Modelling asymmetric sovereign bond yield volatility with univariate GARCH models: Evidence from India

Figures & data

Figure 1. Sovereign bond yield series graphs at first difference.

Table 1. Descriptive statistics (across maturities)

	3-month yield	2-year yield	10-year yield	24-year yield
Observations	1593	1593	1593	1593
Mean	0.053291	0.059938	0.068317	0.073624
Median	0.060800	0.063350	0.067540	0.072530
Maximum	0.073500	0.080250	0.081820	0.085820
Minimum	0.027000	0.038610	0.057600	0.063880
SD	0.013949	0.011328	0.006397	0.005019
Skewness	−0.465886	−0.314492	0.192830	0.270117
Kurtosis	1.595105	1.805048	1.820886	2.096414
Jarque-Bera	188.6329	121.0368	102.1540	73.56480
Probability	0.000000	0.000000	0.000000	0.000000
Sum	84.89260	95.48147	108.8286	117.2830
Sum Sq. Dev.	0.309751	0.204304	0.065144	0.040104

Note: Skewness quantifies the asymmetry of the series’ distribution around its mean. A normal distribution has zero skewness. Kurtosis is a measure of the peakedness or flatness of a series’ distribution. The normal distribution has a kurtosis of 3. If the kurtosis is greater than 3, the distribution is leptokurtic, and if it is less than 3, it is platykurtic in comparison to the normal distribution.

Figure 2. QQ plots of the EGARCH residuals.

Table 2. Descriptive statistics (macro-economic indicators and sovereign yields at quarterly frequency)

	Inflation (CPI)	Interest Rates (LIBOR)	3-month yield	2-year yield	10-year yield	24-year yield
Observations (Quarterly)	28	28	28	28	28	28
Mean	0.47514	0.13616	0.52226	0.59472	0.67892	0.72961
Median	0.49482	0.12041	0.59050	0.63740	0.66840	0.72190
Maximum	0.66234	0.35652	0.70800	0.78670	0.78530	0.82460
Minimum	0.33282	0.01301	0.30700	0.38650	0.58370	0.64470
SD	0.11598	0.09806	0.13917	0.11229	0.06302	0.04933
Skewness	0.3076	0.3658	−0.3838	−0.2851	0.1533	0.2006
Kurtosis	1.9727	2.0797	1.5446	1.9130	1.8273	2.2395
Jarque-Bera	0.4181	1.5549	2.9329	1.6323	1.5303	0.7701
Probability	0.8113	0.4595	0.2307	0.4421	0.4652	0.6804
Sum	33.2595	36.7637	135.79	154.628	169.7300	182.4030
Sum Sq. Dev.	8.0716	25.0025	48.4257	31.5226	9.5340	5.8417

Note: Skewness quantifies the asymmetry of the series’ distribution around its mean. A normal distribution has zero skewness. Kurtosis is a measure of the peakedness or flatness of a series’ distribution. The normal distribution has a kurtosis of 3. If the kurtosis is greater than 3, the distribution is leptokurtic, and if it is less than 3, it is platykurtic in comparison to the normal distribution.

Table 3. Unit root test results

	3-month yield		2-year yield		10-year yield		24-year yield
Test	ADF test	PP test	ADF test	PP test	ADF test	PP test	ADF test	PP test
t-Statistic	−43.348	−43.188	−15.569	−50.529	−40.966	−41.021	−43.877	−43.814
Probability Value	0.0001	0.0001	0.0000	0.0001	0.0000	0.0000	0.0001	0.0001

Note: H0: Sovereign bond yield has unit root one (non-Stationary)

Table 4. Unit root test results for macroeconomic variables

	Inflation (CPI)		Interest (LIBOR)
Test	ADF test	PP test	ADF test	PP test
t-Statistic	−3.6948	−3.6948	−2.9981	−1.0511
Probability Value	0.0018	0.0055	0.0376	0.0197

Note: H0: Sovereign bond yield has unit root one (non-Stationary)

Table 5. ARCH LM test results

	3-month yield	2-year yield	10-year yield	24-year yield
ARCH LM test statistic	130.2611	57.18295	10.03185	47.23850
Probability Value	0.0000	0.0000	0.0016	0.0000

Note: H0: There are no ARCH effects in the sovereign bond yield.

Table 6. EGARCH results

Parameter	Distribution Type	3-month yield	2-year yield	10-year yield	24-year yield
µ (MU)	Normal	−0.000405***	0.000091	−0.000029	0.000080
	Student’s t	−0.000271***	−0.00032***	−0.000052	0.000154
	GED	0.000000	−0.000100***	0.000000	0.00000
Ω (Omega)	Normal	−0.089692***	−0.003373***	−0.112156***	−0.253970***
	Student’s t	−0.627056***	−0.11541***	−0.247236***	−0.453559***
	GED	−0.97271***	−0.110040***	−0.231378***	−0.49158***
α (Alpha)	Normal	−0.076940***	0.035663***	0.002634	0.003083
	Student’s t	−0.136383***	−0.20987	0.001048	−0.028684
	GED	−0.26038***	−0.016005	0.003112	−0.01833
β (Beta)	Normal	0.985170***	0.996215***	0.987394***	0.972437***
	Student’s t	0.932585***	0.98154***	0.975297***	0.928131***
	GED	0.90361***	0.988233***	0.977324***	0.948029***
γ (Gama)	Normal	0.314314***	0.151169***	0.138774***	0.212423***
	Student’s t	0.522368***	1.13358***	0.255290***	0.621886
	GED	0.81360***	0.257037***	0.210311***	0.30120***

