Explaining Central Government’s Tax Revenue Categories through the Bradley-Terry Regression Trunk Model

Figures & data

Table 1 Design matrix with two judges/individuals, three objects, and one continuous subject-specific covariate.

Response	μ	${\lambda }_A^O$	${\lambda }_B^O$	${\lambda }_C^O$	x1
yAB = 1	1	–1	1	0	23
yAB = 0	1	1	–1	0	23
yAC = 1	2	–1	0	1	23
yAC = 0	2	1	0	–1	23
yBC = 1	3	0	1	–1	23
yBC = 0	3	0	–1	1	23
yAB = 0	1	–1	1	0	24
yAB = 1	1	1	–1	0	24
yAC = 0	2	–1	0	1	24
yAC = 1	2	1	0	–1	24
yBC = 0	3	0	1	–1	24
yBC = 1	3	0	–1	1	24

NOTE: The first column indicates if the object i is preferred (y_ij = 1) or not (y_ij = 0) in a certain preference for each pair of objects ij. The second column serves as an index for the $n\times (n-1)/2$ comparisons. Preferences are expressed in the next three columns, and finally, the covariate x₁ is shown in the last column. In this example, the two individuals present opposite rankings, BCA and ACB, respectively.

Table 2 Descriptive statistics for the subject-specific covariates selected by the BTLL model.

Covariates	xp	# missing	Mean	SD	Median	Min	Max	Range	Skew	Kurtosis	SE
Employment rate	x1	2	0.59	0.09	0.59	0.39	0.85	0.45	0.05	0.27	0.01
Government gross debt	x2	1	0.59	0.35	0.50	0.09	2.38	2.29	2.12	6.96	0.03
Savings	x3	5	0.21	0.08	0.21	0.02	0.46	0.44	0.53	0.48	0.01
Exports	x4	13	0.04	0.06	0.05	–0.36	0.17	0.54	–3.59	22.69	0.01
Imports	x5	11	0.04	0.06	0.05	–0.23	0.18	0.41	–1.66	6.12	0.01
Interests	x6	21	0.08	0.05	0.07	0.00	0.29	0.29	1.34	2.98	0.01
Agric added-value	x7	2	0.09	0.09	0.05	0.00	0.45	0.45	1.69	2.79	0.01
Services added-value	x8	21	0.57	0.10	0.57	0.29	0.79	0.50	–0.40	–0.20	0.01
Final consumption	x9	8	0.78	0.11	0.79	0.43	1.12	0.69	–0.49	1.88	0.01
Bank NPloans	x10	16	0.04	0.06	0.03	0.00	0.42	0.42	4.77	25.75	0.01
Claims centr gov	x11	4	0.12	0.18	0.08	–0.14	1.42	1.56	3.85	23.62	0.02
Gov consumption	x12	8	0.16	0.05	0.16	0.04	0.39	0.35	0.63	2.42	0.01
EPI	x13	0	0.59	0.16	0.59	0.00	0.87	0.87	–0.93	2.21	0.02
Military spending	x14	2	0.01	0.01	0.01	0.00	0.05	0.05	1.44	3.72	0.00
Education spending	x15	33	0.04	0.01	0.04	0.01	0.08	0.07	0.72	0.25	0.00
Health spending	x16	0	0.07	0.03	0.07	0.02	0.17	0.15	0.74	0.96	0.00
Other spending	x17	9	0.20	0.10	0.20	0.03	0.76	0.73	1.77	7.89	0.01

NOTE: # missing = n. of missing values; SD = standard deviation; Min = minimum; Max = maximum; Skew = skewness; SE = standard error.

Fig. 1 Pruned regression trunk.

NOTE: In each terminal node $R_1,\dots ,R_6$ and in the root note, the number of countries H, the consensus ordering C, and the correlation coefficient τ_x are shown. Taxes are named as follows: I - Taxes on income, S - Social security contributions, P - Taxes on property, and G - Taxes on goods. When two tax categories are in brackets, it means they are in a tie.

Table 3 Pruned BTRT regression trunk.

	Node n.	$\mathit{\text{Splitting covariate}}$	$\mathit{\text{Split Point}}$	$\mathit{\text{Model Deviance}}$
		main effects (no splits)		313
bestsplit1	1	x15 (Environmental Performance Index)	0.70	271
bestsplit3	2	x4 (Health spending)	0.06	213
bestsplit5	4	x8 (Environmental Performance Index)	0.56	171
bestsplit4	5	x8 (Employment rate)	0.51	190
bestsplit2	3	x2 (GDP % change)	0.40	238

Table 4 Pruned BTRT regression trunk: 10-fold cross-validation results.

