A history of PPP in Europe

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A History of PPP in Europe

Master Thesis

Name: Roald Schuring Student number: 10044892

Email: roald.schuring@hotmail.com Date: July 2014

Supervisor: Franc Klaassen Second reader: Dirk Veestraeten

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5.1 Summary of testing procedure... 34 5.2 Time-series test results... 34-37 5.3 Panel test results... 37 5.4 Half-life of PPP deviations... 37-38 5.5 Comparison of 1535-1650 and 1651-1796... 38 5.6 Analysis... 38.40 6 Conclusion 41-42 7 Bibliography 43-45 8 Appendices 46-47

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1 Introduction

The principle underlying PPP is that, expressed in a common currency, a given basket of goods must have the same price in every nation. PPP can be studied along two dimensions: absolute vs. relative and long-run vs. short-run. Due to difficulties of testing absolute PPP empirically, research more often addresses relative PPP (Taylor & Taylor, 2004). Long-run PPP is less stringent than short-run PPP, allowing for temporary deviations from the long-run equilibrium (Abuaf & Jorion, 1990). As such, this study concerns long-run relative PPP (LRRPPP), which states that a change in the relative price level between two countries must be offset in the long-run by the adjustment of the nominal exchange rate. For instance, higher inflation at home than abroad would require the domestic currency to depreciate vis-à-vis the foreign currency. The choice for LRRPPP is elaborated upon more extensively in subsequent sections.

In order for LRRPPP to hold, the real exchange rate (RER) between two countries must be stationary, converging to a constant equilibrium value over time. Early empirical tests for LRRPPP address relatively short time periods, typically the post-Bretton Woods era. Such periods of 15-30 years contain insufficient observations to reject a null hypothesis of non-stationarity. Additionally, this short time period is insufficient to capture the long-run tendency of the economy. Only large data sets with more observations can yield significant conclusions about LRRPPP. This is achieved by using cross-country/panel or long-time series data.

Research using long-horizon and panel data has yielded some evidence that is largely supportive of LRRPPP (Taylor, 2002). The ability of researchers to investigate long-run PPP is constrained by the limited availability of historical data. Thus, studies of this type remain relatively rare. In an attempt to enrich the stock of research in this field, this thesis employs a long-horizon price data set with annual observations, thereby guaranteeing sufficient statistical power in testing for LRRPPP. Specifically, the purpose of this study is to investigate PPP on a long time horizon in Europe. To this effect, the research question investigated is: ‘how well does PPP hold up historically in Europe?’

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This research is conducted using annual data for several countries in Europe between 1535 and 1796. For this period, data on consumer prices for 15 European cities1 spanning nine countries has been collected by Robert Allen (2008). Using commodity price data in terms of silver, Allen constructs an aggregate Consumer Price Index (CPI). Using the weight of silver as a proxy currency, this CPI can be used to directly compare purchasing power between different locations. The city CPIs can be aggregated to yield a balanced panel of country-level data. A challenge here is to interpret the applicability of the PPP theory in a post-medieval, pre-industrial setting of socioeconomic and political turbulence.

The analysis takes place in four stages. Firstly, the CPIs are used to calculate the RER for each country vis-à-vis an aggregate European currency basket (Allen, 2008). This RER is displayed graphically, followed by a preliminary empirical interpretation of the PPP theory. Secondly, unit root tests are used to test whether LRRPPP holds for individual nations in the panel. Where evidence for stationarity is found, the speed of mean reversion is calculated and compared to the findings of other researchers. Thirdly, panel data testing methods are used to test LRRPPP for the panel of nine nations as a whole. Finally, the time period under investigation is split up into two sections to analyse the applicability of the PPP theory in a period with low versus high inflation.

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These cities are Amsterdam, Antwerp, Augsburg, Gdansk, Krakow, Leipzig, London, Lviv, Madrid, Munich, Paris, Strasbourg, Valencia, Vienna, Warsaw

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2 Literature review

2.1 PPP Theory

The starting point for PPP theory is the law of one price (LOOP), which states that a good, in common currency terms, should have a single price throughout the world (Ardeni, 1989). PPP applies the law of one price to all commodities. Cassel (1922) first formalized this general concept, stating that market exchange rates should equalize relative price levels between countries.

Via a commodity arbitrage process, departures from PPP should be exploited by rational agents (Isard, 1978). If the price level in a domestic country, in common currency terms, is higher than in a foreign country, domestic individuals will import foreign products to exploit the price differential. The subsequent adjustment mechanics depend on the reigning exchange rate regime (Genberg, 1978). If the domestic currency follows a free float, the additional demand for foreign currency causes the nominal exchange rate to depreciate. Greater demand for foreign produce relative to domestic produce may also cause the price level abroad to rise, with the domestic price level falling. This process continues until the purchasing power of both currencies is equalized and no further arbitrage is possible. If the domestic country however, has a fixed exchange rate, the only variable that can adjust to the PPP equilibrium is the relative price level. Since the market exchange rate cannot change, the price level abroad will rise and the domestic price level will fall until no further arbitrage is possible. There are several reasons why the arbitrage process may be imperfect. They are discussed in a subsequent section.

PPP is often considered a cornerstone for many modern economic theories (Dornbusch, 1987). Taylor (2013) exemplifies this point, stating that the importance of PPP as an assumption in monetary models of exchange rate determination makes it worthy of careful scrutiny.

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5 2.1.1 Absolute vs. relative PPP

It is possible to distinguish between the relative and the absolute form of PPP. Absolute PPP requires that the nominal exchange rate level adjusts such that the purchasing power of two currencies is identical (Taylor & Taylor, 2004).

(1) 𝑝𝑝𝑡𝑡∗ = 𝑝𝑝𝑡𝑡+ 𝑠𝑠𝑡𝑡

In the equation above, 𝑝𝑝_𝑡𝑡 denotes the logarithm of the domestic price level, 𝑝𝑝_𝑡𝑡∗ the logarithm of the foreign price level and 𝑠𝑠_𝑡𝑡 the logarithm of the market exchange rate, expressed as the foreign currency price per unit of domestic currency.

Absolute PPP is often disregarded in literature due to empirical problems. Differences in consumer preferences between countries mean that 𝑝𝑝_𝑡𝑡 and 𝑝𝑝_𝑡𝑡∗ measure the price levels of different consumption baskets. This renders a test of absolute PPP invalid (Froot & Rogoff, 1996). A further problem of empirically testing absolute PPP is that many studies use a consumer-price index (CPI) as a proxy for the national price level (Froot & Rogoff, 1996). Price indices have a common base period in which the nominal exchange rate equals the price ratio by construction. It is then no longer possible to make inferences about absolute PPP, because price indices contain no information on absolute PPP.

Relative PPP states that changes in relative prices levels should be offset by changes in the exchange rate (Taylor & Taylor, 2004). Using the same definitions for the variables as in equation (1),

(2) 𝛥𝛥𝑝𝑝𝑡𝑡∗ = 𝛥𝛥𝑝𝑝𝑡𝑡+ 𝛥𝛥𝑠𝑠𝑡𝑡

In past decades, most of the debate on PPP theory has focused on relative PPP (Froot & Rogoff, 1996).

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6 2.1.2 Long-run vs. short-run PPP

The time frame on which PPP is studied must also be specified. Short-run PPP is a stringent doctrine, violated at any point in time when there is a departure from the PPP equations described above (Abuaf & Jorion, 1990). Evidence suggests that such short-run deviations from PPP are large and volatile (Rogoff, 1996). These deviations can be attributed to imperfections in commodity arbitrage.

Imperfections in commodity arbitrage may arise due to the presence of transportation costs (Engel & Rogers, 2001). Benefits from commodity arbitrage may be negated by transportation costs if these are sufficiently high. Rogoff (1996), for instance, estimates transport costs at approximately 10% of total trade. Although there are large variations in this percentage between countries, Rogoff demonstrates that transport costs create a band for the difference in relative price levels between which arbitrage is not profitable. As a result, the transport costs for some goods may be so large that even wide price discrepancies cannot be arbitraged profitably.

