計量學實驗報告 實驗2-多元回歸-多重共線性-預測問題

計量經(jīng)濟學講課提綱PAGEPAGE1大連海事大學交通運輸管理學院實例1——中國糧食生產(chǎn)函數(shù)根據(jù)理論和經(jīng)驗分析，影響糧食生產(chǎn)（Y）的主要因素有農(nóng)業(yè)化肥施用量（X1）、糧食播種面積（X2）、成災面積（X3）、農(nóng)業(yè)機械總動力（X4）和農(nóng)業(yè)勞動力（X5），其中成災面積的符號為負，其余均應(yīng)為正。下表給出了1983——2000中國糧食生產(chǎn)的相關(guān)數(shù)據(jù)，擬建立中國糧食生產(chǎn)函數(shù)。年份糧食產(chǎn)量Y（萬噸）化肥施用量X1（萬千克）播種面積X2（千公頃）成災面積X3（公頃）農(nóng)業(yè)機械總動力X4（萬千瓦）農(nóng)業(yè)勞動力X5（萬人）198338728.01659.8114047.016209.318022.031645.1198440731.01739.8112884.015264.019497.031685.0198537911.01775.8108845.022705.320913.030351.5198639151.01930.6110933.023656.022950.030467.0198740208.01999.3111268.020392.724836.030870.0198839408.02141.5110123.023944.726575.031455.7198940755.02357.1112205.024448.728067.032440.5199044624.02590.3113466.017819.328708.033330.4199143529.02806.1112314.027814.029389.034186.3199244264.02930.2110560.025894.730308.034037.0199345649.03151.9110509.023133.031817.033258.2199444510.03317.9109544.031383.033802.032690.3199546662.03593.7110060.022267.036118.032334.5199650454.03827.9112548.021233.038547.032260.4199749417.03980.7112912.030309.042016.032434.9199851230.04083.7113787.025181.045208.032626.4199950839.04124.3113161.026731.048996.032911.8200046218.04146.4108463.034374.052574.032797.5（1）建立Y對所有解釋變量的回歸模型，結(jié)果如下：Y=-12815.75+6.213*X1+0.421*X2-0.166*X3-0.098*X4-0.028*X5VariableCoefficientStd.Errort-StatisticProb.

C-12815.7514078.90-0.9102800.3806X16.2125620.7408818.3853730.0000X20.4213800.1269253.3199190.0061X3-0.1662600.059229-2.8070650.0158X4-0.0977700.067647-1.4452990.1740X5-0.0284250.202357-0.1404710.8906R-squared0.982798

Meandependentvar44127.11AdjustedR-squared0.975630

S.D.dependentvar4409.100S.E.ofregression688.2984

Akaikeinfocriterion16.16752Sumsquaredresid5685056.

Schwarzcriterion16.46431Loglikelihood-139.5077

F-statistic137.1164Durbin-Watsonstat1.810512

Prob(F-statistic)0.000000

從計算結(jié)果看，R2較大并接近于1，而且F=137.11>F0.05=3.11，故認為糧食生產(chǎn)量與上述所有解釋變量間總體線性相關(guān)顯著。但是，同時，X4、X5前參數(shù)未通過t檢驗，而且符號的經(jīng)濟意義也不合理，故認為解釋變量間存在多重共線性。為了進一步檢驗多重共線性，進行下面操作。（2）計算解釋變量間的兩兩相關(guān)系數(shù)，得到簡單相關(guān)系數(shù)矩陣如下：X1X2X3X4X5X11X20.0118231X30.640175-0.454911X40.960278-0.038480.6895651X50.545450.1823590.35573530.4541691從相關(guān)分析結(jié)果來看，部分解釋變量間確實存在相關(guān)，尤其X1與X4之間高度相關(guān)。為了處理多重共線性，正確選擇解釋變量，進行逐步回歸，首先選擇最優(yōu)的基本方程。（3）分別做糧食生產(chǎn)量對各個解釋變量的回歸，得A．Y對X1回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C30867.311206.36425.587060.0000X14.5761150.39819911.492020.0000R-squared0.891941

Meandependentvar44127.11AdjustedR-squared0.885187

S.D.dependentvar4409.100S.E.ofregression1493.984

Akaikeinfocriterion17.56072Sumsquaredresid35711799

Schwarzcriterion17.65965Loglikelihood-156.0465

F-statistic132.0666Durbin-Watsonstat1.855174

Prob(F-statistic)0.000000

B.Y對X2回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C-33822.4168409.15-0.4944140.6277X20.6988800.6132731.1395900.2712R-squared0.075073

Meandependentvar44127.11AdjustedR-squared0.017265

S.D.dependentvar4409.100S.E.ofregression4370.873

Akaikeinfocriterion19.70775Sumsquaredresid3.06E+08

Schwarzcriterion19.80668Loglikelihood-175.3698

F-statistic1.298665Durbin-Watsonstat0.118043

Prob(F-statistic)0.271231

C.Y對X3回歸結(jié)果VariableCoefficientStd.Errort-StatisticProb.

