Abstract
The paper examines the role and significance of modeling spatially correlated disturbances in the extrapolation of purchasing power parities and real incomes. Alternative specifications for the spatial weighting matrix are experimented within the general econometric framework developed by Rao et al for the purpose of constructing a consistent time-space tableau of PPPs based on the PPPs for benchmarks constructed as a part of the International Comparison Program (ICP) and temporal movements in national price levels. The paper presents a comparative analysis of the effect of alternative specifications of the spatial weights matrix on the PPP extrapolations. A method based on the principal components approach is suggested when several spatial weights matrices are available. The paper presents empirical findings from the application of the methodology to ICP data available including the recently completed 2005 ICP Benchmark comparisons available from the World Bank. The results clearly indicate the need to model and use a spatially correlated error structure especially when the benchmark data are incomplete. The results are very similar when the spatial weights are based on trade data or on principal component weights.
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- 1.
For example, the Human Development Index is computed and published on an annual basis. Similarly, the World Development Indicators publication provides PPP converted real per capita incomes for all the countries in the world for every year.
- 2.
We define “national growth rates” in the next section.
- 3.
This follows because \(\phi\) is strictly bounded by the inverse of the eigenvalues of \(\textbf{\textit{W}}_{t}\).
- 4.
The form of this covariance follows from the definition of PPP. The interested reader is referred to Rao et al. (2009) for details.
- 5.
In the empirical implementation we model \(\sigma_{it}^{2}\) as inversely related to \(GDP_{it}\) per capita measured in $US (exchange rates adjusted). It is well known that exchange rates adjustments accentuate the difference between developed and developing countries and thus provide a suitable measurement of the desired effect.
- 6.
The PPPs between currencies of two countries are invariant to the choice of the base country, which in turn requires the predictions of the reference country to be zero with variance zero in all time periods, \(p_{US,t} = 0\). See Rao et al. (2008) for a proof that the method is invariant to the choice of the reference currency.
- 7.
However, if the distribution of the disturbances in the state-space from are symmetric, it is median-unbiased. We thank an anonymous referee for making this point.
- 8.
The standard errors are computed under the assumption of the lognormality of the predictions.
- 9.
We are indebted to Ms Francette Koechlin (OECD) for providing ICP benchmark data for these years. PPPs for those countries which joined in the Euro zone, the pre-Euro domestic currencies were converted using the 1999 Irrevocable Conversion Rates (Source:http://www.ecb.int/press/date/1998/html/pr981231_2.en.html). The irrevocable conversion rate of the drachma vis a vis the euro was set at GRD 340.750 Source: http://www.bankofgreece.gr/en/euro.
- 10.
This was brought to our attention by Steve Dowrick who attended a seminar on the topic presented at the Australian National University in October 2007.
- 11.
We are conscious of the fact that serious multicollinearity issues may be present here as the variables are potentially correlated. As the main purpose of inclusion of these variables is to improve the quality of the predictions, we decided to leave the variables in the model with the view that the model results in better predictions.
- 12.
We make use of exchange rate converted per capita incomes to overcome the problem of possible endogeneity arising out of the use of PPP converted exchange rates. These data are drawn from the UN sources. Given the systematic nature of the exchange rate deviation index (ratio of PPP to ER), use of exchange rate converted per capita GDP is likely to magnify differences in per capita incomes.
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Appendix: The Construction of the Economic Distance Measure used in the PCW
Appendix: The Construction of the Economic Distance Measure used in the PCW
A principal components approach is used to construct the measure. This technique allows the representation of the variance-covariance (or correlation) structure of a set of variables through a small number of linear combinations called components. Variables are grouped into components by their cross-correlations (Johnson & Wichern, 2002).
1.1 Variables Included in the Construction
The distance between countries is measured using a range of variables. The variables are chosen considering trade closeness, geological proximity, cultural closeness. To provide the reader with an example of the data used, summary statistics for a small group of countries are presented in Appendix DA.4.
