Forecasting VAR processes

Neglecting deterministic terms and exogenous variables the levels VAR form (32.1) is particularly convenient to use in forecasting the variables yt. Suppose the ut are generated by an independent rather than just uncorrelated white noise process. Then the optimal (minimum MSE) one-step forecast in period T is the conditional expectation,

yT+1|T = E(yT+1 | yT, yT-1, . . . ) = A1yT + … + ApyT+1-p. (32.14)

Forecasts for larger horizons h > 1 may be obtained recursively as

yT+h | T = A1 yT+h-1|T + … + ApyT+h-p|T, (32.15)

where yT+;-|T = yT+j for j < 0. The corresponding forecast errors are

yT+h — yT+h | T = UT+h + ®1UT+h-1 + … + ФУ—1uT+1, (32.16)

where it is easy to see by successive substitution that Ф8 = j Aj (s = 1, 2,…)

with Ф0 = IK and Aj = 0 for j > p (see Lutkepohl, 1991, sec. 11.3). Hence, the forecasts are unbiased, that is, the forecast errors have expectation 0 and their MSE matrix is

h-1

£y(h) = E{(yT+h — yT+h|T)(yT+h — yT+h|T) } = ^Ф j^uty. (32.17)

j=0

For any other h-step forecast with MSE matrix X*(h), say, the difference X*(h) — Xy(h) is a positive semidefinite matrix. This result relies on the assumption that ut is independent white noise, i. e. ut and us are independent for s Ф t. If ut is uncorrelated white noise and not necessarily independent over time, these fore­casts are just best linear forecasts in general (see Lutkepohl, 1991, sec. 2.2.2).

The forecast MSEs for integrated processes are generally unbounded as the horizon h goes to infinity. Thus the forecast uncertainty increases without bounds for forecasts of the distant future. This contrasts with the case of I(0) variables for which the forecast MSEs are bounded by the unconditional covariance Xy of yt. This means, in particular, that forecasts of cointegration relations have bounded MSEs even for horizons approaching infinity. The corresponding forecast inter­vals reflect this property as well. Assuming that the process yt is Gaussian, that is, ut ~ iid N(0, Xu), the forecast errors are also multivariate normal. This result may be used to set up forecast intervals in the usual way.

In practice the parameters of a VAR process are usually estimated. Denoting by yT+h|T the forecast based on estimated coefficients corresponding to yT+h|T the associated forecast error is

yT+h — yT+h|T = [yT+h — yT+h | T] + [yT+h|T — yT+h | T]-

At the forecast origin T the first term on the right-hand side involves future residuals only whereas the second term involves present and past variables only, provided only past variables have been used for estimation. Consequently, if ut is independent white noise, the two terms are independent. Moreover, under standard assumptions, the difference yT+h|T – yT+hT is small in probability as the sample size used for estimation gets large. Hence, the forecast error covariance matrix in this case is

£D(h) – E{[yT+h – Ут+ht][yT+h – yT+hT],}

= £ y(h) + o(1),

where o(1) denotes a term which approaches zero as the sample size tends to infinity. Thus, for large samples the estimation uncertainty may be ignored in evaluating the forecast precision and setting up forecast intervals. In small samples, including a correction term is preferable, however. In this case the precision of the forecasts will depend on the precision of the estimators. Hence, if precise forecasts are desired, it is a good strategy to look for precise parameter estimators.

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