3.9 — Logarithmic Regression

# 3.9 — Logarithmic 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

### [Natural Logarithms](#7)

### [Linear-Log Model](#37)

### [Log-Linear Model](#47)

### [Log-Log Model](#58)

### [Comparing Across Units](#72)

### [Joint Hypothesis Testing](#78)

---

# Nonlinearities

- Consider the `gapminder` example
]

]

---

# Nonlinearities

- Consider the `gapminder` example

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

]
]

]

---

# Nonlinearities

- Consider the `gapminder` example

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

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

]
]

]

---

# Nonlinearities

- Consider the `gapminder` example

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

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

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

]

]

---

# Natural Logarithms

---

# Logarithmic Models

.pull-left[
.smallest[
- Another useful model for nonlinear data is the .hi[logarithmic model]<sup>.magenta[†]</sup>
  - We transform either `$X$`, `$Y$`, or *both* by taking the .hi-purple[(natural) logarithm]

- Logarithmic model has two additional advantages
  1. We can easily interpret coefficients as **percentage changes** or **elasticities**
  2. Useful economic shape: diminishing returns (production functions, utility functions, etc)
]

.tiny[<sup>.magenta[†]</sup> Don’t confuse this with a .hi[logistic (logit) model] for *dependent* dummy variables.]

]

---

# The Natural Logarithm

- The .red[exponential function], `$Y=e^X$` or `$Y=exp(X)$`, where base `$e=2.71828...$`

- .blue[Natural logarithm] is the inverse, `$Y=ln(X)$`

]

---

# The Natural Logarithm: Review I

.smallest[
- .hi[Exponents] are defined as
`$$\color{#6A5ACD}{b}^{\color{#e64173}{n}}=\underbrace{\color{#6A5ACD}{b} \times \color{#6A5ACD}{b} \times \cdots \times \color{#6A5ACD}{b}}_{\color{#e64173}{n} \text{ times}}$$`
  - where base `$\color{#6A5ACD}{b}$` is multiplied by itself `$\color{#e64173}{n}$` times
]
--

.smallest[
- .green[**Example**]: `$\color{#6A5ACD}{2}^{\color{#e64173}{3}}=\underbrace{\color{#6A5ACD}{2} \times \color{#6A5ACD}{2} \times \color{#6A5ACD}{2}}_{\color{#e64173}{n=3}}=\color{#314f4f}{8}$`
]

.smallest[
- .hi-purple[Logarithms] are the inverse, defined as the exponents in the expressions above
`$$\text{If } \color{#6A5ACD}{b}^{\color{#e64173}{n}}=\color{#314f4f}{y}\text{, then }log_{\color{#6A5ACD}{b}}(\color{#314f4f}{y})=\color{#e64173}{n}$$`
  - `$\color{#e64173}{n}$` is the number you must raise `$\color{#6A5ACD}{b}$` to in order to get `$\color{#314f4f}{y}$`
]

.smallest[
- .green[**Example**]: `$log_{\color{#6A5ACD}{2}}(\color{#314f4f}{8})=\color{#e64173}{3}$`
]

---

# The Natural Logarithm: Review II

- Logarithms can have any base, but common to use the .hi-purple[natural logarithm] `$(\ln)$` with base `$\mathbf{e=2.71828...}$`
`$$\text{If } e^n=y\text{, then } \ln(y)=n$$`

---

# The Natural Logarithm: Properties

- Natural logs have a lot of useful properties:

1. `$\ln(\frac{1}{x})=-\ln(x)$`

2. `$\ln(ab)=\ln(a)+\ln(b)$`

3.  `$\ln(\frac{x}{a})=\ln(x)-\ln(a)$`

4. `$\ln(x^a)=a \, \ln(x)$`

5.  `$\frac{d  \, \ln \, x}{d \, x} = \frac{1}{x}$`

---

# The Natural Logarithm: Example

`$$\underbrace{\ln(x+\Delta x) - \ln(x)}_{\text{Difference in logs}} \approx \underbrace{\frac{\Delta x}{x}}_{\text{Relative change}}$$`

]

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Let `$x=100$` and `$\Delta x =1$`, relative change is:

`$$\frac{\Delta x}{x} = \frac{(101-100)}{100} = 0.01 \text{ or }1\%$$`

- The logged difference:
`$$\ln(101)-\ln(100) = 0.00995 \approx 1\%$$`
]
]
--

.smallest[
- This allows us to very easily interpret coefficients as **percent changes** or .hi-purple[elasticities]
]

