EDLD 650 Review Session

class: center, middle, inverse, title-slide

.title[
# EDLD 650 Review Session
]
.author[
### David D. Liebowitz
]

---

# SAT score and income

<img src="EDLD_650_Review_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
---
# A closer look

<img src="EDLD_650_Review_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" />
---
# A closer look
<img src="EDLD_650_Review_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />

---
# Graphical fixed effects

```r
df <- df %>% group_by(school) %>% mutate(mean_SAT = mean(SAT))
df <- df %>% mutate(demean_SAT = SAT - mean_SAT)
```

---
# Graphical fixed effects

```r
df <- df %>% group_by(school) %>% mutate(mean_income=mean(income30))
df <- df %>% mutate(demean_income = income30 - mean_income)
```

---
# Graphical fixed effects

```r
df <- df %>% group_by(school) %>% mutate(mean_income=mean(income30))
df <- df %>% mutate(demean_income = income30 - mean_income)
```

---
# A naive regression

```r
summary(lm(income30 ~ SAT, data=df))
```

```
...
#> 
#> Call:
#> lm(formula = income30 ~ SAT, data = df)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -4815.2  -957.9   -64.6   961.1  5934.6 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) 11769.532    573.360   20.53   <2e-16 ***
#> SAT            76.573      1.146   66.83   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
...
```

`$$INCOME_{i} = \beta_{0} + \beta_{1} SAT_{i} + \varepsilon_{i}$$`

---
# Adjust for school

```r
summary(lm(income30 ~ SAT + school, data=df))
```

```
...
#> lm(formula = income30 ~ SAT + school, data = df)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3652.7  -720.8     9.6   701.0  4041.7 
#> 
#> Coefficients:
#>                 Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)    48399.823    843.842  57.357   <2e-16 ***
#> SAT               -1.761      1.777  -0.991    0.322    
#> schoolSchool 2  2494.792     66.184  37.695   <2e-16 ***
#> schoolSchool 3  5078.168    101.285  50.137   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
...
```

`$$INCOME_{ij} = \beta_{0} + \beta_{1} SAT_{ij} + \beta_{2} SCHOOL2_{j} + \beta_{3} SCHOOL3_{j} + \varepsilon_{i}$$`
---
# Cluster-adjusted SEs

```r
summary(fixest::feols(income30 ~ SAT | school, data=df))
```

```
#> OLS estimation, Dep. Var.: income30
#> Observations: 3,000 
#> Fixed-effects: school: 3
#> Standard-errors: Clustered (school) 
#>     Estimate Std. Error   t value Pr(>|t|) 
#> SAT -1.76127    1.81864 -0.968456  0.43498 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 1,076.5     Adj. R2: 0.781517
#>                 Within R2: 3.28e-4
```

**What happened to the standard errors? Is that right? Why or why not?**

---
# Fixed vs. random

### Fixed:
`$$INCOME_{ij} = \beta_{0} + \beta_{1} SAT_{ij} + \beta_{2} SCHOOL2_{j} + \beta_{3} SCHOOL3_{j} + \varepsilon_{ij}$$`

### Random:
`$$INCOME_{ij} = \gamma_{0} + \gamma_{1} SAT_{ij} + (\nu_{j} + \varepsilon_{ij})$$`

---
### Random intercepts
`$$INCOME_{ij} = \gamma_{0} + \gamma_{1} SAT_{ij} + \color{red}{(\nu_{j} + \varepsilon_{ij})}$$`

There is now a composite residual that has parts *for each individual*. A school-level residual `$(\nu_{j})$` and an individual-level residual `$(\varepsilon_{ij})$`. Each residual term can be summarized by examining its variance/SD and correlated with other residual terms.

`$\nu_{j}$`: unique to school *j*; identical for all students in *jth* school; iid for every school

`$\varepsilon_{ij}$`: unique to student i in school j. iid for every student

Can calculate the **intra-class correlation**, *a summary of the proportion of outcome variability attributable to differences across schools.*

`$$\hat{\rho} = \frac{\sigma^{2}_{s}}{\sigma^{2}_{s}+\sigma^{2}_{i}}$$`

.small[Random effects estimation procedures use iterative algorithms (Maximum Likelihood Estimation, restricted MLE, Gradient descent, etc.) to minimize the sum of the square errors at each hierarchical level of the model.]

---
# Estimate random intercept model

```r
summary(lme4::lmer(income30 ~ SAT + (1 | school), data=df))
```

```
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: income30 ~ SAT + (1 | school)
#>    Data: df
#> 
#> REML criterion at convergence: 50411
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.3911 -0.6687  0.0083  0.6518  3.7524 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev.
#>  school   (Intercept) 6440493  2538    
#>  Residual             1160413  1077    
#> Number of obs: 3000, groups:  school, 3
#> 
#> Fixed effects:
#>              Estimate Std. Error t value
#> (Intercept) 50892.981   1713.254  29.705
#> SAT            -1.699      1.776  -0.957
#> 
#> Correlation of Fixed Effects:
#>     (Intr)
#> SAT -0.518
```

---
# Treatment effect?

```
...
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.3911 -0.6687  0.0083  0.6518  3.7524 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev.
#>  school   (Intercept) 6440493  2538    
#>  Residual             1160413  1077    
#> Number of obs: 3000, groups:  school, 3
#> 
#> Fixed effects:
#>              Estimate Std. Error t value
#> (Intercept) 50892.981   1713.254  29.705
#> SAT            -1.699      1.776  -0.957
#> 
#> Correlation of Fixed Effects:
#>     (Intr)
#> SAT -0.518
...
```

**What's different/same compared to fixed effect estimates?**