計量經(jīng)濟學(xué)案例報告 國民經(jīng)濟核算是反映國民經(jīng)濟運行狀況的有效工具；國民經(jīng)濟核算是宏觀經(jīng)濟管理的重要依據(jù)；國民經(jīng)濟核算是制定和檢驗國民經(jīng)濟計劃的科學(xué)方法；國民經(jīng)濟核算是微觀決策的重要依據(jù)。國民經(jīng)濟統(tǒng)計工作是國家整個統(tǒng)計工作的一個重要核心部分，而GNP又是國民經(jīng)濟生產(chǎn)統(tǒng)計中的一個重要目標，GNP是按國民原則計算的國民經(jīng)濟核算中的重要的綜合指標，等于國內(nèi)生產(chǎn)總值與國外凈要素之和。雖然GDP是國民經(jīng)濟的最核心指標，但GNP又有其重要意義，比如，聯(lián)合國根據(jù)連續(xù)六年的國民生產(chǎn)總值和人均國民生產(chǎn)總值來決定一個國家的會費；世界銀行根據(jù)人均國民生產(chǎn)總值來決定一個國家所能享受的硬貸款、軟貸款等優(yōu)惠待遇；國際貨幣基金組織根據(jù)國民生產(chǎn)總值、黃金與外匯儲備、進出口額、出口額占國民生產(chǎn)總值的比例等因素來決定一個國家在基金的份額，進而決定在基金的投票權(quán)、分配特別提款權(quán)的份額及向基金借款的份額等等，在這些方面直接影響到我國的經(jīng)濟利益和政治利益。所以，我們從《中國統(tǒng)計年鑒》(1999)上查找到1987--1998年的GNP，并找出一些變量建立多元線形回歸模型對GNP進行研究。 我們選擇選擇人均主要產(chǎn)品產(chǎn)量作為影響GNP變化的變量，人均主要產(chǎn)品產(chǎn)量有糧食，棉花，油料，糖料，茶葉，水果，豬牛羊肉，水產(chǎn)品，布，機制紙及紙板，紗，原煤，原油，發(fā)電量，鋼，水泥等，經(jīng)過初步考慮，我們決定選用原煤，糧食和棉花作為建立模型所用的三個變量設(shè)為X2，X3，X4，設(shè)GNP為Y，數(shù)據(jù)如下： GNP與人均主要產(chǎn)品產(chǎn)量 年 Y(GNP)/億元 X2(原煤)/噸 X3(糧食)/千克 X4(棉花)/千克 1987 11955 0.86 371.74 3.92 1988 14922 0.89 357.72 3.77 1989 16918 0.94 364.32 3.39 1990 18598 0.95 393.10 3.97 1991 21663 0.94 378.26 4.93 1992 26652 0.96 379.97 3.87 1993 34561 0.98 387.37 3.17 1994 46670 1.04 373.46 3.64 1995 57495 1.13 387.28 3.96 1996 66851 1.15 414.39 3.45 1997 73143 1.12 401.74 3.74 1998 78018 1.01 412.42 3.62 （1） 確定樣本回歸方程：對于中國1987年至1998年國民生產(chǎn)總值及有關(guān)影響因素初步建立多元線形回歸模型。 ^ Y=β1+β2*X2+β3*X3+β4*X4 假設(shè)模型中隨機誤差項ui滿足古典假設(shè)，運用OLS方法估計模型的參數(shù)，利用Eviews計算得出如下輸入結(jié)果： Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 17:40 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -306717.4 89148.80 -3.440511 0.0088 X2 142124.4 51901.06 2.738372 0.0255 X3 570.9199 271.5682 2.102308 0.0687 X4 -4222.676 8323.401 -0.507326 0.6256 R-squared 0.831636 Mean dependent var 38953.65 Adjusted R-squared 0.768499 S.D. dependent var 24406.49 S.E. of regression 11743.08 Akaike info criterion 21.84112 Sum squared resid 1.10E+09 Schwarz criterion 22.00275 Log likelihood -127.0467 F-statistic 13.17199 Durbin-Watson stat 1.425642 Prob(F-statistic) 0.001839 Estimation Command: ===================== LS Y C X2 X3 X4 Estimation Equation: ===================== Y = C(1) + C(2)*X2 + C(3)*X3 + C(4)*X4 Substituted Coefficients: ===================== Y = -306717.449 + 142124.3822*X2 + 570.9199106*X3 - 4222.676239*X4 Correlation Matrix X2 X3 X4 X2 1.000000 0.684966 -0.226024 X3 0.684966 1.000000 -0.167105 X4 -0.226024 -0.167105 1.000000 ^ Y=-306717+142124X2+570.9X3-4223X4 (2.738) （2.102）(-0.5073) R2=0.8316 F=13.17 S=11743 DW=1.426 查表得Fα(r,n-k)=F0.05(4,8)=3.84,tα/2(n-k)=t0.025(8)=2.306,由于F> F0.05(4,8)=3.84,所以拒絕假設(shè)H0:β=0,模型在總體上顯著。