Bruno Thiago Tomio
email: bttomio@furb.br.

Carry trade in developing and developed countries: A Granger causality analysis with the Toda-Yamamoto approach

Carry trade in developing and developed countries : a Granger causality analysis with the Toda-Yamamoto approach

Bruno Thiago Tomio

Introduction

Due to the increased interconnectedness of global financial markets at the end of the 20 th century and beginning of the 21 st century, interest rate differentials among countries have fostered speculative capital flows seeking higher yields. Central bankers worldwide set their base interest rate accordingly with their mission. Every country has its singularity and characteristics, which demands a unique set of monetary policies. In this sense, some countries are obliged to set high interest rates (usually, developing and underdeveloped countries), while others present low interest rates (notably, developed countries). Speculators profit from this type of structure to seek financial gains, contradicting what is expected by the uncovered interest rate parity (UIP), one of the fundamental theories of international finance.

Currency speculation is not a new phenomenon, showing its first institutional developments in the Middle Ages [START_REF] Accominotti | Foreign Exchange Markets and Currency Speculation: Historical Perspectives[END_REF]. Foreign exchange markets (Forex) have augmented their size significantly in recent decades. Financialization of the world economy has led the daily turnover in Forex markets to surpass by 40 times the daily amount of world trade in US Dollars in 2019, as shown in Figure 1. In 1989, the ratio FX to trade was 21, highlighting the strengthening of financialization in Forex markets during the last two decades. Figure 1: Forex daily turnover and ratio between Forex daily turnover and daily trade (ratio FX/Trade), 1989-2019 Source: Bank for International Settlements (BIS) for the Forex daily turnover. International Monetary Fund (IMF) for trade, using the sum of exports and imports of goods and services in current US Dollars, divided by 20 (business days). Both series are daily means for April.

One of the leading financial operations in the Forex market is the currency carry trade. By targeting "international interest differentials", carry traders (investors applying the carry trade investment strategy) "shift their asset holdings from low interest-rate currencies to higher-return currencies" (Greenville 2010, p. 3).

With the speculators' positioning data supplied by the US Commodity Futures Trading Commission (CFTC) Large Trader Reporting Data, I explore the relationship of the carry trade and its related financial variables. The carry trade literature can be divided into two big strands. On the one hand, there is a vast literature exploring carry trade returns with the use of portfolio optimization (e.g. [START_REF] Clarida | Currency Carry Trade Regimes: Beyond the Fama Regression[END_REF][START_REF] Cenedese | Foreign Exchange Risk and the Predictability of Carry Trade Returns[END_REF][START_REF] Doskov | Empirical Evidence on the Currency Carry Trade, 1900-2012[END_REF][START_REF] Kang | A Tale of Two Premiums: The Role of Hedgers and Speculators in Commodity Futures Markets[END_REF]. On the other hand, there is another strand criticizing carry trade and its consequences (e.g. [START_REF] Agrippino | Funding Flows and Credit in Carry Trade Economies[END_REF][START_REF] Goda | Determinants of Real Exchange Rate Movements in 15 Emerging Market Economies[END_REF]. Nonetheless, as shown by [START_REF] Disyatat | Discussion on Funding Flows and Credit in Carry Trade Economies[END_REF], this strand of literature lacks robust empirical analyses.

In this sense, this paper fills a gap in the carry trade literature by trying to approach both strands. [START_REF] Chuffart | The Role of Carry Trades on the Effectiveness of Japan's Quantitative Easing[END_REF] also make use of CFTC data to investigate the effects of carry trade. This is a paper that is close to the main idea explored here: carry trade (proxied by real positioning) impact other financial variables. Meanwhile, their focus is to assess the impacts of carry trade on the real economy during the Quantitative Easing period in Japan.

My results show evidence of the relationship between carry trade and four related financial variables (interest rate differentials, market sentiment, local stock indexes, and the US stock index) in ten currencies (Australian Dollar, Brazilian Real, Canadian Dollar, Euro, British Pound, Japanese Yen, Mexican Peso, New Zealand Dollar, Russian Ruble, and Swiss Franc).

With two different periods based on the US monetary policy (monetary easing and tightening), the Granger causality tests with the [START_REF] Toda | Statistical Inference in Vector Autoregressions with Possibly Integrated Processes[END_REF] technique show relevant differences and similarities in the long-term relationship of these variables for each analysed country.

Methodology and Data

By following the model estimated by [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF], this article focuses on the relationship among carry trade (CT ), nominal exchange rates (ER), interest rates differentials (IRD), market sentiment (V IX), local stock market indices (SM ), and the US stock market index (SM U S).

Methodology

The applied model follows the VAR system as it is similarly proposed by [START_REF] Amiri | Granger Causality Between Total Expenditure on Health and GDP in OECD: Evidence from the Toda-Yamamoto Approach[END_REF]. 1

The null hypothesis of the Granger causality test is that the dependent variable does not Granger cause the independent variable (excluded variable). To find evidence that the other variables Granger cause CT , conditions in Table 1 must hold, as it is shown in Equation (2) 1 . Table 2 shows the conditions for the Granger causality in the direction of other variables to CT , following Equations (1), (3), (4), ( 5), and (6) 1 .

1 See the VAR model equations in the Appendix A of the supplemental material.

Direction ER→CT IRD→CT V IX→CT SM →CT SM U S→CT Condition α 21i = 0∀ i γ 21i = 0∀ i δ 21i = 0∀ i ψ 21i = 0∀ i φ 21i = 0∀ i
Table 2. Conditions for the Granger causality from CT to the other variables

Direction CT →ER CT →IRD CT →V IX CT →SM CT →SM U S Condition β 11i = 0∀ i β 31i = 0∀ i β 41i = 0∀ i β 51i = 0∀ i β 61i = 0∀ i
It is worth highlighting that the ordering of the variables does not change the results from the Granger causality tests.

