Spatial Modelling Of Origin-Destination Commuting Flows In Switzerland Thomas Schatzmann;

Abstract

31 Objective and methodology 32 This paper presents a direct modelling approach for origin-destination public transportation commuting 33 flows with possible endogenous regressors for the case of Switzerland. Its purpose is to improve the 34 gravity modelling approach for OD flows by applying a spatial autoregressive regression model and 35 testing different spatial weighting schemes. To the best of our knowledge, there has been no prior 36 application of such advanced models in the context of transport demand modelling for public transport. 37 Methodologically, in the first step a gravity model is developed and tested for the presence of spatial 38 autocorrelation in its residuals. Subsequently, variants of a spatial lag model with different spatial 39 weighting schemes are developed. Furthermore, we test a variable based on mean income 40 differences on its ability to describe interregional demand patterns. In addition, we treat for the 41 endogenous nature of the newly constructed variable. We are also testing its ability to serve as the 42 basis for the construction of the spatial weight matrix, thus replacing the commonly used travel time / 43 distance metric. On the modelling front, we use an Ordinary least squares (OLS) estimator for the 44 gravity model, while a Generalized Method of Moments and Instrumental Variable (GMM/IV) (IV) 45 estimator for the spatial models is employed in order to obtain unbiased and consistent parameter 46 estimates. Last, we evaluate various models’ goodness-of-fit measures and in-sample predictive 47 accuracies by comparing among each other as well as to those of a state-of-the practice transport 48 model (as provided by national spatial planning bureau (NPVM)). This comparison can allow us to 49 draw solid conclusion with respect to the suitability of the presented method for predicting commuting 50 flows.

52 Case study 53 In brief, a case study for public transport commuting flows in Switzerland is designed to illustrate the 54 concept of OD flow modelling, based on travel-to-work trips data from the Federal Census of 2000. 55 The data cover 2896 Swiss municipalities and contains over 250’000 observations on their initial form. 56 However, it does not fill the whole flow matrix that contains 28962 = 8,386,816 flows. For the remaining 57 OD pairs we assume zero-valued travel flows. An important aspect is the issue with how to deal with 58 zero flows. A large fraction of zero-valued OD flows would definitely point towards a Poisson or a

59 (zero-inflated) negative Binomial interaction model. However, neither a Poisson nor a negative 60 Binomial spatial autoregressive regression model for OD flows has been developed so far. We include 61 income differences between Swiss communes as an explanatory variable in our models, since a 62 higher income gives incentives to commute. In conclusion, we filter the initial flow matrix for inter63 communal travel trips, income data available only in 1595 communes and all zero flows, which gives a 64 final sample size of 46,659 OD flows. Clearly, this is a limitation of our modelling approach but 65 nevertheless the findings can be of apparent value for pointing directions. 66 Modelling commuting behaviour requires a set of relevant explanatory variables that capture 67 the characteristics of origins and destinations, along with the mechanisms that generate the trips 68 among them. The dependent variable, inter-communal travel flows, is regressed on several 69 independent variables obtained or derived from the 2000 Federal Census, the Swiss national transport 70 model ARE (2005), and the Institute for Transport Planning and Systems (IVT) of ETH Zurich. We use 71 the following variables in our framework, which are also common in explaining public transport 72 demand in the literature (e.g. LeSage and Thomas-Agnan, 2015; Farmer, 2011; Axhausen et al., 73 2015).

75 Table 1: Model variables

Statistic Definition Flow Av. daily flows Network dist. minutes Income diff. rel CHF (in 1,000) Population (o) # inhabitants Jobs (d) # jobs Pop. density (d) # pop. / area (in km2) Job density (o) # jobs / area (in km2) Pop. accessibility (d) # accessible pop. Job accessibility (o) # accessible jobs Car (o) # cars / pop. Car (d) # cars / pop. Jobs3rd (o) # jobs in 3rd sector / # jobs Jobs3rd (d) # jobs in 3rd sector / # jobs Workers (o) # workers3 / pop. Workers (d) # workers3 / pop.

Note: (o),(d) = at origin, at destination municipalities

78 The starting point is a log form least-squares gravity model for OD flows in the form of

𝑖𝑛𝑐𝑑 − 𝑖𝑛𝑐𝑜 80 log(𝑦) = 𝛼 log(𝑙𝑁 ) + 𝛽𝑜 log(𝑋𝑜 ) + 𝛽𝑑 log(𝑋𝑑 ) + 𝛿 ( ) + 𝛾 log(𝑔) + 𝜖, 𝑖𝑛𝑐𝑜

82 where Xo and Xd are characteristics of origins and destinations, g denotes the network distance and 83 ((𝑖𝑛𝑐𝑑 − 𝑖𝑛𝑐𝑜 )⁄𝑖𝑛𝑐𝑜 ) reflects the relative difference of income between destination and origin 84 municipalities. Estimation results of the gravity model are showed in Table 2. All parameters are highly 85 significant except those of the share of 3rd sector jobs at origins and the share of cars per origin 86 municipality having p-values lower than 5% and 1% respectively. The network distance decay 87 parameter (-1.537) is within the expected range for commuting patterns, in accordance with previous 88 studies. All other explanatory variables have a much weaker impact on the dependent variable, but 89 this finding is in line with the expectations of existing literature (LeSage and Thomas-Agnan, 2015; 90 Farmer, 2011). Income differences between destinations and origins have a significant and positive 91 effect on travel-to-work trips and should be interpreted as an elasticity, since relative differences are 92 used. This intuitively makes sense. By applying Moran’s I tests we find that the residuals of the 93 aspatial model indeed exhibit remaining spatial dependence and thus justify the need for spatial 94 models. Spatial autoregressive models (SAR) are typically written as

