 Miten tietysti yksi tennös- ja yksi tennös- ja yksi tennös-päivät? On couple of different techniques for figuring out. The set of techniques that I'm talking about applies to our covariance structure models. In these models, all relationships are linear. So there are no interactions, no exponential relationships, no U shapes. And also all observations are independent. And if these two assumptions hold, Täältä voimme esittää koulutuksella, jotta koulutuksessa on koulutuksessa. Ja myös koulutuksessa on koulutuksessa, jotta koulutuksessa on koulutuksessa. Se on tärkeää, että meidän on täydellä informaatiossa. Jos olemme tietysti tämän, niin meidän on myös täydellä. Tämä on tärkeää koulutuksessa, jotta se on hyvää yksityisestä yksityisestä. Sitten on yksi asia, että voimme löydää yksityisen valotus perustelua järjestelmällä. Tämä voi olla yksi asia, jossa on yksi asia, jossa on yksi asia, jossa on yksi asia, jossa on yksityisellä järjestelmällä ja yksi asia, jossa on yksi asia. Ja ei ole aina kutenkaan tiedä, mitä valotus järjestelmällä ja järjestelmällä ovat, mutta vain käyttää informaatiota kovariensiä tai se on semmoinen matka, jossa ei ole kallogorretaisuudesta. Mä näin yksi asia järjestelmällä. Tämä on yksi asia, mutta se on tärkeää, jos valotus järjestelmällä on järjestelmällä. Ja se on myös hyvin käydä, jos valotus on järjestelmällä. Yksi asia, jossa on yksityisen valotus, on järjestelmällä. Järjestelmällä on järjestelmällä, tai maan av cru bodies Solutions, että ovat ol incurienorttia, untuurin että pesticides and applying these rules is a lot more practical solution for establishing identification. S için tavallaan se ei ole enää wnキ salaan sitten, kun ei力un taloja, ja se on tulotteen destail vaginaja, eleg epidemisten antar doppeltuista, joissaeles altaudustaa videoissa. video. My example comes from from book by Paxton and co-authors. I think this is the best source on mathematical identification of covariant structure models. So they have a chapter on identification and they talk about recursive and non-recursive models. They provide examples and it's a math but it's fairly readable if you understand covariance algebra. If you don't then understanding the identification techniques is going to be difficult anyway. So let's go to our example. So we want to prove identification of this model. Our model is a full media sum model x influences m which influences y and there is no end of gen 80 so that the error terms are uncorrelated. So we know based on the rule that all recursive models are identified that they should be identified but we can also prove it and that it's not that difficult to do. How we proceed with proving that this is identified is that we first express these variances and covariances and sample as functions of these these model parameters that we are estimating that we don't know. So this is the information that we know from the sample and this is the information that we need to estimate. So we need to understand if the estimation problem is even principle possible to solve. So we'll start by writing the easy easy part. So the variance of x is simply the variance of x. It doesn't depend on the model because x is exogenous. Then the variance of m is beta x square times variance of x plus these variance of this error term here and the variance of y is calculated the same way. We could of course express the variance of y as a function of variance of x and variance of this error term here and that error term there but it's a lot simpler to express it as a function of variance of m which we know from the from the population or we would know if we had the data. Then the covariance is covariance between x and m is simply a variance of x times beta x, covariance x to y. So we take the variance of x, we multiply by beta x, we multiply by beta m and that gives us the covariance between x and y. So we multiply two paths and the original variance and this covariance between m and y is calculated the same way. So now we have this set of equations based on tracing rules and then we start to solve for these unknown parameters using these equations and high school math tells us that this is an over determined system because we have six equations with five unknowns and the high school math also tells us that there may not be a solution for this but that's a topic of another video. We can start solving this equation so let's start by looking at the recursion coefficients first. So we can solve for beta x using this first equation here, we can solve for beta m for example using the third equation here, we can solve for variance of x, well it does not need to be solved, it's here, it's a simple variance, we can solve for the error term of u using this 50 equation here and the information about beta x2 that we have solved before. So the idea of showing that, the identification using this technique is that after you have shown that one of the parameters has been identified like we identified this beta x first, then we can use that identified parameter to show that another parameter is identified. So here we have beta x square, square can be used here because we have identified it before and variance of y, u y is calculated the same way. So we were able to solve all these parameters from these covariances and covariances and we also noticed that there is this one covariance that we will use. So what about this? What do we do with this covariance? So we had excess information, so this gives us one degree of freedom, so we had one excess unit of information, we had one covariance that was not required for estimating in the model. What we can use this covariance for is to solve for this coefficient beta m because it contains beta m in a different way. We could also solve for beta x, it doesn't matter which one we solve. If we have decrease of freedom excess of zero or positive decrease of freedom, then it means that at least one of the model parameters can be solved in at least two different ways. So we have two different ways of solving for beta m and let's write them here. So we have beta m equals covarance x y divided by covarance x m and covarance my divided by variance of m. So we can actually use this for testing the model. So we can write that these two are equal or we can write that their difference is zero and the latter way is the most useful way. So we can calculate if covarance x y divided by x m minus covarance my divided by variance of m is actually zero. We can calculate this quantity and then we can compare that against there are we can calculate this from the data and then compare it against zero. If our calculation here does not equal zero in the population, then it means that the model is misspecified. So in misspecified model solving for one parameter using two different ways would lead to do different solutions, then we know that the model is not correct for the data. In small samples this will this difference will never be exactly zero and we would use that information and we would ask the question of whether that difference from zero can be attributed to chance only and this is just one constraint so we have one degree of freedom and we would be testing that covarance constraint using the chi-square test with one degree of freedom. If we have more than one of these constraints things that can be solved in multiple different ways then we will use the chi-square with more degrees of