A third-order autoregressive mode is fitted to an annual time series with 17 values and has the following estimated parameters and standard errors:
a0 = 4.5, a1 = 1.8, a2 = .8, a3 = .24, Sa1 = .5, Sa2 = .3, Sa3 = .1.
The three most recent values are Y15 = 23, Y16 = 28, and Y17 = 34.
Forecast the values for the next year and the following year.
To forecast the values for the next year and the following year, we can use the third-order autoregressive model with the estimated parameters and the most recent values.
The autoregressive model is given by:
Y_t = a0 + a1*Y_t-1 + a2*Y_t-2 + a3*Y_t-3 + ε_t
To forecast Y18 (the next year), we substitute the most recent values into the model:
Y18 = 4.5 + 1.8*34 + 0.8*28 + 0.24*23
= 4.5 + 61.2 + 22.4 + 5.52
= 93.62
To forecast Y19 (the following year), we need to use the forecasted value of Y18 and again substitute into the model:
Y19 = 4.5 + 1.8*93.62 + 0.8*34 + 0.24*28
= 4.5 + 168.51 + 27.2 + 6.72
= 206.93
Therefore, the forecast for the next year (Y18) is 93.62 and the forecast for the following year (Y19) is 206.93.