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# Welcome to the Course!!


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## About me

- Instructor: Yuta Toyama

- Research Field: Industrial Organization, Applied Econometrics, Energy and Environmental Economics



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## Data analysis is more important than ever

.center[![Data Scientist](figs/DataScientist.png)]



  
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## Goal of the Course

- We will learn **causal inference (因果推論) ** in economics.


- Goal 1: How to establish **causality** in data analysis.
    - Various tools: RCT, regression, IV, DID, RD, structural estimation
    - Implementation using R language.
    - **Students who took this course before won an award at the Data Science Competition and the thesis competition.**
  
  
- Goal 2: How to critically examine causal statements in research and journalism.
    - **Is my "causal" claim above really true?**



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## Contents today

1. What is causality?

2. Case: Data analysis during the Covid-19 crisis

3. Review of Probability: Conditional Expectation

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# What is Causality?

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## Causal Questions in Economics and Political Science

- Causality (因果): X causes/affects/impacts Y **holding other things fixed *(ceteris paribus)***


- Many questions are causal!
  - How much does an additional year of schooling increase your wage ?
  - How does online advertisement affect sales of products?
  - Do mergers between firms increase product prices?
  - Does democracy cause economic growth?
  - Does higher turnout benefit Democrats in presidential election?

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## Theoretical Models Implies Causal Relationship

- Consumers' Utility maximization:
    - Higher income leads to more consumption (all else being fixed)
    - Higher price (typically) implies less consumption (all else being fixed)


- Educational investment leads to higher income due to 
    - higher human capital (人的資本)
    - signalling effect (シグナリング)

    
- Government spending stimulates GDP by multiplier effect (乗数効果)


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## Data Analysis to Establish Causality

- With data, we want to understand what is going on in reality.

- Ex 1: Test whether theoretical prediction is true.

- Ex 2: Determine the sign of ambiguous theoretical prediction.

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## Descriptive Analysis (記述統計)

- Describe the data (sample) as a first step.
    - College grads earn 98% more per hour than others
    - Income inequality higher now than 30 years ago
    - Health care costs growing more slowly after the Affordable Care Act (ACA)
    - Airline prices higher now than before merger wave 

- Such analysis tells you the **correlation (相関)** between two variables

    

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## Does **correlation** imply **causality**?


- Example 1: Cities with many police officers have more crimes (positive correlation).
   - Does this really mean “X (police officers) causes Y (crimes)”?


- Example 2: Those who went to college earn more money by 10%.
  - Is this difference only due to college? 

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## Three possible channels 

1. Causality
2. Reverse causality
3. Third factor, or **confounder (交絡因子)**
    - often called **spurious correlation (見せかけの相関)**


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## Three Channels in Figure


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## Case: Is Avigan effective treatment for COVID-19?

- Some Japanese celebrities who caught COVID-19 report their own experience that Anti-flu drug Avigan was effective for recovery.
  - [An episode from Junichi Ishida](https://www.chunichi.co.jp/chuspo/article/entertainment/news/CK2020042302100136.html)


- Does this mean Avigan is a silver bullet for COVID-19?


- Why do we care? 
  - Side effects of Avigan, resource allocation of public funds, etc.

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.center[![Data Scientist](figs/news_avigan.png)]


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## Difficulty 1: Counterfactual (反実仮想)

- **Counterfactual outcome** is never observed 

- Even without Avigan, a patient might have gotten better.

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## Difficulty 2: Endogenous selection of treatment

- Suppose we have data on patients who got and did not get Avigan in their treatment.

- Does comparison of those people give you treatment effect of Avigan?

- Issue: Those who get Avigan and who do not get might be quite different in other aspects. 

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## Solution: Clinical Trial

- Collect patients and **randomly** choose patients who are given the treatment.
  - Treatment group and control group.


- Compare the outcome of treatment and control group to see the effectiveness of treatment.


- Since 2020, various institutes have been conducting a clinical trial on Avigan for COVID-19. The results have been not assuring, though. 