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a)
Trade closeness:
We use the trade share between countries which is expressed as the volume of trade (export plus import) with each trading partner in the sample as a percentage of total trade of a particular country. This is constructed for every five year intervals ; 1970, 1975, 1980, 1985, 1990, 1995, 2000 and 2005
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b)
Geographical proximity:
Two dummies are used to consider geographical proximity, regional membership and close neighbours.
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1.
Regional membership: Asia pacific region, Europe, South America, North America, Central America and the Caribbean, Saharan Africa (except North Africa), North Africa and Middle East.
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2.
Close neighbours: We consider both land and sea proximities: e.g. Canada and the US are bordering countries. Sri Lanka and India as well as Singapore and Malaysia are also close neighbours. This dummy captures some aspects of proximity not caputred by the regional dummies. For instance, Venezuela and Trinidad and Tobago are close neighbours, but they are classified in different regions. Further, Japan and Australia are not close in terms of distance, but they are in the same region.
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c)
Cultural and colonial closeness:
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1.
Common language: A dummy is used for each language. In addition, the closeness of the dialect is also considered. For instance, Danish is very close to Swedish and Norwegian.
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2.
Common colonies: This dummy indicates the colonial relationship of countries. We do not totally fix to the standard definition for a colony but include protectorates and other types of foreign rules and consider countries under different types of foreign rules and their colonial occupants (such as UK, Spain etc.)to construct the dummy. Some countries had several colonial relationships over their history. For instance, Sri Lanka was partly or fully under the rules of England, Portugal and Holland. To avoid this complexity, we construct the dummy considering the last colonial power of the country or the colonial rule that made a significant impact on the particular country.
1.2 Estimation
The steps involved in the PC estimation procedure are summarized in the main text and some more details are presented here.
Step 1
A separate principal components model is estimated for each country using the variables for each time period. Therefore, 141 models are estimated. For each country there could be up to nine variables defined: Trade period, region, border, language1, language2,…, language5, colonial. We will refer to the number of variables by m.
Step 2
The m principal components are orthogonal linear combinations of the m variables. The estimate of the weight of a given variable on a principal component is known as the “loading.” We select the first component (corresponding to the largest eigenvalue) as the common factor. During the procedure we drop variables and re-estimate the model if, a) it has a negative loading on the first component, b) its loading is below 0.4. Although this choice of “significance” of the variables is rather ad hoc, it is a common rule of thumb in the principal components literature. For each country we use the same model (that is, the same subset of variables) to estimate the principal components model for all eight time periods.
We present the example of France to illustrate. The factor loadings for common colony and common language are not significant in the first component as the magnitude of the loading is less than 0.4 (see Table). These variables, common colony and Lang 1, are significantly loaded on the second component.
Table
France (all available variables) | ||
|---|---|---|
Components | ||
1 | 2 | |
Trade75 | .865 | .036 |
Border | .906 | –.006 |
Comcol | .005 | .934 |
Lang 1 | .176 | .924 |
Region | .743 | –.258 |
Thus, in this case the principal component model is re-estimated without Comcol and Lang1 and shown on the next table.
Thus, the common factor for France for the period 1975–1979 is given by
Interested readers can consult the authors for the complete set of results.
Step 3
A factor score is computed for each pair of countries using the common factors, for example cfFr7579 is used to compute the scores for France for the years 1975–1979. These factor scores are rescaled to prepare the proximity matrix using the formula presented in (13).
1.3 Descriptive Statistics for the Variables used to Construct the PCW. Selected Countries Shown.
The table below shows descriptive statistics for eleven countries in the sample for each of the variables used to estimate a common factor for each country. The label “Lang #” stands for language number. The maximum number of languages considered for any country was five.
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Rambaldi, A.N., Rao, D.P., Ganegodage, K.R. (2010). Modelling Spatially Correlated Error Structures in the Time-Space Extrapolation of Purchasing Power Parities. In: Biggeri, L., Ferrari, G. (eds) Price Indexes in Time and Space. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2140-6_4
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