Data

As a proxy for carry trade (CT ), the weekly data provided by the US Commodity Futures Trading Commission's (CFTC) Commitments of Traders Report (COTR) is used. This report only provides information for 12 currencies. Excluding the Euro FX/British Pound and the South African Rand, my dataset is composed of 10 of them (Australian Dollar -AUD, Brazilian Real -BRL, Canadian Dollar -CAD, Euro -EUR, British Pound -GBP, Japanese Yen -JPY, Mexican Peso -MXN, New Zealand Dollar -NZD, Russian Ruble -RBL, and Swiss Franc -CHF). The reasons for exclusion is that the former is not a pair with the US dollar, and the latter lacks data.

There are some caveats in the use of this proxy. Usually, exchanges in currency markets are over-the-counter (OTC) operations, complicating the modelling of carry trade activity [START_REF] Galati | Evidence of Carry Trade Activity[END_REF][START_REF] Gubler | Carry Trade Activities: A Multivariate Threshold Model Analysis[END_REF]. Not only CFTC data represents a small fraction of carry trade, but some traders may also be using these contracts for other purposes [START_REF] Curcuru | Measuring Carry Trade Activity[END_REF]. Each contract has information that is not publicly available, leaving space to misinterpretation. Nonetheless, as pointed out by BIS (2015), CFTC data is a reliable indicator of trends in carry trade activity. Also, it is the best publicly available data on speculative traders.

Using the number of contracts of non-commercial traders, I calculate the carry trade as the ratio of positions, as proposed by [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF]. For target currencies, the ratio is calculated by dividing long positions by short positions (CT ). Conversely, short positions over long positions are the ratio for funding currencies (CT F ). As pointed out by Curcuru et al. (2011, p. 438), "engagement in carry trades could be indicated by a net short futures position in the funding currency, paired with a net long futures position in the target currency." Therefore, the use of a specific ratio for each type of currency (target or funding) is more adequate to model its behaviour. As shown in Figure 2, currencies changed their classification accordingly to the movements of the US monetary policy. During the monetary easing (ME) period, only the currencies of Japan and Switzerland are classified as funding currencies, being the US Dollar the target currency. During the monetary tightening (MT) period, there are significant changes. First, currencies of Canada, the Euro area, and the United Kingdom reclassify as funding currencies. Second, currencies of Australia and New Zealand present target and funding classifications, creating respectively two subsamples MTT and MTF. Table 3 shows the number of observations for each country and sample.

Additionally, an exogenous dummy variable for the tapering period (TAPER) in the monetary easing (ME) period is included. TAPER starts in May 2013, with Ben Bernanke mentioning for the first time the possibility of tapering [START_REF] Chari | Taper Tantrums: QE, Its Aftermath and Emerging Market Capital Flows[END_REF]. It ends with the first hike in the US policy interest rate on December 15 2015. This dummy is critical to account for the period wherein the quantitative easing monetary policies started to unwind.

Table 4 shows the detailed description of each variable. Based on Donnelly (2019), nominal exchange rates (ER) are in the same form as used by market practitioners. Market sentiment is given by (V IX). To account for the stock market activity of each country and in the US, main market indexes are used (SM and SM U S, respectively). Overall, I follow the same group of variables proposed by [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF]. 3 Estimation results2

Preliminary procedures

In order to capture long-term impacts, I opt to use all variables in levels. The [START_REF] Toda | Statistical Inference in Vector Autoregressions with Possibly Integrated Processes[END_REF] technique is applied to deal with non-stationary variables.

As pointed out by [START_REF] Amiri | Granger Causality Between Total Expenditure on Health and GDP in OECD: Evidence from the Toda-Yamamoto Approach[END_REF], the [START_REF] Toda | Statistical Inference in Vector Autoregressions with Possibly Integrated Processes[END_REF] approach is useful to circumvent models' misspecification with non-stationary variables. In order to guarantee the usual asymptotic chi-square null distribution of the Wald tests, lagged exogenous variables are added to each non-stationary variable. The number of lags depends on the integration order (d) and the maximum lag length of the VAR model (p). Hence, the number of lags of these variables in the final models is specified by d plus p, generating a modified Wald test (MWald test). Therefore, "it is clearly desirable to have a testing procedure which is robust to the integration and cointegration properties of the process so as to avoid the possible pretest biases." (Toda and Yamamoto 1995, p. 226) First, to find the integration order (d), I apply the unit-roots test with one structural break with unknown breakpoints developed by [START_REF] Clemente | Testing for a Unit Root in Variables with a Double Change in the Mean[END_REF]. In the AO (Additive Outlier) model, equation ( 7) is estimated to remove the deterministic part of the variable:

y t = µ + d 1 DU 1t + y t (1)
In the next step, the test searches for the minimal t-ratio for the unit-root hypothesis (ρ = 1) in the following model Data is gathered using the R package quantmod (function GetSymbols), developed by Ryan and Ulrich (2020). The R package BatchGetSymbols, written by [START_REF] Perlin | BatchGetSymbols: Downloads and Organizes Financial Data for Multiple Tickers[END_REF], was used to confirm that the data collected was clean. Due to problems with data for Russia, data from the Moscow Exchange (MOEX) was used for cleaning.

y t = k i=0 ω 1t DT B 1t-1 + ρ y t-1 k i=0 c i ∆ y t-1 + e t ( 2
)
Table 5 summarizes the results of the unit-roots tests, with details in the supplementary material (Appendix).

Second, tests for the optimal lag length are applied to choose the maximum lag length of the VAR model (p). These tests generate two tests statistics (likelihood-ratio -LR and Akaike's final prediction error -FPE) and three information criteria (Akaike -AIC, Hannan and Quinn -HQIC, and Schwarz's Bayesian -SBIC). I do not rely solely on these tests to choose p because residual autocorrelation may be present. Thus, Lagrange-multiplier (LM) tests for residual autocorrelation are also computed. With d and p, the final robustness check is the stability test, i.e. eigenvalue stability condition (see the Appendix for further details).

