17.4 Delta Method

The Delta Method is a statistical technique for approximating the mean and variance of a function of random variables. It is particularly useful in regression analysis when estimating the standard errors of nonlinear functions of estimated coefficients, such as:

  • Marginal effects in nonlinear models (e.g., logistic regression)
  • Elasticities and risk measures (e.g., in finance)
  • Transformation of regression coefficients (e.g., log transformations)

This method is based on a first-order Taylor Series approximation, which allows us to estimate the variance of a transformed parameter without requiring explicit distributional assumptions.

Let \(G(\beta)\) be a function of the estimated parameters \(\beta\), where \(\beta\) follows an asymptotically normal distribution:

\[ \beta \sim N(\hat{\beta}, \text{Var}(\hat{\beta})). \]

Using a first-order Taylor expansion, we approximate \(G(\beta)\) around its expectation:

\[ G(\beta) \approx G(\hat{\beta}) + \nabla G(\beta) (\beta - \hat{\beta}), \]

where \(\nabla G(\beta)\) is the gradient (also known as the Jacobian) of \(G(\beta)\), i.e., the vector of partial derivatives:

\[ \nabla G(\beta) = \left( \frac{\partial G}{\partial \beta_1}, \frac{\partial G}{\partial \beta_2}, \dots, \frac{\partial G}{\partial \beta_k} \right). \]

The variance of \(G(\beta)\) is then approximated as:

\[ \text{Var}(G(\beta)) \approx \nabla G(\beta) \, \text{Cov}(\beta) \, \nabla G(\beta)'. \]

where:

  • \(\nabla G(\beta)\) is the gradient vector of \(G(\beta)\).

  • \(\text{Cov}(\beta)\) is the variance-covariance matrix of \(\hat{\beta}\).

  • The expression \(\nabla G(\beta)'\) denotes the transpose of the gradient.

Key Properties of the Delta Method

  • Semi-parametric approach: It does not require full knowledge of the distribution of \(G(\beta)\).
  • Widely applicable: Useful for computing standard errors in regression models.
  • Alternative approaches:
    • Analytical derivation: Directly deriving a probability function for the margin.
    • Simulation/Bootstrapping: Using Monte Carlo methods to approximate standard errors.