 Cross-lack panel model is a simple and very commonly used dynamic model. What makes this model dynamic is that there are effects of variables on themselves over time. Let's take a look at what the basic cross-lack panel model does. So if we are interested in estimating the effects of ZO's gender on ROA, we typically would like to collect data over time, and then we would check the effects of ZO's gender on ROA from time one to time two, controlling for past ROA. And this would give us the effect of ZO's gender on ROA under certain assumptions. However, this data set would also allow us to check if there is reverse causality. So we could regress ZO's gender on ROA as well, and this is the idea of cross-lack panel model. So we are testing whether X is the cause of Y, or Y is the cause of X, or whether X and Y are simply correlated initially for reasons that are outside the model, and they tend to persist over time. So we model persistence. This is the autoregressive path, AR path, this is the cross-lack path, CL path. So we have these lags, and they cross, and we have lags that auto-correlate. This model is very commonly used. Let's take a look at the example. This example is of Major Inspector, and they measured work stressors and counterproductive work behavior at five time points, two months apart, and then they were testing whether stressors caused counterproductive work behavior or the other way around. They also had this two-month interval, which allowed us to test different lags, but we'll just look at the two-month lag here. We could also specify the model using four-month lag by leaving out the second and fourth time observation, or we could do a half a year lag or 10 months lag, depending on what is the reason question of interest. So these models allow us to check whether it's the X that causes Y, or Y that causes X, and also how long does it take for X to cause Y. This model of repeated observations causes two concerns. First, if we have repeated observations, then are our standard errors consistent? We know that in normal regression analysis, if we have repeated observations, then standard errors will be underestimated because of unobserted originality. This is not generally a concern in cross-lag panel models, because we estimate these models from the wide format data. So instead of having repeated observations for each time period, we have repeated variables for each time period. So we have five different variables for counterproductive work behavior and five different variables for work stressors, instead of having five different observations for each individual. So standard errors are not of concern because of unobserted originality. What is of concern is what we are actually estimating. Are we estimating the within effect, the between effect, the contextual effect or the population average effect? It turns out that this model actually estimates the population average effect, which will be the same as the within effect if contextual effects are zero. Can we always assume that the contextual effects are zero? We can't. And this is a reason why these kind of models have been criticized in the recent past. So there's this excellent paper by Hamaker and co-authors that argue that instead of having this cross-lag model, where we simply model differences, we model effects over time, we should also model persistence over time. So we should include unobserted originality in some way into the model. And what they propose is this cross-lag panel model with random intercept RICLPM. And basically they have this normal cross-lag model. I'll take a look at this in detail in a few slides from now. And they will add random intercepts. These random intercepts are allowed to be correlated. So this is a correlated random intercept model and we know from econometrics that the correlated random intercept model estimates the within effect. So this model, when we control for these unobserved differences that are on the individual level instead of on the observation level, it allows us to kind of like parcel out the differences between individuals and only focus on the within effect. This model, what it technically kind of does, it cluster means enters the data and then we estimate the panel model. In practice, we can do cluster means entering for reasons that are beyond this video but we can model the within effect by using this kind of model configuration. There is also another model that does the same, that estimates the within effect. So this is a variant of the same model presented by Cypher and co-authors. So this is the normal cross-lag model. So we have x cos y, y cos x and those go over time. There is also persistence over time and we have these late random intercepts affecting x and affecting y. This is a simpler specification than Hamaker specification. So let's take a look at the difference between these models. We'll ignore the means, the triangles for now because they could be easily added to this model but that's not the main difference. The difference in Hamaker's model is that Hamaker introduces latent variables for each observed variable and then the path model or the cross-lag model is specified for those latent variables. The random intercept is specified for the observed variables. What Hamaker's model basically does is that it models there are mean differences independently of this effect. So it kind of like centers the data within cluster and then estimates the cross-lag model. This Cypher's model in a way models the means of these y-variables and means of these x-variables as predictors of y and x-variables. So this is kind of like the correlated random effects uproads which gives you the contextual effect. So both of these cross-lag paths in both models are the within effects. So where is the between effect? Where is the contextual effect? Well the correlation between these two random effects here or latent variables here corresponds to the between correlation. If we switch that correlation to be a regression path then it gives us the between effect. Here if we switch this correlation to be a regression path it gives us the contextual effect. So which model should we apply? If we are only interested in the within effect we should go with Cypher's approach without these added latent variables because it is simpler to specify. So less specification, less room for error. If we are interested in, there are understanding the between effect or the contextual effect then depending on which effect we are interested in we choose the modeling approach based on that. So typically the contextual effect would be more interesting because it has more easier causal interpretation than the between effect. So to summarize, when would these kind of models be useful? We can model the directionality and persistence of effects. We have the autocorrelation coefficients from y to y from x to x and we have the cross-lag coefficients from x to y and from y to x and we have random intercepts which make this a correlated random effect model. These random intercepts are unfortunately not commonly used but they pretty much would always be used because we really cannot assume away constant between individual or between company differences. These models of course can be combined with other modeling features. So this is a dynamic model so with effects over time we can of course add effects of time to the same model. For example, if we combine this model with a latent change model then we would have autoregressive latent trend or ALT model and these different models are simply combinations of different effects in certain different ways. But this basic cross-lag model with random intercepts provide a good foundation on which to start building more complicated models.