Theory 1
Binary choice models: 𝑦𝑖 can take two values (usually 0 or 1)
- OLS (Linear Probability Model (LPM))
- Logit model
- Probit model
Linear probability model: 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝐸[𝑦𝑖 |𝑥𝑖 ] = 𝑥𝑖'β
- Use OLS to estimate
- Problems: probability is not restricted to [0,1] and there is heteroskedasticity
(standard errors of variable are not constant)
Logit and Probit model: 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝐹( 𝑥𝑖'β) and 𝑃(𝑦𝑖 = 0|𝑥𝑖) = 1 − 𝐹( 𝑥𝑖'β)
- Logit: 𝐹 = logistic distribution
- Probit: 𝐹 = standard normal distribution
- These models are fully parametric
- Use maximum likelihood to estimate: consistent but you should use robust standard
errors
δ𝑃(𝑦𝑖=1|𝑥𝑖) δ𝐹( 𝑥𝑖'β)
- δ𝑥𝑖𝑗
= δ𝑥𝑖'β
β𝑗 : effect of x on the probability depends on the observation i and
the direction of the effect is determined by β𝑗
Alternative models:
- Latent variables model
- Random utility model
* *
Latent variable model: 𝑦𝑖 = 𝑥𝑖'β + ϵ𝑖 then 𝑦𝑖 = 1 if 𝑦𝑖 ≥ 0 so 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝑃(ϵ𝑖 ≥ − 𝑥𝑖'β)
*
Panel data: 𝑦𝑖𝑡 = 𝑥𝑖𝑡'β + α𝑖 + λ𝑡 + ϵ𝑖𝑡
- Random effects: no problem but we have to decide on a distribution
- Fixed effects: α𝑖 can be removed in Logit model (but not efficient) and λ𝑡 can be
removed by using time dummies (LSDV) but this doesn’t work
Multinomial choice: 𝑦𝑖 = 𝑗 for j =1,2,...,M
- Bivariate logit and probit models can be generalized
𝑒𝑥𝑝(𝑥𝑖'β𝑗 +γ'𝑧𝑖𝑗+𝑤𝑗'α𝑖)
Multivariate logit: 𝑃(𝑦𝑖𝑗 = 1 | 𝑥𝑖, 𝑧𝑖𝑗, 𝑤𝑗 ) = 𝑀
∑ 𝑒𝑥𝑝(𝑥𝑖'β𝑗 +γ'𝑧𝑖𝑗+𝑤𝑗'α𝑖)
𝑗=1
- 𝑤𝑗 is never present, so we assume α𝑖 = 0
- If γ = 0: Multinomial Logit Model (MNL): only observation specific regressors ->
normalization needed & estimated by maximum likelihood
, - if β𝑗 = 0 for all j: Conditional Logit Model: only observation and alternative specific
regressors -> no normalization needed
- Mixed logit: combination of the above
Revealed vs stated preference data:
- Revealed preference: observe actual choices made: less reliable because
retrospective
- Stated preference: indicate the preferred alternative: less reliable because no actual
choice has been made
Additive Random Utility Models (ARUM): 𝑈𝑖𝑗 = 𝑉𝑖𝑗 + ϵ𝑖𝑗 : errors are split up in a deterministic
part (𝑉𝑖𝑗) and a random part (ϵ𝑖𝑗 )
- If ϵ𝑖𝑗 are independent over i and j and type I extreme value distributed, use
multinomial logit model -> works best for complementary alternatives
- If ϵ𝑖𝑗 are independent over i and multivariate normal distributed, use multinomial probit
model
Nested MNL: tree-structure -> not applied very often
Marhall & Pires (2018)
Introduction
The paper models how consumers (or households, in the paper used interchangeably)
choose where to shop
- Consumers trade off the cost of traveling to different stores and the prices and
product assortment of each store (summarized by the store surplus index)
- Measuring the role of travel costs relative to prices and variety is their main empirical
challenge: in a world without travel frictions, consumers would always choose to shop
at the store offering the greatest surplus to consumers in terms of prices and product
variety, but because of the existence of travel costs this is not a given
With the estimates of their model they measure the extent to which travel costs have an
impact on the surplus earned by consumers (the extent to which travel costs lead consumers
to choose store with higher prices/less variety/both)
Model
Store choice model where consumers trade off the cost of traveling to different stores and
the prices and product assortment of each store
- The prices and product variety of each store are summarized by a measure of
consumer surplus that we call the store surplus index
, - Consumers can place more weight on surplus (i.e. prices and product variety)
