Mediation Analysis for time varying and invariant DV's

Question

I have time-varying independent variables (IV) ($x_1,x_2,x_3...$) and a time varying dependent variable (DV) ($m_1$). I also have a time invariant dependent variable ($m_2$) that is dichotomous. I want to show that my independent variables ($x_1,x_2,x_3...$) impact time invariant DV ($m_2$) through $m_1$.

I tried the following steps:

1. I ran a random parameter model with $m_1$ as DV and all $x_i$ as the IV and find that $x_1$ and $x_2$ have a significant effect on $m_1$.

2. I ran a probit model with $m_2$ as DV and all $x_i$ and $m_1$ as independent variables. I find that, except for $m_1$, all other variables are null.

These results made me think on full mediation. But how to check it? Hayes macro on mediation test or any other SPSS tool doesn't take care of the nesting I do in step 1.

Is there a solution for the above problem?

Answer 1

The question of whether full mediation exists is basically a question of "is my model correct or not". The are many methods to answer this, all of them "looking for specific issues". There is no a "meta-test" to tell whether the model is correct or not. As such, you will never be sure about it.

The way forward is to test for particular problems, which the theory tells you could be affecting your results. Here I will show you one example (omitted variables). Maybe another user comes out with further issues (endogeneity?, spurious regression?).

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Problem: Omitted variable

Consider two cases:

1. There is full mediation. This is:

$$ \{x_1,x_2,x_3\} \rightarrow m_1 \rightarrow m_2 $$

In that case, your conclusion of full mediation is correct.

2. There is a variable omitted which impacts on $m_1$, and it is "significantly" correlated with another $x_i$, which itself is not important for $m_1$. For example, say $x_2$ is not important but $x_4$ is:

$$ \{x_1,x_3,x_4\} \rightarrow m_1 \rightarrow m_2 $$

Then, because of the correlation between $x_2$ and $x_4$, and your omission of $x_4$, your results are confounding the variables which are important in $m_1$.

Now, how to test for this? The first method to distinguish between these two cases is by making sure the model in step 1 is correct. Have you included all "relevant" variables (as per theory)? Secondly, you can perform omitted variable tests. There is no point in going over this here, as there are plenty or related questions (e.g. here). Google is your friend too.