Heterogeneity

]

# Week 12: .fancy[Heterogeneity]

### <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M243.4 2.6l-224 96c-14 6-21.8 21-18.7 35.8S16.8 160 32 160v8c0 13.3 10.7 24 24 24H456c13.3 0 24-10.7 24-24v-8c15.2 0 28.3-10.7 31.3-25.6s-4.8-29.9-18.7-35.8l-224-96c-8-3.4-17.2-3.4-25.2 0zM128 224H64V420.3c-.6 .3-1.2 .7-1.8 1.1l-48 32c-11.7 7.8-17 22.4-12.9 35.9S17.9 512 32 512H480c14.1 0 26.5-9.2 30.6-22.7s-1.1-28.1-12.9-35.9l-48-32c-.6-.4-1.2-.7-1.8-1.1V224H384V416H344V224H280V416H232V224H168V416H128V224zM256 64a32 32 0 1 1 0 64 32 32 0 1 1 0-64z"/></svg> EMSE 6035: Marketing Analytics for Design Decisions
### <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M304 128a80 80 0 1 0 -160 0 80 80 0 1 0 160 0zM96 128a128 128 0 1 1 256 0A128 128 0 1 1 96 128zM49.3 464H398.7c-8.9-63.3-63.3-112-129-112H178.3c-65.7 0-120.1 48.7-129 112zM0 482.3C0 383.8 79.8 304 178.3 304h91.4C368.2 304 448 383.8 448 482.3c0 16.4-13.3 29.7-29.7 29.7H29.7C13.3 512 0 498.7 0 482.3z"/></svg> John Paul Helveston
### <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:white;overflow:visible;position:relative;"><path d="M152 24c0-13.3-10.7-24-24-24s-24 10.7-24 24V64H64C28.7 64 0 92.7 0 128v16 48V448c0 35.3 28.7 64 64 64H384c35.3 0 64-28.7 64-64V192 144 128c0-35.3-28.7-64-64-64H344V24c0-13.3-10.7-24-24-24s-24 10.7-24 24V64H152V24zM48 192H400V448c0 8.8-7.2 16-16 16H64c-8.8 0-16-7.2-16-16V192z"/></svg> November 15, 2023

]

---

# Houskeeping items

- **Final presentations** will be on 12/14 during normal class hours. You can pre-record a video presentation if you want, or you can do it live.

- [**Final Analysis**](https://madd.seas.gwu.edu/2023-Fall/project/6-final-analysis.html) (due 12/10) will also be an html page report.

- I am planning on posting all final analyses (without grades) to the course site as a showcase for future students - **please DM me if you would NOT like your report posted**. ([examples](https://madd.seas.gwu.edu/showcase/2022-fall.html) from last year)

---

# Quiz 5 - last one!

### Download the template from the #class channel

### Make sure you unzip it!

### When done, submit your `quiz5.qmd` on Blackboard

]

]

---

# Week 12: .fancy[Heterogeneity]

### 1. Mixed logit (unobserved heterogeneity)
### 2. Sub-group modeling (observed heterogeneity)

---

# Week 12: .fancy[Heterogeneity]

### 1. .orange[Mixed logit (unobserved heterogeneity)]
### 2. Sub-group modeling (observed heterogeneity)

---

background-color: #EEEDEE
class: center

# Two ways of modeling heterogeneity

]

]

---

background-color: #EEEDEE

## .center[Mixed logit]

### Preference parameters follow a distribution<br>**across sample population**

]

## `$$\tilde{u}_j = \beta_1 x_j + \varepsilon_j$$`

## `$$\beta_1 \sim \mathrm{N} (\mu_1, \sigma_1)$$`

Parameter | Estimate | Standard Error 
----------|----------|-----------------
`$\mu_1$`   | 0.1      | 0.01
`$\sigma_1$` | 0.1      | 0.01

]

---

## Which distribution should I use?

**Normal distribution**

When preferences can be positive or negative

e.g. `brand = "n"`

]

**Log-normal distribution**

When preferences should be strictly positive

e.g. `-1*price = "ln"`

]

