Asymmetric time aggregation and its potential benefits for forecasting annual data

Abstract

For many economic time-series variables that are observed regularly and frequently, for example weekly, the underlying activity is not distributed uniformly across the year. For the aim of predicting annual data, one may consider temporal aggregation into larger subannual units based on an activity timescale instead of calendar time. Such a scheme may strike a balance between annual modeling (which processes little information) and modeling at the finest available frequency (which may lead to an excessive parameter dimension), and it may also outperform modeling calendar time units (with some months or quarters containing more information than others). We suggest an algorithm that performs an approximate inversion of the inherent seasonal time deformation. We illustrate the procedure using two exemplary weekly time series.

Appendix

This appendix contains detailed derivations for the examples of Sect. 2. Most of them rely on the feature that prediction based on data at the generating frequency entails a straightforward evaluation of variances, while first differences of annual data follow first-order moving-average processes. The minimum forecast variance is then slightly smaller than the variance of the first differences. The role model case of the Lemma 1 can be used with little variation in all examples.

Lemma 1

Assume the random walk with independent increments $X_{\tau ,w}=X_{\tau ,w-1}+\varepsilon _t$ and its annual aggregate $Z_{\tau }=\sum _{w=1}^SX_{\tau ,w}$. The forecast error variance using the annual aggregate is given as

$$\begin{aligned} \frac{S(S^2-1)^2\sigma _{\varepsilon }^2}{6\{2S^2+1-S\sqrt{3(S^2+2)}\}}, \end{aligned}$$

denoting $\sigma _{\varepsilon }^2=\mathrm{var}\varepsilon _t$.

Proof

With an insubstantial modification, this is the situation analyzed by Working (1960), who obtained the main result that $Z_{\tau }-Z_{\tau -1}=\eta _{\tau }$ follows a first-order moving-average process $\eta _{\tau }=\xi _{\tau }+\theta \xi _{\tau -1}$ with first-order correlation

$$\begin{aligned} \rho _1=\frac{S^2-1}{2(2S^2+1)} \end{aligned}$$

and variance

$$\begin{aligned} \mathrm{var}(\eta _{\tau })=\frac{S(2S^2+1)}{3}=\sigma _{\eta }^2. \end{aligned}$$

For some of our arguments, it is convenient to note that this expression follows from a triangular weighted sum of errors via

$$\begin{aligned} \sigma _{\eta }^2=\sigma _{\varepsilon }^2\left( \sum _{w=1}^Sw^2+\sum _{w=1}^{S-1}w^2\right) , \end{aligned}$$

a two-sided weighted sum of error variances at the generating frequency. Solving the quadratic equation $\rho _1=\theta /(1+\theta ^2)$ yields

$$\begin{aligned} \theta =\frac{2S^2+1-S\sqrt{3(S^2+2)}}{S^2-1}. \end{aligned}$$

The variance $\mathrm{var}(\xi _{\tau })=\sigma _{\xi }^2$ corresponds to the minimum forecast variance and evolves from evaluating

$$\begin{aligned} \sigma _{\xi }^2=\frac{\sigma ^2_{\eta }}{1+\theta ^2}. \end{aligned}$$

$\square$

Working (1960) remarks that, for larger $S,\, \rho _1$ approaches $0.25$, and even the smallest $S=2$ yields $\rho _1=1/6$. Also note that $\theta \approx \rho _1$, and that $\sigma _{\xi }^2$ is only slightly smaller than $\sigma _{\eta }^2$, the forecast variance due to the naive forecast $Z_{\tau }$ that incorrectly assumes that it follows a random walk.

Example 1. First assume disaggregated data are available. Then

$$\begin{aligned} \hat{Z}_{\tau +1}&= {E}(Z_{\tau +1}|X_{\tau ,s},\ldots ,X_{\tau ,1},\ldots )\\&= { E}\left( \sum _{w=1}^SX_{\tau +1,w}|X_{\tau ,s},\ldots \right) =SX_{\tau ,S}, \end{aligned}$$

and thus

$$\begin{aligned}&{E}(Z_{\tau +1}-\hat{Z}_{\tau +1})^2 \\&= {E}\left\{ SX_{\tau ,S}+\sum _{w=1}^S(S-w+1)\varepsilon _{\tau +1,w}-SX_{\tau ,S}\right\} ^2 \\&= {E}\left\{ \sum _{w=1}^S(S-w+1)\varepsilon _{\tau +1,w}\right\} ^2 = \sigma _{\varepsilon }^2\sum _{w=1}^Sw^2. \end{aligned}$$

If only annual data are available, Lemma 1 can be applied directly. From the proof of the Lemma, observe that $\sigma _{\xi }^2$ considerably exceeds the above expression that is a one-sided weighted sum. The correction factor is too close to one to compensate this discrepancy.

