3.8 — Polynomial Regression

# 3.8 — Polynomial Regression
## ECON 480 • Econometrics • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/metricsF21"><i class="fa fa-github fa-fw"></i>ryansafner/metricsF21</a><br> <a href="https://metricsF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>metricsF21.classes.ryansafner.com</a><br>

---

# Outline

### [The Quadratic Model](#34)
### [The Quadratic Model: Maxima and Minima](#61)
### [Are Polynomials Necessary?](#68)

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

.quitesmall[
`$$\color{red}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i}$$`
]

]
.pull-right[

<img src="3.8-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

.quitesmall[
`$$\color{red}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i}$$`

`$$\color{green}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i+\hat{\beta_2}\text{GDP}_i^2}$$`
]

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

- Get rid of the outliers (>$60,000)

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

- Get rid of the outliers (>$60,000)

.quitesmall[
`$$\color{red}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i}$$`
]

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

- Get rid of the outliers (>$60,000)

.quitesmall[
`$$\color{red}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i}$$`

`$$\color{green}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i+\hat{\beta_2}\text{GDP}_i^2}$$`
]

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" />
]

---

# *Linear* Regression

- OLS is commonly known as “.hi[linear_ regression]” as it fits a **straight line** to data points

- Often, data and relationships between variables may *not* be linear

- Get rid of the outliers (>$60,000)

.quitesmall[
`$$\color{red}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i}$$`

`$$\color{green}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\text{GDP}_i+\hat{\beta_2}\text{GDP}_i^2}$$`

`$$\color{orange}{\widehat{\text{Life Expectancy}_i}=\hat{\beta_0}+\hat{\beta_1}\ln(\text{GDP}_i)}$$`

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" />
]

---

# Nonlinear Effects in Linear Regression

- OLS requires all *parameters* (i.e. the `$\beta$`'s) to be linear, the *regressors* `$(X$`'s) can be nonlinear:

]
--

]

]

.smallest[
- In the end, each `$X$` is always just a number in the data, OLS can always estimate parameters for it; but *plotting* the modelled points `$(X_i, \hat{Y_i})$` can result in a curve!

]
---

# Sources of Nonlinearities

- Effect of `$X_1 \rightarrow Y$` might be nonlinear if:

1. `$X_1 \rightarrow Y$` is different for different levels of `$X_1$`
  - e.g. **diminishing returns**: `$\uparrow X_1$` increases `$Y$` at a *decreasing* rate
  - e.g. **increasing returns**: `$\uparrow X_1$` increases `$Y$` at an *increasing* rate

2. `$X_1 \rightarrow Y$` is different for different levels of `$X_2$`
  - e.g. interaction effects (last lesson)

---

# Nonlinearities Alter Marginal Effects

- **Linear**:
`$$Y=\hat{\beta_0}+\hat{\beta_1}X$$`

- marginal effect (slope), `$(\hat{\beta_1}) = \frac{\Delta Y}{\Delta X}$` is constant for all `$X$`

]
.pull-right[

<img src="3.8-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]

---

# Nonlinearities Alter Marginal Effects

- **Polynomial**:
`$$Y=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2$$`

- Marginal effect, “slope” `$\left(\neq \hat{\beta_1}\right)$` *depends on the value of* `$X$`!

]
.pull-right[

<img src="3.8-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

---

# Sources of Nonlinearities III

- **Interaction Effect**:
`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X_1+\hat{\beta_2}X_2+\hat{\beta_3}X_1 \times X_2$$`

- Marginal effect, “slope” *depends on the value of* `$X_2$`!

- Easy example: if `$X_2$` is a dummy variable:
  - .blue[`\$X_2=0\$` (control)] vs. .pink[`\$X_2=1\$` (treatment)]

]
.pull-right[

<img src="3.8-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" />
]

---

# Polynomial Functions of `$X$` I

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X$$`

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" />
]

---

# Polynomial Functions of `$X$` I

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X$$`

- .green[Quadratic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2$$`

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" />
]

---
# Polynomial Functions of `$X$` I

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X$$`

- .green[Quadratic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2$$`

- .orange[Cubic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2+\hat{\beta_3}X^3$$`

]
]

<img src="3.8-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" />
]