---

# Elasticity

.smallest[
- An .hi[elasticity] between any two variables, `$\epsilon_{Y,X}$` describes the .hi-purple[responsiveness] (in %) of one variable `$(Y)$` to a change in another `$(X)$`

]

`$$\epsilon_{Y,X}=\frac{\% \Delta Y}{\% \Delta X} =\cfrac{\left(\frac{\Delta Y}{Y}\right)}{\left( \frac{\Delta X}{X}\right)}$$`

.smallest[
- .hi-purple[Interpretation]: a 1% change in `$X$` will cause a `$\epsilon_{Y,X}$`% chang in `$Y$`
]

---

# Math FYI: Cobb Douglas Functions and Logs

- One of the (many) reasons why economists love Cobb-Douglas functions:
`$$Y=AL^{\alpha}K^{\beta}$$`

- Taking logs, relationship becomes linear:

`$$\ln(Y)=\ln(A)+\alpha \ln(L)+ \beta \ln(K)$$`

- With data on `$(Y, L, K)$` and linear regression, can estimate `$\alpha$` and `$\beta$`
  - `$\alpha$`: elasticity of `$Y$` with respect to `$L$`
      - A 1% change in `$L$` will lead to an `$\alpha$`% change in `$Y$`
  - `$\beta$`: elasticity of `$Y$` with respect to `$K$`
      - A 1% change in `$K$` will lead to a `$\beta$`% change in `$Y$`

---

# Math FYI: Cobb Douglas Functions and Logs

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.hi-green[Example]: Cobb-Douglas production function:
`$$Y=2L^{0.75}K^{0.25}$$`
]

- Taking logs:

`$$\ln Y=\ln 2+0.75 \ln L + 0.25 \ln K$$`

- A 1% change in `$L$` will yield a 0.75% change in output `$Y$`

- A 1% change in `$K$` will yield a 0.25% change in output `$Y$`

---

# Logarithms in R I

- The `log()` function can easily take the logarithm

```r
gapminder <- gapminder %>%
  mutate(loggdp = log(gdpPercap)) # log GDP per capita

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

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---

# Logarithms in R II

- Note, `log()` by default is the **natural logarithm `$ln()$`**, i.e. base `e`
  - Can change base with e.g. `log(x, base = 5)`
  - Some common built-in logs: `log10`, `log2`

```r
log10(100)
```

```
## [1] 2
```

```r
log2(16)
```

```
## [1] 4
```

```r
log(19683, base=3)
```

```
## [1] 9
```

---

# Logarithms in R III

- Note when running a regression, you can pre-transform the data into logs (as I did above), or just add `log()` around a variable in the regression

```r
lm(lifeExp ~ loggdp,
   data = gapminder) %>%
  tidy()
```

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```r
lm(lifeExp ~ log(gdpPercap),
   data = gapminder) %>%
  tidy()
```

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---

# Types of Logarithmic Models

- Three types of log regression models, depending on which variables we log

1.  .hi-purple[Linear-log model:] `$Y_i=\beta_0+\beta_1 \color{#e64173}{\ln X_i}$`

2. .hi-purple[Log-linear model:] `$\color{#e64173}{\ln Y_i}=\beta_0+\beta_1X_i$`

3. .hi-purple[Log-log model:] `$\color{#e64173}{\ln Y_i}=\beta_0+\beta_1 \color{#e64173}{\ln X_i}$`

---

# Linear-Log Model

---

# Linear-Log Model

- .hi-purple[Linear-log model] has an independent variable `$(X)$` that is logged

`$$\begin{align*}
Y&=\beta_0+\beta_1 \color{#e64173}{\ln X_i}\\
\beta_1&=\cfrac{\Delta Y}{\big(\frac{\Delta X}{X}\big)}\\
\end{align*}$$`

- .hi-purple[**Marginal effect of** `\$\mathbf{X \rightarrow Y}\$`: a **1%** change in `\$X \rightarrow\$` a `\$\frac{\beta_1}{100}\$` **unit** change in `\$Y\$`]

---

# Linear-Log Model in R

```r
lin_log_reg <- lm(lifeExp ~ loggdp,
                  data = gapminder)

library(broom)

lin_log_reg %>% tidy()
```

]

---

# Linear-Log Model in R

```r
lin_log_reg <- lm(lifeExp ~ loggdp,
                  data = gapminder)

library(broom)

lin_log_reg %>% tidy()
```

- A **1% change in GDP** `$\rightarrow$` a `$\frac{9.41}{100}=$` **0.0841 year increase** in Life Expectancy

]
]

---

# Linear-Log Model in R

```r
lin_log_reg <- lm(lifeExp ~ loggdp,
                  data = gapminder)

library(broom)

lin_log_reg %>% tidy()
```

