ECON 3818

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# ECON 3818
## Chapter 26
### Kyle Butts
### 27 September 2021

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## Chapter 26: Regression Inference

---
# Introduction

Chapter 4 and 5 discussed how scatterplots and lines of best fit show us linear relationships, but there are remaining questions

Is there really a linear relationship between x and y, or is the pattern just by chance

- Spurious correlations
  
What is the estimated slope that explains how `$y$` responds to `$x$` .it[in the population]. What is the margin of error for our estimate?

- If we use the least-squares line to predict `$y$` for a given `$x$`, how accurate is that prediction?

- In econometrics, you will discuss when you can answer what is the **effect** on `$y$` of changing `$x$`

---
# Regression Review

We can model the linear relationship between X and Y by thinking of a conditional expectation:

$$ 
E(Y|X)= a + bX 
$$

We want estimates for a and b, `$\hat{a}$` and `$\hat{b}$`, and we find these estimates by minimizing the sum of squared residuals

$$ 
\varepsilon_i = Y_i - \coral{\widehat{Y}_i} \equiv Y_i - (\coral{\hat{a} + \hat{b} X_i})
$$

---
# OLS Estimators

We pick the values of `$\hat{a}$` and `$\hat{b}$` to minimize the sum of least squares, $\sum_{i=1}^n \varepsilon_i^2 $. This yields the Ordinary Least Squares estimators

$$ 
\hat{a}=\bar{Y}-\hat{b}\bar{X} 
$$

$$ 
\hat{b}=r_{XY}\frac{s_Y}{s_X} 
$$

---
# Next Steps
This chapter will answer

- How can I interpret `$\hat{a}$` and `$\hat{b}$`?

- What conditions are necessary for those interpretations?

- Inference from a Regression

---
# Interpreting `$a$` and `$b$`

$$ 
\hat{\coral{\text{Calcification Rate}}} = -12.103 + 0.4615 * \text{Temperature} 
$$

We can now predict how temperature affects the calcification rate. The `$R^2$` will tell us how much of the variation in calcification rate is due to temperature, but it will not tell us whether this relationship is .hi.daisy[statistically significant].

In order for this regression to be meaningful, we must determine whether the results are .hi.daisy[statistically significant]

---
# Estimating the Parameters

When the conditions for the regression are met.super[1]

- The slope `$\hat{b}$` of the least-squares line is an unbiased estimator of the population slope `$b$`
      
- The intercept `$\hat{a}$` of the least-squares line is an unbiased estimator of the population intercept `$a$`

Now we only need to estimate the remaining parameters, `$\sigma$`, the standard deviation of the error term `$\varepsilon_i$`.

.footnote[.super[1] We will discuss the conditions later]

---
# Regression Standard Error

Our regression model is:

$$ 
y = a + X b + \varepsilon
$$

`$\varepsilon$` is the error term that describes why an individual doesn't fall directly on regression line `$a + X b$`.

We denote the variance of `$\varepsilon$` as `$\sigma^2$`. The standard deviation, `$\sigma$`, describes variability of response variable `$y$` about the population regression line ($\pm$).

---
# Estimating Std. Dev. of the Error Term

The least-squares line estimates the population regression line

- The .hi.kelly[residuals] are the deviations of data points from the least-squares line

$$ 
\hat{\varepsilon} \equiv \text{residual} = y - \coral{\hat{y}}
$$

Therefore we estimate `$\sigma$` by the sample standard deviation of the residuals, known as the .hi.kelly[regression standard error]

---
# Regression Standard Error
$$ 
s=\sqrt{\frac{1}{n-2} \sum\text{residual}^2} \equiv \sqrt{\frac{1}{n-2} \sum(y - \coral{\hat{y}})^2}
$$

We use `$s$` to estimate the standard deviation, `$\sigma$`, of responses about the mean given by the population regression line

We will use this error to determine whether our predictions are .daisy[statistically significant]

---
# Testing the Hypothesis of No Linear Relationship
To answer questions about whether associations between two variables are .daisy[statistically significant], we must test a hypothesis about the slope `$b$`:

`$$H_0: \ b = 0$$`
`$$H_1: b \neq 0$$`

If we fail to reject `$H_0$`:

- Regression line with slope 0 is horizontal -- meaning y does not change at all when x changes

- `$H_0$` says that there is no linear relationship between X and Y

If we reject `$H_0$`, and accept `$H_1$`:

- There is some linear relationship between X and Y

---
# Null of No Linear Relationship

If we fail to reject `$H_0$`:

- Regression line with slope 0 is horizontal -- meaning `$y$` does not change at all when `$x$` changes

