## Monday, August 30, 2010

### Computational Modeling...

A nice simple set of lecture slides on computational modeling in the social sciences by Ken Kollman at Michigan (although I have to admit that the uninitiated will probably not find them terribly informative)...also I will have limited to no internet this week...expect fewer than normal posts.

Posting will resume apace from Edinburgh next Monday...

## Friday, August 27, 2010

### Let Them Eat Credit...

A very nice article entitled "Let The Eat Credit" by Raghuram Rajan of Chicago in the New Republic.  Krugman provides a counter to Rajan's assertion that it was predominately government policy mistakes that encouraged the housing bubble...Krugman blames loan originators in the private sector.  I say both are right.  Krugman's data may support the view that private sector housing market players were more at fault...but I do not think that this absolves the government from engaging in the type of short-sighted policies to address income inequality(i.e., credit, credit, and more credit) that Rajan writes about in his article.

### Computational Approaches to Network Formation...

The third major component of my first year of PhD research will focus on computational approaches to financial network formation.  Prof. Leigh Tesfatsion has created an excellent bibliography of the literature in this area.

### Quote of the Day...

"To someone schooled in nonlinear dynamics, economic time series look very far from equilibrium, and the emphasis of economic theories on equilibria seems rather bizarre.  In fact, the use of the word equilibrium in economics appears to be much closer to the notion of attractor as it is used in dynamics rather than any notion of equilibrium used in physics."
-J. Doyne Farmer, 1988

## Thursday, August 26, 2010

### First Section of Lecture Notes on Geometry of Linear Regression...

This is my first cut of lecture notes on the Geometry of Linear Regression...FYI the b and beta are the same...having html issues.  Hopefully I have not made any egregious errors...

The Geometry of Linear Regression

Suppose we have the following system of equations:

y=Xb

Here the dependent variable y is a vector of length m, X is our (m x n) matrix (i.e., m rows and n columns, typically m>n) of independent variables, b is a vector of coefficients of length n.  Why are we going to start by talking about the geometry of solutions to systems of linear equations? Well, because at a fundamental level linear regression is really all about "solving" a system of linear equations when there is no true solution.  Linear regression finds a solution b to our system of equations that is the "best" because it is "closest" in a very specific way to the vector y.

Now our system of m equations with n unknowns (the n coefficients which comprise the vector b) tells us that the vector y (our dependent variable) is a linear combination of the columns of X (our independent variables)….

y= b1x1 + b2x+ … + bnxn

Here xi i=1,…n are the column vectors of length m that make up the matrix X.  This means that the vector y is in the column space, col(X), of our matrix X. In pictures with 2 independent variables…notice that the our independent variable, the vector y, lies in the plane corresponding to the col(X)

Remember from its definition that the col(X) is the vector space spanned by the column vectors of X, which is simply a fancy way of saying that the col(X) includes all linear combinations of the column vectors of  X (which includes y at this point).  If the column vectors, our dependent variables, also happen to be linearly independent of one another then our column vectors form a basis for the col(X).  Normally this will be the case…but it is crucial that our set of dependent variables be independent of one another!

If we have nice case: X is an (m x n) matrix with m>n and that our columns of X, which span the col(X) by definition, are linearly independent of one another and thus also form a basis for the col(X).  This implies that the rank of X (which as you will remember is simply the number of linearly independent columns of X) and the dimension of col(X) (which is simply the number of vectors needed to form the basis of col(X)) are both equal to n.  Our matrix has full column rank! We are off to a good start…

Geometrically, correlation between two variables (which we are representing as vectors) is related to the angle between two variables/vectors via the following formula…

Cosine! Theta! Dot products and Euclidian Norms! Boo! Let’s draw pictures…In this first picture our two independent variables are positively (negatively) correlated because the angle between their two corresponding vectors in the col(X) is acute (obtuse).  I draw the positively correlated case below…

In this second picture, the two vectors are at right angles with one another and are therefore uncorrelated.  This is an extremely important case…when you are learn about OLS, IV and GLS the question of whether or not your error term is uncorrelated with your explanatory (i.e., independent) variables will come up again and again…remember, geometrically, uncorrelated mean vectors at right angles!

