## 1. Exchange Rate Network

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---

### Main Findings
- The evolution of nominal exchange rates can be modeled on a network where the nodes represent individual currencies and the links among them represent weighted returns on a hypothetical investment in each currency.
- A multilateral exchange rate network is represented by multilaterally dependent changes in bilateral exchange rates.
- The currency demand indicator (CDI), an elementary cell of the network model, is defined as weighted log returns on each currency.
- The CDI provides a useful proxy for demand for each currency from other currencies and reflects all underlying balance of payments flows.
- The model identifies the stationary position of the exchange rate network, i.e., the episodes of minimal temporal variety of the CDI when weighted returns on the links are close to zero.
- The stationary position of the exchange rate network points to the equilibrium levels of bilateral exchange rates for each currency pair.
- The model applies mainly to currencies with floating exchange rate regimes, although useful information can also be obtained for currencies with pegged exchange rates.
- Further research is needed to identify the factors that can lead to a switch from a stationary to non-stationary position of the exchange rate network; such factors can be used as leading indicators of currency crises.

### Introduction: motivation and objectives
- Increased volatility of exchange rates is a policy concern linked to advanced economy stimulus, the decline in commodity prices, the slowdown in China, geopolitical tensions, and US monetary policy normalization spillovers.
- Purpose: build a new model of multilaterally equilibrium exchange rates based on complex network theory; introduce a topological component accounting for higher-order interactions among all currencies.
- The paper:
  - proposes the currency demand indicator (CDI),
  - constructs a time-dependent multilateral exchange rate network,
  - finds its dynamically stationary position,
  - identifies multilaterally equilibrium levels of exchange rates.
- Data illustration: applied to daily 1995-2016 exchange rates from the Thomson Reuters Datastream.
- Conceptual distinctions:
  - Multilaterally equilibrium exchange rates: periods when all nominal bilateral exchange rates remain simultaneously relatively stable during a prolonged period of time.
  - Fundamentally equilibrium exchange rates: defined in real effective terms relative to a set of a country’s own fundamentals and partners’ fundamentals (IMF, 2015); assessed relative to macroeconomic fundamentals.
  - Economic implication: high volatility of multilateral exchange rates linked to disequilibria; volatility of fundamental exchange rates can reflect desirable adjustment.
- Contributions:
  - a computable network model of multilaterally equilibrium exchange rates;
  - CDI to assess currency demand across all BoP flows;
  - the concept of multilateral exchange rate network stationarity pointing to equilibrium bilateral rates;
  - illustration using daily 2005-16 exchange rate moves (note: dataset references include both daily 1995-2016 and daily 2005-16 in the text).

### Analytics — Exchange Rate Network concept
- BoP as multilayer network:
  - BoP viewed as a multilayer network reflecting payments across layers (trade, services, remittances, dividends, foreign direct and portfolio investment, credit, reserve assets, etc.).
  - A multilayer BoP network can have any number ݀ of dimensions and a sequence ൌܮ ... of elementary layers, with one set of elementary layers ܮ for each dimension ߙ (Kivelä, 2014).
- Network representation:
  - Nodes: individual currencies (not countries), weighted by multilateral demand from all other currencies.
  - Links: weighted log-returns on each currency pair that represent weighted changes in exchange rates (positive or negative) and indirectly reflect all underlying BoP flows.
  - Appreciation = inflow into a currency node; depreciation = outflow from a node.
- Graph properties:
  - Directed: changes embedded into log returns indicate direction of BoP flow between currencies.
  - Weighted: link weights reflect payment values differing by currency pair.
  - Incomplete: not all currencies are connected directly; many are quoted via cross-rates to the US dollar.
  - Asymmetric: implicit outflows differ from implicit inflows for currency nodes.
- Advantages vs prior approaches:
  - Avoids base currency selection problem by using observed weighted log-return flows rather than correlations quoted to a single base.
  - Contrasts with prior networks that used trade-weighted exchange rates, correlation of log-returns, Pearson correlations, phase synchronization, distance transforms, or tail dependence.
  - Allows indirect representation of BoP flows where bilateral payment data are limited.
- Practical notes on topology and dynamics:
  - Topology evolves daily: not all currency pairs are traded every day; weights and links change with bilateral exchange rate fluctuations.
  - Episodes of USD or other currency appreciation/depreciation show directional inflows/outflows to/from that currency node.
  - High overall exchange rate volatility implies an inherently non-stationary multilateral network; episodes of relatively low volatility correspond to network stationarity.
  - Stationarity implies the network’s evolution is systematically driven by dynamics of chosen constraints (not numerical constancy of rates).
  - During a stationary episode, multilaterally equilibrium bilateral exchange rates can be identified; deviations are interpreted as temporary misalignment expected to disappear in the medium term.

