Lecture 5 - Applied Disaggregate Demand Models

ENCI707: Engineering Demand and Policy Analysis

Outline

• Policy Evaluation
• Forecasting & Prediction
• Confidence Intervals on Predictions
• Model transferability & Parameter Updating

Policy Evaluation Using MNL

Policy Evaluations

• Elasticity:
  • Direct elasticity
  • Cross elasticity
• Consumer surplus & social welfare analysis
• Marginal rate of substitution between two variables (ratio of corresponding parameters) in the choice model:
  • Value of Travel Time Savings (VOTTS) in mode choice
  • Willingness to pay for improved transit reliability in transit choice
  • Willingness to pay for transit accessibility in home location choice

Willingness-to-pay Analysis

• Systematic utility function needs to have cost as a variable \(p\) along with another variable of interest \(x\). \[V_j = V_j(p_j,x_j, \dots) -> p\text{ is a continuous variable & } V_j \text{ is differentiable in x and p}\]
• Assumption: Any changes in \(x\) is balanced by changes in \(p\) to maintain same level of utility: total utility does not change \[V_j = V_j(p_j,x_j, \dots) = V_j((p_j+\delta p_j),(x_j + \delta x_j), \dots)\]
• Using first-order Taylor series expansion \[V_j(p_j,x_j, \dots) = V_j(p_j,x_j, \dots) + \delta p_j \frac{\partial V_j(p_j,x_j, \dots)}{\partial p_j} + \delta x_j \frac{\partial V_j(p_j,x_j, \dots)}{\partial x_j}\]
• After transposing (assuming overall utility change is zero) \[\frac{\delta p_j}{\delta x_j} = - \frac{V_j(p_j, x_j, \dots)/\partial x_j}{V_j(p_j, x_j, \dots)/\partial p_j} \text{; additional price to compensate additional x}\]

Willingness-to-pay Analysis

• For linear-in-parameter utility function \(V_j = \beta_j + \beta_p p_j + \beta_x x_j + \dots\)
• Willing-to-pay for any continuous variable change \(\frac{\partial p_j}{\partial x_j} = - \frac{\beta_x}{\beta_p}\)
• Value of Travel Time Savings (VOTTS): additional amount of money to reduce 1 unit of time \[VOTTS = \frac{\partial p_j}{- \partial t_j} = \frac{\beta_t}{\beta_p}\]
• Value of Reliability (VOR) or Safety (VOS) : additional amount of money to increase 1 level of reliability (r), safety (s) \[VOR = \frac{\partial p_j}{+ \partial r_j} = - \frac{\beta_r}{\beta_p}\text{; } VOS = - \frac{\partial p_j}{+ \partial s_j} = \frac{\beta_s}{\beta_p}\]

Consumer Surplus

\[\Delta CS = \int_{V_j^1}^{V_j^2} Pr_i(j|V_j, V_{k \in C_i, k \neq j}) d V_j = \int_{V_j^1}^{V_j^2} \frac{\exp(\mu V_j)}{\sum_{k \in C_i} \exp(\mu V_k)}d V_j\]

Consumer Surplus

For Changes in only one (independent) alternative (\(j\)): \[\Delta CS = \frac{1}{\mu}\ln\left(\exp(\mu V_j^2) + \sum_{k \in C_i} \exp(\mu V_k)\right) - \frac{1}{\mu}\ln\left(\exp(\mu V_j^1) + \sum_{k \in C_i} \exp(\mu V_k)\right)\] - For changes in multiple alternatives (existence of equal or unequal cross-elasticity, e.g., nested logit, GEV, mixed logit) the integral is path dependent since it becomes conditional upon an income effect - For logit model \[\Delta CS = \frac{1}{\mu}\ln\left(\sum_{j \in C_i} \exp(\mu V_j^2)\right) - \frac{1}{\mu}\ln\left(\sum_{j \in C_i} \exp(\mu V_j^1)\right)\]

