Forecasting Aggregated Moving Average Processes with an Application to the Euro Area Real Interest Rate

Abstract

This paper focuses on the contemporaneous aggregation of moving average processes. It is shown that aggregating across second (or first)-order (integrated) moving average processes leads to a macro process whose parameters are exact functions of the parameters of its generation process. Similar results are obtained at single equation level when a vector moving average framework is considered. In addition, the out-of-sample forecasting properties of aggregate and disaggregate procedures to forecast the aggregate variable are provided. Moreover, it is shown that the condition of equality of aggregate and disaggregate predictors is not necessary for the equality of their mean squared errors. Finally, an application to the euro area real interest rate is presented and discussed.

Explicit Form of Coefficients in any MA(2) Process

“…Sbrana (2011) (2012) asserts that there is only one solution of (10), which he expresses explicitly in terms of γ 0 , γ 1 , γ 2 , which gives (θ 1 , θ 2 ) in (9) corresponding to an invertible process (that is: satisfying (4)-( 6)). His reasoning is graphical (Sbrana, 2012), based on scanning Figure 3 of Stralkowski et al (1974), which is Chart C, p.663 of Box et al (2008).…”

“…Note that Case(4) is impossible. If D is defined to be (30) with strict inequality, then Case(1) and Case (5) can each be split, into Case(1a) and Case(5a), that is ABCD and ABC c D respectively; and Case(1b) and Case(5b), that is ABCD c and ABC c D c respectively. Only Case (1) and Case(5) may be so split, of the possible seven cases.…”

“…We can actually simplify: only four situations are needed to classify all cases based on the constraints of (θ 1 , θ 2 ) in view of ( 7 ), that is Case(1a) and Case(1b), then use x 4 as the correct σ 2 , the largest of (5), that is Case(5a) and Case(5b), or Case (7), then use x 3 as the correct σ 2 , the smallest of x i (the sign of H + in z + exchanged, so the correct σ 2 changes from x 4 (in Case(1)) to 2) or Case(3), then use x 2 as the correct σ 2 , the second largest of x i (the sign of H − in z − exchanged, hence z − , z + exchanged, so the correct σ 2 changes from x 4 (in Case(1)) to 6) or Case(8), then use x 1 as the correct σ 2 , the third largest of x i (the sign of H + in z + , and H − in z − both exchanged, so the correct σ 2 changes from x 4 (in Case(1)) to x 1 ).…”