Untitled.Rmd

# Fixed Effects

Another source of variation is repeated measures of the same unit over time.
This can allow for identification with different identifying assumptions.

Basic idea ... ignorability does not hold conditional on some observed covariates $X_{it}$, but it **may** hold conditional on some unobserved, time-constant, variable ($U_i$),
$$
Y_{it}(d) \perp D_{it} | X_{it}, U_i
$$

Within units, the effect is identified, because even if it is unobserved, it is constant within the unit.
You can think of this as a special kind of control.

### Terminology

There are many different terms for repeated measurement data, including longitudinal, panel, and time-series cross-sectional data.
Generally,

-   **Panel data**: small $T$, large $N$. Examples, longitudinal surveys.
-   **TSCS data**: large $T$, small/medium $N$. Examples: countries over time as seen in international relations or IPE/CPE.

The issues of causality are mostly the **same** for these two types of data.

The esimation methods are different. Estimation methods often rely on asymptotic assumptions about observations going to infinity.
In repeated measurements there are two dimensions: number of units, and number of periods.
Different estimators will work better for small $T$ vs. large $T$, and small $N$ vs. large $N$.

## Fixed effects estimators

### Within Estimator

The within estimator subtracts the mean from the response, treatment, and all the controls:
$$
Y_{it} - \bar{Y} = (\bar{x}_{it} - \bar{x}_i)' \beta + \tau (D_{it} - \bar{D}_i) + (\epsilon_{it} - \bar{\epsilon}_{i})
$$
Note that $\bar{Y}_i$ are unit averages, and
$$
\bar{Y}_i = \bar{x}'_i \beta + \tau \bar{D}_i + U_i + \epsilon_{i} .
$$
Since the unobserved effect is constant over time, subtracting off the mean also subtracts that unobserved effect!
$$
U_i - \frac{1}{T} \sum_{t = 1}^1 U_i = U_i - U_i = 0 .
$$
The assumption of fixed effects being time-constant is essential.

Can use standard robust and sandwich type estimators.

Implications are:

-   Cannot control for or don't need to include time-constant controls.
-   Only removes time-contant unobserved effects in a unit. See diff-in-diff for a method to remove (some types of ) time-varying unobserved values.

## Least Squares Dummy Variable

Dummy variable regression is an alternative way to estimate fixed effects models.
Called the least squared dummy variable (LSDV) estimator.
Include a matrix of indicator variables ($W_i$) for each observation.
$$
Y_{it} = \tau D_{it} + w_i' \gamma + x'_{it} \beta + \epsilon_{it}
$$

-   Within vs. LSDV are equivalent algebraically 
-   LSDV is more computationally demanding.  With $p$ covariates and $G$ groups, within estimator's design matrix has only $p$ columns, whereas the LSDV design matrix has $p + G$ columns.
-   If naively estimate within estimator by demeaning variables and then using OLS standard errors will be incorrect. 
    They need to account for the degrees of freedom due to calculating the group means.
    
## First differences estimation

The first difference model is an alternative to mean-differences.

The model is,
$$
\begin{aligned}[t]
Y_{it} - Y_{i,t-1} &=  (x'_{it} - x'_{i,t-1}) \beta + \tau (D_{it} - D_{i,t-1}) + (\epsilon_{it} - \epsilon_{i,t-1}) \\
\Delta Y_{it} &= \Delta x'_{it} \beta + \tau \Delta D_{it} + \Delta \epsilon_{it}
\end{aligned}
$$

-   If $U_i$ are time-fixed, then first-differences are an alternative to mean-differences
-   If the difference in errors, $\Delta \epsilon_{it}$ are homoskedastic, OLS standard errors work fine.
-   But implies that original errors must have had serial correlation: $\epsilon_{it} = \epsilon_{i,t-1} + \Delta \epsilon_{it}$.
-   If serial correlation, then more efficient than FE
-   Robust/sandwich SEs can be used.


## Extensions

Fixed effects only identifies **contemporaneous effects**.
See other approaches (Blackwell) for dynamic panel data.




# Difference-in-Difference

In causal inference methods we are searching for sources of exogenous variation.
Panel data does not on its own identify an affect, but it does allow us to rely on diffeerent identifying assumptions.

## Basic differences-in-differences model

## Potential Outcomes Approach to DID

What is the takeaway? 


### Constant Effects Linear DID Model

Causal effects are constant across individuals and time,
$$
\E[Y_{it}(1) - Y_{it}(0)] = \tau .
$$
The effects of time $\delta_t$ and individuals $\alpha_g$ are linearly separable,
$$
\E[Y_{it}(0)] = \delta_t + \alpha_{g} .
$$
Then the model is,
$$
Y_{igt} = \delta_{t}
$$

## Threats to identification

-   Treatment independent of idiosyncratic shocks, so variation in outcome is the same for treated and control groups.

