国际经济(英文版)(5).doc

I. First Order Autoregressive Processes
First order autoregressive processes, AR(1), have a structure similar to random walks:
AR(t) = b*AR(t-1) + e(t) .
If the coefficient, b, equals one we have a random walk. If b is greater than one the process is also evolutionary. If b is positive and less than one there is a positive dependence on the past but the process is stationary. Many economic series are characterized with a positive autoregressive structure. If b equals zero, we have a process of order zero, . white noise. If b is between minus one and zero the process depends negatively on the past but is covariance stationary. If b is algebraically less than one, the process is evolutionary. The following plot shows three evolutionary autoregressive processes for values of b of , 1, and -, respectively.
Using the lag operator, an autoregressive process can be expressed as:
[1-bZ] AR(t) = e(t),
so that {1-bZ] is the filter that converts a first order autoregressive process to white noise. Multiplying both sides by the inverse of this filter:
[1-bZ]-1[1-bZ] AR(t) = AR(t) = [1-bZ]-1e(t) ,
or:
AR(t) = [1 + bZ + b2Z2 + b3Z3 + ...] e(t) = e(t) + bZe(t) + b2Z2e(t) ...,
AR(t) = e(t) + be(t-1) + b2e(t-2) + b3e(t-3) ... .
The mean function of this autoregressive process is:
m(t) = E[AR(t)] = Ee(t) + bEe(t-1) + b2Ee(t-2) + b3Ee(t-3) ... = 0,
and the autoco