Microeconometrics - Lecture 6

class: center, middle, inverse, title-slide

# Microeconometrics - Lecture 6
## Endogeneity
### Jonas Björnerstedt
### 2022-03-09

---

## Lecture Content

- Chapter 12 - Instrumental Variable Regression

- Continuation next lecture

- Chapter 7 - Tests

---
class: inverse, center, middle
# Endogeneity

---
## Endogeneity bias

- Assume that regressor `$X_i$` and `$u_i$` are positively correlated

`$$E\left(u_i\left|X_i\right.\right) \neq 0$$`
    
    - Least squares estimation is upward biased!

![](figures/ovarbias.png)

---
## Bias with negative correlation

- Assume that regressor `$X_i$` and `$u_i$` are negatively correlated

- Least squares estimation is biased downward!

![](figures/ovarbiasneg.png)

---
## Types of endogeneity

- Correlation between `$X_i$` and `$u_i$` can arise in different ways

1.  Omitted variable

- Price is correlated with unobserved variables that also affect demand
    
2.  Simultaneity

- Specify other equation
    
3.  Measurement error

- Bias towards zero

- Endogeneity of one variable affects estimates of all!

---
## Omitted variable bias

- Assume that

- population model is: 
        `$Y_i = \beta_{0} + \beta_{1}X_i + \beta_{2}W_i + u_i$`
        
    - `$\mathrm{Cov}\left(X_i,W_i\right) > 0$`

- Estimating the model `$$Y_i = \gamma_{0} + \gamma_{1}X_i + v_i$$` results in
    biased estimate as `$X_i$` is correlated with `$v_i$`:
    `$$\mathrm{Cov}\left(X_i,v_i\right) = \mathrm{Cov}\left(X_i,\beta_{2} W_i + u_i\right) = \beta_{2}\mathrm{Cov}\left(X_i, W_i\right)$$`

- Sign of bias `$\mathrm{E}\left(\hat{\beta_{1}}\right) - \beta_{1}$` depends on the
    sign of `$\beta_{2}$`

- Omission of uncorrelated variables does not cause bias!

- But the standard error of estimates will increase

---
## Illustration: Omitted variable bias

- Correllation `$X$` and `$W$` - parameter for `$X$` includes effect due to `$W$`

![](figures/omitted_variable_bias.png)

---
## Measurement error

- Assume that `$X_i = X^{*}_i + W_i$` is observed but `$Y_i$` is a function of `$X^{*}_i$`

- Measurement error `$W_i$`

- True relationship: `$Y_i = \beta_{0} + \beta_{1}X^{*}_i + u_i$`

- Rewriting: `$$Y = \beta_{0} + \beta_{1}(X_i - W_i) + u_i = \beta_{0} + \beta_{1} X_i  + (-\beta_{1} W_i + u_i)$$`

- Estimate equation: `$Y_i  = \beta_{0} + \beta_{1} X_i  + v_i$`

- `$$Cov(X_i,W_i) = E[(X_i-\bar X_i)W_i] = E[( X^{*}_i + W_i)W_i] = E[(W_i)^2] > 0$$`

- Bias towards zero

- `$W_i$` positively correlated with `$X$` but has negative effect `$-\beta_{1}$` on `$Y_i$`
    
    - With huge error, noise will dominate effect of `$X_i$` on `$Y_i$`

- Measurement error spreads the data horizontally

---
## [Measurement error plot](http://rstudio.sh.se/content/me06-figs/)

```
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
```

![](me06_files/figure-html/unnamed-chunk-2-1.png)

---
## Simultaneity and bias

- Demand `$$q = \beta_{0} + \beta_{1}p + u$$`

- Supply `$$p = \pi_{0} + \pi_{1}q + e$$`

- If `$\beta_{1} < 0$` and `$\pi_{1} > 0$`, simultaneous equations

- There is a unique equilibrium `$\bar p$` and `$\bar q$`

- All variation from equilibrium is unexplained!

- Cannot estimate relationships

---
## Supply and demand shifts

- Solution: other parameters that explain demand or supply
    - Demand and cost shifters

- A cost shifter can identify demand

- A demand shifter can identify costs

---
## Selection bias

- Existence of observation depends on the outcome `$Y_i$`
    - Self selection

- Examples
    - Labor force participation

---
## Solution 1: Proxy variables

- Non-central variables

- ability in explaining wages
    - hard to measure ability (measurement error)
    - important but perhaps not central

- Use other *proxy* variable instead: IQ test scores

- highly correlated with ability
    - smaller measurement errors

- Estimate measures corr between test scores and wages

- Not the effect of different abilities
    - But reduces problem of omitted variable bias

- Proxy variable should

1. be *redundant* - not directly explain anything
    2. Be correlated with endogenous var