---
# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

.pull-left[

.center[**Population**]

]

.pull-right[

.center[**Population relationship**]

$$ y_i = 2.53 + 0.57 x_i + u_i $$

$$ y_i = \beta_0 + \beta_1 x_i + u_i $$

]

---
# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

.pull-left[

.center[**Sample 1:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
<br>
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
<br>
`$\hat{y}_i = 2.36 + 0.61 x_i$`

]

---
# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

.pull-left[

.center[**Sample 2:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
<br>
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
<br>
`$\hat{y}_i = 2.79 + 0.56 x_i$`

]

]
---
# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

.pull-left[

.center[**Sample 3:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
<br>
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
<br>
`$\hat{y}_i = 3.21 + 0.45 x_i$`

]

---
class: clear, center, middle

.center[$1,000$ Samples of size `$30$`]

---
# Population *vs.* sample

.pull-left[
<img src="data:image/png;base64,#ch26_files/figure-html/simulation scatter2-1.png" style="display: block; margin: auto;" />
]

.pull-right[

- On **average**, our regression lines match the population line very nicely.

- However, **individual lines** (samples) can really miss the mark.

- Differences between individual samples and the population lead to **uncertainty** for the econometrician.

]

---
# Sampling Distribution of `$\hat{b}$`

Since `$\hat{b}$` is a function of our data, it has a sampling distribution.

The sampling distribution of `$\hat{b}$` is:
$$
\hat{b} \sim N\left(b, \ \frac{\sigma^2}{\sigma_X^2}\right)
$$
.footnote[Another instance of the sampling distribution being normally distributed!]

`$\sigma^2$` is the variance of `$\varepsilon$` and `$\sigma_X^2$` is the variance of `$X$`.

---
# Significance Test for Regression Slope

To test the hypothesis, `$H_0: b=0$`, compute the t-statistic:

$$ 
t_{n-2} = \frac{\hat{b} - 0}{SE_b} 
$$

Important to note that the degrees of freedom for the t-statistic for testing a regression slope is `$n-2$` (we estimate `$a$` and `$s$`)

In this formula, the standard error of the least-squares slope is our estimate at the sampling distribution's standard deviation:

$$ 
SE_{\hat{b}}=\frac{s}{\sqrt{\sum (x-\bar{x}^2)}} 
$$

---
# Example
We fit a least-squares line to the model, `$\text{Price} = a+b (\text{age})$` with 28 observations from items sold at antiques show. A summary of the output is below:

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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Parameter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Parameter Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Std. Error of Estimate</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_center">$\hat{a}$</td>
<td class="gt_row gt_center">27.730</td>
<td class="gt_row gt_center">34.840</td></tr>
    <tr><td class="gt_row gt_center">$\hat{b}$</td>
<td class="gt_row gt_center">1.893</td>
<td class="gt_row gt_center">0.267</td></tr>
  </tbody>
  
  
</table>
</div>

Suppose we want to test the hypothesis, `$H_0: b=0$` vs. `$H_1: b \neq 0$`. The value of this t-statistic is: 
$$ 
t_{26} = \frac{b}{SE_b} = \frac{1.893 - 0}{0.267} = 7.09 
$$

Using t-table `$\implies p < 0.001$`

---
# Clicker Question

In the previous example we rejected the null hypothesis of `$b=0$`, meaning we claim there is sufficient evidence to say there is a linear relationship between age and price sold of items at a antiques road show.

What type of error would we have committed if it turned out there was no relationship between age and price?

<ol type = "a">
  <li>Type I, reject the null even though its true</li>
  <li>Type II, reject the null even though its true</li>
  <li>Type I, fail to reject a false null</li>
  <li>Type II, fail to reject a false null</li>
</ol>

---
# Additional Example -- Exam Style
My budtender friend Eric did a study on marijuana consumption and hot cheeto consumption. He surveyed 25 of his friends and collected the following regression results. Assume `$\alpha = 0.05$`

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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Cheeto Consumption</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Std. Error</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">t-statistic</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">p-value</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_center">Intercept</td>
<td class="gt_row gt_center">21.0</td>
<td class="gt_row gt_center">12.3</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
    <tr><td class="gt_row gt_center">Joints Smoked</td>
<td class="gt_row gt_center">4.2</td>
<td class="gt_row gt_center">1.8</td>
<td class="gt_row gt_center"></td>
<td class="gt_row gt_center"></td></tr>
  </tbody>
  
  
</table>
</div>

<ol type = "a">
  <li> Fill in the rest of the table </li>
  <li> Is the intercept statistically significant? Why? </li>
  <li> Is the slope coefficient statistically significant? Why? </li>
  <li> Interpret slope coefficient </li>
</ol>

---
# Hypothesis Testing Example

.ex[Example:] Regression analysis provides estimates on the relationship between daily wine consumption on risk of breast cancer. The estimated slope was `$\hat{b} = 0.009$` with a standard error of `$SE_{\hat{b}} = 0.001$` based off 25 observations.