Example 2. In this case, $Z_{\tau }-Z_{\tau -1}$ is independent white noise at the annual frequency, and both the forecast for annual and for disaggregated data are clearly given as $\hat{Z}_{\tau +1}=Z_{\tau }$.

Example 3. Denote the information set formed by past observations $\{X_s,s\le \tau \}$ by $H_{\tau }(X)$. By definition,

$$\begin{aligned} Z_{\tau }=\sum _{w=1}^S(\delta _w+\varepsilon _{\tau ,w}) \end{aligned}$$

and hence

$$\begin{aligned} {E}(Z_{\tau +1}|H_{\tau })=\sum _{w=1}^S\delta _w. \end{aligned}$$

The conditional expectation is identical if data $X_{\tau ,w}$ are available and hence also the concomitant prediction error variance does not change.

Example 4. First consider the annual variable. All calculations closely follow the proof of Lemma 1. Here,

$$\begin{aligned} Z_{\tau +1}-Z_{\tau }=\sum _{w=1}^4\varepsilon _{\tau +1,w}-\sum _{w=1}^2\varepsilon _{\tau ,w}, \end{aligned}$$

an MA(1) process $\eta _{\tau }=\xi _{\tau }+\theta \xi _{\tau -1}$ with variance $6\sigma _{\varepsilon }^2$, first-order covariance $-2\sigma _{\varepsilon }^2$, and hence first-order correlation $\rho _1=-1/3$. The implied MA coefficient follows from equating

$$\begin{aligned} \frac{\theta }{1+\theta ^2}=-\frac{1}{3}, \end{aligned}$$

which yields $\theta =-0.5*(3-\sqrt{5})\approx -0.382$. The MSE is the variance of the implied white noise $\xi _t$ or

$$\begin{aligned} \sigma _{\xi }^2=6\sigma _{\varepsilon }^2/(1+\theta ^2)\approx 5.236\sigma _{\varepsilon }^2. \end{aligned}$$

The conditional expectation of $Z_{\tau +1}$ based on quarterly data is $X_{\tau ,1}+X_{\tau ,2}$, which yields a prediction error of just

$$\begin{aligned} Z_{\tau +1}-X_{\tau ,1}+X_{\tau ,2}=\sum _{w=1}^4\varepsilon _{\tau +1,w}, \end{aligned}$$

with variance $4\sigma _{\varepsilon }^2$.

Example 5. The generating process

$$\begin{aligned} X_{\tau ,w}= \left\{ \begin{array}{l@{\quad }l} X_{\tau -1,2}+\varepsilon _{\tau ,1}, &{} w=1,\\ X_{\tau ,1}+\varepsilon _{\tau ,2}, &{} w=2,\\ \varepsilon _{\tau ,w}, &{} w=3,4,\\ \end{array} \right. \end{aligned}$$

has the forecast based on quarterly data

$$\begin{aligned} {E}(Z_{\tau +1}|X_{\tau ,4},X_{\tau ,3},\ldots )=2X_{\tau ,2} \end{aligned}$$

with error variance

$$\begin{aligned} {E}(Z_{\tau +1}-2X_{\tau ,2})^2&= {E}(2\varepsilon _{\tau +1,1}+\varepsilon _{\tau +1,2}+\varepsilon _{\tau +1,3}+\varepsilon _{\tau +1,4})^2\\&= 7\sigma _{\varepsilon }^2. \end{aligned}$$

By contrast, if only annual data are available, $Z_{\tau }-Z_{\tau -1}=\eta _{\tau }$ again follows a first-order MA process

$$\begin{aligned} \eta _{\tau }=2\varepsilon _{\tau ,1}+\sum _{w=2}^4 \varepsilon _{\tau ,w}+\varepsilon _{\tau -1,2}- \sum _{w=3}^4\varepsilon _{\tau -1,w}, \end{aligned}$$

with variance $\sigma _{\eta }^2=10\sigma _{\varepsilon }^2,\, \rho _1=-0.1$, and concomitant $\theta \approx -0.1010$. This results in a prediction error variance of around $9.899\sigma _{\varepsilon }^2$. The two regrouping variants can be evaluated similarly but calculations become a bit involved. As outlined in the text, they were determined by calculation and confirmed by Monte Carlo at $7.84\sigma _{\varepsilon }^2$ for semesters and at slightly above $9\sigma _{\varepsilon }^2$ for grouping into the first quarter and the remaining quarters as pseudo-semesters.