Real exchange rate in the long run: A multi-resolution analysis

Figures & data

Figure 1. The application of UCM in adjusting Singapore CPI

Notes:Singapore CPI series is constructed by trend, cycle, seasonality and noise

Table 1. Hurst’s exponents

Hurst exponent	Fractional difference parameter	The behaviour of the time series
$H\le 0$	$d\le \frac{-1}{2}$	Non-stationary
$0<H<\frac{1}{2}$	$\frac{-1}{2}<d<0$	Not persistent, mean-reverting
$H=\frac{1}{2}$	$d=0$	No trend, Brownian motion
$\frac{1}{2}<H<1$	$0<d<\frac{1}{2}$	Long-range dependent structure
$H>1$	$d>\frac{1}{2}$	Non-stationary

Source:Vo and Vo (2019a)

Table 2. Descriptive statistics

Country	Mean	Std	Skewness	Kurtosis	DB	LB
SGP	0.374	0.128	−0.162	1.699	.(0.0000)	7060 (0.0000)
VNM	9.893	0.174	0.162	1.418	.(0.0000)	7554 (0.000)
THA	3.584	0.136	0.378	1.922	53.37 (0.0000)	6066 (0.0000)
PHL	3.916	0.158	0.428	1.825	.(0.0000)	6636 (0.0000)
IDN	9.334	0.178	0.654	3.092	13.84 (0.0010)	4767 (0.0000)
MYS	1.288	0.103	0.272	3.887	8.056 (0.0178)	2790 (0.0000)

Figure 2. Real exchange rate

Notes:The real exchange rates for six ASEAN countries

Figure 3. De-noised real exchange rate

Notes:For each subplot, the solid black line demonstrates real exchange rate value which is calculated by the above-mentioned equation ${\mathrm{r}}_t=ln\left(ne\right)+ln{P}^{\ast }-lnP$. While $P^{\ast }$ and $P$ correspond to the CPI values of the US and investigated country, ne stands for the nominal exchange rate of the local currency against 1 US dollar. The red dashed line represents the unruffled (smoothed) real exchange rate value deduced from the wavelet transform method, which is highlighted in the next section.

Table 3. Univariate stationary test results

	Singapore	Vietnam	Thailand	Philippines	Malaysia	Indonesia
ADF	−0.9236 (0.7801)	−0.1406 (0.9452)	−1.0102 (0.7495)	−0.9458 (0.7725)	−3.6322 (0.0052)	−1.8289 (0.3662)
PP	−0.8850 (0.7929)	−0.0966 (0.9498)	−0.8633 (0.7998)	−0.8429 (0.8062)	−3.4461 (0.0095)	−1.7084 (0.4268)
KPSS lags0	2.8485	2.1465	2.9655	2.7107	3.5414	4.1966
KPSS lags1	1.4416	1.2219	1.4888	1.3673	1.8094	2.1332
KPSS lags2	0.9707	0.8232	1.0030	0.9192	1.2285	1.4445
Critical 95	0.146	0.146	0.146	0.146	0.146	0.146
Critical 99	0.216	0.216	0.216	0.216	0.216	0.216
N	249

Notes:Null hypothesis for the ADF and PP tests is the time series is not stationary, whereas the null hypothesis of the KPSS test is that the time series is stationary. All real exchange rate series are assumed to have trend and base level. The lag in which we choose to conduct these tests is based on the order determination method which is comprehensively outlined in Tsay (1984); Paulsen (1984); Nielsen (2001). All corresponding p-values are in parentheses. Critical values of the KPSS test are reported at 99 per cent and 95 per cent.

Table 4. Panel stationary test results

	LLC	HT	BT	IPS	Hadri	Fisher (DF)	Fisher (PP)
Statistic	−1.7549	0.9787	1.4655	1.1919	187.3906	0.3622	0.3622
Critical value	(0.0396)	(0.8524)	(0.9286)	(0.1167)	(0.0000)	(0.6414)	(0.6414)

Notes:Null hypothesis for LLC, HT, BT, IPS, Fisher DF and Fisher PP is the panel data contain unit-roots. The alternative is all panel data are stationary. The null hypothesis for the Hadri test is that all panels are stationary and the alternative hypothesis is that at least one series contains unit-roots.

Figure 4. Multi-level decomposition of SGP/USD real exchange rate

Notes:This figure represents the MRA application on monthly SGP/USD real exchange rate. The investigated time horizon ranges from Dec 1998 to Aug 2019. Our original data is the sum of all detailed coefficients and the first level of averaging coefficients (Level 1). Results from the other pairs of currency are provided upon request.

Table 5. Insights of the MRA mechanism

Decomposition level
1	D1	S1
2	D1	D2	S2
3	D1	D2	D3	S3
4	D1	D2	D3	D4	S4
5	D1	D2	D3	D4	D5	S5
6	D1	D2	D3	D4	D5	D6	S6

Notes:This table depicts the relationship between Detailed coefficients and Averaging coefficients as well as the MRA mechanism. The summation of each row’s components will be exactly equal to the original signal.

