, using observations 1960-1982 (T = 23) Dependent variable: uhat^2 coefficient std. error t-ratio p-value -------------------------------------------------------- const -0.0915587 0.183461 -0.4991 0.6261 l_X2 -0.0105453 0.0696096 -0.1515 0.8819 l_X3 0.0250801 0.147827 0.1697 0.8679 l_X6 0.0327143 0.105413 0.3103 0.7612 sq_l_X2 -0.0497889 0.0394290 -1.263 0.2289 X2_X3 0.00708262 0.0410536 0.1725 0.8657 X2_X4 0.142594 0.125899 1.133 0.2778 sq_l_X3 -0.00549102 0.0405351 -0.1355 0.8943 X3_X4 -0.00530338 0.0737656 -0.07190 0.9438 sq_l_X6 -0.106353 0.104468 -1.018 0.3272 Unadjusted R-squared = 0.238463 Test statistic: TR^2 = 5.484656, with p-value = P(Chi-square(9) > 5.484656) = 0.790181 Fail to reject Ho. Homoskedastic. rho 0.030523 Durbin-Watson 1.910653 5% critical values for Durbin-Watson statistic, n = 23, k = 3

dL = 1.0778 dU = 1.6597 Durbin-Watson 1.910653 > dU = 1.6597 Fail to Reject Ho. Proceed with Breusch-Godfrey Lag 1: Alternative statistic: TR^2 = 0.026446, with p-value = P(Chi-square(1) > 0.0264456) = 0.871 Lag 2: Alternative statistic: TR^2 = 4.991241, with p-value = P(Chi-square(2) > 4.99124) = 0.0824 Lag 3: Alternative statistic: TR^2 = 5.012725, with p-value = P(Chi-square(3) > 5.01273) = 0.171 Lag 4: Alternative statistic: TR^2 = 5.713008, with p-value = P(Chi-square(4) > 5.71301) = 0.222 Lag 5: Alternative statistic: TR^2 = 6.166767, with p-value = P(Chi-square(5) > 6.16677) = 0.29 Lag 6: Alternative statistic: TR^2 = 6.171211, with p-value = P(Chi-square(6) > 6.17121) = 0.404 Lag 7: Alternative statistic: TR^2 = 6.343939, with p-value = P(Chi-square(7) > 6.34394) = 0.5 Lag 8: Alternative statistic: TR^2 = 6.350115, with p-value = P(Chi-square(8) > 6.35012) = 0.608 Lag 9: Alternative statistic: TR^2 = 10.237706, with p-value = P(Chi-square(9) > 10.2377) = 0.332 Lag 10: Alternative statistic: TR^2 = 13.555351, with p-value = P(Chi-square(10) > 13.5554) = 0.194 Lag 11: Alternative statistic: TR^2 = 15.342969, with p-value = P(Chi-square(11) > 15.343) = 0.167 Fail to Reject Ho. No autocorrelation. Jarque-Bera test = 1.07777, with p-value 0.5834 Fail to Reject Ho. Errors are normally distributed. Summary:


ln Y = 2.03 + 0.48 ln X2 - 0.35 ln X3 + 0.06 lnX6

R2 0.98

Adjusted R2 0.98


T Test






Correctly specified

Multicollinearity Test X2 and X6 are collinear l_X2 41.389

l_X3 8.495

l_X6 57.887



DW > dU

All p-values > 0.05

No autocorrelation


2. 10 points. Which demand function would you choose, and why? Interpret results and

recommend policy. Model 1 is the best model. It is correctly specified if we compare to model 3 and does not suffer from multicollinearity issue comparing to models 2, 4, and 5. Lastly, all the assumptions of CLRM are met. ln Y = 2.03 + 0.45 ln X2 - 0.37 ln X3. A 10% increase in real disposable income, demand for chicken increases by 4.5% and a 10% increase in the retail price of chicken its demand decreases by 3.7%. The signs are consistent with the theoretical expectations. Results suggest that the prices of pork and beef do not affect the demand for chicken. This implies that chicken consumption is independent from that of the pork and beef prices. Income elasticity = 0.45 suggests that chicken is a normal good. Furthermore, it is considered necessity good. Own-price elasticity = -0.37 suggests that it is inelastic. Thus, consumers are less responsive to the changes in prices.

B. Quinineza (2010)

1. 30 points. Model the three production functions. In case assumptions are violated, remedy

it if possible.