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# Case: Data Analysis during the COVID-19 Crisis

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# Data Analysis during the COVID-19 Crisis

- Perhaps, you might have started to see data analysis in the news more often than ever.
    - `#` of positive cases each day
    - flow of people (人流) based on mobile phone location (e.g., Agoop, NTT Docomo)
    - Effective reproduction number (実効再生算数)
    - Efficacy of COVID-19 vaccine from clinical trial

    
- Researchers in many fields have been conducting various analysis to tackle the crisis.


- Some examples in Economics
    - Macroeconomic analysis using SEIR (Susceptible-Exposed-Infectious-Recovered) model
    - Nowcasting (ナウキャスティング) using alternative data (オルタナティブデータ)
    - Evaluation of various policies towards the crisis. -> more on later

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# Reference: 経済セミナー別冊「新型コロナ危機に経済学で挑む」




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# Difficulties in Policy Analysis

- 1: **Observational study (観察研究)**: 
    - Data is observation of what people actually do.
    - Unlike clinical trial, cannot randomize treatment of policy in general.


- 2: Data limitation
    - May not be able to observe what researchers need.


- With these obstacles, we still need to **draw the conclusion while clearly acknowledging its limitations**.


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# Case: Special Cash Payment

- Special Cash Payment (特別定額給付金): 100,000 JPY for everyone.


- Question: How much did people actually spend?
    
    
- Why we care?
    - Controversial policy (both supporters and )
    - Many countries have done a similar policy.
    

- Three empirical studies on this issue:
    - Kubota, Onishi, and Toyama (2021): Bank account data
    - Kaneda, Kubota, and Tanaka (2021): Financial app data
    - 宇南山・古村・服部 (2021): Official household survey (「家計調査」)


- I will explain Kubota, Onishi, and Toyama (2021) in detail.

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# Review of Conditional Expectation

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# Conditional Expectation (条件付き期待値)

- Overview
    - We are interested in the relationship between two different variables, say $X$ and $Y$.
    - Conditional expectation is a way to characterize such relationship
    - Quick review of conditional expectation.

-  Reference
    - Angrist and Pischke MHE Chapter 2 and 3.
    - 奥井ら

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# Conditional Distribution (条件付き分布)

- Let $Y$ and $X$ be the random variables (確率変数). 
    - Let's say $Y$ is the outcome variable and $X$ is the explanatory variable. 


- The probability distribution function (確率分布関数)
    - Discrete (離散): $$P(y,x)=Prob(X=x,Y=y)$$ for $y\in\{y_{1},\cdots,y_{L}\}$ and $x\in\{x_{1},\cdots,x_{L}\}$
    - Continuous (連続): the density function (密度関数) $f(y,x)$ for $x\in\mathbb{R}$ and $y\in\mathbb{R}$. 

<!-- - Definition: **Support (台)** of the random variable $X$, denoted by $supp(X)$ is the range within which the random variable $X$ can take a value. -->
<!--     - In the discrete case, $supp(X)=\{x_{1},\cdots,x_{L}\}$ -->

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- The conditional probability mass function for discrete case $$P(y|x)=\frac{P(y,x)}{P(x)}$$ where $P(x)=\sum_{i=1}^{N}P(y_{i},x)$

- The conditional density function for continuous case $f(y|x)=\frac{f(y,x)}{f(x)}$



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# Conditional Expectation (条件付き期待値) 

- The **conditional expectation** function $$E[Y|X]=\sum_{l=1}^{L}y_{l}P(Y=y_{l}|X)$$ for a discrete case and $$E[Y|X]=\int_{-\infty}^{\infty}yf(y|X)dy$$ for a continuous case. 



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## Properties of Conditional Expectation Function

- Proposition CE1 $$E[c(X)|X]=c(X)$$ for a function $c(\cdot)$. 

- Proposition CE2 (linearity) $$E[a(X)Y+b(X)|X]=a(X)E[Y|X]+b(X)$$ for function $a(\cdot)$ and $b(\cdot)$. 

- Proposition CE3. If $X$ and $Y$ are independent, then $E[Y|X]=E[Y]$

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- Proof of CE3 (for a discrete case): 

\begin{eqnarray}
E[Y|X]&=&\sum_{l=1}^{L}y_{l}P(Y=y_{l}|X) \\
	  &=&\sum_{l=1}^{L}y_{l}\frac{P(Y=y_{l},X)}{P(X)}=\sum_{l=1}^{L}y_{l}\frac{P(Y=y_{l})\times P(X)}{P(X)}=E[Y].
\end{eqnarray}

Note that we use $P(Y=y,X=x)=P(X=x)P(Y=y)$ by independence.



