Stock market return predictability: Google pessimistic sentiments versus fear gauge

Figures & data

Table 1. Descriptive statistics

	∆Crash	∆Bear	VIX	∆VIX	S&P 500 Returns	∆VOL
Mean	−0.0004	0.0003	19.3810	0.0022	0.0012	0.0003
Median	−0.0111	0.0045	16.4518	−0.0873	0.0018	−0.0003
Maximum	8.9	2.4785	71.802	17.436	0.1147	0.3546
Minimum	−9.0395	−3.1004	10.015	−15.894	−0.1684	−0.3278
Std. Dev.	0.7484	0.41954	9.24238	2.4367	0.0238	0.0725
Skewness	−0.4104	−0.7324	2.46738	0.8970	−0.5038	−0.0228
Kurtosis	76.4077	12.3025	10.7391	13.8552	9.56648	7.0049
Observations	622	622	622	622	622	622

Notes: This table reports the descriptive statistics of ∆Crash and ∆Bear. Furthermore, the descriptive statistics of weekly first difference S&P 500 returns, volume of S&P stock market and VIX are also reported here.

Table 2. Correlation

	∆Crash	∆Bear	S&P 500 Returns	∆VOL	∆VIX
∆CRASH	1
∆BEAR	0.326351	1
SPRET	0.068136*	−0.0518	1
∆VOL	0.304879***	0.194338***	−0.12791***	1
∆VIX	0.148218***	0.23788***	−0.54209***	0.282728***	1

Notes: Table 2 reports the correlation between the given variables: ∆Crash, ∆Bear, weekly logarithmic change S&P 500 weekly opening prices, weekly first difference of volume of S&P 500, and weekly first difference of VIX.

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table 3. Estimation of symmetric and asymmetric relationship between VIX and S&P500 returns

Without asymmetry (ARDL)		NARDL with LR asymmetry		NARDL with SR asymmetry		NARDL with LR & SR asymmetry
Variable	Coefficient	Variable	Coefficient	Variable	Coefficient	Variable	Coefficient
C	0.023918***	C	0.000444	C	0.023943***	C	0.000409
LOGSP(-1)	−1.15607***	LOGSP(-1)	−1.15871***	LOGSP(-1)	−1.15426***	LOGSP(-1)	−1.15861***
LOGVIX(-1)	−0.00779***	LOGVIXPOS(-1)	−0.00786***	LOGVIX(-1)	−0.00792***	LOGVIXPOS(-1)	−0.00789***
DLOGSP(-1)	−0.05758*	LOGVIXNEG(-1)	−0.00798***	DLOGSP(-1)	−0.05852*	LOGVIXNEG(-1)	−0.00801***
DLOGVIX	−0.10889***	DLOGSP(-1)	−0.05632*	DVIXPOS	−0.10513***	DLOGSP(-1)	−0.0561*
DLOGVIX(-1)	−0.08945***	DLOGVIX	−0.10884***	DVIXPOS(-1)	−0.09011***	DVIXPOS	−0.10658***
		DLOGVIX(-1)	−0.08948***	DVIXNEG	−0.11436***	DVIXPOS(-1)	−0.09141***
				DVIXNEG(-1)	−0.09023***	DVIXNEG	−0.11233***
						DVIXNEG(-1)	−0.08794***
R^2	0.702682	R^2	0.70319	R^2	0.702791	R^2	0.703242
F-statistic	290.2253***	F-statistic	242.0489***	F-statistic	206.399***	F-statistic	180.6932***
AIC	−5.07891	AIC	−5.0774	AIC	−5.07125	AIC	−5.06954
SC	−5.03604	SC	−5.02738	SC	−5.01402	SC	−5.00515
		Wlr	1.049674	Wsr	0.085118	Wsr	0.926483
						Wlr	0.004826

Notes: This table reports the symmetric/asymmetric relationship between VIX and S&P 500 returns. Further asymmetric relationship is expended into three models, one with long-run asymmetry, one with short-run asymmetry, and other with both long-run and short-run asymmetries. Estimated models are given as;

(1) $$\mathrm{\Delta }{\text{SP}}_t=\mathit{\mu }+{\mathit{\rho }}_{\text{sp}}{\text{SP}}_{t-1}+{\mathit{\rho }}_{\text{vix}}{\text{VIX}}_{t-1}+{\sum }_{i=1}^{p=1}{\mathit{\alpha }}_i\mathrm{\Delta }{\text{SP}}_{t-i}+{\sum }_{i=0}^{q=1}{\mathit{\beta }}_i\mathrm{\Delta }{\text{VIX}}_{t-i}+{\mathit{\epsilon }}_t$$(1)