Neoclassical production function of degree 2;1

2 2

0 1 1 2 3 2 4 2y x x x x

Model 2: OLS, using observations 1-120 Dependent variable: PROD coefficient std. error t-ratio p-value -------------------------------------------------------------- const -130.882 487.140 -0.2687 0.7887 FRMAREA 3380.88 998.449 3.386 0.0010 *** TOTAL_NPK -13.5495 28.4854 -0.4757 0.6352 TOTAL_LABOR 0.314997 0.721108 0.4368 0.6631 sq_FRMAREA -834.842 396.459 -2.106 0.0374 ** sq_TOTAL_NPK 1.04733 0.350282 2.990 0.0034 *** sq_TOTAL_LABO 3.33250e-05 0.000143172 0.2328 0.8164 Mean dependent var 1826.023 S.D. dependent var 2853.857 Sum squared resid 5.82e+08 S.E. of regression 2268.575 R-squared 0.399970 Adjusted R-squared 0.368110 F(6, 113) 12.55397 P-value(F) 8.30e-11 Log-likelihood -1093.895 Akaike criterion 2201.790 Schwarz criterion 2221.303 Hannan-Quinn 2209.715 Excluding the constant, p-value was highest for variable 13 (sq_TOTAL_LABO) RESET test for specification (squares and cubes) Test statistic: F = 3.006409, with p-value = P(F(2,111) > 3.00641) = 0.0535 RESET test for specification (squares only) Test statistic: F = 3.788413, with p-value = P(F(1,112) > 3.78841) = 0.0541 RESET test for specification (cubes only) Test statistic: F = 2.157253, with p-value = P(F(1,112) > 2.15725) = 0.145 Fail to Reject Ho. Correctly Specified. Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem FRMAREA 7.846 TOTAL_NPK 5.269 TOTAL_LABOR 6.191

sq_FRMAREA 7.307 sq_TOTAL_NPK 4.987 sq_TOTAL_LABO 5.988 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 1.6204504e+015 Determinant = 2.9447554e+039 Reciprocal condition number = 2.0165205e-015 Note: Some vars are in squared form, no issue of multicollinearity. No multicollinearity. White's test for heteroskedasticity OLS, using observations 1-120 Dependent variable: uhat^2 Omitted due to exact collinearity: sq_FRMAREA sq_TOTAL_NPK sq_TOTAL_LABO coefficient std. error t-ratio p-value -------------------------------------------------------------------- const -9.16483e+06 1.31061e+07 -0.6993 0.4861 FRMAREA 8.48541e+07 5.94950e+07 1.426 0.1571 TOTAL_NPK 1.72210e+06 1.49792e+06 1.150 0.2532 TOTAL_LABOR -5340.16 44919.4 -0.1189 0.9056 sq_FRMAREA 3.50434e+07 7.91599e+07 0.4427 0.6590 sq_TOTAL_NPK -54472.5 74059.5 -0.7355 0.4638 sq_TOTAL_LABO 136.213 59.7019 2.282 0.0247 ** X2_X3 6.21386e+06 3.02101e+06 2.057 0.0424 ** X2_X4 -452929 172488 -2.626 0.0101 ** X2_X5 -5.55173e+07 4.76491e+07 -1.165 0.2469 X2_X6 -116628 56991.7 -2.046 0.0435 ** X2_X7 256.842 105.658 2.431 0.0169 ** X3_X4 -6341.23 2978.62 -2.129 0.0358 ** X3_X5 -2.90596e+06 1.58091e+06 -1.838 0.0692 * X3_X6 580.842 1625.42 0.3573 0.7216 X3_X7 2.02562 1.36852 1.480 0.1421 X4_X5 192685 86105.7 2.238 0.0276 ** X4_X6 132.524 62.2810 2.128 0.0359 ** X4_X7 -0.102022 0.0410704 -2.484 0.0147 ** sq_sq_FRMAREA 1.08107e+07 8.74079e+06 1.237 0.2192 X5_X6 52474.2 28170.7 1.863 0.0656 * X5_X7 -107.691 49.8462 -2.160 0.0333 ** sq_sq_TOTAL_N -4.27062 10.1459 -0.4209 0.6748 X6_X7 -0.0452758 0.0288153 -1.571 0.1195

sq_sq_TOTAL_L 1.20863e-05 5.03290e-06 2.401 0.0183 ** Warning: data matrix close to singularity! Unadjusted R-squared = 0.298696 Test statistic: TR^2 = 35.843505, with p-value = P(Chi-square(24) > 35.843505) = 0.056836 Fail to reject Ho. Homoskedastic. Jarque-Bera test = 2640.44, with p-value 0 Reject Ho. Errors are not normally distributed. But with 120 observations considered large enough, normality assumption is not an issue. Summary:


