05_Panel_DID/Panel1.html

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# Panal Data 1: Framework
### Instructor: Yuta Toyama
### Last updated: 2021-07-09

---


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# Introduction

---
## 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**. 

---
## Course Plan

- Framework

- Implementation in R

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


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# Framework

---
  
## 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. 

---

## 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 `\(l=i\)`. 

---
## 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 `\(\alpha_i\)`.

---

## 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 `\(\alpha_i\)`.


---
## 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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name: logistics
# Estimation

---
## 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**.

---
## 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!.

---
## 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.

---
## 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.

---
## 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. 


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