relative to convenience when shopping for a larger set of goods/when travel is less
costly
The model captures a sequential decision process for 𝐼 consumers who visit one of the
stores in the set 𝐽𝑠𝑡𝑜𝑟𝑒𝑠:
1. Consumer 𝑖 chooses where to shop given two considerations: the need to purchase
𝑋𝑖𝑡 products and a vector of trip-timing characteristics 𝑦𝑖𝑡
- 𝑋𝑖𝑡 : number of items rather than a specific list of products
- Trip-timing characteristics may include e.g. weather conditions, traffic conditions and
the opportunity cost of time of consumer 𝑖 at time 𝑡
2. The consumer chooses which products to purchase given her choice of shopping at
store 𝑘 with product range 𝐽𝑘𝑡 and her need to purchase 𝑋𝑖𝑡 units
- Critique: we don’t know what the choice set of the individuals is: how many
supermarket stores J are there available at time t
Decisions are described in reverse order -> we solve the model by backward induction
So we start with the second step: for each of the 𝑋𝑖𝑡 decisions faced by the consumer, the
consumer chooses the product ℎ that maximizes their indirect utility -> consumer 𝑖 chooses
product ℎ when making choice 𝑗 at store 𝑘 if and only if
ℎ 𝑙
𝑢𝑖ℎ𝑘𝑡 =− α𝑝ℎ𝑘𝑡 + ξℎ𝑘𝑡 + ϵ𝑖𝑗𝑡 ≥ 𝑢𝑖𝑙𝑘𝑡 =− α𝑝𝑙𝑘𝑡 + ξ𝑙𝑘𝑡 + ϵ𝑖𝑗𝑡
- α: marginal utility of income
- 𝑝ℎ𝑘𝑡: price of product h at store k at time t
- ξℎ𝑘𝑡 : captures unobserved product-specific characteristics that may vary over time
and across stores
ℎ
- ϵ𝑖𝑗𝑡 : idiosyncratic taste shock of product h that corresponds to decision number j for
person i at time t -> while consumers face the same observable product
characteristics for each of the 𝑋𝑖𝑡 product choices, the randomness of the unobserved
taste shocks generates variation in the outcome of each of the 𝑋𝑖𝑡 product choices
ℎ
- Later on they use notation ϵ𝑖ℎ𝑗𝑡 instead of ϵ𝑖𝑗𝑡
Mistake: the indirect utility function 𝑢𝑖ℎ𝑘𝑡 =− α𝑝ℎ𝑘𝑡 + ξℎ𝑘𝑡 + ϵℎ𝑖𝑗𝑡 = 𝑉ℎ𝑘𝑡 + ϵℎ𝑖𝑗𝑡 is not actually
an indirect utility function because it does not include the income of the consumer
Binary choice models: 𝑦𝑖 can take two values (usually 0 or 1)
- OLS (Linear Probability Model (LPM))
- Logit model
- Probit model
Linear probability model: 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝐸[𝑦𝑖 |𝑥𝑖 ] = 𝑥𝑖'β
- Use OLS to estimate
- Problems: probability is not restricted to [0,1] and there is heteroskedasticity
(standard errors of variable are not constant)
Logit and Probit model: 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝐹( 𝑥𝑖'β) and 𝑃(𝑦𝑖 = 0|𝑥𝑖) = 1 − 𝐹( 𝑥𝑖'β)
- Logit: 𝐹 = logistic distribution
- Probit: 𝐹 = standard normal distribution
- These models are fully parametric
- Use maximum likelihood to estimate: consistent but you should use robust standard
errors
δ𝑃(𝑦𝑖=1|𝑥𝑖) δ𝐹( 𝑥𝑖'β)
- δ𝑥𝑖𝑗
= δ𝑥𝑖'β
β𝑗 : effect of x on the probability depends on the observation i and
the direction of the effect is determined by β𝑗
Alternative models:
- Latent variables model
- Random utility model
* *
Latent variable model: 𝑦𝑖 = 𝑥𝑖'β + ϵ𝑖 then 𝑦𝑖 = 1 if 𝑦𝑖 ≥ 0 so 𝑃(𝑦𝑖 = 1|𝑥𝑖) = 𝑃(ϵ𝑖 ≥ − 𝑥𝑖'β)
*
Panel data: 𝑦𝑖𝑡 = 𝑥𝑖𝑡'β + α𝑖 + λ𝑡 + ϵ𝑖𝑡
- Random effects: no problem but we have to decide on a distribution
- Fixed effects: α𝑖 can be removed in Logit model (but not efficient) and λ𝑡 can be
removed by using time dummies (LSDV) but this doesn’t work
Multinomial choice: 𝑦𝑖 = 𝑗 for j =1,2,...,M
- Bivariate logit and probit models can be generalized
𝑒𝑥𝑝(𝑥𝑖'β𝑗 +γ'𝑧𝑖𝑗+𝑤𝑗'α𝑖)
Multivariate logit: 𝑃(𝑦𝑖𝑗 = 1 | 𝑥𝑖, 𝑧𝑖𝑗, 𝑤𝑗 ) = 𝑀
∑ 𝑒𝑥𝑝(𝑥𝑖'β𝑗 +γ'𝑧𝑖𝑗+𝑤𝑗'α𝑖)
𝑗=1
- 𝑤𝑗 is never present, so we assume α𝑖 = 0
- If γ = 0: Multinomial Logit Model (MNL): only observation specific regressors ->
normalization needed & estimated by maximum likelihood
, - if β𝑗 = 0 for all j: Conditional Logit Model: only observation and alternative specific
regressors -> no normalization needed
- Mixed logit: combination of the above
Revealed vs stated preference data:
