 Multi-level data are data that have different levels. For example, we have industries, we have companies in industries, and then we have repeated observations of those companies. This is an example of a nested structure. But there is also another kind of data structure that can be used in multi-level modeling, which is the cross structure. Let's take a look at first what is the nested structure. The idea of nested structure is that we have these three levels. For example, we have years nested in firms nested in industries. So we make two important assumptions. First we make the assumption that each level one observation belongs to only one level two unit. And each level two unit belongs to only level one, only one level three unit. So these form kind of like a hierarchy. There is many to one correspondence between these levels. And another important assumption in these models, particularly for estimation, is that the firms in one industry are independent of the firms in another industry. So after controlling for possible common level three effect, then the level two effects are independent, or level one effects within level two effects are independent. So these are two important assumptions in nested data set, which is kind of like a tree structure. So you would have country, which has industries, which have firms, which have repeated observations and so on. But this kind of data structure does not fit well to all possible research questions. So consider, for example, this scenario. We have venture capitalists that make investments in startups. A venture capitalist typically makes multiple investments. So you have a fund you invested in 10, 20 companies, however you get, and depending on the size of the fund. And then you have startups. But the startups typically also, if it's a successful startup, it tries to get multiple investments. So how would you structure this kind of data? How would you structure the investments or startups or venture capitalists? Each startup makes multiple investments. So you can't say that gets multiple investments is venture capitalists makes multiple investments. So there is no clear structure of hierarchy. What we would say is that these are crossed. So we would say that the investments are crossed in startups and venture capitalists. So that each investment has one venture capitalist and one startup that it invested in. But each venture capitalist can have multiple investments. Each startup can have multiple investments. So instead of looking at the data as a tree like structure, like a hierarchy, we will be looking at the data like a cross tabulation of two dimensions. And each cell has one investor, one investment target. And each investment target, its investor can have multiple investments. So how do we deal with this kind of problems in multi-level modeling? This is in a way simple, in a way complicated. Concepts like this simple to distinguish when you have a nested structure and when you have a cross structure. So can you put the observational units into a hierarchy or not? If no, then you have cross structure. If they are a hierarchy so that there is always many to one relationships and they can be used to form a tree, then you have a nested structure. The independent of random effects is assumed in both cases. So if you have two levels, you have investors, you have startups, then the investor random effects and startup random effects are assumed to be independent. And so this independence assumption is assumed for both cases. The cross models are computationally more challenging. So it's fairly straightforward to specify in your software that instead of nested structure, which is the default, you want to have a cross structure for some of the random effects and the computer will do that for you. The calculation is more complicated than I will not get into the details, but it just takes a longer time to estimate. If you have a nested model that runs for 10 seconds, then running the same with cross, depending of course on your software and your computer, could take like 10 minutes or something like that. So there can be several orders of magnitude of more computation, depending on how the computation has been implemented, depending on the sample size and many other factors. Cross models are less common. Perhaps this is because many things in nature and social life form hierarchies and these kind of cross structures are less common. Or perhaps it is because cross models can be often avoided by using fixed effects instead of one of the random effects. So for example if we have let's say 500 startups and 15 investors, instead of estimating a random effect for the investors, we could apply fixed effect for the investors and then use a random effect for the startups and that would take care of the problem. We would no longer have a cross structure. So sometimes you can get away with having cross structure by instead of having a random effect, replacing that with fixed effect or basically dummies for that class of observations. Finally, it's possible to combine cross and nested levels. So for example you could have venture capitalists that specialize in industries. They only invest in one industry and startup belongs to one industry. So you could have these industries that have separate venture capitalists each, separate startups each. So you would have industry as your level three and then venture capitalists and startup would be nested as levels one and two. There's really no difference in crossing. They would be crossed as level one and two and then the actual investments would be the lowest level of services. So it's possible to do these cross defects. They're very common and they're integrated the same way. The only practical issue with these cross random effects models is that they just take a longer time to estimate.