但是通過t值可以看出X3和X4無法通過顯著性檢驗,說明這個模型建立的不是十分理想。我們進而考慮分別建立一個解釋變量和兩個解釋變量的模型，利用Eviews可以得到如下估計結(jié)果： 1）對X2 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 17:43 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -180859.9 42190.69 -4.286725 0.0016 X2 220364.4 42122.47 5.231518 0.0004 R-squared 0.732397 Mean dependent var 38953.65 Adjusted R-squared 0.705637 S.D. dependent var 24406.49 S.E. of regression 13241.81 Akaike info criterion 21.97116 Sum squared resid 1.75E+09 Schwarz criterion 22.05198 Log likelihood -129.8269 F-statistic 27.36878 Durbin-Watson stat 0.716502 Prob(F-statistic) 0.000383 Estimation Command: ===================== LS Y C X2 Estimation Equation: ===================== Y = C(1) + C(2)*X2 Substituted Coefficients: ===================== Y = -180859.8861 + 220364.4473*X2 ^ Y=-180860+220364X2 （a） （5.232） R2=0.7324 F=27.37 S=13242 DW=0.7165 2）對X3 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 15:17 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -386132.4 97641.28 -3.954602 0.0027 X3 1103.697 253.2660 4.357855 0.0014 R-squared 0.655064 Mean dependent var 38953.65 Adjusted R-squared 0.620571 S.D. dependent var 24406.49 S.E. of regression 15033.87 Akaike info criterion 22.22501 Sum squared resid 2.26E+09 Schwarz criterion 22.30583 Log likelihood -131.3501 F-statistic 18.99090 Durbin-Watson stat 1.466938 Prob(F-statistic) 0.001426 Estimation Command: ===================== LS Y C X3 Estimation Equation: ===================== Y = C(1) + C(2)*X3 Substituted Coefficients: ===================== Y = -386132.4019 + 1103.69677*X3 ^ Y=-386132+1104X3 （b） (4.538) R2=0.6551 F=18.99 S=15034 DW=0.7165 3）對X4 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 15:18 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C 96133.60 64802.43 1.483488 0.1688 X4 -15103.66 17013.62 -0.887740 0.3955 R-squared 0.073051 Mean dependent var 38953.65 Adjusted R-squared -0.019644 S.D. dependent var 24406.49 S.E. of regression 24645.04 Akaike info criterion 23.21355 Sum squared resid 6.07E+09 Schwarz criterion 23.29437 Log likelihood -137.2813 F-statistic 0.788082 Durbin-Watson stat 0.214148 Prob(F-statistic) 0.395531 Estimation Command: ===================== LS Y C X4 Estimation Equation: ===================== Y = C(1) + C(2)*X4 Substituted Coefficients: ===================== Y = 96133.60497 - 15103.66409*X4 ^ Y=96134-15104X4 (-0.8877) R2=0.07305 F=0.7881 S=24645 DW=0.2141 4）對X2，X3 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 15:27 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -327701.5 75644.37 -4.332134 0.0019 X2 146213.8 49110.47 2.977242 