Another imperfection in commodity arbitrage is the issue of non-traded inputs (Engel & Rogers, 2001). This is especially relevant at the consumer price level, since the price of a typical consumer good in a supermarket contains a share of fixed costs such as rent, wages and insurance. The fixed costs allocated to consumer goods are likely to differ between nations and cannot be traded internationally. As such, arbitrage cannot ensure complete equality in price levels. A further imperfection in arbitrage arises due to trade restrictions (Rogoff, 1996). Tariffs can drive a wedge between the domestic and foreign price level, similarly to transport costs. Non-tariff barriers can also undermine efficient commodity arbitrage. Quality standards, for instance, increase the cost of exporting for a foreign firm in ensuring that these standards are met. These higher costs translate into higher export prices, again driving a wedge between relative price levels that cannot be equalized by commodity arbitrage.

Rogoff (1996) argues that, given the multiple sources of inefficiency described, short-run commodity arbitrage only has a small effect on equating the market prices of goods internationally. As such, long-run PPP relationships that allow for short-run departures from PPP are more interesting. Long-run PPP simply requires that the economy converges to PPP

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over time. Long-run relative PPP stipulates that the long-run equilibrium real exchange rate (RER) should be a time-invariant constant. The RER is defined as the nominal exchange rate deflated by a ratio of domestic to foreign price levels (Abuaf & Jorion, 1990). Letting 𝑞𝑞_𝑡𝑡 represent the logarithm of the RER, this implies:

(3) 𝑞𝑞𝑡𝑡 = 𝑝𝑝𝑡𝑡+ 𝑠𝑠𝑡𝑡− 𝑝𝑝𝑡𝑡∗

Econometrically, long-run relative PPP states that the relative price level 𝑝𝑝_𝑡𝑡− 𝑝𝑝_𝑡𝑡∗ and the market exchange rate 𝑠𝑠_𝑡𝑡 should move together over time in the sense that the difference between them is stationary. If this is true, the RER will be a stationary process without a unit root. There should be mean reversion such that after any deviation, the RER reverts to its long-run equilibrium value. For this purpose, the half-life of PPP deviations is often calculated. The half-life is defined as the time taken for deviations from PPP to subside by half in response to a unit shock (Kilian & Zha, 1999). Going forward in this thesis, LRRPPP will be used to denote long-run relative PPP.

2.2 The real exchange rate

From the theory on PPP presented above, it is apparent that the RER is a central concept to LRRPPP. In order to understand how LRRPPP is studied empirically, it is important to address how the RER is measured and what its determinants are.

2.2.1 Measuring the RER: the CPI

In calculating the RER, price indices can be used to obtain a measure for 𝑝𝑝_𝑡𝑡 and 𝑝𝑝_𝑡𝑡∗. When such a price index is used to measure the price level, the nominal exchange rate 𝑠𝑠_𝑡𝑡 must also be calculated as an index. Between all variables, there must be a common base period. The resulting value for the RER 𝑞𝑞_𝑡𝑡 must then also be interpreted as an index.

Problems in the construction of price indices can lead to deviations from both short- and long-run PPP being observed. The most popular price index used in PPP tests is the CPI, due to its availability over long time frames (Terra & Vahia, 2005). Formally, the CPI shows the change over time in the price of a representative basket of goods and services purchased by

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urban wage-earners and clerical workers (Hurwitz, 1962). Sjaastad (1998) demonstrates that the use of CPIs leads to significant measurement error in RERs. Accordingly, it is important to acknowledge the limitations of this measure in testing for PPP.

One problem of using CPIs to test for PPP is that they contain information on both tradable and non-tradable goods. For non-tradable goods, the international goods arbitrage process cannot occur and there will be no price convergence. This point is illustrated by Taylor and Taylor (2004), who observe that PPP holds better historically when Producer Price Indices (PPI) are used rather than CPIs. The PPI measures the price level of a basket of inputs commonly bought by producers, and is less contaminated by nontradable components (Burstein, Eichenbaum & Rebolo, 2005). This demonstrates that there is a significant non-tradable component in the CPI that can prevent consumer price convergence between nations.

By construction, CPIs also reflect institutional price distortions passed on to the end consumer. Government taxes and price controls, for instance, increase the price level but cannot be arbitraged internationally. A final limitation of the use of CPIs is the presence of market power (Sjaastad, 1998). If a domestic exporter is a price taker in the international market for traded goods, changes in the exchange rate will not affect its foreign market price. If, however, US firms have significant market power, a change in the exchange rate may lead to an equiproportionate change in the foreign market price. The market power held by domestic exporting firms is a determinant of the price level in the foreign market. This will appear in the CPI, distorting the degree to which PPP applies.

Apart from the PPI, some other alternatives to the CPI are the WPI, the GDP and GNP deflator (Short, Packey & Holt, 1995). The WPI is largely similar to the PPI, but includes taxes. The GDP deflator measures the price level of all goods, services and final products produced within country borders. The GNP deflator does the same for all final goods produced by an economy, going by citizenship rather than country borders.

Since commodity arbitrage can only take place for tradable goods, the ideal price index for PPP tests would only contain tradable goods, with an identical composition between nations (Terra & Vahia, 2005). None of the price indices discussed have these properties. The closest approximation to such an index was made by Xu (2003). Using an export price index, which by definition consists solely of tradables, he was able to construct a price index for traded

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goods. Despite the shortcomings of the CPI, it continues to be used in research on LRRPPP due to its ready availability. In this research, CPI is the index of choice because the data set employed provides price data in this format.

2.2.2 Structural breaks

In a model world in which LRRPPP holds perfectly, one would expect the RER to converge to a certain value and be constant in the long-run. Taylor and Taylor (2004) argue that empirical analysis presupposing a fixed equilibrium RER, if in reality it is moving over time, will lead to biased estimates for mean reversion periods. Theory suggests that there may indeed be movements in the equilibrium RER over time, also termed structural breaks (Taylor & Taylor, 2004). This section serves to explain how structural breaks may arise, and how they can be treated econometrically when testing for PPP.

Balassa-Samuelson (BS) effects can cause the long-run equilibrium RER to be non-constant (Froot & Rogoff, 1996). The starting point of the BS effect is an increase in the productivity of tradables relative to non-tradables. As factor productivity in the domestic tradables sector rises, perfect competition dictates that workers will demand to be rewarded in line with their productivity, leading to upward pressure on their wage. The higher wage entices domestic non-tradables sector workers to move towards the tradables sector. Internal capital and labour mobility causes equalization of wages between sectors. Higher domestic wages expand demand for goods and services, leading to upward pressure on domestic prices. An increase in the price of tradables, however, would lead to these products being imported from abroad. Due to this commodity arbitrage, the price of tradables remains largely unchanged. The price of non-tradables, in the absence of arbitrage for such goods, is free to increase. This causes the overall price level in the economy to increase and the RER to appreciate, without a change in the nominal exchange rate. Over time, sustained productivity differentials can cause persistent movements in the RER.

Despite some controversy, there is empirical evidence for the BS effect. Lothian and Taylor (2004) show that the BS effect may account for up to 40% of variation in the RER between the UK and the US between 1820 and 2001. The direct implication of the BS effect is that the equilibrium RER will continually move over time. The inclusion of a long-term trend using the generalized-least-squares (GLS) version of the Dickey-Fuller (DF) test can help account

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for BS effects. After including a trend in his regression, Taylor (2002) finds lower estimates for the half-lives of PPP deviations than if the trend is omitted. Lothian and Taylor (2004) allow for BS effects by including a nonlinear time trend in their analysis.

Other factors that may influence the RER include wealth effects and productivity effects (Taylor & Taylor, 2004). Alternatively, there may be an effect of current account deficits and government spending on the RER (Rogoff, 1996). These factors can lead to temporary breaks in the RER. Whereas gradual movements in the RER under the BS effect can be accounted for through the inclusion of a trend, temporary breaks require a different econometric approach. Unit root tests such as those by Perron (1989) allow for the inclusion of a temporary break in the RER.

