Persimmons

Yesterday I went jogging with L. and F. in the stroller and B. on his bike. We went to the persimmon tree on Sare Road. All the leaves but only a few fruit had fallen, and they were ripe and delicious. Persimmonpudding.com is a website devoted to persimmons.

Subsidiarity and Hierarchies

The idea of "subsidiarity" came up today at a church meeting. The idea is that affairs ought to be handled at the lowest, most decentralized level. An individual congregation, for example, and not the denomination ought to discipline church members. The term is a Roman Catholic one.

The discussion made me think of the following problem. Suppose we have a worker who is misbehaving. It makes sense for his immediate boss to discipline him, since the boss knows the situation best. His immediate boss, however, likes his employees and is reluctant to bear the emotional cost of intervening. Thus, it may actually work out better to have the top boss-- or some central committee-- begin the discipline process. Perhaps the immediate boss can then handle details, having been positioned as the friend of the worker rather than as the "tough guy".

This reminds me of the style of hierarchy models in economics. I'm not sure whether modelling is useful here or not.

Hydrogen Bonds

H. was asking me why ice floats in water-- that is, why solid water is lighter than liquid water. The answer has to do with hydrogen bonds. When hydrogen and oxygen form a molecule together, the hydrogen's lone electron is pulled toward the oxygen, so the other side of the hydrogen has a positive charge from its lone proton. This positive charge is attracted to the negative charge of another oxygen atom's electrons, forming a hydrogen bond. Well, that's my simple story.

The Edinformatics article on ice is good. It has pictures too. It explains that in ice, the hydrogen bonds hold the water molecules apart in a lattice with lots of space, but in water, the molecules do not have such orderly hydrogen bonds.

Here's what Edinformatics's article on hydrogen bonds says:

As the name "hydrogen bond" implies, one part of the bond involves a hydrogen atom. The hydrogen must be attached to a strongly electronegative heteroatom, such as oxygen, nitrogen or fluorine, which is called the hydrogen-bond donor. This electronegative element attracts the electron cloud from around the hydrogen nucleus and, by decentralizing the cloud, leaves the atom with a positive partial charge. Because of the small size of hydrogen relative to other atoms and molecules, the resulting charge, though only partial, nevertheless represents a large charge density. A hydrogen bond results when this strong positive charge density attracts a lone pair of electrons on another heteroatom, which becomes the hydrogen-bond acceptor.

and

In ice, the crystalline lattice is dominated by a regular array of hydrogen bonds which space the water molecules farther apart than they are in liquid water. This accounts for water's decrease in density upon freezing. In other words, the presence of hydrogen bonds enables ice to float, because this spacing causes ice to be less dense than liquid water.

Someone else says:

The hydrogen bonds that form between water molecules account for some of the essential — and unique — properties of water.

* The attraction created by hydrogen bonds keeps water liquid over a wider range of temperature than is found for any other molecule its size.
* The energy required to break multiple hydrogen bonds causes water to have a high heat of vaporization; that is, a large amount of energy is needed to convert liquid water, where the molecules are attracted through their hydrogen bonds, to water vapor, where they are not.

Two outcomes of this:

* The evaporation of sweat, used by many mammals to cool themselves, achieves this by the large amount of heat needed to break the hydrogen bonds between water molecules.
* Moderating temperature shifts in the ecosystem (which is why the climate is more moderate near large bodies of water like the ocean)

The Risk of Common Investments

I'm frustrated by how we economists have failed to incorporate most of wealth into our theory of asset pricing. The CAPM says that a stock needs a higher expected return if its return is more correlated with the return of the stock market as a whole. That's a good start. It is true, too, that it is possible to hold a diversified portfolio of public stocks, whereas other assets such as private business stock can't be held by everybody.

I worry about other assets. How about bonds? Surely they should be in the CAPM, since they are public and easily diversified into.

How about housing? That's the asset most people hold. And it isn't valued well by economists. It is a hedge against housing consumption risk. If rents rise, then if I own my house, I am insulated. But when I sell my house, I do face risk. Also, if rents rise, maybe I'd like to consume less housing. The optimal house ownership contract isn't what we normally observe. It would involve some insurance against resale capital gains and losses, and adjustment for desired amount of house consumption.

How about labor income? Labor is our greatest wealth-- human capital, and just plain labor endowment. It's risky, and the risk is correlated with the stock market. I'd like a stock that does well when my salary goes down.

The Consumption CAPM is an advance. It notes that we want to have higher stock returns when we want higher consumption. That's odd, though, because consumption is endogenous. Really, we ought to look for the correlation between a stock's return and the return on wealth as a whole, which would be the ratio of GNP to wealth. We should look for the correlation between GNP and stock return, not consumption and stock return. But no, that's not right. Ideally, we'd want the change in total wealth, including labor wealth, which is not the flow of GNP, but a change in a capitalized value of future GNP. I'd like a stock which is uncorrelated with other stocks, and uncorrelated with changes in the value of my labor.