### Model (overview)
- Core elements:
  - Currency Demand Indicator (CDI): defined as weighted log returns on each currency; elementary cell of the network model and proxy for demand from other currencies.
  - Links: weighted log-returns representing hypothetical investor returns and directionality of implicit BoP flows (e.g., USD appreciation relative to EUR -> net BoP flow toward USD from EUR; link arrow from EUR to USD scaled by EUR/USD weight in global FX turnover).
  - Nodes: weighted by share of the currency in international currency turnover (BIS, 2013) reflecting multilateral demand.
- Uses and applicability:
  - Mainly applicable to currencies with floating exchange rate regimes; informative also for pegged regimes.
  - The model can identify dynamically stationary positions of the exchange rate network, which indicate equilibrium levels of bilateral exchange rates.
  - Potential application: estimate magnitudes of under- and overvaluation of individual currencies relative to their multilaterally equilibrium rates.
- Limitations and extensions:
  - Bilateral flow statistics are limited beyond trade and limited bilateral financial flow datasets; the model uses log-returns as an indirect BoP flow representation.
  - Further research needed to identify factors that cause switches from stationary to non-stationary network positions as potential leading indicators of currency crises.

### Methodology: log-returns, weights, and the Currency Demand Indicator (CDI)
- Log-return ߟ௜௝ (itha) on an investment in currency ݅ from currency ݆ is defined as:
  - ߟ௜௝ (ݐ) = ln(ݎ௜௝ (ݐ) / ݎ௜௝ (ݐ−1))  (Equation (1) in source)
- Currency-pair weights ߣ௜௝ (lambda) are defined as the share of each currency pair ݆↔݅ in the global foreign exchange market turnover.
  - For major pairs weights are taken from BIS (2013) statistics.
  - Remaining pairs (representing less than two percent of turnover) are reconstructed under an even distribution assumption.
  - With each currency included on both sides of a pair, total weights of all currency pairs add up to 200 percent.
- Weighted log-return along link ݆↔݅ is the absolute value of the weighted logarithmic return on this link (Equation (2) and (3) in source).
- Bilateral currency demand indicator ߮௜௝ (phi) for currency ݅ from currency ݆ is defined as:
  - ߮௜௝ (ݐ) = ߣ௜௝ ߟ௜௝  (Equation (4))
- Multilateral Currency Demand Indicator (CDI) ߯௜ (chi) for currency ݅ from all other currencies {1,݆...ܰ} is defined as:
  - ߯௜ (ݐ) = ෍௜௝ ߣ௜௝ ߟ௜௝  (Equation (5))
  - The CDI is a novel indicator according to the authors: "To the best of our knowledge, the CDI has not been proposed in the earlier literature."
- Exchange rate network representation:
  - Symmetrical, weighted, and directed graph.
  - Nodes represent CDI ߯௜ (ݐ); node size proportional to average CDI.
  - Links represent ߮௜௝ (ݐ); link width scaled by share of the log-return in the global market; arrow indicates direction of net demand.
  - If ߮௜௝ (ݐ) ≥ 0: excess demand for currency ݅ from currency ݆ (unobserved inflow ݆←݅; appreciation of ݅).
  - If ߮௜௝ (ݐ) ≤ 0: unobserved outflow from currency ݅ to ݆ (݅→݆); depreciation of ݅.