2.2.3 Nonlinearity in the RER

Lothian and Taylor (2004) argue that imperfections in commodity arbitrage such as those discussed in section 2.1.2 can lead to nonlinearity in the RER. Traditional empirical models for the RER assume that the adjustment of the RER is a linear process. This implies that the speed of PPP deviations from parity is uniform, regardless of the magnitude of the deviation (Taylor & Taylor, 2004). Transport costs, tariffs and other nontariff barriers may create a band within which a deviation from PPP is not worth arbitraging. Thus, near the PPP equilibrium there is no commodity arbitrage and the RER is indeterminate. When deviations from PPP are sufficiently large, there will be commodity arbitrage and mean reversion to PPP. This suggests that the speed of mean reversion towards PPP is higher for large deviations. As such, the RER will not be a constant in the long-run, but will fluctuate within a band around some central parity. Non-linear STAR models such as those employed by Taylor, Peel and Sarno (2001) can allow for these dynamics in testing for LRRPPP.

2.2.4 Changing RER volatility

Testing for LRRPPP may require the use of a data set that spans multiple different exchange rate regimes. Frankel and Rose (1995) demonstrate that the volatility of the RER tends to be higher during floating-rate than during fixed-rate regimes. Under fixed exchange rates, deviations from PPP can only be corrected by changes in the domestic price level. Under floating rates, movements in the nominal exchange rate and the relative price level allow for

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convergence to PPP. Thus, in estimating a RER model spanning several exchange rate regimes, it is important to allow for shifts in volatility for the error term (Lothian & Taylor, 2004).

At first sight, this poses a problem econometrically. By definition, a stationary process has a constant probability distribution over time. For LRRPPP to hold, the variance of the RER and thus of the error term must be constant over time, implying that there is no room for shifts in the volatility of the error term. However, closer examination reveals that heteroskedasticity and stationarity may not be mutually exclusive in investigating LRRPPP.

Stationarity is typically used to describe covariance stationarity. Changing volatility in the error term may imply changing covariances, which constitutes a violation of stationarity. On the other hand, heteroskedasticity is concerned with the conditional volatility of a series, whereas stationarity is concerned with unconditional moments. Even if an RER series exhibits conditional heteroskedasticity, the unconditional variance may still be constant. If the covariances are also constant, the series can still be stationary, despite exhibiting heteroskedasticity. Furthermore, although allowing for heteroskedasticity is possible and perhaps desirable in long-horizon studies, the main issue in testing for LRRPPP is still mean reversion to a constant level. PPP theory makes no explicit stipulation about volatility component of the probability distribution of the RER.

Thus, a test for LRRPPP can account for heteroskedasticity when testing for stationarity. One way to do this is the use of heteroskedastic-robust estimation methods. An alternative approach is taken by Lothian and Taylor (2004), who construct a function for the variance that allows for multiple shifts.

2.3 Tests for LRRPPP

Early tests for LRRPPP were largely unsuccessful in rejecting the null hypothesis of a unit root for the RER (Taylor & Taylor, 2004). These tests focused on short time horizons, typically post-Bretton Woods periods of 15-30 years (Bell, Brooks & Moore, 2014). Frankel (1986) argues that this sample is too small to capture mean reversion. This is confirmed by most early tests being unable to find conclusive evidence supporting LRRPPP. As such, up to the 1980s the general conclusion was that LRRPPP did not hold.

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Froot and Rogoff (1996) identify two solutions offered to solve this problem. The first is the use of long-horizon data sets. In this spirit, the span of data used to test for mean reversion has risen in recent decades (Lothian & Devereux, 2011). Increasing horizons of data sets is not straightforward, as it can only be achieved by new data becoming available as time passes, or through the discovery of existing old data. Increasing the frequency of data is not a solution, since adding detail about short-run movements can only provide information about short-run rather than long-run behaviour (Shiller & Perron, 1985). Efforts such as those by Allen (2008) have enriched the stock of existing data. Together with the development of new econometric techniques, this has facilitated a strain of literature to develop that employs long-horizon data sets.

The second way to increase the power of LRRPPP tests is the use of panel data sets. Here, statistical power can be increased for short time periods through the inclusion of many observations. The use of panel data is advantageous since it does not require long-horizon historical data sets.

By the end of the 1990s, the use of these two methods allowed researchers to find more evidence in favour of LRRPPP. As such, existing literature on LRRPPP can be examined in terms of the type of data set employed.

2.3.1 Tests for PPP using long-horizon data sets

Frankel (1986) was among the first to investigate LRRPPP using long-horizon data, conducting a first-order autoregressive AR(1) test on the RER using 116 years of dollar/pound data. Frankel rejects the unit root null, reviving the PPP debate at the time: “From the ashes of absolute PPP, a phoenix has risen” (Frankel, 1986). In his groundbreaking research, he finds a PPP deviation half-life of 4.6 years.

The historical dollar/pound LRRPPP relationship has often been researched, likely due to the availability of data on these currencies. Edison (1987) examines the dollar/pound relationship for 1890-1978 using an error correction model (ECM) to test for cointegration between the relative price level, measured using the GDP price deflator, and the nominal exchange rate. She finds a half-life of PPP deviations of 7.3 years. Taylor and Taylor (2004) plot US and

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UK CPIs for 1820-2001 to give a rudimentary indication of LRRPPP without a formal test. Although price levels did tend to move together over extended periods, substantial short-run deviations from PPP were observed.

Other research using long-horizon data has been conducted for Canadian/US dollar data by Johnson (1990). He uses an Error Correction Model and examines different exchange rate regimes separately. Using 120 years of GNP deflator and exchange rate data, he confirms RER stationarity and finds a half-life of PPP deviations of 3.1 years. He finds that the PPP relationship is maintained in both fixed and floating exchange rate regimes, although the underlying dynamic mechanisms differ. Lothian and Taylor (1996) investigate the franc-pound RER between 1803 and 1990, in addition to the dollar-franc-pound relationship over a similar time period. Using a simple AR(1) model and WPI data, they are able to reject the null hypothesis of a random walk for both RER series. In a more recent effort, Lothian and Devereux (2011) examine the RER for the Dutch guilder and the British Pound on a remarkably long period between 1590 and 2009. They employ annual data on consumer prices. Using a heteroskedastic-robust Phillips-Perron test, they are able to confirm PPP as a long-run equilibrium condition.

As a whole, long-horizon data sets yield overwhelming evidence in favour of mean reversion in RERs. The half-life of PPP deviations is, on average, approximately four years.

Long-horizon data sets allow for more comprehensive analysis of LRRPPP by increasing the number of observations and thus the statistical power of an analysis (Froot & Rogoff, 1996). Additionally, testing LRRPPP over very long time horizons allows for the long-run tendency of the economy to be examined (Edison, 1987). Shorter-horizon data sets may be too short to capture mean reversion to the PPP equilibrium. A further possible advantage of long-horizon studies is that they rely heavily on historical CPIs. These are typically constructed using baskets of subsistence products that may be more homogeneous than modern CPI baskets (Bell, Brooks & Moore, 2014). The introduction of a plethora of new consumer goods in recent centuries has made preferences and thus consumption baskets more heterogeneous.

Long-horizon studies are subject to criticism. Historical price data is often only available in terms of silver. Bell, Brooks and Moore (2014) argue this is a flawed method of calculating PPP, due to a lag between a change in value of the metallic content and adjustment of prices.

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Beveridge (1940) contends that using silver equivalents as prices confuses barter with monetary exchange. Medieval coins in Europe circulated by face value rather than by weight, making silver an inadequate proxy for the actual price level. The use of silver as a proxy for currency requires the implicit assumption that the LOOP holds for silver (Froot & Rogoff, 1996). If this is not true, any results will be distorted. Froot and Rogoff (1996) discuss the issue of survivorship bias. The countries for which long-horizon price data is available typically rank among the world’s wealthiest nations. Developing countries that have experienced vast productivity increases in recent decades are more subject to BS effects, which would cause LRRPPP to fail empirically if no trend parameter is included. Thus, the countries for which data is available are more likely to see LRRPPP hold true than countries for which data may not be available.

