Investor attention and carbon return: evidence from the EU-ETS

Figures & data

Table 1. Descriptive statistics.

Return	Mean	Std. dev	Max	Min	Skewness	Kurtosis
Carbon futures	0.0050	0.0569	0.1697	−0.3045	−0.7219	3.8905
Investor attention	0.0010	0.2596	0.5725	−1.0833	−0.4971	1.0383
Brent oil futures	0.0036	0.0595	0.3682	−0.2523	0.5776	11.7001
Natural gas futures	0.0029	0.0630	0.2440	−0.1880	0.1635	3.7682
Dow Jones industrial average index	0.0028	0.0268	0.1284	−0.1730	−1.1418	15.9684
NASDQ composite index	0.0043	0.0259	0.1059	−0.1264	−0.8189	7.9992
DAX index	−0.0001	0.0065	0.0478	−0.0374	0.7329	19.2279

Source: the Authors.

Table 2. ADF test results.

	Type	t − statistic
Carbon futures return	None	−11.8920***
	Intercept	−11.9582***
	Trent and Intercept	−11.9436***
Investor attention	None	−12.8284***
	Intercept	−12.8011***
	Trent and Intercept	−12.7727***

Note: None, Intercept, Trend and Intercept mean three types of the ADF test, respectively. The null hypothesis of the ADF test assumes that the time series has a unit root.

*** Represents the significance in 1% level.

Source: the Authors.

Table 3. Granger causality test results.

Null Hypothesis	${\chi }^2$-statistic
Investor attention is not the granger cause of carbon futures return	9.3737*
Carbon futures return is not the granger cause of investor attention	5.2019

Note: *represents the 10% level of significance. We set the lag length to be 4 in Granger causality test according to Vozlyublennaia (2014).

Source: the Authors.

Table 4. VAR analysis results.

	$R_t$	${\mathit{Att}}_t$
$R_{t-1}$	0.2413***	−0.2409
	(0.0785)	(0.3627)
$R_{t-2}$	−0.0302	0.6215*
	(0.0808)	(0.3732)
$R_{t-3}$	0.0338	0.3781
	(0.0821)	(0.3796)
$R_{t-4}$	−0.1324*	0.0601
	(0.0779)	(0.3600)
${\mathit{Att}}_{t-1}$	−0.0228	−0.3747***
	(0.0170)	(0.0787)
${\mathit{Att}}_{t-2}$	0.0198	−0.3190***
	(0.0182)	(0.0839)
${\mathit{Att}}_{t-3}$	0.0355**	−0.1438*
	(0.0179)	(0.0829)
${\mathit{Att}}_{t-4}$	0.0284*	−0.1528*
	(0.0169)	(0.0781)
Intercept	0.0071*	−0.0049
	0.0042	(0.0193)
R^2	0.1317	0.1836

Note: The VAR estimation is reported by two columns: the left column is used to present the results of carbon futures return as dependent variable, while the right column is used to present the results of investor attention as dependent variable. The value in the bracket represents the standard error. *,**,*** denote significance at 10%, 5% and 1% level, respectively.

Source: the Authors.

Figure 1. Response of carbon futures return to the shock from investor attention.

Source: the Authors.

Figure 2. Response of investor attention to the shock from carbon futures return.

Note: In the above figures, the blue solid-line is the impulse response to Cholesky one standard deviation innovations, while the red dotted-line is ninety-five percent confidence interval for highest probability density. And the X-axis represents the duration of shock, while the Y-axis represents the magnitude of such shock.

Source: the Authors.

Table 5. Tests for VAR model.