Empirical results

Table 6 shows the results for the target currencies during the ME period. On the one hand, all variables are jointly Granger causing carry trade (CT ) in Australia, Brazil, the Euro area, Mexico, and the United Kingdom. Individually, the exchange rate (ER) is a good predictor for the carry trade in almost all countries (except for Canada, Mexico, and Russia). Additionally, other variables preceding carry trade are interest rate differentials (IRD) for the Euro area, market sentiment (V IX) for Australia, and the local stock market index (SM ) for Brazil. On the other hand, there are statistically significant results showing that carry trade also Granger causes the other variables. Mainly, CT is anticipating movements in market sentiment for almost all countries (excluding the Euro area and Mexico), highlighting the important link with market instability. As other variables are concerned, CT precedes ER, IRD, and SM for Australia. The results for the Australian Dollar show that carry trade impacts go further from the financial market, impacting the real economy. For Mexico, CT predicts movements in the US stock market, showing evidence of their financial market linkages.

As for funding currencies during the ME period, Table 7 illustrates the results. All variables jointly Granger cause carry trade (CT F ) for Japan, showing significant individual results for ER and V IX. For Switzerland, ER is Granger causing CT F . Moreover, there is evidence for bi-directional Granger causality between ERF and CT F for Japan. As for the Swiss case, CT F is preceding movements in the interest rate differential (IRDF ). Last but not least, CT F is a good predictor of local stock prices for both Japan and Switzerland. As highlighted by [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF], speculative investors use funding currencies as leverage to buy other assets. With the upsurge (reduction) in short positions in the futures market, an appreciation (depreciation) in the funding currencies expected. Therefore, a higher (lower) value of the local currency in comparison to the US dollar may increase (decrease) investors' interest in local stock markets. Results for target currencies during the monetary tightening period are in Table 8. Jointly, all variables Granger cause CT for Australia and Russia. For Australia, Brazil, and Mexico, ER and IRD are preceding changes in CT . Also, V IX and SM Granger cause CT for Russia and Australia, respectively. Linkages to the impact of the carry trade in the real economy are supported by the Granger causality of CT to V IX and SM , as shown by the results of Australia, Mexico, and New Zealand. Additionally, CT precedes changes in IRD for Australia and Brazil. For funding currencies, Table 9 shows the results for the MT period. Except for Canada and the Euro area, all variables jointly predict CT F . ER precedes CT for all countries (except for Canada). As [START_REF] Gubler | Carry Trade Activities: A Multivariate Threshold Model Analysis[END_REF] pointed out for the Swiss Franc as a funding currency, nominal exchange rate fluctuations are a good predictor of carry trade with the US Dollar as a target currency. For Switzerland, both IRDF and SM U S Granger cause CT F . With the IRDF peaking at this period for the Swiss economy, carry trade provides excellent opportunities to borrow in Swiss francs and invest with leverage in the US stock market. As pointed out by [START_REF] Vallet | The Role of the Swiss Franc in Switzerland's European Stance[END_REF], Switzerland has an advantage of "interest rates bonus". For the Euro area, IRDF is Granger causing CT F . As for the local stock markets, their movement precedes the carry trade activity. As shown by [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF], with the possibility to use leverage with carry trade, international speculative investors can borrow (in the funding currency) and invest locally (domestic stock market). Additionally, as Table 9 shows, CT F predicts ER for Canada, the Euro area, and the United Kingdom. As far the market sentiment is concerned, CT F Granger causes it for Canada, the Euro area and New Zealand. In terms of stock markets, there is significant evidence of Granger causality from carry trade to (1) SM for Canada, Japan, and Switzerland; (2) SM U S for Canada and the United Kingdom.

Although using different datasets and methodologies, my results are similar to [START_REF] Klitgaard | Exchange Rate Changes and Net Positions of Speculators in the Futures Market[END_REF], [START_REF] Mogford | The Information Content of Aggregate Data on Financial Futures Positions[END_REF], [START_REF] Nishigaki | Relationship Between the Yen Carry Trade and the Related Financial Variables[END_REF], [START_REF] Gubler | Carry Trade Activities: A Multivariate Threshold Model Analysis[END_REF], and [START_REF] Mulligan | Reassessing the Information Content of the Commitments of Traders Positioning Data for Exchange Rate Changes[END_REF].

Conclusion

This paper fills a gap in the carry trade literature with the use of CFTC data to empirically explore target and funding currencies during the periods of monetary easing and tightening in the US. The empirical approach with the Granger causality tests with the [START_REF] Toda | Statistical Inference in Vector Autoregressions with Possibly Integrated Processes[END_REF] technique is also a novelty in this literature. Instead of focusing on the short-term, these new empirical results take into consideration long-term effects to better understand the dynamics of the carry trade in the selected countries.

By focusing on the US futures market, which is one of the largest in the world, my results show evidence of different behaviour of carry traders along with different currencies and US monetary policies. During the monetary easing period, the group of target currencies is much bigger than in the monetary tightening period.

Overall, for both target and funding currencies and both periods, the exchange rate is a good predictor of carry trade. Similarly, when the related financial variables (interest rate differentials, market sentiment, and local and US stock indexes) are jointly considered, there is evidence that they precede movements in the carry trade activity. The Granger causality from carry trade to market sentiment points out to another similarity, which highlights the linkage between speculative investments and market instability.

For target currencies, the bi-directional Granger causality of interest rates differentials and carry trade make clear the importance of the latter for monetary policy. Central banks from Brazil and Mexico deliberately use the interest rate policy to administrate the exchange rate, impacting speculative foreign capital movements (including, carry trade).

Finally, during the monetary tightening period, both CT and CT F Granger cause stock market indexes. For target currencies, there is reasonable evidence that carry trade is pre-ceding the movements of the stock markets of Australia, Mexico, and New Zealand. As far funding currencies are concerned, carry trade predicts fluctuations in the stock market indexes of Canada, Japan, and Switzerland. One of the main difference from target to funding currencies in this period is that the carry trade Granger causes the US stock market, as shown by the results of Canada and the United Kingdom. Additionally, carry trade is Granger causing market sentiment in this period, which indicates a possible linkage of these speculative operations to systemic risk.

Table 1 .

 1 Conditions for the Granger causality from the other variables to CT

  In this case, following[START_REF] Gubler | Carry Trade Activities: A Multivariate Threshold Model Analysis[END_REF], the interest rate differential (IRDF ) is given by the difference between the US policy interest rate and policy country's interest rate. Figure2illustrates the results for the IRD.