- A **1% change in GDP** `$\rightarrow$` a `$\frac{9.41}{100}=$` **0.0841 year increase** in Life Expectancy

- A **25% fall in GDP** `$\rightarrow$` a `$(-25 \times 0.0841)=$` **2.1025 year _decrease_** in Life Expectancy

]
]

---

# Linear-Log Model in R

```r
lin_log_reg <- lm(lifeExp ~ loggdp,
                  data = gapminder)

library(broom)

lin_log_reg %>% tidy()
```

- A **1% change in GDP** `$\rightarrow$` a `$\frac{9.41}{100}=$` **0.0841 year increase** in Life Expectancy

- A **25% fall in GDP** `$\rightarrow$` a `$(-25 \times 0.0841)=$` **2.1025 year _decrease_** in Life Expectancy

- A **100% rise in GDP** `$\rightarrow$` a `$(100 \times 0.0841)=$` **8.4100 year increase** in Life Expectancy

]
]
---

# Linear-Log Model Graph I

```r
ggplot(data = gapminder)+
  aes(x = gdpPercap,
      y = lifeExp)+
  geom_point(color="blue", alpha=0.5)+
* geom_smooth(method="lm",
*             formula=y~log(x),
*             color="orange")+
  scale_x_continuous(labels=scales::dollar,
                     breaks=seq(0,120000,20000))+
  scale_y_continuous(breaks=seq(0,100,10),
                     limits=c(0,100))+
  labs(x = "GDP per Capita",
       y = "Life Expectancy (Years)")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]

---

# Linear-Log Model Graph II

```r
ggplot(data = gapminder)+
* aes(x = loggdp,
      y = lifeExp)+ 
  geom_point(color="blue", alpha=0.5)+
* geom_smooth(method="lm", color="orange")+
  scale_y_continuous(breaks=seq(0,100,10),
                     limits=c(0,100))+
  labs(x = "Log GDP per Capita",
       y = "Life Expectancy (Years)")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]

---

# Log-Linear Model

---

# Log-Linear Model

- .hi-purple[Log-linear model] has the dependent variable `$(Y)$` logged

`$$\begin{align*}
\color{#e64173}{\ln Y_i}&=\beta_0+\beta_1 X\\
\beta_1&=\cfrac{\big(\frac{\Delta Y}{Y}\big)}{\Delta X}\\
\end{align*}$$`

- .hi-purple[**Marginal effect of** `\$\mathbf{X \rightarrow Y}\$`: a **1 unit** change in `\$X \rightarrow\$` a `\$\beta_1 \times 100\$` **%** change in `\$Y\$`]

---

# Log-Linear Model in R (Preliminaries)

.smallest[
- We will again have very large/small coefficients if we deal with GDP directly, again let's transform `gdpPercap` into $1,000s, call it `gdp_t`

- Then log LifeExp
]

```r
gapminder <- gapminder %>%
  mutate(gdp_t = gdpPercap/1000, # first make GDP/capita in $1000s
         loglife = log(lifeExp)) # take the log of LifeExp
gapminder %>% head() # look at it
```

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---

# Log-Linear Model in R

```r
log_lin_reg <- lm(loglife~gdp_t,
                data = gapminder)

log_lin_reg %>% tidy()
```

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]

---

# Log-Linear Model in R

```r
log_lin_reg <- lm(loglife~gdp_t,
                data = gapminder)

log_lin_reg %>% tidy()
```

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]

- A **$1 (thousand) change in GDP** `$\rightarrow$` a `$0.013 \times 100\%=$` **1.3% increase** in Life Expectancy

]
]

---

# Log-Linear Model in R

```r
log_lin_reg <- lm(loglife~gdp_t,
                data = gapminder)

log_lin_reg %>% tidy()
```

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- A **$1 (thousand) change in GDP** `$\rightarrow$` a `$0.013 \times 100\%=$` **1.3% increase** in Life Expectancy

- A **$25 (thousand) fall in GDP** `$\rightarrow$` a `$(-25 \times 1.3\%)=$` **32.5% decrease** in Life Expectancy

]
]

---

# Log-Linear Model in R

```r
log_lin_reg <- lm(loglife~gdp_t,
                data = gapminder)

log_lin_reg %>% tidy()
```

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]

- A **$1 (thousand) change in GDP** `$\rightarrow$` a `$0.013 \times 100\%=$` **1.3% increase** in Life Expectancy

- A **$25 (thousand) fall in GDP** `$\rightarrow$` a `$(-25 \times 1.3\%)=$` **32.5% decrease** in Life Expectancy