(2) $$\mathrm{\Delta }{\text{SP}}_t=\mathit{\mu }+{\mathit{\rho }}_{\text{sp}}{\text{SP}}_{t-1}+{\mathit{\rho }}_{\text{vix}}^{+}{\text{VIX}}_{t-1}^{+}+{\mathit{\rho }}_{\text{vix}}^{-}{\text{VIX}}_{t-1}^{-}+{\sum }_{t=1}^p{\mathit{\alpha }}_i\mathrm{\Delta }{\text{SP}}_{t-1}+{\sum }_{i=0}^q({\mathit{\beta }}_i^{+}\mathrm{\Delta }{\text{VIX}}_{t-1}^{+}+{\mathit{\beta }}_i^{-}\mathrm{\Delta }{\text{VIX}}_{t-1}^{-})+{\mathit{\epsilon }}_t$$(2)

(3) $$\mathrm{\Delta }{\text{SP}}_t=\mathit{\mu }+{\mathit{\rho }}_{\text{sp}}{\text{SP}}_{t-1}+{\mathit{\rho }}_{\text{vix}}{\text{VIX}}_{t-1}+{\sum }_{t=1}^p{\mathit{\alpha }}_i\mathrm{\Delta }{\text{SP}}_{t-1}+{\sum }_{i=0}^q({\mathit{\beta }}_i^{+}\mathrm{\Delta }{\text{VIX}}_{t-1}^{+}+{\mathit{\beta }}_i^{-}\mathrm{\Delta }{\text{VIX}}_{t-1}^{-})+{\mathit{\epsilon }}_t$$(3)

(4) $$\mathrm{\Delta }{\text{SP}}_t=\mathit{\mu }+{\mathit{\rho }}_{\text{sp}}{\text{SP}}_{t-1}+{\mathit{\rho }}_{\text{vix}}^{+}{\text{VIX}}_{t-1}^{+}+{\mathit{\rho }}_{\text{vix}}^{-}{\text{VIX}}_{t-1}^{-}+{\sum }_{t=1}^p{\mathit{\alpha }}_i\mathrm{\Delta }{\text{SP}}_{t-1}+{\sum }_{i=0}^{q=1}{\mathit{\beta }}_i\mathrm{\Delta }{\text{VIX}}_{t-i}+{\mathit{\epsilon }}_t$$(4)

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table 4. OLS estimation results

Model	C	∆Bear(-1)	∆Crash(-1)	∆VIX(-1)	∆VOL(-1)	Adj R-sq
5a (1)	0.00126	−0.006667***			−0.024199*	0.01949
5a (2)	0.00126		−3.02E-05		−0.03162**	0.00613
5b	0.001259			−0.00304***	−0.00279	0.095736
5c (1)	0.001258	−0.003441		−0.002922***	−3.19E-05	0.097701
5c (2)	0.00126		0.000648	−0.003053***	−4.71E-03	0.094648

Notes: Table 4 reports the Ordinary Least Square for Equation (1(1) $$\mathrm{\Delta }{\text{SP}}_t=\mathit{\mu }+{\mathit{\rho }}_{\text{sp}}{\text{SP}}_{t-1}+{\mathit{\rho }}_{\text{vix}}{\text{VIX}}_{t-1}+{\sum }_{i=1}^{p=1}{\mathit{\alpha }}_i\mathrm{\Delta }{\text{SP}}_{t-i}+{\sum }_{i=0}^{q=1}{\mathit{\beta }}_i\mathrm{\Delta }{\text{VIX}}_{t-i}+{\mathit{\epsilon }}_t$$(1) ) given as;

(5a) $$(\mathrm{\Delta }\text{S}\&P_t)={\mathit{\beta }}_0+{\mathit{\beta }}_1(\mathrm{\Delta }{\text{GSVI}}_{t-1})+{\mathit{\beta }}_2(\mathrm{\Delta }{\text{VOL}}_{t-1})+e_t$$(5a)

(5b) $$(\mathrm{\Delta }\text{S}\&P_t)={\mathit{\beta }}_0+{\mathit{\beta }}_1(\mathrm{\Delta }{\text{VIX}}_{t-1})+{\mathit{\beta }}_2(\mathrm{\Delta }{\text{VOL}}_{t-1})+e_t$$(5b)

(5c) $$(\mathrm{\Delta }\text{S}\&P_t)={\mathit{\beta }}_0+{\mathit{\beta }}_1(\mathrm{\Delta }{\text{VIX}}_{t-1})+{\mathit{\beta }}_2(\mathrm{\Delta }{\text{GSVI}}_{t-1})+{\mathit{\beta }}_3(\mathrm{\Delta }{\text{VOL}}_{t-1})+e_t$$(5c)

Here ΔS&P_t shows the weekly logarithmic change of the S&P 500 weekly opening prices, ΔVOL_t−1 defines the weekly logarithmic change for volume of S&P 500 returns with lag of one week, ΔGSVI_t−1 defines the first difference for GSVI.