There's another problem. THe price of stocks is determined by the kind of people who buy them. Poor people don't. The only labor wealth that matters to stock prices is that of the people who buy stocks. So what we want to measure is the value of public stocks, bonds, private companies, housing of people who hold stocks, and labor wealth of people who hold stocks.

Quasiconcavity

Martin Osborne has some good notes on quasiconcavity. I'm still not satisfied, though. It's a basic enough idea that I wish I had better intuition for it, and lots and lots of pictures of functions that are or are not quasiconcave.

October 25: Here are some key features of a quasiconcave function f(x).

• It has convex upper level sets. The set of points x such that f(x) >= a is convex for any number a.
• It has convex indifference curves if it is a utility function. If f(x) is strictly monotonically increasing, the function g(x) such that f(x)=a is a convex function.

Every concave function is quasiconcave, but some quasiconcave functions are not concave. A key feature of quasiconcavity that concavity doesn't have is that if you do an increasing transformation of a qc function, it is still qc. I wonder if the following is true:

Conjecture: Iff function f(.) is quasiconcave, there exists an increasing transformation g(.) such that g(f(.)) is concave.

I'd start to prove the conjecture this way. Let x and y be points in the upper level set of f(.), which means f(x)>=a and f(y)>=a. Since f(.) is quasiconcave, the upper level set is convex, which means that f(mx+ (1-m)y) >=a too. What we need to show first is that there exists some increasing function g() such that
g(f(mx+ (1-m)y)) >= mg(f(x)) + (1-m)g(f(y)). I think we need to start by assuming that f(x) \neq f(y), and that they are both on the boundary of that convex upper level set. Then we can see how g has to affect those two levels of f differently.

If the conjecture is true, then maybe we can think of quasiconcavity as being the equivalent of concavity for functions that are just defined on ordinal, not cardinal spaces.

October 26. Why, though, do we worry about quasi-concavity at all in economics? Why not just assume that utility functions are concave? The conventional answer would be that utility is ordinal, not cardinal. That is a bad answer for three reasons. First, even if it is ordinal, we could say, "It's only the ordinal properties of a utility function that affect decisions. Therefore, for convenience, let's say that whatever function you start with, you have to use a monotonic transformation to make it concave before we start working with it." Second, we might say, "Since only ordinal properties matter, let's assume utility is concave for convenience." Third, we might accept cardinality. Everybody uses von-Neumann Morgenstern cardinal utility in their models anyway, making only a brief nod, if any, to ordinality. But a risk-averse agent has concave utility. For these reasons, I wonder why it's worth making our graduate students learn about quasi-concavity. The opportunity cost is that they're not learning about something more useful such as the CAPM or the Coase Theorem.

Maybe quasi-concavity comes up in enough other contexts to be important. I know Rick Harbaugh has a paper on comparative cheap talk where it comes up. In Varian, it comes up first in production functions, where it allows you to have convex input sets for a given output without requiring diminishing returns to scale, as true concavity would.

October 27. Yet another thought. Margherita Cigola has done work on defining quasiconcavity in ordinal spaces, on lattices. Convexity has to be defined specially there. She uses a different (equivalent in R space) definition of quasiconcavity:
f(mx + (1-m)y) >= mf(x) + (1-m)f(y)

I like that because it is closer to the definition of concavity.

Or another, suitable when the function is differentiable: f is quasiconcave if whenever there is a maximum (i.e., the first derivatives are zero), the matrix of second derivatives is negative definite. MR suggested that, for the single-dimensional x case. I'm not sure it does generalize that way.

Significant Figures

I haven't used this idea since high school, really, but it comes up now and then, so I looked it up in Wikipedia. 100 has one significant figure, as do 20 and 23 and .0001, the article says. The number .00200, however, has three significant figures. The number 1.234 has 4 significant figures. Digits beyond accurate measurement don't count as significant. There is ambiguity, however, in whether 100 feet really has just one significant figure. It may be that you have measured it to the nearest foot, in which case it really has three significant figures.

The real importance of significant figures comes in doing arithmetic. If you run 100 yards in 11.71 seconds, and the 100 has three significant figures, then the speed should be written with three significant figures as 8.54 yards per second, not as 8.53970965 yards per second.

Optical Illusions

U. of Washington has a good optical illusion-experiment site. It has the Muller-Lyon illusion below and lots more that['s harder to post in this blog.

Risky Borrowing

When you save, you have a choice between investing in a risky asset like a stock or a safe asset like a bank account. When you borrow, there is much less choice. Usually, you just borrow at a nominal interest rate, paying back 10% regardless of the return on the stock market or the level of inflation. The only exception I can think of is the variable-rate mortgage, which at least varies with the nominal interest rate.

Why is that? We can imagine a person being able to borrow at a lower interest rate if he agrees to bear risk. His interest rate could be 20% minus the return on the stock market, for example. If the average stock market return is 6%, his average interest rate would be 14% then. Someone who wanted a safe interest rate could borrow at 16% instead.