0.0155 X3 573.3049 260.0840 2.204307 0.0550 R-squared 0.826219 Mean dependent var 38953.65 Adjusted R-squared 0.787601 S.D. dependent var 24406.49 S.E. of regression 11248.17 Akaike info criterion 21.70612 Sum squared resid 1.14E+09 Schwarz criterion 21.82734 Log likelihood -127.2367 F-statistic 21.39463 Durbin-Watson stat 1.247999 Prob(F-statistic) 0.000380 Estimation Command: ===================== LS Y C X2 X3 Estimation Equation: ===================== Y = C(1) + C(2)*X2 + C(3)*X3 Substituted Coefficients: ===================== Y = -327701.5357 + 146213.7762*X2 + 573.304887*X3 ^ Y=-327701+146214X2+573.3X3 （c） (2.977) (2.204) R2=0.8262 F=21.39 S=11248 DW=1.248 5）對X2，X4 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 15:31 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -159025.2 64471.72 -2.466588 0.0358 X2 215651.1 45047.38 4.787206 0.0010 X4 -4525.588 9776.199 -0.462919 0.6544 R-squared 0.738620 Mean dependent var 38953.65 Adjusted R-squared 0.680536 S.D. dependent var 24406.49 S.E. of regression 13794.82 Akaike info criterion 22.11429 Sum squared resid 1.71E+09 Schwarz criterion 22.23552 Log likelihood -129.6858 F-statistic 12.71635 Durbin-Watson stat 0.751053 Prob(F-statistic) 0.002386 Estimation Command: ===================== LS Y C X2 X4 Estimation Equation: ===================== Y = C(1) + C(2)*X2 + C(3)*X4 Substituted Coefficients: ===================== Y = -159025.2067 + 215651.1045*X2 - 4525.587513*X4 ^ Y=-159025+215651X2-4526X4 (4.787) (-0.4629) R2=0.7386 F=12.72 S=13795 DW=0.7511 6）對X3，X4 Dependent Variable: Y Method: Least Squares Date: 12/13/03 Time: 15:43 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C -344553.1 115574.5 -2.981221 0.0154 X3 1072.042 263.3080 4.071439 0.0028 X4 -7762.561 10790.09 -0.719416 0.4901 R-squared 0.673822 Mean dependent var 38953.65 Adjusted R-squared 0.601338 S.D. dependent var 24406.49 S.E. of regression 15410.19 Akaike info criterion 22.33576 Sum squared resid 2.14E+09 Schwarz criterion 22.45699 Log likelihood -131.0146 F-statistic 9.296132 Durbin-Watson stat 1.758444 Prob(F-statistic) 0.006465 Estimation Command: ===================== LS Y C X3 X4 Estimation Equation: ===================== Y = C(1) + C(2)*X3 + C(3)*X4 Substituted Coefficients: ===================== Y = -344553.0724 + 1072.042485*X3 - 7762.560522*X4 ^ Y=-34553+1072X3-7763X4 (4.71) (-0.7194) R2=0.6378 F=9.296 S=15410 DW=1.758 查表得F（2，10）=4.10,F0.05(3,9)=3.86,t0.025(10)=2.228, t0.025（9）=2.262。由以上各樣本回歸方程可以看出X4（人均棉花產(chǎn)量）對Y（GNP）沒有顯著影響，應(yīng)該略去。 