解释变量和误差项存在内生性问题
内生性问题来源
内生性问题主要来自于三个方面，分别为：遗漏变量、联立性以及度量误差
遗漏变量
遗漏变量是指可能与解释变量相关的变量，本来应该加以控制，但却没有控制的变量。这些变量最后进入了误差项，从而导致误差项与解释变量相关，进而导致了内生性问题。联立性
联立性是指一个计量方程中的核心解释变量A对被解释变量B产生影响，反过来，被解释变量B又对A产生影响。
如果B对A有正向的影响，正向冲击就会导致A增加，从而导致核心解释变量A与误差项正相关。
如果B对A有负向的影响，正向冲击就会导致A降低，从而导致核心解释变量A与误差项负相关。
比如：研究犯罪率和警察数量的关系，一般来说，警察数量越多，犯罪率越低；但反过来，犯罪率降低，警察数量也会减少
度量误差
度量误差可以分为解释变量的度量误差和被解释变量的度量误差。
内生性带来的问题
在存在内生性解释变量的情况下，OLS估计量有偏且不一致。只要任何一个解释变量与随机扰动项相关，全部解释变量的系数都会有偏、不一致。解决内生性的变化
通常有两种方法解决内生性问题即使用内生变量的滞后一期和工具变量法。
使用内生变量的滞后一期
一般来说，内生变量的上一期与当期误差项并不存在相关关系，所以可以考虑使用内生变量的滞后一期替代当期的内生变量。这种方法较为简单，并且在直觉上可行，但这种方法的缺点是：不能够回答当期的内生变量对当期的被解释变量的影响程度；而且，上一期的内生变量也可能因为遗漏变量而具有内生性。工具变量法
工具变量(instrumental variable)是指某一个变量与模型中解释变量高度相关，但却不与误差项相关，估计过程中被作为工具使用，以替代模型中与误差项相关的解释变量的变量。
工具变量法则是使用工具变量进行估计的方法。
工具变量法最常用的估计方法为：两阶段最小二乘法（TSLS）。
两阶段最小二乘法
操作：
第一阶段：将内生性变量作为被解释变量，工具变量和方程中的外生变量作为解释变量，来进行最小二乘估计；第二阶段：用第一阶段估计得到的内生变量的预测值替换内生变量，再进行最小二乘估计。
举例：yyy = β1\beta_1β1 + β2x2{\beta_2}{x_2}β2x2 +β3x3{\beta_3}{x_3}β3x3 +β4x4{\beta_4}{x_4}β4x4 + uuu，其中，x2x_2x2是严格外生的，而x3x_3x3x4x_4x4是内生的，则至少需要两个工具变量，设z1z_1z1和z2z_2z2为工具变量。
第一阶段：内生变量工具变量的回归
x3x_3x3 = π1{\pi_1}π1 + π2x2{\pi_2}{x_2}π2x2 +π3z1{\pi_3}{z_1}π3z1 +π4z2{\pi_4}{z_2}π4z2 + u1u_1u1
x4x_4x4 = γ1{\gamma_1}γ1 + γ2x2{\gamma_2}{x_2}γ2x2 +γ3z1{\gamma_3}{z_1}γ3z1 +γ4z2{\gamma_4}{z_2}γ4z2 + u2u_2u2
第二阶段：用预测回归的拟合值进行（代入第一阶段预测值）
yyy = β1\beta_1β1 + β2x2{\beta_2}{x_2}β2x2 +β3x3⋅{\beta_3}{x_3^·}β3x3⋅ +β4x4⋅{\beta_4}{x_4^·}β4x4⋅
STATA实现
regress x3 x1 z1 z2predict vregress x4 x2 z1 z2predict wregress y x2 v w
TSLS的难点不在于估计方法，而在于恰当的工具变量的选择。若存在N个潜在的内生解释变量，则至少需要N个IV。
原理：
第一阶段：消除了潜在内生解释变量的内生性，通过外生变量的预测回归，得到这些变量的外生性部分。第二阶段：利用第一阶段得到外生的预测回归的拟合值进行回归，进而消除偏误。
工具变量法的检验
使用工具变量法进行估计时，我们需要对工具变量进行三项检验，分别为：内生性检验、相关性检验、外生性检验。
内生性检验
内生性检验即检验核心变量是否具有内生性。如果我们关心的核心解释变量不具有内生性，我们就没有必要使用工具变量法进行估计，而如果我们使用了工具变量法虽然得到了一致估计量，但并不是有效估计量。相关性检验
相关性检验是检验工具变量是否与内生变量之间存在强相关关系。如果使用的工具变量是弱工具变量，则会导致内生变量估计的标准系数偏大。外生性检验
外生性检验是检验工具变量是否与误差项不相关。如果工具变量与误差项相关，则不满足外生性条件，那么使用工具变量法（IV）估计很可能会比OLS估计的结果更糟糕。
实操
第一步 模型设定与数据use crime.dta //打开数据集des //查看数据##结果obs: 630vars: 595 Jun 14:32---------------------------------------------------------------------------------------------------------------------storage display valuevariable name type formatlabelvariable label---------------------------------------------------------------------------------------------------------------------countyint%9.0g county identifieryear byte %9.0g 81 to 87crmrtefloat %9.0g crimes committed per personprbarrfloat %9.0g 'probability' of arrestprbconv float %9.0g 'probability' of convictionprbpris float %9.0g 'probability' of prison sentencavgsenfloat %9.0g avg. sentence, dayspolpc float %9.0g police per capitadensity float %9.0g people per sq. miletaxpc float %9.0g tax revenue per