Table 6. Univariate stationary test results on six real exchange rate—Averaging coefficients

	S1	S2	S3	S4	S5	S6
1. SGD/USD
ADF	−0.791	−0.871	−1.252	−2.447	−4.594	−6.500
	(0.822)	(0.797)	(0.651)	(0.129)	(0.000)	(0.000)
PP	−0.949	−1.057	−1.569	−2.524	−3.690	−8.479
	(0.771)	(0.732)	(0.499)	(0.110)	(0.004)	(0.000)
KPSS	8.721	4.226	1.815	0.432	0.199	0.330
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
2. VND/USD
ADF	−0.047	−0.352	−0.955	−2.128	−4.050	−7.999
	(0.954)	(0.918)	(0.769)	(0.234)	(0.001)	(0.000)
PP	−0.338	−0.577	−1.041	−2.065	−3.600	−9.662
	(0.920)	(0.876)	(0.738)	(0.259)	(0.006)	(0.000)
KPSS	11.40	5.775	2.985	1.592	0.786	0.475
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
3. THB/USD
ADF	−0.722	−0.800	−1.016	−1.504	−2.674	−4.096
	(0.841)	(0.819)	(0.747)	(0.531)	(0.079)	(0.001)
PP	−0.999	−1.064	−1.390	−1.841	−2.554	−4.994
	(0.754)	(0.729)	(0.587)	(0.361)	(0.103)	(0.000)
KPSS	8.307	4.045	1.832	0.655	0.156	0.272
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
4. PHP/USD
ADF	−0.799	−1.049	−2.029	−4.295	−8.701	−21.60
	(0.820)	(0.735)	(0.274)	(0.000)	(0.000)	(0.000)
PP	−0.998	−1.282	−2.150	−3.785	−7.142	−33.13
	(0.754)	(0.638)	(0.225)	(0.003)	(0.000)	(0.000)
KPSS	7.779	3.962	2.089	1.042	0.561	0.419
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
5. IDR/USD
ADF	−1.552	−1.638	−3.476	−8.204	−13.74	−16.72
	(0.508)	(0.463)	(0.009)	(0.000)	(0.000)	(0.000)
PP	−1.502	−1.805	−3.686	−6.109	−12.36	−24.13
	(0.532)	(0.378)	(0.004)	(0.000)	(0.000)	(0.000)
KPSS	7.087	3.357	0.850	0.133	0.324	0.372
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
6. MYR/USD
ADF	−3.530	−3.589	−3.538	−3.266	−3.171	−3.386
	(0.007)	(0.006)	(0.007)	(0.017)	(0.022)	(0.011)
PP	−3.424	−3.449	−3.371	−3.240	−3.558	−4.095
	(0.010)	(0.009)	(0.012)	(0.018)	(0.007)	(0.001)
KPSS	2.574	1.452	0.928	0.638	0.544	0.465
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463

Notes:Null hypothesis for ADF and PP test is the time series is not stationary whereas the null hypothesis for the KPSS test is that time series is stationary. All corresponding p-values are in parentheses. Critical values of KPSS test are reported at 99 per cent, 97.5 per cent and 95 per cent.

Table 7. Univariate stationary test results on six real exchange rate detailed coefficients

	D1	D2	D3	D4	D5	D6
1. SGD/USD
ADF	−14.970	−10.379	−9.420	−3.887	−2.729	−1.774
	(0.000)	(0.000)	(0.000)	(0.002)	(0.069)	(0.393)
PP	−16.145	−11.178	−9.779	−3.880	−2.757	−1.763
	(0.000)	(0.000)	(0.000)	(0.002)	(0.065)	(0.399)
KPSS	0.041	0.130	0.131	0.158	0.229	0.190
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
2. VND/USD
ADF	−14.458	−11.533	−7.395	−6.125	−3.854	−2.942
	(0.000)	(0.000)	(0.000)	(0.000)	(0.002)	(0.041)
PP	−16.055	−12.383	−7.326	−6.401	−3.839	−3.112
	(0.000)	(0.000)	(0.000)	(0.000)	(0.003)	(0.026)
KPSS	0.107	0.043	0.149	0.048	0.150	0.099
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
3. THB/USD
ADF	−14.549	−11.549	−10.581	−4.166	−2.690	−2.595
	(0.000)	(0.000)	(0.000)	(0.001)	(0.076)	(0.094)
PP	−15.234	−12.303	−10.512	−4.175	−2.672	−2.739
	(0.000)	(0.000)	(0.000)	(0.001)	(0.079)	(0.067)
KPSS	0.115	0.038	0.040	0.146	0.078	0.095
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
4. PHP/USD
ADF	−14.285	−13.011	−7.717	−5.243	−3.451	−2.898
	(0.000)	(0.000)	(0.000)	(0.000)	(0.009)	(0.046)
PP	−14.891	−13.610	−8.362	−5.206	−3.514	−2.850
	(0.000)	(0.000)	(0.000)	(0.000)	(0.008)	(0.051)
KPSS	0.036	0.046	0.092	0.131	0.160	0.134
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
5. IDR/USD
ADF	−14.374	−9.723	−6.964	−3.185	−2.927	−2.095
	(0.000)	(0.000)	(0.000)	(0.021)	(0.042)	(0.247)
PP	−15.442	−9.913	−7.154	−3.183	−2.952	−2.093
	(0.000)	(0.000)	(0.000)	(0.021)	(0.040)	(0.247)
KPSS	0.045	0.074	0.114	0.155	0.192	0.167
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463
6. MYR/USD
ADF	−14.176	−10.783	−9.239	−3.448	−3.537	−2.869
	(0.000)	(0.000)	(0.000)	(0.009)	(0.007)	(0.049)
PP	−14.605	−11.306	−10.168	−3.405	−3.612	−2.785
	(0.000)	(0.000)	(0.000)	(0.011)	(0.006)	(0.060)
KPSS	0.048	0.076	0.045	0.129	0.137	0.156
Critical Value 1%	0.739	0.739	0.739	0.739	0.739	0.739
Critical Value 2.5%	0.574	0.574	0.574	0.574	0.574	0.574
Critical Value 5%	0.463	0.463	0.463	0.463	0.463	0.463

