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# Introduction

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## Introduction

- Panel data (パネルデータ)
    - combination of crosssection (クロスセクション) and time series (時系列) data


- Examples:
    1. Person $i$'s income in year $t$.
    2. Vote share in county $i$ for the presidential election year $t$.
    3. Country $i$'s GDP in year $t$. 


- Panel data is useful
    1. More variation (both cross-sectional and time series variation)
    2. Can deal with **time-invariant unobserved factors**. 

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## Course Plan

- Framework

- Implementation in R

- **Difference-in-differences (DID, 差の差分法)** 


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# Framework

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## Framework with Panel Data

- Consider the model
$$y_{it} = \beta' x_{it} + \epsilon_{it}, E[\epsilon_{it} | x_{it} ] = 0$$
where $x_{it}$ is a k-dimensional vector

- If there is no correlation between $x_{it}$ and $\epsilon_{it}$, you can estimate the model by OLS **(pooled OLS)**

- A concern here is the omitted variable bias. 

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## Introducing fixed effect (固定効果)

- Suppose that $\epsilon_{it}$ is decomposed as
$$\epsilon_{it} = \alpha_i + u_{it}$$
where $\alpha_i$ is called **unit fixed effect (固定効果)**, which is the time-invariant unobserved heterogeneity. 

- With panel data, we can control for the unit fixed effects by incorporating the dummy variable for each unit $i$!
$$y_{it} = \beta' x_{it} + \gamma_2 D2_i + \cdots + \gamma_n Dn_i + u_{it}$$
where $Dl_i$ takes 1 if $l=i$. 

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## Fixed Effect Model

- Model
$$y_{it} = \beta' x_{it} + \alpha_i + u_{it}$$


- Assumptions:
    1. $u_{it}$ is uncorrelated with $(x_{i1},\cdots, x_{iT})$, that is $E[u_{it}|x_{i1},\cdots, x_{iT} ] = 0$
    2. $(Y_{it}, x_{it})$ are independent across individual $i$.
    3. No outliers
    4. No perfect multicollinarity between explantory variables $x_{it}$ and fixed effects $\alpha_i$.

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## Assumption 1: Mean independence

- Assumption 1 is weaker than the assumption in OLS. 

- Here, the time-invariant unobserved factor is captured by the fixed effect $\alpha_i$.


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## Assumption 4: No Perfect Multicolinearity

- Consider the following model
$$wage_{it} = \beta_0 + \beta_1 experience_{it} + \beta_2 male_{i} + \beta_3 white_{i} + \alpha_i + u_{it}$$
    - $experience_{it}$ measures how many years worker $i$ has worked before at time $t$.

- Multicollinearity issue because of $male_{i}$ and $white_{i}$.

- Intuitively, we cannot estimate the coefficient $\beta_2$ and $\beta_3$ because those **time-invariant variables are captured by the unit fixed effect $\alpha_i$**.

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# Estimation

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## Estimation with Fixed Effects

- Can estimate the model by adding dummy variables for each individual. 
    - **least square dummy variables (LSDV) estimator**.
    - Computationally demanding with many cross-sectional units

- We often use the following **within transformation**.

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## Estimation by within transformation 

- Define the new variable $\tilde{Y}_{it}$ as 
$$\tilde{Y}_{it} = Y_{it} - \bar{Y}_i$$
where $\bar{Y}_i = \frac{1}{T} \sum_{t=1}^T Y_{it}$. 

- Applying the within transformation, can eliminate the unit FE $\alpha_i$
$$\tilde{Y}_{it} = \beta' \tilde{X}_{it} + \tilde{u}_{it}$$

- Apply the OLS estimator to the above equation!.

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## Importance of within variation in estimation

- The variation of the explanatory variable is key for precise estimation. 


- Within transformation eliminates the time-invariant unobserved factor, 
    - a large source of endogeneity in many situations. 


- But, within transformation also absorbs the variation of $X_{it}$. 

- Remember that 
$$\tilde{X}_{it} = X_{it} - \bar{X}_i$$
    - The transformed variable $\tilde{X}_{it}$ has the variation over time $t$ within unit $i$.
    - If $X_{it}$ is fixed over time within unit $i$, $\tilde{X}_{it} = 0$, so that no variation.

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## Various Fixed Effects

- You can also add **time fixed effects (FE)**
$$y_{it} = \beta' x_{it} + \alpha_i + \gamma_t + u_{it}$$

- The regression above controls for both **time-invariant individual heterogeneity** and **(unobserved) aggregate year shock**.

- Panel data is useful to capture various unobserved shock by including fixed effects.

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## Cluster-Robust Standard Errors

- In OLS, we considered two types of error structures:
  1. Homoskedasticity $Var(u_i) = \sigma^2$
  2. Heteroskedasitcity $Var(u_i | x_i) = \sigma(x_i)$


- They assume the independence between observations, that is $Cov(u_{i},u_{i'} ) = 0$.


- In the panel data setting, we need to consider the **autocorrelation (自己相関)**.
  - the correlation between $u_{it}$ and $u_{it'}$ across periods for each individual $i$.


- **Cluster-robust standard error (クラスターに頑健な標準誤差)** considers such autocorrelation.
    - The cluster is unit $i$. The errors within cluster are allowed to be correlated. 


