Panel Threshold Model - Theory and Application

PanelThresholdModel:TheoryandApplicationThresholdeffectPanelthresholdmodelComputationmethodApplicationThresholdeffect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,,(,),(),(,),(',')()iiiiiiiiiiiiiiiiiuqyuqyquIqwhereqIqxβxβxβxxβββxThresholdeffect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。Thresholdeffect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，nuisanceparameter)。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为已知常数，则模型的估计与检验（Chow)与传统方法相同。Thresholdeffect0222221222Thresholdestimate:GridsearchThresholdeffecttest:H:12ˆ~Nonstandarddistributionˆ1()ˆˆmin()sup(),()nnnnnnttinnnnnnBetaBetaFnyxOLSestimatornFFFn22ˆ()ˆ()nnThresholdeffect案例:.sysuselifeexp.scatterlexpgnppc556065707580Lifeexpectancyatbirth010000200003000040000GNPpercapitaThresholdeffect时间序列门限自回归模型TAR、Self-ExcitingTAR等。(1)(1)(1)0111(2)(2)(2)01112()()()0111,,,:,open-loopTARmodeltptpttdtptpttdtkkktptptktdtdtyyuifyyyuifyyyyuifyNoteifyz011011,,::ttktkttdttktkttdyyyuifyyyyuifyddelayparameterthreholdPanelThresholdModel——Specification单门限模型：121212()(),,()()(),(',')()xβxβxβxβxβxxβββxitiitititititiititititiititititiititititititityqquuqyuqyuIqIqPanelThresholdModel——Estimatebetagivengamma1212***1***,,()()()(),(),()ˆ:()'xβxβxxβxxβxxβxββxxxiiiitiiiiititiitiitiitiitiititiitiititititititituqyuqyyuuuuquuqyuGiven***1'()'itititySuuPanelThresholdModel——Estimategamma11******12**1*ˆargmin()[,]ˆˆ()''ˆˆˆ()'ˆˆˆˆ(')/[(1)]()/[(1)]ˆˆˆˆ()()()βxxxxxβitititititititiitiSySuuuuNTSNTuyyPanelThresholdModel——Estimategamma设门限序列Gamma共S个数值以Gamma[i]作为门限值，生成虚拟变量回归模型，记残差平方和为SRR[i]i=i+1令i=1iSMin(RSS)--Th1LR=[RSS-min(RSS)]/Sigsq--CIi=SPanelThresholdModel——EstimategammaPart1:ComputeThresholdseries:Quan=mm_quantile(Thr,1,(Trim\1-Trim))Trunc=Thr:=Quan[1]:&Thr:=Quan[2]Sethr=select(Thr,Trunc)Sethr=uniqrows(Sethr)/*ascendingdistinctvalue*/PanelThresholdModel——EstimategammaPart2:Estimatesinglethresholdfor(i=1;i=rows(Sethr);i++){Thtemp=Sethr[i]dumreg(N,T,THR,Thtemp,y,IX,RX,RSS=.,uf=.,fe)SeRSS[i]=RSS}minindex(SeRSS,1,loc=.,w=.)Thfinal=Sethr[loc[1,1]]RSSfinal=SeRSS[loc[1,1],c]dumreg(N,T,THR,Thfinal,y,IX,RX,RSS=.,uf=.,fe,Sigsq=.)LR=(SeRSS:-RSS)/SigsqPanelThresholdModel——Estimategammar1r2r3r4r5r6r7r8r9r10r11r12r13r14r15r16r17r18r19r20r21r22r23r24r25r26r27r28r29r30r31r32r33r34r35r36r37r38r39r40r41r42r43r44r45r46r47r48r49r50r51r52r53r54r55r56r57r58r59r60r61r62r63r64r65r66r67r68r69r70r71r72r73r74r75r76r77r78r79r80r81r82r83r84r85r86r87r88r89r90r91r92r93r94r95r96r97r98r99r100r101r102r103r104r105r106r107r108r109r110r111r112r113r114r115r116r117r118r119r120380400420440460SumofResidualSquared02468ThresholdPanelThresholdModel——TestThresholdEffectNote:第3步中，F统计量与β没有关系，因此DGP中的β可以任意设定。2012101*1*1ˆH:;F-stat:(())/ComputeP-valuebybootstrap:ˆˆˆˆ(1)stimatemodelunderH()()()ˆ(2)Resamplingwithreplacement(3)GenerateunderHDGP:()(xxβxitiitiitiFSSuyyueyy011)(4)EstimateunderH,H(5)Prob=I()/()xβitiiteFstatFstatFrowsFstatPanelThresholdModel——TestThresholdEffect设自举次数为Iters。提取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Fstat[i]=(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。iItersCriticalValue=Quantile(Fstat,1-alpha)Prob=Num(FstatF)/Itersi=Itersi=1PanelThresholdModel——TestThresholdEffectPart3:Testthresholdeffect*Regresslinearmodel,extactresidualu(residual).*Regresssingle-thresholdmodel,extractyhat,F.for(i=1;i=Iters;i++){u0=u[mm_sample(N,.,Cinfo),.]ynew=yhat+u0/*fixed-effect&single-thresholdregression*/regfe(N,T,ynew,MX,RSS0=.,uf=.,0)ptm(N,T,ynew,MIX,MRX,Thr,Sethr,1,0,0,RSSa=.,Siga=.)Fstat[i]=(RSS0-RSSa)/Siga/*ComputeF-stat*/}Prob=colsum(Fstat:F)/rows(Fstat)Fcrit=mm_quantile(Fstat,1,(0.90\0.95\0.99))PanelThresholdModel——TestThresholdEffectBootstrapnumber=100------------------------------------------------------------------------------------|RSSSig2FstatProbCrit_10Crit_5Crit_1--------------+---------------------------------------------------------------------Threshold_0|464.95053.3692.....Threshold_1|382.94002.735329.98240.00009.388010.158212.6432Threshold_2|373.26082.66613.63040.960015.267418.248220.7824------------------------------------------------------------------------------------PanelThresholdModel——Confidenceinterval211122ˆˆ()()()/Pr()[1exp(/2)]ˆNote:isthestandarderrorunderalternative.LRSSxxPanelThresholdModel——ConfidenceintervalPart4：Confidenceinterval/*findthelocationofthreshold*/for(i=1;i=rows(Sethr);i++){if(Thr[i]==Thfinal)break}loc=i/*Lower:MaximumofThrThreshold*/for(i=loc-1;i=1;