### Stationarity metrics and thresholds
- Temporal volatility of CDI (ߤ(χ)) across a selected period ܶ is calculated as the standard deviation of multilateral demand for each currency:
  - μ஧௧ = √(1/T Σ(χ௧(t) − χത௧)²)  (Equation (6))
  - χത௧ = 1/T Σχ௧(t)  (Equation (7))
  - Temporal volatility measures degree of stationarity over time.
- Sectional variety ν(t) of CDI across N currencies on a given day is the cross-sectional standard deviation:
  - ν஧(t) = √(1/N Σ(χ௧(t) − χത(t))²)  (Equation (8))
  - χത(t) = 1/N Σχ௧(t) = 0  (Equation (9))
  - The sum of sectional CDIs is zero by definition because the global FX market clears daily.
  - Sectional variety measures daily pulsation of the exchange rate network.
- Stationarity definition:
  - Periods when temporal volatility and sectional variety are simultaneously minimal.
  - Practically, minimal sectional variety ν(t) is used to define stationary exchange rate network episodes (static concept relative to each CDI mean).
- Two empirical thresholds to identify stationary episodes:
  - Threshold 1 (floor): minimum variety of the CDI during the sample plus one standard deviation.
    - In the sample: Threshold 1 = 2.66E-04.
      - Computed from minimum variety = 2.0295E-05 plus one standard deviation = 2.46E-04.
    - Deviation from threshold = observed CDI variety − Threshold 1.
    - Negative deviations indicate dates of the lowest CDI variety relative to the threshold.
  - Moving-window aggregation:
    - Apply a 22-day moving average window (average work period in one month, 2,607 days/120 months) to the count of days with negative deviations.
  - Threshold 2: within each 22-day window, at least 80 percent of days must be lowest-variety days.
    - 80 percent of 22 days = 18 days.
  - Dates meeting both thresholds are considered episodes of multilaterally equilibrium exchange rates.

### Data inputs and weight matrix
- Primary inputs:
  - Daily exchange rates for 130 currencies with respect to the US dollar in units of national currency per one USD for 2005-16 (Annex 1).
  - BIS weights of currency pairs in global foreign exchange turnover (Annex 2).
    - Weights available for 21 currency pairs involving USD, 15 involving EUR, and nine involving JPY.
    - Example: weight of EUR/USD pair in global turnover = 24.1 percent.
  - All currency pairs included twice in adjacency matrix, so weights sum to 200.
- The adjacency (weight) matrix dimension: 130 x 130.
  - The share of other currency pairs equal to 1.7 percent is assumed distributed equally among currencies not included in the BIS matrix.
  - The matrix is heavily centered on reserve currencies; other currencies included with much lower weights reflecting relative trade share.
  - Example notes: some pairs (e.g., JPY/GBP) have negligible direct weight because they are commonly exchanged via USD as intermediate currency.

### Empirics and key results
- Sample span: daily observations for 2005-2016.
  - Total daily observations mentioned: 2,610 (and elsewhere 2,607 days referenced for moving-average window).
- Threshold 1 and deviations:
  - Threshold 1 (2.66E-04) computed from minimum variety 2.0295E-05 plus one standard deviation 2.46E-04.
  - In about half of 2,610 daily observations, the deviation (observed CDI variety − Threshold 1) was negative.
- Episodes of exchange rate network stationarity identified during 2005-16:
  - June–July 2007
  - end-November to December 2013
  - February 2014
  - end-April to early September 2014
- The CDI captures high instability during 2008–09 financial crisis via increased temporal volatility and sectional variety.
- Illustration of multilateral equilibrium exchange rates (examples from stationary episodes):
  - 2007 episode:
    - USD/EUR = 0.74 (units of national currency per US Dollar)
    - USD/JPY = 122.39
    - USD/GBP = 0.50
  - 2014 episode:
    - USD/EUR = 0.74
    - USD/JPY = 102.28
    - USD/GBP = 0.59
  - Note: each bilateral rate can be considered in equilibrium only conditional on the levels of other bilateral rates during the same stationary episode.