**Fixed parameter**

When preferences appear to be homogeneous

(e.g. `$\sigma$` is very small)

]

---

### Mixed logits are not equivalent in Preference vs. WTP space

### Preference space

### `$$\tilde{u}_j = \alpha p_j + \beta x_j + \varepsilon_j$$`

### `$$\alpha \sim \ln\mathrm{N} (\mu_1, \sigma_1)$$`

### `$$\beta \sim \mathrm{N} (\mu_2, \sigma_2)$$`

]

---

### Mixed logits are not equivalent in Preference vs. WTP space

### Preference space

### `$$\tilde{u}_j = \alpha p_j + \beta x_j + \varepsilon_j$$`

### `$$\alpha \sim \ln\mathrm{N} (\mu_1, \sigma_1)$$`

### `$$\beta \sim \mathrm{N} (\mu_2, \sigma_2)$$`

### `$$\omega = \frac{\beta}{-\alpha} = \frac{\mathrm{N} (\mu_2, \sigma_2)}{- \ln\mathrm{N} (\mu_1, \sigma_1)}$$`

]

### WTP space

### `$$\tilde{u}_j = \lambda(\omega_1 x_j - p_j) + \varepsilon_j$$`

### `$$\omega_1 \sim \mathrm{N} (\mu_1, \sigma_1)$$`

]

---

# Practice Question 3

a) Use the `logitr` package to estimate the following homogeneous model:

$$
\tilde{u}_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 \delta_j^{\mathrm{feat}} + \beta_3 \delta_j^{\mathrm{dannon}} + \beta_4 \delta_j^{\mathrm{hiland}} + 
\beta_5 \delta_j^{\mathrm{weight}} + \varepsilon_j
$$

where the three `$\delta$` coefficients are dummy variables for Dannon, Hiland, and Weight Watchers brands (Yoplait is the reference level).

b) Use the `logitr` package to estimate the same model but with the following mixing distributions:

- `$\beta_1 \sim \mathrm{N} (\mu_1, \sigma_1)$`
- `$\beta_2 \sim \mathrm{N} (\mu_2, \sigma_2)$`

---

# .center[Estimating mixed logit models with `logitr`]

<br>

## 1. Open `logitr-cars`

## 2. Open `code/8.1-model-mxl.R`

]

---

## Your Turn

As a team, re-estimate the main model you used in your pilot analysis report, but now using a mixed logit model.

Carefully consider which distributions to use (i.e., normal or log-normal) for different variables.

]]

---

# Week 12: .fancy[Heterogeneity]

### 1. Mixed logit (unobserved heterogeneity)
### 2. .orange[Sub-group modeling (observed heterogeneity)]

---

# Two ways of modeling heterogeneity

]

]

---

### .center[Use interactions to model preferences for multiple groups]

### Homogenous model:

### `$\tilde{u}_j = \beta_1 x_j + \varepsilon_j$`

### Two groups: A & B

### `$\tilde{u}_j = \beta_1 x_j + \beta_2 x_j \delta^\mathrm{B} + \varepsilon_j$`

### `$\quad = (\beta_1 + \beta_2 \delta^\mathrm{B}) x_j + \varepsilon_j$`

]

Par.| Meaning 
----|--------
`$\beta_1$` | Effect of `$x_j$` for group A
`$\beta_2$` | _Difference_ in effect of `$x_j$` between groups

]

---

# What's the difference?

## Separate models <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:red;overflow:visible;position:relative;"><path d="M256 48a208 208 0 1 1 0 416 208 208 0 1 1 0-416zm0 464A256 256 0 1 0 256 0a256 256 0 1 0 0 512zM175 175c-9.4 9.4-9.4 24.6 0 33.9l47 47-47 47c-9.4 9.4-9.4 24.6 0 33.9s24.6 9.4 33.9 0l47-47 47 47c9.4 9.4 24.6 9.4 33.9 0s9.4-24.6 0-33.9l-47-47 47-47c9.4-9.4 9.4-24.6 0-33.9s-24.6-9.4-33.9 0l-47 47-47-47c-9.4-9.4-24.6-9.4-33.9 0z"/></svg>

$$
\tilde{u}_j^\mathrm{A} = \beta_1^\mathrm{A} x_j + \varepsilon_j^\mathrm{A}
$$

$$
\tilde{u}_j^\mathrm{B} = \beta_1^\mathrm{B} x_j + \varepsilon_j^\mathrm{B}
$$

]