2.3.2 Panel data tests for LRRPPP

In recent decades, studies have used panel data techniques to investigate LRRPPP. Abuaf and Jorion (1990) were the first to research LRRPPP using a panel data extension of the DF-GLS test. They analyze the RER for eight nations between 1901 and 1972, using annual CPI and exchange rate data. They find strong evidence for LRRPPP, estimating a half-life of PPP deviations of 3.3 years. Frankel and Rose (1995) use a panel of 150 countries to analyse 45 years of post-WWII RER data. In doing so, they use the Levin and Lin (1992) extension of univariate Augmented Dickey-Fuller (ADF) techniques to a panel data setting. Using CPI and exchange rate data, they find evidence in favour of LRRPPP, estimating a half-life of PPP deviations of about four years. This is in line with the conclusions obtained from long-horizon data studies.

Papell (1997) tests for LRRPPP during a period of floating exchange rates using monthly CPI data for 21 countries between 1973 and 1994. He runs DF-GLS tests on individual series and uses a Monte Carlo simulation to generate critical values for the panel data application of these tests. Doing so, he is able to reject the unit root null at the 5% level. He finds a half-life of PPP deviations of two years. Wu (1996) uses panel data analysis to test LRRPPP for 18 countries between 1974 and 1993. Using monthly and quarterly CPIs and WPIs he conducts ADF tests on individual series, generalizing this to multivariate analysis using methods by Levin and Lin (1992). He is able to reject the unit root null in all cases, finding a half-life of PPP deviations of between two and three years.

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In a very recent effort, Bell Brooks and Moore (2014) investigate the applicability of PPP in medieval Europe between 1383 and 1411. They use CPIs to investigate ten European RERs, testing both individual series and panel data sets. They run a battery of panel tests, including those by Levin, Lin and Chu (2002), Im, Perasan and Shin (1995, 2003) and Maddala and Wu (1999). When individual series were tested, only half of the RER series were stationary. Using panel data techniques, there was more evidence in favour of LRRPPP. Taylor (2002) uses panel data techniques on a long-horizon cross-country data set to test RER stationarity for 20 countries between 1850 and 1996. He uses both CPI and GDP deflator data for the price level and conducts a Johansen Likelihood Ratio test. He is unable to find clear-cut evidence for LRRPPP. Even the incorporation of a trend to account for possible BS effects does not provide evidence for LRRPPP.

The advantage of applying a panel unit root test to several RERs simultaneously is that it increases the statistical significance of the results. It is important to note that this will only be true if parameter homogeneity is assumed. If, in reality, the rate of convergence to LRRPPP differs between nations, this benefit does not apply. Another benefit of panel data is that a high degree of statistical significance can be obtained using data over a relatively short time horizon. This avoids problems related to the use of long-horizon data sets, such as the presence of structural breaks in the RER.

Panel data techniques are subject to some criticism (Bell, Brooks & Moore, 2014). Findings of mean reversion tend to be stronger when high inflation countries are included in the analysis (Rogoff, 1996). Indeed, a test may yield evidence for mean reversion, despite only some RERs in the panel being mean-reverting (Taylor & Taylor, 2004). The underlying cause of this problem is the assumption of parameter homogeneity that is frequently made in panel data studies. Many panel data tests such as those by Levin and Lin (1992) and Levin, Lin and Chu (2002) stipulate a single rate of convergence to the long-run equilibrium RER for all countries (Higgins & Zakrajsek, 2000). To solve this problem, Higgins and Zakrasjek (2000) present GLS techniques that can allow for parameter heterogeneity. Alternatively, the tests by Im, Perasan and Shin (1995, 2003) or Maddala and Wu (1999) can allow for parameter heterogeneity.

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16 2.4 Summary of existing research

LRRPPP has remained an area of intense scrutiny in economics throughout the last few decades. Since Frankel (1986) presented the issue of insufficient observations to reject a unit root null, both panel and long-horizon data research have been able to resolve this problem of statistical power. This has yielded evidence that is largely in favour of LRRPPP, with an estimated half-life of PPP deviations of about three to five years. Research on PPP in recent years has largely focused on modelling nonlinearities and investigating the nature of productivity bias in deviations of the RER (Taylor, 2013). This has led to a better understanding of the empirical caveats of the PPP theory, along with the development of increasingly complex econometric testing methods.

Empirical studies that have been conducted have been highly reliant on the availability of historical price data. Many studies only focus on the post-Bretton Woods period. However, the only way to truly understand the long-run tendencies of the economy is to test LRRPPP on a longer time scale. Despite some efforts in recent years, the lack of data availability has only permitted a handful of papers to study LRRPPP over several centuries. As such, work remains to be done in studying LRRPPP on very long time horizons. In addition, the countries for which LRRPPP have been tested are highly subject to survivorship bias. Studies for PPP over multiple centuries typically include the Netherlands, the USA, Great Britain and/or France. There is room for improvement in including a larger variety of nations in these types of tests. The inclusion of nations that are less wealthy, by modern standards, would give greater support for the PPP theory.

Rather than contributing to the development of new econometric techniques that can better account for issues such as structural breaks and nonlinearity, this study seeks to enrich the existing stock of research by testing for LRRPPP using a new data set. LRRPPP is investigated using data on a long time horizon of 262 years. The inclusion of nations such as Poland and the Ukraine, for which no PPP research on such a time scale has been done previously (to my knowledge), constitutes a step towards resolving the issue of survivorship bias. Issues such as BS effects and heteroskedasticity in the RER are addressed, despite not being the primary focus of this thesis. Time-series and panel data techniques developed in existing research are employed to this effect. The problem of nonlinearity is not addressed since this lies outside the scope of my econometric abilities.

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3 Data and preliminary analysis

This section serves to describe the data set that is used in this thesis. Raw city CPI data is translated into national data. This yields the complete data set used for the empirical research in subsequent sections. The properties of this data set are discussed, and the price data is used to perform a preliminary analysis of PPP.

3.1 Construction of the CPI

The starting point for the empirical analysis in this thesis is the data set collected by Allen (2008). He constructs a Laspeyres index that specifies the quantity of several consumer goods, valuing these quantities at the price level prevailing at the time. Goods prices are measured as the amount of silver (in grams) required to purchase a unit of the commodity in question. The reigning market price of silver in each country in the relevant year has been used to convert local-currency prices to prices expressed in terms of silver.

The CPI is constructed using a premodern basket of twelve commodities, shown in table 1, purposefully ignoring goods such as potatoes and coffee that only became available after the era of colonization (Allen, 2001). The spending share on each commodity has been calculated by estimating the annual consumption of a fifteenth century craftsman (Allen, 2001). For food components, the yearly quantity is based on nutritional value. There are some differences in the contents of the CPI between cities. This does not prohibit a study of LRRPPP, as relative PPP does not require the individual components of the goods basket to be identical between countries (Montiel, 2009). As long as each component can be arbitraged and the composition of the bundle remains unchanged, the individual city CPIs can contain different commodities. Bell, Brooks and Moore (2014) argue that this is a useful relaxation, due to international differences in the composition of price indices. Such differences may arise due to heterogeneous consumer preferences, resource availability or accounting conventions.

To allow for heterogeneous consumption preferences between geographical locations, oil and wine are included as possible alternatives to butter and beer, respectively. Another noteworthy difference in CPI composition arises due to the type and quantity of fuel historically used in European cities. Whereas consumers in Amsterdam were assumed to use peat for fuel, those in Krakow were argued to use wood. The relative weight of fuel in the CPI is derived from the amount of energy released by each fuel type. Also, consumers in

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warm climates such as Madrid are assumed to use less fuel than those in Warsaw, for instance.

Allen calibrates his CPI for 17 European cities spanning ten countries, between 1264 and 1913. While the series for London is complete from 1264 to 1830, the series for other cities are only available for spans of between 200 and 500 years, with most series starting around 1500. In some cases such as Naples, there is a gap of almost 100 years in the series. Such gaps remain, despite attempts by Allen (2001) to interpolate and extrapolate certain individual commodity price series to generate a complete CPI series for each city. Most of these gaps concerned goods with small spending shares in the CPI (Allen, 2001). In the case of bread, for which there were large gaps in the series, grain prices and wages (as a proxy for baker income) were used to estimate the price of bread. Allen (2001) argues that interpolations and extrapolations may dampen yearly fluctuations in price levels, but argues that the general trend of the price level remains preserved.