---

# Polynomial Functions of `$X$` I

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X$$`

- .green[Quadratic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2$$`

- .orange[Cubic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2+\hat{\beta_3}X^3$$`

- .purple[Quartic]

`$$\hat{Y}=\hat{\beta_0}+\hat{\beta_1}X+\hat{\beta_2}X^2+\hat{\beta_3}X^3+\hat{\beta_4}X^4$$`
]
]

<img src="3.8-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" />
]

---

# Polynomial Functions of `$X$` I

`$$\hat{Y_i} = \hat{\beta_0} + \hat{\beta_1} X_i + \hat{\beta_2} X_i^2 + \cdots + \hat{\beta_{\color{#e64173}{r}}} X_i^{\color{#e64173}{r}} + u_i$$`
--

- Where `$\color{e64173}{r}$` is the highest power `$X_i$` is raised to
  - quadratic `$\color{e64173}{r=2}$`
  - cubic `$\color{e64173}{r=3}$`

- The graph of an `$r$`<sup>th</sup>-degree polynomial function has `$(r-1)$` bends

- Just another multivariate OLS regression model!

---

# The Quadratic Model

---

# Quadratic Model

`$$\hat{Y_i} = \hat{\beta_0} + \hat{\beta_1} X_i + \hat{\beta_2} X_i^2$$`

- .hi[Quadratic model] has `$X$` and `$X^2$` variables in it (yes, need both!)

- How to interpret coefficients (betas)?
  - `$\beta_0$` as “intercept” and `$\beta_1$` as “slope” makes no sense  🧐
  - `$\beta_1$` as effect `$X_i \rightarrow Y_i$` holding `$X_i^2$` constant??<sup>.magenta[†]</sup>

.footnote[<sup>.magenta[†]</sup> Note: this is *not* a perfect multicollinearity problem! Correlation only measures *linear* relationships!]

- **Estimate marginal effects** by calculating predicted `$\hat{Y_i}$` for different levels of `$X_i$`

---

# Quadratic Model: Calculating Marginal Effects

`$$\hat{Y_i} = \hat{\beta_0} + \hat{\beta_1} X_i + \hat{\beta_2} X_i^2$$`

- What is the .hi[marginal effect] of `$\Delta X_i \rightarrow \Delta Y_i$`?

- Take the **derivative** of `$Y_i$` with respect to `$X_i$`:
`$$\frac{\partial \, Y_i}{\partial \, X_i} = \hat{\beta_1}+2\hat{\beta_2} X_i$$`

- .hi[Marginal effect] of a 1 unit change in `$X_i$` is a `$\color{#6A5ACD}{\left(\hat{\beta_1}+2\hat{\beta_2} X_i \right)}$` unit change in `$Y$`

---

# Quadratic Model: Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$$\widehat{\text{Life Expectancy}_i} = \hat{\beta_0}+\hat{\beta_1} \, \text{GDP per capita}_i+\hat{\beta_2}\, \text{GDP per capita}^2_i$$`
]

- Use `gapminder` package and data

```r
library(gapminder)
```

---

# Quadratic Model: Example II

```r
gapminder <- gapminder %>%
  mutate(GDP_t = gdpPercap/1000)

gapminder %>% head() # look at it
```

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]

---

# Quadratic Model: Example III

```r
gapminder <- gapminder %>%
  mutate(GDP_sq = GDP_t^2)

gapminder %>% head() # look at it
```

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---

# Quadratic Model: Example IV

```r
library(broom)
reg1 <- lm(lifeExp ~ GDP_t + GDP_sq, data = gapminder)

reg1 %>% tidy()
```

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  </script>
</div>
]

---

# Quadratic Model: Example V

- OR use `gdp_t` and add the “transform” command in regression, `I(gdp_t^2)`

```r
reg1_alt <- lm(lifeExp ~ GDP_t + I(GDP_t^2), data = gapminder)

reg1_alt %>% tidy()
```

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]

---

# Quadratic Model: Example VI

.smallest[
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  </script>
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]

.smallest[
`$$\widehat{\text{Life Expectancy}_i} = 50.52+1.55 \, \text{GDP}_i - 0.02\, \text{GDP}^2_i$$`
]

- Marginal effect of GDP on Life Expectancy **depends on initial value of GDP!**
]