96 𝑦 = 𝛼 𝑙𝑁 + 𝜌𝑖 𝑊𝑖 𝑦 + 𝛽𝑜 𝑋𝑜 + 𝛽𝑑 𝑋𝑑 + 𝜖, 𝑤𝑖𝑡ℎ 𝑖 = 𝑜, 𝑑.

98 where the weights for W i are defined as:

−1 1 𝑡𝑟𝑎𝑣𝑒𝑙𝑡𝑖𝑚𝑒𝑖𝑗 100 𝑁𝑒𝑡𝑤. 𝑑𝑖𝑠𝑡. 𝑤𝑒𝑖𝑔ℎ𝑡𝑠: 𝑤𝑖𝑗 = , 𝐸𝑐𝑜𝑛. 𝑑𝑖𝑠𝑡. 𝑤𝑒𝑖𝑔ℎ𝑡𝑠: 𝑤𝑖𝑗 = ( ) 𝑡𝑟𝑎𝑣𝑒𝑙𝑡𝑖𝑚𝑒𝑖𝑗 exp(((𝑖𝑛𝑐𝑑 − 𝑖𝑛𝑐𝑜 )⁄𝑖𝑛𝑐𝑜 ))

102 As it can be seen in Table 2, SAR models relying on origin- and destination-centric network and 103 economic distance weights show positive influence of neighbouring communes on travel-to-work trips. 104 Rho is higher than 1, which is an artefact of using the min-max approach for the spatial weights when 105 building W i, i = (o,d) instead of classic row-normalization (Kelejian and Prucha, 2010). In the transition 106 from the gravity model to the SAR models, variables Car (o), Car (d), and Jobs3rd (o) are not 107 statistically significant anymore and the impact of network distancebecomes smaller. Interestingly, rho 108 for the SAR model relying on economic distance weights has a bigger impact compared to the network 109 distance weighted SAR. It has to be emphasized that parameter estimates of spatial autoregressive 110 regression models can not be interpreted as simple elasticities as in the gravity model, since spatial 111 spillovers complicate the task of interpreting estimates from these models in a direct way. 112 Furthermore, the spatial models yield a higher goodness-of-fit measure than the gravity model. Finally, 113 we employ an IV regression framework to test the endogeneity of income and using a set of valid 114 instruments (in line with Sarlas and Axhausen,2017). Note that pseudo R2 values must be treated with 115 caution, as they are not equivalent to OLS-based R2 measures.

117 Table 2: Gravity model and spatial autoregressive models estimates

Dependent variable: log(commuting flows) Gravity model (OLS) SAR (GMM / 2IV) SAR (GMM / 2IV) Network distance weights Econ. distance weights Estimate Sign. Estimate Sign. Estimate Sign. (Intercept) 4.443 *** 5.765 *** 5.788 *** log(Netw. distance) -1.537 *** -1.250 *** -1.254 *** Rel. Income diff. 0.085 *** 0.047 ** 0.041 ** log(Jobs) (d) 0.473 *** 0.307 *** 0.308 *** log(Pop. density) (d) 0.030 *** 0.036 *** 0.036 *** log(Pop. access.) (d) -0.176 *** -0.207 *** -0.206 *** log(Jobs3rd) (d) 0.102 *** 0.082 *** 0.082 *** log(Car) (d) -0.071 *** -0.007 -0.006 log(Workers) (d) 0.665 *** 0.409 *** 0.417 *** log(Population) (o) 0.440 *** 0.359 *** 0.358 *** log(Job density) (o) -0.043 *** -0.042 *** -0.042 *** log(Job access.) (o) -0.180 *** -0.239 *** -0.239 *** log(Jobs3rd) (o) -0.027 ** -0.019 -0.018 log(Car) (o) -0.023 * -0.003 -0.002 log(Workers) (o) 0.365 *** 0.249 *** 0.254 *** rho 2.387 *** 2.704 *** R2 0.5177 Pseudo adj. R2 0.5898 0.5893 HC robust std. errors yes yes yes Note:

120 The resulting in-sample predictive accuracies outperform those from the current NPVM for different 121 accuracy measures, as can be seen in Table 3.

123 Table 3: In-sample predictive accuracy measures

RMSPE RMdSPE MAPE MdAPE SMAPE SMdAPE NPVM 87.50 6.49 271.76 64.92 74.12 64.24 Gravity model (OLS) 131.20 68.76 913.60 687.62 143.36 154.94 SAR (GMM / 2IV); Netw. dist. weigths 6.81 4.75 51.53 47.52 62.59 55.22 SAR (GMM / 2IV); Econ. dist. weigths 6.40 5.13 51.99 51.34 68.96 63.03