貝葉斯空間計量模型

1、貝葉斯空間計量模型一、采用貝葉斯空間計量模型的原因殘差項可能存在異方差，而ML估計方法的前提是同方差，因此，當(dāng)殘差項存在異方差時，采用ML方法估計出的參數(shù)結(jié)果不具備穩(wěn)健性。二、貝葉斯空間計量模型的估計方法（一）待估參數(shù)對于空間計量模型（以空間自回歸模型為例）y=：Wy假設(shè)殘差項是異方差的，即N（0J2V）V =diag（Vi，V2，vn）上述模型需要估計的參數(shù)有：p<7VI V2Vn共計n+2個參數(shù)，存在自由度問題，難以進(jìn)行參數(shù)檢驗。為此根據(jù)大數(shù)定律，增加了新的假設(shè)：Vi服從自由度為r的卡方分布。如此以來，待估參數(shù)將減少為3個。(二)參數(shù)估計方法采用MCMC(MarkovChainMon

2、teCarlo)參數(shù)估計思想，具體的抽樣方法選擇吉布斯抽樣方法(Gibbssamplingapproach)在隨意給定待估參數(shù)一個初始值之后，開始生成參數(shù)的新數(shù)值，并根據(jù)新數(shù)值生成其他參數(shù)的新數(shù)值，如此往復(fù)，對每一個待估參數(shù)，將得到一組生成的數(shù)值，根據(jù)該組數(shù)值，計算其均值，即為待估參數(shù)的貝葉斯估計值。三、貝葉斯空間計量模型的類型空間自回歸模型far_g()空間滯后模型(空間回歸自回歸混合模型)sar_g()空間誤差模型sem_g()廣義空間模型(空間自相關(guān)模型)sac_g()四、貝葉斯空間模型與普通空間模型的選擇標(biāo)準(zhǔn)首先按照參數(shù)顯著性，以及極大似然值，確定普通空間計量模型的具體類型，之后對于該

3、確定的類型，再判斷是否需要進(jìn)一步采用貝葉斯估計方法。標(biāo)準(zhǔn)一：對普通空間計量模型的殘差項做圖，觀察參數(shù)項是否是正態(tài)分布，若非正態(tài)分布，則考慮使用貝葉斯方法估計。技巧：r=30的貝葉斯估計等價于普通空間計量模型估計，此時可以做出v的分布圖，觀察其是否基本等于1,若否，則應(yīng)采用貝葉斯估計方法。標(biāo)準(zhǔn)二：若按標(biāo)準(zhǔn)一發(fā)現(xiàn)存在異方差，采用貝葉斯估計后，如果參數(shù)結(jié)果與普通空間計量方法存在較大差異，則說明采用貝葉斯估計是必要的。例1:選舉投票率普通SAR與貝葉斯SAR對比：loadelect.dat;loadford.dat;y=elect(:,7)./elect(:,8);x1=elect(:,9)./ele

4、ct(:,8);x2=elect(:,10)./elect(:,8);x3=elect(:,11)./elect(:,8);w=sparse(ford(:,1),ford(:,2),ford(:,3);x=ones(3107,1)x1x2x3;res1=sar(y,x,w);res2=sar_g(y,x,w,2100,100);Vnames=strvcat(Voter','const',educ',home',income");prt(res1);prt(res2);SpatialautoregressiveModelEstimatesDepe

5、ndentVariable=voterR-squared=0.4605Rbar-squared=0.4600sigmaA2=0.0041Nobs,Nvars=3107,4log-likelihood=5091.6196# ofiterations=11minandmaxrho=-1.0000,1.0000totaltimeinsecs=1.0530timeforlndet=0.2330timefort-stats=0.0220timeforx-impacts=0.7380# drawsx-impacts=1000PaceandBarry,1999MClndetapproximationused

6、orderforMCappr=50iterforMCappr=30VariableCoefficientAsymptott-statz-probabilityconst-0.100304-8.4062990.000000educ0.33570421.9010990.000000home0.75406028.2122110.000000income-0.008135-8.5352120.000000rho0.527962335.7243590.000000DirectCoefficientt-statt-problower01ifljper99educ0.36279519.8269370,000

