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10-descriptive-models.qmd
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# Descriptive Model Building
```{r}
#| message: false
#| warning: false
library(tidyverse)
```
{width="80%"}
## Learning Objectives
1. **Understand** that models don't know what they're doing, and it is the role of the data scientist to control and deploy them.
2. **Practice** transforming variables, on the fly, at the time of regression modeling in order to produce an effective, *descriptive* regression model.
3. **Appreciate** that the random variable understanding of the world that we have constructed is useful for thinking of reshaping, or transforming spaces.
## Class Announcements
1. The Regression Lab begins next week.
- Your instructor will divide you into teams.
- As part of the lab, you will perform a statistical analysis using linear regression models.
## Roadmap
**Rearview Mirror**
- Statisticians create a population model to represent the world.
- The BLP is a useful way to summarize the relationship between one outcome random variable $Y$ and input random varibles $X_1,...,X_k$
- OLS regression is an estimator for the Best Linear Predictor (BLP)
- We can capture the sampling uncertainty in an OLS regression with standard errors, and tests for model parameters.
**Today**
- The research goal determines the strategy for building a linear model.
- Description means summarizing or representing data in a compact, human-understandable way.
- We will capture complex relationships by transforming data, including using indicator variables and interaction terms.
**Looking Ahead**
- We will see how model building for explanation is different from building for description.
- The famous Classical Linear Model (CLM) allows us to apply regression to smaller samples.
## Discussion
### Three modes of model building
- Recall the three major modes of model building: Prediction, Description, Explanation.
- What is the appropriate mode for each of the following questions?
1. What is going on?
2. Why is something going on?
3. What is going to happen?
- Think of a research question you are interested in. Which mode is it aligned with?
### The statistical modeling process in different modes
- How does the modeling goal influence each of the following steps in the statistical modeling process?
- Choice of variables and transformation
- Choice of model (ols regression, neural nets, random forest, etc.)
- Model evaluation
## R Activity: Measuring the return to education
- In labor economics, a key concept is *returns to education*.\
- Our goal is description: what is the relationship between education and wages? We will proceed in two steps:
- First, we will discuss what the appropriate specifications are.
- Then we will estimate the different models to answer this question.
- We will use wage1 dataset in the wooldridge package in the following sections.
```{r}
wage1 <- wooldridge::wage1
#names(wage1)
wage1 |>
ggplot() +
aes(x=wage) +
geom_histogram()
```
### Transformations
#### Applying and Interpreting Logarithms
- Which of the following specifications best capture the relationship between education and hourly wage? (Hint: Do a quick a EDA)
- level-level: $wage = \beta_0 + \beta_1 educ + u$
- Level-log: $wage = \beta_0 + \beta_1 \ln(educ) + u$
- log-level: $\ln(wage) = \beta_0 + \beta_1 educ + u$
- log-log: $\ln(wage) = \beta_0 + \beta_1 \ln(educ) + u$
- What is the interpretation of $\beta_0$ and $\beta_1$ in your selected specification?
- Can we use $R^2$ or Adjusted $R^2$ to choose between level-level or log-level specifications?
**Remember**
- **Doing a log transformation for any reason essentially implies a fundamentally different relationship between outcome (Y) and predictor (X) that we need to capture**
#### Applying and Interpreting Polynomials
- The following specifications include two control variables: years of experience (exper) and years at current company (tenure).
- Do a quick EDA and select the specification that better suits our description goal.
- $wage = \beta_0 + \beta_1 educ + \beta_2 exper + \beta_3 tenure + u$
- $\begin{aligned}
wage &= \beta_0 + \beta_1 educ + \beta_2 exper + \beta_3 exper^2 + \\
& \beta_4 tenure + \beta_5 tenure^2 + u
\end{aligned}$
- How do you interpret the $\beta$ coefficients?
#### Applying and Interpreting Indicator variables and interaction terms
- In the following models, first, explain why the indicator variables or interaction terms have been included. Then identify the reference group (if any) and interpret all coefficients.
- $wage = \beta_0 + \beta_1 educ + \beta_2 I(educ \geq 12) + u$
- $wage = \beta_0 + \beta_1 educ + \beta_2 female + u$
- $wage = \beta_0 + \beta_1 educ + \beta_2 female + \beta_3 educ*female + u$
- $\begin{aligned}
wage &= \beta_0 + \beta_1 female + \beta_2 I(educ = 2) + \beta_3 I(educ = 3)\\
&...+ \beta_{20} I(educ = 20) + u\\
\end{aligned}$
### Estimation
**Estimating Returns to Education**
- Answer the following questions using an appropriate hypothesis test.
1. Is a year of education associated with changes to hourly wage? (Include experience and tenure without polynomial terms).
2. Is the association between wage and experience / wage and tenure non-linear?
3. Is there evidence for gender wage discrimination in the U.S.?
4. Is there any evidence for a graduation effect on wage?
- Display all estimated models in a regression table, and discuss the robustness of your results.