再比較不含X4的幾個方程（a），（b），（c），可以看出，式（a）稍微優(yōu)于式（b），在式（c）中， 雖然X3沒有通過顯著性檢驗，但是相應(yīng)的t統(tǒng)計量為2.204，很接近臨界值t0.025（9）=2.262，且式（c）的可決系數(shù)R2明顯高于式（a）中的R2，誤差項的標準差估計值S明顯小于式（a）中的S。因此，最后確定的總體回歸模型為 Y=β1+β2X2+β3X3+u 根據(jù)剛才的輸出結(jié)果，樣本回歸方程為 ^ Y=-327701+146214X2+573.3X3 (2.977) (2.204) R2=0.8262 F=21.39 S=11248 DW=1.248 A．多重共線性的檢驗：由剛才確定總體回歸模型在的分析過程可知:R2很大,F=21.39顯著大于F0.05(3,9)=3.86,而變量X2對應(yīng)的偏回歸系數(shù)t值顯著,X3的t值接近顯著,所以,這個模型是不存在多重共線性的。 B．異方差性的檢驗： 對X2，X3 ARCH Test: F-statistic 0.431993 Probability 0.739434 Obs*R-squared 1.852580 Probability 0.603560 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 12/13/03 Time: 17:55 Sample(adjusted): 1990 1998 Included observations: 9 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 1.97E+08 1.13E+08 1.748049 0.1409 RESID^2(-1) -0.481687 0.671468 -0.717364 0.5053 RESID^2(-2) -0.138626 0.701171 -0.197706 0.8511 RESID^2(-3) -0.610979 0.667046 -0.915948 0.4017 R-squared 0.205842 Mean dependent var 1.20E+08 Adjusted R-squared -0.270653 S.D. dependent var 1.64E+08 S.E. of regression 1.85E+08 Akaike info criterion 41.21034 Sum squared resid 1.71E+17 Schwarz criterion 41.29800 Log likelihood -181.4465 F-statistic 0.431993 Durbin-Watson stat 1.241293 Prob(F-statistic) 0.739434 從輸出的輔助回歸函數(shù)中得到R2，計算（n-P）R2=（9-3）*0.7282=4.3692,查Χ2分布表,給定α=0.05,自由度為P=3,得臨界值Χ20.05(3)=7.815, （n-P）R2=4.3692<Χ20.05(3)=7.815,所以接受H0,表明模型中隨機誤差項不存在異方差性。從下面的White也可得出相同結(jié)果，模型中不存在異方差。 White Heteroskedasticity Test: F-statistic 4.688229 Probability 0.037155 Obs*R-squared 8.738234 Probability 0.067986 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 12/13/03 Time: 17:56 Sample: 1987 1998 Included observations: 12 Variable Coefficient Std. Error t-Statistic Prob. C 1.51E+10 1.41E+10 1.068697 0.3207 X2 1.34E+10 8.16E+09 1.636286 0.1458 X2^2 -7.01E+09 3.99E+09 -1.755779 0.1226 X3 -1.18E+08 77269611 -1.523985 0.1713 X3^2 161943.7 99399.14 1.629226 0.1473 R-squared 0.728186 Mean dependent var 94890959 Adjusted R-squared 0.572864 S.D. dependent var 1.48E+08 S.E. of regression 96545494 Akaike info criterion 39.90326 Sum squared resid 6.52E+16 Schwarz criterion 40.10531 Log likelihood -234.4196 F-statistic 4.688229 Durbin-Watson stat 2.218332 Prob(F-statistic) 0.037155 C．自相關(guān)性的檢驗： 從圖中可以看出殘差et沒有呈線形自回歸，表明隨機誤差項不存在自相關(guān)性 （2） 預(yù)測：因為GNP具有非常重要的國民經(jīng)濟統(tǒng)計意義，對它的預(yù)測也具有現(xiàn)實意義。首先，可以通過相關(guān)部門指定的在預(yù)測期內(nèi)的變量計劃生產(chǎn)值來對這一期的GNP數(shù)值作出定量的估計；其次還可以在實際統(tǒng)計中，根據(jù)已統(tǒng)計出的變量實際生產(chǎn)值來估計當期的GNP將在一個范圍內(nèi)達到多少。這樣，根據(jù)預(yù)測值制定經(jīng)濟發(fā)展政策或判斷相關(guān)經(jīng)濟政策的可行性以及了解已實施的經(jīng)濟政策取得的成果都具有重要意義。 結(jié)束語 至此，我們完成了對國民經(jīng)濟生產(chǎn)總值滿足古典假定的多元線性回歸模型的建立及分析。進一步強化了所學(xué)知識及處理實際應(yīng)用問題的能力，由于能力有限，如果這份報告中存在缺陷和不足，請老師諒解。