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table 5. VAR estimation results

Variable	S&P 500	∆Bear	∆VOL	S&P 500	∆Crash	∆VOL	S&P 500	∆VIX	∆VOL
C	0.001488	−0.00683	0.001043	0.001443	−0.01322	0.001153	0.002092**	−0.07722	0.001144
SPRET(-1)	−0.09885**	1.713983***	0.142839	−0.07805*	3.049641***	0.098164	−0.3912***	25.58728***	0.231274
SPRET(-2)	0.033089	1.199125*	−0.16266	0.028682	2.220856**	−0.1703	−0.0316	12.14456**	−0.20385
SPRET(-3)	−0.0716*	1.174379*	−0.05983	−0.07322*	2.124862*	−0.08503	−0.14211***	12.35746**	−0.20346
SPRET(-4)	−0.01361	1.483207**	0.215174*	−0.00471	2.151778*	0.221636*	−0.10546**	11.04582**	0.222862
∆Index(-1)	−0.00925***	−0.41209***	0.016482**	−0.00081	−0.6268***	−0.00161	−0.00491***	0.091026*	0.001954
∆Index(-2)	−0.00248	−0.40597***	0.011664	−0.00094	−0.43463***	0.004326	−0.00135***	0.123535**	0.000866
∆Index(-3)	−0.00558**	−0.32531***	0.013044*	−0.00226	−0.32181***	0.013704***	−0.00071	0.132201**	−0.00221
∆Index(-4)	−0.00167	−0.1465***	0.009926	−0.00673***	−0.19544***	0.006153	−0.00149***	0.045141	−0.00101
∆VOL(-1)	−0.03683**	0.055663	−0.40311***	−0.03772**	0.5529	−0.38231***	−0.00971	2.614023*	−0.39573***
∆VOL(-2)	−0.02449	−0.3213	−0.32661***	−0.02216	−0.68722	−0.32488***	−0.00406	1.055653	−0.32571***
∆VOL(-3)	−0.02286	−0.1493	−0.30645***	−0.01746	−0.79775*	−0.34446***	−0.00972	1.332646	−0.29326***
∆VOL(-4)	−0.00887	−0.40317*	−0.22333***	0.012526	−0.2376	−0.23614***	0.013656	0.258714	−0.21263***
Adj. R-sq	0.034904	0.226435	0.180547	0.04386	0.287155	0.185074	0.196271	0.037617	0.177522
F-statistic	2.859571	16.05047	12.32847	3.358594	21.71217	12.67701	13.55601	3.009718	12.09764
AIC	−4.64923	0.865901	−2.58361	−4.65855	1.944709	−2.58915	−4.8322	4.606784	−2.57992

Notes: Table 5 reports the vector auto-regression results of Equation (6(6) $$(\mathrm{\Delta }\text{S}\&P_t)={\mathit{\beta }}_0+{\sum }_{s=1}^i{\mathit{\beta }}_s(\mathrm{\Delta }\text{S}\&P_{t-s})+{\sum }_{i=1}^i{\mathit{\beta }}_i(\mathrm{\Delta }{\text{Index}}_{t-i})+{\sum }_{v=1}^i{\mathit{\beta }}_v(\mathrm{\Delta }{\text{VOL}}_{t-v})+e_t$$(6) ) given as;

(6) $$(\mathrm{\Delta }\text{S}\&P_t)={\mathit{\beta }}_0+{\sum }_{s=1}^i{\mathit{\beta }}_s(\mathrm{\Delta }\text{S}\&P_{t-s})+{\sum }_{i=1}^i{\mathit{\beta }}_i(\mathrm{\Delta }{\text{Index}}_{t-i})+{\sum }_{v=1}^i{\mathit{\beta }}_v(\mathrm{\Delta }{\text{VOL}}_{t-v})+e_t$$(6)

Here, ΔS&P_t−s defines the weekly logarithmic change of S&P 500 with different weekly lags, ΔGSVI_t−i defines the first difference of GSVI with different weekly lags (GSVI refers to “Market Crash” and “Bear Market”) and ΔVOL_t−v defines the weekly logarithmic change for weekly S&P 500 volume with different weekly lags.