Notes 1: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

2: In the EGARCH results, the term ‘Omega’ represents the constant part: a kind of ‘Ambient Volatility’, Alpha represents the adjustment to past shocks, the Beta term represents an adjustment to past volatility and long-run memory of the current volatility pattern, Gama represents ‘leverage coefficient’ or the influence of good and bad news on the current volatility.

Table 7. Distribution selection under EGARCH

	Distribution Type	3-month yield	2-year yield	10-year yield	24-year yield
AIC (Akaike)	Normal	−7.0532	−6.7158	−7.3634	−7.5603
	Student’s t	−7.4912	−7.2615	−7.5876	−7.8800
	GED	−8.1640	−7.2935	−7.5749	−7.9078
BIC (Bayes)	Normal	−7.0362	−6.6988	−7.3465	−7.5434
	Student’s t	−7.4675	−7.2378	−7.5639	−7.8563
	GED	−8.1437	−7.2732	−7.5546	−7.8875
Log-Likelihood	Normal	5594.626	5327.234	5840.498	5996.537
	Student’s t	5943.768	5761.774	6020.156	6251.921
	GED	6476.008	5786.079	6009.13	6272.946

Note: Student’s t and GED are non-normal distributions. The AIC and BIC values determine the best distribution for the data set. Distribution with the lowest AIC/BIC value can be regarded as the best suitable distribution. Similarly, the log-likelihood determines the best asymmetric GARCH models for the data set considered in the study, and we have followed the maximum likelihood approach. Thus, the EGACRH model with GED turns out to be the best model for our data set (Refer to appendix for the results of other models).

Table 8. EGARCH results during pre-COVID-19

Parameter	Distribution Type	3-month yield	2-year yield	10-year yield	24-year yield
α (Alpha)	GED	−0.02656 ***	−0.029940	−0.018020	−0.060160
β (Beta)	GED	0.864455 ***	0.968846***	0.984466***	0.976944***
Ln(0.5)		−0.69315	−0.69315	−0.69315	−0.69315
Ln(β)		−0.14566	−0.03165	−0.01566	−0.02333
Mean Reversion in days		4.7588	21.9006	44.2738	29.7157

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table 9. EGARCH results during the COVID-19 era

Parameter	Distribution Type	3-month yield	2-year yield	10-year yield	24-year yield
α (Alpha)	GED	−0.23636***	−0.308765	−0.354528	0.005480
β (Beta)	GED	0.914978***	0.951023***	0.965075***	0.655224***
Ln(0.5)		−0.69315	−0.69315	−0.69315	−0.69315
Ln(β)		−0.08886	−0.05022	−0.03555	−0.42278
Mean Reversion in days		7.8008	13.8030	19.4981	1.6395

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table 10. Granger casualty test for the linkage between sovereign bond yield volatility and inflation

	3-month yield	2-year yield	10-year yield	24-year yield
	F-Statistic	F-Statistic	F-Statistic	F-Statistic
Inflation does not Granger cause Bond yield volatility	4.19187**	5.35855**	4.61866**	3.62401**
Bond yield volatility does not Granger cause Inflation	2.30873	1.11131	2.71625	1.04665

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table 11. Granger casualty test for the linkage between sovereign bond yield volatility and money market volatility

	3-month yield	2-year yield	10-year yield	24-year yield
	F-Statistic	F-Statistic	F-Statistic	F-Statistic
Money Market Volatility does not Granger cause Bond yield volatility	3.12562***	3.385210**	3.98426**	3.36984**
Bond yield volatility does not Granger cause Money Market Volatility	4.32622***	3.64875**	3.78358**	3.81264**

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table

	3-month yield		2-year yield		10-year yield		24-year yield
Test	ADF test	PP test	ADF test	PP test	ADF test	PP test	ADF test	PP test
t-Statistic	−2.9918	−2.9918	−2.9918	−2.9918	−2.9980	−2.9980	−2.9980	−2.9980
Probability Value	0.0067	0.0097	0.0326	0.0370	0.0022	0.0021	0.0003	0.0003

Note: H0: Sovereign bond yield has unit root one (non-Stationary)

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
µ	Normal	−0.000125	−0.000179	−0.000125
	Student’s t	−0.000140	−0.000186	−0.000140
	GED	0.000000	0.000000	0.000000
Ω	Normal	0.000001	0.000001	0.000001
	Student’s t	0.000005***	0.000005***	0.000005***
	GED	0.000000	0.000000	0.000000
α	Normal	−0.124864***	0.091242***	0.124582***
	Student’s t	−0.286370***	0.161999***	0.286682***
	GED	−0.084712***	0.052954***	0.06464***
β	Normal	0.874136***	0.872288***	0.875418
	Student’s t	0.712630***	0.710942***	0.713318***
	GED	0.897382***	0.890587***	0.93536***
γ	Normal		0.070938***
	Student’s t		0.260461***
	GED		0.081369***