capitawest byte %9.0g =1 if in western N.C.central byte %9.0g =1 if in central N.C.urban byte %9.0g =1 if in SMSApctmin80 float %9.0g perc. minority, 1980wcon float %9.0g weekly wage, constructionwtuc float %9.0g wkly wge, trns, util, communwtrd float %9.0g wkly wge, whlesle, retail tradewfir float %9.0g wkly wge, fin, ins, real estwser float %9.0g wkly wge, service industrywmfg float %9.0g wkly wge, manufacturingwfed float %9.0g wkly wge, fed employeeswsta float %9.0g wkly wge, state employeeswloc float %9.0g wkly wge, local gov empsmix float %9.0g offense mix: face-to-face/otherpctymle float %9.0g percent young maled82 byte %9.0g =1 if year == 82d83 byte %9.0g =1 if year == 83d84 byte %9.0g =1 if year == 84d85 byte %9.0g =1 if year == 85d86 byte %9.0g =1 if year == 86d87 byte %9.0g =1 if year == 87lcrmrte float %9.0g log(crmrte)lprbarr float %9.0g log(prbarr)lprbconv float %9.0g log(prbconv)lprbpris float %9.0g log(prbpris)lavgsen float %9.0g log(avgsen)lpolpcfloat %9.0g log(polpc)...xtset county year //设置面板数据格式##结果panel variable: county (strongly balanced)time variable: year, 81 to 87delta: 1 unitxtdes //查看数据##结果county: 1, 3, ..., 197n = 90year: 81, 82, ..., 87T =7Delta(year) = 1 unitSpan(year) = 7 periods(county*year uniquely identifies each observation)Distribution of T_i: min5%25% 50% 75%95%max7 7 7 7 7 7 7Freq. Percent Cum. | Pattern---------------------------+---------90 100.00 100.00 | 1111111---------------------------+---------90 100.00 | XXXXXXX
第二步 描述性统计与作图
sum lcrmrte lprbarr lprbconv lprbpris lavgsen lpolpc ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban## 结果Variable | Obs Mean Std. Dev. Min Max-------------+---------------------------------------------------------lcrmrte | 630 -3.609225 .5728077 -6.31355 -1.808895lprbarr | 630 -1.274264.415897 -2.833214 1.011601lprbconv | 630 -.6929193 .6095949 -2.682732 3.610918lprbpris | 630 -.8786315 .2305144 -1.904239 -.3877662lavgsen | 630 2.153344 .2737295 1.439835 3.251537-------------+---------------------------------------------------------lpolpc | 630 -6.490637 .5266539 -7.687507 -3.336024ldensity | 630 -.0159271 .7747352 -1.62091 2.177889lwcon | 630 5.462869 .2481783 4.183905 7.751303lwtuc | 630 5.915883 .3702186 3.362377 8.020257lwtrd | 630 5.232423 .2143915 2.82576 7.715457-------------+---------------------------------------------------------lwfir | 630 5.579433 .2772037 1.257233 6.233362lwser | 630 5.364625 .3600984 .6118253 7.685734lwmfg | 630 5.615181 .2727473 4.623305 6.472115lwfed | 630 5.988757 .1587609 5.542831 6.393507lwsta | 630 5.677787 .1761313 5.153407 6.306275-------------+---------------------------------------------------------lwloc | 630 5.540139 .1596908 5.097363 5.961237lpctymle | 630 -2.443015 .1967842 -2.77808 -1.29332lpctmin | 630 2.913361 .9546147 .2497076 4.164309west | 630 .2333333 .423288701central | 630 .3777778 .485216901-------------+---------------------------------------------------------urban | 630 .0888889 .284809401twoway (scatter lcrmrte lprbarr) (lfit lcrmrte lprbarr) //关键变量与被解释变量的散点图并画出回归直线
