Analysis and forecasting of the dry bulk shipping market: structural VAR models using FFA-spot-time charter rates

Figures & data

Figure 1. Determination of spot and FFA rates.

Source: Authors

Table 1. Summary of yearly dataset.

Cargo volume	Fleet	BDI (Freight rate)
World seaborne dry bulk trade (unit: ton)	Total bulk carrier fleet (unit: DWT)	Baltic Dry Index (unit: index)
Sample period: 1990 to 2020. Number of observations: 31

Source: Clarksons.

Figure 2. Movements of demand (cargo), supply (fleet), and price (BDI) in dry bulk shipping market.

Note: The values of the three variables in 1990 are set to be 100.

Source: Clarksons

Table 2. Results of the unit root test for log-level and log-differenced data in yearly dataset.

log-level	Cargo volume	t-value	−0.46
		p-value	0.88
	Fleet	t-value	−0.31
		p-value	0.91
	BDI	t-value	−1.93
		p-value	0.31
log-differenced	Cargo volume	t-value	−4.57
		p-value	0.00***
	Fleet	t-value	−2.63
		p-value	0.09*
	BDI	t-value	−2.46
		p-value	0.13

Notes1) *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

2) The t-statistic refers to the Augmented Dickey–Fuller test statistic.

3) For the log-level and log-differenced data, the test has a constant.

Source: Authors.

Table 3. Results of cointegration tests and VECM estimation for yearly dataset.

Null hypothesis	Number of cointegration Relationship: none	Number of cointegration Relationship: at most 1
Cointegration rank test (trace)
Trace statistic	36.69	16.01
p-value	0.00***	0.04**
Cointegrating vector $\left[\begin{array}{cc}1& {\beta }_{\mathrm{\_}}\mathit{c}\end{array}\mathit{\beta }\_\mathit{f}\mathit{c}\right]$ such that $\left[1{\beta }_c{\beta }_f\mathit{c}\right]\left[\begin{array}{c}{b}_{\mathit{t}-1}\\ c_{\mathit{t}-1}\\ f_{\mathit{t}-1}\\ 1\end{array}\right]$	[1–3.73 0.34 2.03] (0.70) (0.55)
Adjustment coefficients
$b_t$	$c_t$	$f_t$
−0.73(0.27) ***	−0.07(0.02) ***	0.01(0.01)

Notes: 1) b_t means BDI, means cargo volume, and means fleet.

2) standard errors in parentheses.

3) *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Source: Authors.

Table 4. Granger–causality test results of log-level FFA, spot, and TC rates.

-	Chi-sq. statistic	p-value
Spot → FFA	13.49	0.00***
TC → FFA	3.05	0.21
FFA → Spot	367.83	0.00***
TC → Spot	38.32	0.00***
FFA → TC	112.46	0.00***
Spot → TC	4.27	0.11

Note: *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Source: Authors.

Table 5. Results of cointegration tests for weekly dataset.

Null hypothesis	Number of cointegration Relationship: none	Number of cointegration Relationship: at most 1	Number of cointegration Relationship: at most 2
Cape
Trace statistic	89.44	31.64	4.03
p-value	0.00***	0.00***	0.04**
Panamax
Trace statistic	106.18	41.61	1.77
p-value	0.00***	0.00***	0.18
Supramax
Trace statistic	94.93	24.04	0.24
p-value	0.00***	0.00***	0.61

Note: *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Source: Authors.

Table 6. Summary of the weekly dataset.

–	Spot rate	FFA rate	TC rate	Sample period	Number of observations
Capesize	5TC Average of 5 freight rates for 180,000 DWT	1-month FFA for 5TC	6-month TC for 180,000 DWT	4^th Jul. 2014 ~ 24^th Sep. 2021	373
Panamax	4TC Average of 4 freight rates for 74,000 DWT	1-month FFA for 4TC	6-month TC for 75,000 DWT	7^th Jan. 2011 ~ 24^th Sep. 2021	552
Supramax	10TC Average of 10 freight rates for 58,000 DWT	1-month FFA for 10TC	6-month TC for 58,000 DWT	28^th Oct. 2016 ~ 24^th Sep. 2021	254

Source: Baltic Exchange and Clarksons Research.