### Limitations of the model and CDI
- The CDI is more likely to capture stability in reserve currencies, rather than stability across all currencies in the sample.
- Weights used in the analysis rely heavily on the share of currencies in the international currency turnover; the weights exhibit fat tails in the sense that reserve currencies dominate global foreign exchange markets.
- The CDI has to rely on incomplete triennial data sources, such as BIS. Therefore, the adjacency matrix that yields the weights on bilateral flows is remarkably sparse.
  - Example: the adjacency matrix suggests that the U.K. and Japan have no links, as the currency weight is missing from the BIS and may be misspecified.
- The model rests on strong assumptions, such as the balance of demand and supply of currencies being delinked from countries of their issuances.
  - Countries issuing reserve currencies may often affect them via their monetary and exchange rate policies.
  - Example: the normalization of U.S. monetary policy and monetary policy divergence among countries issuing reserve currencies is likely to have an impact on the CDI.
- Embedded empirical snippets and sample statistics (as presented):
  - Episode dates: 5/29/2007 11/19/2012 1/27/2014 4/21/2014 7/24/2007 1/1/2013 2/25/2014 9/5/2014
  - Duration (working days) 413 22 100
  - EURO EUR 0.74 0.77 0.73 0.74
  - JAPANESE YEN JPY 122.39 83.36 102.14 102.28
  - SWISS FRANC CHF 1.22 0.92 0.90 0.90
  - RUSSIAN ROUBLE RUB 25.75 30.83 35.15 35.19
  - CHINESE YUAN CNY 7.61 6.23 6.07 6.21
  - CANADIAN DOLLAR CAD 1.06 0.99 1.11 1.09
  - INDIAN RUPEE INR 40.63 54.85 62.38 60.09
  - BRAZILIAN REAL BRL 1.91 2.08 2.40 2.24
  - AUSTRALIAN DOLLAR AUD 1.17 0.96 1.12 1.07
  - SOUTH AFRICA RAND ZAR 7.07 8.69 11.03 10.61
  - MEXICAN PESO MXN 10.81 12.91 13.31 13.02
  - SOUTH KOREAN WON KRW 924.03 1,078.52 1,072.41 1,023.28
  - NORWEGIAN KRONE NOK 5.91 5.63 6.14 6.09
  - SWEDISH KRONA SEK 6.84 6.62 6.49 6.75
  - INDONESIAN RUPIAH IDR 9,002.21 9,629.00 12,021.43 11,685.97
  - UK POUND GBP 0.50 0.62 0.60 0.59

### Policy implications and applications
- Exchange rate assessment:
  - The model can find ex-post periods of multilaterally equilibrium exchange rates and calculate deviations of observed bilateral exchange rates from their latest multilaterally equilibrium levels.
  - This can help detect potential over- and undervaluation of individual currencies that may not be evident from bilateral exchange-rate moves or reserve pressures alone.
- Early warning indicators:
  - Temporal currency demand volatility and sectional currency variety increase before crises; the model can contribute to early warning of possible currency crises.
- Role monitoring:
  - The weighted CDI allows study of the evolution of the role of individual currencies and balance of payments flows denominated in those currencies.
- Operational use:
  - Timely detection and prediction of episodes of multilateral exchange rate network stationarity and non-stationarity may be useful for policymakers to ensure exchange rate stability during periods of increased temporal variety of exchange rates.
- Possible extensions and research directions:
  - Further exploration of the CDI, its properties and dynamics.
  - Additional studies of the exchange rate network and its early warning properties to allow signaling of switching between stationary and non-stationary regimes; these may be important predictors of currency crises.
  - Additional exploration of the multilaterally equilibrium bilateral exchange rates to assess current levels relative to the new metric of the multilateral network equilibrium.
  - Calculations of the observed bilateral exchange rates deviations from their multilaterally stationary levels to signal overvaluation or undervaluation, which could become an important input for the policy dialogue with IMF member countries.

*Source: _wp16130 - 18. The balance of payment flows among currency nodes can be indirectly measured*

### 1. Exchange Rate Network ...............................................................................................

