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Okay, so want to take a lot at, at a structural model and fitting structural

Â models if, of network formation, I, and that combine aspects of both strategic

Â formation and chance meetings. And, the idea here is that, you know we

Â can build these models to explore the fact that in a lot of settings there's

Â going to be some choice involved but also some chance involved and we might want to

Â estimate some things like relative roles...

Â And you know, the the random models can be too extreme, the strategic models can

Â be too extreme. We seen the beginnings in terms of the

Â exponential random graph models of ways to combine some of these things but we

Â can also in particular instances fit models that are more precise to the

Â setting involved and more directed at asking a very specific question.

Â And so for instance, let's ask a question of, when we see homophily how much of

Â that was due to the choices of the individuals and how much of that was due

Â to the fact that you're more likely just to be meeting individuals of your own

Â type rather than choosing to interact with individuals of your own type...

Â So if we want to ask a question like that.

Â Can we build a simple model to address that?

Â And so here, what I want to emphasize is really the techniques for doing this

Â rather than a specific model. So this is going to be a very specific

Â and stylus model. But what the what I want to do is just

Â illustrate that you can use, you can do similar things to where you build what

Â you think is the right model for a particular application.

Â And then use that to generate networks. Look at the networks that come out.

Â Try and match them up with the data and that will allow you to fit parameters to

Â the model that best match the data and then do statistical tests to see whether,

Â you know, certain things are really going on.

Â how much choice is really going on. how much chance is really there.

Â How much noise is in the data and so on, so forth.

Â So that's the idea here. And so I want to emphasis basically an

Â approach rather then taking so seriously the specifics of this particular model.

Â It's more as an illustatration or an example then as as to be taken seriously

Â as the model. So in terms of application of homophily

Â Let's suppose that we've got two types of, two groups, Group A and Group B and

Â they form fewer cross say race relationships than would be expected

Â given their population mix. So if we go back to our add health data

Â and look at one of those high schools. And we see that, then we see a

Â segregation by race. We could ask is, is this due to

Â structure? So maybe they just don't meet each other

Â very often. They don't meet each other very often

Â because In the school, there's certain kinds of structural patterns in terms of

Â the way the courses are organized or the way that people will take extra circular

Â that don't allow for many meetings between different races, or is it due

Â maybe to the preferences of group A or the preferences of group B or both of

Â their preferences and so forth... So can we begin to sort these things out.

Â So, I'm going to just take a look at, at a couple of papers the techniques from a

Â couple of papers that are with Sergio Quarini and Paulo Pin, from nine 2009 and

Â 10. And what we'll do is just, we'll specify

Â how much utility a given individual gets as a function of the friendships they

Â have. And then we'll allow a meeting process

Â that has randomness in terms of who you're going to meet.

Â And we'll allow this, both the utilities and the meeting process to depend on your

Â type, so in this case, say your race or your gender, or your age, or your

Â profession. Whatever, whatever it might be.

Â and then begin to see what comes out of that and, and try and match up the

Â parameters to the to the data. Okay, so let me say a little bit about

Â the idea here and, and, you know, when we're, when we're thinking about trying

Â to estimate strategic formation models Generally, what we end up seeing is, is

Â the result of some choices that were made.

Â And there's something that's known as revealed preference theory in economics.

Â Which refers to the fact that you know, we might see say a consumer buying

Â certain products. And then, based on the fact that they

Â bought one product at a given price and not another product at a given price.

Â We begin to try and infer what their preferences over different product

Â attributes are. So what do they really want if they ended

Â up buying something and not buying something else?

