受约束的回归及检验例题

Y K L

3722.73078.22113

1442.521684.4367

1752.372742.7784

1451.291973.8227

5149.35917.01327

2291.161758.77120

1345.17939.158

656.77694.9431

370.18363.4816

1590.362511.9966

616.71973.7358

617.94516.0128

4429.193785.9161

5749.028688.03254

1781.372798.983

1243.071808.4433

812.71118.8143

1899.72052.1661

3692.856113.11240

4732.99228.25222

2180.232866.6580

2539.762545.6396

3046.954787.9222

2192.633255.29163

5364.838129.68244

4834.685260.2145

7549.587518.79138

867.91984.5246

4611.3918626.94218

170.3610.9119

325.531523.1945

Unrestricted regression

lnY lnK lnL

8.2222048.032107 4.727388

7.2741477.429183 4.204693

7.4687247.916724 4.430817

7.2802087.587726 3.295837SUMMARY OUTPUT

8.5466168.685587 5.78996

7.7368147.472374.787492回归统计

7.204276 6.844922 4.060443Multiple R0.899958

6.487334 6.543826 3.433987R Square0.809925

0.796348

5.913989 5.895724 2.772589Adjusted R Squ

7.3717167.828831 4.189655标准误差0.425538

6.424399 6.881134 4.060443观测值31

6.426391 6.246126 3.332205

8.3959728.239042 4.110874方差分析

8.6567859.069701 5.537334df SS MS F

7.4851387.936982 4.418841回归分析221.6049310.8024659.65501 7.1253397.50022 3.496508残差28 5.0703030.181082

6.700362

7.020021 3.7612总计3026.67523

7.5494517.626648 4.110874

8.2141548.718191 5.480639Coefficients标准误差t Stat P-value

8.4622939.130025 5.402677Intercept 1.1539940.727611 1.5860040.123969

7.6871867.960899 4.382027X Variable 10.6092360.176378 3.4541490.001776

7.8398257.842133 4.564348X Variable 20.3607960.201591 1.7897410.084317

8.0218968.473847 5.402677

7.6928578.088037 5.09375

8.587629.003277 5.497168lnY=1.153994+0.609236lnK+0.360796lnL

8.483578.567924 4.976734

8.9292478.925164.927254

6.766088 6.892154 3.828641

8.4362859.832364 5.384495

5.137562

6.414952.944439

5.7854557.328562 3.806662

Restrict: α+β=1

代入，化简，移项，右边只保留lnA和有回归系数的项，得到约束条件下新的回归模型为

Y'=lnY-lnL=lnA+αln(K/L)

lnY-lnL ln(K/L)

3.494817 3.304719

3.069454 3.22449SUMMARY OUTPUT

3.037908 3.485907

3.984371

4.291889回归统计

2.756656 2.895626Multiple R0.545313

2.949322 2.684878R Square0.297366

0.273138

3.143833 2.784479Adjusted R Squ

3.053347 3.109838标准误差0.418891

3.141401 3.123136观测值31

3.182061 3.639176

2.363956 2.820691方差分析

3.094187 2.913922df SS MS F

4.285098 4.128168回归分析1 2.153586 2.15358612.27328

3.11945 3.532367残差29 5.0886130.175469

3.066297 3.518141总计307.242199

3.628832

4.003712

2.939162

3.258821Coefficients标准误差t Stat P-value

3.438577 3.515774Intercept 1.0260480.596769 1.7193390.096211

2.733515

3.237552X Variable 10.6081410.17359 3.5033240.001511

3.059616 3.727347

3.305159 3.578873

3.275477 3.277785lnY-lnL=1.026048+0.608141ln(K/L)

2.619219

3.07117lnY=1.026048+0.608141lnK+0.391859lnL

2.599107 2.994286

3.090452 3.506109(RSS R-RSS U)/k U-k R

F==0.101118

3.506836 3.591191RSS U/n-k U-1

4.001994 3.997907

2.937447

3.063513F0.05(1,28)=

4.195972

3.05179

4.447869

2.193123

3.470511可见计算出的F统计量明显小于临界值，不拒绝原约束条件表示α+β=1是1.978792 3.5219