Abstract:
The paper presents unified analytical solution for combining high-frequency and low-frequency economic time-series by additive and proportional Denton methods with parametrical dependence on the initial values of variable and indicator in evident form. This solution spans Denton’s original and Cholette’s advanced benchmarking initial conditions as the subcases. Computational complexity of the obtained solution is associated with inversion of a square matrix of the order that is equal to the number of low-frequency observations available. Practical applying the proposed solution under data revisions allows to construct suboptimal concatenation of frozen and newly revised parts of benchmarked time-series by using the last benchmarked-to-indicator ratio (or benchmarked and indicator difference in additive case) from the range of data fixed as initial condition for benchmarking or re-benchmarking the newly revised data by the proportional (or additive) Denton method.
Machine summary:
A Generalization of Initial Conditions in Benchmarking of Economic Time-Series by Additive and Proportional Denton Methods Vladimir Motorin1 Received: 2015/12/19 Accepted: 2016/01/09 Abstract he paper presents unified analytical solution for combining high- frequency and low-frequency economic time-series by additive andproportional Denton methods with parametrical dependence on theinitial values of variable and indicator in evident form.
3. Objective functions in additive and proportional Denton methods For algebraic notation convenience, let us formally expand the vectors q, d, x, and y by attaching to them the initial scalar values associated with sub period 0 before the range of observations available.
Sqˆ for the proportional Thus, the generalized time-series benchmarking problem within Denton first difference approach is to minimize the unified quadratic objective function (8) subject to linear constraints (9).
4. Initial conditions for benchmarking of time-series The unified objective function (8) of variable z with parameter z0 can be written in matrix notation by two following ways: 2 and f (z; z0 ) = z¢Δ¢WΔ z - 2w1z1z0 + w1z0 f (z; z0 ) = z¢Δ1¢ W1Δ1z + w1z2 - 2w z z + w z2 (10) (11) where W is nonsingular diagonal matrix of order T with weight (relative reliability) coefficient w ={wt½ t = 1¸T} on its main diagonal and W1 is its nonsingular square submatrix of order T–1 obtained by deleting the first column and the first row from matrix W.