Basics of Demand Estimation

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# Basics of Demand Estimation
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### Ian McCarthy | Emory University
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### Econ 771, Fall 2022
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# Discrete Choice

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# Setup

Indirect utility of person `$i$`, `$$u_{ij} = x_{ij}\beta + \epsilon_{ij},$$` where `$x_{ij}$` denotes person (and perhaps product) characteristics and `$\epsilon_{ij}$` denotes an error term.

- Standard logit: one choice, `$j=0,1$`
- Multinomial logit: many possible choices, `$j=0,1,...,J$`

---
# Logit terminology

A few different terms for very similar models:

- Multinomial Logit
- Conditional Logit
- "Mixed" Logit

---
# Multinomial

- Individual covariates only
- Alternative-specific coefficients

--
`$$u_{ij}=x_{i}\beta_{j} + \epsilon_{ij},$$` such that `$$p_{ij} = \frac{e^{x_{i}\beta_{j}}}{\sum_{k} e^{x_{i}\beta_{k}}}$$`

---
# Conditional

Allow for alternative-specific regressors, such that `$$u_{ij}=x_{ij}\beta + \epsilon_{ij}$$`

---
# Mixed

Allow for individual and alternative-specific regressors, such that `$$u_{ij}=x_{ij}\beta + w_{i} \gamma_{j} + \epsilon_{ij}$$`

<br>
--
*but* people sometimes use "mixed" to refer to random-coefficients logit

---
# Does it matter?

These are really all the same and it's just a matter of specification (e.g., interact individual covariates with product characteristics or with product dummies). I'll refer to them as "multinomial" logit.

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# Basic Multinomial Logit

With utility specification:
`$$u_{ij}=x_{ij}\beta + \epsilon_{ij} = V_{ij} + \epsilon_{ij}$$`

The probability of selecting choice `$j$` is:
`$$\begin{align}
p_{ij} & = Pr(u_{ij} > u_{ik} \forall k \neq j) \\
       & = Pr(V_{ij} + \epsilon_{ij} > V_{ij} + \epsilon_{ik} \forall k \neq j) \\
       & = Pr(\epsilon_{ij} - \epsilon_{ik} > V_{ik} - V_{ij} \forall k \neq j)
\end{align}$$`

---
# Basic Multivariate Logit

Impose some distributional assumptions on `$\epsilon_{i}$`, with
`$$p_{ij} = \int I(\epsilon_{ij} - \epsilon_{ik} > V_{ik} - V_{ij} ) f(\epsilon_{i}) d \epsilon_{i}$$`

- Multivariate normal: `$\epsilon_{i} \sim N(0, \Omega)$` yields multinomial probit
- Gumbel or Type I Extreme Value: `$f(\epsilon_{i}) = e^{-\epsilon_{ij}} e^{-e^{-\epsilon_{ij}}}$` and `$F(\epsilon_{i}) = 1 - e^{-e^{-\epsilon_{ij}}}$` yields multinomial logit

---
# Identification

- Only differences in utility matter (not scale)
- Adding constant is irrelevant
- Can only include `$J-1$` alternatives and need to normalize one to 0
- Can't identify individual specific factors that don't vary across choices

---
# Identification

- Since only relative utility differences are identified (not scale), `$u_{ij}^{0} = V_{ij} + \epsilon_{ij}$` and `$u_{ij}^{1} = \lambda V_{ij} + \lambda \epsilon_{ij}$` are equivalent
- In logit, normalize variance to `$\pi^{2}/6$` (comes from constant of integration)

---
# Basics of Multinomial Logit

With this setup, we can write

`$$p_{ij} = \frac{e^{V_{ij}}}{\sum_{k} e^{V_{ik}}},$$`

where we tend to approximate `$V_{ij}$` with some linear (in parameters) specification

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# Inclusive Value

- Nice property of this framework...
- Expected maximum has a close form `$$E[\max_{j} u_{ij}] = \log \left(\sum_{j} e^{V_{ij}} \right) + C$$`
  - Expected utility of best option does not depend on `$\epsilon_{ij}$`
  - Allows simple computation of the change in consumer surplus, `$\Delta CS$`

---
# The Indepenence of Irrelevant Alternatives

Fundamental issue with logit models...the ratio of choice probabilities for `$j$` and `$k$` does not depend on any other alternatives: `$$\frac{p_{ij}}{p_{ik}} = \frac{e^{V_{ij}}}{e^{V_{ik}}}.$$`

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# IIA

Well known critique of multinomial logit is the imposition of IIA

- Famous red bus blue bus example...introduction of red bus should only affect probability of selecting the blue bus (not car), but logit would predict all probabilities of `$1/3$`
- Imposes that more popular products have higher markups (mechanical relationship between elasticity and share)
- Imposes proportional substitution...as option `$k$` improves, it proportionally reduces the shares of all other choices

---
# Relaxing IIA

Relax assumptions on the error term with:
- nested logit (choose first between groups and then, within group, choose product)
- random-coefficient logit (random coefficients, `$\beta_{i}$`, on product characteristics)
- multinomial probit (completely different error structure, but more difficult to estimate)

---
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# Discrete Choice with Market Data

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# Setup

Utility of individual `$i$` from selecting product `$j$` is `$$U_{ij}=\delta_{j}+\epsilon_{ij},$$`
where `$\delta_{j}=x_{j}\beta + \xi_{j}$`, and `$\xi_{j}$` represents the mean level of utility derived from unobserved characteristics.

---
# Estimating with market shares

With normalization of the outside option to `$\delta_{0}=0$`, we can write:

`$$\begin{align}
s_{j} & =e^{\delta_{j}}/(1+\sum e^{\delta_{k}}) \\
s_{0} & =e^{0}/(1+\sum e^{\delta_{k}})
\end{align}$$`

Taking logs:

`$$\begin{align}
\ln (s_{0}) & = -\log ( 1 + \sum e^{\delta_{k}}) \\
\ln (s_{j}) & = \delta_{j} -\log ( 1 + \sum e^{\delta_{k}}) \\
\ln (s_{j}) - \ln (s_{0}) = \delta_{j}
\end{align}$$`

---
# Estimation in practice

1. "Transform" the data, `$\tilde{s} = \ln s_{j} - \ln s_{0}$`
2. IV regression of `$\tilde{s}$` on `$x$`, `$p$`, with instruments for `$p$`

---
# Nested logit with market shares (Berry, 1994)

A bit more work, but Berry (1994) shows that the analogous nested logit form is:

`$$\begin{align}
\ln (s_{j}) - \ln (s_{0}) - \rho \ln (s_{j|g}) & = \delta_{j}\\
\ln (s_{j}) - \ln (s_{0})  & = \delta_{j} + \rho \ln (s_{j|g})
\end{align}$$`

where `$g$` is the nesting structure or group, and `$\rho$` is the nesting parameter

- Note: **must** instrument for `$s_{j|g}$`
- common instrument is number of products in the nest

---
# Random coefficients with market shares (BLP)

- More complicated but still doable
- Much harder to estimate and more sensitive to specification (convergence can be an issue)
- See Chris Conlon and Jeff Gortmaker's "Best Practices for BLP" for a good guide, linked [here](https://chrisconlon.github.io/site/pyblp.pdf)