- A **$100 (thousand) rise in GDP** `$\rightarrow$` a `$(100 \times 1.3\%)=$` **130% increase** in Life Expectancy 
]
]

---

# Linear-Log Model Graph I

```r
ggplot(data = gapminder)+
* aes(x = gdp_t,
*     y = loglife)+
  geom_point(color="blue", alpha=0.5)+
* geom_smooth(method="lm", color="orange")+
  scale_x_continuous(labels=scales::dollar,
                     breaks=seq(0,120,20))+
  labs(x = "GDP per Capita ($ Thousands)",
       y = "Log Life Expectancy")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]

---

# Log-Log Model

---

# Log-Log Model

- .hi-purple[Log-log model] has both variables `$(X \text{ and } Y)$` logged

`$$\begin{align*}
\color{#e64173}{\ln Y_i}&=\beta_0+\beta_1 \color{#e64173}{\ln X_i}\\
\beta_1&=\cfrac{\big(\frac{\Delta Y}{Y}\big)}{\big(\frac{\Delta X}{X}\big)}\\
\end{align*}$$`

- .hi-purple[**Marginal effect of** `\$\mathbf{X \rightarrow Y}\$`: a **1%** change in `\$X \rightarrow\$` a `\$\beta_1\$` **%** change in `\$Y\$`]

- `$\beta_1$` is the .hi-turquoise[elasticity] of `$Y$` with respect to `$X$`!

---

# Log-Log Model in R

```r
log_log_reg <- lm(loglife ~ loggdp,
                  data = gapminder)

log_log_reg %>% tidy()
```

<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":"2.864177","3":"0.02328274","4":"123.01718","5":"0"},{"1":"loggdp","2":"0.146549","3":"0.00282126","4":"51.94452","5":"0"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

]

---

# Log-Log Model in R

```r
log_log_reg <- lm(loglife ~ loggdp,
                  data = gapminder)

log_log_reg %>% tidy()
```

<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":"2.864177","3":"0.02328274","4":"123.01718","5":"0"},{"1":"loggdp","2":"0.146549","3":"0.00282126","4":"51.94452","5":"0"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

.smallest[
`$$\widehat{\text{ln Life Expectancy}}_i=2.864+0.147 \, \text{ln GDP}_i$$`
- A **1% change in GDP** `$\rightarrow$` a **0.147% increase** in Life Expectancy

]
]

---

# Log-Log Model in R

```r
log_log_reg <- lm(loglife ~ loggdp,
                  data = gapminder)

log_log_reg %>% tidy()
```

<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":"2.864177","3":"0.02328274","4":"123.01718","5":"0"},{"1":"loggdp","2":"0.146549","3":"0.00282126","4":"51.94452","5":"0"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

- A **1% change in GDP** `$\rightarrow$` a **0.147% increase** in Life Expectancy

- A **25% fall in GDP** `$\rightarrow$` a `$(-25 \times 0.147\%)=$` **3.675% decrease** in Life Expectancy

]
]

---

# Log-Log Model in R

```r
log_log_reg <- lm(loglife ~ loggdp,
                  data = gapminder)

log_log_reg %>% tidy()
```

<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":"2.864177","3":"0.02328274","4":"123.01718","5":"0"},{"1":"loggdp","2":"0.146549","3":"0.00282126","4":"51.94452","5":"0"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

- A **1% change in GDP** `$\rightarrow$` a **0.147% increase** in Life Expectancy

- A **25% fall in GDP** `$\rightarrow$` a `$(-25 \times 0.147\%)=$` **3.675% decrease** in Life Expectancy

- A **100% rise in GDP** `$\rightarrow$` a `$(100 \times 0.147\%)=$` **14.7% increase** in Life Expectancy

]
]
---

# Log-Log Model Graph I

```r
ggplot(data = gapminder)+
* aes(x = loggdp,
*     y = loglife)+
  geom_point(color="blue", alpha=0.5)+
* geom_smooth(method="lm", color="orange")+
  labs(x = "Log GDP per Capita",
       y = "Log Life Expectancy")+
  ggthemes::theme_pander(base_family = "Fira Sans Condensed",
           base_size=16)
```
]
]

---

# Comparing Models I

| Model | Equation | Interpretation |
|-------|----------|----------------|
| Linear-.hi[Log] | `$Y=\beta_0+\beta_1 \color{#e64173}{\ln X}$` | 1.hi[%] change in `$X \rightarrow \frac{\hat{\beta_1}}{100}$` **unit** change in `$Y$` |
| .hi[Log]-Linear | `$\color{#e64173}{\ln Y}=\beta_0+\beta_1X$` | 1 **unit** change in `$X \rightarrow \hat{\beta_1}\times 100$`.hi[%] change in `$Y$` |
| .hi[Log]-.hi[Log] | `$\color{#e64173}{\ln Y}=\beta_0+\beta_1\color{#e64173}{\ln X}$` | 1.hi[%] change in `$X \rightarrow \hat{\beta_1}$`.hi[%] change in `$Y$` |

- Hint: the variable that gets .hi[logged] changes in .hi[percent] terms, the variable not logged changes in **unit** terms
  - Going from units `$\rightarrow$` percent: multiply by 100
  - Going from percent `$\rightarrow$` units: divide by 100

---

# Comparing Models II

