計量經濟學模型分析方法

1、計量經濟學上機模型分析方法總結一、隨機誤差項的異方差問題的檢驗與修正模型一：Dependent Variable: LOG(Y)Method: Least SquaresDate: 07/29/12 Time: 09:03Sample: 1 31Included observations: 31VariableCoefficientStd. Errort-StatisticProb.  C1.6025280.8609781.8612880.0732LOG(X1)0.3254160.1037693.1359550.0040LOG(X2)0.5070780.04859910.4

2、33850.0000R-squared0.796506    Mean dependent var7.448704Adjusted R-squared0.781971    S.D. dependent var0.364648S.E. of regression0.170267    Akaike info criterion-0.611128Sum squared resid0.811747    Schwarz criterion-

3、0.472355Log likelihood12.47249    F-statistic54.79806Durbin-Watson stat1.964720    Prob(F-statistic)0.000000（一）異方差的檢驗1、GQ檢驗法模型二：Dependent Variable: LOG(Y)Method: Least SquaresDate: 07/29/12 Time: 09:19Sample: 1 12Included observations: 12VariableCoefficientStd

4、. Errort-StatisticProb.  C3.7446261.1911133.1438040.0119LOG(X1)0.3443690.0829994.1490770.0025LOG(X2)0.1689040.1188441.4212280.1890R-squared0.669065    Mean dependent var7.239161Adjusted R-squared0.595524    S.D. dependent var0.133581S.E. of regressio

5、n0.084955    Akaike info criterion-1.881064Sum squared resid0.064957    Schwarz criterion-1.759837Log likelihood14.28638    F-statistic9.097834Durbin-Watson stat1.810822    Prob(F-statistic)0.006900模型三：Dependent Variable

6、: LOG(Y)Method: Least SquaresDate: 07/29/12 Time: 09:20Sample: 20 31Included observations: 12VariableCoefficientStd. Errort-StatisticProb.  C-0.3533811.607461-0.2198380.8309LOG(X1)0.2108980.1582201.3329420.2153LOG(X2)0.8565220.1086017.8868560.0000R-squared0.878402    Me

7、an dependent var7.769851Adjusted R-squared0.851381    S.D. dependent var0.390363S.E. of regression0.150490    Akaike info criterion-0.737527Sum squared resid0.203824    Schwarz criterion-0.616301Log likelihood7.425163    

8、;F-statistic32.50732Durbin-Watson stat2.123203    Prob(F-statistic)0.000076進行模型二和模型三兩次回歸，目的僅是得到出去中間7個樣本點以后前后各12個樣本點的殘差平方和RSS1和RSS2，然后用較大的RSS除以較小的RSS即可求出F統(tǒng)計量值進行顯著性檢驗。2、懷特檢驗法（White）模型一的懷特殘差檢驗結果：White Heteroskedasticity Test:F-statistic4.920995    Probability0.00

9、4339Obs*R-squared13.35705    Probability0.009657Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 05/29/13 Time: 09:04Sample: 1 31Included observations: 31VariableCoefficientStd. Errort-StatisticProb.  C3.9821372.8828511.3813190.1789LOG(X1)-0.5792890.91

10、6069-0.6323640.5327(LOG(X1)20.0418390.0668660.6257100.5370LOG(X2)-0.5636560.203228-2.7735140.0101(LOG(X2)20.0402800.0138792.9021730.0075R-squared0.430873    Mean dependent var0.026185Adjusted R-squared0.343315    S.D. dependent var0.038823S.E. of regression0.0

11、31460    Akaike info criterion-3.933482Sum squared resid0.025734    Schwarz criterion-3.702194Log likelihood65.96898    F-statistic4.920995Durbin-Watson stat1.526222    Prob(F-statistic)0.004339 一方面，根據(jù)上面的Obs*R2=31*0.4308

12、73=13.357052(4)，說明存在顯著的異方差問題；另一方面，根據(jù)下面的輔助回歸模型可以看出LOG(X2) 與(LOG(X2)2均通過了t檢驗，說明異方差的形式可以用LOG(X2) 與(LOG(X2)2的線性組合表示，權變量可以簡單確定為1/LOG(X2)。（二）加權最小二乘法（WLS）修正1、方法原理：具體參見教材。2、回歸結果分析模型四：Dependent Variable: LOG(Y)Method: Least SquaresDate: 07/29/12 Time: 09:06Sample: 1 31Included observations: 31Weighting serie

13、s: 1/LOG(X2)VariableCoefficientStd. Errort-StatisticProb.  C1.4780850.8176101.8078110.0814LOG(X1)0.3779150.0969253.8990440.0006LOG(X2)0.4734710.0483989.7828640.0000Weighted StatisticsR-squared0.872646    Mean dependent var7.423264Adjusted R-squared0.863550  &#