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
AIC	Normal	−7.0177	−7.0271	−7.0199
	Student’s t	−7.4705	−7.4779	−7.4718
	GED	−7.9678	−7.9469	−8.0218
BIC	Normal	−7.0042	−7.0101	−7.0097
	Student’s t	−7.4502	−7.4542	−7.4549
	GED	−7.9508	−7.9266	−8.0082
LogLikelihood	Normal	5565.549	5573.946	5566.261
	Student’s t	5926.387	5933.226	5926.433
	GED	6319.45	6303.9	6361.248

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
µ	Normal	−0.000142	−0.000144	−0.000145
	Student’s t	−0.000258***	−0.000259	−0.000145***
	GED	−0.000100***	−0.000200***	−0.000100***
Ω	Normal	0.000000	0.000000	0.000000
	Student’s t	0.000002	0.000001	0.000002
	GED	0.000001	0.000000	0.000001
α	Normal	−0.052424***	0.072798***	0.052936***
	Student’s t	−0.184439***	0.119272***	0.182315
	GED	−0.104348***	0.054199***	0.105391***
β	Normal	0.946576***	0.945179***	0.947064
	Student’s t	0.814561***	0.835069***	0.817685
	GED	0.894544***	0.911427***	0.894609***
γ	Normal		−0.037954***
	Student’s t		0.087720***
	GED		0.025154

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
AIC	Normal	−6.7251	−6.7355	−6.7287
	Student’s t	−7.2199	−7.2213	−7.2214
	GED	−7.2821	−7.2468	−7.2836
BIC	Normal	−6.7116	−6.7186	−6.7185
	Student’s t	−7.1995	−7.1976	−7.2044
	GED	−7.2652	−7.2264	−7.2701
LogLikelihood	Normal	5333.67	5342.894	5335.457
	Student’s t	5727.734	5729.855	5727.934
	GED	5776.092	5749.063	5776.284

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
µ	Normal	−0.000033	−0.000016	−0.000035
	Student’s t	−0.000023	−0.000016	−0.000023
	GED	0.000000***	0.000000	0.000000
Ω	Normal	0.000000	0.000000	0.000000
	Student’s t	0.000001	0.000001	0.000001
	GED	0.000001	0.000000	0.000000
α	Normal	−0.056860***	0.064419***	0.058089***
	Student’s t	−0.119837***	0.126086***	0.120626***
	GED	−0.097748***	0.105957***	0.09602***
β	Normal	0.942137***	0.942077***	0.941911
	Student’s t	0.879163***	0.880148***	0.879374***
	GED	0.900983***	0.902721***	0.90398***
γ	Normal		−0.015014
	Student’s t		−0.014587
	GED		−0.019362***

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
AIC	Normal	−7.3546	−7.3543	−7.3559
	Student’s t	−7.5823	−7.5812	−7.5836
	GED	−7.5698	−7.5689	−7.5710
BIC	Normal	−7.3410	−7.3373	−7.3457
	Student’s t	−7.5620	−7.5575	−7.5667
	GED	−7.5529	−7.5486	−7.5575
LogLikelihood	Normal	5832.513	5833.27	5832.526
	Student’s t	6014.968	6015.072	6015.039
	GED	6004.103	6004.355	6004.036

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
µ	Normal	0.000049	0.000061	0.000044
	Student’s t	0.000080	0.000076	0.000080
	GED	0.000000	0.000000	0.000000
Ω	Normal	0.000001	0.000001	0.000001
	Student’s t	0.000003	0.000003	0.000003
	GED	0.000000	0.000001	0.000001
α	Normal	0.095711***	0.096498***	0.099220***
	Student’s t	0.208429***	0.197801***	0.208740***
	GED	0.083868***	0.149875***	0.167508***
β	Normal	0.898964***	0.899192***	0.900780
	Student’s t	0.790570***	0.786656***	0.791260
	GED	0.910881***	0.800982***	0.802731***
γ	Normal		−0.002485
	Student’s t		0.028666
	GED		0.015400

Note: *, **, ***: Statistically significant at the 10%, 5%, and 1% level of significance, respectively.

Table

		GARCH (1,1)	GJR-GARCH	IGARCH
AIC	Normal	−7.5433	−7.5420	−7.5442
	Student’s t	−7.8617	−7.8606	−7.8631
	GED	−7.8526	−7.8967	−7.8992
BIC	Normal	−7.5297	−7.5251	−7.5341
	Student’s t	−7.8414	−7.8369	−7.8461
	GED	−7.8357	−7.8764	−7.8857
LogLikelihood	Normal	5982.035	5982.041	5981.807
	Student’s t	6236.417	6236.511	6236.472
	GED	6238.195	6264.163	6264.14