Stephen Gordon asks a question I continually asked myself as a Ph.D. student at the University of Rochester:
Why does anyone care about the distinction between convergence in probability and almost sure convergence?
I'm teaching two econometrics classes this term (master's and PhD), and I just covered the parts on asymptotic theory. In both lectures, I stumbled badly at explaining the difference between these two forms of convergence: my heart simply isn't in it...
I've never heard of empirical study where the distinction between the two forms of convergence made a difference...
Why do we force our students to learn these things?
As an aside, undergraduates in economics are often surprised how little economics there is in graduate programs outside of fields courses such as Labor Economics and Public Finance.
Although from an empirical perspective it probably doesn't make sense in the slightest to worry about the distinction between the two types of convergence, I am quite certain it makes a difference in theory papers. Unlike most economists, I tend to use 'theory paper' in a pejorative sense, because too many theory papers:
- Make assumptions that cannot possibly be true and are not reasonable simplifications of the 'real world'.
- Are non-falsifiable, do not make predictions, or have so many free parameters they can be used to predict anything.
You need to know this math in order to be able to publish 'theory papers'. The larger question should be, "why does the economics profession put any value on this work?"
Comments
Amen.
Amen X2. I have a sneaking hope that hopefully our younger generation of economists will push out doing ‘math for math’s sake.’
Almost sure convergence is something you expect to happen, but has not yet happened. Convergence in probability is something that has happened; you have the results that show it.
I eat a somewhat random amount of food every day. Eventually (when I die) that amount will converge at zero. It hasn’t happened yet, but it almost certainly will.
If I note the amount of food someone else eats every day, until after that person is dead, the amount *has* converged at zero. Empirical results.
First things first. The level of math done by US economics PhDs in economics is baby math. Undergraduates in engineering and sciences are far, far beyond this by their undergrad junior year.
Second: the math used and taught is wholly inadequate and misplaced. Without differential equations up to and including nonlinear dynamical systems and systems theory along with information theory and control systems, there is really nothing of value in an economics PhD in terms of mathematical training. An economics PhD is utterly unprepared to actually discover anything of value or significance in the field of economics.
The entire fiasco of academic economics completely missing the bubble and its crash is completely explained by the background of those who didn’t miss it or even saw it coming years ahead of time. These people are all either entrepreneurs and small businessman who were not tiny cogs in the corporate machine but rather who live economics personally and thus have an intuition lacking in academia, or they are all engineers who had enough math and empirical acuity to see the obvious relationships and inevitable outcomes of our economy in terms of the mathematics they’d already learned as undergrads.
Jeff,
I’d have to disagree with your categorical remark about the level of mathematics being “baby math”. I’ll be up front and admit that I study mathematics and so I don’t really know how economics uses math, but I have met several economics grad students in math courses that would, without a doubt, make a 3rd year engineering student spoil his pants. (The classes were differential topology & stochastic analysis, the latter was of interest to people wanting to work in some areas of finance).
With that said, I do imagine that those students are the exception and not the rule. My general impression is that the average electrical engineering student knows tremendously more mathematics than the average economics student. The significance of this obviously depends on the actual field in which the economist is working, but e.g. it is not uncommon for big banks to hire signal processing PhD’s to work in trading groups. I know some math finance people and I do see a bit of a problem in the fact that the current economics training is woefully inadequate to actually say much about something as simple as an idealized market, much less figure out how the real world works and make predictions with enough confidence to back them up with dollars.
‘The larger question should be, “why does the economics profession put any value on this work?”‘
Perhaps because that sort of worthless, non-falsifiable garbage work is what enables economists to get good (read: high paying) jobs at banks and dream up economy-destroying “investment vehicles” based on these ridiculous abstractions from their ‘theory papers’. Another reason is that they can get jobs at newpapers where they pretend to know what they’re talking about while cheerleading the investment banks madness from the sidelines (see Krugman).