At this stage, it is important to note that this data set would also allow for absolute PPP to be studied, using the price level rather than an index. However, for reasons outlined in section 2.1.1, relative PPP is a more active field of research than absolute PPP. Also, absolute PPP has largely been disproven in existing research (Taylor & Taylor, 2004). As such, the concept of relative PPP seems to have more empirical relevance than PPP in the absolute form. To abide by the conventions of existing relative PPP research, the data must be calibrated as a price index. To this effect, the denominator of the CPI is the average cost of the basket of goods in Strasbourg between 1745 and 1754. This is shown in table 1, where the annual price of this consumption bundle over this period is estimated at about 415 grams of silver. Although this choice of base period is relatively arbitrary, the CPIs used in this thesis abide by this convention for lack of a better alternative. The value of the Strasbourg CPI for 1745-1754 would thus, by construction, be one.

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19 Table 1: CPI composition

Item Quantity

per person per year

Price g silver per unit Weight (in %)

Bread 182 kg 0.69 30.42 Beans/peas 52 liter 0.48 5.98 Meat 26 kg 2.21 13.87 Butter 5.2 kg 3.47 4.35 Cheese 5.2 kg 2.84 3.56 Eggs 52 each 0.10 1.25 Beer 182 liter 0.47 20.64 Soap 2.6 kg 2.88 1.80 Linen 5 m 4.37 5.27 Candles 2.6 kg 4.98 3.12

Lamp oil 2.6 liter 7.55 4.73

Fuel 5.0 M BTU 4.16 5.02

TOTAL 414.90 100%

3.2 Aggregation of city data

As stated in the previous section, many of the city-level CPI series contain gaps. This is true for Krakow, Naples, Northern Italy (Milan and Rome), Paris and Warsaw. By aggregating the city CPIs, according to modern-day borders, country-level data can be obtained. Since many of the gaps of the city-level series overlap each other chronologically, such an aggregation yields a more balanced panel.

For nations with one city in the data set, the city CPI has been taken as a proxy for the national price level. For nations featuring multiple cities in the dataset, city data has been averaged to yield a country CPI. For Italy, a gap in the time series remains for the period 1646-1700. To allow for testing methods that require a balanced panel, this country has been omitted from the analysis. Once the city data for the remaining nine countries has been compiled, the longest time period for which a balanced panel exists is between 1535 and 1796. This is the time period that will be investigated in this study.

3.3 The time period 1535-1796

The time period between 1535 and 1796 in Europe was characterized by underdeveloped monetary systems, diseases and political conflict. Prior to any preliminary analysis about LRRPPP, it is important to understand the historical context of this study. As such, this section serves a twofold purpose. In the first part, it is explained why the RER must be

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studied without using nominal exchange rate data. Secondly, the consequences of historical turmoil for LRRPPP theory are presented.

3.3.1 The omission of nominal exchange rate data

Typically, an analysis of LRRPPP investigates whether the nominal exchange rate and the relative price level between two countries are cointegrated. This study, however, investigates LRRPPP without the use of nominal exchange rate data. This section explains why this is necessary from an historical point of view. A subsequent section justifies this omission from a theoretical point of view also.

Pre-industrial Europe was characterised by a highly complex monetary system without a documented system of currency exchange. With the discovery of silver deposits in Europe at the end of the 15th century, the first large coinages were minted (Lannoye, 2011). Individual territories began issuing their own versions of this coinage, called Thalers. The Thaler became the new standard for international commerce (Denzel, 2010). Nevertheless, there remained a large number of versions of this coin. During the Eighty Years War (1568-1648), the Netherlands had a total of fourteen mints, each following their own policy of debasements (Carlos & Neal, 2011). Monetary unification of the country was only completed by 1659. In addition to the multitude of different currencies, many people in more rural areas still engaged in barter trade. With similar issues arising across other parts of Europe, large-scale trade in national currencies that could yield reliable exchange rate data remained sparse. The absence of large foreign exchange institutions to manage this scattered monetary system further reduced the scope for a well-documented currency exchange. Some early attempts were made: in the 15th century, the Italian Medici family converted currencies on behalf of merchants (Roover, 1963). To provide stability to the payments system in the rapidly growing foreign trade sector, the Amsterdamsche Wisselbank was established in 1609. This institution provided monetary exchange at established rates that were documented (Carlos & Neal, 2011). As such, some exchange rate data is available, but this does not cover all of the countries in the panel of this research.

A final argument explaining the omission of nominal exchange rate data is changing geographical borders (Persson, 2010). Germany, for instance, was not unified until 1871. Augsburg and Munich were cities in the state of Bavaria, whereas Leipzig belonged to the electorate of Saxony. Similarly, Italy only became a unified nation in the late 19th century,

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meaning that the cities featured in the data set belonged to different countries. With changing country borders, the area over which a certain currency is legal tender may have changed also. Omitting nominal exchange rate data resolves complications that could arise due to this issue.

3.3.2 Historical turmoil and LRRPPP

To provide context to this research, two more historical issues must be addressed. The first of these issues is the underdeveloped nature of markets (Persson, 2010). In the absence of fast means of communication and cheap transportation of goods between locations, there is reason to believe that there may have been significant barriers to commodity arbitrage. Persson (2010) highlights this, arguing that the LOOP was violated more seriously before efficient communication means became available. Commodity arbitrage that did take place took the form of pan-European trade fairs such as those held in Champagne, France, from the 12th century onwards. However, most trade fairs were smaller in nature. This will have prohibited full equalization of prices, facilitating departures from PPP.

A further issue specific to the time period being studied is the plethora of historical events that fall within it. Major historical events can yield real shocks to the economy and thus structural breaks in the RER (Lothian & Devereux, 2011). The period encompasses the “price revolution”, a period of high inflation in Western Europe from the first half the 16th century to the first half of the 17th (Fisher, 1989). The period also encompasses several recurring epidemics of the Black Death, leading to dramatic fluctuations in labour supply and thus in the supply and demand (and price) for food (Persson, 2011). Throughout this period, Western European nations began to exploit economic opportunities abroad through colonization. The period encompasses the rise of Amsterdam as the seat of world finance, and its subsequent eclipse by London (Lothian & Devereux, 2011). The 16th, 17th and 18th centuries were all subject to both small and large military conflicts within Europe. There were even wars between countries featured in the data set of this research.

A redeeming feature of the time period 1535-1796 is that it precedes the industrial revolution. The industrial revolution, argued to have begun in the late 1700s, led to rapid productivity increases that are unlikely to have been homogeneous among all European nations (Persson, 2011). As such, the industrial revolution may have been subject to significant BS effects that have been argued to induce structural breaks in the RER. By investigating the period 1535-1796, the likelihood of structural breaks due to the industrial revolution is lower.

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Despite omitting most of the industrial revolution, it appears that there are many possible reasons why PPP may not hold. This can seemingly be attributed to barriers to commodity arbitrage and the variety of reasons for structural breaks in the RER. Although these issues may pose a challenge econometrically, they also provide an opportunity for a rigorous testing procedure. If it appears that RERs are stationary, even amidst undeveloped commodity markets and the economic and political chaos of 16th-18th century Europe, this would provide strong testimony of the robustness of the PPP theory.

3.4 Complete data set and comparison with other studies

Now that the CPI has been compiled and historical issues have been addressed, it is possible to generate a series for the RER and obtain the full data set for this study. Since the commodity prices contained in Allen’s CPI have the same base period and denomination, it is possible to compare the resulting indices directly. As such, silver (in grams) becomes a de facto “currency”. For silver to be a proxy for currency, the LOOP must be assumed to hold for silver (Froot & Rogoff, 1996). Kim, Froot and Rogoff (1995) demonstrate that between the Netherlands and Great Britain from the 1500s onwards, the LOOP is a very good approximation for the silver market. After justifying the omission of nominal exchange rate data from a historical point of view, its omission can thus be justified from a theoretical point of view also. The shortcomings of silver as a proxy currency, explained in section 2.3.1, do not outweigh the practicality of its use in this research.

