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.

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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.