加入交乘项后符号变了!?

https://mp.weixin.qq.com/s/lqfTDF0xM5V9DrtX94Ejsw

1. 简介

为何加入交乘项后主变量变得不显著了，甚至符号都变掉了？

简单的解释是：此一时，彼一时！

因为，加入交乘项前后，主变量的系数含义发生了实质性的变化，二者不具可比性。本文的目的在于澄清这种差异，并介绍一种让主变量系数在加入交乘项前后不会发生大幅变化(具有可比性)的方法。

2.为何加入交乘项后主变量符号会变化？

3.如何尽力保证模型间的系数可比性？

若想让加入交乘项前后的模型 (1) 和模型 (2) 中主变量 (x1)的系数具有可比性，可以采用如下模型设定形式(见BalliHO,SørensenBE.Interactioneffectsineconometrics[J].EmpiricalEconomics,2013,45(1):583-603.[PDF])：

4.部分离差还是全部离差？

*-Source: /* Balli, H. O., B. E. Sørensen, 2013, Interaction effects in econometrics, EmpiricalEconomics,45(1):583-603. */ /* Table 1 The true model is Y = 3X1 + 5X2 + 8X1X2 + e where X1 = 1 + e1 and X2 = 1 + e2, ei~N(0,1) for i = 1, 2 (X1 and X2 are not correlated) and e~N(0,100). A constant is included but not reported. The sample size is 500 and the number of simulations is 20,000. Averages of estimated t statistics are shown in parentheses */ clear set obs 500 set seed 135 local rhox = 0 gen x = 1 + rnormal() gen z = 1 + rnormal() + `rhox'*x gen e = rnormal(0,10) gen y = 10 + 3*x + 5*z + 8*x*z + e pwcorr y x z center x z, prefix(c_) *-模型 (0) reg y x est store m0 *-模型 (1) reg y x z est store m1 *-模型 (2) reg y x z c.x#c.z est store m2 *-模型 (3) reg y x z c.c_x#c.c_z // Balli2013, Eq.(3) est store m3 *-模型 (4) reg y c_x c_z c.c_x#c.c_z est store m4 *-结果对比 local m "m0 m1 m2 m3 m4" local m "m1 m2 m3 m4" esttab `m' `s', nogap replace order(x z c_x c_z) /// b(%6.3f) s(N r2_a) drop(`drop') /// star(* 0.1 ** 0.05 *** 0.01) /// addnotes("*** 1% ** 5% * 10%")

---------------------------------------------------------------------------- Model (1) (2) (3) (4) ---------------------------------------------------------------------------- x 9.979*** 2.904*** 10.047*** (17.66) (4.48) (21.68) z 12.898*** 5.450*** 13.101*** (22.53) (8.14) (27.90) c_x 10.047*** (21.68) c_z 13.101*** (27.90) x#z 7.479*** (15.59) c_x#c_z 7.479*** 7.479*** (15.59) (15.59) _cons 3.024*** 9.792*** 2.485*** 25.275*** (3.12) (10.81) (3.12) (53.61) ---------------------------------------------------------------------------- N 500.000 500.000 500.000 500.000 r2_a 0.629 0.751 0.751 0.751 ---------------------------------------------------------------------------- t statistics in parentheses *** 1% ** 5% * 10% . sum y x z Variable | Obs Mean Std. Dev. Min Max -------------+--------------------------------------------------------- y | 500 25.55029 21.10401 -33.97629 103.5704 x | 500 1.023 1.018471 -2.755543 3.808831 z | 500 .9550672 1.005523 -1.766887 3.902355

参考文献

• BalliHO,SørensenBE.Interactioneffectsineconometrics[J].EmpiricalEconomics,2013,45(1):583-603.[PDF]这篇讲的非常清楚，还做了MC分析。
• BunMJG,HarrisonTD.OLSandIVestimationofregressionmodelsincludingendogenousinteractionterms[J].EconometricReviews,2018:1-14.-PDF-
• Ebbes,P.,D.Papies,H.vanHeerde.2016,Dealingwithendogeneity:Anontechnicalguideformarketingresearchers[C],Handbookofmarketresearch1-37.从IV讲起，很细致，进一步讨论了交乘项内生以及多个内生变量的情形。[PDF]
• Papies, D., P. Ebbes, H. J. Van Heerde. 2017, Addressing endogeneity in marketing models[C], Advanced methods for modeling markets (Springer, 581-627. [PDF], [web]