Expressed in common currency terms, the RER between two currencies, in logarithms and as an index, can be defined as the difference between the two price levels:

(4) 𝑞𝑞𝑡𝑡 = 𝑝𝑝𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑐𝑐 1− 𝑝𝑝𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑐𝑐 2

In order to generate an RER process, it is important to decide relative to which currency the RER will be calculated. Kalyoncu (2009) demonstrates that the validity of PPP in Turkey is influenced by the choice of base country. O’Connell (1996) explains how cross-sectional dependence between RERs with the same country as the numéraire can adversely affect the size and power of tests for stationarity. Following the example of Taylor (2002), the RER will thus be generated relative to a “world” (N=9) basket of currencies. This avoids problems that may arise due to the choice any one nation as the base country.

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23 (6) 𝑝𝑝_{𝑖𝑖𝑡𝑡}𝑊𝑊 = 1

𝑁𝑁−1∑ 𝑝𝑝9𝑗𝑗≠𝑖𝑖 𝑗𝑗𝑡𝑡

The full data set for this thesis then consists of annual observations for the RER between 1535 and 1796 for nine countries. The parameter i = 1, ...., 9 covers the set of countries Austria, Belgium, France, Germany, Great Britain, the Netherlands, Poland, Spain and the Ukraine. The parameter t runs over the set of years from 1535 to 1796, a time period that yields a balanced panel of 9 x 262 observations.

The dataset used in this study is unique due to its combination of size and scope. Existing research investigating LRRPPP over several centuries has often been in the form of bilateral studies (Edison, 1987, Lothian & Devereux, 2011). Here, the time series under investigation were long, but only two countries were investigated. Other studies use wide panels, but these lack length. Although Taylor (2002) uses a panel of 20 nations between 1892 and 1996, this time period is significantly shorter than the period being considered in this analysis. To my knowledge, there are no existing studies with a panel that is both as wide (nine countries) and as long (262 years of annual observations) as the one being used in this research. The large number of observations in this thesis may improve statistical significance of the results found. Another special feature of this study is the inclusion of countries for which little or no PPP research on long time horizons has been done previously. The inclusion of countries such as the Ukraine enriches the existing stock of research on LRRPPP.

A further unique feature of this data set is that inferences drawn from it will be able to shed light on the degree of market integration for this time period. Since data on trade volumes is often hard to come by for historical time periods, price convergence is used as a measure of commodity market integration (Bell, Brooks & Moore, 2014). This study will be able to uncover new evidence on how well commodity arbitrage worked in post-medieval Europe. 3.5 Preliminary analysis

Using the complete data set, it is now possible to conduct a preliminary analysis of LRRPPP. This can be done by graphing the RERs of the countries in the panel. For LRRPPP to hold the RER should be a stationary process exhibiting mean reversion.

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24 -.2 0 .2 .4 .6 R ER 1550 1600 1650 1700 1750 1800 Year Austria

Figure 1: RERs for European countries, 1535-1796

The graphs in figure 1 seem to exhibit reversion to some mean level. On first glance, this is most apparent in the case of Austria, for which the RER fluctuates around a relatively constant average level. Noteworthy here is the spike in the Austrian series in 1623-1624. This spike is also visible in the RER series for Germany, and can probably be attributed to a spell of hyperinflation at the start of the Thirty Years’ War (Kindleberger, 1991). In an attempt to finance expenditure on arms, existing coins were debased, causing the RER to depreciate for both Germany and Austria, who were members of the Holy Roman Empire at the time. Similar spikes are visible in the RER series for other countries such as Great Britain and Poland. These can likely also be attributed to real shocks resulting from historical events. For this study, a relevant observation here is that the RER series for all countries seem to revert to a long-run mean level after such shocks take place.

For some countries, visual inspection of the series suggests the possibility of a deterministic time trend. In the case of Great Britain, the RER series exhibits an upward-sloping tendency. In existing literature, the inclusion of a linear deterministic time trend is often motivated by BS effects (Lothian & Taylor, 2000). An issue here is whether the mechanism of the BS

-.2 -.1 0 .1 .2 .3 R ER 1550 1600 1650 1700 1750 1800 Year Belgium 0 .1 .2 .3 .4 R ER 1550 1600 1650 1700 1750 1800 Year France -.2 0 .2 .4 .6 R ER 1550 1600 1650 1700 1750 1800 Year Germany -.8 -.6 -.4 -.2 0 .2 R ER 1550 1600 1650 1700 1750 1800 Year Great Britain -.2 -.1 0 .1 .2 .3 R ER 1550 1600 1650 1700 1750 1800 Year Netherlands -.5 -.4 -.3 -.2 -.1 R ER 1550 1600 1650 1700 1750 1800 Year Poland .1 .2 .3 .4 .5 R ER 1550 1600 1650 1700 1750 1800 Year Spain -.6 -.4 -.2 0 .2 R ER 1550 1600 1650 1700 1750 1800 Year Ukraine

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effect worked between 1535 and 1796, a time period that is largely pre-industrial. In the post-industrial period between 1820 and 2001, Lothian and Taylor (2004) argue that the BS effect accounts for 40% in the variation of the RER for the US, the UK and France. It is unclear whether the BS effect will have such significance for the time period under consideration in this study.

To understand whether the inclusion of a deterministic time trend is justified for the RER processes in figure 1, it is important to review the goal of this study. In essence, the aim of this research is to assess whether LRRPPP can explain RER behaviour in Europe between 1535 and 1796. As argued by Lothian and Taylor (2000), the presence of a linear trend will not undermine any conclusions drawn about LRRPPP. If a trend is present and statistically significant, this may indicate that the RER is a trend stationary process. Such a process also exhibits mean reversion, albeit to a mean level that is changing over time. Given the vast scope for real shocks within the time period under consideration, many of which may have had permanent components, any form of mean reversion will serve as evidence for LRRPPP. As such, a linear deterministic time trend will be allowed for in the formal tests for LRRPPP presented in the subsequent section.

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4 Methodology

Using the RER series presented in the previous section, it is possible to conduct a test for stationarity using both time-series as well as panel data techniques. The following sections will outline the testing procedures for time-series and panel data testing, respectively.

4.1 Time-series testing for LRRPPP

The time-series testing procedure for LRRPPP in this thesis will use an ADF as well as a Phillips-Perron (PP) unit root test, following the example of Lothian and Devereux (2011). The ADF and the PP test will both allow for a constant and a time trend. The PP test can also account for heteroskedasticity, which otherwise requires the more complex calculation of White standard errors in the ADF configuration. Modelling heteroskedasticity is desirable since it can allow for the effect of changes in the exchange rate regime over time, as explained in section 2.2.4. A further advantage of the PP test is that the lag length for the test regression does not have to be specified. However, the PP test does suffer from size distortions, rejecting the null hypothesis of non-mean reversion too frequently.

The time period under investigation is so long that there are likely to be several breaks. However, it is unclear how many breaks are present in the series being investigated, and when these take place. Accordingly, the issue of temporary breaks in the RER will not be addressed by these tests for reason of simplicity.

4.1.1 The ADF test

The ADF test is a standard unit root test used in LRRPPP studies (Taylor, 2013). If the RER reverts to a constant long-run mean, the RER should have a pth order autoregressive representation with the form:

(7) 𝑞𝑞𝑡𝑡= 𝛼𝛼0+ � 𝛼𝛼𝑗𝑗𝑞𝑞𝑡𝑡−𝑗𝑗+ 𝜀𝜀𝑡𝑡 𝑝𝑝

𝑗𝑗=1

Here, 𝜀𝜀_𝑡𝑡 represents white-noise disturbance. Imagine a situation in which there are no shocks to the RER, i.e. 𝜀𝜀_𝑡𝑡 = 0. In this case, the RER would, given enough time, settle down to a long-run equilibrium level 𝑞𝑞∗. In terms of the parameters of regression (7), this requires ∑𝑝𝑝_𝑗𝑗=1𝛼𝛼_𝑗𝑗 < 1. The situation ∑𝑝𝑝_𝑗𝑗=1𝛼𝛼_𝑗𝑗 > 1 is not possible since it implies explosive behaviour in the RER.

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When ∑𝑝𝑝_𝑗𝑗=1𝛼𝛼_𝑗𝑗 = 1, there is a unit root in the process driving 𝑞𝑞_𝑡𝑡 and there will be no mean reversion.

