计量经济学中"交互项"相关的5个问题和回应

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欢迎优秀社科研究学者到计量经济圈社群交流访问.务必是优秀的社科研究学者, 非诚勿扰.

计量经济圈的圈友，现在是凌晨0点，我是咱们社群的小编之一阿柯。以下分享的是几个关于“交互项”的问题，水平有限，有错误接受批评指正。如果想要彻底搞清楚这些，可以到计量经济圈社群交流讨论。

1.最近在研究交互项问题。关于计量经济圈的文章《计量回归中的交互项到底什么鬼? 捎一本书给你》我有一个问题，户籍与教育交互作用不显著，做联合检验显著说明什么？如果颠倒过来，交互项显著，联合检验不显著，往下再怎么分析？说到底就是联合检验有什么用？因为在看文献的时候很少有提到联合检验的。请赐教🙏🏻

阿柯: 也不知道你是不是计量经济圈社群群友，可以推荐二篇文章你看看：

https://www.cscu.cornell.edu/news/statnews/stnews40.pdf

http://www.uh.edu/~bsorense/Ozer-Balli_Sorensen.pdf

关于为什么应该做一个wald联合检验，你可以看看这个，主要看看是不是存在交互效应(这是针对所有交互项)。这里的Wald检验是相当于线性中的F检验，检验模型整体的情况显著情况，比如，这里是检验模型是不是存在age的交互项。你在《计量回归中的交互项到底什么鬼? 捎一本书给你》文章里可以看到，作者是针对整个edu的交互效应讨论的，因此建议用wald进行联合检验下。

如果我们只有一个交互项的话，那么不需要做Wald检验，因为交互项前面的系数的显著性是由t检验自动完成的，此时，只要这个交互项显著(即p值<0.05)，那么这个交互效应就是存在的。

2.圈内还有一篇涉及交互项的文章《实证研究中交叉项的使用和解读策略指南案例》引用了一篇社会学研究的文章。个人认为这篇文章对主效应和交互项解释不通。比如说第135页，第二段关于性别与收入关系的那段解释难以理解（按照分组后差分解释不通）。如果有时间还请您再check一下，谢谢！

阿柯: 至于《实证研究中交叉项的使用和解读策略指南案例》文章的解释，很多伙伴都认为与咱们通常见到的不一样。但是不是错了，这个还不能直接这样说。我们注意到，一旦有交互项出现，那么我们的主效应sex或者income的系数都发生了变化，他们现在变成了conditional效应。比如，在当income接近于0的时候(此时sex*income=0)，即该样本群体是低收入群体的时候，咱们的sex系数=-0.53。那咱们解释的时候就需要在低收入这个群体基础上进行，即低收入男性相对于低收入女性的再婚比率要低很多。

在习以为常的实证研究中，我们倾向于这样解释：sex前面的系数为负数——男性相对于女性再婚比率低很多(女性作为参照组)，sex*income前面的系数为正——收入越高那么男性再婚比率相对于女性而言就提高了。这就是收入对性别具有调节作用的体现，这与上面的解释其实是殊途同归的。整体而言，上面的那个解释相对更加具体，更加像一个分组回归的解释，这也恰恰是交互效应的本质。你可以以找大众接受度比较高的方式进行解释，这样评审人可能不会去抠字眼，不必理会上面这篇社会学文章的解释嘛。

3.计量经济圈小组您好！在阅读了您的《多期双重差分法和三重差分法指南》一文后，受益匪浅。但是关于其中did中介机制的检验，您用了一个类似ddd的方法，我想请问一下该种方法的出处，谢谢！感激不尽！

阿柯: did中介机制检验，在文献中确实有使用ddd方法进行的，这相当于是用了交互项，因为did, ddd本质上都是交互项的体现。而我们又知道，交互项可以作为一种机制分析，这个你可以在文献中寻得，即获得调节效应或中介效应，我们都称之为机制分析。

至于使用的具体文献，你可以到计量经济圈社群(微信群)问问，一下子太多记不太清楚了，下次咱们发文的时候会注意附上一些参考文献，感谢提醒我们。

4. 能不能用一些英文Notes解读一下interaction terms?

https://www.cscu.cornell.edu/news/statnews/stnews40.pdf

Adding interaction terms to a regression model can greatly expand understanding of the relationships among the variables in the model and allows more hypotheses to be tested. A previous newsletter, StatNews #39, discussed how to interpret coefficients in regression models. This newsletter will extend those ideas to explain how to interpret the coefficients of interaction terms.