Prod = -131 + 3381FA - 14NPK + 0.31Lbr - 835FA^2 + 1.05NPK^2 + 0.00003Lbr^2 R2 0.4

Adjusted R2 0.37 F Test 8.30E-11 Model is valid T Test const 0.7887 Not Significant FA 0.0010 *** Significant NPK 0.6352 Not Significant Lbr 0.6631 Not Significant FA^2 0.0374 ** Significant NPK^2 0.0034 *** Significant Lbr^2 0.8164 Not Significant RAMSEY'S RESET All p-values > 0.05 Correctly specified Multicollinearity Test No multicollinearity FA 7.846 NPK 5.269 Lbr 6.191 FA^2 7.307 NPK^2 4.987 Lbr^2 5.988 Heteroskedasticity Test 0.056836 Homoskedastic Autocorrelation Test Cross-section data N/A Normality Test

0 Not Normally Distributed Errors; T=120

Cobb-Douglas like production function; 1 2

1 2y Ax x

Model 3: OLS, using observations 1-120 (n = 110)

Missing or incomplete observations dropped: 10 Dependent variable: l_PROD coefficient std. error t-ratio p-value ------------------------------------------------------------- const 4.07748 1.06432 3.831 0.0002 *** l_FRMAREA 0.404932 0.0909299 4.453 2.10e-05 *** l_TOTAL_NPK 0.272977 0.0816036 3.345 0.0011 *** l_TOTAL_LABOR 0.412605 0.162591 2.538 0.0126 ** Mean dependent var 6.705082 S.D. dependent var 1.453283 Sum squared resid 137.4626 S.E. of regression 1.138779 R-squared 0.402885 Adjusted R-squared 0.385986 F(3, 106) 23.84012 P-value(F) 7.20e-12 Log-likelihood -168.3412 Akaike criterion 344.6824 Schwarz criterion 355.4843 Hannan-Quinn 349.0637 Log-likelihood for PROD = -905.9 RESET test for specification (squares and cubes) Test statistic: F = 1.326946, with p-value = P(F(2,104) > 1.32695) = 0.27 RESET test for specification (squares only) Test statistic: F = 1.746025, with p-value = P(F(1,105) > 1.74602) = 0.189 RESET test for specification (cubes only) Test statistic: F = 1.593000, with p-value = P(F(1,105) > 1.593) = 0.21 Fail to Reject Ho. Correctly Specified. Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_FRMAREA 1.275 l_TOTAL_NPK 1.157 l_TOTAL_LABOR 1.138 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 7007.9536

Determinant = 2.1075696e+008 Reciprocal condition number = 0.00013665496 No multicollinearity. White's test for heteroskedasticity OLS, using observations 1-120 (n = 110) Missing or incomplete observations dropped: 10 Dependent variable: uhat^2 coefficient std. error t-ratio p-value ------------------------------------------------------------ const 12.6141 12.0515 1.047 0.2978 l_FRMAREA 0.586472 2.04832 0.2863 0.7752 l_TOTAL_NPK 0.241717 1.14422 0.2113 0.8331 l_TOTAL_LABOR -3.02615 3.52514 -0.8584 0.3927 sq_l_FRMAREA 0.209283 0.0889927 2.352 0.0206 ** X2_X3 -0.268857 0.125804 -2.137 0.0350 ** X2_X4 0.108659 0.301761 0.3601 0.7195 sq_l_TOTAL_NP 0.0342725 0.0585846 0.5850 0.5599 X3_X4 -0.132929 0.177492 -0.7489 0.4557 sq_l_TOTAL_LA 0.223080 0.258269 0.8637 0.3898 Unadjusted R-squared = 0.165291 Test statistic: TR^2 = 18.181970, with p-value = P(Chi-square(9) > 18.181970) = 0.033120 Reject Ho. Heteroskedastic. Proceed with Heteroskedasticity-corrected model Model 4: Heteroskedasticity-corrected, using observations 1-120 (n = 110) Missing or incomplete observations dropped: 10 Dependent variable: l_PROD coefficient std. error t-ratio p-value ------------------------------------------------------------- const 4.39476 1.04713 4.197 5.64e-05 *** l_FRMAREA 0.451822 0.0952527 4.743 6.59e-06 *** l_TOTAL_NPK 0.254008 0.0814835 3.117 0.0023 *** l_TOTAL_LABOR 0.371294 0.156083 2.379 0.0192 ** Statistics based on the weighted data: Sum squared resid 291.2692 S.E. of regression 1.657656 R-squared 0.393884 Adjusted R-squared 0.376730 F(3, 106) 22.96136 P-value(F) 1.57e-11 Log-likelihood -209.6405 Akaike criterion 427.2809

