This paper calculates inter-district inequality among West Bengal’s 17 districts and then highlights the disparity in physical and social infrastructure among them. The latter traces the ranking of districts over time. Though this does not conclusively prove what the main determining factors for the movement of inequality are, a rank correlation analysis of per capita incomes with their physical and social infrastructure ranks gives sufficient hints about the causal relations between the two.

The authors are grateful for the comments of the participants, especially those of Anjan Mukherjee and Atul Sharma. The authors gratefully acknowledge computational help from Poulomi Roy.

Ajitava Raychaudhuri (ajitava1@gmail.com) and Sushil Kr Haldar (haldarsk2002@yahoo.co.in) are at the Department of Economics, Jadavpur University, Kolkata.

D

1 Aty A

it

• Hoover coefficient (HC) = 2∑ n | −− |

i=1 Ay A

tot tot

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INTER AND INTRA-STATE DISPARITIES

nn

• Gini (U) (unweighted) = 12 ∑∑| yi −yj |2ny i=1 j =1

nn

1 ij

• Gini (W) (population weighted) = ∑∑AA | yi −yj |

2y AA

i=1 j=1 tot tot

• Population-weighted coefficient of variation

1

1 ⎡n 2 At ⎤2

(Williamson index-WI) = (y −y)

⎢∑i ⎥

y ⎣i=1 Atot ⎦

1−e

⎡ n

⎡⎤ ⎤ 1

1 y

• Atkinson index (AT) = 1−⎢ t ⎥

∑⎢⎥

ny

⎢t=1 ⎣⎦ ⎥

⎣⎦

1 ⎡ 2 ⎤

⎛⎞2

⎢1 n ⎜At.y ⎟⎥

A

• Coulter coefficient (CC) = ⎢∑⎜ − i − t ⎟⎥ ⎢2 i=1 ⎜ Atot ⎟⎥ ⎢⎝Atoty ⎠⎥

⎣⎦

Here, Ai and Aj are the population in regions i and j respectively, and A the national population. Yi and Yj are the per capita

tot

income of regions i and j respectively, Xi refers to mean per capita income of region i, μ is a state’s per capita mean income, and Pi refers to the population in region i. And n is the number of regions and e stands for inequality aversion parameter; e is the standard Atkinson’s preference for equality parameter whose value is taken to be 0.5 for our calculation.

According to neoclassical growth theories, on account of competitive forces and inter-regional migrations, regional inequalities will gradually vanish because factor prices will tend to be equalised (Solow 2000). The new growth theory, led by Lucas and Romer, combined with the new theories of economic geography propounded by Krugman, Venables and Fujita, tend to show the opposite (Barro and Sala-i-Martin 1995). This is because for Lucas human capital is the driving force whereas for Romer, it is research and development (R&D). All regions may not be blessed with these factors evenly. The new economic geography utilises the scale economies to justify the advantages of economic concentration outweighing the effects of congestion. One may add the uneven spread of physical infrastructure as exacerbating the problems just mentioned.

In this paper, we will calculate inter-district inequality according to the measures discussed above and then highlight the disparity in physical and social infrastructure among the districts. The latter will trace the ranking of districts over time. However, this will not conclusively prove what the main determining f actors for the movement of inequality over time are. But a rank correlation analysis of districts’ per capita incomes with their physical and social infrastructure ranks will give sufficient hint about the causal relations between the two.

As far as infrastructural variables are concerned, one has to note that there are a large number of variables in both categories, especially in physical infrastructure. However, we have to reduce the variables to an aggregate index so as to relate it to other measures. For this we have used the Principal Component Analysis (PCA) technique, with some innovation to calculate the weights since otherwise they would make no intuitive economic sense. The steps we have followed to calculate PCA are described below.

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Before running the PCA, the following procedure is adopted to convert raw data into a normalised form. This is done to make the raw data unit free as well as to get the relative position of each district in respect of infrastructure. First, the best and worst values in a particular indicator (infrastructure) are identified. In case of a positive indicator, the highest value is treated as the best value and the lowest, as the worst value. In the present case, we consider all the positive indicators.

