 Latent variables are commonly used in statistical modeling. People normally associate the concept of latent variable with factor analysis models, but this is actually a more general concept. Let's take a look at what latent variables are. The variables in a statistical model can be divided into latent variables and observed variables. The difference between observed variable and latent variable is that for observed variables we have scores for individual cases. So if we have our data set, which is like an actual sheet, then observed variables are columns in the data, then we have rows to observations and each variable observation pair identifies a value, which can be missing, but most of the cases want to have data for those cases. In latent variables we assume that the variable exists, but we don't have data for it. For example, if we study intelligence we assume that intelligence exists, but we don't have data for intelligence, we have data for some imperfect measures of intelligence. Normally we want to say something about these latent variables by analyzing the observed variables. So we assume typically that the latent variables exist and they cause variation in the observed variables and that variation in the observed variables allows us to say something about latent variables that are unobserved sources of various. Latent variables are also sometimes referred to as random effects. So this builds a link between a factor analysis and structural ecosystem models and econometrics, which uses the random effects term. These latent variables, we don't have the case values, they can be estimated, but typically the estimation of case values cannot be done perfectly, so during estimating a statistical model we actually don't use the case values. Instead we assume a distribution for a variable, typically we assume that the distribution is normal and then we estimate the variance of that normal distribution assuming that the mean is zero. Let's take a look at different models that include latent variables. The first thing that comes to mind is the factor analysis model. So we assume that we have our three indicators, growth willingness indicators, and then we fit a factor analysis model, where we say that all these indicators are affected by one common factor, which is presented by a latent variable in the model, and then we want to estimate how much of the variation in the indicators can be attributed to this one unobserved common source. Another commonly used model, which contains latent variables or actually a latent variable, is the normal Richardson analysis. So we can see here that error term is not observed, it is a latent variable. So whenever a variable is not observed, then we don't have values for it, then it's latent. So that's the definition. If it has case values, then it's not the latent variable, but it's an observed variable. If we take some of these X indicators, we have a specific value for its company, that sum is an observed variable, not a latent variable. Another useful latent variable model is the multi-level model. So in this multi-level model we have our three latent variables, the error term, and then we have this level two effect. So we have the random interest and the random slope. They are latent variables. We estimate their variances, but we don't estimate our specific case values. And this is the equation in the mixed model format. This same equation can also be expressed as a path diagram, in which case we would be talking about a latent growth model. We have a latent variable, intercept latent variable, slope, and this is actually exactly the same model as the one before, but instead it is shown as a path diagram. So these latent variables, even if you don't do factor analysis, they are very commonly used in different estimation techniques and modelling approaches. For example, you can take the HECMAN Selection model and it's actually a latent variable model as well. So there's a latent variable, an omitted variable, that we estimate during the analysis process, that affects both wages and whether a person chooses to work. This is of course HECMAN's classic example.