Abstract
In the modifiable areal unit problem (MAUP) literature, it is known that in measuring spillover effects, the magnitudes of spillovers are different depending on the scale of the temporal/spatial units. However, the research has not addressed whether the magnitude of the spillover increases or decreases as the unit of measurement increases. From an exercise using a constructed regional economic system, it is shown that depending on how researchers make assumptions about the data generating process of regional economies, the magnitude of the measured spillover may be smaller or larger as the unit of measurement increases. It is also argued that with a reasonable assumption that there is a common factor affecting the data generating processes of regional units (i.e., the regional economic system is characterized by a multi-level structure), the larger the unit of scale, the smaller the amount of spillover.
Keywords
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The research was performed while resident in the Regional Economics Applications Laboratory and Department of Economics, University of Illinois, Urbana, IL, 61801-3671
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- 1.
- 2.
Also, since cumulative impulse response function (CIRF) measures the cumulative effect of a regional shock on the future values of regional values, the “direction” of a spillover can be defined as the sign of the long-term cumulative response. An exercise on the CIRFs is also considered here, but it is found that CIRF results do not exhibit any pattern that can be generalized on a geographically defined economic system. The results are presented in Appendix 1. All appendices are available at www.real.illinois.edu/d-paper/15/Aggregation%20Appendix.pdf
- 3.
Contrary to the multi-level structure regional economic system, a single-level structure regional economic system does not consider the existence of the higher-level common factor. More details can be found in Chung and Hewings (2015).
- 4.
Almost exogenous means regional shock does not have a significant impact on the region common economic behaviors. More practically, in the model structure, the coefficients associated with the effect from regional shock to the region common behavior should be close to zero so that the local impact on the global behavior decays very fast throughout time. Conceptually, a multi-level structure model with endogenous region common factor can be regarded as a single-level structure model in our exercise.
- 5.
Brewer (1973), Wei (1981), and Weiss (1984) tackle the issues related to temporal aggregation in empirical studies. Marcellino (1999) reviewing the literature on this issue showed that impulse response functions and forecast error variance decompositions, along with other properties such as Granger-causality and cointegration, change with the level of aggregation. On the other hand, the spatial aggregation problem, or the modifiable areal unit problem (MAUP), has also been an interest of spatial analysts for a long time since the pioneering work of Gehlke and Biehl (1934). The effects on standard regression estimators are addressed in Barker and Pesaran (1990), Okabe and Tagashira (1996), Tagashira and Okabe (2002), and Griffith et al. (2003). Their main findings are that “the GLS estimators of regression’s parameters are BLUE with a sampling variance greater than that obtained using GLS on the original data” (Arbia and Petrarca 2011). Arbia and Petrarca (2011) also explored the efficiency loss of the estimators in the presence of spatial dependency.
- 6.
Since the spillover effects varies not only with space but also with time, the study of spillover effects should be conducted in terms both spatial and temporal scale.
- 7.
Bayesian inference using Gibbs sampling for Windows.
- 8.
Only VAR(1) structure model estimation results are presented in Sect. 3.2 since, according to the deviations information criteria (DIC), any higher lag order does not outperform VAR(1) structure of equations introduced in Sect. 3.1. Also, the Bayesian inference relies on priors, but for this exercise, uninformative priors were used. More detailed procedures can be found in Chung and Hewings (2015).
- 9.
It is named after the interstate highway I74, because those counties are located along this highway.
- 10.
Except for the regional division level spatial units, the selection of the regional units at all other levels suffer borderline problem since there is a possibility that some relevant regional unit could have been omitted. However, further consideration of solving this borderline problem was not tried here since this section aims to sketch how the aggregation of spatial temporal unit affects the observed spillover effects in the real-world data and does not aim to exactly identify the regional economic system.
- 11.
The results for single-level model are also available in Appendix 2.
- 12.
If a time series is stock data, the aggregation of the time series can be a point-in-time sampling, for example, Aggregated Observation at t = Last Disaggregated Observation during t. For more detail, see Marcellino (1999).
- 13.
For example, \( {\displaystyle \begin{array}{l}\ln \left({a}_t+{b}_t+{c}_t\right)-\ln \left({a}_{t-1}+{b}_{t-1}+{c}_{t-1}\right)\approx \frac{a_t-{a}_{t-1}+{b}_t-{b}_{t-1}+{c}_t-{c}_{t-1}}{a_{t-1}+{b}_{t-1}+{c}_{t-1}}\\ {}\kern1em \approx \frac{1}{3}\left(\frac{a_t-{a}_{t-1}}{a_{t-1}}+\frac{b_t-{b}_{t-1}}{a_{t-1}}+\frac{c_t-{c}_{t-1}}{a_{t-1}}\right)\approx \frac{1}{3}\left(\frac{a_t-{a}_{t-1}}{a_{t-1}}+\frac{b_t-{b}_{t-1}}{b_{t-1}}+\frac{c_t-{c}_{t-1}}{c_{t-1}}\right)\\ {}\kern1em \approx \frac{1}{3}\left\{\left(\ln {a}_t-\ln {a}_{t-1}\right)+\left(\ln {b}_t-\ln {b}_{t-1}\right)+\left(\ln {c}_t-\ln {c}_{t-1}\right)\right\}\end{array}} \)
- 14.
In February 2013, Beverly 18, a movie theater in Champaign, IL, closed, and shortly after, Savoy 16, a neighborhood movie theater in Savoy, IL, opened a new I-Max theater, which provides an example of a negative spatial spillover effect at the disaggregated level data.
- 15.