Â Okay, and so here what we'll do is we'll be basically inferring preferences by

Â saying, okay, this person formed these friendships and not the other, another

Â set of friendships. That gives us some insight into what

Â their preferences might be. Why did they form these friendships and

Â not those? Well, it tells us something about the,

Â what they preferred to form in terms of friendships now again that could be due

Â to what they have available. And just as in consumer theory you might

Â have a budget which says okay look these were the things I could've afforded, and

Â I bought this and not that. here what we're going to have to do is

Â just sort of infer what are the, what is the rate at which you had opportunities

Â to form different types of friendships? And so, the chance part is going to be

Â fitting what were the opportunities that were coming along and then what choices

Â were made as a function of those, and that'll give us information about what's

Â actually the, the preferences and, and what were, were the relative

Â opportunities that they had. So that's the idea.

Â One thing to emphasize here is this gives us, say, a different kind of look at

Â things than just direct surveys. So you might, for instance, ask people,

Â what's your attitude on race or would you like to form friendships across races and

Â so forth. And the difficulty with asking people

Â directly is that people often answer in ways that aren't necessarily congruent

Â with the choices that they make. So this takes seriously what did you

Â actually do, not what you would say on a survey.

Â And sometimes there can be differences about this and so this is a different way

Â of sort of measuring attitudes towards, you know things like race, or gender, or

Â age, or whatever it might be in that particular context.

Â Okay, so a simple model. what we're going to have is some set of

Â types 1 through k, so this might be ethnicity, it might be the, the age of

Â the individual, it might be a combination of their age, their religion, their

Â gender, etc. and what we'll have is a very simple

Â model in terms of the preferences that people have.

Â So, this is going to be a simple independent link formation model.

Â So, it's going to be simple in that dimension.

Â It's not going to be trying to recreate richer parts of the network but it's

Â going to allow to separate out some of the preference aspects from, from some

Â other aspects. And so what people value, is they care

Â about how many same-type friendships they have, and how many different-type

Â friendships they have. So, really simple model.

Â You just care about how many friendships do I have with t-, people that look like

Â me, how many friendships that I have of people that are of a different type, and

Â I get some benefit from just that. Okay, so very, the simplest possible

Â formulation you can imagine. And in particular, what you get in terms

Â of utility, is then some number of, of same and different type friendships

Â weighted by a parameter, gamma i, where gamma i is capturing how much do you

Â weight a different friendship compared to a same type friendship, okay?

Â So, it's a preference bias. If this was 1, then all I care about is

Â the total number of friendships. I don't care what their mix is.

Â If this is bigger than 1, then I actually care for diversity.

Â I care more to have friendships with other types than same types.

Â If it's less than 1, then I get a higher benefit from same-type friendships than

Â different-type friendships, right? So.

Â Gama i is going to be the critical perimeter.

Â In terms of representing preference bias. And then we also have this other

Â perimeter. Alpha.

Â And what is Alpha going to keep track of. Alpha, is going to be generally less than

Â one. Is going to be some diminishing returns

Â to friendships. So my first friendship might be very

Â valuable to me. My second one additional value and so

Â forth. By the time I get to my 10th 12th

Â etcetera these friendships are becoming less valuable and so the fact that alpha

Â might be less than one would give a concave function.

Â So as you look at at the utility as a function of total numbers the utility's

Â going to tend to be concave if alpha is less than 1.

Â So we've got a situation where, as alpha's less than 1, then we've got

Â curvature in that utility function. Okay.

Â so let's let t i be the total number of, of friendships that we're forming.

Â And, so basically, people are socializing, they have an opportunity to

Â form friendships. They meet people of different types, and,

Â in this model let's let qi be the fraction of own types that you're going

Â to meet and be able to form friends with. And then 1 minus qi is the relative

Â number of other types that you're going to form.

Â And so if you spend, if TI is the total number of friends that you form, then the

Â relative number, this is going to be your S size, is going to be, the fraction that

Â were of same type and, the DI is going to be the fraction that were different type,

Â times your total friendship. And so here in this model is a very

Â simple model I just have opportunities coming and the cost is just going to be

Â its going to be costly for me to form some number of friendships I'll cut off

Â that total number but then the mix I get is just going to depend on the relative

Â meeting rate. So I, I meet people at some rates, and I

Â take whatever friendships come, but it's expensive for me to form friendships, and

Â so after some time period I stop socializing or trying to find new

Â friends. Okay?

