1 Introduction

1.1 Contents

  • Framework
  • Cluster-Robust Standard Errors
  • Implementation in R: felm command

1.2 Introduction

  • Panel data
    • \(n\) cross-sectional units at \(T\) time periods
    • Dataset \((X_{it}, Y_{it})\)
  • 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 temporal variation)
    2. Can deal with time-invariant unobserved factors.
    3. (Not focus in this course) Dynamics of individual over time.

2 Framework

2.1 Framework

  • 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 natural concern here is the omitted variable bias.
  • We now consider that \(\epsilon_{it}\) is written as \[ \epsilon_{it} = \alpha_i + u_{it} \] where \(\alpha_i\) is called unit fixed effect, 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 \(l=i\).
    • Notice that we cannot do this for the cross-section data!
  • We write the model with unit FE as \[ y_{it} = \beta' x_{it} + \alpha_i + u_{it} \]

2.2 Framework

  • The fixed effects 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

2.3 Assumption 1: Mean independence

  • Assumption 1 is weaker than the assumption in OLS, because the time-invariant factor \(\alpha_i\) is captured by the fixed effect.
    • Example: Unobserved ability is caputured by \(\alpha_i\).

2.4 Assumption 4: No Perfect Multicolinear.

  • Consider the following regression with unit FE \[ wage_{it} = \beta_0 + \beta_1 experience_{it} + \beta_2 male_{i} + \beta_3 white_{i} + \alpha_i + u_{it} \]
    • \(experience_{it}\) 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 completely captured by the unit fixed effect \(\alpha_i\).

3 Estimation

3.1 Estimation (within transformation)

  • 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.
  • 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 \(\alpha_i\) \[ \tilde{Y}_{it} = \beta' \tilde{X}_{it} + \tilde{u}_{it} \]
    • Then use the OLS estimator to the above equation!.

3.2 Importance of within variation in estimation

  • The variation of the explanatory variable is key for precise estimation.
    • Remember the lecture “Regression 3”
  • 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.

3.3 Other things to note

  1. 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.
  2. You can use IV regression with panel data.
    • The argument for the conditions of instruments should consider the presence of fixed effects.
    • Correlation (or uncorrelatedness) after controlling for the fixed effects.

4 Inference

4.1 Cluster-Robust Standard Errors

  • So far, we considered the two cases on the error structure
    1. Homoskedasticity \(Var(u_i) = \sigma^2\)
    2. Heteroskedasitcity \(Var(u_i | x_i) = \sigma(x_i)\)
  • In the above case, we still assume the independence between observations, that is \(Cov(u_{i},u_{i'} ) = 0\).
  • In the panel data setting, we need to consider the autocorrelation.
    • the correlation between \(u_{it}\) and \(u_{it'}\) across periods for each individual \(i\).
  • Cluster-robust standard error considers such autocorrelation.
    • The cluster is unit \(i\). The errors within cluster are allowed to be correlated.
  • For a more discussion, see Chapter 8 in “Mostly Harmless Econometrics”.