	Monetary easing (ME)	Monetary tightening (MT)
	Interest rate differentials
	Australia	Canada	Japan	New Zealand	Switzerland
	Brazil*	Euro area	Mexico	Russia*	United Kingdom

The interest rate differential gives the classification of target and funding currencies. If the difference between the country's policy interest rate and the US policy interest rate (IRD) is positive, the country's currency is classified as a target currency. Contrariwise, a negative value for IRD indicates a funding currency. 0% 5% 10% 15% Dec-08 Nov-09 Oct-10 Sep-11 Aug-12 Jul-13 Jun-14 May-15 Apr-16 Mar-17 Mar-18 Jan-19 Figure 2: Difference between the country's policy interest rate and the US policy interest rate, in per cent *: Gaps are present due to lack of data in other variables.

Table 3 .

 3 Sample description Period differs from other countries due to lack of data.

	Sample

*:

Table 4 .

 4 Description of variables

	Variable Definition

*:

Table 5 .

 5 Stationary variables, I(0)

	Country	ME	MT	MTT	MTF
	Australia	CT , V IX		V IX, SM ER, V IX
	Brazil	V IX, SM U S	*
	Canada	CT , V IX	*
	Euro area	IRD, V IX	CT F , V IX, SM
	Japan	V IX	ER, SM
	Mexico	ER, IRD, V IX	ER, IRD
	New Zealand	V IX		ER, V IX	*
	Russia	V IX, SM	*
	Switzerland	CT F , ER, V IX	ER, SM
	United Kingdom CT , IRD, V IX	SM
	*: All variables are I(1).

Table 6 .

 6 Granger causality tests for target currencies, ME period

				Variable Granger causes CT				CT Granger causes variable
	Australia	0.0690*	0.3713	0.0440**	0.9281	0.7600	0.0237** 0.0107**	0.0177**	0.0579*	0.0032***	0.3430
	Brazil	0.0138**	0.8548	0.3617	0.0072***	0.8281	0.0136**	0.7520	0.3516	0.0003***	0.8471	0.3479
	Canada	0.3964	0.9922	0.7171	0.7768	0.7653	0.6446	0.6495	0.1778	0.0668*	0.4429	0.9260
	Euro area	0.0018***	0.0648*	0.9997	0.7238	0.5735	0.0193**	0.4133	0.4627	0.9528	0.8781	0.9076
	Mexico	0.4464	0.1380	0.0284	0.9543	0.7690	0.0841*	0.8426	0.8025	0.6783	0.7541	0.0986*
	New Zealand	0.0075***	0.8657	0.4763	0.6224	0.6691	0.1286	0.2216	0.9520	0.0002***	0.8175	0.5921
	Russia	0.8960	0.9999	0.2621	0.8321	0.8158	0.9648	0.6354	0.9972	0.0131***	0.5677	0.8950
	United Kingdom 0.0001***	0.4233	0.7454	0.1410	0.5969	0.0008***	0.3816	0.3306	0.0000***	0.3244	0.8850
	Notes: '***', '**', and '*' denote significance at 1%, 5%, and 10% respectively.

ER→CT IRD→CT V IX→CT SM →CT SM U S→CT All→CT CT →ER CT →IRD CT →V IX CT →SM CT →SM U S

Table 7 .

 7 Granger causality tests for funding currencies, ME periodVariable Granger causes CT F CT F Granger causes variableER→CT F IRDF →CT F V IX→CT F SM →CT F SM U S→CT F All→CT F CT F →ER CT F →IRDF CT F →V IX CT F →SM CT F →SM U S

	Japan	0.0001***	0.9882	0.0538*	0.7254	0.8043	0.0031***	0.0219**	0.7318	0.8856	0.0009***	0.5406
	Switzerland	0.0315**	0.2420	0.7194	0.1945	0.3151	0.1662	0.2932	0.0559*	0.0022***	0.0002***	0.4742
	Notes: '***', '**', and '*' denote significance at 1%, 5%, and 10% respectively.

Table 8 .

 8 Granger causality tests for target currencies, MT period IX→CT SM →CT SM U S→CT All→CT CT →ER CT →IRD CT →V IX CT →SM CT →SM U S For Australia and New Zealand, MTT period. '***', '**', and '*' denote significance at 1%, 5%, and 10% respectively.

				Variable Granger causes CT				CT Granger causes variable
	ER→CT IRD→CT V Australia* 0.0036*** 0.0783*	0.5150	0.5115	0.0144**	0.0133**	0.1550	0.0080*** 0.0016***	0.0850*	0.2292
	Brazil	0.0938*	0.0429**		0.7410	0.4735	0.6990	0.2299	0.4449	0.0340**	0.7168	0.1873	0.6518
	Mexico	0.0075*** 0.0142**		0.4638	0.8785	0.6085	0.1219	0.5304	0.3040	0.0023***	0.0545*	0.3163
	New Zealand*	0.4363	0.7138		0.1302	0.6977	0.1170	0.1943	0.2292	0.4252	0.0001*** 0.0351**	0.5152
	Russia	0.5043	0.2712	0.0001***	0.4899	0.2214	0.0002***	0.4339	0.6712	0.6069	0.2167	0.8777
	Notes:

Table 9 .

 9 Granger causality tests for funding currencies, MT period Variable Granger causes CT F CT F Granger causes variable ER→CT F IRDF →CT F V IX→CT F SM →CT F SM U S→CT F All→CT F CT F →ER CT F →IRDF CT F →V IX CT F →SM CT F →SM U S For Australia and New Zealand, MTF period. '***', '**', and '*' denote significance at 1%, 5%, and 10% respectively.