Substituting 𝑞𝑞∗ in for the values for the RER in equation (7), it is possible to solve for 𝑞𝑞∗ as: (8) 𝑞𝑞∗₌ 𝛼𝛼0

1 − ∑𝑝𝑝𝑗𝑗=1𝛼𝛼𝑗𝑗

Equation (7) can be rewritten as follows to yield the typical format of an ADF test. This is done to split non-stationary components from stationary components. This derivation is first done for the more complex case of an ADF including a time trend, whereafter the test without a time trend is presented.

The starting point for this derivation is to add a trend to each q in (7).

(9) 𝑞𝑞𝑡𝑡− 𝜃𝜃𝜃𝜃 = 𝛼𝛼0+ � 𝛼𝛼𝑗𝑗(𝑞𝑞𝑡𝑡−𝑗𝑗 − 𝜃𝜃(𝜃𝜃 − 𝑗𝑗)) + 𝜀𝜀𝑡𝑡 𝑝𝑝

𝑗𝑗=1

Next, the terms are multiplied out:

(10) 𝑞𝑞𝑡𝑡 = 𝛼𝛼0+ � 𝛼𝛼𝑗𝑗𝑞𝑞𝑡𝑡−𝑗𝑗 − 𝜃𝜃𝜃𝜃 � 𝛼𝛼𝑗𝑗+ 𝜃𝜃𝜃𝜃 𝑝𝑝 𝑗𝑗=1 + 𝜃𝜃 � 𝑗𝑗𝛼𝛼𝑗𝑗 𝑝𝑝 𝑗𝑗=1 + 𝜀𝜀𝑡𝑡 𝑝𝑝 𝑗𝑗=1

Like terms are collected:

(11) 𝑞𝑞𝑡𝑡 = �𝛼𝛼0+ 𝜃𝜃 � 𝑗𝑗𝛼𝛼𝑗𝑗 𝑝𝑝 𝑗𝑗=1 � + � 𝛼𝛼𝑗𝑗𝑞𝑞𝑡𝑡−𝑗𝑗+ �𝜃𝜃(1 − � 𝛼𝛼𝑗𝑗 𝑝𝑝 𝑗𝑗=1 )� 𝜃𝜃 𝑝𝑝 𝑗𝑗=1 + 𝜀𝜀𝑡𝑡

Subtracting 𝑞𝑞_𝑡𝑡−1 and reparametrizing, this yields (12) 𝑞𝑞𝑡𝑡− 𝑞𝑞𝑡𝑡−1 = �𝛼𝛼0+ 𝜃𝜃 � 𝑗𝑗𝛼𝛼𝑗𝑗 𝑝𝑝 𝑗𝑗=1 � + �� 𝛼𝛼𝑗𝑗− 1 𝑝𝑝 𝑗𝑗=1 � 𝑞𝑞𝑡𝑡−1 + � 𝛼𝛼𝑗𝑗∆𝑞𝑞𝑡𝑡−𝑗𝑗+ �𝜃𝜃 �1 − � 𝛼𝛼𝑗𝑗 𝑝𝑝 𝑗𝑗=1 �� 𝜃𝜃 𝑝𝑝−1 𝑗𝑗=1 + 𝜀𝜀𝑡𝑡

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28 Which can be rewritten as:

(13) ∆𝑞𝑞𝑡𝑡= 𝜑𝜑 + 𝛽𝛽𝑞𝑞𝑡𝑡−1+ � 𝛼𝛼𝑗𝑗∆𝑞𝑞𝑡𝑡−𝑗𝑗+ 𝛾𝛾𝜃𝜃 𝑝𝑝−1

𝑗𝑗=1

+ 𝜀𝜀𝑡𝑡

Where 𝜑𝜑 = �𝛼𝛼₀+ 𝜃𝜃 ∑𝑝𝑝_𝑗𝑗=1𝑗𝑗𝛼𝛼_𝑗𝑗�, 𝛽𝛽 = �∑𝑝𝑝_𝑗𝑗=1𝛼𝛼_𝑗𝑗− 1� and 𝛾𝛾 = �𝜃𝜃�1 − ∑𝑝𝑝_𝑗𝑗=1𝛼𝛼_𝑗𝑗�� In the case without a linear time trend,

(14) ∆𝑞𝑞𝑡𝑡= 𝜑𝜑 + 𝛽𝛽𝑞𝑞𝑡𝑡−1+ � 𝛼𝛼𝑗𝑗∆𝑞𝑞𝑡𝑡−𝑗𝑗 𝑝𝑝−1

𝑗𝑗=1

+ 𝜀𝜀𝑡𝑡

Now, 𝜑𝜑 = 𝛼𝛼₀. In conducting tests (13) and (14), the hypotheses have the following form: 𝐻𝐻0: 𝛽𝛽 = 0

𝐻𝐻1: 𝛽𝛽 < 0

For both (13) and (14), rejecting the null hypothesis means rejecting the hypothesis that the RER is not mean reverting.

The regressions (13) and (14) are run on all the individual RER series in the data set. Critical values are obtained from the distribution of the ADF statistic, with and without the inclusion of a trend. In both tests, the number of lags for each country is determined using Schwartz’s Bayesian Information criterion.

An ADF specification can also be used to calculate the half-life of PPP deviations. To this effect, a first-order autoregression model must be specified (Choi, Mark, & Sul, 2004).

(15) ∆𝑞𝑞𝑡𝑡= 𝜑𝜑 + 𝛽𝛽𝑞𝑞𝑡𝑡−1+ 𝜀𝜀𝑡𝑡

The time required for a divergence from PPP to dissipate by one half can be calculated as follows:

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29 4.1.2 The Phillips-Perron test

In order to allow for heteroskedasticity, a PP test is also run. This test also allows for the inclusion of a linear trend and is robust to serial correlation. The PP test involves estimating:

(17) ∆𝑞𝑞𝑡𝑡 = 𝛿𝛿 + 𝜇𝜇𝑞𝑞𝑡𝑡−1+ 𝑢𝑢𝑡𝑡

The PP test corrects for serial correlation and heteroskedasticity by modifying two test statistics. The first, 𝜃𝜃_𝜇𝜇, is a transform of the standard t-statistic and is calculated as:

(18) 𝜃𝜃𝜇𝜇 = �𝜎𝜎 2 𝜆𝜆2� 1 2 ∙ 𝜃𝜃𝜇𝜇=0−1_{2 �}𝜆𝜆 2_{− 𝜎𝜎}2 𝜆𝜆2 � ∙ �𝑇𝑇 ∙ 𝑆𝑆𝑆𝑆(𝜇𝜇̂)� 𝜎𝜎2

Where 𝜎𝜎2 = lim_{𝑇𝑇→∞}𝑇𝑇−1∑𝑇𝑇_𝑡𝑡=1𝑆𝑆[𝑢𝑢_𝑡𝑡2] and 𝜆𝜆2= lim_{𝑇𝑇→∞}∑𝑇𝑇_𝑡𝑡=1𝑆𝑆[𝑇𝑇−1𝑆𝑆_𝑇𝑇2], 𝑆𝑆_𝑇𝑇 = ∑𝑇𝑇_𝑡𝑡=1𝑢𝑢_𝑡𝑡. Here, T denotes the sample size.

It is used to test the hypothesis: 𝐻𝐻0: 𝜇𝜇 = 0

The second test statistic, 𝑍𝑍(𝜑𝜑), is a transform of the standard F-statistic and is calculated as:

(19) 𝑍𝑍(𝜑𝜑) = 𝑇𝑇𝜇𝜇̂ −1_{2 ∙}(𝜆𝜆2_{− 𝜎𝜎}2_{) ∙}�𝑇𝑇2∙ 𝑆𝑆𝑆𝑆(𝜇𝜇̂)�

𝜎𝜎2

It is used to test the hypothesis: 𝐻𝐻0: (𝛿𝛿, 𝜇𝜇) = (0, 0).

In both cases, the alternative hypothesis states that 𝑞𝑞_𝑡𝑡 is stationary. Further details on the test statistics can be found in Perron (1988).