Â So ti is just going to maximize, it's going to be a maximizer of this overall

Â utility function. Where you've got, same type firend,

Â different type friend, and so forth. And the rate at which the come is Qi for

Â same type, 1 minus Qi for different types.

Â And that's going to be coming out of the random part of the process.

Â And right now, then what we could do, is say, we can figure out if we knew what

Â gamma was, and was alpha was, and what Q is, and C, and so forth We could solve

Â this function and say, how many total friendships would a given individual like

Â to have? And how would that depend on those

Â relative parameters, okay? Okay, so to maximizes this function.

Â If you solve that, you can get an expression for what ti is in terms of the

Â over the other parameters. So, maximizing that function.

Â take the derivative with respect to TI, set it equal to zero.

Â Alpha's less than one. This is necessary and sufficient for the

Â solution. and then we'll also add some noise to the

Â given decision. So it might be that a given individual

Â for whatever reason has more or fewer opportunities or more or less values.

Â So they're going to, we're just going to add noise in terms of the, the

Â friendships that have given individual forms so a person a of type i is going to

Â have an extra error term, epsilon a and so the total number of friendships of any

Â given individual forms is just going to be some noisy thing about this solution.

Â Okay. So very simple model in terms of the the

Â formation. But now we can see if we write down a

Â simple model of How much utility you get from some aspect of, of the network.

Â we maximize that. And we get a solution for what.

Â How many in this case. What degree.

Â So we can think of this really as the degree of aging eye.

Â This is the degree that they would like to have.

Â In terms of this model. and then what they end up with is some

Â noisy variation on what they would like to have.

Â given the parameters of the model. Okay so how do we actually identity the

Â parameters from the data. so what we can do is in the data we'll

Â actually observe the ti's the tai's so we'll see, how many we'll use the add

Â health data. So when we look at the actual networks of

Â friendships in these high schools we can see how many friendships did each

Â individual form. And so we observe this directly in the

Â data, and that's going to vary with the qi's, so as a function of qi, the

Â function of the alphas, the gamma i and so forth.

Â That gives us a tai. And so one thing to notice, is that when

Â we look at this expression for tai, This is increasing in q i if gamma is less

Â than one. Right?

Â So, if gamma is less than one, then you've got a plus one q i and then you've

Â got minus gamma. So, you've got q i Times 1 minus gamma i,

Â in here. And so if gamma i is less than 1, then

Â you've got a positive expression for t a, t is a function of q.

Â So, more of my if the fraction of people I'm meeting is more of my own type, I

Â should form more friendships And so that's what's going to allow us to begin

Â to fit what gamma i is, right? So the idea is t i should be a function

Â of q i, and how quickly it varies with q i is going to be dependent on what gamma

Â i is. Okay?

Â And, in particular, if you actually look, this is a picture of the add health data.

Â So these high schools here there are 84 schools.

Â And each dot here represents a certain race.

Â Group within a particular school. So for instance, this dot here is a group

Â of white students that formed. So it was a particular school, and in

Â that school the white students formed about a little over between 60 and 65% of

Â the population. this school right here, this is a, a

Â group of black students in a high school where their groups' size is a fraction of

Â the school was just below, between 20 and 30 percent, a little closer to 30 percent

Â and this then tells us on average how many friendships did they form?

Â They formed on average about eight friendships.

Â This group formed on average you know about three and a little bit of change,

Â and so forth. What we see here is that indeed if you do

Â just the slope between here, you see that the slope is 2.3.

Â So, there's an increase as a function of your group size.

