 Excellent. Please start to get away. Thank you. Okay. Thank you. Yeah, so thank you. Isaac. So, yeah, we'll be discussing the evolution of income or wealth distribution with higher order of the regressive process and I'm Sam and I work with Pritika and as I mentioned our supervisors were Jonathan and Ravi. So to introduce what we'll be discussing. Our main question is, so how does intergenerational mobility affect inequality. And in particular, we're talking about in economic mobility and economic inequality. And we shall be investigating this using auto regressive process models. And another interesting question is the reverse. So if you had a measure of inequality, how would you then estimate mobility between generations, however, we'll be investigating the first first one so how does mobility effect inequality. And just to illustrate what mobility means so I'm sure many of you have heard of this phrase so from shirt sleeves to shirt sleeves in three generations so essentially you can have. Like someone was poor and then their children like made it in the economy say and became rich but then their children again went down and became poor again so the old term for this is shirt sleeves. So that would be high mobility. And then I was trying to find a quote for the reverse of this but I couldn't really find one with shirts leaves anyway so I just made one up so. If you say shirt sleeve stays shirt sleeve over many many generations so this would be like low mobility so you know those who are poor stay poor for many generations or you can look at the reverse of rich stay rich for many generations, etc. Okay, so to intro. And we let x t be the log of income or health as well but for our purposes of this talk we're going to focus on income. And we let epsilon. cultural and and or genetic. Endowment of generation at generation T. And this, and we assume this following model, which has some micro or economic foundations, which was from so long, which he summarizes in 18. And this was adapted from an earlier paper in 1979 by Becker in terms. So the model is essentially. So I'll just read it so it's x t equals alpha plus beta x t minus one plus epsilon t and epsilon t equals theta epsilon t minus one plus each. So this alpha part is the trend in average incomes across the entire population. And this beta is the relationship between generations of income. Yeah, as I mentioned before this silent T which is serially correlated is a measure of the some other factor that's not income so you can say cultural or genetic endowment so how does parents genes I guess path, or how does their involvement past the next generation, you could say, and this is just a random noise from a normal distribution with zero mean zero and variant sigma squared. And we notice this epsilon t is an AR one process. And if we zero then x t itself would be an AR one process. And so this model if you just substitute for epsilon t, you can rearrange. And so, and get express that same model as a AR two process. So you have this x t minus two so the income of grandparents as well, or log of income of grandparents. What's interesting about this is that you would expect beta and theta to be greater than zero so pause essentially positive relationship between income of parents and children and some sort of cultural or genetic factor between parents and children expect both both of those to be positive, which then implies the coefficient of the grandparents is negative so the minus beta theta would then be negative. So, would you then think that this would imply a negative relationship between children's and grandparents income. And expect that but we're going to come back to this point later on. Okay. So now just some general results of AR two process. So AR two process so this is up here. Hopefully you can see the mouse. So, and in particular we want to look at when it's stationary so when it's roughly when the mean and variance and are independent. And this happens for these conditions on beta on beta one and beta two. And which can be represented as within this triangle in the beta one beta two space. And so under the stationary condition. The mean as this the variance as this and the autocorrelation or real one as this formula. And also you have that x t become follows an in stationarity is a normal distribution with the mean mu and the variance gamma naught. And that implies that. So that log of income is normal so that implies income is log normal under this model. And now we're introduced to quantities this intergenerational income elasticity and variants of logs. So, first assume an AR one process as follows. And then, given this process and AR one process you can estimate beta beta hat is at the OS ordinary least squares estimator of beta beta hat is the is the autocorrelation of x t. However, assuming instead that the model was actually the original so long model introduced, then setting. So, from the previous slide here we have an AR two with beta one as beta plus beta and beta two has minus beta theta. And substituting that in here. So beta one beta plus theta and beta two is minus beta theta. We get exactly this, which was the autocorrelation of an AR two. And this we call the intergenerational income elasticity. So if we substitute beta one as beta plus theta and beta two is minus beta theta into the variance of the AR one AR two. And of the solar model we get this which is the variance of logs, and the variance of logs isn't inequality measure. And we notice in both these formulas the symmetry of beta and theta so replacing beta and theta. And you get the same formula. Okay. And now say you say we had from data we inferred that it was an AR one process so say you had the data is x t and x t minus one. Then you could estimate beta hat and gamma not hat exactly. But then say you thought it was actually the solar model, then you have these representations of beta, beta hat in terms of beta and theta and gamma not in terms of beta and theta. And then you could then sort so