	D	D^cv	SE^cv
mod0	313.0671	0.3775	0.0010
mod1	271.9198	0.3682	0.0010
mod2	238.8955	0.3659	0.0011
mod3	213.5469	0.3617	0.0011
mod4	190.8353	0.3594	0.0013
mod5	171.2580	0.3567	0.0013
mod6	136.5742	0.3565	0.0014

NOTE: $D=$ model deviance for a Poisson regression; $D^{\mathit{\text{cv}}}=$ casewise cross-validation deviance (eq. (14) Baldassarre et al. 2022); $S{E}^{\mathit{\text{cv}}}=$ standard error of D^cv (eq. (15) Baldassarre et al. 2022)

Fig. 2 Countries’ distribution map in each region (terminal node) $R_1,\dots ,R_6$, where R₁ is the reference region in the BTRT model.

Table 5 BTRT estimated coefficients.

	${\widehat{\lambda }}_{o_1,h}$	${\widehat{\lambda }}_{o_2,h}$	${\widehat{\lambda }}_{o_3,h}$	Interaction effect
${\widehat{\lambda }}_i$	13.15${}^{*}$ (5.33)	–33.63${}^{**}$ (11.83)	–50.09 (115.79)
${\widehat{\beta }}_{i,x1}$	–4.16 (2.77)	1.24 (4.62)	8.73 (4.54)
${\widehat{\beta }}_{i,x2}$	0.71 (0.93)	–5.28${}^{**}$ (1.94)	–2.99 (1.98)
${\widehat{\beta }}_{i,x3}$	5.45 (3.68)	20.95${}^{**}$ (7.56)	15.78 (8.15)
${\widehat{\beta }}_{i,x4}$	–16.68${}^{**}$ (5.58)	–5.02 (9.13)	–32.94${}^{*}$ (12.96)
${\widehat{\beta }}_{i,x5}$	14.89${}^{*}$ (5.43)	–11.30 (7.38)	5.96 (8.51)
${\widehat{\beta }}_{i,x6}$	–15.22${}^{*}$ (7.38)	–2.61 (10.94)	4.06 (12.10)
${\widehat{\beta }}_{i,x7}$	–10.12${}^{*}$ (4.48)	2.40 (10.70)	–15.69 (12.72)
${\widehat{\beta }}_{i,x8}$	–8.05${}^{*}$ (3.78)	16.25${}^{*}$ (6.72)	11.85 (7.51)
${\widehat{\beta }}_{i,x9}$	2.82 (3.39)	5.23 (5.50)	7.94 (6.69)
${\widehat{\beta }}_{i,x10}$	–4.54 (4.09)	16.92${}^{*}$ (7.60)	–0.19 (10.47)
${\widehat{\beta }}_{i,x11}$	0.33 (1.60)	7.91${}^{**}$ (3.05)	9.32${}^{*}$ (4.04)
${\widehat{\beta }}_{i,x12}$	–11.06 (7.49)	–28.28${}^{*}$ (12.56)	–51.68${}^{**}$ (16.17)
${\widehat{\beta }}_{i,x13}$	–6.54${}^{*}$ (2.65)	22.30${}^{***}$ (6.52)	24.17${}^{**}$ (7.61)
${\widehat{\beta }}_{i,x14}$	–25.63 (25.38)	6.32 (33.37)	137.84${}^{**}$ (53.16)
${\widehat{\beta }}_{i,x15}$	–3.98 (21.42)	–65.68 (38.44)	–45.98 (43.92)
${\widehat{\beta }}_{i,x16}$	10.26 (13.88)	20.80 (17.44)	5.51 (25.97)
${\widehat{\beta }}_{i,x17}$	–4.15 (2.63)	22.81${}^{**}$ (7.01)	33.23${}^{***}$ (8.64)
${\widehat{\beta }}_{i,R_2}$	–3.21${}^{*}$ (1.32)	0.83 (1.88)	14.30 (115.09)	$R_2=I(\text{EPI}\le 0.56,\text{Healthsp}.\le 0.06)$
${\widehat{\beta }}_{i,R_3}$	–2.25${}^{*}$ (1.06)	0.50 (1.42)	11.58 (115.09)	$R_3=I(0.56<\text{EPI}\le 0.7,\text{Healthsp}.\le 0.06)$
${\widehat{\beta }}_{i,R_4}$	–0.77 (1.32)	4.87${}^{*}$ (2.22)	19.19 (115.10)	$R_4=I(\text{EPI}\le .7,\text{Healthsp}.>.06,\text{Empl}.\text{rate}\le .51)$
${\widehat{\beta }}_{i,R_5}$	–3.24${}^{***}$ (0.96)	3.31${}^{*}$ (1.29)	6.00 (153.15)	$R_5=I(\text{EPI}\le .7,\text{Healthsp}.>.06,\text{Empl}.\text{rate}>.51)$
${\widehat{\beta }}_{i,R_6}$	7.93 (201.22)	–4.26${}^{*}$ (1.73)	8.35 (115.10)	$R_6=I(\text{EPI}>0.7,\text{Gross debt}\le 0.4)$