Figure 3. Movements of the weekly spot, FFA, and TC rates of panamax.

Source: Authors’ calculations from Baltic Exchange and Clarksons Research data.

Table 7. Correlations among the spot, FFA, and TC rates of the three bulk markets.

–	Capesize			Panamax			Supramax
	FFA	Spot	TC	FFA	Spot	TC	FFA	Spot	TC
FFA	1	–	–	1	–	–	1	–	–
Spot	0.92	1	–	0.96	1	–	0.98	1	–
TC	0.96	0.92	1	0.96	0.95	1	0.99	0.99	1

Source: Authors’ calculations from Baltic Exchange and Clarksons Research data.

Table 8. Results of the unit root test for log-level and log-differenced data.

Capesize	log-level	FFA	t-value	−2.11
			p-value	0.23
		Spot	t-value	−3.94
			p-value	0.00***
		TC	t-value	−2.20
			p-value	0.20
	log-differenced	FFA	t-value	−18.44
			p-value	0.00***
		Spot	t-value	−12.28
			p-value	0.00***
		TC	t-value	−15.51
			p-value	0.00***
Panamax	log-level	FFA	t-value	−2.26
			p-value	0.18
		Spot	t-value	−2.24
			p-value	0.18
		TC	t-value	−1.63
			p-value	0.46
	log-differenced	FFA	t-value	−24.31
			p-value	0.00***
		Spot	t-value	−12.66
			p-value	0.00***
		TC	t-value	−17.22
			p-value	0.00***
Supramax	log-level	FFA	t-value	0.05
			p-value	0.96
		Spot	t-value	−0.95
			p-value	0.76
		TC	t-value	0.20
			p-value	0.97
	log-differenced	FFA	t-value	−14.80
			p-value	0.00***
		Spot	t-value	−7.56
			p-value	0.00***
		TC	t-value	−8.09
			p-value	0.00***

Notes: 1) *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

2) The t-statistic refers to the Augmented Dickey–Fuller test statistic.

3) For the log-level data, the test has a constant. But for the log-differenced data, the test does not have a constant or trend.

Source: Authors.

Figure 4. Movements of the log-differenced data of weekly panamax spot, FFA, and TC rates.

Source: Authors’ calculations from Baltic Exchange and Clarksons Research data.

Table 9. Correlations of log-differenced data for the spot, FFA, and TC rates in the three bulk markets.

–	Capesize			Panamax			Supramax
	FFA	Spot	TC	FFA	Spot	TC	FFA	Spot	TC
FFA	1	–	–	1	–	–	1	–	–
Spot	0.46	1	–	0.28	1	–	0.25	1	–
TC	0.72	0.57	1	0.58	0.64	1	0.50	0.56	1

Source: Authors’ calculations from Baltic Exchange and Clarksons Research data.

Table 10. Optimal lag length selected by AIC, SC, and HQ criteria.

Capesize			Panamax			Supramax
AIC	SC	HQ	AIC	SC	HQ	AIC	SC	HQ
9	1	1	6	2	2	2	2	2

AIC means Akaike information criterion, SC means Schwarz information criterion, HQ means Hannan-Quinn information criterion.

Source: Authors.

Table 11. LM tests for no autocorrelation in error terms.

Capesize		Panamax		Supramax
Lag length = 1		Lag length = 2		Lag length = 2
LM-stat	p-value	LM-stat	p-value	LM-stat	p-value
12.98	0.16	31.23	0.00	12.31	0.19

Source: Authors.

Table 12. White heteroskedasticity (null hypothesis: no heteroskedasticity) tests in covariance matrix.