	Australia*	0.0323**	0.1012	0.8550	0.0715*	0.1909	0.0656*	0.1040	0.3168	0.7488	0.9954	0.4580
	Canada	0.6226	0.6228	0.4557	0.7414	0.1372	0.3900	0.0185**	0.4607	0.0000***	0.0747**	0.0289**
	Euro area	0.0759*	0.0279**	0.7845	0.3672	0.6791	0.1033	0.0994*	0.4433	0.0342**	0.7538	0.9454
	Japan	0.0412**	0.5194	0.3446	0.0001***	0.1677	0.0000***	0.8205	0.3287	0.6732	0.0451**	0.1237
	New Zealand*	0.0134**	0.1053	0.8448	0.2399	0.7215	0.0718*	0.2880	0.2086	0.0242**	0.7772	0.7968
	Switzerland	0.0295**	0.0913*	0.1613	0.3978	0.0066***	0.0052***	0.6266	0.1635	0.3141	0.0136**	0.1481
	United Kingdom 0.0008***	0.9577	0.3457	0.5971	0.8893	0.0411**	0.0972*	0.5902	0.9149	0.2962	0.0423**
	Notes:

 and data to replicate the results are available as supplemental material, as well as an Appendix providing detailed results.

no Exterior/Edital no 48-2017 and FURB (Universidade de Blumenau) are gratefully acknowledged. I profited from


Piero Baraldi

Francesco Di Maio

Enrico Zio

Ikram Najeh
email: ikram.najeh@vedecom.fr

Laurent Bouillaut
email: laurent.bouillaut@ifsttar.fr

Dimitri Daucher
email: dimitri.daucher@ifsttar.fr

Maxime Redondin
email: maxime.redondin@vedecom.fr

Maintenance Strategy for the Road Infrastructure for the Autonomous Vehicle

Keywords: Grouping maintenance actions, Road infrastructure maintenance, Autonomous vehicle, Pavement cracks, Road marking degradation

The quality of the road infrastructure plays a major role in the road safety, especially for the autonomous vehicles (AV). The AV contains cameras and lidars able to detect the road markings, obstacles and the other vehicles. The road markings help the AV to identify the path runway and to understand their localization. Thus, the AV must interact with the road infrastructure to drive around without any human interactions with a high automatization level. The failures of the components of the road infrastructure (pavement and road markings) are incompatible phenomena with the operation of AV. To ensure a good evolution of this kind of vehicles, the infrastructure and the vehicles must coordinate, each providing a certain level of service. Thus, an efficient road maintenance must be considered. The proposed paper suggests a maintenance policy for the road infrastructure by grouping the maintenance strategies of the road markings and the pavement. This strategy considers the road infrastructure as serial system. A genetic algorithm is used to group the maintenance activities. This methodology is applied to feedback datasets from both the French National Road 4 and the American pavement using the Long-Term Pavement Performance (LTPP) database.

Introduction

If autonomous vehicles are considered as a potential economic, social and environmentally friendly solution to some of our current mobility needs, their operation is strongly correlated with maintaining a high level of quality of the infrastructure. Indeed, the erasure of the road marking, the appearance of longitudinal and transverse cracks are incompatible phenomena with the operation of autonomous vehicles. Those infrastructure failures can be responsible for some bias in the estimation of the distance between objects, vehicles stability and users' comfort. Moreover, the higher automatization level will be, the stronger expectations on the availability of the road infrastructure components will be.

The figure (1) represents an example of failure of the infrastructure in an autonomous vehicle context. Here, the vehicle must detect the road marking to identify the runway. In this case, the right-side marking (marked in red) is detected by the vehicle when the central line (indicated by the dot yellow line) is not. Then, the autonomous vehicle considers this road as a single-lane road.

From this example we conclude that an efficient infrastructure maintenance should be considered to optimize the operation. In the same context, M. Zouch et al (2009) proposed a maintenance strategy for the longitudinal cracks to minimize the total maintenance cost. The degradation model was presented as a two-phases process. During the initial phase the inferior road layers are deteriorating even if the surface is free of cracks. The first deterioration step leads to the propagation phase during which the cracks are observable, and the deterioration of inferior road layers continues. For the road marking, M. Redondin et al (2019) proposed two maintenance strategies using the feedback data of the French National Road 4 (NR4): the first one systematic according to the age and the second one conditional according the degradation level.

In order to propose an efficient preventive maintenance policy, the road infrastructure is considered as a multi-component system. It consists on the road markings (emergency line, center line and median strip line) and the pavement. Each component has its own maintenance strategy and the aim of this paper is to propose a grouping maintenance policy. The failure of one component makes the system is failed. Thus, the road infrastructure is presented as a serial system with 4 components.

In the literature, H.VU et al ( 2014) proposed a maintenance policy for systems with a complex structure using the Genetic Algorithm (GA). This strategy is about taking advantage of the maintenance action on one component to maintain other components and thus, limit the logistic consequences and optimize the maintenance organization. In the same context, Faddoul et al (2018) proposed a grouping maintenance strategy for the serial systems as an extension of the individual maintenance.

Then in this paper, the considered datasets and their preprocessing will be first introduced. Then, the clustering process of the considered infrastructure, based on the degradation model, will be detailed.

Finally, before some conclusions and prospects, maintenance actions will be optimized firstly with an individual approach and the considering a grouping strategy.

Introducing of the considered databases and pre-processing

At present, we have not identified any accessible database containing degradation monitoring for both marking and pavement, which are the two components considered in this study. It was therefore necessary to consider different database for each component. However, the approach introduced could easily be implemented on a single database the day such a feedback database will be available.

Road marking

Retroreflective marking has the particularity of reflecting the light emitted from the headlights of a vehicle back to the driver. A marking with a high level of retro-reflection indicates that it is white and perfectly visible. The level of retro-reflection decreases over time and the marking turns grey. A road marking must ensure a minimum level of retro-reflection of 150 mcd/m²/lx. This threshold is proposed by M. Redondin et al ( 2017) and can be modified in order to optimize the detection of markings by the autonomous vehicles. Thus, the road marking is considered as failed when the retro-reflection level is less than 150 mcd/m²/lx. In this work, the feedback data of the French National Road 4 (RN4), that relies Paris to Strasbourg, is used. This road is inspected every September and the road markings are replaced in the following March.

Pavement

Long Term Pavement Performance (LTPP) is a program founded by the Federal Highway Administration (FHWA) since 1988. It collects the data of the quality of the pavement performance from test sections across the United States and Canada.

