1、Panel Threshold Model:Theory and ApplicationThreshold effectPanel threshold modelComputation methodApplication Threshold effect门限效应：变量之间的关系取决于门限变量的状态，即当门限变量低于门限值或者高于门限值时，回归方程的系数也不同。以单门限模型为例。1212,(,),(),(,),(,)()iiiiiiiiiiiiiiiiiuqyuqyquI qwhereqI q x x xxxxThreshold effect门限效应由于其直观的含义，门限模型在金融市场和宏观经济
2、政策研究中得到广泛应用。金融约束对企业投资的影响。是否负债率较高、较低的企业其金融约束对投资的影响存在明显差异。一体化程度对经济增长的影响。是否收入越低的国家其经济一体化对经济增长的影响越大。央行利率政策对通货膨胀的响应。是否低、高通货膨胀的国家其央行对通胀率的响应存在明显差异。Threshold effect对于门限值的检验：beta1=beta2。但是，在备择假设下多了一个未知参数gamma（称作冗余参数，nuisance parameter)。这种情况下传统的检验统计量不再有效。这种情况下的模型也一直没有得到很好的应用，直至Andrews（1983）等给出统计量的分布。注：如果gamma
3、为已知常数，则模型的估计与检验（Chow)与传统方法相同。Threshold effect 0222221222Threshold estimate:GridsearchThreshold effect test:H:12 Nonstandard distribution1()min()sup(),()nnnnnnttinnnnnnBetaBetaFnyxOLS estimatornFFFn22()()nnThreshold effect案例:.sysuse lifeexp.scatter lexp gnppc556065707580Life expectancy at birth010000
4、200003000040000GNP per capitaThreshold effect时间序列门限自回归模型TAR、Self-Exciting TAR等。(1)(1)(1)0111(2)(2)(2)01112()()()0111,:,open-loop TAR modeltptptt dtptptt dtkkktptptkt dt dtyyuifyyyuifyyyyuifyNote if yz 011011,:ttkt ktt dttkt ktt dyyyuif yyyyuif yd delay parameterthrehold Panel Threshold ModelSpecific
5、ation单门限模型：121212()(),()()(),(,)()xxx x xxxxitiitititititiititititiititititiititititititityqquuqyuqyuI qI qPanel Threshold ModelEstimate beta given gamma 1212*1*,()()()(),(),():()x x xxxxxxxxxxiiiitiiiiititiitiitiitiitiititiitiititititititituqyuqyyuuuuquuqyuGiven*1()itititySu uPanel Threshold ModelE
6、stimate gamma 11*12*1*argmin(),()()()/(1)()/(1)()()()xxxxxitititititititiitiSySu uu uN TSN Tuyy Panel Threshold ModelEstimate gamma设门限序列Gamma共S个数值 以Gammai作为门限值，生成虚拟变量回归模型，记残差平方和为SRRii=i+1令i=1i Th1LR=RSS-min(RSS)/Sigsq-CIi=SPanel Threshold ModelEstimate gammaPart 1:Compute Threshold series:Quan=mm_qu
7、antile(Thr,1,(Trim1-Trim)Trunc=Thr:=Quan1:&Thr:=Quan2Sethr=select(Thr,Trunc)Sethr=uniqrows(Sethr)/*ascending distinct value*/Panel Threshold ModelEstimate gammaPart 2:Estimate single thresholdfor(i=1;i=rows(Sethr);i+)Thtemp=Sethridumreg(N,T,THR,Thtemp,y,IX,RX,RSS=.,uf=.,fe)SeRSSi=RSS minindex(SeRSS,
8、1,loc=.,w=.)Thfinal=Sethrloc1,1RSSfinal=SeRSSloc1,1,cdumreg(N,T,THR,Thfinal,y,IX,RX,RSS=.,uf=.,fe,Sigsq=.)LR=(SeRSS:-RSS)/SigsqPanel Threshold ModelEstimate gamma r 1r 2r 3r 4 r 5r 6r 7r 8r 9r 1 0r 1 1r 1 2r 1 3r 1 4r 1 5r 1 6r 1 7r 1 8r 1 9r 2 0r 2 1r 2 2r 2 3 r 2 4r 2 5r 2 6r 2 7r 2 8 r 2 9r 3 0r
