Delta Method

The delta method is used to estimate the variance of a non-linear function of a set of already estimated parameters. The basic result is that if \(\theta\) are the parameters and we have

\begin{equation} \sqrt T \left( {\hat \theta - \theta } \right) \xrightarrow{d} N(\left( 0,\Sigma _\theta \right) \label{eq:stats_deltabase} \end{equation}

and if \({f(\theta )}\) is continuously differentiable, then, by using a first order Taylor expansion

\begin{equation} \left( {f(\hat \theta ) - f(\theta )} \right) \approx f'(\theta )\left( {\hat \theta - \theta } \right) \end{equation}

Reintroducing the \(\sqrt T\) scale factors and taking limits gives

\begin{equation} \sqrt T \left( {f(\hat \theta ) - f(\theta )} \right) \xrightarrow{d} N\left( {0,f'(\theta )\Sigma _\theta f'(\theta )'} \right) \end{equation}

In practice, this means that if we have

\begin{equation} \hat \theta \approx N(\theta ,{\bf{A}}) \label{eq:stats_deltarebase} \end{equation}

then

\begin{equation} f\left( {\hat \theta } \right) \approx N\left( {f(\theta ),f'(\hat \theta ){\bf{A}}f'(\hat \theta )'} \right) \label{eq:stats_deltareresult} \end{equation}

\eqref{eq:stats_deltabase} is the type of formal statement required, since the \(\bf{A}\) in \eqref{eq:stats_deltarebase} collapses to zero as \(T \to \infty \). It's also key that \eqref{eq:stats_deltabase} implies that \(\hat \theta \xrightarrow{p} \theta \), so \(f'(\hat \theta ) \xrightarrow{p} f'(\theta )\) allowing us to replace the unobservable \(f'(\theta )\) with the estimated form in \eqref{eq:stats_deltareresult}. So the point estimate of the function is the function of the point estimate, at least as the center of the asymptotic distribution. If \(\hat \theta\) is unbiased for \(\theta\), then it's almost certain that \(f(\hat \theta)\) will not be unbiased for \(f(\theta)\) so this is not a statement about expected values.

To compute the asymptotic distribution, it's necessary to compute the partial derivatives of \(f\). For scalar functions of the parameters estimated using a RATS instruction, that can usually be most easily done using the instruction SUMMARIZE.