Once the best and worst values are identified, the following formula is used to obtain normalised values:

NVij = 1 –[best Xij – observed Xij]/R, where R=best Xij – worst Xij, i= ith observation and j=jth district

Normalised values always lie between 0 and 1.

Algebraically, PCs are particular linear combinations of the p random variables X1, X2,…….X. Following Johnson and Wichern

p

(2006), PCs depend solely on the covariance matrix (Σ) or correlation matrix (ρ) of X1, X2,…….X Let the random vector X ′=[ X1,

p.

X2,…X] have the covariance matrix Σ with eigenvalues

p

λ≥λ≥……….≥λ≥0

12p

Consider the linear combinations:

′

Y1=a1 X=a11 X1 + a12 X2…………………… a1p X

p

′

Y2=a2 X=a21 X1 + a22 X2…………...……… a2p X

p

.. .. …. ………… …….

(A) .. .. …. ………… .……

′

Y=aX=ap1 X1 + ap2 X2…………....……… a X

pp ppp

Now using the property of mean vector and variancecovariance matrix (Σ), we have:

′

Var(Yi)= ai Σai ……….(B) i=1,2…………..p

′

Cov(Yi, Yk)= ai Σak ………..(C) i,k=1,2…..……p

The PCs are those uncorrelated linear combinations Y1, Y2….YP whose variances in (B) are as large as possible. The first PC is the linear combination with maximum variance that is, it maximises

′′

Var(Y1)= a1 Σa1. It is clear that Var(Y1)= a1 Σa1 can be increased by multiplying any a1 by some constant. To eliminate this indeterminacy, it is convenient to restrict attention to coefficient vectors of unit length.

We therefore define:

′′

First PC = linear combination a1 X that maximises Var (a1 X)

′

subject to a1 a1=1

′′

Second PC = linear combination a1 X that maximises Var (a2 X)

′ ′′

subject to a2 a2=1 and Cov (a1 X, a2 X )=0. Similarly at the i-th step,

′

Similarly at the i-th step, i-th PC = linear combination ai X that max

′′ ′′

imises Var (ai X) subject to ai ai=1 and Cov (ai X, ak X)=0 for k<i.

Once the normalised values are obtained for all the infrastructural variables across the districts, the next step is to assign factor loadings and weights. The PCA is used to compute the factor loadings and weights of these indicators (infrastructures).

To understand the process, let us take the following example.

Table A: Description of the Indicators

MI = Medical institution per 10,000 population

MB = Medical beds per 10,000 population

FWC = Family welfare centre per 10,000 population

SP = Number of institutions (schools including primary, secondary and higher secondary) per 10,000 population

TS = School teacher ratio

TP = Number of teachers per 1,000 population

To find out the social infrastructure index across the 15 districts Where I is the index, Xi is the i-th Indicator; Lij is the factor of West Bengal in 1990-91, we consider the following indicators loading of the i-th variable on the j-th factor; Ej is the Eigen value (infrastructures). of the j-th factor. The following is an example for Barddhaman

The normalised values of the indicators across the districts are district. given in Table B. The total weight of the indicators is 14.4587.

2.8439 3.1598 3.1328 1.1952 2.0364 2.0905 14.4587= Birbhum 0.208 0.202 0.303 0.422 0.324 0.465

0.3641 Bankura 0.36 0.39 0.338 1 0.225 1 Midnapur 0.137 0.104 0.175 0.526 0.286 0.497 In the same way, the infrastructure indices of the rest of the

Howrah 0.297 0.301 0.178 0.05 1 0.766 districts have been estimated. The explanatory power of the prin-Hoogli 0.135 0.293 0.128 0.164 0.817 0.635 cipal components through the PCA taken to calculate the social 24 Parganas© 0 0.05 0.07 0.072 0.621 0.266

Table D: Number of Principal Components with Variance Explained in each Dinajpur© 0.304 0 0.082 0.248 0.24 0.076 Infrastructure Sub-Group

Malda 0.243 0.075 0 0.155 0.441 0.183 Dimension of Name of the Indicators Number of Principal Percentage Infrastructure Components (PCs) Variance Explained

Physical Infrastructure V1, V2, V3, V4, V5, 4 PCs for 1990-91 81.983+ Running the PCA in the software package SPSS, we have identi-(including Financial) V6, V7, V8 and V9 4 PCs for 1994-95 82.391++ 3 PCs for 2000-01 76.708+++

