Introduction
Formulation of hypotheses
Cost Damping
Operating on | Entire function | Separate components |
---|---|---|
Fixed function ‘differential scaling’ | I | II |
Dependent on costs ‘transformed variables’ | III | IV |
Interpretation for ticket sales data analyses
-
the probabilities p j|m and p mj are so small they can be neglected (or p mj is small and θ is close to 1) and
-
we neglect also any difference between the distributions of the error term.
Model formulation for analysis
Type | Constraints | |
---|---|---|
0 | log T = B + λ
1
c + λ
2
t
|
λ < 0 |
I | log T = B + (λ
1
c + λ
2
t)d
−α
|
λ < 0 0 < α < 1 |
II |
\(\log T = B + \lambda_{1} cd^{{ - \alpha_{1} }} + \lambda_{2} td^{{ - \alpha_{2} }}\)
|
λ < 0 0 < α < 1 |
III | log T = B + λ
0(c + λ
2
t)
α
|
λ
0 < 0, λ
2 > 0 0 < α < 1 |
IIIA | log T = B + λ
0 log (c + λ
2
t) |
λ
0 < 0, λ
2 > 0 |
IV |
\(\log T = B + \lambda_{1} c^{{\alpha_{1} }} + \lambda_{2} t^{{\alpha_{2} }}\)
|
λ < 0 0 < α < 1 |
IVA | log T = B + λ
1 log c + λ
2 log t
|
λ < 0 |
IVB |
\(\log T = B + \left( \begin{aligned} \lambda_{1} \alpha_{1} c + \lambda_{1} q_{1} \left( {1 - \alpha_{1} } \right)\log \,c \hfill \\ + \lambda_{2} \alpha_{2} t + \lambda_{2} q_{2} \left( {1 - \alpha_{2} } \right)\log \,t \hfill \\ \end{aligned} \right)\)
|
λ < 0 0 < α < 1
q constant > 0 |
Type | Cost elasticity | Time elasticity | Value of Time ν
|
---|---|---|---|
0 |
λ
1
c
|
λ
2
t
|
\({\raise0.7ex\hbox{${\lambda_{2} }$} \!\mathord{\left/ {\vphantom {{\lambda_{2} } {\lambda_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda_{1} }$}}\)
|
I |
\({\raise0.7ex\hbox{${\lambda_{1} c}$} \!\mathord{\left/ {\vphantom {{\lambda_{1} c} {d^{\alpha } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${d^{\alpha } }$}}\)
|
\({\raise0.7ex\hbox{${\lambda_{2} t}$} \!\mathord{\left/ {\vphantom {{\lambda_{2} t} {d^{\alpha } }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${d^{\alpha } }$}}\)
|
\({\raise0.7ex\hbox{${\lambda_{2} }$} \!\mathord{\left/ {\vphantom {{\lambda_{2} } {\lambda_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda_{1} }$}}\)
|
II |
\({\raise0.7ex\hbox{${\lambda_{1} c}$} \!\mathord{\left/ {\vphantom {{\lambda_{1} c} {d^{{\alpha_{1} }} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${d^{{\alpha_{1} }} }$}}\)
|
\({\raise0.7ex\hbox{${\lambda_{2} t}$} \!\mathord{\left/ {\vphantom {{\lambda_{2} t} {d^{{\alpha_{2} }} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${d^{{\alpha_{2} }} }$}}\)
|
\({\raise0.7ex\hbox{${\lambda_{2} }$} \!\mathord{\left/ {\vphantom {{\lambda_{2} } {\lambda_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda_{1} }$}}d^{{\alpha_{1} - \alpha_{2} }}\)
|
III |
αλ
0
cG
α−1
|
αλ
0
λ
2
tG
α−1
|
λ
2
|
IIIA |
