# RDP 2001-04: Measuring the Real Exchange Rate: Pitfalls and Practicalities 2. Calculating a Real Exchange Rate Index

As mentioned above, a real exchange rate index is generally calculated as a weighted average of bilateral exchange rates that have been adjusted for relative price levels. These real bilateral rates are calculated as:

where e is the nominal bilateral rate, expressed as the number of foreign currency units per home currency unit, p is the price level of the home country, and p* is the price level in the foreign country. In this presentation, an appreciation is recorded as an increase in the exchange rate index. Some literature uses the alternative presentation of the exchange rate as the number of home currency units per foreign currency units, where an appreciation is recorded as a fall in the index. It should be noted that standard market quotations for currencies may be in either format. Therefore in some cases it will be necessary to invert the published bilateral series to be compatible with the other bilateral rates comprising the index.

## 2.1 The Averaging Process

There are two ways to calculate the weighted average. Suppose there are N components en to be averaged, with weights wn, . Then the arithmetic average of the e's is calculated as:

while the geometric average is calculated as:

The arithmetic average is probably more familiar, but there are strong theoretical and statistical reasons to prefer the geometric average. Percentage movements in an arithmetic index will differ in magnitude depending on whether the bilateral rates are expressed as units of home currency per foreign currency unit, or the other way around. Exchange rate indices based on arithmetic averages can also be distorted when the base period is changed. Because geometrically averaged indices treat movements in exchange rates symmetrically, they do not have these undesirable properties (see Rosensweig (1987) for a more detailed discussion). In addition, the logarithm of a geometric average is the arithmetic average of the logs of the bilateral rates. This is a useful feature, as a linear representation in logarithms greatly simplifies many econometric models.

Although the geometric averaging process is theoretically preferable, it can complicate the process of calculating average exchange rate indices for particular periods (month, quarter). To derive the average value of the index over a quarter, for example, one could take daily readings of the index (calculated as a geometric average across all the bilateral exchange-rate pairs) and form the (arithmetic) average of these readings over the quarter. Alternatively, one could form the (arithmetic) average over the quarter of each bilateral real exchange rate, and then generate the geometric average across these quarterly average bilateral pairs. These two approaches do not, however, produce the same outcome in general. In research papers and for the Bank's other published material, the Reserve Bank staff use the second method, because inflation data are only published at monthly or quarterly frequency and it is therefore not possible to derive the daily real bilateral exchange rates that comprise a daily real exchange rate index.

For the same reason, real exchange rate indices cannot be constructed by deflating the nominal exchange rate index by the ratio of the domestic price level to the weighted arithmetic-average world price level. Instead, they must be constructed by averaging bilateral real exchange rates using the geometric averaging procedure described above.

## 2.2 Dealing with Changing Weights

Fixed-weight indices of the real exchange rate are frequently used because they are easy to calculate. There are, however, a number of reasons why it might be preferable to allow the weights to change. For example, if some countries have become more important trading partners over time, a trade-weighted exchange rate index should reflect this. Otherwise, if actual trade shares move too far away from the shares embodied in the weights used, the exchange rate index will give a misleading picture of the net effect of movements in particular bilateral exchange rates.

Indeed, we should expect that weights would change over time as trade patterns adapt to exchange rate movements. If the home currency appreciates relative to another currency, imports coming from that country will become cheaper relative to domestic production or imports from elsewhere. Therefore it is likely that the share of total imports sourced from that country would increase, which will directly change weights based on import shares. The converse will be true for exports. Similar arguments apply to other bases for weighting schemes, such as shares of world GDP or capital flows.

If weights are allowed to vary, the index must be spliced together at every period that the weights are changed. Otherwise, movements in the index will be misleading; in the periods in which the weights change, it would not be clear if the movement in the index reflected changes in the underlying exchange rates, or changes in the weights. We can see this using a simple numerical example. Suppose that the home country trades with two other countries, A and B. The bilateral exchange rates and weights are shown in Table 1.

Table 1: Exchange Rate Index for a Country with Two Trading Partners
Period Bilateral exchange rate Weight on exchange rate Exchange rate index
Country A Country B Country A Country B Not spliced Spliced
1 100 100 0.5 0.5 100.00 100.00
2 110 90 0.5 0.5 99.50 99.50
3 110 90 0.6 0.4 101.50 99.50

If the exchange rate index has not been spliced together, it gives the erroneous impression that the home country's exchange rate has appreciated between periods 2 and 3, even though neither bilateral exchange rate has changed. If the weights are updated on a regular basis, these small errors will tend to compound rather than offset each other. This is because trade shares (and other bases for weights) tend to move in the same direction for a number of years.