Notes:Null hypothesis for ADF and PP test is the time series is not stationary whereas the null hypothesis for the KPSS test is that the time series is stationary. All corresponding p-values are in parentheses. Critical values of KPSS test are reported at 99 per cent, 97.5 per cent and 95 per cent.

Table 8. The multivariate test results on six averaging coefficients and detailed coefficients

1. Averaging coefficients
	S1	S2	S3	S4	S5	S6
LLC	−1.080	−0.835	−0.061	0.557	−3.727	.
	(0.140)	(0.202)	(0.476)	(0.711)	(0.000)	.
HT	0.854	0.336	−1.943	−3.184	−1.072	0.906
	(0.804)	(0.632)	(0.026)	(0.001)	(0.142)	(0.818)
IPS	−1.045	−1.395	−3.411	−7.232	−10.527	−8.181
	(0.146)	(0.085)	(0.003)	(0.000)	(0.016)	.
BT	1.275	0.831	0.528	0.562	0.799	1.091
	(0.899)	(0.797)	(0.701)	(0.713)	(0.788)	(0.862)
Hadri	90.643	40.369	13.519	4.210	3.092	1.611
	(0.000)	(0.000)	(0.000)	(0.000)	(0.001)	(0.054)
Fisher DF	0.238	0.804	0.872	−0.408	−1.186	.
	(0.594)	(0.789)	(0.808)	(0.342)	(0.118)	.
Fisher PP	0.621	0.118	−1.826	−5.015	−9.472	−15.861
	(0.733)	(0.547)	(0.034)	(0.000)	(0.000)	(0.000)
2. Detailed coefficients
	D1	D2	D3	D4	D5	D6
LLC	−27.538	−15.201	−7.215	−5.150	−6.878	.
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	.
HT	−75.122	−38.554	−20.529	−7.407	−4.985	−2.939
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.002)
IPS	−36.966	−27.285	−19.888	−7.728	−5.420	−20.999
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	.
BT	−21.404	−15.489	−11.022	−6.409	−4.647	−3.271
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.001)
Hadri	−2.141	−2.543	−1.411	−0.349	−1.082	−1.393
	(0.984)	(0.995)	(0.921)	(0.636)	(0.860)	(0.918)
Fisher DF	−16.417	−12.338	−6.733	−4.821	−3.705	.
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	.
Fisher PP	−19.904	−19.904	−17.940	−8.228	−5.084	−3.088
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.001)

Notes:Null hypothesis for LLC, HT, BT, IPS, Fisher DF and Fisher PP is the panel data contain unit-roots. The alternative is that all panel data are stationary. The null hypothesis for the Hadri test is that all panels are stationary. The alternative hypothesis is that at least one series contains unit-roots. Time series are presumed to have trend, and the lag specifications are based on the AIC.

Figure 5. ARIMA on six real exchange rate series

Figure 6. The estimated alpha and beta coefficients

Vo, H. L., & Vo, H. D. (2019a). Application of wavelet-based maximum likelihood estimator in measuring market risk for fossil fuel. Sustainability, 11(10), 2843. https://doi.org/10.3390/su11102843

 Web of Science ®Google Scholar

Tsay, R. S. (1984). Order Selection in Nonstationary Autoregressive Models. Annals of Statistics, 12(4), 1425–1433. https://doi.org/10.1214/aos/1176346801

 Web of Science ®Google Scholar

Paulsen, J. (1984). Order determination of multivariate autoregressive time series with unit roots. Journal of Time Series Analysis, 5(2), 115–127. https://doi.org/10.1111/j.1467-9892.1984.tb00381.x

 Google Scholar

Nielsen, B. (2001). Order determination in general vector auto regressions. Working paper, Department of Economics, University of Oxford and Nuffi eld College.

 Google Scholar