 Structural regression model with latent variables is always nested in an unconstrained confirmatory factor analysis model. To understand why that is the case and what does it mean and why the decrease of freedom changes when we move from confirmatory factor analysis to a structural regression model, let's take a look at this example from Esquid and Lazarini. Their estimate is fairly complex model with 10 theoretical variables. Some of the variables that they have are directly observed boxes. I will use single indicator factors for them because of reasons that I explained in the next slide. But basically this is the model that we want to do. So let's just look at the latent variable part. And here's the latent variable part. So we have, for example, investment affects horizontal, affects innovation, so that's a mediation relationship, and all kinds of direct effects as well. Interestingly, the competition variable here is constrained to be uncorrelated with all the other exogenous variables. Why is that the case? This is because that is a directly observed variable. It is not a latent variable, so they have a single indicator measure, and they just apply that directly as an exogenous observed variable in the model. This is a bad idea because in some SCM software, all the latent variables that are exogenous are constrained to be uncorrelated with all the exogenous predictors. Even in software that allow those correlations, some software constrained those correlations to be zeros initially. So you have to remember to free the correlations. And I've seen a few papers that actually have these control variables that are constrained to be uncorrelated with interesting variables. And of course, then in that case, you wouldn't do any controlling because a control variable needs to be correlated with interesting variables and explain the dependent variable. So this is a specific case and error that could have been avoided by using latent variables to represent all theoretical concepts. And if there's a single indicator measure, then I'll use a single indicator factor for that measure, which I'm doing here. So let's go now from the conforter factor analysis towards the structural regression model. So this is the initial case in the conforter factor analysis model. So we have 10 factors that are all allowed to be freely correlated. And then we start to add constraints to the model, and we will see that the decrease of freedom goes up. So we have now 55 covariances and variances of the factors, and this number will decrease as we take these factors away from the exogenous variables, make them endogenous by adding regression paths. So let's first add the constraints that they have for the competition that it needs to be uncorrelated with all the other control variables. So we add three constraints. Three correlations are not estimated, but they are constrained to be zero. That gives us three degrees of freedom more. So this is a total decrease of freedom, and this is the change between the current model and the conforter factor analysis model, where all the factors are freely correlated. Then we proceed and we look at each variable at a time. So we look at horizontal. Horizontal was an endogenous variable, so we take it away from the matrix of exogenous variables. So we lose all these correlations, and that gives us eight degrees of freedom more. Then we add one regression path, and well that gives us eight degrees of freedom. So we take away nine correlations and add one regression path. So we gain eight degrees of freedom, because we take away nine things and add one thing. Then we proceed to vertical, and we'll take away all correlations involving vertical, which is eight correlations, and we add one regression path, so we gain seven degrees of freedom. We're now at plus 18. And we do this for all the variables. Manufacturing productivity has a few more paths, but there are some missing paths here, and that gives us more degrees of freedom. Innovation, same thing, not all paths are present, so the degrees of freedom goes up, and so on collective sourcing global markets. Eventually, when we add global markets as a dependent variable, we actually are allowed to be correlated with some of the other endogenous variables that we previously constrained to be uncorrelated, so the degrees of freedom went from 20 to 20 up from the base model. So the degrees of freedom difference between the confrontational factor analysis model and the theoretical model is 20. So we took things away, correlation away, correlations away, and we put regression coefficients back. And the difference in the number of correlations that we took away minus the number of coefficients that we put there is 20, and that is shown in the paper. So they have this table, and the degrees of freedom for the measurement model, including the factor law and its indicator errors, is 147, and the theoretical model 167, which is plus 20 more. So you can do these degrees of freedom calculations to understand how the model was specified. And there is also an article by Cortene and co-authors that provide a calculator online that allows you to do this just by you typing how many variables the model has, and then the calculator will give you the number of the degrees of freedom of the model and how it's calculated. One nice thing about this particular table in this paper is that it contains a nice footer. So this footer here provides us the degrees of freedom calculation for the best model, which is their final model. Unfortunately, they don't do that for the theoretical model, but this is the final model. And they also tell here that they provide the data covariance matrix in the paper, and you can replicate everything that they do. And that is actually true, because if you run these models that they do from the correlation matrix, you will get the same exact results, which is nice.