### 1. Exchange Rate Network

### Main Findings
- The evolution of nominal exchange rates can be modeled on a network where the nodes represent individual currencies and the links among them represent weighted returns on a hypothetical investment in each currency.
- For the purposes of this paper, a multilateral exchange rate network is represented by multilaterally dependent changes in bilateral exchange rates.
- The currency demand indicator (CDI), an elementary cell of the network model, is defined as weighted log returns on each currency.
- The CDI provides a useful proxy for demand for each currency from other currencies and reflects all underlying balance of payments flows.
- The model identifies the stationary position of the exchange rate network, i.e., the episodes of minimal temporal variety of the CDI when weighted returns on the links are close to zero.
- The stationary position of the exchange rate network points to the equilibrium levels of bilateral exchange rates for each currency pair.
- The model applies mainly to currencies with floating exchange rate regimes, although useful information can also be obtained for currencies with pegged exchange rates.
- Further research is needed to identify the factors that can lead to a switch from a stationary to non-stationary position of the exchange rate network; such factors can be used as leading indicators of currency crises.

### Introduction: motivation and objectives
- Increased volatility of exchange rates is a policy concern linked to advanced economy stimulus, the decline in commodity prices, the slowdown in China, geopolitical tensions, and US monetary policy normalization spillovers.
- Purpose: build a new model of multilaterally equilibrium exchange rates based on complex network theory; introduce a topological component accounting for higher-order interactions among all currencies.
- The paper:
  - proposes the currency demand indicator (CDI),
  - constructs a time-dependent multilateral exchange rate network,
  - finds its dynamically stationary position,
  - identifies multilaterally equilibrium levels of exchange rates.
- Data illustration: applied to daily 1995-2016 exchange rates from the Thomson Reuters Datastream.
- Conceptual distinctions:
  - Multilaterally equilibrium exchange rates: periods when all nominal bilateral exchange rates remain simultaneously relatively stable during a prolonged period of time.
  - Fundamentally equilibrium exchange rates: defined in real effective terms relative to a set of a country’s own fundamentals and partners’ fundamentals (IMF, 2015); assessed relative to macroeconomic fundamentals.
  - Economic implication: high volatility of multilateral exchange rates linked to disequilibria; volatility of fundamental exchange rates can reflect desirable adjustment.
- Contributions:
  - a computable network model of multilaterally equilibrium exchange rates;
  - CDI to assess currency demand across all BoP flows;
  - the concept of multilateral exchange rate network stationarity pointing to equilibrium bilateral rates;
  - illustration using daily 2005-16 exchange rate moves (note: dataset references include both daily 1995-2016 and daily 2005-16 in the text).

### Analytics — Exchange Rate Network concept
- BoP as multilayer network:
  - BoP viewed as a multilayer network reflecting payments across layers (trade, services, remittances, dividends, foreign direct and portfolio investment, credit, reserve assets, etc.).
  - A multilayer BoP network can have any number ݀ of dimensions and a sequence ൌܮ
ሼ
ܮ
ఈ
ሽ
ఈୀଵ
ௗ
 of elementary layers, with one set of elementary layers ܮ
ఈ
 for each dimension ߙ (Kivelä, 2014).
- Network representation:
  - Nodes: individual currencies (not countries), weighted by multilateral demand from all other currencies.
  - Links: weighted log-returns on each currency pair that represent weighted changes in exchange rates (positive or negative) and indirectly reflect all underlying BoP flows.
  - Appreciation = inflow into a currency node; depreciation = outflow from a node.
- Graph properties:
  - Directed: changes embedded into log returns indicate direction of BoP flow between currencies.
  - Weighted: link weights reflect payment values differing by currency pair.
  - Incomplete: not all currencies are connected directly; many are quoted via cross-rates to the US dollar.
  - Asymmetric: implicit outflows differ from implicit inflows for currency nodes.
- Advantages vs prior approaches:
  - Avoids base currency selection problem by using observed weighted log-return flows rather than correlations quoted to a single base.
  - Contrasts with prior networks that used trade-weighted exchange rates, correlation of log-returns, Pearson correlations, phase synchronization, distance transforms, or tail dependence.
  - Allows indirect representation of BoP flows where bilateral payment data are limited.
- Practical notes on topology and dynamics:
  - Topology evolves daily: not all currency pairs are traded every day; weights and links change with bilateral exchange rate fluctuations.
  - Episodes of USD or other currency appreciation/depreciation show directional inflows/outflows to/from that currency node.
  - High overall exchange rate volatility implies an inherently non-stationary multilateral network; episodes of relatively low volatility correspond to network stationarity.
  - Stationarity implies the network’s evolution is systematically driven by dynamics of chosen constraints (not numerical constancy of rates).
  - During a stationary episode, multilaterally equilibrium bilateral exchange rates can be identified; deviations are interpreted as temporary misalignment expected to disappear in the medium term.