\({\raise0.7ex\hbox{${\lambda_{0} c}$} \!\mathord{\left/ {\vphantom {{\lambda_{0} c} G}}\right.\kern-0pt} \!\lower0.7ex\hbox{$G$}}\)
|
\({\raise0.7ex\hbox{${\lambda_{0} \lambda_{2} t}$} \!\mathord{\left/ {\vphantom {{\lambda_{0} \lambda_{2} t} G}}\right.\kern-0pt} \!\lower0.7ex\hbox{$G$}}\)
|
λ
2
|
IV |
\(\lambda_{1} \alpha_{1} c^{{\alpha_{1} }}\)
|
\(\lambda_{2} \alpha_{2} t^{{\alpha_{2} }}\)
|
\({\raise0.7ex\hbox{${\lambda_{2} \alpha_{2} t^{{\alpha_{2} - 1}} }$} \!\mathord{\left/ {\vphantom {{\lambda_{2} \alpha_{2} t^{{\alpha_{2} - 1}} } {\lambda_{1} \alpha_{1} c^{{\alpha_{1} - 1}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda_{1} \alpha_{1} c^{{\alpha_{1} - 1}} }$}}\)
|
IVA |
λ
1
|
λ
2
|
\({\raise0.7ex\hbox{${\lambda_{2} c}$} \!\mathord{\left/ {\vphantom {{\lambda_{2} c} {\lambda_{1} t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda_{1} t}$}}\)
|
IVB |
\(\lambda_{1} \left( {\alpha_{1} c + q_{1} \left( {1 - \alpha_{1} } \right)} \right)\)
|
λ
2(α
2
t + q
2(1 − α
2)) |
\(\frac{{\lambda_{2} c\left( {\alpha_{2} t + q_{2} \left( {1 - \alpha_{2} } \right)} \right)}}{{\lambda_{1} t\left( {\alpha_{1} c + q_{1} \left( {1 - \alpha_{1} } \right)} \right)}}\)
|
The role of income
-
First, the level of income affects the overall level of demand; as described below, this is accounted for by estimating a coefficient that links the demand to the income (adjusted for inflation) per capita for each region and year.
-
Second, the impact of fare is mitigated by income; that is, regions and years with higher incomes are more willing to accept higher fares. In the modelling, this is accommodated by multiplying the fares by the factor
Analysis results
Data chosen for analysis
Classical rail models
Variable | Estimate |
t value |
---|---|---|
λ
1
| −0.4875 | −53.56 |
λ
2
| −1.2193 | −69.79 |
β
1 (log Income) | 0.5292 | 54.18 |
β
2 (Hat2000) | −0.0357 | −10.25 |
β
3 (Hat2001) | −0.0008 | −0.22 |
β
4 (Hat2002) | 0.0106 | 2.95 |
Observations | 48,015 | |
\(\hat{\sigma }^{2}\)
| 0.034676 | |
Log likelihood | 14,235.26 | |
AIC | −22,054.5 |
-
the mean squared error \(\hat{\sigma }^{2} = SSE/\left( {R - h} \right)\), where R is the number of observations and h is the number of estimated parameters;
-
the Akaike Information Coefficient (AIC) is calculated by$${\text{AIC}} = 2\left( {h + 1} \right) - 2log\left( L \right)$$
Undamped model
Variable | Estimate |
t value |
---|---|---|
λ
1
| −0.0045 | −12.48 |
λ
2
| −0.0034 | −44.81 |
β
1 (log Income) | 0.8832 | 98.94 |
β
2 (Hat2000) | −0.0206 | −5.64 |
β
3 (Hat2001) | 0.0113 | 3.05 |
β
4 (Hat2002) | 0.0164 | 4.37 |
Observations | 48,015 | |
\(\hat{\sigma }^{2}\)
| 0.038297 | |
Log likelihood | 11,850.77 | |
AIC | −17,285.5 |
Distance-damped models
Variable | Type I | Type II | ||
---|---|---|---|---|
Estimate |