We want to test whether these results are strong enough to reject the null hypothesis 
`$$H_0: b = 0$$`

in favor or the alternative hypothesis

`$$H_1: b > 0$$`

---
# Hypothesis Testing Example

So we have `$\hat{b}$`=0.009 and `$SE_{\hat{b}}$`=0.001. Solving hypothesis test:

- Find t-stat

$$ 
t=\frac{0.009}{0.001}=9 
$$

- Use t-table to find p-value

$$ 
25 \text{ observations } \implies t_{n-2} = t_{23} 
$$

$$
t_{23}^{0.0005} = 3.8 \implies p<0.0005 
$$

- Interpret p-value 
$$ 
p < 0.0005 \implies p < 0.05 \implies \textbf{Reject `$H_0$`} 
$$

---
# Regression Results

```r
# Hourly Earnings ($) on Years of Education
summary(lm(wage ~ educ, data = wage1))
```

```
#> 
#> Call:
#> lm(formula = wage ~ educ, data = wage1)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -5.3396 -2.1501 -0.9674  1.1921 16.6085 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.90485    0.68497  -1.321    0.187    
#> educ         0.54136    0.05325  10.167   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 3.378 on 524 degrees of freedom
#> Multiple R-squared:  0.1648,	Adjusted R-squared:  0.1632 
#> F-statistic: 103.4 on 1 and 524 DF,  p-value: < 2.2e-16
```

---
# Confidence Interval for Regression Slope

The slope, `$b$`, of the population regression is usually the most important parameter in a regression problem

- The slope is the rate of change of the mean response as the explanatory variable increases

- The slope explains how changes in x affect outcome variable y

A confidence interval is useful because it shows us <span class="it">how accurate the estimate of `$b$` is likely to be</span>.

---
# Confidence Interval for Regression Slope

A level `$C$` confidence interval for the slope `$b$` of the population regression line is

$$ 
\hat{b} \pm t^* \cdot SE_{b}, 
$$

where `$t^* = t^{\frac{1-C}{2}}_{n-2}$`

---
# Confidence Interval for Regression Slope

.ex[Example:] Recall our regression results looking at the relationship of temperature on coral calcification. The estimated slope was `$\hat{b} = 0.4615$` and a standard error `$SE_{\hat{b}} = 0.07394$`. Note this was based off a sample of 12 observations.

12 observations mean our `$t_{n-2}$` distribution has 12-2=10 degrees of freedom and that critical `$t$`-stat is `$2.23$` when `$(1-C)/2 = 0.05/2 = 0.025$`

--
If we want to construct a 95% confidence interval:

$$
\hat{b} \pm t^* SE_{\hat{b} }= 0.4615 \pm (2.23)(0.07394) 
$$

The 95% confidence interval for population slope `$b$` is `$[0.297, 0.626]$`.

---
# Clicker Question

A random sample of 19 companies were selected and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by a regression, `$profits = a + b \cdot sales$`. The following results were obtained from statistical software:

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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Parameter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Parametere Estimate</th>
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    <tr><td class="gt_row gt_center">$\hat{a}$</td>
<td class="gt_row gt_center">-176.6440</td>
<td class="gt_row gt_center">61.1600</td></tr>
    <tr><td class="gt_row gt_center">$\hat{b}$</td>
<td class="gt_row gt_center">0.0925</td>
<td class="gt_row gt_center">0.0075</td></tr>
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An approximate 90% confidence interval for the slope `$b$` is:

---
# Confidence Intervals

R will spit out a 95% confidence interval associated with slope estimates with `confint`:

```r
# Hourly Earnings ($) on Years of Education
confint(lm(wage ~ educ, data = wage1))
```

```
#>                  2.5 %    97.5 %
#> (Intercept) -2.2504719 0.4407687
#> educ         0.4367534 0.6459651
```

95% confident that an additional year of schooling increases average hourly earnings between $0.44 and $0.65

---
# Significance and Margin of Error

Conducting a hypothesis test on `$\hat{b}$` tells you about the .hi.purple[significance] of your result

- `$p$`-value `$< \alpha$`, we can say our coefficient is statistically different from zero

A confidence interval says something about the precision of the coefficient

- What are the ranges of coefficient values we expect the true-value to be in between

- Confidence interval is also the only points you will fail to reject the null.

