CLAIM C1 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C1

DEFINITIONS
D1 [method] ETS(M,*,*): Pure multiplicative exponential smoothing state space model family written in logarithmic state space form as $$\begin{aligned} \log y_t &= w' \log(v_{t-l}) + \log(1 + \epsilon_t) \\ \log v_t &= F \log v_{t-l} + \log(1_k + g \epsilon_t) \end{aligned}$$ with conventional multi-step point forecast $\hat{y}_{t+h} = \exp\left((w_1' F_1^{h-1} + w_m' F_m^{\lceil h/m \rceil - 1}) \log v_t\right)$. [source B7]

CLAIMS
C1 [formal; principal; extension; support: formal_proof E1, cited B3, at pp. 21-23; assumes C12, C13; limitations L1; verification unchecked; verifiability open; planning exploratory]
  In general, the conventional multi-step point forecast $\hat{y}_{t+h}$ of a pure multiplicative [[D1:ETS(M,*,*)]] model does not coincide with the conditional arithmetic expectation $\mu_{y,t+h}$ or the conditional geometric expectation $\tilde{y}_{t+h}$, satisfying $\hat{y}_{t+h} \le \tilde{y}_{t+h} \le \mu_{y,t+h}$ when smoothing parameters are non-zero.
  scope: conditions: pure multiplicative ETS models with non-zero smoothing parameters and forecast horizon $h > 1$
  extends isbn 978-3-540-71918-2, at Chapter 2, pp. 21-23: Derives the explicit ordering between point forecasts, geometric conditional expectations, and arithmetic conditional expectations using logarithmic state space recursion.
C12 [formal; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$.
C13 [formal; supporting; assumption (methodological, tested); verifiability not_applicable]
  Smoothing parameters $\alpha, \beta, \gamma$ in pure multiplicative ETS models lie within $[0, 1]$ so that state update terms $(1 + g\epsilon_t)$ remain strictly positive whenever $1 + \epsilon_t > 0$.

EVIDENCE
E1 [formal; foundation classical_probability] Analytical derivation of recursive conditional expectations and point forecasts for pure multiplicative ETS models in logarithmic state space form (Section 2, Eqs. 2.1-2.11).
  assumes C12, C13
  E1.1 Reformulate pure multiplicative ETS in logarithmic state space form: $\log y_t = w' \log(v_{t-l}) + \log(1 + \epsilon_t)$ and $\log v_t = F \log v_{t-l} + \log(1_k + g \epsilon_t)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, .,  , (, 2, ., 1, ); logarithmic_transformation)
  E1.2 Expand $h$-steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation $E(\log y_{t+h}|t)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, s, .,  , (, 2, ., 5, ), -, (, 2, ., 6, ); recursive_substitution)
  E1.3 Apply Jensen's inequality and non-negativity of $g$ to show point forecast $\hat{y}_{t+h} \le \tilde{y}_{t+h} \le \mu_{y,t+h}$, with equality only at $h=1$, $h \le m$ without trend, or for $ETS(M,N,N)$. (by S, e, c, t, i, o, n,  , 2, ,,  , E, q, s, .,  , (, 2, ., 8, ), -, (, 2, ., 1, 1, ); jensens_inequality)

LIMITATIONS
L1 [model_specification; affects C1; reduces_transferability; mitigation none] Conditional expectations for pure multiplicative ETS models other than $ETS(M,N,N)$ lack closed-form analytical expressions and require Monte Carlo simulation.

REFERENCES
B3 978-3-540-71918-2 (isbn)
B7 10.1201/9781003452652 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C1, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1#C1. https://entrylib.org/p/2026.000006/C1
The claimset it is from: Svetunkov, I. (guarantor) (2026). Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1. https://entrylib.org/p/2026.000006@v1