Modelling volatility in job loss during the COVID-19 pandemic: The Malaysian case

Figures & data

Figure 1. Daily number of job loss and number of COVID-19 new cases.

Figure 2. Daily number of job loss and number of COVID-19 new deaths.

Figure 3. Daily log and log changes in number of job loss, number of COVID-19 new cases and new deaths.

Figure 4. Q-Q plots of log job loss, COVID-19 new cases and new deaths, and log changes in loss of employment, COVID-19 new cases and new deaths.

Figure 5. Q-Q plot of job loss regression residuals.

Table 1. Descriptive statistics

Series	No. obs	Mean	Max	Min	Std. Dev.	Skewness	Kurtosis	Jarque-Bera	Anderson-Darling
Job loss	363	6.11	8.03	2.64	0.80	−0.84	4.46	75.28***	3.59***
New cases	363	4.30	8.53	0.00	2.55	−0.16	2.02	16.11***	4.46***
New deaths	363	0.69	3.18	0.00	0.92	0.92	2.43	56.30***	41.63***
$\mathrm{\Delta }\mathrm{J}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}$	363	0.00	2.74	−3.09	0.72	0.15	5.14	70.35***	7.02***
$\mathrm{\Delta }\mathrm{N}$ew cases	363	0.02	4.56	−3.69	0.84	0.51	8.98	556.0***	10.53***
$\mathrm{\Delta }\mathrm{N}$ew deaths	363	0.01	2.31	−3.09	0.67	−0.09	6.22	157.2***	29.27***

Notes: Asterisks *** denotes statistically significant at 1% level. Series of job loss, new cases and new deaths are in natural logarithm.

Table 2. Regression estimates for loss of employment with COVID-19 new cases and new deaths

Dependent/Independent variables	$\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}}$	$\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}}$
$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$	6.0145***	0.0025
	(70.168)	(0.0688)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$	0.0486**
	(2.0663)
$\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$		−0.0291
		(−0.6508)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}{\mathrm{s}}_{\mathrm{t}}$	−0.1620**
	(−2.4659)
$\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}{\mathrm{s}}_{\mathrm{t}}$		−0.0469
		(−0.8325)
${\mathrm{R}}^2$	0.0170	0.0030
SER	0.7919	0.7181
LM${\chi }^2\left(1\right)$	[0.000]	[0.5742]
ARCH${\chi }^2\left(1\right)$	[0.000]	[0.0536]

Notes: Asterisks ***,**,* denote statistically significant at 1%, 5% and 10% level, respectively. LM ${\chi }^2\left(1\right)$ and ARCH${\chi }^2\left(1\right)$ denote the Lagrange multiplier test for serial correlation of order one and heteroscedasticity of order one in the OLS equations, respectively. Figures in round brackets (…) are t-statistics, while figures in square brackets […] are p-values. ${\mathrm{R}}^2$ and SER denote R-squared and standard error of regression, respectively. $\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}}$ refers to log in job loss; while $\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}}$.denotes log changes in job loss. All variables are in logarithm.

Table 3. Results of unit root tests

Types	Level		First-difference
	Intercept	Intercept+trend	Intercept	Intercept+trend
Job loss	−2.1006 (12)	−3.0452 (12)	−9.0972 (12)***	−9.1143 (12)***
New cases	−1.1899 (4)	−1.7475 (4)	−14.398 (3)***	−14.378 (3)***
New deaths	−2.2756 (4)	−2.9340 (4)	−21.935 (1)***	−21.910 (1)***

Notes: Asterisks *** denotes statistically significant at 1% level. The critical values are referred to MacKinnon (1996); Figures in round bracket (…) denote optimal lag length chosen using SBC criteria.