Â So the more prevalent your group is in the population ,that's going to lead to

Â higher qis. And indeed, we see that there's a higher

Â group of friends as, higher friendship, as a function of the the size of the

Â group. So the easier it is to meet your own type

Â the more friends different groups are forming, and so we will be able to

Â actually identify that gamma perimeter from this data, and in particular when

Â you look at this thing, you know the slope here is 2.3, the T statistic on

Â that is 7.3, so... You're you're quite a number of standard

Â deviations away from from zero so so we're actually seeing a highly

Â significant slope here. So we will be able to identify the fact

Â that groups that have higher proportions are forming more friendships which would

Â indicate that they're getting higher utility under this model.

Â and we can then estimate what the gammas are based on that.

Â And in particular I'll just sort of you know, show you the best fit lines.

Â If you look at the best fit lines for different parameters, you'll for

Â different races you'll end up seeing different slopes, and that'll allow us to

Â back out gamma-ise the gammas for different Races, because each one of them

Â has, having a different relationship between how big their size is, their

Â group size, and then how many friendships they are forming.

Â Okay. So the last part of the puzzle in terms

Â of figuring out the randomness in this kind of model is where do the Qis come

Â from. So we've got the, how many friendships

Â each... Person would want to form as a function

Â of the parameters of the utility function and the rate at which they meet different

Â individuals, but now we want to ask, what's the rate at which they meet

Â different individuals? Okay, and the important thing here is

Â that the rate at which they're going to meet different individuals is going to

Â depend on the decisions of the other agents.

Â Agents, okay? So if everybody was trying to form the

Â same number of friendships, and we're just sort of mixing in the population,

Â then if my group formed 30% of the population and some other group formed

Â 70% of the population, then I would meet my own group at, at a rate 30%.

Â And I would meet agents of different types at a rate 70%.

Â But if, the other types, imagine if the other types are actually trying to form,

Â they form many more friendships. They're spending more time circulating

Â and mixing, then they're going to be easier to meet and my type is going to be

Â relatively less likely to meet, and so it's not just...

Â a function of the relative sizes, it's also a function of how many friendships

Â different groups are trying to form. And so we need to solve this overall as

Â an equilibrium given that the, the, that the t's are going to be determined by

Â these relative rating, the q's, and the q's are going to be determined also by

Â the actual decisions of the agents. So in particular let's think of the

Â meeting process and we'll think of this as a giant party.

Â So we can think of this like a cocktail party.

Â So let's think of a different given individual say is a green agent.

Â And this green agent is bouncing around in a party where there are green agents

Â and red agents. And so what's going to happen.

Â imagine that the incoming proportion of reds is 80%.

Â And green's it is 20%. But if the red's spend more time, trying

Â to form friendships. And are generally forming more

Â friendships. It's going to be easier to form

Â friendships with red's then green's. And so even though it say let's say .8.

Â 0.2 coming in. The mixture in here could be, say, 90%

Â 10%, or, or even more skewed than that if the reds are spending, say, twice as much

Â time in the, in this party than, than the greens are.

Â So, the rate at which they come in and, and go out, is not necessarily going to

Â be the same as what the relative stock of people is, if the greens are exiting much

Â more rapidly than the reds are. So, as we go through this process then,

Â you know, this group, given green node bounces in to somebody, meets one

Â friendship, meets two. Three, four, so it's got three red

Â friends and one green friend, and it decides, okay, that's enough, I'm

Â satiated. You know, I formed four friendships, and

Â that's enough for me. And then it decides to exit.

Â a red might find this to be, if, if gamma i is less than 1, then reds are meeting

Â reds at a higher rate. They might want to stay longer.

Â And that's basically the idea of the model.

Â Ok? So we've got this q i is the rate at

Â which i meets i. one minus q i the rate at which you meet

Â the different types and the way in which this is going to be modeled is the q i

Â the rate at which you meet your own type in terms of this process Is going to be

Â dependent on the stock, how many of those individuals are actually in a room.

Â But will also allow this to be biased, so that even when I'm in the room, it might

Â be that that I'm biased in terms of meeting my own type.

Â So maybe I'm in this large room, but I actually look for greens and try and find

Â greens. In which case I'm going to meet greens at

Â a faster rate than actually they're, they're in the room.