```r
library(huxtable)
huxreg("Life Exp." = lin_log_reg,
       "Log Life Exp." = log_lin_reg,
       "Log Life Exp." = log_log_reg,
       coefs = c("Constant" = "(Intercept)",
                 "GDP ($1000s)" = "gdp_t",
                 "Log GDP" = "loggdp"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
```
]

- Models are very different units, how to choose? 
  - Compare `\$R^2\$`’s
  - Compare graphs
  - Compare intution

]

.pull-left[
.tiny[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-29">
<col><col><col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Life Exp.</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Log Life Exp.</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Log Life Exp.</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-9.10 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.97 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">2.86 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1.23)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.02)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">GDP ($1000s)</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.01 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.00)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Log GDP</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">8.41 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.15 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.15)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.00)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1704       </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1704       </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1704       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.65    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.30    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.61    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">7.62    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.19    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.14    </td></tr>
<tr>
<th colspan="4" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]

---

# Comparing Models III

.smallest[
| Linear-.hi[Log] | .hi[Log]-Linear | .hi[Log]-.hi[Log] |
|:----------:|:----------:|:-------:|
| ![](3.9-slides_files/figure-html/unnamed-chunk-17-1.png) | ![](3.9-slides_files/figure-html/unnamed-chunk-23-1.png) | ![](3.9-slides_files/figure-html/unnamed-chunk-28-1.png) |
| `$\hat{Y_i}=\hat{\beta_0}+\hat{\beta_1}\color{#e64173}{\ln X_i}$` | `$\color{#e64173}{\ln Y_i}=\hat{\beta_0}+\hat{\beta_1}X_i$` | `$\color{#e64173}{\ln Y_i}=\hat{\beta_0}+\hat{\beta_1}\color{#e64173}{\ln X_i}$` |
| `$R^2=0.65$` | `$R^2=0.30$` | `$R^2=0.61$` |
]

---

# When to Log?

.smaller[
- In practice, the following types of variables are logged:
  - Variables that must always be positive (prices, sales, market values)
  - Very large numbers (population, GDP)
  - Variables we want to talk about as percentage changes or growth rates (money supply, population, GDP)
  - Variables that have diminishing returns (output, utility)
  - Variables that have nonlinear scatterplots

]
--

.smaller[
- Avoid logs for:
  - Variables that are less than one, decimals, 0, or negative
  - Categorical variables (season, gender, political party)
  - Time variables (year, week, day)
]

---

# Comparing Across Units

---

# Comparing Coefficients of Different Units I

- We often want to compare coefficients to see which variable `$X_1$` or `$X_2$` has a bigger effect on `$Y$`

- What if `$X_1$` and `$X_2$` are different units?

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:
`$$\begin{align*}
\widehat{\text{Salary}_i}&=\beta_0+\beta_1\, \text{Batting average}_i+\beta_2\, \text{Home runs}_i\\
\widehat{\text{Salary}_i}&=-\text{2,869,439.40}+\text{12,417,629.72} \, \text{Batting average}_i+\text{129,627.36}\, \text{Home runs}_i\\
\end{align*}$$`
]
]

---

# Comparing Coefficients of Different Units II

- An easy way is to .hi[standardize]<sup>.magenta[†]</sup> the variables (i.e. take the `$Z$`-score)

`$$X_Z=\frac{X_i-\overline{X}}{sd(X)}$$`

---

# Comparing Coefficients of Different Units: Example

.smallest[
| Variable | Mean | Std. Dev. |
|----------|------|-----------|
| Salary | $2,024,616 | $2,764,512 |
| Batting Average | 0.267 | 0.031 |
| Home Runs | 12.11 | 10.31 |

]

.quitesmall[
`$$\begin{align*}\scriptsize  
	\widehat{\text{Salary}_i}&=-\text{2,869,439.40}+\text{12,417,629.72} \, \text{Batting average}_i+\text{129,627.36} \, \text{Home runs}_i\\ 
	\widehat{\text{Salary}_Z}&=\text{0.00}+\text{0.14} \, \text{Batting average}_Z+\text{0.48} \, \text{Home runs}_Z\\
	\end{align*}$$`
]
--

.quitesmall[
- .hi-purple[Marginal effects] on `$Y$` (in *standard deviations* of `$Y$`) from 1 *standard deviation* change in `$X$`:

- `$\hat{\beta_1}$`: a 1 standard deviation increase in Batting Average increases Salary by 0.14 standard deviations

$$0.14 \times \$2,764,512=\$387,032$$

- `$\hat{\beta_2}$`: a 1 standard deviation increase in Home Runs increases Salary by 0.48 standard deviations
  
$$0.48 \times \$2,764,512=\$1,326,966$$
]
---

# Standardizing in `R`

| Variable | Mean | SD |
|----------|-----:|---:|
| `LifeExp` | 59.47 | 12.92 |
| `gdpPercap` | $7215.32 | $9857.46 |

]
.quitesmall[
- Use the `scale()` command inside `mutate()` function to standardize a variable

]