### Model (overview)
- Core elements:
  - Currency Demand Indicator (CDI): defined as weighted log returns on each currency; elementary cell of the network model and proxy for demand from other currencies.
  - Links: weighted log-returns representing hypothetical investor returns and directionality of implicit BoP flows (e.g., USD appreciation relative to EUR -> net BoP flow toward USD from EUR; link arrow from EUR to USD scaled by EUR/USD weight in global FX turnover).
  - Nodes: weighted by share of the currency in international currency turnover (BIS, 2013) reflecting multilateral demand.
- Uses and applicability:
  - Mainly applicable to currencies with floating exchange rate regimes; informative also for pegged regimes.
  - The model can identify dynamically stationary positions of the exchange rate network, which indicate equilibrium levels of bilateral exchange rates.
  - Potential application: estimate magnitudes of under- and overvaluation of individual currencies relative to their multilaterally equilibrium rates.
- Limitations and extensions:
  - Bilateral flow statistics are limited beyond trade and limited bilateral financial flow datasets; the model uses log-returns as an indirect BoP flow representation.
  - Further research needed to identify factors that cause switches from stationary to non-stationary network positions as potential leading indicators of currency crises.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16130.pdf*

### 18. The balance of payment flows among currency nodes can be indirectly measured

### 18. The balance of payment flows among currency nodes can be indirectly measured

### Methodology: log-returns, weights, and the Currency Demand Indicator (CDI)
- Log-return ߟ
௜௝ (itha) on an investment in currency ݅ from currency ݆ is defined as:
  - ߟ
௜௝ (ݐ) = ln(ݎ
௜௝ (ݐ) / ݎ
௜௝ (ݐ−1))  (Equation (1) in source)
- Currency-pair weights ߣ
௜௝ (lambda) are defined as the share of each currency pair ݆↔݅ in the global foreign exchange market turnover.
  - For major pairs weights are taken from BIS (2013) statistics.
  - Remaining pairs (representing less than two percent of turnover) are reconstructed under an even distribution assumption.
  - With each currency included on both sides of a pair, total weights of all currency pairs add up to 200 percent.
- Weighted log-return along link ݆↔݅ is the absolute value of the weighted logarithmic return on this link (Equation (2) and (3) in source).
- Bilateral currency demand indicator ߮
௜௝ (phi) for currency ݅ from currency ݆ is defined as:
  - ߮
௜௝ (ݐ) = ߣ
௜௝ ߟ
௜௝  (Equation (4))
- Multilateral Currency Demand Indicator (CDI) ߯
௜ (chi) for currency ݅ from all other currencies {1,݆...ܰ} is defined as:
  - ߯
௜ (ݐ) = ෍
௜௝ ߣ
௜௝ ߟ
௜௝  (Equation (5))
  - The CDI is a novel indicator according to the authors: "To the best of our knowledge, the CDI has not been proposed in the earlier literature."
- Exchange rate network representation:
  - Symmetrical, weighted, and directed graph.
  - Nodes represent CDI ߯
௜ (ݐ); node size proportional to average CDI.
  - Links represent ߮
௜௝ (ݐ); link width scaled by share of the log-return in the global market; arrow indicates direction of net demand.
  - If ߮
௜௝ (ݐ) ≥ 0: excess demand for currency ݅ from currency ݆ (unobserved inflow ݆←݅; appreciation of ݅).
  - If ߮
௜௝ (ݐ) ≤ 0: unobserved outflow from currency ݅ to ݆ (݅→݆); depreciation of ݅.