14、160; S.D. dependent var0.436598S.E. of regression0.161276    Akaike info criterion-0.719639Sum squared resid0.728274    Schwarz criterion-0.580866Log likelihood14.15440    F-statistic49.27256Durbin-Watson stat2.036239   

15、 Prob(F-statistic)0.000000Unweighted StatisticsR-squared0.789709    Mean dependent var7.448704Adjusted R-squared0.774688    S.D. dependent var0.364648S.E. of regression0.173088    Sum squared resid0.838862Durbin-Watson stat2.028211加權修正

16、以后的模型四懷特檢驗結果如下：White Heteroskedasticity Test:F-statistic6.555091    Probability0.000870Obs*R-squared15.56541    Probability0.003661可以看出并沒有消除異方差性，加權修正無效。下面采用1/abs(e)權變量進行WLS回歸，結果如下：模型五：Dependent Variable: LOG(Y)Method: Least SquaresDate: 07/29/12 Time: 09:10Sam

17、ple: 1 31Included observations: 31Weighting series: 1/ABS(E)VariableCoefficientStd. Errort-StatisticProb.  C1.2279290.2972684.1307080.0003LOG(X1)0.3757480.0568306.6117340.0000LOG(X2)0.5101200.01778128.688470.0000Weighted StatisticsR-squared0.999990    Mean dependent var

18、7.558578Adjusted R-squared0.999989    S.D. dependent var12.31758S.E. of regression0.041062    Akaike info criterion-3.455703Sum squared resid0.047210    Schwarz criterion-3.316930Log likelihood56.56339    F-statistic1960

19、.131Durbin-Watson stat2.487309    Prob(F-statistic)0.000000Unweighted StatisticsR-squared0.794514    Mean dependent var7.448704Adjusted R-squared0.779836    S.D. dependent var0.364648S.E. of regression0.171099    Sum squ

20、ared resid0.819694Durbin-Watson stat2.007122對加權以后的模型五進行懷特檢驗如下：White Heteroskedasticity Test:F-statistic0.199645    Probability0.936266Obs*R-squared0.923778    Probability0.921125可以看出，模型已經不再存在異方差問題，模型五可以作為修正以后的最終模型。二、隨機誤差項序列相關性問題的檢驗與修正 模型一：Dependent Variable: Y

21、Method: Least SquaresDate: 07/29/12 Time: 09:48Sample: 1991 2011Included observations: 21VariableCoefficientStd. Errort-StatisticProb.  C178.975555.064213.2503050.0042X0.0200020.00113417.641570.0000R-squared0.942463    Mean dependent var922.9095Adjusted R-squared0.93943

22、5    S.D. dependent var659.3491S.E. of regression162.2653    Akaike info criterion13.10673Sum squared resid500270.3    Schwarz criterion13.20621Log likelihood-135.6207    F-statistic311.2248Durbin-Watson stat0.658849

23、0;   Prob(F-statistic)0.000000 初始回歸模型一經濟意義合理，統(tǒng)計指標較為理想，但DW值偏低，模型可能存在序列相關性。（一）序列相關性的檢驗方法1、自回歸模型檢驗法Dependent Variable: EMethod: Least SquaresDate: 07/29/12 Time: 09:49Sample (adjusted): 1992 2011Included observations: 20 after adjustmentsVariableCoefficientStd. Errort-StatisticProb. 

24、; E(-1)0.7170800.2018523.5524970.0021R-squared0.398929    Mean dependent var2.801737Adjusted R-squared0.398929    S.D. dependent var161.7297S.E. of regression125.3870    Akaike info criterion12.54939Sum squared resid298716.2  

25、;  Schwarz criterion12.59918Log likelihood-124.4939    Durbin-Watson stat1.080741說明模型一的隨機誤差項至少存在一階正序列相關性，結合該自回歸模型的DW值為1.08，懷疑存在更高階的序列相關，繼續(xù)引入e(-2)如下：Dependent Variable: EMethod: Least SquaresDate: 07/29/12 Time: 09:49Sample (adjusted): 1993 2011Included observations: 19

26、after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  E(-1)1.0949740.1787686.1251080.0000E(-2)-0.8150100.199977-4.0755130.0008R-squared0.692885    Mean dependent var7.790341Adjusted R-squared0.674819    S.D. dependent var164.5730S.E. of reg