7、0000.3123650.4112B3honeD.81307027.4488S40.0000000.74234Z0.8St366income-0.0088368.5OS3820.000000-0.0LJS71-0.0Q&258IndirectCoefficierrtt-statt-problower01ifljper99educ349S8013.5555790.0000000.30372&0+3$8939hone78486314,5658870.0040000.6534110+943592inccune-0.0QB5U-8.3202210.000000-0.0L1I42-0.0

8、0620STotalCoefficientt-statt-problower01呼per99educ0.712*7522.0+0000000.630TS20.785895L59793321.5123550.0040001.406309L3091QSincojne-0.017348-8.9415620.000000-0.022571-0.012650檢驗是否存在異方差是否存在遺漏變量：貝葉斯對列向量做柱狀圖。bar(res.vmean);BayesianspatialautoregressivemodelHeteroscedasticmodelDependentVariable=voterR-s

9、quared=0.4425Rbar-squared=0.4419meanofsigedraws=0.0023sige,epe/(n-k)=0.0065r-value=4Nobs,Nvars=3107,4ndraws,nomit=2100,100totaltimeinsecs=20.6420timeforlndet=0.2370timeforsampling=19.2790PaceandBarry,1999MCIndetapproximationusedorderforMCappr=50iterforMCappr=30minandmaxrho=-1.0000,1.0000PosteriorEst

10、imatesVariableCoefficientStdDeviationp-levelconst-0.1078630.0127290.000000educ0.3484160.0180720.000000home0.7277990.0264160.000000income-0.0096030.0010500.000000rho0.5610540.0133130.000000Directlower01lower05Coefficientupper95upper99educ0.3282560.3411550.3804310.4188020.431102home0.7216090.7362560.7

11、946800.8523520.865360income-0.013430-0.012833-0.010486-0.008285-0.007535Indirectlower01lower05Coefficientupper95upper99educ0.3401420.3548940.4140590.4770480.496021home0.7301080.7595030.8649660.9853691.023184income-0.015080-0.014207-0.011413-0.008841-0.008030Totallower01lower05Coefficientupper95upper

12、99educ0.6716150.7023640.7944900.8896450.918055home1.4646461.5109931.6596461.8226761.878911income-0.028288-0.026981-0.021899-0.017254-0.015491對遺漏變量的測量:loadelect.dat;lat=elect(:,5);lon=elect(:,6);lonsli=sort(lon);lats=lat(li,1);elects=elect(li,:);y=elects(:,7)./elects(:,8);x1=elects(:,9)./elects(:,8);

13、x2=elecrs(:,10)./elects(:,8);x2=elects(:,10)./elects(:,8);x3=elects(:,11)./elects(:,8);x=ones(3107,1)x1x2x3;w1ww2=xy2cont(lons,lats);vnames=strvcat('voters','const','educ','home','income');res=sar(y,x,w,2100,100);res=sar_g(y,x,w,2100,100);prt(res,vnames);Bayes

14、ianspatialautoregressivemodelHeteroscedasticmodelDependentVariable=votersR-squared=0.4402Rbar-squared=0.4396meanofsigedraws=0.0022sige,epe/(n-k)=0.0065r-value=4Nobs,Nvars=3107,4ndraws,nomit=2100,100totaltimeinsecs=20.3230timeforlndet=0.2460timeforsampling=18.9770PaceandBarry,1999MClndetapproximation

15、usedorderforMCappr=50iterforMCappr=30minandmaxrho=-1.0000,1.0000*PosteriorEstimatesVariableCoefficientStdDeviationp-levelconst-0.1331820.0126330.000000educ0.3006530.0179860.000000home0.7252020.0259440.000000income-0.0082190.0010090.000000rho0.6284070.0141160.000000DirectlowerQIcjirer05Coefficientupp

16、er95upper99educ0.278S640.2916610.3308810.39910.3S0129hone0,7197880.7409180.7981160.8546850.871987income-0,011823-0,011255-fl.OO0O4B-0.006SB2-0+006056Indirect1。*“QIIcrflT05Coefficientupper95upper39educ0.3753000.4013400.4793530.5572330.582751hans0.&642160.9994991.1562391.3198061,378247income-0.017674-0.016599-0.013104-0.009689-0,OOB661Totallow«r01lower