Reconciliation of systems of time series according to a growth rates preservation principle

Abstract

We propose new simultaneous and two-step procedures for reconciling systems of time series subject to temporal and contemporaneous constraints according to a growth rates preservation (GRP) principle. The techniques exploit the analytic gradient and Hessian of the GRP objective function, making full use of all the derivative information at disposal. We apply the new GRP procedures to two systems of economic series, and compare the results with those of reconciliation procedures based on the proportional first differences (PFD) principle, widely used by data-producing agencies. Our experiments show that (1) the nonlinear GRP problem can be efficiently solved through an interior-point optimization algorithm, and (2) GRP-based procedures preserve better the growth rates than PFD solutions, especially for series with high temporal discrepancy and high volatility.

Appendix: Gradient and Hessian of the global GRP criterion

In this appendix we present the analytical expressions of the gradient vector and of the Hessian matrix for the function $F_{\textit{GRP}}$ as defined in (5), which can be exploited by Newton-type nonlinear programming optimization procedures. The derivation is a straightforward extension of the expressions shown by Di Fonzo and Marini (2013b) for the univariate GRP criterion.

Let us denote with $y_{j,t}$ and $p_{j,t}$, respectively, the $j$th target and preliminary series of the system observed in the period $t$, with $j=1,\ldots ,m,\, t=1,\ldots ,n$, where $m$ is the number of variables and $n$ the number of the high-frequency periods. The value $\{y_{j,t}\}$ is the $[i+(j-1)n]$th element in the stacked $nm$-dimensional vector $\mathbf{y}$. The gradient vector of $F_\textit{GRP}$ is the $(nm \times 1)$ vector

$$\begin{aligned} {\nabla } F_\textit{GRP}( \mathbf{{y}} ) = \mathbf{g}( \mathbf{{y}} ) = \left\{ {{g_i}} \right\} _{i = 1}^{nm}, \end{aligned}$$

where

$$\begin{aligned} {{g_{1+(j-1)n}}}= & {} - 2\displaystyle \frac{{{y_{j,2}}}}{{y_{j,1}^2}}\left( {\frac{{{y_{j,2}}}}{{{y_{j,1}}}} - \frac{{{p_{j,2}}}}{{{p_{j,1}}}}} \right) \\ {{g_{t+(j-1)n}}}= & {} \displaystyle \frac{2}{{{y_{j,t - 1}}}}\left( {\frac{{{y_{j,t}}}}{{{y_{j,t - 1}}}} - \frac{{{p_{j,t}}}}{{{p_{j,t - 1}}}}} \right) - 2\frac{{{y_{j,t + 1}}}}{{y_{j,t}^2}}\left( {\frac{{{y_{j,t + 1}}}}{{{y_{j,t}}}} - \frac{{{p_{j,t + 1}}}}{{{p_{j,t}}}}} \right) \\ {{g_{n+(j-1)n}}}= & {} \displaystyle \frac{2}{{{y_{j,n - 1}}}}\left( {\frac{{{y_{j,n}}}}{{{y_{j,n - 1}}}} - \frac{{{p_{j,n}}}}{{{p_{j,n - 1}}}}} \right) \\ \end{aligned}$$

for $j=1,\ldots ,m$ and $t = 2, \ldots ,n - 1$.

Let us denote the elements of the Hessian matrix, ${\nabla }^2 F_{\textit{GRP}}( \mathbf{{y}} ) = \mathbf{H}(\mathbf{y})$, as

$$\begin{aligned} {h_{r,s}} = \frac{{{\partial ^2}F_{\textit{GRP}}\left( \mathbf{{y}} \right) }}{{\partial {y_r}\partial {y_s}}} = \frac{{\partial {g_r}}}{{\partial {y_s}}},\quad r,s = 1, \ldots ,nm. \end{aligned}$$

Notice that the Hessian matrix is both symmetric and tri-diagonal, that is its non-zero items are $h_{s,s},\, s=1,\ldots ,nm,\, h_{s-1,s},\, s=2,\ldots ,nm$, and $h_{s+1,s},\, s=1,\ldots ,nm-1$. After some calculations, for $j=1,\ldots ,m$ we find:

$$\begin{aligned}&\displaystyle h_{1+(j-1)n, 1+(j-1)n} = 2\displaystyle \frac{{{y_{j,2}}}}{{y_{j,1}^3}}\left( {3\frac{{{y_{j,2}}}}{{{y_{j,1}}}} - 2\frac{{{p_{j,2}}}}{{{p_{j,1}}}}} \right) \\&\displaystyle h_{i+(j-1)n,i+(j-1)n} = \displaystyle \frac{2}{{y_{j,t - 1}^2}} + 2\frac{{{y_{j,t + 1}}}}{{y_{j,t}^3}}\left( {3\frac{{{y_{j,t + 1}}}}{{{y_{j,t}}}} - 2\frac{{{p_{j,t + 1}}}}{{{p_{j,t}}}}} \right) \quad t=2,\ldots , n-1 \\&\displaystyle h_{n+(j-1)n,n+(j-1)n} = \displaystyle \frac{2}{{y_{j,n - 1}^2}} \\&\displaystyle h_{i+(j-1)n,k+(j-1)n} = - \displaystyle \frac{2}{{y_{j,i}^2}}\left( {2\frac{{{y_{j,k}}}}{{{y_{j,i}}}} - \frac{{{p_{j,k}}}}{{{p_{j,i}}}}} \right) \\&\displaystyle i=k+1, k=1,\ldots , n-1 \quad \vee \quad i=k-1, k=2,\ldots ,n. \\ \end{aligned}$$