CLAIM C3 of entrylib:2026.000006 Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models
licence CC-BY-4.0; contributions C3

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]
D6 [concept] Inverse Gaussian error distribution: Probability distribution on $(0, \infty)$ applied to $1+\epsilon_t$ with location $\mu = 1$ and scale $\sigma^2$ so that $E(1+\epsilon_t) = 1$ and $V(1+\epsilon_t) = \sigma^2$.
D8 [variable] percentage difference (PD): Relative deviation of the conventional point forecast $\hat{y}_{t+h}$ from the conditional expectation $\mu_{y,t+h}$, defined as $$PD = \frac{\mu_{y,t+h} - \hat{y}_{t+h}}{\mu_{y,t+h}}$$ lying in $[0, 1)$. [unit 1]
D13 [method] ETS(M,Md,N): Pure multiplicative exponential smoothing model with damped multiplicative trend and no seasonality. [a kind of D1] [source B3]
D14 [data] Simulation experiment 1 synthetic series: 10,000 iterations of 100-observation time series simulated from $ETS(M,Md,N)$ with Inverse Gaussian errors across parameter grids $\sigma \in \{0.01, 0.05, 0.1, 0.5, 1\}$, $\alpha, \beta \in \{0, 0.1, 0.2, 0.5\}$, $\phi \in \{0.5, 0.9, 0.95, 1\}$, and horizons $h \in \{10, 50, 100\}$.
D19 [design] Box-Cox odds regression procedure: Transformation of percentage difference $PD$ into odds $PD/(1-PD)$ followed by Box-Cox transformation ($\lambda = 0.146$) and linear regression on scaled parameters $\sigma$, horizon, $\alpha$, $\beta$, and $\phi$ using `greybox::alm` (`distribution='dbcnorm'`).

CLAIMS
C3 [descriptive; principal; support: simulation E2 (E2.T1) artefacts A3; assumes C12, C13; limitations L2; strength high (author); verifiability unverifiable; planning exploratory]
  For [[D13:ETS(M,Md,N)]] simulated on [[D14:Simulation experiment 1 synthetic series]] with an [[D6:Inverse Gaussian error distribution]], the [[D8:percentage difference (PD)]] between the conditional expectation and the point forecast increases with the damping parameter $\phi$, the trend smoothing parameter $\beta$, the forecast horizon $h$, the scale parameter $\sigma$, and the level smoothing parameter $\alpha$, with $\phi$ having the largest effect and $\alpha$ having the smallest effect among the scaled predictors.
  scope: conditions: $ETS(M,Md,N)$ simulated over $10,000$ iterations with Inverse Gaussian errors across parameter grids
  contradicts isbn 978-3-540-71918-2, at Chapter 2, p. 23: Shows that the difference between point forecasts and conditional expectations is non-negligible and grows substantially with horizon and parameter values.
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)
E2 [simulation; runs 10000] Monte Carlo simulation of $10,000$ iterations of $100$ observations from $ETS(M,Md,N)$ with Inverse Gaussian errors using `smooth::sim.es()`, followed by Box-Cox odds regression (`greybox::alm` with `distribution='dbcnorm'`) of percentage difference $PD$ on scaled parameters $\sigma$, horizon, $\alpha$, $\beta$, and $\phi$ (Section 3, Table 1).
  setup: studied D13, D6; cases D14; procedure D19; lens D8
  E2.T1 Box-Cox odds regression output for percentage difference $PD$ in Simulation Experiment 1 ($\lambda = 0.146$, $R^2 = 0.986$, $\text{df} = 534,884$).
  | row | variable | estimate (1) | std_error (1) | lower_0.05pct (1) | upper_99.95pct (1) |
  | r1 | (Intercept) | -9.222 | 0.006 | -9.242 | -9.202 |
  | r2 | sigma | 2.755 | 0.003 | 2.745 | 2.765 |
  | r3 | horizon | 3.656 | 0.003 | 3.646 | 3.667 |
  | r4 | alpha | 1.578 | 0.007 | 1.555 | 1.6 |
  | r5 | beta | 3.739 | 0.007 | 3.716 | 3.762 |
  | r6 | phi | 6.167 | 0.006 | 6.148 | 6.187 |

LIMITATIONS
L2 [model_specification; affects C3; reduces_transferability; mitigation none] Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters.

ARTEFACTS
A3 [table] Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV.

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

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C3, in: Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models. EntryLib. entrylib:2026.000006@v1#C3. https://entrylib.org/p/2026.000006/C3
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