Finally what does it look like if the two vectors are perfectly positively (negatively) correlated with one another?  Although I will leave it up to you to draw your own picture, for the perfectly positively correlated case look at the picture of the acute case and think about what happens as the angle gets really, really small.  Once you figure that out and get your picture, the perfectly negatively correlated case is simply the 180-degree (hint) opposite…

To be continued in the near future with a simple linear model y=Xb + e!

### Still Watching Linear Algebra...

For those interested in an introduction to networks, graph theory, and linear algebra lectures 11 and 12 from Gilbert Strang at MIT are brilliant...

## Wednesday, August 25, 2010

### Idea About Regulatory Policy Design...

Fair use warning!!!  Totally speculative blog post...

In my last post I mentioned that a major difficulty for policymakers in designing regulatory policy was how to develop a framework to mitigate systemic risk in an environment where there is a trade-off between mitigating individual risk and systemic risk.  I have an idea...

Study the evolution of communities of agents within the financial sector using network data and some set of risk measures.  In theory, at least, the agents within these evolving community structures should be affected by some common key drivers of individual risk (given that they are in the same community).  Then try to develop a regulatory framework that encourages the individuals within communities to mitigate community risk...the idea is that this difficult coordination task would be made easier given that the agents are affected by a common set of risk drivers.

This idea implicitly assumes that, in terms of mitigating systemic risk, mitigating risk individually at the community level is somehow better than mitigating risk individually at the level of the individual agent.  But is there any basis for this belief?  I have no idea...perhaps this idea is only a good one insofar as it makes the policymakers job easier...but I am not even convinced it does...

### Major Insight of the Literature on Financial Networks...

Keshav passed along some links to papers on financial networks that are being presented at the EEA conference that he is attending.  I have glanced over the abstracts from the papers that and they all remind/reinforce what I think is the major insight from the networks literature on systemic risk in the financial sector that was extremely under appreciated until very recently:

Individuals agents in the financial sector who are rationally pursuing strategies that minimize their individual idiosyncratic financial risks can actually cause substantial increases in overall/systemic risk in the financial sector.  If individual agents and their risk mitigation strategies existed independently of one another (i.e., did not interact) then minimizing their risks individually would surely (I think) lead to a minimization of systemic risk as well.  The key issue that the network literature focuses nicely on is that financial agents and their strategies do interact with one another and that these interactions can create both stabilizing and destabilizing (depending on circumstances) feedback effects.  Oftentimes the destabilizing feedback effects dominate...

Generally speaking I think most policy makers would have assumed that a regulatory/incentive structure in the financial system that encouraged all players to mitigate individual risk would also be a good regulatory framework for mitigating systemic risk.  Based on recent empirical evidence and research I would say that this notion should be discredited...Given that there seems to be a trade-off between mitigating individual and systemic risk, the hard part from a policy perspective is how do you a design a regulatory regime that encourages mitigation of some weighted average of individual vs. collective/systemic financial risk.

Different papers in the financial networks literature look at a variety of specific issues that relate to systemic risk, but ultimately it is the trade-off between mitigating individual risk and mitigating systemic risk that continues to emerge.  Maybe I am wrong about my interpretation of the literature...but for me at least, this has been my major takeaway...

## Tuesday, August 24, 2010

### Watching More Linear Algebra Lectures...

In the original link I sent out with the 35 MIT lectures on Linear Algebra, lecture 8 on Solving Ax=b and the Row Reduced Form R is corrupted.  You can find the whole (or at least most of the series) on Google Video.  Here is a working link to for lecture 8...

### The Visa Process, Cont'd...

Passport and UK Student Visa arrived in the mail today from British Consulate.  I am now good to go...