An initial analysis of this database shows that the state of Texas is the one with the most information on the percent of pavement cracking. In this work, Texas data were therefore considered for our study.

It is assumed that the Texas sections were inspected periodically every two years between 1991 and 2001 and the decision parameter is the longitudinal cracks percentage.

A section is down if the percent of cracks is greater than 5%. This threshold can be modified if more information is available on the expectations of the autonomous vehicle with respect to the infrastructure.

Censored data

In all this study, we unfortunately have to face with censored date. Indeed, in this case of study, the road infrastructure is periodically inspected, thus the exact time of the failure (or the observed degradation state) is unknown.

Thus, when the failure happens before the first inspection, the data is left censored. In case the failure happens between two inspections 𝑟 and 𝑙 the data is interval censored. Finally, when the road infrastructure is still in good condition after the last inspection, the monitoring data are right censored.

To deal with this kind of data, McLachlan et al (2007) proposed to use the EM algorithm to estimate the date of failure. M Redondin et al ( 2018) proposed an extension of this algorithm for highly censored data.

The considered EM algorithm for censured data estimation

In this paper, the Weibull distribution 𝑊(𝛼, 𝛽) is defined by its associated probability density function given by equation ( 1) where 𝛼 and 𝛽 are respectively the scale and shape parameters:

𝑓(𝑥) = { 𝛽 𝛼 ( 𝑡 𝛼 ) 𝛽-1 𝑒 ( 𝑡 𝛼 ) 𝛽 , 𝑡 > 0 0, 𝑡 ≤ 0 (1)
In order to estimate the parameters of a Weibull density, the EM algorithm is used to find the maximum likelihood. In this work, the observed data (t 1 , 𝑡 2 , … , 𝑡 𝑛 ) is independent identically distributed. Redondin et al ( 2018) proposed a censor detector is associated for each observation as follows:

∀ 𝑖 = 1, … , 𝑛 𝛿 𝑖 = { 0 𝑖𝑓 𝑡 𝑖 𝑖𝑠 𝑢𝑛𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 1 𝑖𝑓 𝑡 𝑖 𝑖𝑠 𝑙𝑒𝑓𝑡 𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 2 𝑖𝑓 𝑡 𝑖 𝑖𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 -𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 3 𝑖𝑓 𝑡 𝑖 𝑖𝑠 𝑟𝑖𝑔ℎ𝑡 𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 (2)
From now, the observed data is defined by the detector as The EM algorithm is an iterative process. The initial Weibull distribution 𝑊(𝛼 0 , 𝛽 0 ) is given arbitrary. In each iteration 𝑘, an estimation of 𝛼 𝑘 𝑎𝑛𝑑 𝛽 𝑘 is given. The algorithm is stopped when |𝛽 𝑘 -𝛽 𝑘-1 | < 10 -4 .

Clustering

To simplify the process, the data are classified according to their degradation model. In data mining, clustering is the act and the process of grouping objects into classes (clusters). Each cluster contains objects that have similar characteristics. Many algorithms are available in the literature such as: k-means, Mean-shift clustering, Hierarchical clustering and Multi-Layer clustering.

During his works, M. Redondin et al ( 2018) suggested to classify markings into 9 clusters using Hierarchical clustering.

The same clustering methodology is used to classify the pavement sections.

Hierarchical clustering

In literature, there are two types of hierarchical clustering: Divisive and Agglomerative according to Sasirekha, K., & Baby, P (2013).

In this work, the agglomerative method is used. Each reference point or pavement section is considered as an individual cluster. In each iteration, the clusters with the same characteristics (cracks percentage or the retro reflexion level) merge together in order to create a new cluster. The process is still working until having one cluster that group all individuals. To calculate the similarity between two clusters, there are certain approaches which are used to calculate the distance such as Min, Max and Ward.

In this case study, the Ward distance defined by Eq (8) in where 𝑋 and 𝑌 are both clusters, 𝑥 and 𝑦 their average profiles, 𝑛 𝑥 , 𝑛 𝑦 their numbers.

𝑑(𝑋, 𝑌) = 𝑑(𝑥,𝑦) 1 𝑛 𝑥 + 1 𝑛 𝑦 (8)
The hierarchical clustering technique can be visualized using a Dendrogram. Based on its shape, the number of clusters is defined using the semi-partial 𝑅² criterion. It calculates the loss of between-class inertial. The loss must be as small as possible. The figure 2 represents the dendrogram the pavement sections. Based on it, the pavement section can be grouped into 8 clusters. The number of clusters is defined using the semipartial criterion. As illustration, let's consider an example of a cluster that contains 106 sections. The table 1 represents the result of EM algorithm. It indicates that (68,6%) of the data is right censored and (30%) are interval censored. Particularly, the main failure is between 72 and 120 months the data are right censored.

Individual maintenance strategy optimization

The reference points of the road markings are clustered into n groups and the pavement sections are clustered into m groups. For each cluster 𝑖, a Weibull analysis is applied to identify the parameters 𝛼 𝑖 and 𝛽 𝑖 . In this section, an age-based maintenance strategy is presented. This strategy consists in identifying a path 𝑇 * of preventive maintenance time for each component. Based on the Weibull parameters, the preventive maintenance cost 𝐶 𝑝 and the corrective maintenance cost 𝐶 𝑐 , the critical operational age 𝑥 𝑖 * is defined. The average maintenance cost per unit of time over an infinite horizon defined in the Eq (9) is calculated as given by Rausand M and A Hoyland (2004). The optimum path of maintenance is defined as showed in the equation ( 10). If a failure happens between two planning maintenance actions, a corrective maintenance is performed, and the age of the component is initialled on 0.

Grouping maintenance actions

This section is devoted to proposing a grouping maintenance policy. A multi-component system is a structured combination of components or subsystems that interact in an organized manner to accomplish a common functionality. In this paper, a maintenance grouping strategy for multicomponent systems is proposed. The proposed strategy considers the structure of the system. For systems with complex structure, the components are classified into 2 types: critical if the its shutdown leads to the shutdown of the system and non-critical if the system is still functioning even if the component fails.