---

# Quadratic Model: Example VII

.smallest[
`$$\begin{align*}
\frac{\partial \, Y}{\partial \; X} &= \hat{\beta_1}+2\hat{\beta_2} X_i\\
\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} &\approx 1.55+2(-0.02) \, \text{GDP}\\
 &\approx \color{#e64173}{1.55-0.04 \, \text{GDP}}\\
\end{align*}$$`
]

---

# Quadratic Model: Example VIII

`$$\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} = 1.55-0.04 \, \text{GDP}$$`

Marginal effect of GDP if GDP `$=5$` ($ thousand):

`$$\begin{align*}
\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} &= 1.55-0.04\text{GDP}\\
&= 1.55-0.04(5)\\
&= 1.55-0.20\\
&=1.35\\
\end{align*}$$`

- i.e. for every addition $1 (thousand) in GDP per capita, average life expectancy increases by 1.35 years

---

# Quadratic Model: Example IX

`$$\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} = 1.55-0.04 \, \text{GDP}$$`

Marginal effect of GDP if GDP `$=25$` ($ thousand):

`$$\begin{align*}
\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} &= 1.55-0.04\text{GDP}\\
&= 1.55-0.04(25)\\
&= 1.55-1.00\\
&=0.55\\
\end{align*}$$`

- i.e. for every addition $1 (thousand) in GDP per capita, average life expectancy increases by 0.55 years

---

# Quadratic Model: Example X

`$$\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} = 1.55-0.04 \, \text{GDP}$$`

Marginal effect of GDP if GDP `$=50$` ($ thousand):

`$$\begin{align*}
\frac{\partial \, \text{Life Expectancy}}{\partial \, \text{GDP}} &= 1.55-0.04\text{GDP}\\
&= 1.55-0.04(50)\\
&= 1.55-2.00\\
&=-0.45\\
\end{align*}$$`

- i.e. for every addition $1 (thousand) in GDP per capita, average life expectancy *decreases* by 0.45 years

---

# Quadratic Model: Example XI

.smallest[
`$$\begin{align*}\widehat{\text{Life Expectancy}_i} &= 50.52+1.55 \, \text{GDP per capita}_i - 0.02\, \text{GDP per capita}^2_i \\
\frac{\partial \, \text{Life Expectancy}}{d \, \text{GDP}} &= 1.55-0.04\text{GDP} \\ \end{align*}$$`

]

| *Initial* GDP per capita | Marginal Effect<sup>.magenta[†]<sup> |
|----------------|-------------------:|
| $5,000 | `$1.35$` years |
| $25,000 | `$0.55$` years |
| $50,000 | `$-0.45$` years |

---

# Quadratic Model: Example XII

```r
ggplot(data = gapminder)+
  aes(x = GDP_t,
      y = lifeExp)+
  geom_point(color="blue", alpha=0.5)+
* stat_smooth(method = "lm",
*             formula = y ~ x + I(x^2),
*             color="green")+
  geom_vline(xintercept=c(5,25,50),
             linetype="dashed",
             color="red", size = 1)+
  scale_x_continuous(labels=scales::dollar,
                     breaks=seq(0,120,10))+
  scale_y_continuous(breaks=seq(0,100,10),
                     limits=c(0,100))+
  labs(x = "GDP per Capita (in Thousands)",
       y = "Life Expectancy (Years)")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]

---

# The Quadratic Model: Maxima and Minima

---

# Quadratic Model: Maxima and Minima I

- For a polynomial model, we can also find the predicted **maximum** or **minimum** of `$\hat{Y_i}$`

- A quadratic model has a single global maximum or minimum (1 bend)

- By calculus, a minimum or maximum occurs where:

`$$\begin{align*}
		\frac{ \partial \, Y_i}{\partial \, X_i} &=0\\
		\beta_1 + 2\beta_2 X_i &= 0\\
		2\beta_2 X_i&= -\beta_1\\
		X_i^*&=-\frac{\beta_1}{2\beta_2}\\
		\end{align*}$$`

---

# Quadratic Model: Maxima and Minima II

.quitesmall[
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]
]

`$$\begin{align*}
GDP_i^*&=-\frac{\beta_1}{2\beta_2}\\
GDP_i^*&=-\frac{(1.55)}{2(-0.015)}\\
GDP_i^*& \approx 51.67\\ \end{align*}$$`

]

---

# Quadratic Model: Maxima and Minima III