Welfare Analysis

• Compensating variation analysis: normalized changes in expected maximum utility of a particular choice before/after any changes (policy/infrastructure): \[U_j = \beta_I(I-p_j) + \sum \beta x + \epsilon_j\]
• where \(I\) = income, \(p_j\) = price/cost, \(\beta_I\) = marginal utility of income = coefficient of income normalized cost
• Consumer surplus (CS): \[CS = \frac{1}{\mu}\left(\ln \sum_j \exp \left(\mu (\beta_I(I-p_j) + \sum \beta x)\right) \right)\]
• Compensting variation (CV) in monetary value: \[CV = \frac{1}{|\beta_I|}\left( CS_{after-change} - CS_{before-change} \right)\]

Forecasting & Prediction Using Disaggregate Choice Models

Applying Disaggregate Model for Forecasting

• Aggregate forecasting \[\sum_{individuals} \text{Individual Behaviour} = \text{Aggregate(collective) Outcome}\]
• Can synthesize people with attributes from census data - i.e., 100% microsimulation with synthetic population

Applying Disaggregate Choice Model for Forecasting

• We do aggregate forecasts, so why do we need disaggregate models?
  • Aggregation suppresses information -> misleading information after data aggregation (i.e., Simpson’s Paradox & Ecological Fallacy)
  • Statistical inefficiency of aggregate approach
  • Although we aggregate the forecast, we can explicitly capture information at individual level
  • We can track many sensitive policy issues
  • Aggregation of forecast is possible at any level we want, as the model is disaggregate
  • Sometimes modelling at disaggregate level makes the model independent of artificial zonal structures and the model may be transferrable

Aggregation Methods for Forecasting

• Microsimulation
• Sample enumeration
• Naïve aggregation
• Classification with naïve aggregation

Microsimulation

Synthetic Population	Choice Alternatives
	1	2	…	J	Sum
1	\(Pr(1|x_1,\beta)\)	\(Pr(2|x_1,\beta)\)	.	\(Pr(J|x_1,\beta)\)	1
2	\(Pr(1|x_2,\beta)\)	\(Pr(2|x_2,\beta)\)	.	\(Pr(J|x_2,\beta)\)	1
…	.	.	.	.	.
N	\(Pr(1|x_N,\beta)\)	\(Pr(2|x_N,\beta)\)	.	\(Pr(J|x_N,\beta)\)	1
Sum	N(1)	N(2)		N(J)	N
Market share	N(1)/N	N(2)/N		N(J)/N	1

Sample Enumeration

• Rather than 100% population, use a representative sample
  • If we have a representative sample, \(N_S\), of the target population \(N_T\) then: \[\overset{\sim}{W_i} = \text{Predicted sample share of j} = \frac{1}{N_S}\sum_{s=1}^{N_S} Pr(j|X_S,\beta)\]

Representative Sample	Choice Alternatives
	1	2	…	J	Sum
1	\(Pr(1|x_1,\beta)\)	\(Pr(2|x_1,\beta)\)	.	\(Pr(J|x_1,\beta)\)	1
2	\(Pr(1|x_2,\beta)\)	\(Pr(2|x_2,\beta)\)	.	\(Pr(J|x_2,\beta)\)	1
…	.	.	.	.	.
N	\(Pr(1|x_S,\beta)\)	\(Pr(2|x_S,\beta)\)	.	\(Pr(J|x_S,\beta)\)	1
Sum	n(1)	n(2)		n(J)	\(N_S\)
Market share	n(1)/\(N_S\)	n(2)/\(N_S\)		n(J)/\(N_S\)	1

Sample Enumeration

• Although there may be sampling error, the expected value should match the population average

• For a ‘G’ stratified sample: \[\overset{\sim}{W_i} = \sum_{g=1}^G \left(\frac{N_g}{N_T}\right) \frac{1}{N_{sg}}\sum_{t=1}^{N_{sg}} Pr(j|X_t,\beta)\]

• Here, \(N_g\) is the population in segment \(g\), \(N_{sg}\) is the sample population in segment \(g\), and \(N_T\) is the target population

• Sample for estimation should be representative

• Sample enumeration can generally be used to test policies in the short-run during which the base population sample can be assumed to remain representative

Naïve Aggregation

• Considering mean value of \(X_t\) as the unique value for whole population gives crude estimate

  Population Strata	Choice Alternatives
  	1	2	…	J	Sum
  Market share	\(Pr(1|x_1,\beta)\)	\(Pr(2|x_1,\beta)\)	.	\(Pr(J|x_1,\beta)\)	1