xtline lcrmrte //关键变量的时间序列图
第三步 模型选择
xtivreg lcrmrte lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban d82 d83 d84 d85 d86 d87 (lprbarr lpolpc= ltaxpc lmix), fe //双向固定效应的两阶段最小二乘估计 ##结果Variable | Obs Mean Std. Dev. Min Max-------------+---------------------------------------------------------lcrmrte | 630 -3.609225 .5728077 -6.31355 -1.808895lprbarr | 630 -1.274264.415897 -2.833214 1.011601lprbconv | 630 -.6929193 .6095949 -2.682732 3.610918lprbpris | 630 -.8786315 .2305144 -1.904239 -.3877662lavgsen | 630 2.153344 .2737295 1.439835 3.251537-------------+---------------------------------------------------------lpolpc | 630 -6.490637 .5266539 -7.687507 -3.336024ldensity | 630 -.0159271 .7747352 -1.62091 2.177889lwcon | 630 5.462869 .2481783 4.183905 7.751303lwtuc | 630 5.915883 .3702186 3.362377 8.020257lwtrd | 630 5.232423 .2143915 2.82576 7.715457-------------+---------------------------------------------------------lwfir | 630 5.579433 .2772037 1.257233 6.233362lwser | 630 5.364625 .3600984 .6118253 7.685734lwmfg | 630 5.615181 .2727473 4.623305 6.472115lwfed | 630 5.988757 .1587609 5.542831 6.393507lwsta | 630 5.677787 .1761313 5.153407 6.306275-------------+---------------------------------------------------------lwloc | 630 5.540139 .1596908 5.097363 5.961237lpctymle | 630 -2.443015 .1967842 -2.77808 -1.29332lpctmin | 630 2.913361 .9546147 .2497076 4.164309west | 630 .2333333 .423288701central | 630 .3777778 .485216901-------------+---------------------------------------------------------urban | 630 .0888889 .284809401. twoway (scatter lcrmrte lprbarr) (lfit lcrmrte lprbarr). xtline lcrmrte. xtivreg lcrmrte lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle l> pctmin west central urban d82 d83 d84 d85 d86 d87 (lprbarr lpolpc= ltaxpc lmix), feFixed-effects (within) IV regression Number of obs= 630Group variable: countyNumber of groups = 90R-sq: Obs per group:within = 0.3587 min =7between = 0.4442 avg = 7.0overall = 0.4431 max =7Wald chi2(22)= 368612.24corr(u_i, Xb) = -0.1867 Prob > chi2 =0.0000------------------------------------------------------------------------------lcrmrte |Coef. Std. Err.z P>|z|[95% Conf. Interval]-------------+----------------------------------------------------------------lprbarr |-0.5760.802 -0.72 0.473 -2.148 0.997lpolpc |0.6580.8470.78 0.438 -1.002 2.317lprbconv |-0.4230.502 -0.84 0.399 -1.407 0.561lprbpris |-0.2500.279 -0.90 0.371 -0.798 0.297lavgsen |0.0090.0490.19 0.853 -0.087 0.105ldensity |0.1391.0210.14 0.891 -1.862 2.141lwcon |-0.0290.054 -0.54 0.591 -0.134 0.076lwtuc |0.0390.0311.27 0.205 -0.021 0.100lwtrd |-0.0180.045 -0.39 0.695 -0.107 0.071lwfir |-0.0090.037 -0.26 0.798 -0.081 0.062lwser |0.0190.0390.48 0.632 -0.057 0.095lwmfg |-0.2430.420 -0.58 0.562 -1.065 0.579lwfed |-0.4510.527 -0.86 0.392 -1.484 0.582lwsta |-0.0190.281 -0.07 0.947 -0.569 0.532lwloc |0.2630.3120.84 0.399 -0.349 0.876lpctymle |0.3511.0110.35 0.728 -1.631 2.333lpctmin |0.000 (omitted)west |0.000 (omitted)central |0.000 (omitted)urban |0.000 (omitted)d82 |0.0380.0620.61 0.540 -0.083 0.159d83 |-0.0440.042 -1.05 0.295 -0.127 0.039d84 |-0.0450.055 -0.82 0.410 -0.153 0.062d85 |-0.0210.074 -0.28 0.777 -0.166 0.124d86 |0.0060.1280.05 0.961 -0.245 0.257d87 |0.0440.2160.20 0.840 -0.380 0.467_cons |2.9432.6941.09 0.275 -2.337 8.223-------------+----------------------------------------------------------------sigma_u | .41829289sigma_e | .14923885rho | .88708121 (fraction of variance due to u_i)------------------------------------------------------------------------------F test that all u_i=0:F(89,518) = 13.93Prob > F = 0.0000------------------------------------------------------------------------------Instrumented: lprbarr lpolpcInstruments: lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfirlwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west centralurban d82 d83 d84 d85 d86 d87 ltaxpc lmix------------------------------------------------------------------------------est store FE2SLSxtivreg lcrmrte lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfir lwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west central urban d82 d83 d84 d85 d86 d87 (lprbarr lpolpc= ltaxpc lmix), ec2sls //随机效应的两阶段最小二乘估计##结果EC2SLS random-effects IV regression Number of obs= 630Group variable: countyNumber of groups = 90R-sq: Obs per group:within = 0.4521 min =7between = 0.8158 avg = 7.0overall = 0.7840 max =7Wald chi2(26)=575.73corr(u_i, X) = 0 (assumed)Prob > chi2 =0.0000------------------------------------------------------------------------------lcrmrte |Coef. Std. Err.z P>|z|[95% Conf. Interval]-------------+----------------------------------------------------------------lprbarr |-0.4130.097 -4.24 0.000 -0.604-0.222lpolpc |0.4350.0904.85 0.000 0.259 0.611lprbconv |-0.3230.054 -6.03 0.000 -0.428-0.218lprbpris |-0.1860.042 -4.44 0.000 -0.269-0.104lavgsen |-0.0100.027 -0.38 0.706 -0.063 0.043ldensity |0.4290.0557.82 0.000 0.322 0.537lwcon |-0.0070.040 -0.19 0.850 -0.085 0.070lwtuc |0.0450.0202.30 0.022 0.007 0.084lwtrd |-0.0080.041 -0.20 0.844 -0.089 0.073lwfir |-0.0040.029 -0.13 0.900 -0.060 0.053lwser |0.0060.0200.28 0.780 -0.034 0.045lwmfg |-0.2040.080 -2.54 0.011 -0.362-0.046lwfed |-0.1640.159 -1.03 0.305 -0.476 0.149lwsta |-0.0540.106 -0.51 0.609 -0.261 0.153lwloc |0.1630.1201.36 0.173 -0.071 0.398lpctymle |-0.1080.140 -0.77 0.439 -0.382 0.166lpctmin |0.1890.0414.56 0.000 0.108 0.270west |-0.2270.100 -2.28 0.023 -0.422-0.032central |-0.1940.060 -3.24 0.001 -0.311-0.077urban |-0.2250.116 -1.95 0.052 -0.452 0.001d82 |0.0110.0260.42 0.677 -0.040 0.061d83 |-0.0840.031 -2.73 0.006 -0.144-0.024d84 |-0.1030.037 -2.79 0.005 -0.176-0.031d85 |-0.0960.049 -1.94 0.053 -0.193 0.001d86 |-0.0690.060 -1.16 0.248 -0.186 0.048d87 |-0.0310.071 -0.45 0.656 -0.170 0.107_cons |-0.9541.284 -0.74 0.458 -3.470 1.563-------------+----------------------------------------------------------------sigma_u | .2145596sigma_e | .14923885rho | .67394424 (fraction of variance due to u_i)------------------------------------------------------------------------------Instrumented: lprbarr lpolpcInstruments: lprbconv lprbpris lavgsen ldensity lwcon lwtuc lwtrd lwfirlwser lwmfg lwfed lwsta lwloc lpctymle lpctmin west centralurban d82 d83 d84 d85 d86 d87 ltaxpc lmix------------------------------------------------------------------------------est store EC2SLShausman FE2SLS EC2SLS //hausman检验##结果---- Coefficients ----|(b)(B) (b-B)sqrt(diag(V_b-V_B))|FE2SLS EC2SLS DifferenceS.E.-------------+----------------------------------------------------------------lprbarr | -.5755052 -.4129264 -.1625788 .7962526lpolpc |.657526.4347488 .2227773 .8421081lprbconv | -.423144 -.3228871 -.1002569 .4990749lprbpris | -.2502547 -.1863195 -.0639352 .2762967lavgsen | .0090987 -.0101765 .0192752 .0408606ldensity |.139412.4290282 -.2896162 1.019765lwcon | -.0287308-.007475 -.0212558 .0360199lwtuc | .0391292.0454451 -.0063158 .0236726lwtrd | -.0177536 -.0081411 -.0096124 .0184617lwfir | -.0093443 -.0036395 -.0057048 .0223483lwser | .0185855.0056098 .0129756 .0331904lwmfg | -.2431675 -.2041395 -.039028 .411768lwfed | -.4513386 -.1635112 -.2878273 .5024337lwsta | -.0187447 -.0540496 .0353049 .2601761lwloc | .2632589.1630526 .1002062 .2885798lpctymle | .3511095 -.1081064 .4592159 1.001351d82 |.037856.0107451 .0271109 .0560526d83 | -.0443806 -.0837946 .039414 .0292202d84 | -.0451873 -.1034999 .0583125 .040481d85 | -.020942-.095702.07476 .0548511d86 | .0063223 -.0688986 .0752209 .1133461d87 | .0435043 -.0314075 .0749118 .2039854------------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtivregB = inconsistent under Ha, efficient under Ho; obtained from xtivregTest: Ho: difference in coefficients not systematicchi2(22) = (b-B)'[(V_b-V_B)^(-1)](b-B)= 19.50Prob>chi2 =0.6140
第四步 报告计量结果
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