you have two equations to unknowns and where the unknowns are beta and theta and you could solve this. However, we weren't looking in particular any data so to look at this numerically just to simulate for some beta hat and gamma not hats which we don't know. We just fixed beta hat and then estimated gamma not hat as a as the variant stationary variants of an AR one process. And to note that these two equations actually have analytical solutions but they're very complicated to write down it. There's lots of terms going on there in square roots and things so it's a bit ugly to look at. So anyway, I'm not going to look at the numerical results for this so fixing beta hats and estimating gamma not hat as from this. And also with Sigma squared equal to what set Sigma squared equal to one. To do this, I'm, we minimize this cost function. And so, essentially, when this is at the minimum point you would expect the beta hat to be closest to the, the IGE representation and the gamma not hat to be closest to the variants of logs representation and beta and theta. So we set the parameters for beta to be between zero and one and theta to between minus one and one. And, yeah, so then you solve this numerically and it gave the following results. And so the left graph. So the beta going from not point one to about not point nine and the theta on the why and which are all roughly zero. And so that's the result you get so it seems like the beta roughly tracks the beta hat. And the theta is roughly zero, although there's potentially an interesting sign change going on there. And you can see that the cost function so how well it works essentially this optimization. It seems to work steadily better from low beta to around not point eight and goes up slightly. And so. So this is essentially find assuming AR one but then assuming the Solon model and finding the beats and theta. However, this was just an initial exploration and we haven't really found the, like real world effects or what this means exactly. Like why should the beta be near beta hat why should the data be almost insignificant. Okay. So after these influential analysis presented by Sam, I'd like to talk about few qualitative results that emerged out of a study. So here I have formula of variance as function of beta and theta for time being we have fixed sigma square as well. First thing that we notice is asymptotic behavior of variance, where either theta or beta 10 towards plus or minus one. So because if it is plus or minus one, we have a strong direct relationship between children and parents well and genetic pattern, which creates a strict immobility across generations and which restrains the possibility of equality. The second thing that we observe is a perfect symmetry between beta and theta, which implies be it conscious investment choices made by parents in human and non human capital of children, or be it unconscious inheritance in form of genetic traits or social prestige. Both of them hold equal footing and determining the eventual income of the children. Third point is inequality for appreciable range of beta and theta going from zero to one, where one means direct relation and zero means no link whatsoever. The highest inequality is observed for higher values of beta and theta and the lowest inequality point is observed for theta equals to zero and beta equals to zero. In reality, we would encounter the positive stimulus of income by parents, we desire a negative draw on genes to counter the effect and gain maximum equality. But in real world, we know there's a positive transmission of wealth and genes. So in that case, theta equals to zero and beta equals to zero gives lowest inequality and highest equality. In next slide, this is the graphical representation of points that I made about. So in the left plot, we have plotted variance as function of theta going from minus one to one. For various positive constant beta, we observe the minimum point in the negative space of theta almost roughly equals to minus beta. In the right plot, we have plotted theta for only practically appreciable range going from zero to one. And here we can see minima is lying around zero and for higher beta, we observe higher variance. Second point that we see, the second result that we see is negative grandparent coefficient relating between grandparent's income and children's income for first order income model and first order genetic model. It might seem paradoxical in first plans that richer grandparent's will lead to poorer children, but we miss out a subtle implication in framing this equation. It says that there is a negative impact given that all other factors remain constant, which also include parents' income. So despite increasing grandparent's income, if parents couldn't increase their income, it signals that maybe they received a poor genetic transmission, which was also received by their children and hence the overall impact of grandparent's is negative on their children. However, time and again, our studies have shown a positive coefficient for grandparent's and children, which is minus beta theta. So does that mean this analysis is wrong? Not wrong, but I'd say incomplete. What if the genetic endowment is not as simple as AR1 model? We tried, in fact, we observed a higher order endowment model in genetic transmission. So we have also considered a factor by grandparent's genes as a direct addition to children's genes. In this case, we observe an AR3 model for lower income. Now the sign of grandparent's coefficient is positive, but we observe a new factor, a negative, even smaller factor for great grandparent's coefficient. We can extend this to p-order AR model in genetic transmission and we would always receive this last incessant coefficient to be negative. So with advent of data, if we consistently get a positive coefficient for grandparent, it does mean that they have a direct role in genetic transmission to their children. We still don't have gone through all the possibilities of reasons why