Capesize		Panamax		Supramax
Chi-square	p-value	Chi-square	p-value	Chi-square	p-value
115.96	0.00	132.08	0.00	131.65	0.00

Source: Authors.

Figure 5. Accumulated impulse response analyses of structural VAR model 1.

The blue lines are responses to permanent shock, the dotted red lines to seasonal shock, and the dashed black lines to post—two-period permanent shock. The horizontal axis means the considered periods and the vertical axis measures the responses of logarithmic values of the considered variables.

Figure 6. Forecast error variance decomposition of structural VAR model 1.

The blue lines are the fraction of the forecast error due to permanent shock, the dotted red lines are the error due to seasonal shock, and the dashed black lines are the error due to post—two-period permanent shock. The horizontal axis means the considered periods and the vertical axis measures the percentage portions of the considered shocks in the forecast errors.

Figure 7. Accumulated impulse response analyses from structural VAR model 2.

The blue lines are for responses to permanent shock, the dotted red lines, to seasonal shock, and the dashed black lines, to post—one-period permanent shock. The horizontal axis means the considered periods and the vertical axis measures the responses of logarithmic values of the considered variables

Table 13. Standard deviations for each structural shock under structural VAR model 1.

–	Capesize		Panamax		Supramax
	standard deviation	% to A	standard deviation	% to A	standard deviation	% to A
Permanent shock	0.1440 (A)	100%	0.0911 (A)	100%	0.0657 (A)	100%
Seasonal shock	0.1226	85%	0.0571	63%	0.0394	60%
Post—two-period shock	0.0660	46%	0.0352	39%	0.0296	45%

Source: Authors.

Table 14. Standard deviations for each structural shock under structural VAR model 2.

–	Capesize		Panamax		Supramax
	standard deviation	% to A	standard deviation	% to A	standard deviation	% to A
Permanent shock	0.9865 (A)	100%	0.7951 (A)	100%	0.0397 (A)	100%
Seasonal shock	0.1257	13%	0.0868	11%	0.0357	90%
Post—one-period shock	0.1456	15%	0.3926	49%	0.0104	26%

The sizes of standard deviation of permanent shocks in Capesize and Panamax cases are noticeably large. However, the total influence of the structural shock depends on both of the size of standard deviation and the contemporaneous correlations measured by the elements of $B^{-1}$. So, this apparent anomaly of large variance can be resolved in the aspects of total influence of structural shocks.

Source: Authors.

Table 15. In-sample comparisons of forecasting accuracy between VAR models.

	RMSE	MAE	MASE	Theil’s coefficient
Capesize
FFA-Spot-TC VAR (A)	0.1424	0.1025	0.6810	0.4568
FFA-Spot VAR (B)	0.1461	0.1066	0.7080	0.4754
Improvement: (A-B)/B	−2.5%	−3.8%	−3.8%	−3.9%
TC-Spot VAR(C)	0.1505	0.1097	0.7289	0.4990
Improvement: (A-C)/C	−5.4%	−6.6%	−6.6%	−8.5%
Panamax
FFA-Spot-TC VAR (A)	0.0581	0.0443	0.5536	0.3113
FFA-Spot VAR (B)	0.0589	0.0449	0.5615	0.3170
Improvement: (A-B)/B	−1.5%	−1.4%	−1.4%	−1.8%
TC-Spot VAR(C)	0.0747	0.0576	0.7194	0.4341
Improvement: (A-C)/C	−22.3%	−23.1%	−23.1%	−28.3%
Supramax
FFA-Spot-TC VAR (A)	0.0359	0.0261	0.6308	0.2890
FFA-Spot VAR (B)	0.0367	0.0268	0.6477	0.2970
Improvement: (A-B)/B	−2.3%	−2.6%	−2.6%	−2.7%
TC-Spot VAR(C)	0.0421	0.0315	0.7597	0.3511
Improvement: (A-C)/C	−14.7%	−17.0%	−17.0%	−17.7%

Note: RMSE means root mean square error, MAE means mean absolute error, MASE means mean absolute scaled error, and Theil’s coefficient means Theil’s inequality coefficient.