The PP test is conducted with and without a linear time trend. A deterministic time trend is included by adding a linear trend component to (17). DF critical values can be used in testing. The presence of a trend requires the use of adjusted ADF critical values.

4.2 Panel tests for LRRPPP

In existing literature, there are three main panel data techniques that are used to study stationarity of the RER. Of these tests, the LLC test by Levin, Lin and Chu (2002) is not employed because of its restriction of parameter homogeneity. Although it is possible that the

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speed of mean reversion is constant for every nation in the panel, it is argued by Koedijk, Tims and van Dijk (2010) that this is probably a poor depiction of reality. They suggest that the speed of mean reversion depends on factors such as relative proximity, mutual trade regulations and economic openness. Tests falsely assuming parameter homogeneity are argued to exhibit serious biases and have low statistical power.

To avoid such problems, the IPS test by Im, Pesaran and Shin (1995, 2003) and the MW test by Maddala and Wu (1999), are run. These tests do allow for parameter heterogeneity. As such, the speed of convergence to LRRPPP under the alternative hypothesis can differ between countries. These techniques share the characteristic that their outcomes are dependent on results from independent unit root tests.

4.2.1 The Im, Perasan and Shin Test

The IPS test is a panel data test that uses the average of test statistics from the ADF unit root test. The average test statistic that is obtained does not follow a normal distribution. The test statistic must be centred and scaled using the mean and variance of each country such that it becomes asymptotically normal.

Following the explanation of Ekpithakdamrong (2007), the overall model for the IPS test without a trend is a regression with the form:

(20) ∆𝑞𝑞𝑖𝑖,𝑡𝑡 = 𝜑𝜑𝑖𝑖 + 𝛽𝛽𝑖𝑖𝑞𝑞𝑖𝑖,𝑡𝑡−1+ � 𝛼𝛼𝑖𝑖,𝑗𝑗∆𝑞𝑞𝑖𝑖,𝑡𝑡−𝑗𝑗+ 𝜀𝜀𝑖𝑖,𝑡𝑡 𝑝𝑝−1

𝑗𝑗=1

𝜃𝜃 = 1, 2, … . 𝑇𝑇

Here, T must be the same for all cross-section units. As such, only balanced panel data can be used to conduct the IPS test. Note that regression (20) is the same as (14), but adjusted for the use in panel data research. Accordingly, the subscript i has been added to denote each country. The null and alternative hypotheses for the IPS panel unit root test can be formulated as follows:

𝐻𝐻0: 𝛽𝛽𝑖𝑖 = 0, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖

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By formulating the alternative hypothesis in this way, the IPS test allows for the possibility that each some series have unit roots, while others may not.

In the IPS test, separate unit root tests are conducted for the N cross-section units. To this effect, the t-statistics testing the null hypothesis 𝛽𝛽 = 0 in (14) must be used. The individual t-statistics obtained can be used to calculate the test statistic for the IPS test.

Let 𝜃𝜃_{𝑖𝑖,𝑇𝑇}(𝑖𝑖 = 1, 2, … 𝑁𝑁) denote the t-statistic for testing unit roots in individual series i. Then

(21) 𝜃𝜃̅𝑁𝑁,𝑇𝑇 =_{𝑁𝑁 � 𝜃𝜃}1 𝑖𝑖,𝑇𝑇 𝑁𝑁 𝑖𝑖=1

The average t-statistic calculated in (14) can be adjusted to yield the IPS test statistic:

(22) 𝜋𝜋𝑡𝑡̅𝑁𝑁,𝑇𝑇 =

√𝑁𝑁 �𝜃𝜃̅𝑁𝑁,𝑇𝑇− 1𝑁𝑁 ∑ 𝑆𝑆�𝜃𝜃𝑁𝑁𝑖𝑖=1 𝑖𝑖,𝑇𝑇��

��1𝑁𝑁�∑ 𝑉𝑉𝑎𝑎𝑓𝑓�𝜃𝜃𝑁𝑁𝑖𝑖=1 𝑖𝑖,𝑇𝑇�

~ 𝑁𝑁(0,1)

Here, 𝑆𝑆�𝜃𝜃_{𝑖𝑖,𝑇𝑇}� and 𝑉𝑉𝑎𝑎𝑓𝑓�𝜃𝜃_{𝑖𝑖,𝑇𝑇}� are the mean and variance of the t-statistic for country i, respectively. A Monte Carlo simulation must be conducted to calculate these parameters. This simulation takes into account the number of observations T, the number of lag lengths and the inclusion of a linear time trend in the original regression specification. Im, Pesaran and Shin (1997, 2003) have tabulated values for 1

𝑁𝑁∑𝑁𝑁𝑖𝑖=1𝑆𝑆�𝜃𝜃𝑖𝑖,𝑇𝑇� and � 1

𝑁𝑁� ∑𝑁𝑁𝑖𝑖=1𝑉𝑉𝑎𝑎𝑓𝑓�𝜃𝜃𝑖𝑖,𝑇𝑇� for

different lag lengths. To make use of their tables, the same lag lengths must be used for all ADF regressions on individual series. As such, the IPS test is run for 1, 2 and 3 lags.

IPS test statistics can be tested using the t-distribution table, using standard significance levels.

The IPS test is also run using the panel data variant of regression (13), which includes a linear time trend. Once the IPS test statistic is calculated, the t-distribution table can again be used to obtain critical values.

4.2.2 The Maddala-Wu Test

Similarly to the IPS test, the MW test uses evidence from several independent tests to make inferences about the panel as a whole. Unlike the IPS test, however, the MW test is non-parametric and can be performed for results from any type of unit root test, and not just the

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ADF test. The main difference between the two tests is that the IPS test on combining test statistics, while the MW test is based on combining significance levels of individual tests. The MW panel unit root test is based on the 𝑝𝑝_𝜆𝜆 test by Fisher (1932). The starting point for this test is to obtain the p-values of independent unit root tests. For this thesis, the MW test is conducted for individual ADF test statistics, with and without the inclusion of a deterministic time trend. The null and alternative hypotheses for the MW test have the same form as those used for the IPS test. Again, parameter heterogeneity is allowed for.

Maddala and Wu (1999) suggest that the critical values and thus the p-values of the individual ADF tests must be calculated using Monte Carlo simulation methods. It is argued that this is necessary to account for correlation among cross-sectional units (Ekpithakdamrong, 2007). For this purpose, the simulation-based critical values for the ADF test as calculated by MacKinnon (2010) are employed.

The significance levels 𝜋𝜋_𝑖𝑖(𝑖𝑖 = 1, 2, … , 𝑁𝑁) are taken to be independent uniform (0,1) variables (Maddala & Wu, 1999). It is shown that −2𝑎𝑎𝑓𝑓𝑙𝑙_𝑒𝑒𝜋𝜋_𝑖𝑖 then follows a 𝜒𝜒2 distribution with 2𝑁𝑁 degrees of freedom. Under the additive property of 𝜒𝜒2_variables,_{𝜆𝜆 = −2 ∑} _{𝑎𝑎𝑓𝑓𝑙𝑙}

𝑒𝑒𝜋𝜋𝑖𝑖 𝑁𝑁

𝑖𝑖=1 has

a 𝜒𝜒2 distribution with 2𝑁𝑁 degrees of freedom. The results of the MW test can be testing using the 𝜒𝜒2 table of critical values. As with the IPS test, the MW test is conducted for a range of different lag lengths.

4.3 The price revolution

A final issue that will be addressed in testing for LRRPPP is the period of high inflation that swept through Europe between about 1500 and 1650 (Munro, 1999). This period is often labelled the ‘price revolution’ and is introduced in section 3.3.2. It is argued that factors such as a south-German mining boom, changes in credit institutions and an increase in silver imports from the Americas led to sustained increases in the price level for countries in Western Europe. By 1650, prices in Great Britain, for instance, had increased sevenfold as compared to 1500. Harsh currency debasements in countries such as France and Spain led to similarly high inflation rates in these countries. It is likely that many of the spikes in the RER series for individual nations, as seen in figure 1, can be attributed to factors related to the price revolution. A glance at the RER graphs confirms that the largest spikes seem to take place before 1650.

A history of PPP in Europe