```r
gapminder <- gapminder %>%
* mutate(life_Z = scale(lifeExp),
*        gdp_Z = scale(gdpPercap))

std_reg <- lm(life_Z ~ gdp_Z, data = gapminder)
tidy(std_reg)
```

```
## # A tibble: 2 × 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept) 1.10e-16    0.0197  5.57e-15 1.00e+  0
## 2 gdp_Z       5.84e- 1    0.0197  2.97e+ 1 3.57e-156
```

]

.quitesmall[
- A 1 standard deviation increase in `gdpPercap` will increase `lifeExp` by 0.584 standard deviations `$(0.584 \times 12.92 = = 7.55$` years)

]

---

# Joint Hypothesis Testing

---

# Joint Hypothesis Testing I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Return again to:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i+\hat{\beta_2}Northeast_i+\hat{\beta_3}Midwest_i+\hat{\beta_4}South_i$$`

]

- Maybe region doesn't affect wages *at all*?

- `$H_0: \beta_2=0, \, \beta_3=0, \, \beta_4=0$`

- This is a .hi[joint hypothesis] to test

---

# Joint Hypothesis Testing II

- A .hi[joint hypothesis] tests against the null hypothesis of a value for **multiple** parameters:
`$$\mathbf{H_0: \beta_1= \beta_2=0}$$`
the hypotheses that **multiple** regressors are equal to zero (have no causal effect on the outcome)

- Our .hi-purple[alternative hypothesis] is that:
`$$H_1: \text{ either } \beta_1\neq0\text{ or } \beta_2\neq0\text{ or both}$$`
or simply, that `$H_0$` is not true

---

# Types of Joint Hypothesis Tests

.smallest[
1) `$H_0$`: `$\beta_1=\beta_2=0$`
  - Testing against the claim that multiple variables don't matter
  - Useful under high multicollinearity between variables
  - `$H_a$`: at least one parameter `$\neq$` 0
]

.smallest[
2) `$H_0$`: `$\beta_1=\beta_2$`
  - Testing whether two variables matter the same
  - Variables must be the same units
  - `$H_a: \beta_1 (\neq, <, \text{ or }>) \beta_2$`
]
--

.smallest[
3) `$H_0:$` ALL `$\beta$`'s `$=0$`
  - The "**Overall F-test"**
  - Testing against claim that regression model explains *NO* variation in `$Y$`
]

---

# Joint Hypothesis Tests: F-statistic

- The .hi-turquoise[F-statistic] is the test-statistic used to test joint hypotheses about regression coefficients with an .hi-turquoise[F-test]

- This involves comparing two models:
  1. .hi[Unrestricted model]: regression with all coefficients
  2. .hi-purple[Restricted model]: regression under null hypothesis (coefficients equal hypothesized values)

- `$F$` is an .hi-turquoise[analysis of variance (ANOVA)]
  - essentially tests whether `$R^2$` increases statistically significantly as we go from the restricted model$\rightarrow$unrestricted model

- `$F$` has its own distribution, with *two* sets of degrees of freedom

---

# Joint Hypothesis F-test: Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Return again to:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i+\hat{\beta_2}Northeast_i+\hat{\beta_3}Midwest_i+\hat{\beta_4}South_i$$`

]

- `$H_0: \beta_2=\beta_3=\beta_4=0$`

- `$H_a$`: `$H_0$` is not true (at least one `$\beta_i \neq 0$`)

---

# Joint Hypothesis F-test: Example II
.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Return again to:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i+\hat{\beta_2}Northeast_i+\hat{\beta_3}Midwest_i+\hat{\beta_4}South_i$$`

]

- .hi[Unrestricted model]:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i+\hat{\beta_2}Northeast_i+\hat{\beta_3}Midwest_i+\hat{\beta_4}South_i$$`
]

--
.smallest[
