Week 03 - Observational Studies

# Week 03 - Observational Studies
## Causal Inference without Randomisation
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### Danilo Freire
### 6th February 2019

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# Today's Agenda

* Observational studies vs RCTs (QSS 2.5-2.6)

* Homework: finish `CAUSALITY01` and our first assignment 
]
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# Revision

.font150[
* What is the fundamental problem of causal inference?
]
--
.font150[
* How can we (try to) solve that problem?
]
--
.font150[
* .orange[Randomisation]
]

---

# Randomisation

* Randomisation .orange[generates balance] between treatment and control groups

* Treatment and control groups .orange[identical on average]

* Allows us to estimate the average treatment effect simply as a .orange[difference in means]: `$\frac{1}{n} \sum_{i=1}^n \{Y_i(1) - Y_i(0)\}$`
]

---

# Social Pressure Example

* Send postcards with different, randomly-assigned messages
  - no message (control group)
  - civic duty message
  - "you are being studied" message (Hawthorne effect)
  - neighbourhood social pressure message
]

---

# Social Pressure Example

```r
tapply(social$yearofbirth, social$messages, mean) # Year of Birth
```

```
## Civic Duty    Control  Hawthorne  Neighbors 
*##   1956.341   1956.186   1956.295   1956.147
```

```r
tapply(social$hhsize, social$messages, mean) # Household size
```

```
## Civic Duty    Control  Hawthorne  Neighbors 
*##   2.189126   2.183667   2.180138   2.187770
```

```r
tapply(social$primary2006, social$messages, mean) # Outcome
```

```
## Civic Duty    Control  Hawthorne  Neighbors 
*##  0.3145377  0.2966383  0.3223746  0.3779482
```

```r
tapply(social$primary2006, social$messages, mean)[-2] -
    mean(social$primary2006[social$messages == "Control"]) # SATE for each group
```

```
## Civic Duty  Hawthorne  Neighbors 
*## 0.01789934 0.02573631 0.08130991
```
---

# Observational Studies

.font150[
* Often, it is unethical or infeasible to randomise the treatment. 
  - Smoking and cancer 
  - Gender and salary
  - Death penalty and crime

* .orange[Observational data]: the treatment is naturally assigned
]

---

# Observational Studies

--
.font150[
* .orange[Confounders/confounding variables:] variables that are associated both with the treatment and the outcome 
]
--

.font150[
* If the treatment is not independent from pre-treatment variables, we cannot be sure about what causes what
]

---

# Examples of Confounders:

.font150[
* Are incumbents more likely to win elections?
  - Maybe, but they also receive more campaign donations
]
--

.font150[
*  Are democratic countries more peaceful than authoritarian ones?
   - Maybe, but they also tend to be richer
]

---

# Observation Studies

.font150[
* Pre-treatment variables have to be .orange[controlled for], that is, _held constant_ in the analysis

* .orange[Statistical control]: we use statistical methods to create balance between treatment and control and emulate an experiment as best as we can. E.g:
  - Instrumental variables (IV)
  - Regression discontinuity designs (RDD)
  - Differences-in-differences (DD/DiD/Diff-in-diff)
]

---

# Differences-in-Differences (DiD)

* Controls for the initial conditions

* Requires data for two time periods (before and after treatment) and at least two units

* .orange[Parallel time trends assumption]
]

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# Differences-in-Differences (DiD)

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# Minimum Wage and Unemployment

* Economists believe the effect is positive: higher wages lead to higher unemployment

* Difficult to randomise minimum wage legislation

* In 1992, NJ minimum wage increased from $4.25 to $5.05...

