6.
Process or Product Monitoring and Control
6.6. Case Studies in Process Monitoring 6.6.2. Aerosol Particle Size

AR(2) Model Parameter Estimates 
The following parameter estimates were computed for the AR(2)
model based on the differenced data.
Parameter Standard 95 % Confidence Source Estimate Error Interval     Intercept 0.0050 0.0119 AR1 0.4064 0.0419 (0.4884, 0.3243) AR2 0.1649 0.0419 (0.2469, 0.0829) Number of Observations: 558 Degrees of Freedom: 558  3 = 555 Residual Standard Deviation: 0.4423 Both AR parameters are significant since the confidence intervals do not contain zero. The model for the differenced data, \(Y_t\), is an AR(2) model, $$ Y_{t} = 0.4064 Y_{t1}  0.1649 Y_{t2}  0.0050 \, , $$ with \(\sigma = 0.4423\). It is often more convenient to express the model in terms of the original data, \(X_t\), rather than the differenced data. From the definition of the difference, \(Y_t = X_t  X_{t1}\), we can make the appropriate substitutions into the above equation, $$ X_{t}  X_{t1} = 0.4064 (X_{t1}  X_{t2})  0.1649 (X_{t2}  X_{t3})  0.0050 \, , $$ to arrive at the model in terms of the original series, $$ X_{t} = 0.5936 X_{t1} + 0.2415 X_{t2} + 0.1649 X_{t3}  0.0050 \, . $$ 
MA(1) Model Parameter Estimates 
Alternatively, the parameter estimates for an MA(1) model
based on the differenced data are the following.
Parameter Standard 95 % Confidence Source Estimate Error Interval     Intercept 0.0051 0.0114 MA1 0.3921 0.0366 (0.4638, 0.3205) Number of Observations: 558 Degrees of Freedom: 558  2 = 556 Residual Standard Deviation: 0.4434 The model for the differenced data, \(Y_t\), is an ARIMA(0,1,1) model, $$ Y_{t} = a_{t}  0.3921 a_{t1}  0.0051 \, , $$ with \(\sigma = 0.4434\). It is often more convenient to express the model in terms of the original data, \(X_t\), rather than the differenced data. Making the appropriate substitutions into the above equation, $$ X_{t}  X_{t1} = a_{t}  0.3921 a_{t1}  0.0051 \, , $$ we arrive at the model in terms of the original series, $$ X_{t} = X_{t1} + a_{t}  0.3921 a_{t1}  0.0051 \, . $$ 