# 1041 forecasts from the regression model with first

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10.4.1 FORECASTS FROM THE REGRESSION MODEL WITH FIRST-ORDER AUTOREGRESSIVE ERRORS In order to simplify the notation, we consider the case of a single explanatory variable. The model is given as y t = β 0 + β 1 x t + ε t and ε t = φε t 1 + a t (10.33) Suppose that we have data available for time periods t = 1 , 2 ,..., n . The obser- vation at the next time period, n + 1, can be written as y n + 1 = φ y n + ( 1 φ)β 0 + β 1 ( x n + 1 φ x n ) + a n + 1 The value of the explanatory variable at time n + 1 , x n + 1 , is assumed known. However, the random error a n + 1 is unknown. Since random errors are assumed independent, a n + 1 is not predictable from the previous errors a n , a n 1 ,..., a 1 , and E ( a n + 1 | a n , a n 1 ,..., a 1 ) = E ( a n + 1 ) = 0. Hence, the one-step-ahead predic- tion is y n ( 1 ) = φ y n + ( 1 φ)β 0 + β 1 ( x n + 1 φ x n ) (10.34) and the one-step-ahead prediction error is y n + 1 y n ( 1 ) = a n + 1 , with variance σ 2 a . A 95% prediction interval for the future observation y n + 1 is given by y n ( 1 ) ± ( 1 . 96 a or φ y n + ( 1 φ)β 0 + β 1 ( x n + 1 φ x n ) ± ( 1 . 96 a The observation two-steps-ahead is y n + 2 = φ y n + 1 + ( 1 φ)β 0 + β 1 ( x n + 2 φ x n + 1 ) + a n + 2 = φ [ φ y n + ( 1 φ)β 0 + β 1 ( x n + 1 φ x n ) ] + ( 1 φ)β 0 + β 1 ( x n + 2 φ x n + 1 ) + a n + 2 + φ a n + 1 Future random shocks have mean zero. The two-step-ahead forecast is y n ( 2 ) = φ [ φ y n + ( 1 φ)β 0 + β 1 ( x n + 1 φ x n ) ] + ( 1 φ)β 0 + β 1 ( x n + 2 φ x n + 1 ) = φ y n ( 1 ) + ( 1 φ)β 0 + β 1 ( x n + 2 φ x n + 1 ) (10.35) The two-step-ahead forecast error is y n + 2 y n ( 2 ) = a n + 2 + φ a n + 1 , with variance ( 1 + φ 2 2 a .
Abraham Abraham ˙ C10 November 8, 2004 12:5 322 Regression Models for Time Series Situations The r -step-ahead forecast of y n + r can be calculated from the difference equation y n ( r ) = φ y n ( r 1 ) + ( 1 φ)β 0 + β 1 ( x n + r φ x n + r 1 ) (10.36) The r -step-aheadforecasterror y n + r y n ( r ) = a n + r + φ a n + r 1 + ··· + φ r 1 a n + 1 has variance ( 1 + φ 2 + ··· + φ 2 ( r 1 ) 2 a = 1 φ 2 r 1 φ 2 σ 2 a . A 95% prediction interval is given by y n ( r ) ± 1 . 96 σ a [ ( 1 φ 2 r )/( 1 φ 2 ) ] 1 / 2 (10.37) If the parameters are estimated from past data, we must replace the parameters β 0 , β 1 , and φ with their estimates. In the prediction interval, we replace σ 2 a by s 2 a in Eq. (10.27). However, note that the resulting interval is optimistic and too narrow because the substitution approach does not incorporate the uncertainty from the parameter estimation. One can also incorporate this uncertainty, but the analysis becomes tedious. 10.4.2 FORECASTS FROM THE REGRESSION MODEL WITH ERRORS FOLLOWING A NOISY RANDOM WALK In this case, the observation at time n + 1 can be written as y n + 1 = y n + β 1 ( x n + 1 x n ) + a n + 1 θ a n (10.38) wherea n + 1 isafutureerror,anditsmean E ( a n + 1 | a n , a n 1 ,..., a 1 ) = E ( a n + 1 ) = 0. However, a n is an error that has been realized already. Assuming that β 1 and θ are known, the forecast is y n ( 1 ) = y n + β 1 ( x n + 1 x n ) θ a n (10.39) The term a n is the last component in the recursions of Eq. (10.32), which for given values of the parameters is easy to calculate. Note that the model in Eq. (10.38) in-