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# Law of iterated expectation (繰り返し期待値の法則)

- Proposition CE4 (**law of iterated expectation**)$$E[Y]=E[E[Y|X]]$$

- Expectation of conditional expectation $E[Y|X]$ is unconditional expectation $E[Y]$




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- Proof for a discrete case

\begin{eqnarray}
E[Y]&=&\sum_{l=1}^{L}y_{l}P(y_{l}) \\
	&=&\sum_{l=1}^{L}y_{l}\left[\sum_{l'=1}^{L} P(y_{l},x_{l'})\right] \\
	&=&\sum_{l=1}^{L}y_{l}\left[\sum_{l'=1}^{L}P(y_{l}|x_{l'})P(x_{l'})\right] \\
	&=&\sum_{l'=1}^{L}P(x_{l'})\underbrace{\left[\sum_{l=1}^{L}P(y_{l}|x_{l'})y_{l}\right]}_{E[Y|X]}=E[E[Y|X]]
\end{eqnarray}

- Note: A similar proof for the continuous case, though it requires some technical conditions for the exchange of integrals (Fubini's theorem).



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<br/><br/>
- Proposition CE4' $$E[Y|X]=E\left[E[Y|X,Z]|X\right]$$
    - Conditional on $X$, can apply the law of iterated expectation.
    
    
- Proposition CE5: If $E[Y|X]=E[Y]$, then $Cov(X,Y)=0$.
    - We often say $E[Y|X]=E[Y]$ is **mean independence**
    - Remember: The opposite is not true.


<!-- <br/><br/> -->
<!-- - Proposition CE6: Suppose that $E[Y^{2}]<\infty$ and $E[g(X)]<\infty$ for some function $g$. Then $$E[(Y-\mu(X))^{2}]\leq E[(Y-g(X))^{2}]$$ where $\mu(X)=E[Y|X]$.  -->

<!-- - Interpretation:  -->
<!--     - Suppose that we explain $Y$ using some function $g$ and data $X$. -->
<!--     - The function $g$ that minimizes **the mean squared error** is conditional expectation.  -->



<!-- --- -->

<!-- - Proof:  -->

<!-- \begin{eqnarray} -->
<!-- E[(Y-g(X))^{2}]	&=&E[\left\{ \left(Y-\mu(X)\right)+\left(\mu(X)-g(X)\right)\right\} ^{2}]\\ -->
<!-- 	&=&E\left[\left(Y-\mu(X)\right)^{2}\right]+E\left[\left(\mu(X)-g(X)\right)^{2}\right]\\ -->
<!-- 	&+& 2E\left[\left(Y-\mu(X)\right)\left(\mu(X)-g(X)\right)\right].  -->
<!-- \end{eqnarray} -->

<!-- Using the law of iterated expectation,  -->

<!-- \begin{eqnarray} -->
<!-- E\left[\left(Y-\mu(X)\right)\left(\mu(X)-g(X)\right)\right]	&=&E\left\{E\left[\left(Y-\mu(X)\right)\left(\mu(X)-g(X)\right)|X\right]\right\} \\ -->
<!-- 	&=&E\left\{ \left(\mu(X)-g(X)\right)\left(E[Y|X]-\mu(X)\right)\right\} \\ -->
<!-- 	&=&0 -->
<!-- \end{eqnarray} -->

<!-- Thus,  -->
<!-- $$E[(Y-g(X))^{2}]=E\left[\left(Y-\mu(X)\right)^{2}\right]+E\left[\left(\mu(X)-g(X)\right)^{2}\right],$$ which can take its minimum when $g(X)=\mu(X)$.  -->



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# Another useful properties 

- Law of iterated probability $$P(Y)=\sum_{l=1}^{L}P(Y|x_{l})P(x_{l})$$ for a discrete random variable $X$
- Law of total variance $$Var(Y)=E[V(Y|X)]+V[E(Y|X)]$$