Â And so, if beta i is exactly equal to 1, then the rate at which I meet people is

Â just, what's the stock of these people in this party.

Â 20:18

If beta is greater than 1, then you're going to meet your own types, at a rate

Â faster than than you would just milling around.

Â You're actually going to meet your own types.

Â at a, at a faster rate so, so this particular formulation says the weight at

Â which you're going to meet people is dependant first of all on how many people

Â are in this party and then also can be skewed by this extra parameter which

Â represents some viscosity in this meeting process so own types are going to tend

Â to, to meet own types. So this is going to be the bias in

Â meetings, right? This is going to be the parameter beta i,

Â where beta i greater than one means that you're meeting your own type at a rate of

Â above what you should be meeting them relative to how they're mixing in this

Â population setting, okay? So we've got Qi equals to the, equal to

Â the, this stock thing, so if, if, if I was 50% of the population and beta was 1,

Â then I would meet my own type at a rate of 1 out of 2.

Â If we set beta to 2, then the, the chance I would meet my own type to be about 71%

Â And if beta was as high as seven then, you know, the chance that I would meet my

Â own type would be, would be about 91%. So, as you begin to, you know, this would

Â all be with, with a half, and, and sticking in whatever my relative size is,

Â what this does is sort of buy us this relative rate at which I'm going to make

Â own type friendships compared to other type friendships.

Â Relative to what the mixing, the total number of friendships are, where the

Â stock is going to be just you know, based on the sum of the ti's from my type,

Â compared to the sum of the tj's overall. The, the j's right?

Â So it's keeping track of sort of what's the relatives size of meeting the

Â population compared to the others, and then we raise that to some power.

Â Okay, so what does this all work out to be?

Â Then we've got the ti maximizing this function.

Â The stocks are going to be relative to the relative number of meetings that

Â different groups want, rated by their relative sizes.

Â And then the meetings are going to be determined by what the stock is raised to

Â these bias parameters. And so, if the b-, if the s- The, the,

Â the fact that the stocks have to add up to one.

Â tells us that we have a balance equation. In terms of what these qi's have to look

Â like. When you sum across the i's.

Â The qi race to the beta eyes, have to equal one.

Â Okay, so what we end up with in terms of having balance on the meetings that the,

Â you know, if one group's meeting the other groups at a certain rate, they have

Â to match up, that's going to give us an equation which will help us solve for

Â this beta i parameter. So we're going to be able to solve for

Â the beta i parameter from this. Okay, so a simple model where we maximize

Â utilities, we have a meeting process, we estimate the meeting process, we put all

Â these pieces together, and we'll be able to estimate both the beta i is from here,

Â and the gamma i s from here, and then see what it looks like in the data.

Â Okay, so we've got these two conditions. This, maximizing this, this will help us

Â identify the, these parameters. we've got this, which will help us

Â identify these parameters. The qi's we'll actually observe in the

Â data, so what's the relative proportion of own type friendships to other type

Â friendships for each group? So we've basically can identify these

Â perimeters by fitting this model to data. the only perimeter we got left .Is, we've

Â still got this cost of forming friendships.

Â That we don't know exactly what that is. and so when we look at the equations that

Â we have For the t's and the betas I'm putting in some errors.

Â Then what do we end up with? We end up with these two equations we

Â have to fit. We've still got this c out here.

Â and so, what we can do is when we look at you know, solving this out For two

Â different groups we can say that the, the relative weight at which is should be

Â forming friendships compared to js forming friendships including the errors

Â should be a ratio here, where now this ratio is going to divide the c out.

Â So the c we can factor out. By just looking at relative numbers of

Â friendships because the c scales everybodys friendships up or down.

Â And so if we look at relatives numbers of friendships formed by one group compared

Â to another then that factors out the c and then we don't have to estimate the c

Â directly, we can just estimate the alphas and gammas.