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table 6. VAR results of VIX and GSVI

	∆VIX	∆Bear	∆VIX	∆Crash
C	0.002173	−0.00037	0.002324	−0.00201
∆VIX(-1)	−0.03138	−0.01567**	−0.02455	−0.03861***
∆VIX(-2)	0.001839	−0.01671***	0.013371	−0.03849***
∆VIX(-3)	0.056588	−0.01959***	0.04395	−0.04447***
∆VIX(-4)	−0.01977	−0.00906	−0.02783	−0.03022***
∆GSVI(-1)	0.676919**	−0.40308***	0.414559***	−0.61225***
∆GSVI(-2)	0.444782	−0.40101***	0.274525	−0.44879***
∆GSVI(-3)	0.499864*	−0.29791***	0.441663***	−0.32014***
∆GSVI(-4)	0.009859	−0.14253***	0.420603***	−0.15851***
Adj. R-sq.	0.005144	0.230907	0.013155	0.318257
F-statistic	1.398811	24.15545	2.028086	37.00419
AIC	4.633614	0.853747	4.625529	1.893743

Notes: Table 6 shows the vector auto-regression results of following equations:

(7) $$(\mathrm{\Delta }{\text{VIX}}_t)={\mathit{\beta }}_0+{\sum }_{i=1}^i{\mathit{\beta }}_i(\mathrm{\Delta }{\text{VIX}}_{t-i})+{\sum }_{v=1}^i{\mathit{\beta }}_v(\mathrm{\Delta }{\text{GSVI}}_{t-v})+e_t$$(7)

(8) $$(\mathrm{\Delta }{\text{GSVI}}_t)={\mathit{\beta }}_0+{\sum }_{v=1}^i{\mathit{\beta }}_v(\mathrm{\Delta }{\text{GSVI}}_{t-i})+{\sum }_{i=1}^i{\mathit{\beta }}_i(\mathrm{\Delta }{\text{VIX}}_{t-v})+e_t$$(8)

Here ∆VIX_t defines the daily first difference for VIX with the lag of one week and ∆GSVI_t defines the first difference for GSVI with the lag of one week.

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table 7. Granger causality results of VIX and GSVI

Panel A				Panel B
Dependent variable: ∆VIX				Dependent variable: ∆VIX
Excluded	χ^2	df	Prob.	Excluded	χ^2	df	Prob.
∆Bear	8.023793	4	0.0907	∆Crash	13.03229	4	0.0111
All	8.023793	4	0.0907	All	13.03229	4	0.0111
Dependent variable: ∆Bear				Dependent variable: ∆Crash
Excluded	Chi-sq	df	Prob.	Excluded	Chi-sq	df	Prob.
∆VIX	25.88565	4	0.000	∆VIX	55.57464	4	0.000
All	25.88565	4	0.000	All	55.57464	4	0.000

Notes: This table presents the Granger causality results. ∆VIX, ∆Bear, and ∆Crash define the change in VIX and Google indices. Table is divided into two columns. Each with the individual Google index.

Table 8. Results of time series regression

Model	Dep. var.	C	∆Crash	∆Bear	∆VIX	R^2	Adj. R^2
5a	∆Bear	0.000227			0.040954***
5b	∆Crash	−0.00054			0.045521***
6a	∆VIX	0.002	0.4826***			0.021969	0.020391
6b	∆VIX	0.001		1.3817***		0.056587	0.055065
7	∆VIX	0.001	0.257226*	1.2319***		0.062163	0.059133

Notes: This table shows the time series regression results of given models.

(9) $$\mathrm{\Delta }\text{GSVI}=\propto +\mathit{\beta }\mathrm{\Delta }\text{VIX}+{\mathit{\epsilon }}_{it}$$(9)

(10) $$\mathrm{\Delta }\text{VIX}=\propto +\mathit{\beta }\mathit{\delta }\text{GSVI}+{\mathit{\epsilon }}_{it}$$(10)

(11) $$\mathrm{\Delta }\text{VIX}=\propto +{\mathit{\beta }}_1\mathrm{\Delta }\text{Crash}+{\mathit{\beta }}_2\mathrm{\Delta }\text{Bear}+{\mathit{\epsilon }}_{it}$$(11)

* Coefficient significance level of 10%.

** Coefficient significance level of 5%.

*** Coefficient significance level of 1%.

Table

	Augmented Dickey-Fuller (ADF) test statistic		Phillips-Perron (PP) test statistic
	t-statistic	Prob.	t-statistic	Prob.
Spret	−26.5031	0.000	−26.5015	0.0000
VIX	−3.30681	0.015	−3.36388	0.0126
dVIX	−24.8473	0.000	−24.8473	0.0000
dvixPOS	−21.8636	0.000	−21.8636	0.0000
dvixNEG	−24.7383	0.000	−25.1603	0.0000

Table

Lag	SC	HQ
0	−6.617847	−6.62666
1	−6.757831*	−6.784259*

Table

Unrestricted co-integration rank test (Trace)
Hypothesized trace 0.05
No. of CE(s)	Eigen value	Statistic	Critical value	Prob.**
None*	0.256122	193.5412	15.49471	0.0001
At most 1*	0.017645	10.9842	3.841466	0.0009