freedom like if we have three different tests we use chi-square distribution of three with three degrees of freedom for testing if those constraints hold in there in the population. So in practice this is not a practical solution for establishing identification practice but there are a couple of good rules of thumb and heuristics. First if your decrease of freedom is negative then the model is going to be unidentified no matter what. Some part of the model can be identified but generally our estimating model with negative decrease of freedom is not a good idea because at least the beginner would not have an easy way of knowing which parameters are identified and which are not. Recurssing models are always identified that's a good thing to know so if there are no feedback loops no correlations between the error terms then the model will be always identified and there are various heuristics that can be applied that are explained by explaining another video that applied to non recursive models and particularly rank and order conditions are something that many books and structural equations model explain. There are also empirical checks so it may be possible to establish identification but in some scenarios you might not be sure if your model is identified and this identification is something that you generally should check before data collection because if you collect data and then you realize that your model that you wanted to estimate is not identified then our estimation with that data or any other data would be futile so that's like worthless exercise. So how do you know that the model is under identified or non identified based on empirical checks? First some statistical software provide you checks so there are checks programmed into software for example M plus that's a pretty good job at telling you whether when the model is identified or not and also pointing out which parameter it thinks is not identified. So if your software gives you warnings use it pay attention. Another indication of non identification is missing or extremely large standard errors sometimes confidence intervals can be missing or but if your results are really weird that is an indication of non identification. It can be also indication of some other problems so when you have a weird result then you need to start doing diagnostics and to understand what's going on. Establishing identification is one of the first things that you should use to check if your estimation does not seem to go through smoothly and then there are empirical checks and you can try estimating the model with different starting values. So the estimation of these models is an iterative procedure so the computer guesses some starting values typically are using some kind of instrument of variable technique and then it then it starts to optimize those starting values to make the model fit better and you can also try your own starting values. If two different starting values lead you to two different solutions for the model parameters then the model is not identified at least in this class of models. There are other classes of models where there may be scenarios where two different set of starting values are converged to the two different solutions but that would not immediately imply identification problem. So identification problem in this class of models can be easily checked by running the model with different starting values. If you choose five different sets and you always get to the same estimates then you can be pretty guaranteed that pretty certain that the model is actually identified. Then you can re-estimate the model with the model implied covariance matrix so you can ask the computer to generate their covariance matrix that is implied by the model and estimate it again if you get the same result then that is an indication that the model may be identified. If you get a different result then for sure the model is not identified. Final strategy is to re-estimate the model with simulated data and this is a good strategy because you actually don't need to have the data at hand so this can be done before data collection. So if you're not sure about whether the model is identified or not generate the data set from a large sample like 100 000 observations or a million observations something like that where you know that your model holds then estimate the model if you get the correct estimates then the model probably is identified. If not then the model is not identified for sure. Sometimes generating data from the correct model may be difficult because generating random numbers for for example models of feedback loop is challenging. In that case you can just generate any data any correlated data and then apply this estimate model with different starting value strategy. So these are some empirical strategies for establishing identification. Let's take a look at that non-identified model and what it looks like in a statistical software so this is data and we are estimating a structural ecosystem model with SCM command. We have two variables X and Y and we are estimating the effect of X on Y and Y on X that of course is not identified because we have just one covariance and we try to estimate two relationships with the variables. You cannot estimate two things from one thing but nevertheless data provides estimates. So the problem with data here is that data does not do a good job in warning about identification problems. If you are to estimate this same model with LaVan in R or M+, you would get a warning that the model is not identified. Say that does not give you that unfortunately. But there are a couple of signs that we can look at that tell us that this is actually a really problematic analysis that we shouldn't trust at all. First of all the negative increase of freedom so we have chi-square with minus one decrease of freedom of course that can be calculated because it's not defined but that already indicates that this model has serious problems. There are two other indications. One is that the standard errors are very large here so that means the computer tells that there's this great uncertainty about estimates and that's a bit of an understatement because we don't know about the correct estimates at all given that the model is not identified. And the final thing that is the easiest spot is that the confidence intervals are missing here. So sometimes standard errors are missing, sometimes confidence intervals are missing. And now the problem of course is that if we have a researcher who has heard about the fact that structural equation model can be used to estimate these two directional relationships and a researcher does not understand much about identification, this model still gives the results that the researcher is looking at. So typically we want to know the p-values for the structural paths x, y, y, x and we get those p-values. So we would say that they are not significant, they are not related but that would be an incorrect conclusion because these x and y are actually related because they are actually highly correlated in the data. But it is very easy to overlook these identification issues and because of that you need to pay attention to the decrease of freedom, you need to pay attention to missing results even if you're not using them and you need to pay attention to our standard errors even if you were looking at the p-values eventually.