9、3 1r 3 2r 3 3r 3 4r 3 5r 3 6r 3 7r 3 8r 3 9 r 4 0r 4 1r 4 2r 4 3r 4 4r 4 5r 4 6r 4 7r 4 8r 4 9r 5 0r 5 1r 5 2r 5 3 r 5 4r 5 5r 5 6r 5 7r 5 8r 5 9r 6 0r 6 1r 6 2r 6 3r 6 4r 6 5r 6 6r 6 7r 6 8r 6 9r 7 0r 7 1r 7 2r 7 3r 7 4r 7 5r 7 6r 7 7r 7 8r 7 9r 8 0r 8 1r 8 2r 8 3r 8 4r 8 5r 8 6r 8 7r 8 8r 8 9r 9 0
10、r 9 1r 9 2r 9 3r 9 4r 9 5r 9 6r 9 7r 9 8r 9 9r 1 0 0r 1 0 1r 1 0 2r 1 0 3r 1 0 4r 1 0 5r 1 0 6r 1 0 7r 1 0 8r 1 0 9r 1 1 0r 1 1 1r 1 1 2r 1 1 3r 1 1 4r 1 1 5r 1 1 6r 1 1 7r 1 1 8r 1 1 9r 1 2 0380400420440460Sum of Residual Squared02468ThresholdPanel Threshold ModelTest Threshold EffectNote:第3步中，F统计量
11、与没有关系，因此DGP中的可以任意设定。2012101*1*1H:;F-stat:()/Compute P-value by bootstrap:(1)stimatemodelunder H()()()(2)Resamplingwith replacement(3)Generateunder H DGP:()(xxxitiitiitiFSSuyyueyy011)(4)Estimateunder H,H(5)Prob=I()/()xitiiteFstatFstatFrows FstatPanel Threshold ModelTest Threshold Effect设自举次数为Iters。提取
12、u的自举样本u*，计算y*=yhat+u*利用自举样本y*，回归H1和H0模型。计算Fstati=(RSS1-RSS0)/Sigsq1i=i+1估计H1模型，提取残差u、RSS1、Sigsq1。估计H0模型，计算yhat、RSS0。计算F=(RSS1-RSS0)/Sigsq1。iF)/Itersi=Itersi=1Panel Threshold ModelTest Threshold EffectPart 3:Test threshold effect*Regress linear model,extact residual u(residual).*Regress single-thresh
13、old model,extract yhat,F.for(i=1;iF)/rows(Fstat)Fcrit=mm_quantile(Fstat,1,(0.900.950.99)Panel Threshold ModelTest Threshold EffectBootstrap number=100-|RSS Sig2 Fstat Prob Crit_10 Crit_5 Crit_1-+-Threshold_0|464.9505 3.3692 .Threshold_1|382.9400 2.7353 29.9824 0.0000 9.3880 10.1582 12.6432 Threshold
14、_2|373.2608 2.6661 3.6304 0.9600 15.2674 18.2482 20.7824-Panel Threshold ModelConfidence interval 211122()()()/Pr()1exp(/2)Note:is thestandard error under alternative.LRSSxxPanel Threshold ModelConfidence intervalPart 4：Confidence interval/*find the location of threshold*/for(i=1;i=rows(Sethr);i+)if
15、(Thri=Thfinal)breakloc=i/*Lower:Maximum of Thr=1;i-)if(LRicrit)breaklow=Thri+1/*Upper:Minimum of ThrThreshold*/for(i=loc+1;icrit)breakupp=Thri-1Panel Threshold ModelConfidence intervalThreshold estimation:Alpha=0.0500-|Threshold Lower Upper-+-Th_1|4.4565 4.4135 4.4860-r 1r 2r 3r 4 r 5r 6r 7r 8r 9r 1
16、 0r 1 1r 1 2r 1 3r 1 4r 1 5r 1 6r 1 7r 1 8r 1 9r 2 0r 2 1r 2 2r 2 3r 2 4r 2 5r 2 6r 2 7 r 2 8r 2 9r 3 0r 3 1r 3 2r 3 3 r 3 4r 3 5r 3 6r 3 7r 3 8r 3 9 r 4 0r 4 1r 4 2 r 4 3r 4 4r 4 5r 4 6r 4 7r 4 8r 4 9r 5 0r 5 1r 5 2r 5 3 r 5 4r 5 5 r 5 6r 5 7r 5 8r 5 9r 6 0r 6 1r 6 2r 6 3r 6 4r 6 5r 6 6r 6 7r 6 8r
