Nonlinearity and efficiency dynamics of foreign exchange markets: evidence from multifractality and volatility of major exchange rates

Figures & data

Table 1. Descriptive statistics for the returns of exchange rates in major exchange markets.

Currency	Sample period	Number of observations	Mean (%)	Standard deviation	Skewness	Kurtosis
EUR	2005/01/03 ∼ 2019/03/29	3663	−0.0053	0.6044	0.1356	5.3118
GBP	2005/01/03 ∼ 2019/03/29	3663	−0.0106	0.5953	−0.5923	9.2681
CAD	2005/01/03 ∼ 2019/03/29	3713	0.0028	0.5817	−0.1118	5.8700
JPY	2005/01/03 ∼ 2019/03/29	3663	−0.0021	0.6417	0.1200	7.4159

Notes: ‘EUR’, ‘GBP’, ‘CAD’ and ‘JPY’ denote daily returns of exchange rates of the EUR, GBP, CAD, and JPY against the USD, respectively.

Data source: RESSET database (www.resset.cn).

Figure 1. The daily return series of EUR. Source: Author calculations.

Figure 2. The daily return series of GBP. Source: Author calculations.

Figure 3. The daily return series of CAD. Source: Author calculations.

Figure 4. The daily return series of JPY. Source: Author calculations.

Figure 5. The Hurst exponent and multifractal spectrum of the daily returns of four exchange rates. (Notes: Figure 5 describes the Hurst exponent and multifractal spectrum of EUR, GBP, CAD and JPY, respectively. H(q), hq and Dq denote order value, q-order Hurst exponent, singularity strength and singularity spectrum, respectively.). Source: Author calculations.

Table 2. Multifractal properties of four exchange rate returns.

Index	Δα	Δf	R
EUR	1.4407	2.1874	−0.5219
GBP	1.3539	2.2285	−0.5725
CAD	1.0508	1.3789	−0.3799
JPY	0.8456	1.0322	−0.3001

Notes: ‘EUR’, ‘GBP’, ‘CAD’ and ‘JPY’ denote the daily returns based on exchange rates of EUR, GBP, CAD and JPY, respectively. Δα, Δf and R denote multifractal spectrum width, distribution proportion of high values and low values, and asymmetric exponent, respectively. Source: Author calculations.

Table 3. MD values of four exchange rate returns in different periods.

	Whole sample	Period I	Period II	Period III
EUR	1.4407	0.7069	0.5751	0.5994
GBP	1.3539	0.5650	0.4975	0.7617
CAD	1.0508	1.0410	0.3348	1.1123
JPY	0.8456	0.4681	0.5099	0.6607

Notes: ‘EUR’, ‘GBP’, ‘CAD’ and ‘JPY’ denote the daily returns based on exchange rate of EUR, GBP, CAD and JPY, respectively. Period I, Period II, and Period III denote the period before the 2008 financial crisis, between the financial crisis and the Federal Reserve’s announcement of its withdrawal from quantitative easing (QE) policy in 2014, and after the announcement, respectively.

Figure 6. The Hurst Exponent and Multifractal Spectrum of exchange rate return series during Period I, II, III. Source: Author calculations.