Note: *Cumulative value (that is first PC explains 47.035%, second PC explains 25.60% and values above one varies from data to data. Incidentally, the three third PC explains 18.781%), **Cumulative value (that is, first PC explains 46.291%, second PC explains 34.334%), ***Cumulative value (that is first PC explains 63.208%, second PC components explain 91.4% variance of the variables included in explains 16.777%), ****Cumulative value (that is first PC explains 59.734%, second PC explains 22.560%), + Cumulative value (that is first PC explains 38.666%, second PC explains 18.202%,

PC explains 26.007%, third PC explains 12.78% ). The variables like MI, MB, FWC, SP, TS and TP for Variables Principal Component Factor social category and the variables like V1,……..V9 for infrastructure group are defined earlier.

MB .909 .235 .200 Let us first look at the district income and population profile of FWC .811 .346 -.281 West Bengal. One may note that there are currently 17 districts in SP 8.670E-02 .780 -.527

TP .163 .959 .143 Table 1: District Per Capita Income over time for West Bengal (in Rs) Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser District 1990-91 1994-95 2000-01 2004-05 Normalisation. Rotation converged in 4 iterations. Then we have multiplied 1st Eigen value

(2.822) with 1st Extracted Component Column (0.928, 0.909, 0.811, 0.0867, -0.0239, 0.163), 2nd Barddhaman 5,972.47 8988.58 17537.98 25,397.78 Eigen value (1.536) with 2nd Component column

(-0.0997, 0.235, 0.346, 0.780, -0.018, 0.959) and 3rd Eigen value (1.127) with 3rd Component column (-0.0639, 0.200, -0.281, -0.527, 0.985, 0.143). We have considered absolute values Bankura 4,608.68 7,369.1 15,741.64 19,994.67 (irrespective of sign, negative values are treated as positive). Finally, we have summed up the

2.822 x 0.928 + 1.536 x 0.0997 + 1.127 x 0.0639 = 2.8439 (Here the negative values of the 2nd and Midnapore (west) 4,446.46 7,413.39 15,526.01 22,458.08 3rd component matrix are treated as positive).

Murshidabad 4,153.34 6,493.98 13,392.39 18,162.13 MB = 3.1598 Dinajpur (N) 6,027.21 5,366.86 11,182.86 13,817.83 FWC = 3.1328 Dinajpur (S) 6,027.21 6,522.49 14,579.19 18,461.43 Malda 3,830.43 5,913.91 14,777.2 19,763.99 Jalpaiguri 4,776.43 6,822.68 16,749.07 21,389.56

Purulia 3,613.32 6,224.61 13,044.67 16,901.32 I = Σ Xi ( Σ|Lij|.Ej)/Σ( Σ|Lij|.Ej) West Bengal 4,672.81 7,435.53 16,072.26 22,496.62 i=1 j=1 i j=1 Source: Economic Review, various years, Government of West Bengal.

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of 24 Parganas, Midnapore and Dinajpur over time. So there are some problems regarding population and income data for comparison purposes. Second, the inter-decadal population figures are interpolated using the decadal growth rates. Third, the figures for district per capita income as well those for infrastructure

Table 2: District Population Figures for West Bengal

District 1991 1995 2001 2005

Barddhaman 60,50,605 64,83,223 69,19,698 74,16,532

Birbhum 25,55,664 27,83,118 30,12,546 32,81,867

Bankura 28,05,065 29,97,211 31,91,822 34,10,461

Midnapur@ 83,31,912 89,81,801 96,38,473 103,94,129

Howrah 37,29,644 40,01,908 42,74,010 45,86,012

Hoogli 43,55,230 46,97,115 50,40,047 54,36,194

24 Parg (N) 72,81,881 81,04,733 89,30,295 99,41,204

24 Parg (S) 57,15,030 63,09,393 69,09,015 76,30,661

Nadia 38,52,097 42,27,676 46,03,756 50,52,852

Murshidabad 47,40,149 53,01,856 58,63,717 65,58,567

Dinajpur (N) 18,97,045 21,69,270 24,41,824 27,92,469

Dinajpur (S) 12,30,608 13,66,590 15,02,647 16,68,764

Malda 26,37,032 29,62,705 32,90,160 36,97,646

Jalpaiguri 28,00,543 31,01,601 34,03,204 37,69,388

Darjeeling 12,99,919 14,52,659 16,05,900 17,94,914

CBR 21,71,145 23,24,210 24,78,280 26,53,618

Purulia 22,24,577 23,79,185 25,35,233 27,12,192

West Bengal 6,80,77,965 6,96,44,254 8,02,21,171 8,27,97,470

@ Denotes the combined population of Midnapore district. Source: Census of India, 1991 and 2001.

