## 4 Model Formulation and Specification
### Aggregate Hourly Football Model Specifications  
The actual footfall is denoted by the variable \(Y_{2 st}\) for station \(s\) during the 1 h time period \(t\). The linear regression model is shown in Equation (6). The subscript \(s\) refers to the station location, whereas \(t\) corresponds to the hourly time periods of observed footfall. It is to be noted that since these variables are in vector form, the corresponding coefficients \(\gamma_{1}\) to \(\gamma_{6}\) also represent vectors. \[Y_{2 st}=\gamma_{0}+\gamma_{1}TP+\gamma_{2}TRO_{s}+\gamma_{3}MMI_{s}+\gamma_{4 }FLM_{s}+\gamma_{5}SOC_{s}+\gamma_{6}IP_{s}+\epsilon_{2 st} \tag{6}\]  
where the independent variables are grouped into the following set of variables:  
1. TP is a vector that captures the variation in the hourly footfall by dividing the 24 h period into five discrete time periods (before the AM peak, AM peak, afternoon off-peak, PM peak, and after the PM peak). 2. Transit operational characteristics (TROs): headway of metro, and fare/km for metro and bus. 3. Multimodal network integration (MMI): Intermodal connectivity variables, such as the no. of train routes passing through the station, as binary indicators for a hub or terminal station, the presence of an airport in the vicinity, and the quality of access roads, including the proportion of primary and residential roads. 4. First-Last-Mile (FLM) connectivity and walkability: FLM mode availability variables, such as no. of parking areas and cycles for hire near the station, and FLM service quality variables, such as out-of-vehicle travel time by train (waiting time, etc. ), as well as user perception of footpath usability and adequacy. 5. Socio-demographic and land-use (SOC): population and average income in the buffer area. 6. IP represents the vector of an instrumental variable based on the consideration probability and is explained, subsequently, in Equation (10). Besides the linear model, two non-linear models are also specified and evaluated, as shown in Equation (7), obtained using log transformation and Box-Cox transformation. In addition, two count models of footfall (Poisson and Negative Binomial) are also evaluated. The most suitable model is selected based on the goodness of fit, as well as the signs, magnitudes, and significance of coefficients of variables affecting footfall. \[{Y^{\prime}}_{2 st}=\left\{ \begin{array}{c}Y_{2 st},\ for\ linear\ regression\\ \frac{(Y^{\lambda}_{2 st}-1)}{\lambda},\ for\ box-cox\ transformation\\ \log(Y_{2 st}),\ for\ log-transformation\end{array} \right. \tag{7}\]  
For the above non-linear models in Equation (7), it is necessary to retransform the dependent back to the original scale to obtain unbiased predictions. For this purpose, [PERSON]'s smearing estimator technique is used for back transformation [37]. Potential endogeneity due to self-selection is captured by using the disaggregate consideration probability from the first model as an instrumental variable in the footfall model. This instrumental variable used in the hourly footfall model is derived by averaging the predicted probabilities of considering the train across all the respondents whose work or home location is within 1.5 km of the metro station (represented by \(\overline{P_{s}}\) in Equation (8) below). The number of individuals who are present in the buffer area of stations is represented by ns in the equation below. \[\overline{P_{s}}(Y_{1}=1)=\frac{\sum_{i=1}^{n_{s}}\delta_{si}*P(Y_{1i}=1)}{n_{s}} \tag{8}\]  
\[\delta_{si}= \begin{cases}1,\text{ }if\text{ }individual\text{ }belongs\text{ }to\text{ }the\text{ }in\text{ }\text{ }\text{fluence\ area\ of\ metro\ station\ s}\\ 0,\text{ }otherwise\end{cases}  \tag{9}\]  
The statistical tests for endogeneity (Durbin-Wu-Hausman) [38], instrument validity (Sargan's test) [39], and relevance were also carried out, which confirmed the presence of self-selection bias by neglecting the instrumental variable.