	$R_{t-1}$	$R_{t-2}$	$R_{t-3}$	$R_{t-4}$	${\mathit{Att}}_{t-1}$	${\mathit{Att}}_{t-2}$	${\mathit{Att}}_{t-3}$	${\mathit{Att}}_{t-4}$
Panel A: 1/VIF
	0.8770	0.8239	0.8026	0.8734	0.8234	0.7225	0.7393	0.8313
Panel B: Correlation matrix
$R_{t-1}$	1.0000
$R_{t-2}$	0.2628	1.0000
$R_{t-3}$	0.0149	0.2644	1.0000
$R_{t-4}$	−0.0492	0.0338	0.2849	1.0000
${\mathit{Att}}_{t-1}$	−0.0741	0.0555	0.1739	0.0784	1.0000
${\mathit{Att}}_{t-2}$	−0.1879	−0.0815	0.0466	0.1630	−0.2472	1.0000
${\mathit{Att}}_{t-3}$	−0.0049	−0.1860	−0.0801	0.0470	−0.1624	−0.2471	1.0000
${\mathit{Att}}_{t-4}$	0.1177	−0.0092	−0.1896	−0.0881	0.0033	−0.1582	−0.2474	1.0000
Panel C: VAR residual heteroscedasticity test
			${\mathrm{\chi }}^2$				${\mathrm{\chi }}^2$
No cross terms			47.5303		With cross terms		147.4948
			(0.4920)				(0.1687)

Note: This table reports the results of VIF test, correlation analysis, and heteroscedasticity test.

Source: the Authors.

Table 6. The effects of high order moment.

	Coefficient	Standard Error
$R_{t-1}$	0.2245***	0.0820
$R_{t-2}$	−0.0301	0.0838
$R_{t-3}$	0.0143	0.0845
$R_{t-4}$	−0.1311	0.0810
${\mathit{Att}}_{t-1}$	−0.0130	0.0186
${\mathit{Att}}_{t-2}$	0.0278	0.0208
${\mathit{Att}}_{t-3}$	0.0397*	0.0207
${\mathit{Att}}_{t-4}$	0.0371**	0.0186
${\mathit{Att}}_{t-1}^2$	0.1068**	0.0504
${\mathit{Att}}_{t-2}^2$	−0.0442	0.0500
${\mathit{Att}}_{t-3}^2$	0.0230	0.0492
${\mathit{Att}}_{t-4}^2$	0.0585	0.0474
Intercept	−0.0028	0.0069
R^2	0.1700

Note: The table contains the estimation results of lagged carbon return ($R_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$), lagged investor attention (${\mathit{Att}}_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$) and lagged squared term of investor attention (${\mathit{Att}}_{t-i}^2,\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$).

Source: the Authors.

Table 7. The joint impact results.