have been collected from state government sources, mainly A nnual Economic Reviews published by the department of finance as well as State Statistical Abstracts published by the B ureau of Applied

Table 3: Coefficient of Variation Measures

Economic Research
Year	CV	CD
of the West Bengal	1990-91	0.1858	0.1535
government. It is to	1994-95	0.1374	0.1010
be noted that in	a	2000-01	0.1246	0.0992
couple of cases the	2004-05	0.1652	0.1338

data for the year concerned was not available and are proxied by the year i mmediately preceding or following that year. This is mainly for some infrastructural variables which should not create a significant difference to our result.

Let us first present the data for district per capita income and popu lation for four years, 1990-91, 1994-95, 2000-01 and 2004-05.3

Table 2 gives the population figures for the districts. The intercensus year data are calculated by the decadal growth rates, e xcept for 2004-05, which is extrapolated.

We have estimated some of the inequality measures using the STATA 9.0 and DAD software, as well as by calculating some others manually. The results are shown in Table 3. The coefficient of v ariation declined steadily for the initial years and then increased rapidly in the last four or five of our sample years. It will be further strengthened by the results of the other indices which show very similar results.

Table 3 confirms the results obtained from coefficient of variation – the inequality among districts has taken a U-turn after falling uninterruptedly during the 1990s. This result is interesting since the beginning of our analysis coincided with the opening

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up of the Indian economy. Somewhat contrary to what Sen and Himanshu (2004) have found, inter-district inequality of income declined in West Bengal if we go by any accepted index, although Sen and Himanshu concentrated on monthly per capita expenditure. As far as income is concerned, inequality did decline. But the inequality started widening from the beginning of the 21st century. All the indices show a rising inequality among districts. The rise in the coefficient of variation measures clearly exhibits more volatility in income among districts – some of the richer districts have become wealthier while the poorer have increasingly fallen behind. This asymmetry of i nter-district inequality movement over time in West Bengal is interesting. It is difficult to u nderstand the case prima facie – one needs more in-depth study to unravel the truth behind such peculiarity.

What we have done next is to initiate some investigation into what might have caused this. Although this study is part of a larger planned project and we will be able to test the robustness of our findings later, we have tried to analyse the same from the angle of infrastructural development in the districts. As is well

Table 4: Inequality Indices of District Per Capita Income

Year	Gini	Gini (W)	Theil (0)	Theil (1)	Atkinson	WI	HC	CC
1990-91	0.1057	.081	0.0179	0.012	0.0080	.161	.066	.036
1994-95	0.0756	.082	0.0096	0.009	0.0042	.142	.061	.031
2000-01	0.0586	.051	0.0055	0.006	0.0037	.095	.04	.018

2004-05 0.0825 .085 0.0109 0.011 0.0064 .142 .062 .029

The Gini and Theil are calculated by Bootstrap method using STATA 9.0 and Atkinson is calculated by DAD assuming inequality aversion parametre ε = 0.5. The other measures are estimated manually.

known, physical infrastructure is not only a strong complement to physical capital but also a big obstacle to full utilisation of existing resources. Similarly, lack of proper social capital prevents appropriate skill development among the labour force, thereby making the human capital qualitatively weak and less productive. What we will do here is understand the movement of ranking of the districts in terms of physical and social infrastructure indices and get the correlation of the same with the rankings of the districts in terms of district per capita income. We have also created inequality indices for social and infrastructure indices and have compared their ranks with the ranks of district incomes for the years under consideration.