## Monday, August 23, 2010

### A Visual Approach to OLS...

As I am prepping for my teaching requirements for the upcoming year, I have been thinking about employing a more visual/geometric approach to OLS.  I remember sitting in a lecture on GMM where the professor was talking about projection matrices and all of the sudden something just clicked.  I recall thinking: "Wow, the different versions of regression analysis (i.e., OLS, GLS, IV, etc) can be thought of as projections of the dependent variable (y) onto the column space spanned by the regressors (Xs)."  When you multiply the vector y by the projection matrix P = X(X′X)−1X′ you project the vector y onto the space spanned by the columns of X and the result is shown in the diagram below.
The matrix is simply a linear combination of the columns of X (i.e., a linear
combination of the regressors) and as such lies in the column space of X.  The idea
behind OLS is to then simply choose the right β so that the linear combination will
minimize the distance between the vector y and its projection in the column space of X.
Additionally it is straightforward to extend this framework to talk about issues related
to correlation between regressors, errors, etc.

I would like to emphasize this projection interpretation to my introductory statistics and econometrics students.  Although this approach does use some basic jargon borrowed from linear algebra (most of which they are already required to know)...I think/hope that it will be more intuitive for the students.

### Cobb-Douglas Applet...

Useful JAVA applet that helps visualize Cobb-Douglas production function in 3-D.   Would be useful for students trying to understand the relationship between increasing, decreasing, or constant returns to scale, and the shape of marginal returns...

## Sunday, August 22, 2010

### Watching Linear Algebra Lectures...

I was watching MIT Linear Algebra lectures 3-Multiplication and Inverses, and 4-Factorization into A=LU and 10 minutes into lecture 4...the tape quit!  Damn!  Factoring a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U is a pretty useful thing to know how to do.  It makes calculating things like trace, determinants, and inverses of A really easy...

Alas...it is a good thing that I already new how to do the factorization...but here is a link to the video lecture that will work...the link takes you to a series of written notes based on the MIT lecture series which is also quite useful...

## Saturday, August 21, 2010

### Bibliography of Networks...

For anyone else interested in networks I have found a fairly extensive bibliography courtesy of the Stern School at NYU.

## Friday, August 20, 2010

### Teach Yourself Linear Algebra in a Day and a Half...

35 lectures on linear algebra from MIT Professor Gilbert Strang.  Start with "The Geometry of Linear Equations" and end with "Final Course Review."  I am on lecture two: "Elimination with Matrices."  I had two course on linear algebra as an undergrad and the subject played a starring role in my MSc.  I am prepping for TAing Quantitative Methods courses this year, and I am finding these lectures to be a nice refresh...

## Thursday, August 19, 2010

### More on my PhD Research Agenda...

The following papers are going to form my point of departure for my PhD research:
1. Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk-This paper by Battiston et al has all of the ingredients that I am interested in studying (specifically credit networks and their relationship to systemic risk).  An excellent slide deck that covers the key concepts and conclusions of the paper can be found here.  In their model, as the network becomes more dense (i.e., more inter-connected) individual risk decreases because more connections allow each individual to diversify idiosyncratic shocks.  However, increasing network density also increases the ability of negative shocks to propogate through the entire network (instead of being locally contained) and thus leads to an increase in systemic risk.
2. Financially Constrained Fluctuations in an Evolving Network Economy-This paper by Delli Gatti et al is a nice model of credit networks between firms and banks.  It includes both inside or trade credit as well as outside or bank credit.  The interaction effects of the firms financial positions generates business cycles (similar to Minsky's financial instability hypothesis).  Model simulations also replicates power law distribution of firm sizes and Laplace distribution of firms' growth rates.
3. Econometric Measures of Systemic Risk in the Finance and Insurance Sectors-This paper by Billio et al suggests some possible practical applications of the network approach to systemic risk analysis.

## Wednesday, August 18, 2010

### Even More Economics 101 by Brad DeLong...

Another must read for those of us teaching/tutoring intro macroeconomics...

### The Hidden Hazards of Adaptive Behavior...

James passed me this interesting paper on the stability of multivariate systems where agents use adaptive behavior via email.  I did a quick read...heavy on the maths (although I think it is mostly just multivariate calculus and linear algebra). They also went all out on the notation...I haven't seen some of those symbols outside of formal advanced maths texts. It would take quite a bit of work for me to fully grasp the paper as I would need to follow through on the calculations etc. Although plowing through the maths would be a good refresher...