Table 4. Results of cointegration tests

Dependent/Independent variables	ARDL(3,3,0)	Conditional ARDL-ECM model	Long-run model	Error-correction Model (short-run)
$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$	2.6648***	2.6648***	5.9215***
	(8.2453)	(8.2453)	(38.472)
$\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-1}$	0.7313***	−0.4500***
	(13.875)	(−8.5025)
$\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-2}$	−0.2851***
	(−4.4952)
$\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-3}$	0.1038**
	(1.9793)
$\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-1}$		0.1813***		0.1813***
		(3.4176)		(3.4360)
$\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-2}$		−0.1038**		−0.1038**
		(−1.9793)		(−1.9915)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$	−0.0780*		0.0916**
	(−1.7348)		(2.1109)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-1}$	−0.0752	0.0412**
	(−1.5999)	(2.0817)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-2}$	0.1248***
	(2.6463)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-3}$	0.0698
	(1.5328)
$\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$		−0.0780*		−0.0780*
		(−1.7348)		(−1.7527)
$\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-1}$		−0.1946***		−0.1946***
		(−3.7858)		(−3.8903)
$\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-2}$		−0.0698		−0.0698
		(−1.5328)		(−1.5474)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}{\mathrm{s}}_{\mathrm{t}}$	−0.1211**		−0.2692**
	(−2.2455)		(−2.2841)
$\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}{\mathrm{s}}_{\mathrm{t}-1}$		−0.1211**
		(−2.2455)
$\mathrm{E}\mathrm{C}{\mathrm{M}}_{\mathrm{t}-1}$				−0.4500***
				(−8.5643)
${\mathrm{R}}^2$	0.4275	0.2847	-	0.2883
LM${\chi }^2\left(1\right)$	[0.5692]	-	-	[0.7519]
ARCH${\chi }^2\left(1\right)$	[0.0335]	-	-	[0.0248]
Bound F-statistic	-	18.182***	-	-

Notes: Asterisks ***,**,* denote statistically significant at 1%, 5% and 10% level, respectively. Figures in round brackets (…) are t-statistics, while figures in square brackets […] are p-values. ${\mathrm{R}}^2$ denotes R-squared. $\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}}$ refers to job loss. LM ${\chi }^2\left(1\right)$ and ARCH${\chi }^2\left(1\right)$ denote the Lagrange multiplier test for serial correlation of order one and heteroscedasticity of order one in the ARDL equation, respectively ARDL(p,q1,q2) denotes optimal lag length chosen using AIC criteria.

Table 5. Parameter estimates of ARDL(3,3,0)-GARCH(1,1)-M and ARDL(3,3,0)-EGARCH(1,1)-M

Independent variables	ARDL(3,3,0)-GARCH(1,1)-M:			ARDL(3,3,0)-EGARCH(1,1)-M:
	Normal	Student’s t	GED	Normal	Student’s t	GED
Mean equation
${\delta }_0\hspace{0.17em}\left(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\right)$	1.0537***	1.0472	0.8440	−0.2274***	0.7190***	0.2130***
	(2.5872)	(1.5831)	(1.6096)	(−3.4730)	(3.1519)	(3.0880)
$\pi \mathrm{E}\mathrm{C}{\mathrm{M}}_{\mathrm{t}-1}$	−0.3845***	−0.3608***	−0.2974***	−0.4035***	−0.0543**	−0.0565**
	(−5.5116)	(−6.6739)	(−7.8204)	(−8.7297)	(−2.1631)	(−2.5519)
${\delta }_1\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-1}$	0.1759***	0.1153**	0.0360	−0.0934**	0.1619*	0.0900**
	(2.3363)	(2.0842)	(0.9431)	(−2.4910)	(1.7642)	(1.9750)
${\delta }_2\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-2}$	−0.1364**	−0.1029**	−0.0062	−0.1001***	−0.1211**	0.0285
	(−2.2593)	(−2.1387)	(−0.2204)	(−3.8742)	(−2.5704)	(0.9398)
${\delta }_3\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$	−0.0812**	−0.0680*	−0.1016***	−0.0836***	−0.0948***	−0.0937***
	(−2.0340)	(−1.8561)	(−3.6049)	(−3.5104)	(−2.7601)	(−4.8875)
${\delta }_4\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-1}$	−0.1770***	−0.1557***	−0.1595***	−0.1629***	−0.1504***	−0.1088***
	(−3.7755)	(−3.8670)	(−5.0559)	(−7.8241)	(−4.3637)	(−4.8202)
${\delta }_5\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-2}$	−0.0651	−0.0820**	−0.0682**	−0.0913***	−0.0101	−0.0301
	(−1.2857)	(−1.9703)	(−2.4835)	(−4.5609)	(−0.2516)	(−1.3809)
${\lambda \mathrm{l}\mathrm{o}\mathrm{g}\sigma }_{\mathrm{t}}^2$	1.0035***	1.0438	0.8882	−0.1650***	0.5994***	0.2179***
	(2.6340)	(1.5250)	(1.6071)	(−4.2421)	(3.0789)	(3.7818)
Variance equation
$\mathrm{c}$ (constant)	0.1884	0.1889**	0.2370***	−0.3508***	−0.7744***	−0.4843***
	(1.5693)	(2.0965)	(3.4193)	(−3.5821)	(−3.9977)	(−3.6169)
$\alpha$ (ARCH effect)	0.0542*	0.0542	0.0658	−0.6828***	0.1026*	0.0301
	(1.6501)	(1.2259)	(1.3543)	(−5.7886)	(1.7633)	(1.2895)
$\beta$ (GARCH effect)	0.4153	0.4639**	0.3384**	0.3026***	0.4032***	0.5813***
	(1.1694)	(2.0538)	(2.2615)	(4.0766)	(3.0049)	(5.9841)
$\gamma$ (Leverage effect)				−0.7794***	−0.4119***	−0.5990***
				(−9.3706)	(−4.2888)	(−5.2422)
$\alpha +\beta$	0.4695	0.5181	0.4042	−0.3802	0.5058	0.6114
${\mathrm{R}}^2$	0.2988	0.2893	0.2562	0.2702	0.2850	0.1689
SER	0.6053	0.6094	0.6234	0.6175	0.6112	0.6590
Q-stat:	[0.9480]	[0.3880]	[0.5690]	[0.0940]	[0.1810]	[0.4230]
ARCH test:	[0.6456]	[0.6914]	[0.2761]	[0.9136]	[0.1299]	[0.0164]
Criteria:
AIC	1.8607	1.7768	1.7670	1.6066	1.6717	1.6412
SC	1.9792	1.9061	1.8962	1.7359	1.8118	1.7812
HQC	1.9078	1.8282	1.8184	1.6580	1.7274	1.6968