### Stationarity metrics and thresholds
- Temporal volatility of CDI (ߤ(χ)) across a selected period ܶ is calculated as the standard deviation of multilateral demand for each currency:
  - μ஧௧ = √(1/T Σ(χ௧(t) − χത௧)²)  (Equation (6))
  - χത௧ = 1/T Σχ௧(t)  (Equation (7))
  - Temporal volatility measures degree of stationarity over time.
- Sectional variety ν(t) of CDI across N currencies on a given day is the cross-sectional standard deviation:
  - ν஧(t) = √(1/N Σ(χ௧(t) − χത(t))²)  (Equation (8))
  - χത(t) = 1/N Σχ௧(t) = 0  (Equation (9))
  - The sum of sectional CDIs is zero by definition because the global FX market clears daily.
  - Sectional variety measures daily pulsation of the exchange rate network.
- Stationarity definition:
  - Periods when temporal volatility and sectional variety are simultaneously minimal.
  - Practically, minimal sectional variety ν(t) is used to define stationary exchange rate network episodes (static concept relative to each CDI mean).
- Two empirical thresholds to identify stationary episodes:
  - Threshold 1 (floor): minimum variety of the CDI during the sample plus one standard deviation.
    - In the sample: Threshold 1 = 2.66E-04.
      - Computed from minimum variety = 2.0295E-05 plus one standard deviation = 2.46E-04.
    - Deviation from threshold = observed CDI variety − Threshold 1.
    - Negative deviations indicate dates of the lowest CDI variety relative to the threshold.
  - Moving-window aggregation:
    - Apply a 22-day moving average window (average work period in one month, 2,607 days/120 months) to the count of days with negative deviations.
  - Threshold 2: within each 22-day window, at least 80 percent of days must be lowest-variety days.
    - 80 percent of 22 days = 18 days.
  - Dates meeting both thresholds are considered episodes of multilaterally equilibrium exchange rates.

### Data inputs and weight matrix
- Primary inputs:
  - Daily exchange rates for 130 currencies with respect to the US dollar in units of national currency per one USD for 2005-16 (Annex 1).
  - BIS weights of currency pairs in global foreign exchange turnover (Annex 2).
    - Weights available for 21 currency pairs involving USD, 15 involving EUR, and nine involving JPY.
  - Example: weight of EUR/USD pair in global turnover = 24.1 percent.
  - All currency pairs included twice in adjacency matrix, so weights sum to 200.
- The adjacency (weight) matrix dimension: 130 x 130.
  - The share of other currency pairs equal to 1.7 percent is assumed distributed equally among currencies not included in the BIS matrix.
  - The matrix is heavily centered on reserve currencies; other currencies included with much lower weights reflecting relative trade share.
  - Example notes: some pairs (e.g., JPY/GBP) have negligible direct weight because they are commonly exchanged via USD as intermediate currency.

### Empirics and key results
- Sample span: daily observations for 2005-2016.
  - Total daily observations mentioned: 2,610 (and elsewhere 2,607 days referenced for moving-average window).
- Threshold 1 and deviations:
  - Threshold 1 (2.66E-04) computed from minimum variety 2.0295E-05 plus one standard deviation 2.46E-04.
  - In about half of 2,610 daily observations, the deviation (observed CDI variety − Threshold 1) was negative.
- Episodes of exchange rate network stationarity identified during 2005-16:
  - June–July 2007
  - end-November to December 2013
  - February 2014
  - end-April to early September 2014
- The CDI captures high instability during 2008–09 financial crisis via increased temporal volatility and sectional variety.
- Illustration of multilateral equilibrium exchange rates (examples from stationary episodes):
  - 2007 episode:
    - USD/EUR = 0.74 (units of national currency per US Dollar)
    - USD/JPY = 122.39
    - USD/GBP = 0.50
  - 2014 episode:
    - USD/EUR = 0.74
    - USD/JPY = 102.28
    - USD/GBP = 0.59
  - Note: each bilateral rate can be considered in equilibrium only conditional on the levels of other bilateral rates during the same stationary episode.

### Model assumptions and limitations
- Assumptions:
  - Exchange rates already reflect all fundamental factors affecting each currency.
  - Exchange rate movements represent underlying balance of payments flows for payments in different currencies.
  - Exchange rate reflects balance of global demand and supply of a currency.
  - Demand and supply of currencies are delinked from countries of issuance (e.g., USD demand may be presented by traders in Saudi Arabia and China).
  - Equilibrium is assumed to be static: the model finds minimal CDI and periods when CDI does not fluctuate much around its own mean.
- Practical data/weight limitations:
  - Weights are concentrated on major reserve currencies; many cross pairs have negligible direct weight because of indirect trading via intermediate currencies (e.g., USD).