C35712.864926.5837.2490120.0000X30.3499780.2008021.7429060.1005R-squared0.159563

Meandependentvar44127.11AdjustedR-squared0.107036

S.D.dependentvar4409.100S.E.ofregression4166.457

Akaikeinfocriterion19.61196Sumsquaredresid2.78E+08

Schwarzcriterion19.71089Loglikelihood-174.5076

F-statistic3.037721Durbin-Watsonstat0.935587

Prob(F-statistic)0.100533

D.Y對X4回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C31918.721828.71517.454180.0000X40.3799670.0544486.9785870.0000R-squared0.752707

Meandependentvar44127.11AdjustedR-squared0.737252

S.D.dependentvar4409.100S.E.ofregression2260.060

Akaikeinfocriterion18.38861Sumsquaredresid81725964

Schwarzcriterion18.48754Loglikelihood-163.4975

F-statistic48.70067Durbin-Watsonstat1.109488

Prob(F-statistic)0.000003

E.Y對X5回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C-28260.0227240.49-1.0374270.3150X52.2396140.8423522.6587620.0172R-squared0.306429

Meandependentvar44127.11AdjustedR-squared0.263081

S.D.dependentvar4409.100S.E.ofregression3784.948

Akaikeinfocriterion19.41989Sumsquaredresid2.29E+08

Schwarzcriterion19.51882Loglikelihood-172.7790

F-statistic7.069018Durbin-Watsonstat0.357079

Prob(F-statistic)0.017160

（4）逐步回歸，A、Y對X1、X4回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C31164.921137.21927.404510.0000X16.9259381.3315025.2015970.0001X4-0.2211780.120350-1.8377920.0860R-squared0.911800

Meandependentvar44127.11AdjustedR-squared0.900040

S.D.dependentvar4409.100S.E.ofregression1394.000

Akaikeinfocriterion17.46875Sumsquaredresid29148555

Schwarzcriterion17.61715Loglikelihood-154.2188

F-statistic77.53409Durbin-Watsonstat1.992572

Prob(F-statistic)0.000000

從回歸結(jié)果看，擬合優(yōu)度雖然上升，但X4的系數(shù)不顯著，因此，存在共線性，而相比較而言，X1更重要，因此剔除X4（從相關(guān)分析也有助于這個結(jié)論）。B、Y對X1、X5回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C24133.8412406.481.9452610.0707X14.4315590.4858839.1206250.0000X50.2212890.4057060.5454420.5935R-squared0.894042

Meandependentvar44127.11AdjustedR-squared0.879914

S.D.dependentvar4409.100S.E.ofregression1527.902

Akaikeinfocriterion17.65219Sumsquaredresid35017273

Schwarzcriterion17.80059Loglikelihood-155.8697

F-statistic63.28281Durbin-Watsonstat1.839712

Prob(F-statistic)0.000000

擬合優(yōu)度升高不顯著，修正的擬合優(yōu)度略微下降，且X5系數(shù)不顯著，因此，剔除X5.C、Y對X1、X3回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C35065.011064.61232.936880.0000X15.6543300.31219918.111320.0000X3-0.3045460.056452-5.3948030.0001R-squared0.963248

Meandependentvar44127.11AdjustedR-squared0.958348

S.D.dependentvar4409.100S.E.ofregression899.8443

Akaikeinfocriterion16.59333Sumsquaredresid12145797

Schwarzcriterion16.74173Loglikelihood-146.3400

F-statistic196.5723Durbin-Watsonstat1.728340

Prob(F-statistic)0.000000從回歸結(jié)果看，擬合優(yōu)度提高，X1和X3的系數(shù)顯著，因此接受X3.

D、Y對X1、X2、X3回歸結(jié)果：VariableCoefficientStd.Errort-StatisticProb.

C-11978.1814072.92-0.8511510.4090X15.2559350.26859519.568280.0000X20.4084320.1219743.3485220.0048X3-0.1946090.054533-3.5686370.0031R-squared0.979593

Meandependentvar44127.11AdjustedR-squared0.975220

S.D.dependentvar4409.100S.E.ofregression694.0715

Akaikeinfocriterion16.11616Sumsquaredresid6744293.