- .hi-purple[Restricted model]:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i$$`
]

--
.smallest[

- `$F$`-test: **does going from .hi-purple[restricted] to .hi[unrestricted] model statistically significantly improve `$R^2$`?**
]
---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(R^2_u-R^2_r)}{q}\right)}{\left(\displaystyle\frac{(1-R^2_u)}{(n-k-1)}\right)}$$`
]

]

---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(\color{#e64173}{R^2_u}-R^2_r)}{q}\right)}{\left(\displaystyle\frac{(1-\color{#e64173}{R^2_u})}{(n-k-1)}\right)}$$`
]

.smallest[
- `$\color{#e64173}{R^2_u}$`: the `$R^2$` from the .hi[unrestricted model] (all variables)
]
]
---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(\color{#e64173}{R^2_u}-\color{#6A5ACD}{R^2_r})}{q}\right)}{\left(\displaystyle\frac{(1-\color{#e64173}{R^2_u})}{(n-k-1)}\right)}$$`
]

.smallest[
- `$\color{#e64173}{R^2_u}$`: the `$R^2$` from the .hi[unrestricted model] (all variables)

- `$\color{#6A5ACD}{R^2_r}$`: the `$R^2$` from the .hi-purple[restricted model] (null hypothesis)
]
]
---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(\color{#e64173}{R^2_u}-\color{#6A5ACD}{R^2_r})}{q}\right)}{\left(\displaystyle\frac{(1-\color{#e64173}{R^2_u})}{(n-k-1)}\right)}$$`
]

.smallest[
- `$\color{#e64173}{R^2_u}$`: the `$R^2$` from the .hi[unrestricted model] (all variables)

- `$\color{#6A5ACD}{R^2_r}$`: the `$R^2$` from the .hi-purple[restricted model] (null hypothesis)

- `$q$`: number of restrictions (number of `$\beta's=0$` under null hypothesis)
]
]
---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(\color{#e64173}{R^2_u}-\color{#6A5ACD}{R^2_r})}{q}\right)}{\left(\displaystyle\frac{(1-\color{#e64173}{R^2_u})}{(n-k-1)}\right)}$$`
]

.smallest[
- `$\color{#e64173}{R^2_u}$`: the `$R^2$` from the .hi[unrestricted model] (all variables)

- `$\color{#6A5ACD}{R^2_r}$`: the `$R^2$` from the .hi-purple[restricted model] (null hypothesis)

- `$q$`: number of restrictions (number of `$\beta's=0$` under null hypothesis)

- `$k$`: number of `$X$` variables in .hi[unrestricted model] (all variables)
]
]
---

# Calculating the F-statistic

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(\color{#e64173}{R^2_u}-\color{#6A5ACD}{R^2_r})}{q}\right)}{\left(\displaystyle\frac{(1-\color{#e64173}{R^2_u})}{(n-k-1)}\right)}$$`
]

.smallest[
- `$\color{#e64173}{R^2_u}$`: the `$R^2$` from the .hi[unrestricted model] (all variables)

- `$\color{#6A5ACD}{R^2_r}$`: the `$R^2$` from the .hi-purple[restricted model] (null hypothesis)

- `$q$`: number of restrictions (number of `$\beta's=0$` under null hypothesis)

- `$k$`: number of `$X$` variables in .hi[unrestricted model] (all variables)

- `$F$` has two sets of degrees of freedom:
  - `$q$` for the numerator, `$(n-k-1)$` for the denominator
]
]
---

# Calculating the F-statistic II

.pull-left[
`$$F_{q,(n-k-1)}=\cfrac{\left(\displaystyle\frac{(R^2_u-R^2_r)}{q}\right)}{\left(\displaystyle\frac{(1-R^2_u)}{(n-k-1)}\right)}$$`
]

- .hi-purple[Key takeaway]: The bigger the difference between `$(R^2_u-R^2_r)$`, the greater the improvement in fit by adding variables, the larger the `$F$`!

- This formula is (believe it or not) actually a simplified version (assuming homoskedasticity)
  - I give you this formula to **build your intuition of what F is measuring**
  
]

---

# F-test Example I

- We'll use the `wooldridge` package's `wage1` data again

