process with the distribution 
e(t) = N(O, Ge) 
The model of an ARMA process can be subdivided into two 
basic models, the autoregressive - (i.e. AR-) and the 
moving average - (i.e. MA-) model. 
The AR-model of order p defines the current value x(t) as a 
linear combination of the p previous values of the time 
series and the current innovation e(t). 
AR(p) : x(t) = -a,+ x(t-L) = Lis > a, x(t-p) + e(t) 
The MA-model of order q defines the current value of the 
time series x(t) as a weighted sum of the current and q 
previous innovations e(t). 
MA(q) : x(t) - e(t) + b,*e(t-1) + Jo. + b -e(t-q) 
The combination of an AR(p) and a MA(q) model leads to the 
ARMA(p,q) model which expresses the current value x(t) as a 
weighted sum of the p previous values of the time series 
and a weighted sum of the current and q previous 
innovations. 
p q 
ARMA(p,q) : x(t) + = a,-x(t-i) - e(t) * Z b,-e(t-j) 
iz 121 
Introducing the z-transform which can be interpreted as the 
unit delay operator 
z" x(t) - x(t-m) 
the ARMA model may be written 
A(z):x(t) - B(z)*e(t) 
with the polynomials in z 
A(z)- d13, 2 ho Vp 
B(z) - 1b, z^ i + 
Stationarity is an important property of many stochastic 
processes. A stochastic process is said to be strictly 
stationary if its statistical distribution function is not 
affected by a shift along the time axis. A less restrictive 
condition, called weak stationarity, is satisfied if the 
expectation, the variance and the autocorrelation function 
of a stochastic process are independent of time t 
E- (x(t)) - 
E. (x 6C). -. 1) ?4 - 
E {x(t)->x(t + n)} = 
- constant 
- constant 
p 
g 2 
x 
R, (m) only a function in m. 
- 428 -