Schwarzcriterion16.31402Loglikelihood-141.0454

F-statistic224.0086Durbin-Watsonstat1.528658

Prob(F-statistic)0.000000

從回歸結(jié)果看，擬合優(yōu)度提高，X1、X2和X3的系數(shù)顯著，因此接受X2.即，回歸方程為：Y=-11978.18057+5.255935121*X1+0.408432175*X2-0.1946087795*X3實例2我國1988年-1998年的城鎮(zhèn)居民人均全年耐用消費品支出、人均全年可支配收入以及耐用消費品價格指數(shù)的統(tǒng)計資料如下表，試建立城鎮(zhèn)居民人均全年耐用消費品支出Y關(guān)于人均全年可支配收入x1和耐用消費品價格指數(shù)X2的回歸模型，并進行回歸分析。根據(jù)經(jīng)驗和對經(jīng)濟現(xiàn)實的分析，設(shè)定模型為二元線性回歸模型，理論形式為：，（1）數(shù)據(jù)如下表，年份人均全年可支配收入X1（元）耐用消費品價格指數(shù)X2（1987）人均耐用消費品支出Y（元）19881181.4115.96137.1619891375.7133.35124.5619901501.2128.21107.9119911700.6124.85102.9619922026.6122.49125.2419932577.4129.86162.4519943496.2139.52217.4319954283140.44253.4219964838.9139.12251.0719975160.3133.35285.8519985425.1126.39327.26（2）Eviews的輸出結(jié)果（下表）寫出回歸方程為：EXPENSEY=158.5398355+0.04940379666*INCOMEX1-0.911684216*PRINDEX2VariableCoefficientStd.Errort-StatisticProb.

C158.5398121.80711.3015640.2293INCOMEX10.0494040.00468410.547860.0000PRINDEX2-0.9116840.989546-0.9213160.3838R-squared0.947989

Meandependentvar190.4827AdjustedR-squared0.934986

S.D.dependentvar79.29127S.E.ofregression20.21757

Akaikeinfocriterion9.077982Sumsquaredresid3270.001

Schwarzcriterion9.186499Loglikelihood-46.92890

F-statistic72.90647Durbin-Watsonstat1.035840

Prob(F-statistic)0.000007

（3）檢驗①從經(jīng)濟意義來看，可支配收入前的系數(shù)為0.0494，正的，介于0和1之間，符號、大小與理論符合；價格指數(shù)前的系數(shù)為-0.91，大小和符號符合經(jīng)濟理論；②從統(tǒng)計角度看，R-squared=0.947989，AdjustedR-squared=0.934986，從多元回歸修正的判定系數(shù)看，回歸方程較好地擬合了散點，被解釋變量的變異中有93%以上可以由方程來解釋；從F統(tǒng)計量的結(jié)果來看，F(xiàn)=72.90647>F0.05(2,8)=4.46，而且F=72.90647>F0.01(2,8)=8.65，可見方程總體來看，無論在0.05還是0.01水平上都顯著，即在我國城鎮(zhèn)居民人均全年耐用消費品支出與人均全年可支配收入和耐用消費品價格指數(shù)之間存在顯著的線性關(guān)系。這一點結(jié)論由F統(tǒng)計量的精確顯著性水平Prob=0.000007也可得到。t統(tǒng)計量結(jié)果來看，可支配收入incomex1的系數(shù)，t值=10.54786>t0.05(8)=2.306，系數(shù)顯著，可支配收入對耐用消費品支出有顯著影響，變量x1保留；而對于耐用消費品價格指數(shù)prindex2的系數(shù)，t值=-0.921316，其絕對值小于t0.05（8），可以接受系數(shù)為零的原假設(shè)，剔除X2。以上結(jié)論由Eviews輸出結(jié)果中系數(shù)的精確顯著性水平Prob也可以直接得到。（4）預測：點預測和區(qū)間預測若已知2000年，我國城鎮(zhèn)居民家庭人均可支配收入為5800元，耐用消費品價格指數(shù)為135，對2000年我國城鎮(zhèn)居民家庭人均耐用消費品支出進行預測。①點預測，將x1=5800，X2=135代入估計方程，EXPENSEY=158.5398+0.0494*5800-0.9117*135，得到Y(jié)的估計值=321.9803（教材中是按照小數(shù)點后保留4位數(shù)字后的樣本回歸方程計算得到的，而Eviews軟件的估計值是322.0045）。（也可可以在Eviews中調(diào)整擴大數(shù)據(jù)范圍至2000，再將解釋變量的數(shù)據(jù)輸入，再利用Equation窗口下的forecast進行預測，估計值即保存在EXPENSEYf序列中。）②區(qū)間預測E(Y0)的區(qū)間預測Y的估計值的Y0的方差估計為，已知X2000矩陣為：，則X2000矩陣的轉(zhuǎn)置矩陣為：，而解釋變量X矩陣為：11181.4115.9611375.7133.3511501.2128.2111700.6124.8512026.6122.4912577.4129.8613496.2139.5214283140.4414838.9139.1215160.311×11×315425.1126.3