The example from StatNews #39 was a model of the height of a shrub (Height) based on the amount of bacteria in the soil (Bacteria) and whether the shrub is located in partial or full sun (Sun). Height is measured in cm,Bacteria is measured in thousand per ml of soil, and Sun = 0 if the plant is in partial sun and Sun = 1 if the plant is in full sun. The regression equation was estimated as follows:

Height = 42 + 2.3Bacteria + 11Sun

It would be useful to add an interaction term to the model if we wanted to test the hypothesis that the relationship between the amount of bacteria in the soil on the height of the shrub was different in full sun than in partial sun. One possibility is that in full sun, plants with more bacteria in the soil tend to be taller, whereas in partial sun, plants with more bacteria in the soil are shorter. Another possibility is that plants with more bacteria in the soil tend to be taller in both full and partial sun, but that the relationship is much more dramatic in full than in partial sun.

The presence of a significant interaction indicates that the effect of one predictor variable on the response variable is different at different values of the other predictor variable. It is tested by adding a term to the model in which the two predictor variables are multiplied. The regression equation will look like this: 

Height = B0 +B1Bacteria + B2Sun + B3BacteriaSun

Adding an interaction term to a model drastically changes the interpretation of all of the coefficients. If there were no interaction term, B1 would be interpreted as the unique effect of Bacteria on Height. Since the interaction indicates that the effect of Bacteria on Height is different for different values of Sun, the unique effect of Bacteria on Height is not limited to B1, but also depends on the values of B3 and Sun. The unique effect of Bacteria is represented by everything that is multiplied by Bacteria in the model: B1 + B3Sun. B1 can now be interpreted as the unique effect of Bacteria on Height only when Sun = 0.

In our example, once we add the interaction term, our model looks like the following:

Height = 35 + 4.2Bacteria + 9Sun + 3.2BacteriaSun

Notice that adding the interaction term changed the values of B1 and B2. The effect of Bacteria on Height is now 4.2 + 3.2Sun. For plants in partial sun, Sun = 0, so the effect of Bacteria is 4.2 + 3.20 = 4.2. So for two plants in partial sun, a plant with 1000 more bacteria/ml in the soil would be expected to be 4.2 cm taller than a
plant with less bacteria. For plants in full sun, however, the effect of Bacteria is 4.2 + 3.21 = 7.4. So for two plants in full sun, a plant with 1000 more bacteria/ml in the soil would be expected to be 7.4 cm taller than a plant with less bacteria.

Because of the interaction, the effect of having more bacteria in the soil is different if a plant is in full or partial sun. Another way of saying this is that the slopes of the regression lines between height and bacteria count are different for the different categories of sun. B3 indicates how different those slopes are.

Interpreting B2 is more difficult. B2 is the effect of Sun when Bacteria = 0. Since Bacteria is a continuous variable, it is unlikely that it equals 0 often, if ever, so B2 can be virtually meaningless by itself. Instead, it is more useful to understand the effect of Sun, but again, this can be difficult. The effect of Sun is B2 + B3*Bacteria, which is different at every one of the infinite values of Bacteria. For that reason, often the only way to get an intuitive understanding of the effect of Sun is to plug a few values of Bacteria into the equation to see how Height, the response variable, changes. A subsequent newsletter will illustrate this approach.

5.接下来，我们准备给圈友引荐一篇实践性比较强的论文《interaction terms in Econometrics》，若有兴趣可以到后面下载下来读一读。

完整版pdf《interaction terms in Econometrics》

链接: 

https://pan.baidu.com/s/1HVQnMqOQcjYckCQyC1VRcQ 密码: qedr

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欢迎优秀社科研究学者到计量经济圈社群交流访问.务必是优秀的社科研究学者, 非诚勿扰. 

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