In the case of weights based on import or export shares alone, the bias will naturally compound. As mentioned above, import shares tend to rise if the home currency has appreciated against that trading partner's currency, and conversely currency depreciation tends to expand the share of home country exports to that destination. Therefore, unless the index is properly spliced, changing weights can impart an appreciation bias to an import-weighted exchange rate index, or a depreciation bias to an export-weighted index.

The conceptually correct method for calculating an index with changing weights can now be described. Assume that at time t=τ, weights change from their previous values w(i,τB), which had been set at time τB, to new values w(i,τ). Then for tτ,

where rert is the index, breri,t is the bilateral real exchange rate with country i at time t, as defined in Equation (1) above, and w(i,τ) is the weight for the ith exchange rate at time τ. Qτ is a splicing adjustment calculated as:

Therefore, this method of calculation can be characterised as a spliced Laspeyres index.

With some rearrangement, we can see that the real exchange rate is the product of the real exchange rate's level prior to the introduction of new weights, and a Paasche index of bilateral real exchange rates in that base period and in the current period. These are weighted according to the weights that apply in the current period:

That is, the real exchange rate is calculated as the ratio of geometrically weighted bilateral real exchange rates in the base period and the current period, using current weights, spliced onto the base period level of the real exchange rate. If the composition of countries in the basket has changed for the new set of weights, the set of bilateral exchange rates used for both parts of the second term in this expression are those included in the current-period set of weights.

By splicing together the series in this way, weighting schemes can be updated to reflect changing trade patterns. Although this method complicates the calculations, it avoids biasing the results.

This spliced Laspeyres index results in a measure of the effective exchange rate that gives similar results to the Törnqvist index, which like the Fisher index, is a superlative index. In the current context, where the weights are trade shares, the Törnqvist index can be calculated as the geometric average of a Paasche index and a Laspeyres index constructed with trade (expenditure) weights (Diewert 1976; Caves, Christensen and Diewert 1982).

There are practical reasons for preferring the spliced Laspeyres to the Törnqvist index in policy work. The first is that the Törnqvist index requires next-period weights, for example trade shares for the following year. These data will not be available for the latest period since the next year has not happened yet; the latest period's index will therefore inevitably be revised when the data become available. Therefore although a Törnqvist index is suitable for econometric estimation using historical data, it induces some uncertainty into real-time analysis.

Secondly, the Törnqvist index still requires that the weights are updated at least as frequently as typically occurs for a spliced Laspeyres index – it is not sufficient to choose two base periods a long way apart. Otherwise, if trends in the weights used are not monotonic, the level of the exchange rate in the intervening period between the base and end periods can be distorted.

For a simple numerical example of this second point, consider again the case of a country with two trading partners, A and B. Its trade share with A rises then falls, while the share with B falls then rises. The home country's currency is appreciating against A's currency and depreciating against B's currency, both by 5 per cent per period. As Table 2 shows, if the weights in the beginning and end period are the same, the Törnqvist index will give the same result as its components, the Laspeyres and Paasche indices. These fixed-weight indices all imply that the home country's effective exchange rate has not changed. In fact, the increased importance of country A in the intermediate periods should be reflected in the effective exchange rate, and some appreciation should occur over those periods. A spliced Laspeyres index captures this effect, provided the splicing occurs sufficiently frequently. On the other hand, it does not then return to the original index level once weights have returned to theirs. A Törnqvist index comprised of spliced Laspeyres and spliced Paasche indices (not shown) will also capture this effect. However, it is not clear that it is necessarily an improvement on the spliced Laspeyres index, given that the next-period weights required for the spliced Paasche component are not available in real time.