Classification with Naïve Aggregation

• Classify population into a number of strata and consider population within a particular segment as homogenous (having same attributes)
• When sample enumeration infeasible, can use classification with naïve aggregation
• Categorization is based on attributes / market segmentation

Population Strata	Choice Alternatives
	1	2	…	J	Sum
1	\(Pr(1|x_1,\beta)\)	\(Pr(2|x_1,\beta)\)	.	\(Pr(J|x_1,\beta)\)	1
2	\(Pr(1|x_2,\beta)\)	\(Pr(2|x_2,\beta)\)	.	\(Pr(J|x_2,\beta)\)	1
…	.	.	.	.	.
n	\(Pr(1|x_S,\beta)\)	\(Pr(2|x_S,\beta)\)	.	\(Pr(J|x_S,\beta)\)	1
Sum	n(1)	n(2)		n(J)	n
Market share	n(1)/n	n(2)/n		n(J)/n	1

Confidence Interval on Predictions

Confidence Interval on Predictions

• Estimated parameters are MVN distributed by central limit theorem
• Same approach is applicable for any choice model

Confidence Interval of Predictions: Elasticity, Market Share, & Revenue

Confidence Interval of Predictions: Elasticity, Market Share, Revenue

Confidence Interval of Predictions: Elasticity, Market Share, & Revenue

Sampling & Discrete Choice Models

Sampling

• Sampling of choice makers from population using RP or SP survey:
  • Exogenous Sample: Sample share of alternatives are:
    • Identical to those of population shares if representative
    • Not-identical if not representative
  • Endogenous/Choice-based Sample: Respondents are sampled based on observed choices
    • Sample share of alternatives are different from those of population shares
• Sampling of choice alternatives: irrespective of sampling of respondents:
  • Choice alternatives are collectively exhaustive
  • When choice alternative space is large, then sample alternatives

Sampling of Choice Makers

\[\text{Choice Model: }Pr(j|x,\beta)\]

• Exogenous sampling of choice makers (Stratified Random Sampling)
  • Population size, \(N\)
  • Stratification of population into total \(G\) groups
  • Share (%) individual in a particular group, g in population, \(W_g\)
  • Sample size, \(n\)
  • Share (%) individual in the particular group, \(g\) in sample, \(w_g\)
  • Probability of an individual being selected in the sample, \(q_g = \frac{w_g n}{W_g N}\)
  • Sampling probability is not a function of parameter, \(\beta\)

Sampling of Choice Makers

\[\text{Choice Model: }Pr(j|x,\beta)\]

• Exogenous sampling of choice makers (Simple Random Sampling)
• Probability of an individual being selected in the sample, \(q_g = \frac{n}{N}\)
• Sampling probability is not a function of parameter, \(\beta\)
• For representative exogenous sample (stratified/simple random sampling) no correction to choice model estimates is needed
  • For non-representative sample, parameters except the Alternative Specific Constants (ASC) do not require correction
  • ASC needs to corrected/updated to match actual market share

Sampling of Choice Makers

\[\text{Choice Model: }Pr(j|x(j),\beta)\]

• Endogenous/Choice-based sampling of choice makers (randomly select choice makers of specific choices)
  • Selection of respondent of specific choice maker group (e.g. transit users, driver, bicyclists) still should be random in some way
  • Choice model with choice-based sample produces unbiased estimates of all parameters except the Alternative Specific Constants (ASC)
  • ASCs need to corrected/updated to match actual market share

Sampling of alternatives: Large Choice Set

• Large number of alternatives (e.g., destination location choice model), a random sample of alternatives can be used to estimate the model - where \(C = \text{full choice set}\) and \(G = \text{subset of alternatives drawn from C}\)
• Conditional probability of selecting a subset of alternatives \[q(G|j) > 0 \text{; if G includes the chosen alt. j}\] \[q(G|j) = 0 \text{; if G excludes the chosen alt. j}\]
• Joint probability that \(J\) is drawn with non-zero probability and \(j\) is chosen by an individual \(i\) \[Pr(G,j) = q(G|j)Pr(G) = Pr(j|G)Q(G) \text{; (Bayes Theorem)}\] \[\text{Marginal probability: } Q(G) = \sum_{k \in C} Pr(k)q(G|k) \text{; (Sum over all possible subsets of j')}\]

