11/24/2005
Difference Equations
The study of difference equations is one of the more rewarding experiences I can recommend to those who like numbers and their applications. Their have particularly far-reaching and deep applications in biology, demography, economics, mathematics and physics.
Difference equations show the path that a phenomenon has taken over discrete time intervals. A sample first-order equation is
Y(t) = a Y(t-1) + W (t)
I have found the best book to study difference equations is Elaydi, "An Introduction to Difference Equations." On p. 3 of the book, the very first formula given shows how to solve this for all situations were a and w are functions of t.
It's possible to understand that formula readily by working out a problem like
Y(n+1) - 1/2 Y(n) = 2 with Y(0) = 1.
The standard, excellent book "Time Series Analysis" by James Hamilton opens with a chapter on difference equations which restates the above-mentioned formula and then shows how to use simple methods to solve a second-order difference equation where a Y(n+2) is a function of Y(n+1) and Y(n).
The next chapter shows how these simple formulas explain all lagged time series and moving average aplications in econometrics with the subsequent chapter discussing forecasting.
Such are the joys of a little review of arithmetic extensions.