```r
ggplot(data = gapminder)+
  aes(x = GDP_t,
      y = lifeExp)+
  geom_point(color="blue", alpha=0.5)+
* stat_smooth(method = "lm",
*             formula = y ~ x + I(x^2),
*             color="green")+
  geom_vline(xintercept=51.67, linetype="dashed", color="red", size = 1)+
  geom_label(x=51.67, y=90, label="$51.67", color="red")+
  scale_x_continuous(labels=scales::dollar,
                     breaks=seq(0,120,10))+
  scale_y_continuous(breaks=seq(0,100,10),
                     limits=c(0,100))+
  labs(x = "GDP per Capita (in Thousands)",
       y = "Life Expectancy (Years)")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]
.pull-right[
<img src="3.8-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" />
]

---

# Are Polynomials Necessary?

---

# Determining Polynomials are Necessary I

- Is the quadratic term necessary?

]
--

.smallest[
- Determine if `$\hat{\beta_2}$` (on `$X_i^2)$` is statistically significant:
  - `$H_0: \hat{\beta_2}=0$`
  - `$H_a: \hat{\beta_2} \neq 0$`
]

.smallest[
- Statistically significant `$\implies$` we should keep the quadratic model
  - If we only ran a linear model, it would be incorrect!
]
---

# Determining Polynomials are Necessary II

.quitesmall[
`$$\color{#6A5ACD}{\widehat{\text{Life Expectancy}_i} = \hat{\beta_0}+\hat{\beta_1} GDP_i+\hat{\beta_2}GDP^2_i+\hat{\beta_3}GDP_i^3}$$`
]
]

]

---

# Determining Polynomials are Necessary III

- In general, you should have a .hi-purple[compelling theoretical reason] why data or relationships should .hi-purple[“change direction”] multiple times

- Or clear data patterns that have multiple “bends”

- Recall, [we care more](https://metricsf21.classes.ryansafner.com/slides/3.1-slides#3) about accurately measuring the causal effect between `$X$` and `$Y$`, rather than getting the most accurate prediction possible for `$\hat{Y}$`
]

---

# A Second Polynomial Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How does a school district's average income affect Test scores?
]

]

<img src="3.8-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" />
]

---

# A Second Polynomial Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How does a school district's average income affect Test scores?
]

.smallest[
`$$\color{red}{\widehat{\text{Test Score}_i}=\hat{\beta_0}+\hat{\beta_1}\text{Income}_i}$$`
]
]

---

# A Second Polynomial Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How does a school district's average income affect Test scores?
]

.quitesmall[
`$$\color{green}{\widehat{\text{Test Score}_i}=\hat{\beta_0}+\hat{\beta_1}\text{Income}_i+\hat{\beta_1}\text{Income}_i^2}$$`
]

]

---

# A Second Polynomial Example II

.pull-left[
.tiny[
<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"607.30173501","3":"3.046219282","4":"199.362449","5":"0.000000e+00"},{"1":"avginc","2":"3.85099474","3":"0.304261693","4":"12.656850","5":"2.690099e-31"},{"1":"I(avginc^2)","2":"-0.04230846","3":"0.006260061","4":"-6.758474","5":"4.713383e-11"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

]

---

# A Second Polynomial Example III

.pull-left[
.tiny[
<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"6.000790e+02","3":"5.8295880342","4":"102.936774","5":"4.611745e-298"},{"1":"avginc","2":"5.018677e+00","3":"0.8594537744","4":"5.839379","5":"1.056874e-08"},{"1":"I(avginc^2)","2":"-9.580515e-02","3":"0.0373591998","4":"-2.564433","5":"1.068452e-02"},{"1":"I(avginc^3)","2":"6.854842e-04","3":"0.0004719549","4":"1.452436","5":"1.471343e-01"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

]

---

# Strategy for Polynomial Model Specification

1. Are there good theoretical reasons for relationships changing (e.g. increasing/decreasing returns)?

2. Plot your data: does a straight line fit well enough?

3. Specify a polynomial function of a higher power (start with 2) and estimate OLS regression

4. Use `$t$`-test to determine if higher-power term is significant

5. Interpret effect of change in `$X$` on `$Y$`

6. Repeat steps 3-5 as necessary