17、6 9r 7 0r 7 1r 7 2r 7 3r 7 4r 7 5r 7 6r 7 7r 7 8r 7 9r 8 0r 8 1r 8 2r 8 3r 8 4r 8 5r 8 6r 8 7r 8 8r 8 9r 9 0r 9 1r 9 2r 9 3r 9 4r 9 5r 9 6r 9 7r 9 8r 9 9r 1 0 0r 1 0 1r 1 0 2r 1 0 3r 1 0 4r 1 0 5r 1 0 6r 1 0 7r 1 0 8r 1 0 9r 1 1 0r 1 1 1r 1 1 2r 1 1 3r 1 1 4r 1 1 5r 1 1 6r 1 1 7r 1 1 8r 1 1 9r 1 2 0
18、0102030LR Statistics02468ThresholdPanel Threshold ModelModel estimation=Linear regression(fixed effect):Sum of Squared Residual:RSS=464.9505Standard error of regression:Se=3.3692R-squared:R2=0.4783-|Coef Std t prob-+-drgdp|-0.1100 0.0468 -2.3524 0.0201 dcpi|0.6049 0.0545 11.0948 0.0000-=Threshold Re
19、gression:threshold number=1.00Sum of Squared Residual:RSS=382.9400Standard error of regression:Se=2.8157R-squared:R2=0.5703-|Coef Std t prob-+-drgdp_1|-0.0895 0.0519 -1.7259 0.0866 dcpi_1|-0.0703 0.1388 -0.5067 0.6132 drgdp_2|-0.0682 0.0659 -1.0349 0.3026 dcpi_2|0.5504 0.0683 8.0532 0.0000-Panel Thr
20、eshold ModelMultiple thresholds以双门限模型为例，其他模型依此类推。111222311212321122()()(),()()()(),()xxxx x x xxxxxitiitititititititiititititiitititiititititiititititititititityqqquuqyuquqyuI qIqI q123(,)Panel Threshold ModelEstimate thresholds如果利用普通的格点搜寻，需要迭代计算（NT）2。显然，实践当中这不太可行。根据(Chong,1994;Bai,1997;Bai and Perr
21、on,1998)，序贯估计具有一致性。21111121222222211221Step 1:Estimatesingle threshold model:,()Step Given,Estimatesecond threshold:(,),(),argmin()(,),Step 3iseffecient,but rrrSSSSS 11212111112112isnt.(,),(),argmin()(,),rrrrrrrSSSS Panel Threshold ModelMultiple thresholds双门限模型：Step 1:估计单门限模型-Th1Step 2:给定Th1，估计第二个门限
22、值-Th21，CIStep 3:给定Th21，重新估计第一个门限值-Th22，CI三门限模型：Step 1:估计双门限模型-Th21，Th22Step 2:给定Th21、Th22，估计第三个门限值-Th31,CIStep 3:给定Th31、Th22，重新估计第二个门限值-Th32,CIStep 4:给定Th31、Th32，重新估计第一个门限值-Th33,CIPanel Threshold ModelTest double threshold effectNote:在第3步中，DGP中的为单门限模型的估计量。0122212222*112H:Single threshold model;H:Dou
23、ble threshold model.F-stat:()()/,()/(1)Compute P-value by bootstrap:(1)stimatemodelunder H()(,)xxrrrritiitiFSSSN Tuyy 12*101(,)(2)Resamplingwith replacement(3)Generateunder H DGP:()()()(4)Estimateunder H,H(5)Prob=I()/()xxDitiitiSitueyyeFstatFstatFrows Fstat Panel Threshold ModelConfidence interval 2
24、2222222111111()()()/()()()/rrrrrrrrLRSSLRSSPanel Threshold ModelExample Hansen(1999).cd d:stata10panel.use hansen1999,clear.xtptm i q1-qd1,rx(c1)thrvar(d1)trim(0.01)grid(393)thnum(3)iters(300)Panel Threshold ModelExampleWu and Wang(2008).cd d:stata10panel.use res4,clear.xtptm depr,rx(drgdp dcpi)thrvar(dcpi)thnum(2)trim(0.10)iters(100)其他扩展（1）多门限变量（2）动态面板门限模型谢谢！
