Aggregation of time-series variables: a survey

The first three sections of this chapter will consider three types of aggregation that occur in the context of the analysis of time series: (a) ‘small-scale’ aggregation, (b) ‘large-scale’ aggregation, and (c) temporal aggregation. An example of small-scale aggregation is if X_t is generated by an AR(1) equation, such as https://www.w3.org/1998/Math/MathML"> X t = α 1 X t − 1 + ε 1 t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315822549/a1c36de0-fe94-4936-be4b-07495fc3034a/content/math_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and similarly Y_t is AR(1) https://www.w3.org/1998/Math/MathML"> Y t = α 2 Y t − 1 + ε 2 t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315822549/a1c36de0-fe94-4936-be4b-07495fc3034a/content/math_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where ε_{1 t} ε_{2 t} are each white noises and if X _t, Y _t are independent, so that ε_{1 t} ε_{2 t} are independent, then what model does https://www.w3.org/1998/Math/MathML"> S t = X t + Y t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315822549/a1c36de0-fe94-4936-be4b-07495fc3034a/content/math_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> obey? The answer is found usually to be ARMA(2,1). Small-scale aggregation involves sums of a few time-series variables, which are not necessarily independent. A clear practical conclusion is that mixed models, i.e. ARMA (p, q), are likely to arise from small-scale aggregation.