-   Example: Ashenfelter's Dip is an empirical phenomena in which people who enroll in job training programs see their earnings decline .
    
-   It may be possible to condition on covariates (control) in order to make treatment and shocks independent.

## Robustness Checks 

-   Lags and Leads

    -   If $D_{igt}$ causes $Y_{igt}$, then current and lagged values should have an effect on $Y_{igt}$, but future values of $D_{igt}$ should not.
    
-   Time Trends

    -   If more than two time periods, add unit specific linear trends to regression DID model.
        $$
        Y_{igt} = \delta_{t} + \tau G_{i} + \alpha_{0g} + \alpha_{1g} \times t + \epsilon_{igt} ,
        $$
        where $\alpha_{0g}$ are group fixed effects,  $\delta_t$ is the overall (not necessarily linear) time trend,
        and $\alpha_{1g}$ is the group linear time trend.
        
    -   Helps detect if varying trends when estimated from pre-treatment data.
    
    
## Extensions

The general DID model relies on linear-separability and constant treatment effects.

The **parallel trends** assumption is the important assumption:
$$
\E[ Y_{i1}(0) - Y_{i0} | X_i, G_i = 1] = \E[ Y_{i1}(0) - Y_{i0} | X_i, G_i = 0].
$$
It says that the potential trend under control is the same for the control and treated groups, conditional on covariates.

With the parallel trends assumption unconditional ATT is,
$$
\E[Y_{i1}(1) - Y_{i1}(0) | G_i = 1] = \E_{X}[\E[Y_{i1}(1) - Y_{i1}(0) | G_i = 1]] =  
, 𝐺𝑖 = 1]].
$$
What we need is an estimator of each CEF.
This doesn't need to be linear or parametric.

However, cannot estimate ATE because $\E(Y_{i1}(1) | X_i, G_i = 0)$ could be anything.

With covariates we can estimate conditional DID in sevearl ways.

-   Regression DID
-   Match on $X_i$ and then use regular DID
-   Weighting approaches Abadie (2005)

Regression DID includes $X_i$ in a linear, additive manner,
$$
Y_{it} = \mu + x'_i \beta_t + \delta I(t = 1) + \tau(I(t = 1) \times G_i) + \epsilon_{it}
$$
If there are repeated observations, take difference between $t = 0$ and $t = 1$,
$$
Y_{i1} - Y_{i0} = \delta + x'_i \beta + \tau G_i +
(\epsilon_{i1} - \epsilon_{i0})
$$
Have $\beta = \beta_1 - \beta_0$. 
Because everyone is untreated in first period, $D_{i1} - D_{i0} = D_{i1}$.

For panel data, regress changes on treatment.

Depends on constant effects and linearity in $X_i$.
Matching could reduce model dependence.

## Standard Error Issues

### Serial Correlation

$$
Y_{igt} = \mu_g + \delta_t + \tau (I_{it} \times G_i) + \nu_{gt} + \epsilon_{igt}
$$

Problem is that $\nu_{gt}$ can be serially correlated
$$
Cor(\nu_{gt}, \nu_{gs}) \neq 0 \text{ for } s \neq t .
$$
An example called $AR(1)$ serial correlation is when each $\nu_t$ is a function of its lag,
$$
\nu_t = \rho \nu_{t - 1} + \eta_t \text{ where } \rho \in (0, 1).
$$
Since errors are usually positvely correlated, the outcomes are correlated over time and effectively there are fewer independent observations in the sample; it's almost as if the same observation was simply copy and pasted over time with a little error added.
This will mean that the standard errors will likely be too optimistic (too narrow).
See Bertrand et al (2004).

There are a couple of solutions:

-   Placebo tests
-   Clustered standard errors at the **group** level
-   Clustered bootstrap (resample groups, not inidividual observations)
-   Aggregated to $g$ units with two time periods each: pre- and post-intervention.

All these solutions depend on larger numbers of groups.
The problem is that the serial correlation makes the panel data closer to simply a two-period DID.

## Other DID Approaches

Changes in Changes (Athey and Imbens 2006) generalizes DID to allow for different changes in the distribution of $Y_{it}$, not just the mean.
This allows for estimating ATT or any changes in distribution (quantiles, variance, etc.).
Unfortunately requires more data than estimating the mean.

Synthetic controls is used when there is one treated group, but many controls. (Abadie and Gardeazabel)

The basic idea is to compare the time series of the outcome in the treated group to a control.

-   But what if there are many control group? 
-   What if they aren't comparable to the treated?

Synthetic control uses a weighted average of different controls.

        
## Extensions

-   Imai etc.
-   Matching


## References

-   Matthew Blackwell, "Gov 2002: 9. Differences in
Differences" October 30, 2015 [URL](http://www.mattblackwell.org/files/teaching/s09-diff-in-diff-handout.pdf)