Our borrower would be hedging someone else against stock market risk. This would be useful if the borrower were less risk averse than some saver.

Yet we don't see that. Is it because borrowers are usually more risk averse than savers? Is it because borrowers are too likely to default if they have a risky payment to make?

October 21. At lunch I figured out a new twist. Suppose I want to borrow $100 for consumption, and I am risk-loving. I could borrow an extra $50, and invest it in the stock market. That way, I have to pay back $150 in cash, but my later wealth will be $(150-value of stocks I bought) lower as a result. Thus, even a poor person could take on risk if it weren't for the possibility of going bankrupt.

Cygnus

From the Wikipedia entry on the constellation Cygnus, the Northern Cross, one of the easiest to find this time of year:

The constellation also contains the X-ray source Cygnus X-1, which is now known to be caused by a black hole accreting matter in a binary star system. The system is located close to the star Eta Cygni on star charts.

Eta Cygni is in the long part of the neck of Cygnus, about halfway from the wings to the head. Cygnus X1 isn't visible to the naked eye, though, and Eta Cygni is very faint.

Beta Cygni is Albireo, a double star, at the swan's head. Alpha Cygni, the tail, is the brightest star, named Deneb.

Cygnus contains several variable stars too, as this website describes.

Wald, LR, and Score Tests

From Cornell, "Econ 620: Three Classical Tests; Wald, LM(Score), and LR tests" is a good description of the Wald, likelihood ratio, and score tests. The Hausman test seems more like an LR test, since it estimates both the restricted and unrestricted equations. I found the statalist post below on the Wald test for exogeneity of regressors:

This test is mentioned along with the theory behind -ivprobit- in Wooldridge's "Econometric Analysis of Cross Section and Panel Data" (2002, pp. 472-477). For the maximum likelihood variant with a single endogenous variable, the test is simply a Wald test that the correlation parameter rho is equal to zero. That is, the test simply asks whether the error terms in the structural equation and the reduced-form equation for the endogenous variable are correlated. If there are multiple endogenous variables, then it is a joint test of the covariances between the k reduced form equations' errors and the structural equation's error. In the two-step estimator, in the second stage we include the residuals from the first-stage OLS regression(s) as regressors. The Wald test is a test of significance on those residuals' coefficients.

Learning Hebrew

John Parsons has a good Biblical Hebrew site, Hebrew for Christians. The page I've linked has the ABCD song with Hebrew letters, nicely done, and if you click on the letters on that page, you get interesting tidbits of information

The Scriptures begin with the book of Genesis, but in Hebrew this book is named after its first word: [Hebrew omitted] (bereshit). The first letter of revelation from the LORD, then, was the Bet found in this word.... Note: The sole difference between the letter Bet and the letter Vet is the presence or absence of the dot in the middle of the letter (called a dagesh mark). When you see the dot in the middle of this letter, pronounce it as a "b"; otherwise, pronounce it as a "v."

M.R. sent me a link to a page describing a $125,000 bicycle. It looks like it lacks fenders or kickstand, so I'm not enthusiastic. My other question is this:

What kind of lock does it come with?

Conditional Logit

I was trying to understand how conditional logit and fixed effects in multinomial logit worked, to explain to someone who asked, and I failed. Greene's text was not very helpful. The best thing I found was some notes from Penn: "Conditional Logistic Regression (CLR) for Matched or Stratified Data". The bottom line seems to be that conditional logit (clogit in Stata) chooses its parameter estimates to maximize the likelihood of the variation we see within the strata, while ignoring variation across strata. Thus, if we have data on 30 people choosing to travel by either car or bus over 200 days, we could use 30 dummies for the people, but in conditional logit we don't. Also, in conditional logit, unlike logit with dummies, if someone always travels by car instead of varying, that person is useless to the estimation.

Reuleaux Triangle

This Reuleaux Triangle from Wolfram/Mathematica is a nice idea for a shape. It is the shape a Wankel engine takes, perhaps because you can rotate this triangle inside a square as shown at the Wolfram site.

Mortgage-backed Securities

Via Marginal Revolution, I found Gary Gorton's The Panic of 2007, a long paper on the institutional details of the financial crisis. It gives particular examples of subprime mortgage bonds, tells how subprime mortgages work in detail, talks about the collateralized debt obligations that buy the bonds and issue their own securities, and so forth. It also talks about why there is a liquidity crisis. Prof. Gorton assigns blame entirely to subprime mortgages, because they are especially sensitive to housing prices (as opposed to interest rates). I only read a little of the article, and it makes difficult reading. The main impression I get is that these mortgage-backed assets are far more complicated than I had thought, and it is no wonder that they are hard to value. I'm also amazed anyone would buy them, for that very reason. It seems that they must just have trusted the rating agencies, which should not have assigned ratings to such complicated securities.