t value | Estimate |
t value | |
λ
1
| −4.7632 | −14.73 | −35.70 | −12.52 |
λ
2
| −0.7292 | −17.12 | −0.1132 | −9.69 |
α
| 1.1111 | 79.22 | n.a. | |
α
1
| n.a. | 1.5424 | 79.33 | |
α
2
| n.a. | 0.6690 | 29.86 | |
β
1 (log Income) | 0.6835 | 73.00 | 0.6299 | 66.68 |
β
2 (Hat2000) | −0.0349 | −9.83 | −0.0322 | −9.17 |
β
3 (Hat2001) | −0.0027 | −0.74 | 0.0015 | 0.42 |
β
4 (Hat2002) | 0.0046 | 1.25 | 0.0094 | 2.61 |
Observations | 48,015 | 48,015 | ||
\(\hat{\sigma }^{2}\)
| 0.035993 | 0.035275 | ||
Log likelihood | 13,340.93 | 13,825.40 | ||
AIC | −20,263.9 | −21,230.8 |
‘Dynamic’ functional forms
Variable | Type IIIa
| Type IIIA | ||
---|---|---|---|---|
Estimate |
t value | Estimate |
t value | |
λ
0
| −9.3867 | −74.66 | −1.5352 | −76.11 |
λ
2
| 24.5461 | 35.22 | 0.2231 | 36.66 |
α
| 0.08 | Fixed | n.a. | |
β
1 (log Income) | 0.6364 | 67.08 | 0.6216 | 65.52 |
β
2 (Hat2000) | −0.0327 | −9.26 | −0.0337 | −9.57 |
β
3 (Hat2001) | 0.0024 | 0.68 | 0.0011 | 0.32 |
β
4 (Hat2002) | 0.0124 | 3.41 | 0.0112 | 3.08 |
Observations | 48,015 | 48,015 | ||
\(\hat{\sigma }^{2}\)
| 0.035690 | 0.035508 | ||
Log likelihood | 13,543.67 | 13,666.50 | ||
AIC | −20,671.3 | −20,917.0 |
Variable | Type IV | Type IV fixed α
| ||
---|---|---|---|---|
Estimate |
t value | Estimate |
t value | |
λ
1
| −1.0 | Fixed | −14.6403 | −52.47 |
λ
2
| −1.0 | Fixed | −34.7155 | −69.51 |
α
1
| 0.2006 | 64.29 | 0.03 | Fixed |
α
2
| 0.2705 | 155.84 | 0.03 | Fixed |
β
1 (log Income) | 0.6340 | 66.10 | 0.5360 | 54.83 |
β
2 (Hat2000) | −0.0313 | −8.87 | −0.0355 | −10.18 |
β
3 (Hat2001) | 0.0037 | 1.03 | −0.0004 | −0.12 |
β
4 (Hat2002) | 0.0137 | 3.78 | 0.0109 | 3.04 |
Observations | 48,015 | 48,015 | ||
\(\hat{\sigma }^{2}\)
| 0.035563 | 0.034773 | ||
Log likelihood | 13,629.10 | 14,168.40 | ||
AIC | −20,842.2 | −21,920.8 |
Variable | Type IVB | |
---|---|---|
Estimate |
t value | |
λ
1
| −0.1073 | −63.55 |
λ
2
| −0.0289 | −47.85 |
α
1
| −0.1801 | −44.69 |
α
2
| −0.0216 | −5.03 |
β
1 (log Income) | 0.5109 | 52.74 |
β
2 (Hat2000) | −0.0331 | −9.63 |
β
3 (Hat2001) | −0.0012 | −0.35 |
β
4 (Hat2002) | −0.0091 | 2.57 |
Observations | 48,015 | |
\(\hat{\sigma }^{2}\)
| 0.033655 | |
Log likelihood | 14,954.25 | |
AIC | −23,488.5 |
Discussion and conclusions
Type | Fit (log-likelihood) | Range issues | Cost elasticity (at mean) | Time elasticity (at mean) | Value of time (p/minute, at mean) |
---|---|---|---|---|---|
0 | 11,851 | No | −0.09 | −0.82 | 75.6 |
I | 13,341 |
α > 1 | −0.39 | −0.76 | 15.3 |
II | 13,825 |
α
1 > 1 | −0.35 | −1.03 | 23.0 |
III | 13,544 |
α fixed | −0.37 | −1.16 | 24.5 |
IIIA | 13,667 | No | −0.40 | −1.13 | 22.3 |
IV | 14,168 |
α fixed | −0.36 | −1.19 | 26.0 |
IVA | 14,235 | No | −0.49 | −1.22 | 19.7 |
IVB | 14,954 |
α < 0 | −0.52 | −1.19 | 17.9 |