B Physical Infrastructural Variables and Theil indices of inequality as described for the income ine-V1 = Road length per sq km. quality case. This will help us compare the concurrent movement V2 = Post and telegraph offices per 1,000 population. of inequality indices for the two variables. Also, we will report V3 = Number of banks per 1,00,000 population. the rank correlation coefficients between income and SII or PII, so V4 = Percentage of villages electrified. as to get an idea about the possibility of district disparity in indi-V5 = Number of minor irrigation units per thousand net area sown. ces leading to disparity in income. V6 = Number of registered factories per 10,000 population. Table 7: Social and Physical Infrastructure Index V7 = Number of primary agricultural credit society per 10,000

population. Social Infrastructure Index (PCA Result) Barddhaman 0.3641 3 0.4358 4 0.3734 7 0.4331 6

Hoogli 0.3007 7 0.3949 6 0.3523 8 0.3231 10Table 5: Weights of the Variables in Respect of Social c alculated. 24 Par© 0.1294 15 0.2189 13 0.2431 11 0.2657 13Infrastructure Estimated by Using PCA

From Table 5 Nadia 0.2801 8 0.2681 12 0.1871 14 0.2758 12

among social infra- Murshid 0.1463 13 0.1682 15 0.1811 15 0.2577 14MI 2.844 2.734 3.167 3.689

structural variables, Dinajpur© 0.1433 14 0.2871 11 0.3011 10 0.2547 15MB 3.159 2.645 2.202 1.543

CBR 0.2066 11 0.3293 7 0.3371 9 0.4548 5TP 2.090 2.742 2.742 2.140 and higher second

Purulia 0.3115 5 0.5413 2 0.6807 2 0.6034 2 Total (∑) 14.458 15.937 18.096 16.70 ary schools), TS

Physical Infrastructure Index (PCA Result)(teacher student ra- Barddhaman 0.2887 10 0.432 5 0.5369 2 0.4492 4

Table 6: Weights of the Variables in Respect of Social Infrastructure Estimated by Using PCA tio) and FWC (family Birbhum 0.4318 6 0.5226 2 0.4356 4 0.4622 3Variables 1990-91 1994-95 2000-01 2004-05 welfare centres) Bankura 0.5923 2 0.2695 13 0.3316 9 0.3002 9

V1 3.11 3.03 2.80 4.08 play a significant Midnapur© 0.3073 8 0.3011 11 0.3653 7 0.3825 6 V 2.57 2.68 3.69 3.27 role over the years Howrah 0.7214 1 0.5294 1 0.6004 1 0.6558 1

ical infrastructure Darjeeling 0.4503 5 0.3972 7 0.3357 8 0.4039 5index (PII), it is CBR 0.3374 7 0.3977 6 0.4329 5 0.3744 7found that road length per sq km, working capital per primary Purulia 0.2202 15 0.3574 9 0.3253 10 0.2778 10 © implies they are combined values for districts which were bifurcated later.

agricultural credit society and working capital per member of primary agricultural credit society play very important roles. From Table 8, it is clear that the physical inequality indices moved Thus financial devolution to the disaggregated level with reduc-almost the same way as the income inequality indices. They declined tion in disparity as the goal must have these variables in mind. monotonically from 1990-91 to 2000-01 but then rose in 2004-05.

The above might create impression that the other variables like But social infrastructure took a different course. The index fell medical beds in case of social variables and irrigation and banks b etween 1990-91 and 1994-95, then rose in the next five years but in the case of physical variables are not important for district-declined again in 2004-05. Thus physical infrastructure seems to have level economic well-being. This is not true because the above data a concurrent movement with Table 8: Inequality Indices for Infrastructure

reflect the variables which are most important in causing inter-income disparity while social

district disparity in income and not the ones which primarily de-infrastructure does not have a

Physical Infrastructuretermine growth of income in the districts. We are concerned here similar movement. 1990-91 0.213 0.072with the issue of disparity and medical beds or minor irrigation A rank correlation exercise 1995-96 0.157 0.038projects are not the variables which are available in significantly also supported the above con- 2000-01 0.145 0.033 2004-05 0.189 0.056

different quantities across districts. They may be important clusions. The following shows

d eterminants for the growth of district income but are not that the Spearman rank correla-

significant in causing income disparity among the districts. tion between physical infra

1990-91 0.274 0.132From Table 7, one gets a bird’s eye view of the rank and move-structure and per capita dis- 1995-96 0.228 0.087ment of SII and PII. However, such spread-out tables cannot con-trict income for the four years 2000-01 0.256 0.103 2004-05 0.191 0.058

vey things in a summary fashion. So we have calculated the Gini under consideration.