27、ression93.84710    Akaike info criterion12.02051Sum squared resid149723.7    Schwarz criterion12.11993Log likelihood-112.1949    Durbin-Watson stat1.945979由于e(-2)的t檢驗顯著，說明模型一的隨機誤差項確實存在二階正序列相關性，結合該二階自回歸模型的DW值為1.95，基本確定不存在更高階的序列相關。Breusch-God

28、frey Serial Correlation LM Test:F-statistic0.888958    Probability0.431668Obs*R-squared1.998924    Probability0.368077可以看出二階自回歸模型的隨機誤差項不存在序列相關性，論證了原模型僅存在二階序列相關。2、DW檢驗法0<DW<dL 存在正自相關(趨近于0) DL<DW<dU 不能確定 DU<DW<4dU 無自相關（趨近于2）3、LM檢驗法原理：一方面，根據(jù)上面的假

29、設檢驗結果判斷是否存在序列相關性，即根據(jù)(n-p)*R2統(tǒng)計量值與卡方檢驗臨界值2(P)進行比較，其中n為原模型樣本容量，P為選擇的滯后階數(shù)，R2為下面輔助回歸模型的可決系數(shù)。若(n-p)*R22(P)，則拒絕不序列相關的原假設，說明模型存在顯著的序列相關性；另一方面，結合下面的輔助回歸模型中殘差滯后變量是否通過t檢驗及DW值判斷序列相關的具體階數(shù)，方法與上面的自回歸模型檢驗法相同。選擇滯后一階檢驗：Breusch-Godfrey Serial Correlation LM Test:F-statistic13.15036    Probability0

30、.001931Obs*R-squared8.865308    Probability0.002906Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:51Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C-14.2447243.18361-0.3298640.7453

31、X0.0007140.0009070.7866170.4417RESID(-1)0.7632630.2104773.6263420.0019R-squared0.422158    Mean dependent var1.30E-13Adjusted R-squared0.357953    S.D. dependent var158.1566S.E. of regression126.7275    Akaike info criterion12.65352Sum squa

32、red resid289077.4    Schwarz criterion12.80274Log likelihood-129.8619    F-statistic6.575179Durbin-Watson stat1.159275    Prob(F-statistic)0.007183說明原模型確實存在一階序列相關性，結合該輔助回歸模型的DW值為1.16，懷疑存在更高階的序列相關，引入滯后二階檢驗如下：Breusch-Godfrey Serial Correlatio

33、n LM Test:F-statistic20.49152    Probability0.000030Obs*R-squared14.84303    Probability0.000598Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:51Presample missing value lagged residuals set to zero.VariableCoefficientStd. E

34、rrort-StatisticProb.  C14.0646332.409870.4339610.6698X-0.0006280.000742-0.8463030.4091RESID(-1)1.1084880.1761276.2936960.0000RESID(-2)-0.9181750.226004-4.0626430.0008R-squared0.706811    Mean dependent var1.30E-13Adjusted R-squared0.655072    S.D. de

35、pendent var158.1566S.E. of regression92.88633    Akaike info criterion12.07027Sum squared resid146673.8    Schwarz criterion12.26923Log likelihood-122.7379    F-statistic13.66102Durbin-Watson stat1.950263    Prob(F-stati

36、stic)0.000087由于e(-2)的t檢驗顯著，說明模型一的隨機誤差項確實存在二階正序列相關性，結合該二階自回歸模型的DW值為1.95，基本確定不存在更高階的序列相關。當然可以繼續(xù)引入滯后三階檢驗如下：Breusch-Godfrey Serial Correlation LM Test:F-statistic12.85743    Probability0.000157Obs*R-squared14.84303    Probability0.001956Test Equation:Dependent Var

37、iable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:52Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C14.0646733.407340.4210050.6794X-0.0006280.000765-0.8209340.4237RESID(-1)1.1082060.2713274.0844010.0009RESID(-2)-0.9175590.499523-1

38、.8368700.0849RESID(-3)-0.0006010.431119-0.0013950.9989R-squared0.706811    Mean dependent var1.30E-13Adjusted R-squared0.633514    S.D. dependent var158.1566S.E. of regression95.74504    Akaike info criterion12.16551Sum squared resid146673.