## Single model <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:green;overflow:visible;position:relative;"><path d="M438.6 105.4c12.5 12.5 12.5 32.8 0 45.3l-256 256c-12.5 12.5-32.8 12.5-45.3 0l-128-128c-12.5-12.5-12.5-32.8 0-45.3s32.8-12.5 45.3 0L160 338.7 393.4 105.4c12.5-12.5 32.8-12.5 45.3 0z"/></svg>

`$$\tilde{u}_j = \beta_1 x_j + \beta_2 x_j \delta^\mathrm{B} + \varepsilon_j$$`

]

---

# .center[Accounting for scale differences]

## .center[Separate models <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:red;overflow:visible;position:relative;"><path d="M256 48a208 208 0 1 1 0 416 208 208 0 1 1 0-416zm0 464A256 256 0 1 0 256 0a256 256 0 1 0 0 512zM175 175c-9.4 9.4-9.4 24.6 0 33.9l47 47-47 47c-9.4 9.4-9.4 24.6 0 33.9s24.6 9.4 33.9 0l47-47 47 47c9.4 9.4 24.6 9.4 33.9 0s9.4-24.6 0-33.9l-47-47 47-47c9.4-9.4 9.4-24.6 0-33.9s-24.6-9.4-33.9 0l-47 47-47-47c-9.4-9.4-24.6-9.4-33.9 0z"/></svg>]

$$
\tilde{u}_j^\mathrm{A} = \alpha^\mathrm{A} p_j + \beta_1^\mathrm{A} x_j + \varepsilon_j^\mathrm{A}
$$

$$
\tilde{u}_j^\mathrm{B} = \alpha^\mathrm{B} p_j + \beta_1^\mathrm{B} x_j + \varepsilon_j^\mathrm{B}
$$

Imagine you got the following results

- `$\hat{\alpha}^\mathrm{A}$` = 100
- `$\hat{\beta}^\mathrm{A}$` = 200
- `$\hat{\alpha}^\mathrm{B}$` = 1
- `$\hat{\beta}^\mathrm{B}$` = 2

]

## .center[Single model <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:green;overflow:visible;position:relative;"><path d="M438.6 105.4c12.5 12.5 12.5 32.8 0 45.3l-256 256c-12.5 12.5-32.8 12.5-45.3 0l-128-128c-12.5-12.5-12.5-32.8 0-45.3s32.8-12.5 45.3 0L160 338.7 393.4 105.4c12.5-12.5 32.8-12.5 45.3 0z"/></svg>]

`$$\tilde{u}_j = \alpha_1 p_j + \alpha_2 p_j \delta^\mathrm{B} + \beta_1 x_j + \beta_2 x_j \delta^\mathrm{B} + \varepsilon_j$$`

`$$\quad = (\alpha_1 + \alpha_2 \delta^\mathrm{B}) p_j + (\beta_1 + \beta_2 \delta^\mathrm{B}) x_j + \varepsilon_j$$`

]

---

# .center[Accounting for scale differences]

## .center[Preference Space <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:red;overflow:visible;position:relative;"><path d="M256 48a208 208 0 1 1 0 416 208 208 0 1 1 0-416zm0 464A256 256 0 1 0 256 0a256 256 0 1 0 0 512zM175 175c-9.4 9.4-9.4 24.6 0 33.9l47 47-47 47c-9.4 9.4-9.4 24.6 0 33.9s24.6 9.4 33.9 0l47-47 47 47c9.4 9.4 24.6 9.4 33.9 0s9.4-24.6 0-33.9l-47-47 47-47c9.4-9.4 9.4-24.6 0-33.9s-24.6-9.4-33.9 0l-47 47-47-47c-9.4-9.4-24.6-9.4-33.9 0z"/></svg>]

$$
\tilde{u}_j^\mathrm{A} = \alpha^\mathrm{A} p_j + \beta_1^\mathrm{A} x_j + \varepsilon_j^\mathrm{A}
$$

$$
\tilde{u}_j^\mathrm{B} = \alpha^\mathrm{B} p_j + \beta_1^\mathrm{B} x_j + \varepsilon_j^\mathrm{B}
$$