Table 2: Exchange Rate Index for a Country with Two Trading Partners
Period Bilateral exchange rate Weight on exchange rate Exchange rate index
Country A

Country B

Country A

Country B

Törnqvist

Laspeyres
spliced each period
1 100 100 0.5 0.5 100 100.00
2 105 100/(1.05) 0.6 0.4 100 100.98
3 100*(1.05)2 100/(1.05)2 0.7 0.3 100 102.97
4 100*(1.05)3 100/(1.05)3 0.6 0.4 100 103.98
5 100*(1.05)4 100/(1.05)4 0.5 0.5 100 103.98

Non-monotonic movements in trade shares will distort the index in intermediate periods between re-weightings for any index, including the spliced Laspeyres. Therefore if the weights used move significantly, they should be updated frequently. A Törnqvist index is not a means of avoiding the requirement for regular updating of weights. Ideally, weights should be updated for every period – quarterly in the case of a real exchange index, since this is the highest frequency at which price data are available for Australia. However, the slow speed at which trade shares move mean that quarterly updating provides little advantage over the annual updating approach pursued by the Reserve Bank. Since quarterly trade data by country are only available on a non-seasonally adjusted basis, any such measurement advantage would furthermore be outweighed by the seasonal volatility introduced by using a quarterly weighting scheme. If the weights were updated less frequently, such as at three to five-year intervals, some distortions could potentially arise.

## 2.3 Choice of Which Bilateral Rates to Include

As long as the bilateral exchange rates are not wildly divergent, it seems reasonable to include all currencies with ‘significant’ weights in the index. For example, in constructing the (nominal) trade-weighted index of the Australian dollar, the Reserve Bank includes enough countries to account for at least 90 per cent of Australia's total international merchandise trade. However, there are occasions when large exchange rate movements specific to a single currency may result in the index giving a misleading indication of overall competitiveness. For example, the dramatic depreciation of some east Asian currencies, particularly the Indonesian rupiah, in 1997 and 1998 resulted in the published TWI remaining at roughly the same level in June 1998 as it had been a year earlier, despite the A\$'s depreciation against other currencies (RBA 1998).

One possible response to an exceptionally large depreciation in a single currency would be to exclude the currency from the index entirely. The difference between the published TWI and a trade-weighted index for the A\$ excluding the rupiah is shown in Figure 1. This response could be justified on the grounds that any country experiencing such a massive depreciation would no longer be a potential export market, and so further movements in its exchange rate do not impinge on other countries' competitiveness. However, it has the disadvantage of being possible only after the fact. Nonetheless, as a general rule, exclusion of a minor trading partner from the index should not affect the end result much. If it does, it indicates that this exchange rate is biasing the results, and should be excluded (Rosensweig 1987). Since its weight in the TWI was around 3.5 per cent at the time, this suggests that the ex post exclusion of Indonesia is justifiable for some purposes.

## 2.4 Choice of Price Level

Most of the issues discussed up until now apply to the construction of both nominal and real exchange rate indices. When constructing real indices, however, there are additional issues relating to the price series used to deflate the bilateral nominal exchange rates.

The most commonly used price series for this purpose are consumer price indices (CPIs). Although there are theoretical reasons to prefer other types of price index when measuring competitiveness (Rosensweig 1987), CPIs have the advantage of being timely and available for a wide array of countries over a long time period. Other classes of price or cost index, such as producer price or unit labour cost indices, are often difficult to obtain on a comparable basis across more than a few countries.

Ideally, the price series used should be comparable across countries, representative of price conditions in those countries, and relatively free from measurement error.

Although CPIs are not perfect on these criteria, they come closer than other candidate series, especially for indices covering many currencies. In its own work, the Reserve Bank uses ‘core’ or underlying price measures where available. These measures generally abstract from food and energy prices, which can impart unnecessary volatility into relative price measures. Even so, data availability may still prevent the inclusion of some currencies in a real exchange rate index. For example, the Soviet Union and Russian roubles have been included in the RBA's published nominal TWI from time to time, but are excluded from the real TWI used in this paper because of the difficulties in obtaining price data for the Soviet Union and its successors. Similarly, the United Arab Emirates does not currently publish domestic consumer price data; the RBA uses the IMF's Middle East CPI series as a proxy in calculation of real exchange rate indices.

Inclusion in the index of countries experiencing hyperinflation can also create problems. Since these countries' currencies generally depreciate rapidly, other countries' nominal exchange rate indices can be distorted. For real exchange rate indices, some of this distortion is offset because most of the nominal depreciation captures relative inflation rates rather than a real depreciation. However, the measurement error in measures of domestic price levels for countries experiencing hyperinflation can be large relative to that in other countries. Therefore, although real exchange rate indices abstract from the large nominal depreciations of hyperinflating currencies, some measurement error may remain when these currencies are included.