---
## Regression with correlated regressor

- Population equation: `$E(wage|educ, ability) = educ + ability$`

- Let education depend on ability: `$educ = iq + v$`

- Let IQ score be a part of ability `$ability = 0.6 iq + e$`

```r
obs = 1000
iq = 2 + rnorm(obs)
ability = 0.6*iq  + 0.3 * rnorm(obs)
educ = iq + 0.5 * rnorm(obs)
wage = educ + ability + rnorm(obs)
lm( wage ~ educ )
```

```
## 
## Call:
## lm(formula = wage ~ educ)
## 
## Coefficients:
## (Intercept)         educ  
##      0.1713       1.5138
```

---
## Regression with correlated regressor

- IQ does not explain wages

- employer does not know IQ score result

- IQ is a _proxy variable_ (or _control variable_)

- IQ coefficient has no interpretation 
    
    - Helps to account for the correlation between educ and ability

---
## Proxy variable estimation

```r
m1 = lm( wage ~ educ)
m2 = lm( wage ~ educ + iq)
m3 = lm( wage ~ educ + ability)
m4 = lm( wage ~ educ + ability + iq)

huxreg(m1, m2, m3, m4)
```

<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-4">
<col><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;">(1)</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;">(2)</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;">(3)</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;">(4)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(Intercept)</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;">0.171 *  </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;">-0.093    </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;">-0.067    </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;">-0.059    </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;">(0.070)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.073)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.066)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.069)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">educ</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.514 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.960 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.974 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.991 ***</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;">(0.031)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.067)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.046)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.063)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">iq</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;">0.689 ***</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;">-0.037    </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;">(0.075)   </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;">(0.096)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">ability</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;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.105 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.131 ***</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;">        </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;">(0.074)   </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;">(0.101)   </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;">1000        </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;">1000        </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;">1000        </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;">1000        </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R2</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.711    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.734    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.764    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.764    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">logLik</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1482.327    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1441.830    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1382.230    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1382.156    </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;">AIC</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;">2970.653    </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;">2891.660    </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;">2772.460    </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;">2774.312    </td></tr>
<tr>
<th colspan="5" 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>

---
## Definition of omitted variable bias

If `$W_i$` is not included, we get _omitted variable bias_ if

1. `$X_i$` and `$W_i$` are correlated

2. `$W_i$` is a determinant of `$Y_i$`

- Equation (6.1) on page 231 is not very intuitive

---
## A formula for omitted variable bias

- If `$X$` and `$W$` are correlated, then `$\delta_1 \neq 0$` in
`$$W_i = \delta_0 + \delta_1 X_i + v_i$$`
Omitting `$W_i$` in the regression 
`$$Y_i = \beta_{0} + \beta_{1}X_{i} + u_i$$`

- gives a bias
`$$E(\hat\beta_1) = \beta_1 + \beta_2 \delta_1$$`

---
## Omitted variable bias in education

- Let: 
`$$wage_i = \beta_0 + \beta_1 educ_i + \beta_2 ability_i + u_i$$`
- If ability is correlated with educ, we have `$$ability = \delta_0 + \delta_1 educ_i + v_i$$`

- and thus
`$$E(\hat\beta_1) = \beta_1 + \beta_2 \delta_1$$`

---
## Omitted variable experiment

```r
obs = 1000
educ = 2 + rnorm(obs)
ability = 1 + 0.5 * educ  +  rnorm(obs)
wage = educ + ability + rnorm(obs)
lm( wage ~ educ)
```

```
## 
## Call:
## lm(formula = wage ~ educ)
## 
## Coefficients:
## (Intercept)         educ  
##       1.083        1.484
```

---
## Good and bad proxy variables

- If `$$ability_i = \pi_0 + \pi_1 iq_i + v_i$$`
then
`$$wage_i = \beta_0 + \beta_1 educ_i + \beta_2 (\pi_0 + \pi_1 iq_i + v_i) + u_i$$`
- Rewriting:
`$$wage_i = (\beta_0 + \beta_2 \pi_0) + \beta_1 educ_i + \beta_2 \pi_1 iq_i + (\beta_2 v_i + u_i)$$`

- If `$educ$` is correlated with `$iq$`, but not with other aspects of ability `$v$`, then it is uncorrelated with 
the error term `$\beta_2 v + u$`
 
    - Is this assumption valid?

