 Cross-level interactions are sometimes presented as an advanced topic in multi-level modeling. However, cross-level interactions are actually pretty simple once you understand a couple of basics and how these models relate to normal interaction model and the random slope model. Let's use these models as our starting point. So we have the model from Holcomb that I've used in other videos and the model has effects of year, so basically a time trend and some dependent variable, then we say that the trend depends on firm, it depends on industry, it varies in slope, it varies in intercept. So what's the starting point varies, to which direction the trend goes varies and so this is just different regression lines for different companies of time. Let's expand this to our cross-level interaction model. For simplicity, I'm just going to be focusing on the first two levels. So we have year observations level one, firm observations level two. To understand what the cross-level interaction is, we need to understand what are level one variables, what are level two variables. So level one variable is a variable that varies within level one. So if we have something that varies within companies, for example profitability number of employees, how many new products they launch and so on, then that is a level one variable. If we have something that does not vary within companies, it only varies between companies, then it's a level two variable. For example founding year where the company was founded, in some cases industry, that kind of variables are level two variables. And now a cross-level interaction model is a model where the effect of a level one variable depends on a level two variable. Of course if we have more levels we could have interactions between level one and level three and so on. To understand what it means that variables from different levels interact and how do we model that in the multi-level modeling framework, we need to understand that interaction means that the effect of one variable on the dependent variable depends on the value of another variable. So here we have the level one regression coefficient and to do a cross-level interaction model, we would simply say that this level one interaction coefficient depends on some level two variable. But as explained in a previous video, these slope and intercepts, they should always be allowed to be correlated. So it's unrealistic to assume that if a variable affects the slope, it would not affect the intercept. So in practice when you do a cross-level interaction, then you need to add the variable as a predictor to the slow intercept as well. So you always predict the intercept and the slope if you want to predict the slope. Now when we write this equation in the mixed format, we can see that the random part is as before so we have the random slope, we have the random intercept and then we have the error term of the model and this is simply a normal interaction model. So nothing special here, the level two variable enters separately, level two variable and a year, they are multiplied together and each of those receives a regression coefficient. So this is just a normal interaction model that we test for moderation and this is just a normal random slope model random part. So nothing special about that. So how do we then work with these cross-level interactions? Well this is nothing more than an interaction model with random slopes. So you can just as well define the interaction in the level one model and don't care about the level two models with your level two variable. The interpretation of these models is a bit more complicated. Most of the time when you have these extra seats or built in procedures for doing an interaction graph of a normal interaction model or regression model, they don't actually support multi-level models. So you need to do some predictions by hand but that's not very complicated to do. So there are basically two strategies that you can use to model, to visualize these kind of modeling results. One is to take the spaghetti plot that I discussed in another video as a starting point and then introduce groups to that spaghetti plot. For example take different values, take high value, low value and medium value of the level two variable and then calculate predicted lines and plot those using different colors depending on the value of that level two variable. So that's one strategy. One strategy is to use a marginal prediction plot where you plot just the fixed part and then you use the bands that you would normally use for confidence intervals to indicate where the predicted lines would go based on the predicted randomity. So the only thing that is complicated here is the plotting but that's more of a statistical software used kind of complexity instead of a conceptual complexity. Conceptually multi-level or cross-level interactions are very simple. They are just plain interaction models with random slopes.