Comparing results from structural equation model and OLS regressions Are results obtained from a structural equation model (SEM) comparable to those from a series of separate regressions?
Or, conversely, can I obtain direct, indirect, and total effect of an independent variable of interest by running three separate regressions (e.g. OLSs)?
Let us say that IV=independent variable, MV = mediation variable, and DV = dependent variable. I conduct three separate regressions.

*

*DV = B0 + B1*MV + B2*IV + Controls
B1 is the direct effect of MV on DV, while B2 is the direct effect of IV on DV.


*MV = K0 + K1*IV + Controls
K1 is the direct effect of IV on MV.


*DV = P0 + P1*IV + Controls
P1 is the indirect effect of IV on DV.
The notation used in the mediation analysis below is based on that used in this handbook:
Hayes, A. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press, New York, US.

*

*K1 = a is the direct effect of IV on MV

*B1 = b is the direct effect of MV on DV

*B2 = c' is the direct effect of IV on DV

*B2+B1*K1 = c'+a*b is the total effect of IV on DV

In turn, the total effect, that it, c'+a*b, should equal the coefficient E1 from the following regression:
DV = E0 + E1*IV + Controls
To my understanding, a*b is a measure of omitted variable bias. Thus, the entire SEM could be seen as an application of the Frisch-Waugh-Lovell theorem (Greene, 2003; p. 148-149).
However, when I compare the results obtained from this series of OLSs and the results from an ad-hoc package (i.e. I use the built-in GSEM in Stata), that results do not compare. In particular, while "a" and "b" are similar, "c'" is very different.
Here is a related post on Statalist, where I discuss the script. While the question is more focused on the script per se in Statalist, here I focus more on the methodology.
[I am new to SEM]

The answers to my questions are "yes".
This was the issue on which I have wasted a couple of days. The independent variable is endogenous, so I have regressed the first stage and obtained the predicted values of this variable (I had omitted this detail from the post).
Then, I have used the (predicted) IV so obtained in both methods, that is, series of regressions versus GSEM; however, the first stage to obtain the IV used a slightly larger sample. The reason is that the mediator variable (MV) has some missing values.
My solution was first to run a rubbish OLS: Y IV MV Controls, and flag the sample.
Then, I have regressed the first stage to obtain the predicted values of IV.
Finally, I have used these predicted values in both the regressions and in GSEM. Now the results are identical across the two methods.
The lesson learned is: make sure that the samples used across methods and different stages are the same.