---
class: inverse, center, middle
# Hypothesis tests and confidence intervals

---
## Correlation and OLS estimators

- Let  `$Y = \beta_0 + \beta_1 X + \beta_2 W + u$`

- Higher slope means lower intercept

- If `$X$` and `$W$` are _positively_ correlated then

- `$\hat\beta_1$` and `$\hat\beta_2$` are negatively correlated
    
    - If a high `$X$` explains more of `$Y$`, then the high value of `$W$` explains less

- If `$X$` and `$W$` are _negatively_ correlated then

- `$\hat\beta_1$` and `$\hat\beta_2$` are positively correlated

---
## Standard errors for the OLS estimators

.pull-left[

- Jointly normal distribution with means `$\hat\beta_0$`, `$\hat\beta_1$`

- joint variance specified by _variance covariance matrix_

]
.pull-right[

![](figures/pdf.png)
]

---
## Hypothesis tests for a single coefficient

- Test null hypothesis `$H_0$` that the coefficient of str is 2.1

- Compute `$$t^{act}= \frac{\hat\beta_j-2.1}{SE(\hat\beta_j)}$$`

- Reject at 5 percent level if

- `$|t^{act}| > 1.96$`

- Alternatively calculate the p-value 
`$$-2\Phi(-|t^{act}|)$$`

---
## Confidence interval
.pull-left[

- 90% confidence interval for `$\hat\beta_0$`

- `$\hat\beta_1$` can take any value
]
.pull-right[

![](figures/pdf_interval.png)
]
---
## Confidence sets
.pull-left[

- Most likely combinations of slopes and intercepts
]
.pull-right[

![](figures/pdf_region.png)

]
---
## Geometry of confidence sets

.pull-left[

- If the parameters were independent, the set would be a circle

- Set is parameter pairs within a certain distance (circle)
]
.pull-right[
![](figures/pdf_iid.png)

]

---
## Test of joint hypothesis on two independent variables

- Testing hypotheses on two or more coefficients

- What is the distribution of the distance?
    
    - For standard normal the distance is given by: `$t_1^2+t_2^2$`
    
    - Has `$\chi^2_2$` (or `$F_{2,\infty}$`) distribution

- The F-statistic

- Set critical level to exclude 5% of outcomes with greatest distance

- The F-value is a "distance measure"

- We can reject `$H_0$` if the F-value is large

---
## F-test

- Test if the squared sum is close to 0

- after rescaling the variance

- Estimates `$\beta_1$` and `$\beta_2$` are approximately normal

- Squared sum of `$\beta_1$` and `$\beta_2$` will be `$\chi^2$` distribution, 2 degrees of freedom

- Rescaling with variance gives F distribution
    
    - Rescaling implies dividing by `$\chi^2$` distributed variable, `$n - k$` degrees of freedom

---
## Testing several parameters

- Test `$q$` restrictions

- For example `$H_0: \beta_1 = \beta_2=0$`

- Can construct test statistic with

- F distribution wih `$q$` and `$n-k$` degrees of freedom

- Measures a 'distance' from

- higher F value makes `$H_0$` less likely
    
    - p value measures probability of test value given `$H_0$`

---
## F-test of regression

- Test of hypothesis: **all** estimated slopes are zero

- A high F value desirable in this case as we want to reject null hypothesis

- Corresponds to low probability of accepting the null hypothesis

- Joint significance different than individual

- F-test is very close to the `$R^2$` statistic

- When one is low the other is high

---
## Joint significance of correlated regressor

- Population equation: `$E(Y_i|X_i,W_i) = 1 + X_i + W_i$`

```r
library(AER)
obs = 100
X = rnorm(obs)
W = X + 0.1 * rnorm(obs)
Y = 1 + X + W + rnorm(obs)
model = lm( Y ~ X + W)
model
```

```
## 
## Call:
## lm(formula = Y ~ X + W)
## 
## Coefficients:
## (Intercept)            X            W  
##       1.123        2.920       -1.017
```

---
## Joint test

* Can specify more complex test with `linearHypothesis()`

```r
linearHypothesis(model, c("X=0", "W=0"))
```

---
## Regression statistics

* Regression statistics - test of all coefficients included

```r
summary(model)
```

```
## 
## Call:
## lm(formula = Y ~ X + W)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.3209 -0.6792  0.0605  0.5844  2.6535 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.12288    0.09921  11.319  < 2e-16 ***
## X            2.91952    0.91304   3.198  0.00187 ** 
## W           -1.01723    0.89701  -1.134  0.25958    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9883 on 97 degrees of freedom
## Multiple R-squared:  0.7928,	Adjusted R-squared:  0.7886 
## F-statistic: 185.6 on 2 and 97 DF,  p-value: < 2.2e-16
```

---
## Regression statistics with broom package

```r
library(broom)
tidy(model)
```

```r
glance(model)
```

<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-9">
<col><col><col><col><col><col><col><col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0.4pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">r.squared</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">adj.r.squared</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">sigma</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">statistic</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">p.value</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">df</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">logLik</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">AIC</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">BIC</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">deviance</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">df.residual</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0.4pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: bold;">nobs</th></tr>
<tr>
<td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0.4pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.793</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.789</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.988</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">186</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">6.92e-34</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">2</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-139</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">286</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">297</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">94.7</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">97</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0.4pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">100</td></tr>
</table>

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## Next Lecture

- Chapter 12 - Instrumental Variable Regression