	Brent	Gas	Dow	NASDQ	DAX
	$R_t$	$R_t$	$R_t$	$R_t$	$R_t$
$R_{t-1}$	0.2588***	0.2390***	0.2544***	0.2548***	0.2303***
	（0.0828)	(0.0822)	(0.0824)	(0.0823)	(0.0834)
$R_{t-2}$	−0.0196	−0.0166	−0.0354	−0.0201	−0.0264
	（0.0853）	(0.0846)	(0.0863)	(0.0850)	(0.0863)
$R_{t-3}$	0.0269	0.0321	0.0429	0.0461	0.0529
	（0.0868）	(0.0854)	(0.0878)	(0.0852)	(0.0868)
$R_{t-4}$	−0.1459*	−0.1232	−0.1606*	−0.1508*	−0.1111
	(0.0818)	(0.0813)	(0.0856)	(0.0830)	(0.0825)
${\mathit{Att}}_{t-1}$	−0.0163	−0.0175	−0.0242	−0.0303	−0.0316*
	(0.0181)	(0.0176)	(0.0186)	(0.0184)	(0.0181)
${\mathit{Att}}_{t-2}$	0.0185	0.0247	0.0268	0.0242	0.0096
	(0.0192)	(0.0188)	(0.0198)	(0.0197)	(0.0200)
${\mathit{Att}}_{t-3}$	0.0357*	0.0353*	0.0294	0.0270	0.0243
	(0.0189)	(0.0186)	(0.0194)	(0.0194)	(0.0197)
${\mathit{Att}}_{t-4}$	0.0315*	0.0293*	0.0350*	0.0400*	0.0235
	(0.0179)	(0.0175)	(0.0186)	(0.0187)	(0.0184)
${\mathit{Control}}_{t-1}$	0.0561	0.0690	−0.1792	−0.2121	0.0103
	(0.1054)	(0.0697)	(0.2516)	(0.2150)	(0.2181)
${\mathit{Control}}_{t-2}$	0.0993	0.0212	0.2473	0.4197*	0.4069*
	(0.1054)	(0.0692)	(0.2511)	(0.2093)	(0.2153)
${\mathit{Control}}_{t-3}$	−0.1270	−0.0517	−0.2014	0.0811	0.1358
	(0.1051)	(0.0689)	(0.2517)	(0.2143)	(0.2173)
${\mathit{Control}}_{t-4}$	0.0535	0.1209*	−0.0636	−0.1285	−0.0146
	(0.1037)	(0.0700)	(0.2529)	(0.2142)	(0.2140)
${\mathit{Att}}_{t-1}\mathrm{*}{\mathit{Control}}_{t-1}$	−0.6694*	−0.1612	−0.1338	0.1750	−1.3803
	(0.3581)	(0.2671)	(0.9737)	(0.7615)	(0.9189)
${\mathit{Att}}_{t-2}\mathrm{*}{\mathit{Control}}_{t-2}$	0.1762	0.4532*	−0.6685	−0.4968	−0.6331
	(0.3619)	(0.2608)	(0.8322)	(0.6904)	(0.9326)
${\mathit{Att}}_{t-3}\mathrm{*}{\mathit{Control}}_{t-3}$	0.2641	0.3664	1.2235	1.1463	0.2304
	(0.368)	(0.2651)	(0.8275)	(0.6852)	(0.9222)
${\mathit{Att}}_{t-4}\mathrm{*}{\mathit{Control}}_{t-4}$	0.1504	0.1333	−0.9382	−1.1251	−0.4905
	(0.3617)	(0.2651)	(0.9338)	(0.7316)	(0.8957)
Intercept	0.0067	0.0070	0.0077	0.0065	0.0053
	(0.0045)	(0.0043)	(0.0047)	(0.0046)	(0.0044)
R^2	0.1738	0.1904	0.1655	0.1958	0.1721

Note: The table contains the estimation results of lagged carbon return ($R_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$), lagged investor attention (${\mathit{Att}}_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$), lagged control variable (${\mathit{Control}}_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$) and the interaction term between lagged investor attention and control variable (${\mathit{Att}}_{t-i}\mathrm{*}{\mathit{Control}}_{t-i},\mathrm{}i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4$). The returns of Brent oil futures (Brent), Natural gas futures (GAS), Dow Jones Industrial Average Index (DOW), NASDQ Composite Index (NASDQ) and DAX Index (DAX) represent the control variables.

Source: the Authors.

Table 8. Out-of-sample prediction results with forecast horizon equal to one.

	$R_{\mathit{oos}}^2$	MSFE	MSFE-adj
VAR	−0.0478	0.0035	1.9286
High	−0.0943	0.0037	0.0036
Brent	0.1446	0.0029	2.1843**
Gas	−0.0038	0.0034	1.6457
Dow	0.0296	0.0032	1.5904*
NASDQ	−0.0207	0.0034	2.1349
DAX	−0.0123	0.0034	1.2897

Note: VAR and High represent the models (4(4) $$\widehat{R_{t+h}}=\widehat{{\alpha }_0}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}$$(4) ) and (5(5) $$\widehat{R_{t+h}}=\widehat{{\alpha }_1}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\eta }_i}{\mathit{Att}}_{t+h-i}^2$$(5) ), respectively. Brent, Gas, Dow, NASDQ and DAX refer to the cases when the control variable in Equation (6)(6) $$\widehat{R_{t+h}}=\widehat{{\alpha }_2}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\gamma }_i}{\mathit{control}}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\delta }_i}{{\mathit{Att}}_{t+h-i}\mathrm{*}\mathit{control}}_{t+h-i}$$(6) is controlled by the corresponding return, respectively. As indicated by Welch and Goyal (2008), it is difficult for an individual model to significantly outperform the historical average model in the out-of-sample forecasts. Thus, we set the historical average model as our benchmark model. A positive $R_{\mathit{oos}}^2$ indicates that the forecasting performance of the predictive models outperforms the benchmark model.