Â Right. So basically what that tells us is that

Â the we're going to end up with ti minus tj equaling some error, and now we, and

Â this is cross multiplying. so we end up with an expression which no

Â longer has a season because we're comparing relative ts to each other

Â rather than absolute ti. so that is one way of just factoring out

Â one of the parameters. Okay, so that's a technical detail in

Â terms of estimation, which will make our life a little easier.

Â Now we just have three perimeters to estimate.

Â We estimate Alpha, the Gamma Is, and the Beta Is.

Â Alright. So these are the parameters that are

Â left, and we factored out that, that C parameters.

Â Okay. Fitting technique, very simple...

Â What we'll do is, we'll just build a grid of Alphas, Beta i's, Gamma i's.

Â So, we've got a grid over all these things for each network and each school

Â and each specification of biases. We can see what's the actual number of

Â total friendships that would be predicted for each group What's the realized number

Â and so we can calculate an error in terms of how, how big the error is in, in total

Â friendships compared to what it was.What's the error in terms of actual

Â group relative group meeting rates, the qi's.

Â So we can, we, these predict ti's and qi's Right?

Â 26:39

So for each one of these it predicts ti's and qis, and then we can look at what the

Â actual ones in the data are. And, sum the squared errors across all

Â the networks. So for each one of these we're going to

Â have say if we have four races, we'll have four sets of tis, four sets of qis.

Â And we can sum the squared errors, for each school we'll have a set of eight

Â errors, sum all those up, and then choose the biases, to minimize the weighted sum

Â of the squared errors, okay? So we, we just choose those things to

Â minimize these. Okay, so what do you get when you fit

Â this? So you can go through there's actually

Â five categories of students because there's the Asians, blacks, Hispanics,

Â whites, and there's also some that are miscoded or, or didn't indicate race.

Â Okay, so we have some others, and then we have the fit.

Â Alpha comes out to be about 0.55, so roughly like a square root in terms of

Â diminishing returns. When we look at gammas, what do we get?

Â We get Asian, they get different type friendships are worth about 0.9 of same

Â type friendship. Blacks 0.55, Hispanics 0.65, white's

Â 0.75. So we get different fits of that

Â parameter, all of them are less than 1, but they're varying in terms of at what

Â rate they would like to form or they get a value of different type friendship

Â compared to same type friendship. And then the second thing that we have

Â are these beta parameters. And the beta parameters indicate that for

Â asians and blacks we're seeing a high rate of bias towards meeting owned types.

Â so a good portion of the bias that they actually observe is actually due to the

Â fact taht they're meeting themselves at a much higher rate.

Â whereas for Hispanics and whites, these parameters are much lower and in fact the

Â Whites see a mixing rate which is roughly 1, Hispanics about 2.5 and then Asians

Â and blacks a factor of 7 higher. Now, one thing we can do is then ask you

Â know, are these statistically significant numbers?

Â Do we have any idea whether these could be, you know maybe all these numbers are

Â just noisily different from one and in fact the model isn't, isn't all that

Â different. so are these you know, truly different in

Â terms of some statistical sense? And what you can do, is we can test a

Â hypothesis. So what we could do for instance is look

Â at the sum of squared errors. So this is the residual sum of squared

Â errors. So this is the sum of squared errors that

Â we get, by looking say just at the preference biases.

Â So look all the, all of the gammas and look at the ti's that are generated, see

Â what's the errors that you actually see in the data.

Â and then say, let's suppose that we restricted all of these to be equal to 1.

Â So we, we've forced all of the gamma parameters to be equal to 1.

Â Okay, so you force those to be equal to 1 and then you do the best fit of the

Â model. What you end up with, you, you'd end up

Â with, a-alpha would drop to .2. But the error would go up to 17000

Â compared to 4000 when you allow these parameters to vary.

Â Then you can do an F test. And what this says is that this is, the F

Â value here is 42. The F threshold for even a 99% confidence

Â level is 3.3. This thing is way, I mean, the,the size

Â of the square areas you're getting its so much larger, its a factor of four larger,

Â so you are getting basically, you know, a huge amount of the error is actually

Â being explained by allowing these, gammas to differ.

