Lecture 10

IV Estimation in Simple Regression

• Regression Model:
  yt = \alpha + \beta xt + u_t

  • Assumption: {Cov}(xt, ut) ≠ 0

  • Existence of Instrument: Variable $z_t$ such that:

  • {Cov}(xt, zt) ≠ 0 (Relevance)

  • {Cov}(zt, ut) = 0 (Validity)

• IV Estimators:
  

  

Properties of the IV Estimator

• Consistent Estimators (TS1 - TS3):

  • $\hat{\alpha}_{IV}$ is a consistent estimator of $\alpha$

  • $\hat{\beta}_{IV}$ is a consistent estimator of $\beta$

• Statistical Inference Result:
  $\sqrt{T}(\hat{\beta}<em>{IV} - \beta) \sim N(0, Var((z</em>t - E(z<em>t))u</em>t)/Cov(x<em>t, z</em>t)^2) \text{ approximately}$

• Conditional Variance:

  • If $Var(u<em>t|z</em>t) = E(u<em>t^2|z</em>t) = \sigma^2$ (Homoskedasticity),

  • $Var((z<em>t - E(z</em>t))u<em>t) = \sigma^2 Var(z</em>t)$

    • In this case,
      $\sqrt{T}(\hat{\beta}<em>{IV} - \beta) \xrightarrow{d} N(0, \frac{\sigma^2 Var(z</em>t)}{Cov(x<em>t, z</em>t)^2})$

Other Issues Related to Stochastic Regressors

• Model Specification:

  • Regression Model:
    $y<em>t = \alpha + \beta x</em>t + u_t$

• Assumptions Violations:

  1. $E[u<em>t|x</em>t] = 0$

  2. $Var(u<em>t|x</em>t) = \sigma^2$

  3. (xt, ut) is iid

• If $E[u<em>t|x</em>t] \neq 0$ (Exogeneity Violation):

  • OLS Estimator fails.

Conditional Heteroskedasticity

• If $Var(u<em>t|x</em>t) = \sigma^2<em>t \neq \sigma^2$, the variance depends on $x</em>t$.

• Implications for OLS Estimator:

  • Unbiased:
    $\hat{\beta} = \frac{\sum<em>{t=1}^T (x</em>t - \bar{x})(y<em>t - \bar{y})}{\sum</em>{t=1}^T (x_t - \bar{x})^2}$

  • Error:
    $\hat{\beta} - \beta = \frac{\sum<em>{t=1}^T (x</em>t - \bar{x})(u<em>t - \bar{u})}{\sum</em>{t=1}^T (x_t - \bar{x})^2}$

  • Therefore, $E[\hat{\beta} - \beta|x_t] = 0$

• Variance of the OLS Estimator:

  • When no conditional heteroskedasticity:
    $Var(\hat{\beta}|x<em>t) = \sigma^2 \frac{1}{\sum</em>{t=1}^T (x_t - \bar{x})^2}$

  • Estimated by $s^2$ instead of $\sigma^2$

Statistical Inference under Heteroskedasticity

• Use of Robust Standard Errors:

  • Remedies for heteroskedasticity: use White’s or HAC Standard Errors.

  • Hypothesis Testing and Confidence Intervals are then valid.

Conditional Autocorrelation

• Definition:

• Correlation of Errors:
  $Cor(u<em>t, u</em>s) = \frac{Cov(u<em>t, u</em>s)}{\sqrt{Var(u<em>t)Var(u</em>s)}}$

• Implies OLS Estimator is unbiased but requires a careful derivation for the standard error.

Basics on Time Series

• Importance in Economics:

• Key Models: - Autocorrelation in linear regression framework must be understood.

Autocovariance and Autocorrelation

• Definitions:

  • Autocovariance:
    $\gamma<em>{t,s} = E[(X</em>t - \mu)(X_s - \mu)]$

  • Autocorrelation:
    $\rho<em>{t,s} = \frac{\gamma</em>{t,s}}{\sigma^2}$

Stationary Time Series

• Definitions of Stationarity:

  • Strictly Stationary: Joint distributions invariant under time shifts.

  • Weakly Stationary: Mean, Variance constant, Autocovariance depends only on lag.

• Autocovariance Function:
  $\gamma(h) = E((X<em>h - \mu)(X</em>0 - \mu))$

Examples of Time Series Models

• White Noise:
  $E(X<em>t) = 0 and Var(X</em>t) = \sigma^2$

• Moving Average Processes (MA(1)):
  $X<em>t = \epsilon</em>t - \theta \epsilon_{t-1}$

AR(p) Models

• Model Specification:
  $X<em>t = \mu + \phi</em>1 X<em>{t-1} + \ldots + \phi</em>p X<em>{t-p} + \epsilon</em>t$

AR(1) Model and Covariance Stationary

• Condition for Stationarity: |\phi1| < 1 \Rightarrow Cov(Xt) \text{ stationary}

  • If $|\phi_1| \geq 1$, series is generally not stationary, exemplified by random walk behaviour.