Let's consider π 𝑖 an indicator function defined as:

𝜋 𝑖 = { 1 𝑖𝑓 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖 𝑖𝑠 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 0 𝑖𝑓 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖 𝑖𝑠 𝑛𝑜𝑡

Planning horizon

To group the maintenance activities, a planning horizon must be defined. It begins at 𝑇 𝐵𝑒𝑔𝑖𝑛 and ends at 𝑇 𝐸𝑛𝑑 .

To guarantee that all the components are maintained at least once time, H. Vu et al (2014) proposed to define the 𝑇 𝐸𝑛𝑑 as follows: Where 𝑖 𝑗 the 𝑗 𝑡ℎ maintenance action if the component 𝑖 and 𝑘 the number of the group that contains 𝑖 𝑗 . For 𝑁 maintenance actions, the maximum number of groups is 𝑁 -1.

The initial population is randomly defined and contains 𝑛 𝑝𝑜𝑝 solutions. In each iteration, the GA phases (selection, crossover, mutation and reparation) determines the optimal solution.

During the selection step, the best candidate solutions are selected from population. In this paper, Roulette Wheel Selection is used. The solutions are sorted in ascending order in s groups according to the economic profit. A selection likelihood is associated for each solution proportional to its performance. The candidate solutions (the parent solutions) are used to crossover. A crossover point is randomly chosen to combine selected solutions to generate better solutions in the next generation. In order to generate a new solution, a component of a group is moved to another one. The solutions obtained during the selection and the mutation phases may, sometimes, contains defects for example: empty group, incoherence in actions execution times... During the repair phase, some modifications are made.

• If two maintenance activities 𝑖 𝑘 and 𝑖 𝑙 belong to the group 𝑢 where 𝑙 < 𝑘, 𝑖 𝑘 is moved, randomly, to another group 𝑣. • If 𝑋(𝑖 𝑗 ) = 𝑢 and 𝑋(𝑖 𝑙 ) = 𝑣 where 𝑗 < 𝑙 and 𝑡 𝐺 𝑣 > 𝑡 𝐺 𝑢, the maintenance actions are permuted. • Empty groups are deleted, and the others are renumbered.

Mathematic formulation of economic profit

Let 𝐺 𝑘 be a group of preventive maintenance. The economic profit of a group is divided into three parts as follows:

Setup cost

The preventive maintenance actions are grouped, only one setup cost is required and is calculated as:

𝑈 𝐺 𝑘 = [𝐶𝑎𝑟𝑑(𝐺 𝑘 ) -1]. 𝑆 (14)
Where 𝐶𝑎𝑟𝑑(𝐺 𝑘 ) the number of the preventive maintenance activities and 𝑆 the setup cost.

Penalty cost

When the maintenance activities are grouped, the maintenance execution time 𝑡 𝑖 𝑗 is changed to 𝑡′ 𝑖 𝑗 = 𝑡 𝑖 𝑗 + ∆𝑡 𝑖 𝑗 where ∆𝑡 𝑖 𝑗 > -𝑥 𝑖 * (see the figure 4). This modification leads to a penalty cost ∆𝐻 1 𝐺 𝑘 because of reducing service life or increasing likelihood of failure.

Let ℎ 𝑖 (Δ𝑡 𝑖 𝑗 ) be the penalty cost when 𝑖 𝑗 is executed at 𝑡 𝑖 𝑗 defined as follow: the average asymptotic cost. The penalty cost of the group 𝐺 𝑘 is computed as:

ΔH G k 1 = ∑ ℎ(Δ𝑡 𝑖 𝑗 ) 𝑖 𝑗 ∈𝐺 𝑘 (16)
The optimal execution time of the group 𝐺 𝑘 is found by minimizing Δ𝐻 𝐺 𝑘 1 .

(15) Where 𝐶 𝐺 𝑘 ̅̅̅̅ is the total planned shutdown cost if the maintenance activities are performed separately:

𝐶 𝐺 𝑘 ̅̅̅̅ = ∑ [π 𝑖 . 𝐶 𝑠𝑦𝑠 𝑖 + (1 -π 𝑖 ) . 𝐶 𝑖 𝑙 ] 𝑖 𝑗∈𝐺 𝑘 (18)
𝐶 𝐺 𝑘 is the planned shutdown cost when the maintenance activities are performed together:

𝐶 𝐺 𝑘 = 𝐶 𝑠𝑦𝑠 𝑝 ( 19
)
Where 𝐶 𝑠𝑦𝑠 𝑝 planned shutdown cost of the system

Economic profit

For each group 𝐺 𝑘 , the economic profit EP is defined as:

𝐸𝑃(𝐺 𝑘 ) = 𝑈 𝐺 𝑘 -∆𝐻 1 𝐺 𝑘 -∆𝐻 2 𝐺 𝑘 (20)
The total economic profit 𝑇𝐸𝑃 of the grouping structure 𝑆𝐺𝑀 is the calculated as:

𝑇𝐸𝑃(𝑆𝐺) = ∑ 𝐸𝑃(𝐺 𝑘 𝐺 𝑘 ∈ 𝑆𝐺𝑀 ) (21)
A grouping structure is optimal if it maximizes the total economic profit.

𝑆𝐺𝑀 𝑜𝑝𝑡 = 𝑚𝑎𝑥 𝑆𝐺𝑀 𝑇𝐸𝑃(𝑆𝐺𝑀) (22)

Study case

The grouping strategy is applied to the feedback data of the road markings of the French National Road 4 (NR4) and the Texas's pavement sections introduced in section 2. The failure of one component leads the "shutdown" of the system. Thus, the road infrastructure is presented as a serial system with 4 components. As per the definition of a critical component, all the components are critical, so π 𝑖 = 1, ∀𝑖 ∈ [1,4].

It assumed that a road section is composed with pavement and 3 marking lines (Median strip line, Emergency line and skip centre line). For the road markings, for each cluster, the maintenance strategy presented in the section 4 is applicated. The critical age 𝑥 𝑖 * of replacement is defined. The same strategy is used for the pavement's sections. The table 2 is an example of simulated road.

