Thus in order to identify causal effects we need to condition on variables that

Thus in order to identify causal effects we need to

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Thus, in order to identify causal effects, we need to condition on variables that are effectively causes and not on consequences. This criterion is called the Back- door criterion . 29 It asserts that a causal effect is identified if all back-door paths are blocked. A path is any sequence of connections in any direction between two variables. For example, in Figure 3, the sequence INC t INC t - 1 PG t CR t - 1 S t S t +1 is the path connecting the levels of income at the t-th period and segregation at the next period. Moreover, a back-door path is a path where a directed edge points towards the initial variable. It is important to control for these back-door paths because they generate bias in the estimation of the true causal effect, in our example INC t S t +1 . The previous example is a back-door 27 Pearl[45] discusses this point at length. 28 Cholecystitis is the inflammation of the gallbladder. It was believed to cause or aggravate diabetes (see Berkson[7]). 29 See Pearl[45] and, Morgan and Winship[37] for more details on this point. 10
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path for instance. We will further say that a path is blocked if we can find a set of control variables Z to condition on which are either the middle variable of a chain of mediation or the common cause of two consequences. If a control variable Z is a consequence of two independent causes, it will spoil the analysis due to the spurious correlation mentioned earlier. However, if we can additionally control for another variable Z’ which is not a consequence of two causes, we will thus still block this back-door path. Then the back-door criterion identifies causal effects by controlling for all the other potential causes. Our causal graph in Figure 3 has a lot of back-door paths potentially threat- ening the identification of causal effects. In fact, every path is a back-door path because a determinant is itself caused by another determinant at the previous period. Then, a directed link is pointing towards our starting variable, thus cre- ating a back-door path by definition. But we can block all these back-door paths by conditioning by all the determinants at the previous period. Due to the def- inition of a cause, we have this nice autoregressive structure that allows only descendent links from the previous period to the actual one. Thus, we have only two types of back-door paths, those with common consequences and those with- out. For instance, between CR t S t +1 we can have the following back-door paths, without common consequences CR t CR t - 1 S t S t +1 and with CR t CR t - 1 PG t INC t - 1 SD t PG t +1 S t S t +1 . For the first back-door path, we can block it by conditioning either on S t or CR t - 1 as S t is in the chain of mediation CR t - 1 S t S t +1 , and CR t - 1 is the common cause of S t and CR t . For the second back-door path, there are two common consequences on which we should not condition, PG t in CR t - 1 PG t INC t - 1 and PG t +1 in SD t PG t +1 S t . However, it can be blocked by conditioning on all the other variables, for instance S t .
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