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OEAR Diffusion-Scale Estimator

oear_sigma2_hac.Rd

Compute the OEAR (observation-error-and-autocovariance-robust) diffusion-scale estimate \(\tilde{\sigma}^2\) by estimating the long-run variance (LRV) of standardized growth increments using AR(1) pre-whitening and a Bartlett (Newey-West) HAC estimator, followed by recoloring.

Arguments

mu

Numeric scalar. Estimated growth rate \(\hat{\mu}\) used to center increments.

delta_log_n

Numeric vector of length q. Log-scale increments \(\Delta Y_i\).

tau

Numeric vector of length q. Sampling intervals \(\tau_i > 0\).

Value

A list with components:

sigma2_tilde

Numeric scalar. OEAR diffusion-scale estimate \(\tilde{\sigma}^2\).

rho_tilde_pw

Numeric scalar. Estimated AR(1) pre-whitening coefficient \(\tilde\rho_{\mathrm{pw}}\).

j

Integer. Selected Bartlett truncation lag $J$.

Details

The estimator targets the LRV of the standardized centered increment series $$U_i = (\Delta Y_i - \hat{\mu}\,\tau_i)/\sqrt{\tau_i}, \qquad i=1,\ldots,q,$$ which is interpreted as an effective environmental variance per unit time.

Implementation steps:

  1. Center \(u_i\) by subtracting \(\bar u\) to obtain \(\tilde u_i\).

  2. Estimate the AR(1) pre-whitening coefficient \(\tilde\rho_{\mathrm{pw}}\) by OLS in \(\tilde u_i = \rho_{\mathrm{pw}}\,\tilde u_{i-1} + \varepsilon_i\) and form pre-whitened residuals \(\tilde\varepsilon_i\).

  3. Compute residual autocovariances and the Bartlett HAC LRV \(\widetilde{\mathcal C}^{(\varepsilon)}_{\mathrm{NW}}(J)\).

  4. Choose the truncation lag \(J\) via the Andrews (1991) AR(1) plug-in rule specialized to the Bartlett window.

  5. Recolor to the original scale to obtain $$\tilde{\sigma}^2 = \widetilde{\mathcal C}_{\mathrm{NW}}(J) = \widetilde{\mathcal C}^{(\varepsilon)}_{\mathrm{NW}}(J)/ (1-\tilde\rho_{\mathrm{pw}})^2.$$

This construction is designed to be robust to short-run autocovariance and, under additive observation error on the log scale, to the cancellation property of the LRV when increments are appropriately centered.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858.

Newey, W. K. and West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708.