In this case study, the planning horizon started at 𝑇 𝐵𝑒𝑔𝑖𝑛 = 0 : it assumed that the road markings have just been replaced and the pavement is still in a good state (the percent cracks is less than 5%) and it ends 132 months after (T = [0,132]). Thus, the maintenance activities of the road markings are grouped in this planning horizon.

According to M. Redondin et al ( 2019), the annual preventive maintenance cost per marking is 𝐶 𝑝 = 940€ and the corrective maintenance is supposed 𝐶 𝑐 = 3 * 𝐶 𝑝 . For the pavement, the exact preventive maintenance cost is still unknown and is assumed equal to 2000 € for a section of 152m. Those parameters can be modified if more cost data are available. According to the table 2, the road markings must be replaced each 11, 12, and 19 months. The aim of this section is to identify an optimal time to maintain the road markings simultaneously. By applying the grouping strategy (presented at section 5), the maintenance activities of the road markings can be grouped every 13 months. Using this policy, the economic profit is about 10.3%.

Conclusions and prospects

This paper presents a grouping maintenance actions for road infrastructure to optimize the operation of the autonomous vehicles. It assumed that the infrastructure is a serial system with 4 components (3 road markings and the pavement). The policy proposed is applied to the feedback data of both French National Road 4 and the American database LTPP. To simplify the study, the road lines and the pavement sections are divided into clusters using the Agglomerative Hierarchical Clustering. The datasets are censored; thus, a Weibull analysis is applied to each cluster to estimate the failure dates and the Weibull parameters. Then, an individual maintenance strategy is suggested to each cluster. In order to minimize the maintenance costs and ensure a good state of the road infrastructure, a grouping maintenance actions is proposed using a genetic algorithm.

Future works will consider the dynamic context: changing pavement conditions during the planning horizon, maintenance cost…. The Remaining Useful Lifetime of the pavement will be considered to analyse the pavement state.

Fig. 1 .

 1 Fig. 1. Example of a disturbed detection due to a severely degraded infrastructure (Revilloud et al 2016 b)

  τ = {𝑡 ∈ ((𝑡 1 , 𝛿 1 ), … , (𝑡 𝑛 , 𝛿 𝑛 ))/𝛿 𝑖 = 0} (3) X = {𝑥 ∈ ((𝑡 1 , 𝛿 1 ), … , (𝑡 𝑛 , 𝛿 𝑛 ))/𝛿 𝑖 = 1} (4) Y = {𝑦 ∈ ((𝑡 1 , 𝛿 1 ), … , (𝑡 𝑛 , 𝛿 𝑛 ))/𝛿 𝑖 = 2} (5) Z = {𝑧 ∈ ((𝑡 1 , 𝛿 1 ), … , (𝑡 𝑛 , 𝛿 𝑛 ))/𝛿 𝑖 = 3} (6) In this case, the maximum likelihood is defined as follow: As detailed in McLachlan et al (2007), the EM algorithm is a two-phases process: • The Expected Step estimates censored data according to a given Weibull distribution 𝑊(𝛼, 𝛽). • The Maximization Step estimates a Weibull distribution 𝑊(𝛼, 𝛽) by the Maximum Likelihood Estimator according to both uncensored and completed data.

Fig. 2 .

 2 Fig. 2. Dendrogram associated to the AHC of the pavement sections

Fig 3 .

 3 Fig 3. Example of age-based maintenance replacement The figure (3) illustrates the age replacement maintenance strategy for a component 𝑖.

  date of the first maintenance action of the component 𝑖, where 𝑎𝑔𝑒 𝑖 0 the age of the component 𝑖 at 𝑇 𝐵𝑒𝑔𝑖𝑛 . As an illustration, the figure 4 is an example of a system with 4 components. The first maintenance execution time of the component 2 is the biggest, thus, in this case the planning horizon is defined by 𝑇 = [𝑇 𝐵𝑒𝑔𝑖𝑛 , 𝑡 2 1 ].

Fig 4 .

 4 Fig 4. Example of planning horizon for system with 4 components 5.2 Genetic algorithm To identify the best grouping strategy, a Genetic Algorithm (GA) is used in this work. The GA is part of the stochastic optimization algorithms based on the mechanisms of natural selection and genetics. The process starts with an initial population. In a maintenance grouping problem, each candidate solution is encoded as a vector 𝑋 with 𝑁 elements corresponding to 𝑁 maintenance activities. According to H. Vu et Al (2014) 𝑋 is structured as: 𝑖 𝑗 ∈ 𝐺 𝑘 → 𝑋(𝑖 𝑗 ) = 𝑘, 𝑘 ∈ [1, … , (𝑁 -1)] (13)

  ℎ 𝑖 (Δ𝑡 𝑖 𝑗 ) = 𝐸[𝑀𝐶 𝑖 (𝑥 𝑖 * + Δ𝑡 𝑖 𝑗 )] -𝐸[𝑀𝐶 𝑖 (𝑥 𝑖 * total corrective maintenance cost in the interval time [0, 𝑥] for the component 𝑖 and 𝐶𝐴 * = Δ𝑡 𝑖 𝑗 .

Fig 6 .

 6 Fig 6. Example of changing execution time of a component 𝑖

  ∆𝐻 2 𝐺 𝑘 = 𝜋 𝐺 𝑘. (𝐶 𝐺 𝑘 ̅̅̅̅ -𝐶 𝐺 𝑘) (17)

Table 1 :

 1 Weibull analysis of the cluster 1 of pavement sections.

	Age Failure	Interval	Right
	estimated	Censored	Censored
	48	2(1,88 %)	0
	72	7(6,16%)	0
	96	17 (16,03%)	0
	120	7(6,16%)	73(68,8%)

Table 2 .

 2 Result of application of the EM algorithm and the maintenance strategy

	Components	𝛼	β	𝑥 𝑖 *
	Pavement	128.7	7.25	132
	Median strip line	21.84	2.01	11
	Emergency line	23.8	2.02	12
	Skip centre line	30.34	6.64	19

Acknowledgment

VEDECOM and IFSTTAR thank the DIR EST and CEREMA for inspections data of the marking lines of the French National Road 4. Thanks to the Federal Highway Administration for making the LTPP database accessible.