Schwarz criterion 438.0828 Hannan-Quinn 431.6622 Statistics based on the original data: Mean dependent var 6.705082 S.D. dependent var 1.453283 Sum squared resid 137.9045 S.E. of regression 1.140608 Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_FRMAREA 1.275 l_TOTAL_NPK 1.157 l_TOTAL_LABOR 1.138 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables No multicollinearity. Jarque-Bera test = 0.970613, with p-value 0.615508 Fail to reject Ho. Errors are normally distributed. Summary:


Prod = exp(4.07748)*FA^0.40*NPK^0.27*Lbr^0.41

R2 0.4

Adjusted R2 0.38


T Test

const 0.0002 *** Significant

l_FA 2.10e-05 *** Significant

l_NPK 0.0011 *** Significant

l_Lbr 0.0126 ** Significant


Correctly specified

Multicollinearity Test No multicollinearity l_FA 1.275

l_NPK 1.157

l_Lbr 1.138

Heteroskedasticity Test 0.03312 Heteroskedastic

Prod = exp(4.39476)*FA^0.45*NPK^0.25*Lbr^0.37

R2 0.39

Adjusted R2 0.38


T Test


l_FA 6.59e-06 *** Significant

l_NPK 0.0023 *** Significant

l_Lbr 0.0192 ** Significant


l_NPK 1.157

l_Lbr 1.138

Autocorrelation Test Cross-section data

N/A


Transcendental production function; 1 2

1 2 1 1 2 2exp( )y Ax x x x

Model 5: OLS, using observations 1-120 (n = 110) Missing or incomplete observations dropped: 10 Dependent variable: l_PROD coefficient std. error t-ratio p-value ------------------------------------------------------------- const 4.13301 1.79119 2.307 0.0230 ** l_FRMAREA 0.305242 0.126520 2.413 0.0176 ** l_TOTAL_NPK 0.164503 0.117612 1.399 0.1649 l_TOTAL_LABOR 0.357553 0.305824 1.169 0.2450 FRMAREA 0.301405 0.282062 1.069 0.2878 TOTAL_NPK 0.0113142 0.00982531 1.152 0.2522 TOTAL_LABOR 6.40693e-05 0.000282174 0.2271 0.8208 Mean dependent var 6.705082 S.D. dependent var 1.453283 Sum squared resid 133.5030 S.E. of regression 1.138484 R-squared 0.420085 Adjusted R-squared 0.386304 F(6, 103) 12.43538 P-value(F) 1.70e-10 Log-likelihood -166.7336 Akaike criterion 347.4673 Schwarz criterion 366.3707 Hannan-Quinn 355.1346 Log-likelihood for PROD = -904.293 Excluding the constant, p-value was highest for variable 5 (TOTAL_LABOR) RESET test for specification (squares and cubes) Test statistic: F = 0.594812,

with p-value = P(F(2,101) > 0.594812) = 0.554 RESET test for specification (squares only) Test statistic: F = 0.521271, with p-value = P(F(1,102) > 0.521271) = 0.472 RESET test for specification (cubes only) Test statistic: F = 0.709329, with p-value = P(F(1,102) > 0.709329) = 0.402 Fail to Reject Ho. Correctly Specified. Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem l_FRMAREA 2.470 l_TOTAL_NPK 2.405 l_TOTAL_LABOR 4.029 FRMAREA 2.364 TOTAL_NPK 2.371 TOTAL_LABOR 3.693 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 1.1564449e+008 Determinant = 7.7468179e+020 Reciprocal condition number = 2.8699396e-009 No multicollinearity. White's test for heteroskedasticity OLS, using observations 1-120 (n = 110) Missing or incomplete observations dropped: 10 Dependent variable: uhat^2 coefficient std. error t-ratio p-value -------------------------------------------------------------- const -17.3266 323.616 -0.05354 0.9574 l_FRMAREA -17.8431 11.3510 -1.572 0.1198 l_TOTAL_NPK -3.47278 5.92890 -0.5857 0.5597 l_TOTAL_LABOR -8.45506 138.288 -0.06114 0.9514 FRMAREA 28.1622 20.5250 1.372 0.1738 TOTAL_NPK 2.94260 1.84528 1.595 0.1146

