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...

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

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 + b₂x₂  + … + b_nx_n

Here x_i 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…

Let’s Talk About Correlation…

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...

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...