- Revealed preference: observe actual choices made: less reliable because
retrospective
- Stated preference: indicate the preferred alternative: less reliable because no actual
choice has been made
Additive Random Utility Models (ARUM): 𝑈𝑖𝑗 = 𝑉𝑖𝑗 + ϵ𝑖𝑗 : errors are split up in a deterministic
part (𝑉𝑖𝑗) and a random part (ϵ𝑖𝑗 )
- If ϵ𝑖𝑗 are independent over i and j and type I extreme value distributed, use
multinomial logit model -> works best for complementary alternatives
- If ϵ𝑖𝑗 are independent over i and multivariate normal distributed, use multinomial probit
model
Nested MNL: tree-structure -> not applied very often
Marhall & Pires (2018)
Introduction
The paper models how consumers (or households, in the paper used interchangeably)
choose where to shop
- Consumers trade off the cost of traveling to different stores and the prices and
product assortment of each store (summarized by the store surplus index)
- Measuring the role of travel costs relative to prices and variety is their main empirical
challenge: in a world without travel frictions, consumers would always choose to shop
at the store offering the greatest surplus to consumers in terms of prices and product
variety, but because of the existence of travel costs this is not a given
With the estimates of their model they measure the extent to which travel costs have an
impact on the surplus earned by consumers (the extent to which travel costs lead consumers
to choose store with higher prices/less variety/both)
Model
Store choice model where consumers trade off the cost of traveling to different stores and
the prices and product assortment of each store
- The prices and product variety of each store are summarized by a measure of
consumer surplus that we call the store surplus index
, - Consumers can place more weight on surplus (i.e. prices and product variety)
relative to convenience when shopping for a larger set of goods/when travel is less
costly
The model captures a sequential decision process for 𝐼 consumers who visit one of the
stores in the set 𝐽𝑠𝑡𝑜𝑟𝑒𝑠:
1. Consumer 𝑖 chooses where to shop given two considerations: the need to purchase
𝑋𝑖𝑡 products and a vector of trip-timing characteristics 𝑦𝑖𝑡
- 𝑋𝑖𝑡 : number of items rather than a specific list of products
- Trip-timing characteristics may include e.g. weather conditions, traffic conditions and
the opportunity cost of time of consumer 𝑖 at time 𝑡
2. The consumer chooses which products to purchase given her choice of shopping at
store 𝑘 with product range 𝐽𝑘𝑡 and her need to purchase 𝑋𝑖𝑡 units
- Critique: we don’t know what the choice set of the individuals is: how many
supermarket stores J are there available at time t
Decisions are described in reverse order -> we solve the model by backward induction
So we start with the second step: for each of the 𝑋𝑖𝑡 decisions faced by the consumer, the
consumer chooses the product ℎ that maximizes their indirect utility -> consumer 𝑖 chooses
product ℎ when making choice 𝑗 at store 𝑘 if and only if
ℎ 𝑙
𝑢𝑖ℎ𝑘𝑡 =− α𝑝ℎ𝑘𝑡 + ξℎ𝑘𝑡 + ϵ𝑖𝑗𝑡 ≥ 𝑢𝑖𝑙𝑘𝑡 =− α𝑝𝑙𝑘𝑡 + ξ𝑙𝑘𝑡 + ϵ𝑖𝑗𝑡
- α: marginal utility of income
- 𝑝ℎ𝑘𝑡: price of product h at store k at time t
- ξℎ𝑘𝑡 : captures unobserved product-specific characteristics that may vary over time
and across stores
ℎ
- ϵ𝑖𝑗𝑡 : idiosyncratic taste shock of product h that corresponds to decision number j for
person i at time t -> while consumers face the same observable product
characteristics for each of the 𝑋𝑖𝑡 product choices, the randomness of the unobserved
taste shocks generates variation in the outcome of each of the 𝑋𝑖𝑡 product choices
ℎ
- Later on they use notation ϵ𝑖ℎ𝑗𝑡 instead of ϵ𝑖𝑗𝑡
Mistake: the indirect utility function 𝑢𝑖ℎ𝑘𝑡 =− α𝑝ℎ𝑘𝑡 + ξℎ𝑘𝑡 + ϵℎ𝑖𝑗𝑡 = 𝑉ℎ𝑘𝑡 + ϵℎ𝑖𝑗𝑡 is not actually
an indirect utility function because it does not include the income of the consumer
