Usual notation including BDA3 is what you write, but here more specifically

N_eff = N / (1 + 2\sum_[t=1}^T \hat{\rho}_t),

where T is the last lag to include and \hat{\rho}_t is the empirical autocorrelation for lag t.

By definition \rho_0=1.

In the above example we have

\rho_0=1

\hat{\rho}_1 \approx -0.055

\hat{\rho}_2 \approx 0.038

\hat{\rho}_3 \approx -0.007

\hat{\rho}_4 \approx -0.011

\hat{\rho}_5 \approx 0.005

Geyer’s paired sums are

\rho_0+\hat{\rho}_1 \approx 0.945 (positive, include in the sum)

\hat{\rho}_2+\hat{\rho}_3 \approx 0.031 (positive, include in the sum)

\hat{\rho}_4+\hat{\rho}_5 \approx -0.005 (negative, stop, don’t include in the sum)

Thus we include lags 1,2,3

\hat{\rho}_1+\hat{\rho}_2+\hat{\rho}_3 \ approx -0.024

Then

1 + 2\sum_[t=1}^T \hat{\rho}_t \approx 0.953

and

N_eff = N / (1 + 2\sum_[t=1}^T \hat{\rho}_t) \ approx 4198 > 4000

Geyer uses autocovariances instead of autocorrelations, but that’s same up to scaling. Geyer also writes alternative form

-1 + 2\sum_[t=0}^T \hat{\rho}_t

where summing starts from t=0, but then 1 is changed to -1

We have now two different sums

- \sum_[t=1}^T \hat{\rho}_t can be negative
- \sum_[t=0}^T \hat{\rho}_t is always positive, but can be smaller than 1

Starting the summing from lag 0 has the benefit that it’s enough that we store the paired sum values during the computation and then sum them together. Starting the summing from lag 1 has the benefit that it’s the more commonly used form in articles and textbooks.