Notes: Notes: Asterisks ***,**,* denote statistically significant at 1%, 5% and 10% level, respectively. Figures in round brackets (…) are z-statistics, while figures in square brackets […] are p-values. ${\mathrm{R}}^2$ and SER denote R-squared and standard error of regression, respectively; while Q-stat and ARCH test are test for autocorrelation and heteroscedasticity, respectively. AIC, SC and HQ denote Akaike information criteria, Schwarz criteria and Hannan-Quinn criteria, respectively.

Table 6. Parameter estimates of ARDL(3,3,0)-TGARCH(1,1)-M and ARDL-PGARCH(1,1)-M

Independent variables	ARDL(3,3,0)-TGARCH(1,1)-M:			ARDL(3,3,0)-PGARCH(1,1)-M:
	Normal	Student’s t	GED	Normal	Student’s t	GED
Mean equation
${\delta }_0\hspace{0.17em}\left(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\right)$	0.0803	0.4747**	0.6918***	−0.1023**	0.6587	0.6140***
	(0.5306)	(2.4329)	(2.7632)	(−2.2053)	(1.4117)	(4.3756)
$\pi \mathrm{E}\mathrm{C}{\mathrm{M}}_{\mathrm{t}-1}$	−0.2492***	−0.1576***	−0.1143***	−0.3901***	−0.4725***	−0.0701**
	(−4.0611)	(−3.5593)	(−3.8741)	(−9.2235)	(−6.1104)	(−2.3634)
${\delta }_1\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-1}$	0.2549***	0.1811**	0.0828	0.0700***	0.1639***	−0.0253
	(3.8225)	(2.4954)	(1.4921)	(4.2602)	(3.0377)	(−0.5276)
${\delta }_2\mathrm{\Delta }\mathrm{j}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{s}{\mathrm{s}}_{\mathrm{t}-2}$	−0.1511***	−0.0995**	−0.0010	−0.1118***	−0.0597	−0.0374
	(−2.9828)	(−2.1611)	(−0.0414)	(−3.8089)	(−1.1557)	(−1.0865)
${\delta }_3\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}}$	−0.0641*	−0.0564*	−0.0980***	−0.0383	−0.0645*	−0.0701***
	(−1.7951)	(−1.7217)	(−4.3801)	(−1.6435)	(−1.7730)	(−3.0091)
${\delta }_4\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-1}$	−0.1308***	−0.1128***	−0.1245***	−0.1294***	−0.1605***	−0.1030***
	(−3.2193)	(−3.1794)	(−5.3889)	(−7.7116)	(−3.9082)	(−3.7481)
${\delta }_5\mathrm{\Delta }\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}{\mathrm{s}}_{\mathrm{t}-2}$	−0.0343	−0.0177	−0.0112	−0.0372	−0.0734*	−0.0102
	(−0.7491)	(−0.4287)	(−0.4609)	(−1.2953)	(−1.8667)	(−0.3801)
${\lambda \mathrm{l}\mathrm{o}\mathrm{g}\sigma }_{\mathrm{t}}^2$	0.0811	0.4150***	0.6740***	−0.0932**	0.6917	0.4905***
	(0.6723)	(2.5804)	(2.9043)	(−2.2564)	(1.4190)	(4.4770)
Variance equation
$\mathrm{c}$ (constant)	0.1548***	0.1962***	0.2038***	0.5394***	0.0984***	0.3666***
	(10.364)	(3.4248)	(4.3116)	(6.8302)	(10.169)	(8.8817)
$\alpha$ (ARCH effect)	−0.1452***	−0.1005**	−0.0333	0.2081***	0.0299*	0.1996***
	(−10.669)	(−2.3643)	(−1.1723)	(3.7644)	(1.7022)	(3.4812)
$\beta$ (GARCH effect)	0.3899***	0.2883**	0.3453***	0.2301**	0.8529***	0.3350***
	(41.094)	(2.2572)	(3.4170)	(2.3569)	(138.43)	(3.8049)
$\gamma$ (Leverage effect)	0.6966***	0.6257**	0.3066**	0.9372***	−0.3995	0.9691***
	(4.1254)	(2.4138)	(2.1725)	(12.446)	(−0.7352)	(10.677)
$\mathrm{d}$ (Power)				0.2343*	0.5462	0.6948***
				(1.6496)	(1.4317)	(3.8487)
$\alpha +\beta$	0.2447	0.1878	0.3120	0.4382	0.8828	0.5346
${\mathrm{R}}^2$	0.2563	0.2985	0.2583	0.2739	0.2832	0.2505
SER	0.6234	0.6054	0.6225	0.6160	0.6120	0.6258
Q-stat:	[0.1770]	[0.5430]	[0.4100]	[0.1110]	[0.7940]	[0.0430]
ARCH test:	[0.1726]	[0.1021]	[0.0634]	[0.0525]	[0.1114]	[0.1892]
Criteria:
AIC	1.7763	1.7176	1.6942	1.7395	1.7987	1.6725
SC	1.9056	1.8576	1.8343	1.8796	1.9495	1.8233
HQC	1.8277	1.7732	1.7499	1.7952	1.8586	1.7324