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As Table 9 shows, the rank correlation coefficient b etween physical infrastructure and district income shows s ignificant dependence only in 2004-05, which

Table 9: Rank Correlation between PII and corroborates the fact that the Per Capita District Income

rise in inequality in the districts may have been precipitated by a similar movement in the physical infrastruc-	Year 1990-911994-952000-01	Spearman Coefficient 0.2500 0.1571 0.2179	Prob > |t| for H0 = VariablesAre Independent 0.3688 0.5760 0.4354
ture. Such a movement is	2004-05	0.5500	0.0337
not visible for the social in
frastructure though.

The findings are represented in Charts 1-3 which show it in unambiguous terms.

Chart 1: Movement of Income Inequality Indices

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00 1990-91 1994-95 2000-01 2004-05

Values

Chart 2: Movement of Physical Infrastructure Index Inequality Coefficients

(Gini and Theil (0) Inequality Indices for PII)

0.157 0.145 0.189 0.072 0.038 0.033 0.056 Gini (PII) TI(0) PII

1990-91 1995-96 2000-01 2004-05

Chart 3: Movement of Social Infrastructure Index Inequality Coefficients

(Gini and Theil (0) Inequality Indices for SII)

Gini (SII) TI(0) SII 0.228 0.274 0.256 0.191 0.058 0.103 0.087 0.132

1990-91 1995-96 2000-01 2004-05

Charts 1 to 3 clearly show the collinear movement of income inequality across districts and inequality of physical infra structure index inequality coefficients over the districts. Such a c ollinearity is absent in the case of the social infrastructure index.

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Conclusions

This paper has investigated the nature and possible reasons for disparity among West Bengal’s districts in the last 15 years or so. The findings clearly show a rising disparity among the districts in the first half of the present decade after a continuous decline in the last decade of the last century. A similar and concurrent movement is noticed in the composite physical infrastructure index of the districts although the social infrastructure index of the districts does not show a similar movement. The physical infrastructure plays an important role in f acilitating production and sale of output, while social infrastructure helps to build human capital. Perhaps physical infrastructure has a greater influence on income distribution among West Bengal’s districts than social infrastructure. We are currently looking into the causality aspect among growth, income distribution and infrastructure indices and perhaps there the social infrastructure plays an important role in the growth prospect of West Bengal’s districts.

Notes

1 Recent empirical findings give us two distinct types of results of convergence of income levels across the states. Cashin and Sahay (1996) find weak and conditional convergence of income levels across the 20 states for 1961-91. Considering a period of time from 1970 to 1994 and taking 17 states, Nagaraj, Varoudakis and Veganzones (2000) find conditional convergence. For a sample of 19 states during 1971-96, Aiyar (2001) finds conditional convergence. Singh and Srinivasan (2002) do not find any clear evidence of conditional convergence or divergence. Their study is based on 14 major states for 1991-99. In contrast with these findings, Rao, Shand and Kalirajan (1999) find absolute and conditional divergence across 14 states for the period of 1965-95. The same is observed by Sachs, Bajpai and Ramiah (2002) for 1980-98. Recently, Nayyar (2008) in a panel study of 16 major states for 1978-2003 does not find any convergence.

2 The PCA technique reduces the dimensionality, but the numbers of principal components to be taken into consideration for creating the SII or PII depends on how much variability they explain of the variables used to create the principal components.

3 The reason for taking the data from 1990-91 is that some of the district-level data like irrigation, cooperative credit society lending, health and school-level indicators are not available in a comparable format before 1990-91. Also, data are taken in five-year intervals from then onwards because social infrastructure variables (like school and health indicators) are not available for each year. Similarly some of the physical infrastructure indicators (like post and telegraph offices and road length) do not show any variation within the five years. To keep the set of included variables large as well as comparable (without using techniques like interpolation), five-year intervals appear very reasonable.

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