39、8    Schwarz criterion12.41421Log likelihood-122.7379    F-statistic9.643071Durbin-Watson stat1.950030    Prob(F-statistic)0.000363 可以看出并不存在三階序列相關。（二）廣義差分法修正1、方法原理參考教材自己推導二元線性回歸模型存在二階序列相關時的廣義差分模型。2、上機實現(xiàn)結果分析 模型二：Dependent Variable: YMethod:

40、Least SquaresDate: 07/29/12 Time: 09:55Sample (adjusted): 1992 2011Included observations: 20 after adjustmentsConvergence achieved after 8 iterationsVariableCoefficientStd. Errort-StatisticProb.  C160.0892182.89170.8753230.3936X0.0214690.0030726.9889750.0000AR(1)0.7300780.2033523.5902230.0

41、023R-squared0.964570    Mean dependent var958.0450Adjusted R-squared0.960402    S.D. dependent var655.9980S.E. of regression130.5388    Akaike info criterion12.71870Sum squared resid289686.3    Schwarz criterion12.86806L

42、og likelihood-124.1870    F-statistic231.4107Durbin-Watson stat1.116066    Prob(F-statistic)0.000000Inverted AR Roots      .73 由于AR(1)通過t檢驗，說明模型一確實至少存在一階序列相關，結合DW值為1.12，懷疑存在更高階序列相關性， LM檢驗結果如下： Breusch-Godfrey Serial Correlation LM

43、 Test:F-statistic6.380262    Probability0.009885Obs*R-squared9.193288    Probability0.010086Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 07/29/12 Time: 09:57Presample missing value lagged residuals set to zero.VariableCoefficientStd. Error

44、t-StatisticProb.  C80.86347145.26430.5566650.5860X-0.0035540.002602-1.3655560.1922AR(1)-0.5728410.437314-1.3099090.2099RESID(-1)1.0291570.3395413.0310220.0084RESID(-2)-0.1879230.598223-0.3141360.7577R-squared0.459664    Mean dependent var-7.24E-11Adjusted R-squared0.315

45、575    S.D. dependent var123.4773S.E. of regression102.1528    Akaike info criterion12.30313Sum squared resid156527.8    Schwarz criterion12.55207Log likelihood-118.0313    F-statistic3.190131Durbin-Watson stat2.021319&#

46、160;   Prob(F-statistic)0.043963說明模型一在一階廣義差分修正后仍然存在序列相關性。繼續(xù)引入AR(2)進行修正。模型三：Dependent Variable: YMethod: Least SquaresDate: 07/29/12 Time: 09:58Sample (adjusted): 1993 2011Included observations: 19 after adjustmentsConvergence achieved after 5 iterationsVariableCoefficientStd. Errort-S

47、tatisticProb.  C210.523342.671174.9336180.0002X0.0189160.00098719.173600.0000AR(1)1.0954460.1852545.9131940.0000AR(2)-0.9453840.250542-3.7733570.0018R-squared0.981385    Mean dependent var998.3158Adjusted R-squared0.977662    S.D. dependent var648.07

48、72S.E. of regression96.86089    Akaike info criterion12.16909Sum squared resid140730.5    Schwarz criterion12.36792Log likelihood-111.6064    F-statistic263.6012Durbin-Watson stat2.002336    Prob(F-statistic)0.000000Inve

49、rted AR Roots .55+.80i     .55-.80i由于AR(1)和AR(2)都通過t檢驗，說明模型一確實至少存在二階序列相關，結合DW值為2.00，確定不存在更高階序列相關性，LM檢驗結果如下：Breusch-Godfrey Serial Correlation LM Test:F-statistic0.880914    Probability0.437745Obs*R-squared2.267656    Probability0.

50、321799 可以看出，二階廣義差分修正后的模型三不再存在序列相關性，可以作為最終選擇模型。三、多元線性回歸模型分析中解釋變量的選取問題多重共線性的檢驗與修正假設用解釋變量x1、x2、x3、x4來解釋Y。模型一：Dependent Variable: YMethod: Least SquaresDate: 07/29/12 Time: 10:35Sample: 1994 2011Included observations: 18VariableCoefficientStd. Errort-StatisticProb.  C-43872.2714512.82-3.023002

51、0.0086X14.5610550.24699318.466320.0000X20.6704910.1300225.1567600.0001R-squared0.961029    Mean dependent var44127.11Adjusted R-squared0.955833    S.D. dependent var4409.100S.E. of regression926.6166    Akaike info criterion16.65197Sum squa

52、red resid12879274    Schwarz criterion16.80036Log likelihood-146.8677    F-statistic184.9504Durbin-Watson stat2.014913    Prob(F-statistic)0.000000模型二：Dependent Variable: YMethod: Least SquaresDate: 07/29/12 Time: 10:36Sample: 1994 2011Incl

53、uded observations: 18VariableCoefficientStd. 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    Mean dependent var44127.11Adjusted R-squared0.9752

54、20    S.D. dependent var4409.100S.E. of regression694.0715    Akaike info criterion16.11616Sum squared resid6744293.    Schwarz criterion16.31402Log likelihood-141.0454    F-statistic224.0086Durbin-Watson stat1.528658    Prob(