Notes: Notes: Asterisks ***,**,* denote statistically significant at 1%, 5% and 10% level, respectively. Figures in round brackets (…) are z-statistics, while figures in square brackets […] are p-values. ${\mathrm{R}}^2$ and SER denote R-squared and standard error of regression, respectively; while Q-stat and ARCH test are test for autocorrelation and heteroscedasticity, respectively. AIC, SC and HQ denote Akaike information criteria, Schwarz criteria and Hannan-Quinn criteria, respectively.

Table 7. Evaluation of in-sample and out-of-sample forecasting for (log) loss of employment

Models	In-sample forecasting:			Out-of-sample forecasting:
	RMSE	MAE	Theil	RMSE	MAE	Theil
ARDL(3,3,0)-GARCH(1,1)-M:
Normal	11.077	9.5431	0.4857	1.5093	1.1759	0.1214
Student-t	21.050	20.222	0.6346	2.2873	1.6472	0.1708
G.E.D.♠	9.8860	9.0964	0.4520	1.7008	1.2738	0.1334
ARDL(3,3,0)-EGARCH(1,1)-M:
Normal♠	7.9783	7.2856	0.4839	1.4866	1.2614	0.1248
Student-t	4.8027	4.5479	0.2834	0.8902	0.6964	0.0716
G.E.D.	5.8756	4.7158	0.5713	0.6621	0.5371	0.0551
ARDL(3,3,0)-TGARCH(1,1)-M:
Normal	4.8022	3.5811	0.3064	1.0855	0.8670	0.0890
Student-t	19.200	17.676	0.6164	1.8321	1.4039	0.1388
G.E.D. ♠	10.421	9.9366	0.4623	1.1710	0.9052	0.0939
ARDL(3,3,0)-PGARCH(1,1)-M:
Normal	7.5315	5.6032	0.4192	1.8989	1.7559	0.1666
Student-t	22.223	21.191	0.6480	2.3617	1.6665	0.1752
G.E.D. ♠	12.288	11.314	0.5064	1.3290	1.0608	0.1082

Notes: RMSE, MAE and Theil refer to root mean square error, mean absolute error, mean absolute percent error and Theil inequality coefficient, respectively.

MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11(6), 601–618. https://doi.org/10.1002/(SICI)1099-1255(199611)11:6<601:AID-JAE417>3.0.CO;2-T

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