Source: the Authors.

Table 9. Out-of-sample prediction results with different forecast horizons.

	Panel A: h = 2			Panel B: h = 4
	$R_{\mathit{oos}}^2$	MSFE	MSFE-adj	$R_{\mathit{oos}}^2$	MSFE	MSFE-adj
VAR	−0.0496	0.0034	1.1385	−0.0079	0.0033	1.6596
High	−0.1054	0.0036	1.1665	−0.0638	0.0035	1.7264
Brent	0.1985	0.0026	2.0574**	0.2180	0.0026	2.1107**
Gas	−0.0086	0.0033	1.5114	0.0179	0.0032	1.7203**
Dow	0.0331	0.0032	1.5925*	−0.0600	0.0035	1.5942
NASDQ	−0.0157	0.0033	2.1058	−0.0859	0.0036	2.3528
DAX	−0.0104	0.0033	1.2607	0.0476	0.0032	1.3345*
	Panel C: h = 8			Panel D: h = 12
	$R^2$	MSFE	MSFE-adj	$R^2$	MSFE	MSFE-adj
VAR	−0.0100	0.0033	1.4207	0.0074	0.0034	1.3186*
High	−0.0759	0.0035	1.4108	−0.0403	0.0035	1.5331
Brent	0.1985	0.0026	2.2288**	0.2834	0.0024	1.9062**
Gas	0.0276	0.0032	1.5886*	0.0648	0.0032	1.5504*
Dow	0.1066	0.0029	1.3908*	0.1086	0.0030	1.4503*
NASDQ	0.0236	0.0032	2.1243**	−0.0189	0.0024	2.4527
DAX	0.1239	0.0029	1.3906*	0.2153	0.0027	1.4387*

Note: VAR and High represent the models (4(4) $$\widehat{R_{t+h}}=\widehat{{\alpha }_0}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}$$(4) ) and (5(5) $$\widehat{R_{t+h}}=\widehat{{\alpha }_1}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\eta }_i}{\mathit{Att}}_{t+h-i}^2$$(5) ), respectively. Brent, Gas, Dow, NASDQ and DAX refer to the cases when the control variable in Equation (6)(6) $$\widehat{R_{t+h}}=\widehat{{\alpha }_2}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\gamma }_i}{\mathit{control}}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\delta }_i}{{\mathit{Att}}_{t+h-i}\mathrm{*}\mathit{control}}_{t+h-i}$$(6) is controlled by the corresponding return, respectively.

Source: the Authors.

Table 10. Portfolio performance measures.

Cost	$\gamma$	Indicator	Benchmark	VAR	High	Brent	Gas
bps = 0	3	utility	−0.0012	0.0002	0.0006	0.5860	0.7430
		SR	0.0205	0.1097	0.1024	0.2820	0.1115
	6	utility	−0.0006	−0.0015	−0.0011	0.5649	0.6637
		SR	0.0205	0.0942	0.0844	0.2983	0.1371
	9	utility	−0.0005	−0.0010	−0.0012	0.4987	0.5504
		SR	0.0205	0.0987	0.0732	0.3043	0.1512
bps = 10	3	utility	−0.0012	−0.0005	−0.0002	0.5860	0.7430
		SR	0.0202	0.1004	0.0904	0.2712	0.0982
	6	utility	−0.0006	−0.0021	−0.0017	0.5649	0.6637
		SR	0.0204	0.0800	0.0682	0.2870	0.1231
	9	utility	−0.0005	−0.0015	−0.0017	0.4987	0.5504
		SR	0.0204	0.0824	0.0544	0.2918	0.1378
bps = 20	3	utility	−0.0012	−0.0012	−0.0009	0.5860	0.7430
		SR	0.0198	0.0912	0.0784	0.2603	0.0850
	6	utility	−0.0006	−0.0027	−0.0023	0.5649	0.6637
		SR	0.0202	0.0657	0.0519	0.2756	0.1092
	9	utility	−0.0005	−0.0020	−0.0022	0.4987	0.5504
		SR	0.0204	0.0661	0.0357	0.2793	0.1244
			Benchmark	Dow	NASDQ	DAX
bps = 0	3	utility	−0.0012	0.6498	0.6264	0.5989
		SR	0.0205	0.2489	0.2176	0.1922
	6	utility	−0.0006	0.6311	0.6256	0.5814
		SR	0.0205	0.2083	0.2361	0.2015
	9	utility	−0.0005	0.5289	0.5234	0.5063
		SR	0.0205	0.1924	0.2215	0.2013
bps = 10	3	utility	−0.0012	0.6498	0.6264	0.5989
		SR	0.0202	0.2368	0.2064	0.1814
	6	utility	−0.0006	0.6311	0.6256	0.5814
		SR	0.0204	0.1939	0.2231	0.1890
	9	utility	−0.0005	0.5289	0.5234	0.5063
		SR	0.0204	0.1787	0.2086	0.1880
bps = 20	3	utility	−0.0012	0.6498	0.6264	0.5989
		SR	0.0198	0.2248	0.1952	0.1706
	6	utility	−0.0006	0.6311	0.6256	0.5814
		SR	0.0202	0.1795	0.2102	0.1764
	9	utility	−0.0005	0.5289	0.5234	0.5063
		SR	0.0204	0.1650	0.1957	0.1747