I think his (i.e., James') characterization of the paper in the email is pretty much on target. They seem to formally develop general conditions under which multivariate systems with adaptive expectations are not stable. They find that for the most part, such systems are not stable.  And they stress that their results provide further evidence that equilibrium stability results from models where homogeneous agents are using adaptive behavior (or that are otherwise one-dimensional) should not be generalized to high dimensional or heterogeneous agent models where agents are using adaptive behavior.  The study of multivariate (and heterogeneous agent) systems, would seem to be a study in disequilibrium behavior...

Their results support the other research on multivariate and complex adaptive systems that I have read. From my reading of the literature, multivariate systems (including heterogeneous agent systems), generally speaking, are not stable.  At least in the sense that it is rare that such complex systems settle down to some static equilibrium.  On the other hand many multivariate and complex adaptive systems do exhibit endogenous self-organizing behavior (which I like to think of as a type of stable disequilibrium behavior...although this may not be the best choice of words to capture the phenomenon), punctuated equilibrium dynamics, etc.

The more I learn about economics, the more I am becoming convinced that economics is not an equilibrium science.  Economics is fundamentally a science of disequilibrium behavior.  I will go out on a limb here and say that perhaps one of the reasons that economics, particularly macroeconomics, has struggled so much is because we as economists are trying to force our (for the most part) linear equilibrium models to describe a non-linear disequilibrium world...

There is also the (I'd like to think small) possibility that I am cocooned in a world of complex adaptive systems, and am suffering from a massive case of confirmation bias...

## Tuesday, August 17, 2010

### Quote of the Day...

"That's you...drops of water.  And you're on top of the mountain of success.  But one day you start sliding down the mountain and you think wait a minute I'm a mountain top water drop and I don't belong in this valley...in this river...in this little dark ocean with all these other drops of water.  Then one day it gets hot, and you slowly evaporate into the air.  Way up.  Higher than any mountain top.  All the way to the heavens.  Then you understand that it was when you were at your lowest that you were closest to god.  Life's a journey that goes round and round and the end is closest to the beginning.  So if its change you need...relish the journey."
-Some guy

### Linking Liquidity Constraints to Network Formation...

For some time now I have been struggling to develop a mechanism to link liquidity constraints with the agent's network formation decision.  I suspect that this has a lot to do with the fact that while I have read quite a lot on network theory, I have not yet got around to reading much of anything having to do with liquidity constraints.  All of my knowledge of liquidity constraints has come from the two weeks we spent talking about them during my MSc.  What follows is my first attempt to link the two concepts together.  The idea follows closely to the textbook treatment of consumption decisions of agents facing liquidity constraints.

The agent has two choice variables each time period: consumption, ct, and the number of neighbors in the network, nt.  The agent has wealth wt and anticipates some uncertain future income yt+1. This agent's savings can be defined to be st=wt-ct.  Typically an agent then maximizes something like the sum of his current utility plus the expected discounted sum of future utility subject to the constraint that next period's wealth wt+1=Rt+1(st+yt+1), where Rt+1 is the interest rate.  A liquidity constraint in this scenario would require that the agent's savings st be non-negative (i.e., agents can not borrow)

What I want to do is allow agents to borrow funds from neighbors in the network (assuming that they have excess savings to lend).  Agents would become liquidity constrained if neither themselves nor any of their neighbors had funds to lend them.  In this scenario, an agent's consumption decisions over time are affected by his position in the network (i.e., his access to credit from his neighbors).  Clearly there are a number of issues to work out with the framework.  Such as how to specify the interest rate, the income stream, the appropriate utility function, the information set, etc.   Also this is definitely not going to be analytically tractable (but them neither are more traditional liquidity constraint problems). Ideally it would also be nice to be able to model agent default.  Maybe this could be done by specifying some type of stochastic income stream where there is some positive probability of the agent unexpectedly ending up in the "low" income state and is thus unable to repay his loan.

This is pretty much wild speculation at this point...I just wanted to get these thoughts down on digital paper...

## Monday, August 16, 2010

### The Terrible Flooding in Pakistan...

Some sobering photos .  The severity of the situation would seem to call for aggressive international response...