Growth rate formulae

Halcyon: an enhancement of X-13ARIMA-SEATS 2023年1月9日

The original X-13ARIMA-SEATS has subroutine “change” to calculate month-to-month changes of original and adjusted series. The formula is ordinary percent change calculation for multiplicative model and log-additive model. If \(X_t\) is the value for period \(t\) and \(Y_t\) is the result growth rate,

\(\quad Y_t = X_t / X_{t-1} – 1 = \frac{X_t – X_{t-1}}{X_{t-1}}, \quad when\ X_{t-1} > 0,\)

\(\quad Y_t = -999, \quad when\ X_{t-1}\leq 0.\)

If the mode is additive, subroutine “change” automatically selects difference of the series.

\(\quad Y_t = X_t – X_{t-1}.\)

Our enhanced version has several other formulae.

1. Annualised change

\(\quad Y_t = \left( \frac{X_t}{X_{t-1}} \right) ^{Sp}-1, \quad Sp = number\ of\ period\ within\ a\ year.\)

2. Logarithmic change

\(\quad Y_t = \log{X_t} – \log{X_{t-1}}, \quad where \log is\ natural\ logarithm.\)

3. Annualised logarithmic change

\(\quad Y_t = \left(\log{X_t} – \log{X_{t-1}}\right) \times Sp.\)

4. Midpoint change

\(\quad Y_t = \frac{2 \left( X_t – X_{t-1}\right) }{\left( X_t + X_{t-1}\right)}.\)

5. Annualised midpoint change

\(\quad Y_t = \frac{2 \left( X_t \left(\frac{X_t}{X_{t-1}}\right)^{Sp-1} – X_{t-1}\right) }{\left( X_t + X_{t-1}\right)}.\)