Note: This table reports the portfolio performance measurement for a risk-average investor who allocates assets between carbon futures and risk-free asset by using either an historical average or the predictive models incorporated with attention. Utility is the net CER gains by assuming a proportional transactions cost. We compare the different proportional transaction costs of 0, 10 and 20 basis points (bps) per transaction to assure the robustness. SR is the Sharpe ratio, and $\gamma$ is the risk-aversion degree. VAR and High represent the models (4(4) $$\widehat{R_{t+h}}=\widehat{{\alpha }_0}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}$$(4) ) and (5(5) $$\widehat{R_{t+h}}=\widehat{{\alpha }_1}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\eta }_i}{\mathit{Att}}_{t+h-i}^2$$(5) ), respectively. Brent, Gas, Dow, NASDQ and DAX refer to the cases when the control variable in Equation (6)(6) $$\widehat{R_{t+h}}=\widehat{{\alpha }_2}+{\sum }_{i=1}^4\widehat{{\alpha }_i}{R}_{t+h-i}+{\sum }_{i=1}^4{\widehat{{\beta }_i}Att}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\gamma }_i}{\mathit{control}}_{t+h-i}+{\sum }_{i=1}^4\widehat{{\delta }_i}{{\mathit{Att}}_{t+h-i}\mathrm{*}\mathit{control}}_{t+h-i}$$(6) is controlled by the corresponding return, respectively.

Source: the Authors.

Table 11. Granger causality and VAR analyses for robustness checks.

Panel A: Granger causality test result for robustness check
Investor attention is not the granger cause of carbon futures return		5.3586*
Carbon futures return is not the granger cause of investor attention		0.1533
Panel B: VAR analysis for robustness checks
	VAR analysis
	$R_t$	${\mathit{Att}}_t$
$R_{t-1}$	0.4548***	0.0961
	(0.0893)	(0.3529)
$R_{t-2}$	−0.3145***	−0.1272
	(0.0887)	(0.3505)
${\mathit{Att}}_{t-1}$	−0.0416	−0.3170***
	(0.0259)	(0.1022)
${\mathit{Att}}_{t-2}$	−0.0565**	−0.0763
	(0.0268)	(0.1061)
Intercept	0.0034	−0.0136
	(0.0058)	(0.0230)
R^2	0.2663	0.0900

Source: the Authors.

Vozlyublennaia, N. (2014). Investor attention, index performance, and return predictability. Journal of Banking & Finance, 41, 17–35. https://doi.org/10.1016/j.jbankfin.2013.12.010

 Web of Science ®Google Scholar

Welch, I., & Goyal, A. (2008). A comprehensive look at the empirical performance of equity premium prediction. The. Review of Financial Studies, 21(4), 1455–1508. https://doi.org/10.1093/rfs/hhm014

 Web of Science ®Google Scholar