* ... but neighbouring PA stays at $4.25

* NJ and (eastern) PA are similar, and fast food chains are similar too: wages, prices, products, etc

* [David Card and Alan Krueger (1994)](http://davidcard.berkeley.edu/papers/njmin-aer.pdf) see this change as a natural experiment
]

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# Minimum Wage and Unemployment

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# Minimum Wage and Unemployment

```r
minwage <- read.csv("https://raw.githubusercontent.com/pols1600/pols1600.github.io/master/datasets/causality/minwage.csv")
str(minwage)
```

```
## 'data.frame':	358 obs. of  8 variables:
##  $ chain     : Factor w/ 4 levels "burgerking","kfc",..: 4 4 1 1 2 2 3 1 1 1 ...
##  $ location  : Factor w/ 5 levels "centralNJ","northNJ",..: 3 3 3 3 3 3 3 3 3 3 ...
##  $ wageBefore: num  5 5.5 5 5 5.25 5 5 5 5 5.5 ...
##  $ wageAfter : num  5.25 4.75 4.75 5 5 5 4.75 5 4.5 4.75 ...
##  $ fullBefore: num  20 6 50 10 2 2 2.5 40 8 10.5 ...
##  $ fullAfter : num  0 28 15 26 3 2 1 9 7 18 ...
##  $ partBefore: num  20 26 35 17 8 10 20 30 27 30 ...
##  $ partAfter : num  36 3 18 9 12 9 25 32 39 10 ...
```

---

# Minimum Wage and Unemployment

```r
# Subset the data into NJ and PA
minwageNJ <- subset(minwage, subset = (location != "PA"))
minwagePA <- subset(minwage, subset = (location == "PA"))
```

```r
# Compute the proportion of restaurants whose wage is less than $5.05
mean(minwageNJ$wageBefore < 5.05) # NJ before
```

```
*## [1] 0.9106529
```

```r
mean(minwageNJ$wageAfter < 5.05) # NJ after
```

```
*## [1] 0.003436426
```

```r
mean(minwagePA$wageBefore < 5.05) # PA before
```

```
*## [1] 0.9402985
```

```r
mean(minwagePA$wageAfter < 5.05) # PA after
```

```
*## [1] 0.9552239
```

---

# Minimum Wage and Unemployment

```r
# Are the NJ and PA Restaurants Comparable?
mean(minwageNJ$wageBefore)
```

```
*## [1] 4.609966
```

```r
mean(minwagePA$wageBefore)
```

```
*## [1] 4.651343
```

```r
# Compute the proportion of full-time employees after the increase
minwageNJ$fullPropBefore <- minwageNJ$fullBefore / (minwageNJ$fullBefore + minwageNJ$partBefore)
minwagePA$fullPropBefore <- minwagePA$fullBefore / (minwagePA$fullBefore + minwagePA$partBefore)
mean(minwageNJ$fullPropBefore) # Proportion full-time workers NJ
```

```
*## [1] 0.2965262
```

```r
mean(minwagePA$fullPropBefore) # Proportion full-time workers PA
```

```
*## [1] 0.3099657
```

---

# Minimum Wage and Unemployment

```r
# Compute the proportion of full-time employees after the increase
minwageNJ$fullPropAfter <- minwageNJ$fullAfter / (minwageNJ$fullAfter + minwageNJ$partAfter)
minwagePA$fullPropAfter <- minwagePA$fullAfter / (minwagePA$fullAfter + minwagePA$partAfter)
mean(minwageNJ$fullPropAfter) # Proportion full-time workers NJ
```

```
## [1] 0.320401
```

```r
mean(minwagePA$fullPropAfter) # Proportion full-time workers PA
```

```
## [1] 0.2722821
```

---

# Minimum Wage and Unemployment

```r
# Compare NJ before and after the change
NJdiff <- mean(minwageNJ$fullPropAfter) - mean(minwageNJ$fullPropBefore)
NJdiff
```

```
*## [1] 0.02387474
```
.font120[
However, the region as a whole could have changed too. So we use PA as a control case
]

```r
# PA before/after difference to control for possible changes over time
PAdiff <- mean(minwagePA$fullPropAfter) - mean(minwagePA$fullPropBefore)
PAdiff
```

```
*## [1] -0.03768357
```

```r
# Difference in difference
NJdiff - PAdiff
```

```
*## [1] 0.06155831
```

---

# Minimum Wage and Unemployment

.font150[
* .orange[Parallel trends assumption:] NJ would have followed the same trend as PA had it not received the treatment. However, _we can't test that_.
]

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# Animation of DiD

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# Questions?

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# Homework

* `swirl()` `CAUSALITY02`
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# See you on Friday!