Since real exchange rates are intended to capture movements in competitiveness, it would be conceptually preferable to deflate the nominal exchange rates with some measure of producer prices or costs rather than consumer prices. Therefore, if data series of sufficient quality are available, it makes sense to use them instead of CPIs when assessing movements in competitiveness. An index constructed on this basis is shown in Figure 2, along with a consumer-price-based real exchange rate index calculated on the same basis. As the figure shows, the quarterly profile of real exchange rate measures is not greatly affected by the choice of deflator. However, wedges between the levels can persist for some time. Figure 2: Australian Real Exchange Rate against the G7 GDP-weighted, March 1995=100

A final consideration in the selection of a price index is the treatment of changes in indirect taxation. A Goods and Services Tax (GST) was introduced in Australia on 1 July 2000, which increased the price level by a few percentage points. For a given nominal exchange rate, this resulted in an increase in the relative price level and thus an apparent real appreciation of the A\$ relative to other currencies. However, this does not represent a genuine deterioration in competitiveness. Imports into Australia attract GST on the same basis as domestically produced goods at the retail level; Australian exports are zero-rated – that is, they do not attract GST. Therefore the introduction of the GST should not have had a deleterious effect on the relative competitiveness of Australian goods and services in either domestic or overseas markets.

This suggests that there is a case for excluding the effects of changes in indirect taxes on measured consumer prices when constructing real exchange rate indices. Accordingly, the real exchange rate measures presented in this paper incorporate an approximate adjustment for the introduction of the GST in Australia. However, similar adjustments were not made to allow for the price effects of indirect tax changes in other countries, as the effects of these on a A\$ exchange rate are likely to be small, once they are weighted by those countries' weights in the various exchange rate measures.

## Footnotes

Another way of looking at real exchange rates is as relative price levels adjusted for exchange rate movements (Rosensweig 1987). 

For the Pound sterling, New Zealand dollar, Australian dollar, PNG kina, the euro and the currencies of most Pacific nations, market quotations are generally for number of US dollars per home currency unit (denoted, for example, AUD/USD). For all other currencies, the quotation is usually in the form of home currency units per US dollar (eg USD/JPY). 

Since an appreciation will tend to result in a fall in exports from the home country to the foreign country, the effect on weights based on total trade (exports plus imports) is ambiguous. The response of total trade shares to a movement in one bilateral exchange rate (keeping all other bilateral exchange rates involving the home currency unchanged) depends in part on the bilateral trade balance with that country, the types of goods and services traded, and the price responsiveness of demand for imports in both countries. 

Rosensweig (1987) identifies this problem. The splicing procedure described in this paper eliminates the bias Rosensweig describes. 

Omitting the splicing adjustment creates only a fairly small bias in the short run. Cox (1986) calculates both fixed-weight and (unspliced) moving-weight nominal exchange rate indices for the US dollar, and finds estimates of the depreciation between March 1985 and September 1986 only ranging between 4½ and 6 per cent. This indicates that the bias in the moving-weight index must have been small, despite the divergence in bilateral exchange rates affecting the US dollar at the time. On the other hand, the rationale for using moving weights is to ensure weights remain consistent over the longer term. Therefore, if the moving-weight method is justified, the index should also be properly spliced. 

The trade shares or other sources for weights for the current period may not be available in real time, let alone for the future period. For its own purposes, the Reserve Bank of Australia uses trade shares for the previous year in calculating current-period exchange rate indices. Although this introduces some measurement error compared with the conceptually correct contemporaneous weights, trade shares move sufficiently slowly that the distortion is small, and arguably the disadvantages of this approach are more than offset by the advantage of enabling real-time analysis. Moreover, current-dated weights may not be appropriate for some purposes. Estimation of trade equations could be distorted if the right-hand side variable (the exchange rate) is constructed using current trade shares, as these incorporate the endogenous response to exchange rate movements that is being estimated. The right-hand side variable would therefore be correlated with the equation's error term. 

In addition, the RBA uses merchandise trade by country to determine weights in the TWI, thereby excluding the effects of trade in services. For some applications, total (goods and services) trade weights might be preferable. These data are available disaggregated by country on an annual basis, and so a total-trade weighted exchange rate index with weights updated annually could be constructed. 

Similar considerations apply to countries with multiple exchange rates for different purposes or market participants. See Rosensweig (1987) for a discussion. 

On the other hand, it could be argued that imposition of a GST does affect tourism flows and therefore should be included. These adjustments result from the difference between CPIs and the theoretically preferable producer price indices for tradables, and are not straightforward. The Commonwealth Treasury (2000) argued that the tax changes would result in a nominal exchange rate appreciation of 3–3½ per cent relative to what it would otherwise have been. This would have offset the effect of the GST on the relative price of imports to exports.