```r
# load in data from wooldridge package
library(wooldridge)
wages <- wage1

# run regressions
unrestricted_reg <- lm(wage ~ female + northcen + west + south, data = wages)
restricted_reg <- lm(wage ~ female, data = wages)
```

---

# F-test Example II

- .hi[Unrestricted model]:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i+\hat{\beta_2}Northeast_i+\hat{\beta_3}Northcen+\hat{\beta_4}South_i$$`

- .hi-purple[Restricted model]:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

- `$H_0: \beta_2 = \beta_3 = \beta_4 =0$`

- `$q = 3$` restrictions (F numerator df)

- `$n-k-1 = 526-4-1=521$` (F denominator df)

---

# F-test Example III

.smallest[
- We can use the `car` package's `linearHypothesis()` command to run an `$F$`-test:
  - first argument: name of the (unrestricted) regression
  - second argument: vector of variable names (in quotes) you are testing

]

```r
# load car package for additional regression tools
library(car) 
# F-test
linearHypothesis(unrestricted_reg, c("northcen", "west", "south")) 
```

```
## Linear hypothesis test
## 
## Hypothesis:
## northcen = 0
## west = 0
## south = 0
## 
## Model 1: restricted model
## Model 2: wage ~ female + northcen + west + south
## 
##   Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
## 1    524 6332.2                                
## 2    521 6174.8  3    157.36 4.4258 0.004377 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]
]

]

---

# Second F-test Example: Are Two Coefficients Equal?

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:

`$$\widehat{wage_i}=\beta_0+\beta_1 \text{Adolescent height}_i + \beta_2 \text{Adult height}_i + \beta_3 \text{Male}_i$$`

]
]
--

`$$H_0: \beta_1=\beta_2$$`
]

`$$\widehat{wage_i}=\beta_0+\beta_1(\text{Adolescent height}_i + \text{Adult height}_i )+ \beta_3 \text{Male}_i$$`

- `$q=1$` restriction
]

---

# Second F-test Example: Data

```r
# load in data
heightwages <- read_csv("../data/heightwages.csv")

# make a "heights" variable as the sum of adolescent (height81) and adult (height85) height

heightwages <- heightwages %>%
  mutate(heights = height81 + height85)

height_reg <- lm(wage96 ~ height81 + height85 + male, data = heightwages)
height_restricted_reg <- lm(wage96 ~ heights + male, data = heightwages)
```

---

# Second F-test Example: Data

- For second argument, set two variables equal, in quotes

```r
linearHypothesis(height_reg, "height81 = height85") # F-test 
```

```
## Linear hypothesis test
## 
## Hypothesis:
## height81 - height85 = 0
## 
## Model 1: restricted model
## Model 2: wage96 ~ height81 + height85 + male
## 
##   Res.Df     RSS Df Sum of Sq      F Pr(>F)
## 1   6591 5128243                           
## 2   6590 5127284  1     959.2 1.2328 0.2669
```

- Insufficient evidence to reject `$H_0$`!

- The effect of adolescent and adult height on wages is the same

---
  
# All F-test I

```r
summary(unrestricted_reg)
```

```
## 
## Call:
## lm(formula = wage ~ female + northcen + west + south, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.3269 -2.0105 -0.7871  1.1898 17.4146 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.5654     0.3466  21.827   <2e-16 ***
## female       -2.5652     0.3011  -8.520   <2e-16 ***
## northcen     -0.5918     0.4362  -1.357   0.1755    
## west          0.4315     0.4838   0.892   0.3729    
## south        -1.0262     0.4048  -2.535   0.0115 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.443 on 521 degrees of freedom
## Multiple R-squared:  0.1376,	Adjusted R-squared:  0.131 
## F-statistic: 20.79 on 4 and 521 DF,  p-value: 6.501e-16
```
]
]

.pull-right[
.smallest[
- Last line of regression output from `summary()` is an **All F-test**
  - `$H_0:$` all `$\beta's=0$` 
      - the regression explains no variation in `$Y$`
  - Calculates an `F-statistic` that, if high enough, is significant (`p-value` `$<0.05)$` enough to reject `$H_0$`
]
]

---

# All F-test II

- Alternatively, if you use `broom` instead of `summary()`:
  - `glance()` command makes table of regression summary statistics
  - `tidy()` only shows coefficients

```r
library(broom)
glance(unrestricted_reg)
```

```
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1     0.138         0.131  3.44      20.8 6.50e-16     4 -1394. 2800. 2826.
## # … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
```
]

- `statistic` is the All F-test, `p.value` next to it is the p-value from the F test