That is, the root of |IR − A z| = 0 from Eq. (3.5) falls outside the unit circle.
- 16.
A more general version of this kind of structure can be expressed as equation (∗):
$$ {Y}_t={\sum \limits}_{i=1}^p{A}_i{Y}_{t-i}+{\varepsilon}_t,\kern0.5em t=1,\dots, T $$(∗)where
-
\( {Y}_t={\left({y}_t^1,{y}_t^2,\dots {y}_t^R\right)}^{\prime } \) is an R × 1 dependent variable,
-
{Ai| i = 1, .., p} are R × R coefficient matrices, \( E\left({\varepsilon}_t\right)=0,E\left({\varepsilon}_t{\varepsilon_t}^{\prime}\right)=\Sigma\ \left(\mathrm{a}\ \mathrm{positive}\ \mathrm{definite}\ \mathrm{coviariance}\ \mathrm{matrix}\right),\kern0.5em E\left({\varepsilon}_t{\varepsilon_{t^{\prime}}}^{\prime}\right)=0\forall t\ne {t}^{\prime } \)
For stationarity, all roots of \( \left|{I}_R-{\sum}_{i=1}^p{A}_i{z}^i\right|=0 \) fall outside the unit circle. Also, (∗) can be expressed as VMA form as in equation (∗∗):
$$ {Y}_t={\sum}_{i=1}^{\infty }{\Phi}_i{\varepsilon}_{t-i} $$(∗∗)where
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\( {\Phi}_i={\sum}_{j=1}^p{A}_j{\Phi}_{i-j},i=1,2,\dots \), Φ0 = IR, and Φi = 0 ∀ i < 0. Since in this general case where the error term structure is not diagonal, the time profile of the shock affects the FEVD and CIRF; thus in our case, the error term structure is set to be diagonal for simplicity.
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- 17.
The generalized versions of FEVDs and CIRFs with different spatiotemporal scale considering the contemporaneous shocks are also provided in the footnotes of the next subsection, in case the readers of this chapter wants to conduct a simulation with more complex regional economic system.
- 18.
A graphical example of a CIRF of a spatially non-stationary process is provided in Appendix 4.
- 19.
Temporally/spatially aggregated forms of CIRFs are provided in Appendix 5.
- 20.
For example, ϵτ = ⋯ + (I + A + A2)εt − 2 + ⋯. This hinders researchers from performing efficient Monte Carlo type of simulation study.
- 21.
For example, when aggregating two units into one aggregated level unit (n = 2) where there are four spatial units, \( G=\left(\begin{array}{cccc}1& 1& 0& 0\\ {}0& 0& 1& 1\end{array}\right). \)
- 22.
More specifically, when a regional shock spills over to region-sharing borders (rook contiguous), then region #1 spills over to regions #2 and #33, whereas region #34 spills over to regions #2, #33, #35, and #66.
- 23.
For example, aggregating at the quarterly interval, FEVDs are calculated assuming that the same amounts of shocks are given for the first 3 months. One can also simulate and visualize the case that shocks are unevenly distributed across within an aggregated time period, but since there are infinitely many cases of uneven distributions, and since even distribution is representative, only evenly distributed case is visualized here.
- 24.
For more general case, as in equation (∗), FEVD can be derived as \( {\theta}_{rs}(h)=\frac{\sum_{l=0}^h{\left({e}_r^{\prime }{\Phi}_l{Pe}_s\right)}^2}{\sum_{l=0}^h\left({e}_r^{\prime }{\Phi}_l\Sigma {\Phi}_l^{\prime }{e}_r\right)} \). Thus, a temporally aggregated version of FEVD should be \( {\theta}_{rs}(H)=\frac{\sum_{m=0}^{n-1}{\sum}_{l=m}^{h+m}{\left({e}_r^{\prime }{\Phi}_l{Pe}_s\right)}^2}{\sum_{m=0}^{n-1}{\sum}_{l=m}^{h+m}\left({e}_r^{\prime }{\Phi}_l\Sigma {\Phi}_l^{\prime }{e}_r\right)} \).
- 25.
Likewise, more generalized version of the spatially aggregated version of FEVD can be expressed as \( {\theta}_{RS}(h)=\frac{\sum_{l=0}^h{\left({e}_R^{\prime }{\Phi}_l{Pe}_S\right)}^2}{\sum_{l=0}^h\left({e}_R^{\prime }{\Phi}_l\Sigma {\Phi}_l^{\prime }{e}_R\right)} \).
- 26.
Other values of autoregressive coefficient for region common factor, effect from the region common factor, and autoregressive coefficient of regions own are also tried, but not presented here, because the conclusions drawn from the results are same.
- 27.
As it is already shown in the real-world data example, the portion of neighborhood effect varies depending on the region. For our constructed regional economy, the variance of a region located at the border has larger portion of its own innovations.
- 28.
The results do not change much when the neighborhood coefficients are negative. Appendix 6 provides the results with negative neighborhood effects.
- 29.
This phenomenon also appears when we assign the autoregressive coefficients larger value such as ai, i = 0.8.
- 30.
The results do not change much when the neighborhood coefficients are negative. Appendix 7 provides the results.
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Chung, S., Hewings, G.J.D. (2019). A Short Exercise to Assess the Effects of Temporal and Spatial Aggregation on the Amounts of Spatial Spillovers. In: Franklin, R. (eds) Population, Place, and Spatial Interaction. New Frontiers in Regional Science: Asian Perspectives, vol 40. Springer, Singapore. https://doi.org/10.1007/978-981-13-9231-3_3
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