Econometricians/Time Series Analysts refer to linear difference equations with constant coefficient as Autoregressive models. The first and second order difference equations are called the first and second order autoregressive models and generally denoted by AR(1) and AR(2). AR(1) and AR(2) are widely used for estimating economic time series, notably macroeconomic series, with upward or downward trend. The following is an example of AR(1) estimate of US money supply.

The narrowest measure of money that the Federal Reserve Bank reports is M1. M1 includes currency, checking account deposits, and the traveler's checks. A broader measure of monetary aggregate is M2. M2 consists of M1 plus saving deposits, small denomination (less than $100,000) time deposits, money market mutual fund deposits, and several other liquid financial assets. Using monthly data from January 2005 to September 2006, the following AR(1) model for M2 was estimated
M2t = 1.0043 M2t -1 - 5.05

(a) Solve the first order difference equation for the money supply and analyze its behavior over time.

(b) If in January 2005 the money supply was $6,415.1 billion, what is your estimate of October, November, and December 2006 money supply?

(c) If the actual money supply for October, November, and December 2006 are
$6,936.2, $6,977.0, and $7,021.0 billion respectively, what is the models average estimation error for the three month?