TOTAL_LABOR -0.257498 0.639582 -0.4026 0.6883 sq_l_FRMAREA 0.182104 0.740198 0.2460 0.8063 X2_X3 -0.721723 0.412032 -1.752 0.0836 * X2_X4 3.52847 1.54240 2.288 0.0247 ** X2_X5 -12.7402 13.3466 -0.9546 0.3426 X2_X6 0.0663691 0.0791524 0.8385 0.4042 X2_X7 -0.00724960 0.00411750 -1.761 0.0820 * sq_l_TOTAL_NP -0.754892 0.441101 -1.711 0.0908 * X3_X4 -0.112128 0.940608 -0.1192 0.9054 X3_X5 0.965556 0.991340 0.9740 0.3329 X3_X6 -0.620836 0.354777 -1.750 0.0839 * X3_X7 6.21719e-05 0.00139869 0.04445 0.9647 sq_l_TOTAL_LA 2.67788 16.4153 0.1631 0.8708 X4_X5 -5.12446 2.66130 -1.926 0.0576 * X4_X6 -0.0403050 0.104306 -0.3864 0.7002 X4_X7 0.0292329 0.0712596 0.4102 0.6827 sq_FRMAREA 4.92378 3.88880 1.266 0.2090 X5_X6 -0.100749 0.129488 -0.7781 0.4388 X5_X7 0.00938909 0.00512880 1.831 0.0708 * sq_TOTAL_NPK 0.00579912 0.00377690 1.535 0.1285 X6_X7 6.14741e-05 0.000147803 0.4159 0.6786 sq_TOTAL_LABO -4.24088e-06 7.29466e-06 -0.5814 0.5626 Warning: data matrix close to singularity! Unadjusted R-squared = 0.322849 Test statistic: TR^2 = 35.513425, with p-value = P(Chi-square(27) > 35.513425) = 0.126331 Fail to reject Ho. Homoskedastic. Jarque-Bera test = 0.53486, with p-value 0.765344 Fail to reject Ho. Errors are normally distributed. Summary:


Prod = exp(4.13301)*FA^0.31*NPK^0.16*Lbr^0.38*exp(0.30FA+0.01NPK+0.00006Lbr)

R2 0.42

Adjusted R2 0.39


T Test

const 0.0230 ** Significant

l_FA 0.0176 ** Significant

l_NPK 0.1649 Not Significant

l_Lbr 0.245 Not Significant

FA 0.2878 Not Significant

NPK 0.2522 Not Significant

Lbr 0.8208 Not Significant


Correctly specified


l_NPK 2.405

l_Lbr 4.029

FA 2.364

NPK 2.371

Lbr 3.693


Autocorrelation Test Cross-section data

N/A


2. 10 points. Which production function would you choose, and why? Interpret results and

recommend policy. The best production function is the Cobb-Douglas like. It has normally distributed error terms compared to the neo-classical and it has all significant vars compared to Transcendental function. Lastly, all the assumptions of CLRM are met. Prod = exp(4.39476)*FA^0.45*NPK^0.25*Lbr^0.37. A 10% increase in farm area increases production by 4.5%, a 10% increase in NPK application increases production by 2.5%, and a 10% increase in labor increases production by 3.7%. Results suggest that there is a 0.45 + 0.25 + 0.37 = 1.07 returns to scale (close to constant). Among the three inputs to production, farm area has the highest returns to scale = 0.45. This implies that farm managers/producers must concentrate on farm expansion in order to increase more the production of vegetables, secondary would be the increase in labor use and last is the increase in NPK use.

C. 10 pts Bonus. Summarize the procedure on how to carry out the average econometric methodology using a flowchart taking close attention to the assumptions of CLRM, how to tests and remedy. The most comprehensive will have a score of 10, followed by 9, 8, 7, 6 and 5. >Comidoy, Marimon, Avenir

1le solution

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