Source: Authors.

Table 16. Out-of-sample comparisons of forecasting accuracy between VAR models.

	RMSE	MAE	MASE	Theil’s coefficient
Capesize
FFA-Spot-TC VAR (A)	0.1450	0.1014	0.6737	0.4483
FFA-Spot VAR (B)	0.1495	0.1058	0.7026	0.4681
Improvement: (A-B)/B	−3.0%	−4.1%	−4.1%	−4.2%
TC-Spot VAR(C)	0.1511	0.1102	0.7319	0.4918
Improvement: (A-C)/C	−4.0%	−7.9%	−7.9%	−8.9%
Panamax
FFA-Spot-TC VAR (A)	0.0630	0.0468	0.5847	0.3069
FFA-Spot VAR (B)	0.0640	0.0476	0.5949	0.3113
Improvement: (A-B)/B	−1.5%	−1.7%	−1.7%	−1.4%
TC-Spot VAR(C)	0.0756	0.0570	0.7130	0.4430
Improvement: (A-C)/C	−16.7%	−18.0%	−18.0%	−30.7%
Supramax
FFA-Spot-TC VAR (A)	0.0441	0.0330	0.7965	0.2447
FFA-Spot VAR (B)	0.0442	0.0327	0.7896	0.2516
Improvement: (A-B)/B	−0.2%	0.9%	0.9%	−2.7%
TC-Spot VAR(C)	0.0511	0.0384	0.9266	0.2981
Improvement: (A-C)/C	−13.7%	−14.0%	−14.0%	−17.9%

Notes: 1) RMSE means root mean square error, MAE means mean absolute error, MASE means mean absolute scaled error, and Theil’s coefficient means Theil’s inequality coefficient.

2) The period for out-of-sample forecasting in Capesize and Panamax markets is from Jan. 2018 to Sep. 2021; for the Supramax market, it is Jan. 2019 to Sep. 2021.

Source: Authors.

Table 17. Comparisons of forecast directions with the VAR and VAR B-N methods.

–		Capesize	Panamax	Supramax
FFA-Spot VAR (A)	Right	146	164	117
	%	75.3%	84.5%	82.4%
TC-Spot VAR (B)	Right	143	153	103
	%	73.7%	78.9%	72.5%
FFA-Spot-TC VAR (C)	Right	151	166	116
	%	77.8%	85.6%	81.7%
B-N decomposition of FFA-Spot-TC VAR (D)	Right	150	159	107
	%	77.3%	82.0%	75.4%
C & D	Same	193	181	123
	%	99.5%	93.3%	86.6%
Total		194	194	142

Notes: 1) The period for out-of-sample forecasting in Capesize and Panamax markets is from Jan. 2018 to Sep. 2021; for the Supramax market, it is Jan. 2019 to Sep. 2021.

2) B-N method = Beveridge-Nelson decomposition.

3) C&D means the cases in which the FFA-Spot-TC VAR model and B-N method of tri-variate yield the same direction.

Source: Authors.

Table 18. Comparisons of forecasting directions based on signals conditional on the forecasts generated by the VAR model.

–		Capesize	Panamax	Supramax
Three variables have the same direction as the forecast	Total (A)	114	99	71
	Right (B)	95	91	62
	B/A	83%	92%	87%
Two variables have the same direction as the forecast	Total (A)	65	63	48
	Right (B)	46	52	37
	B/A	71%	83%	77%
One variable has the same direction as the forecast	Total (A)	15	32	23
	Right (B)	10	22	16
	B/A	67%	69%	70%

Note: The forecasting period in Capesize and Panamax markets is from Jan. 2018 to Sep. 2021; for the Supramax market, it is Jan. 2019 to Sep. 2021.