we might receive a positive grandparent's coefficients and measurements. It might be because of group effects. Various ethnic groups leads to different intercepts in our equation, which leads to a mission of fixed effects. And bias analysis have shown that it gives a positive push to all the coefficients of all the generations. It might also be due to measurement error, where fitting a regression model on our data gives positive coefficients for all the generations. It leads me to my final and concluding slide. So overall, in our work, we were able to show the relationship between equality and mobility. Higher values of beta implies lower mobility, which is reflected in society as higher inequality. Now one might ask, why is it important to ask this question? Why is it important to pursue this research? It is crucial because it puts us in the center of debate of process policies versus outcome policies. To outline the contour of this debate, I'd like to talk a few minutes about process policies and outcome policies. So outcome policies aim to manipulate the leaders of government, that is subsidy, tax, regulations, and expenditures to achieve a final certain outcome. For example, direct cash transfers to keep children school. The success of these policies is measured by to what extent these transfer encouraged people to satisfy that particular condition. Process policies, on the other hand, try to shift the trajectory of change. For example, giving political reservations to certain class of people. In these cases, the approximate impact is too subtle to notice, but they are longer lasting. In this case, we tried to change the power dynamic of people by shifting the process of decision making in favor of lesser privilege. So coming back to the wealth part of this debate, outcome-based approach argues that maximizing economic growth would allow all the individuals to benefit from this and create a more equal society. So we should only investigate more deeply about final wealth distributions and variants of log and similar representations. Policy process, on the other hand, recognize the importance of opportunity and constraints on choices, and hence solicit affirmative action on details and thicas of the world. So overall, this work provided as a space to rethink the diagnosis of inequality to present the case for both process policies and outcome policies and how to include them in mainstream, which brings me to my final point. To reinforce ME theory, we must make our ideas legitimate by methods of measurement and statistics. We still have a lot of diligent work to do to test our theory against data, given that the variables of this model are social influence, wealth or genetic inheritance. Our noisy indicators themselves. It is difficult to quantify them and even more difficult to track them over a time spanning several generations. But we hope that with more data collection and data analysis, with refined data, our understanding will be better and improve, and we'll be able to find a more solid model of multi-generational wealth distribution. With this, we'd end our slide, we'd end our presentation, and we'll be happy to receive your views or questions. So thank you very much, Sam, and I'm critical for this very nice, very neat presentation. I would like to actually well congratulate you for this very nice presentation project to you and also to your supervisors. So there are a few time for questions. Actually, we have like seven minutes for questions. So if any of us have some questions, so please unmute your microphone and go ahead. I think Rhino may have had his hand raised. I don't know if that's true, but I have a question, so I will, I will just ask you. So go ahead, but let me point out a couple of things. I'm trying to control the whole thing with a desktop and a laptop. If somebody raises their hand with a video, I cannot see it. Okay. So, so maybe do it with the with the icon or just simply a medium microphone. So Alex, please go ahead. I was just wondering, I think this is this potentially this this negative coefficient of grandparent and great grandparent rental log income is sort of curious and counterintuitive at first glance. I was just wondering if you just look at the regional correlation say between child and grandparent log incomes. Presumably you find that to be positive. And actually this is really just a, if you like a correction term, because the effect of the grand parental log income is maybe overstated in the parental log income. Does that sound right or have I got that wrong. And yeah, I would say so yeah I think I did look at the auto correlation of lag too and I think yeah for the beaters and Peters positive I think it is positive. Yeah. Yeah, so yeah you're right about that. I think it was just the interesting point where you go from you always seem to get this factor come out so when you increase the endowment then you get the negative in the, then the great grandparent, whereas the grandparent can be positive. It's either it's like an interesting effect but yeah maybe the auto correlations are still positive so in effect it doesn't. There's still a positive transmission even though you see negative effect. It's just interesting you always get this negative factor at the last generation, when you when you increase the lag of the epsilon T. Thanks. I mean I would need to think about it a lot more but my guess is it's really just reflecting the fact that this is an autoregressive process. And that and that this is basically a correction because you've already got contributions within the the lower order terms. Yeah. Anyway, it is an interesting point. Thank you. So, Alex and thank you for the answer. More questions colleagues, please directly unmute your microphone and go ahead. Or write it in the chat if you have some technical problems back on with it. Okay, so if not, shall we thank Sam critic and the supervisors for this fantastic presentation and I'm pretty thank you very much guys. Thank you. So let me.