Sampling of alternatives: Large Choice Set

• Choice probability of alternative \(j\) from subset: \[Pr(j|G)Q(G) = q(G|j)Pr(j)\] \[Pr(j|G)Q(G) = \frac{Pr(j)q(G|j)}{Q(G)} = \frac{Pr(j)q(G|j)}{\sum_{k \in C} Pr(k)q(G|k)} = \frac{\exp(V_j)q(G|j)}{\sum_{k \in C} \exp(V_k)q(G|k)}\]
• For any subset, \(q\), that does not include \(k\), \(q(G|k)=0\), so: \[Pr(j|G) = \frac{\exp(V_j)q(G|j)}{\sum_{k \in G} \exp(V_k)q(G|k)} \text{; within C, any G without k drops out}\] \[Pr(j|G) = \frac{\exp(V_j)\exp(\ln(q(G|j)))}{\sum_{k \in G} \exp(V_k)\exp(\ln(q(G|j)))} = \frac{\exp(V_j + \ln(q(G|j)))}{\sum_{k \in G} \exp(V_k + \ln(q(G|j)))}\]

Sampling of alternatives: Large Choice Set

• Choice model with sampling from the alternatives: \[Pr(j|G) = \frac{\exp(V_j + ln(q(G|j)))}{\sum_{k \in G} \exp(V_k + ln(q(G|k)))} \text{; } q(G|j) = \prod_{l=1}^{G-1}Pr(l)\]
• where \(Pr(l)\) is the binary probability of drawing a choice
• Under simple random sampling, samples are drawn uniformly with equal probability
• This makes the additional \(ln(q(G|j))\) redundant (same for all alternative and so drops out: uniform conditional property) - MNL result becomes standard form
• Not true for other models such as nested logit, GEV, or mixed logit

Managing Large Choice Sets

• Reducing alternatives:
  • Use various feasibility constraints to reduce number of alternatives
  • Random sampling from large choice set
  • Wrong choice set may generate distorted predictions

Transferability and Updating Model Parameters: MNL

Model Transferability

• Transferability issues:
  • Systematic bias.
  • Local conditions.
• Model Updating Issues:
  • Systematic bias of the model is captured by model constants. - Alternative Specific Constants (ASC) are error basket - At a minimum level, the constant terms change from place-to-place
  • Scale parameter of random error term changes from place-to-place
  • Model specification changes from place-to-place

Updating MNL Model Parameters

• MNL predictions matches the market shares of sample data used to estimate the model parameter
• Updating ASC of MNL for matching population shares if those are different (if the sample is not representative or model is transferred to a different spatial/temporal context) in sample share:
• Iterative ASC adjustment for the utility functions of the choice alternatives (including the based alternative for which the ASC is estimated to be 0) \[𝐴𝑆𝐶_{𝑢𝑝𝑑𝑎𝑡𝑒𝑑, j} = 𝐴𝑆𝐶_{𝑡𝑜−𝑏𝑒−𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑, j} + \ln\left(\frac{\text{actual market share, j}}{\text{predicted market share, j}} \right) \]

Updating MNL Model Parameters

• Population market share of \(𝑗= 𝐻_𝑗\)
• Sample share of \(j = h_j\)
• Adjustment proprtion of \(j = s_j - \frac{H_j}{h_j}\)
• Adjusted choice probability \(Pr(j) = \frac{s_j \exp(\beta_{0,j}+\sum \beta x)}{\sum_k s_k \exp(\beta_{0,k}+\sum \beta x)}\) \[Pr(j) = \frac{\exp(\beta_{0,j}+\sum \beta x + \ln(s_j))}{\sum_k \exp(\beta_{0,k}+\sum \beta x + \ln(s_k))} = \frac{\exp(\beta_{0,j}+\sum \beta x + \ln(H_j/h_j))}{\sum_k \exp(\beta_{0,k}+\sum \beta x + \ln(H_k/h_k))}\] \[\text{Updated } \beta_{0,j} = (\beta_{0,j})_{estimated} + ln(H_j/h_j)\]

Further topics (among many others)

• Logit model with probabilistic choice set formation
• Logit model with multiple overlapping choice sets
• Logit model with myopic preferences regarding specific alternatives