Â So, if you allow the gammas to differ across race you are actually explaining a

Â huge amount of the error. The error blows up by a factor of four

Â when you, you force all of these gamma parameters to be equal to 1.

Â So you know, you can, you can reject, so the, the ones red here indicate that you

Â rejecting these things this particular hypothesis.

Â So they're certainly not all equal to one statistically under this particular

Â model. Are they all equal well the error goes up

Â to 61.75, if you forced them all to be equal, the best guess would be that

Â they're all .8. Okay.

Â 31:10

And then you can ask, okay, is it, is it true that Asians and Blacks have the same

Â preference parameter bias? If you fit a model where you force those

Â 2 things to be the same, and re-estimate the model, you know, you'd end up with a,

Â an estimate of alpha to be 0.7. The gammas for those two races that are

Â forced to be the same, the Asians and blacks will be 0.8, and then so forth.

Â What would the error be there? Well it would go from 4700 to 5300.

Â It actually has an F value of 9.93 still highly significant.

Â So, it looks like Asians and blacks have different parameters.

Â The reduction in the error is not just due to randomness.

Â So, using these kinds of models, you can go through and do F tests and other kinds

Â of statistical tests, by looking at the errors you observe under the model and

Â the errors that you would observe if you forced.

Â or if you work with some null hypothesis or some alternative hypothesis, then,

Â then the one the one that allows all of the parameters to be fit, that gives you,

Â a new set of estimations. You can compare the errors that you get

Â under the two and then ask whether that reduction in error came up at random or

Â not. A standard statistical test.

Â In this case an F test tells you which ones.

Â So you can't reject the, the hypothesis that Asians and whites have the same

Â preference bias. You can reject the hypothesis that Asians

Â and blacks have the same, and so forth. So you can go through and do, a, so

Â blacks and Hispanics are not distinguishable here.

Â but blacks and whites are distinguishable, right?

Â So when you look at these F tests. which ones are statistically significant,

Â you get certain differences you can say are statistically significant, and other

Â ones are, are not. Okay.

Â You can do the same thing for the meeting bias, you can go through and, you know,

Â same kind of tests. And indeed, the meeting biases are also

Â highly significant, so it really appears that there's bias both in preferences and

Â In meetings. And what this again.

Â What I want to emphasize here. Is not this particular model, but this

Â approach of. If your careful about writing down a

Â structural model. And you can began then, to derive

Â implications of that model. That model then generates certain

Â observed patterns. Match those patterns up with the data, so

Â in this case what it was generating was total number...the degrees of all the

Â agents and the relative fractions of friends of the different types they

Â should have. And then we can look at the degrees and

Â fractions of different friends that they have in the data, try and best match

Â those parameters up That gives us estimates for preference parameters an so

Â forth, and then we can test whether they're significant and learn something

Â about the relative choices that were made.

Â Here it appears that both choice and chance were present, if you believe the

Â model then it looks like people have biased preferences towards own type and

Â that's accounting for the fact that you're forming more friendships when

Â you're put in a school that has more of your own type.

Â and, and so you know we, we end up with estimates, there.

Â And the you know, the kind of thing that that allows one to do is then do analysis

Â where you can go to look at say, counter factuals.

Â What would happen in a school if we change the way in which people meet.

Â And so we try and eliminate that beta parameter and move that towards one.

Â So we want to make sure that everybody meets each other.

Â How much of an impact is that going to have on friendship formation.

Â Using this model you could begining to estimate something like that.

Â So it allows you to, to look at different policies or, as opposed to a policy that

Â tries to influence a preference parameters that would have a different

Â impact on, on what would happen. And so using a model like this, you can

Â begin to sort those things out. And so this is just an idea of one

Â particular model that marries strategic formation with some randomness.

Â Very specific model, but it's a technique that can be used much more generally.

Â