Imagine you got the following results

- `$\hat{\alpha}^\mathrm{A}$` = 100
- `$\hat{\beta}^\mathrm{A}$` = 200
- `$\hat{\alpha}^\mathrm{B}$` = 1
- `$\hat{\beta}^\mathrm{B}$` = 2

]

## .center[WTP Space <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:green;overflow:visible;position:relative;"><path d="M438.6 105.4c12.5 12.5 12.5 32.8 0 45.3l-256 256c-12.5 12.5-32.8 12.5-45.3 0l-128-128c-12.5-12.5-12.5-32.8 0-45.3s32.8-12.5 45.3 0L160 338.7 393.4 105.4c12.5-12.5 32.8-12.5 45.3 0z"/></svg>]

$$
\tilde{u}_j^\mathrm{A} = \lambda^\mathrm{A}(\omega_1^\mathrm{A} x_j - p) + \varepsilon_j^\mathrm{A}
$$

$$
\tilde{u}_j^\mathrm{B} = \lambda^\mathrm{B}(\omega_1^\mathrm{B} x_j - p) + \varepsilon_j^\mathrm{B}
$$

$$
\omega = \frac{\beta}{- \alpha}
$$

- `$\hat{\omega}^\mathrm{A} = 200 / (-100) = -2$`
- `$\hat{\omega}^\mathrm{B} = 2 / (-1) = -2$`

]

---

# Practice Question 1

Suppose we estimate the following utility model describing preferences for cars:

$$
\tilde{u}_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 x_j^{\mathrm{mpg}} + \beta_3 x_j^{\mathrm{elec}} + \varepsilon_j
$$

a) Using interactions, write out a model that accounts for differences in the effects of `$x_j^{\mathrm{price}}$`, `$x_j^{\mathrm{mpg}}$`, and `$x_j^{\mathrm{elec}}$` between two groups: A and B.

b) Write out the effects of `$x_j^{\mathrm{price}}$`, `$x_j^{\mathrm{mpg}}$`, and `$x_j^{\mathrm{elec}}$` for each group.

---

# Practice Question 2

Suppose we estimate the following utility model describing preferences for chocolate bars between two groups: A & B

$$
\tilde{u}_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 x_j^{\mathrm{caco}} + \beta_3 x_j^{\mathrm{price}}\delta_j^{\mathrm{B}} + \beta_4 x_j^{\mathrm{cacao}}\delta_j^{\mathrm{B}} + \varepsilon_j
$$

The estimated model produces the following coefficients and hessian:

`$\beta$` = [-0.7, 0.1, 0.2, 0.8]

$$
H =
`\begin{bmatrix}
-6000 & 50 & 60 & 70
\\
50 & -700 & 50 & 100
\\
60 & 50 & -300 & 20
\\
70 & 100 & 20 & -6000
\end{bmatrix}`
$$

]

a) Use the `mvrnorm()` function from the `MASS` library to generate 10,000 draws of the model coefficients.

b) Use the draws to compute the mean WTP and 95% confidence intervals of the effects of `$x_j^{\mathrm{price}}$` and `$x_j^{\mathrm{cacao}}$` for each group (A & B).

]

---

# .center[Estimating mixed logit models with `logitr`]

<br>

## 1. Open `logitr-cars`

## 2. Open `code/8.2-model-mnl-groups.R`

---

## Your Turn

### Do this individually, and compare with your teammates:

- Examine the demographic and other variables in your pilot data and specify a model that estimates differences between different groups.
- Write code to estimate that model (or multiple models, e.g. WTP space models).
- Compute and compare WTP across the different groups.

]]