NOTE: Estimated coefficients associated to the objects taxes on income o₁, social security contributions o₂, taxes on property o₃, and taxes on goods and services o₄ (reference category, that is, coefficients equal to zero). Standard errors are in parenthesis and the stars $\mathrm{`}\mathrm{`}*$” associated with some estimated coefficients indicate they are different from zero with a p-value $\le 0.001$ (“$***$”), $\le 0.01$ (“$**$”), and $\le 0.05$ (“$*$”), respectively.

Table 6 Mean worth parameters π_i for each tax revenue category in each terminal node of the pruned regression trunk.

	πincome	πsocial	πproperty	πgoods
R2	0.04713	0.00000	0.00000	0.95287
R3	0.12403	0.00011	0.00001	0.87585
R4	0.96217	0.00000	0.00000	0.03783
R5	0.02498	0.00276	0.00000	0.97226
R6	1.00000	0.00000	0.00000	0.00000
R1	0.40127	0.39267	0.00000	0.20605

Notes: Each π_i corresponds to the average value of the worth parameters associated with each country in the specific terminal node.

Table 7 OLS estimated coefficients with interaction effects.

	Estimate	Std. Error	t value	Pr($>|$t$|$)
Intercepts	–2.10	0.21	–9.85	0.00
Employment rate	0.01	0.10	0.10	0.92
Government gross debt	0.03	0.04	2.06	0.04
Savings	–0.11	0.16	–0.68	0.50
Exports	0.11	0.21	0.52	0.60
Imports	–0.03	0.20	–0.42	0.68
Interests	–0.29	0.21	–1.39	0.17
Agriculture value added	–4.58	0.81	–5.62	0.00
Services	–0.22	0.14	–1.54	0.13
Final consumption	–0.20	0.14	–1.40	0.17
Bank NP loans	–0.27	0.20	–1.40	0.17
Claims centr gov	–0.16	0.07	–2.19	0.03
Gov consumption	0.23	0.57	0.41	0.68
EPI	0.35	0.08	4.32	0.00
Military spending	6.08	5.14	1.18	0.24
Education spending	2.84	0.83	3.42	0.00
Health spending	2.42	0.45	5.32	0.00
Other spending	2.80	0.12	24.17	0.00
Agriculture VA:Gov consumption	30.15	5.84	5.16	0.00
Agriculture VA:Military sp	190.34	36.34	5.24	0.00
Gov cons:Military sp	–8.01	27.54	–0.29	0.77
Agriculture VA:Gov cons:Military sp	–1313.86	260.58	–5.04	0.00

Fig. 3 Mean worth parameters π_i conditioned to the share of agricultural sector to GDP.

NOTE: The worth parameters are computed for $R_1,\dots ,R_6$ by fixing all covariates (except for the share of the agricultural sector) to the average level of the group. The order of graphs within the figure follows the trunk in Figure 1: R₂ top left, R₃ top right, R₄ center left, R₅ center right, R₆ bottom left, R₁ bottom right. The x-axis shows simulated values of the covariate agricultural share within the standardized range $[0,1]$. Finally, for each plot, the mean values of the covariates resulting from the splitting procedure of the regression trunk are reported. For the sake of clarity, the mean values of all the other country-specific covariates are not reported.

Table 8 Log-linear Bradley-Terry model without subject-specific covariates: results.

	Estimate	Std. Error	z value	Pr($>|$z$|$)
(Intercept)	3.61	0.05	67.24	0.00
o1	–0.20	0.07	–2.72	0.00
o2	–0.73	0.08	–9.24	0.00
o3	–1.26	0.08	–15.37	0.00
o4	0.00	0.00	0.00	0.00

Fig. 4 Output of BTtree.

NOTE: H is the number of countries, I = taxes on income, S = social security contributions, P = taxes on property, and G = taxes on goods and services. The plot shows the estimated worth parameters in both terminal nodes.

Baldassarre, A., Dusseldorp, E., D’Ambrosio, A., Rooij, M. d., and Conversano, C. (2022), “The Bradley–Terry Regression Trunk Approach for Modeling Preference Data with Small Trees,” Psychometrika, 88, 1443–1465. DOI: 10.1007/s11336-022-09882-6.

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