---
# Significance and Margin of Error

Do we reject null that education has no effect on wage?

---
# Categorical Variable inside Regression

In that previous example, the explanatory variable was categorical. Let's see how that changes interpretation.

```r
# Hourly Earnings ($) on HS Degree
summary(lm(wage ~ hs_deg, data = wage1))
```

```
#> 
#> Call:
#> lm(formula = wage ~ hs_deg, data = wage1)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -5.8865 -2.4165 -0.9267  1.1734 18.5635 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   4.0567     0.3309  12.258  < 2e-16 ***
#> hs_deg        2.3598     0.3748   6.296 6.48e-10 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 3.564 on 524 degrees of freedom
#> Multiple R-squared:  0.07032,	Adjusted R-squared:  0.06854 
#> F-statistic: 39.63 on 1 and 524 DF,  p-value: 6.485e-10
```

---
# Categorical Variable inside Regression

This regression implies the relationship between HS Degree and hourly earnings is:

$$ 
\coral{\hat{Income}} = \$4.06 + \$2.36 \cdot \text{HS Degree}
$$

The takeaways here would be:

- Without a HS degree, predicted wage is $4.06

- With a PhD, predicted wage is $4.06 + $2.36

The coefficient on an indicator represents the *difference* in averages of `$Y$` between the `$= 0$` and `$= 1$` groups.

---
# Conditions for Regression Inference

Say we have `$n$` observations regarding explanatory variable `$x$` and response variable `$y$`.

- The mean response `$E(Y | X)$` has a .hi.purple[straight-line relationship] with x, given by a population regression line `$$E(Y|X) = a + b X$$`

- For any fixed value of `$x$`, the response variable `$y$` varies according to a normal distribution

- Repeated responses `$y$` are independent of each other

- The .hi.daisy[standard deviation] of `$\varepsilon$`, `$\sigma$`, is the same for all values of x.

---
# Intuition about Conditions

<span class="it">The mean response `$E(Y \ \vert \ X)$` has a .hi.purple[straight-line relationship] with `$x$`, given by a population regression line</span>

- In practice, we observe `$y$` for many different values of `$x$`. Eventually we see an overall linear pattern formed by points scattered about the population line.

---
# Intuition about Conditions

<span class="it">For any fixed value of `$x$`, the response variable `$y$` varies according to a normal distribution</span>

- We cannot observe the entire population regression line. The values of `$y$` that we do observe vary about their means according to a normal distribution. If we hold x constant and take many observations of y, the Normal pattern will eventually appear in a histogram.

---
# Intuition about Conditions

<span class="it">The .hi.purple[standard deviation] of `$\varepsilon$`, `$\sigma$`, is the same for all values of x. The value of `$\sigma$` is unknown.</span>

- The standard deviation determines whether the points fall close to the population regression line (small `$\sigma$`) or are widely scattered (large `$\sigma$`)
  
- If `$\sigma$` changes depending on `$x$`, then our sample distribution would be wrong.

---
# Intuition about Conditions

- For each possible value of `$x$`, the mean of the responses moves along the population regression line

- For a fixed `$x$`, the responses `$y$` follow a normal distribution with std. dev `$\sigma$`
      
- The normal curve shows how `$y$` will vary when `$x$` is held constant

---
# Checking Conditions for Inference

Remember, all of this discussion about inferences hinges on the data meeting certain conditions.

- The relationship is linear in the population

- The response varies normally about the regression line

- Observations are independent

- The standard deviation of the responses is the same for all values of x

---
# Checking Conditions for Inference

In order to check these conditions, it can be helpful to look at a residual plot. A .hi.purple[residual plot] plots the residuals against the explanatory variable `$x$`, with a horizontal line at the "residual =0" position. The "residual =0" line represents the position of the least-squares line in the scatterplot of y against `$x$`.

<div class = "pull-left">
<h3 style="margin-bottom: 5px;">
  Regression Plot
</h3>

<img src="data:image/png;base64,#scatter.png" width="100%" style="display: block; margin: auto;" />
</div>
<div class = "pull-right">
<h3 style="margin-bottom: 5px;">
  Residual Plot
</h3>

---
# Checking Conditions for Inference

- .hi[The relationship is linear]. Look for curved patterns or other deviations from an overall straight line pattern in residual plot

- .hi[The response varies normally about regression line]. Check for departures from normality in your stemplot or histogram of residuals.

- .hi[Observations are independent]. Signs of dependence in the residual plot are subtle, so usually use common sense.

- .hi[Standard deviation of responses is same for all values of] `$x$`. Look at the scatter of residuals above and below the "residual =0" line. The scatter should be roughly the same from one end to the other.