### Policy implications and applications
- Exchange rate assessment:
  - The model can find ex-post periods of multilaterally equilibrium exchange rates and calculate deviations of observed bilateral exchange rates from their latest multilaterally equilibrium levels.
  - This can help detect potential over- and undervaluation of individual currencies that may not be evident from bilateral exchange-rate moves or reserve pressures alone.
- Early warning indicators:
  - Temporal currency demand volatility and sectional currency variety increase before crises; the model can contribute to early warning of possible currency crises.
- Role monitoring:
  - The weighted CDI allows study of the evolution of the role of individual currencies and balance of payments flows denominated in those currencies.
- Operational use:
  - Timely detection and prediction of episodes of multilateral exchange rate network stationarity and non-stationarity may be useful for policymakers to ensure exchange rate stability during periods of increased temporal variety of exchange rates.

*Source: _wp16130 - 18. The balance of payment flows among currency nodes can be indirectly measured*

### 38. The model has a number of important limitations. First, the CDI is more likely to

### 38. The model has a number of important limitations. First, the CDI is more likely to

### Limitations of the model and CDI
- The CDI is more likely to capture stability in reserve currencies, rather than stability across all currencies in the sample.
- Weights used in the analysis rely heavily on the share of currencies in the international currency turnover; the weights exhibit fat tails in the sense that reserve currencies dominate global foreign exchange markets.
- The CDI has to rely on incomplete triennial data sources, such as BIS. Therefore, the adjacency matrix that yields the weights on bilateral flows is remarkably sparse.
  - Example: the adjacency matrix suggests that the U.K. and Japan have no links, as the currency weight is missing from the BIS and may be misspecified.
- The model rests on strong assumptions, such as the balance of demand and supply of currencies being delinked from countries of their issuances.
  - Countries issuing reserve currencies may often affect them via their monetary and exchange rate policies.
  - Example: the normalization of U.S. monetary policy and monetary policy divergence among countries issuing reserve currencies is likely to have an impact on the CDI.

### Embedded empirical snippets and sample statistics (as presented)
- Epis ode  date s5/29/200711/19/20121/27/20144/21/2014 7/24/20071/1/20132/25/20149/5/2014
- Duration (working days)413222100
- EUROEUR  0.740.770.730.74
- JAPANESE YENJPY122.3983.36102.14102.28
- SWISS FRANCCHF1.220.920.900.90
- RUSSIAN ROUBLERUB25.7530.8335.1535.19
- CHINESE YUANCNY  7.616.236.076.21
- CANADIAN DOLLARCAD1.060.991.111.09
- INDIAN RUPEEINR40.6354.8562.3860.09
- BRAZILIAN REALBRL   1.912.082.402.24
- AUSTRALIAN DOLLARAUD1.170.961.121.07
- SOUTH AFRICA RANDZAR7.078.6911.0310.61
- MEXICAN PESOMXN10.8112.9113.3113.02
- SOUTH KOREAN WONKRW924.031,078.521,072.411,023.28
- NORWEGIAN KRONENOK5.915.636.146.09
- SWEDISH KRONASEK6.846.626.496.75
- INDONESIAN RUPIAHIDR9,002.219,629.0012,021.4311,685.97
- UK POUNDGBP0.500.620.600.59

### Possible extensions and policy-relevant research directions
- Further exploration of the CDI, its properties and dynamics.
- Additional studies of the exchange rate network and its early warning properties to allow signaling of switching between stationary and non-stationary regimes; these may be important predictors of currency crises.
- Additional exploration of the multilaterally equilibrium bilateral exchange rates to assess current levels relative to the new metric of the multilateral network equilibrium.
- Calculations of the observed bilateral exchange rates deviations from their multilaterally stationary levels to signal overvaluation or undervaluation, which could become an important input for the policy dialogue with IMF member countries.

*Source: Excerpt from provided IMF content unit.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16130.pdf_