Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models: claim C3

The claim

Support 36 C3 · descriptive · novelty: new, by the authors

For ETS(M,Md,N) simulated on Simulation experiment 1 synthetic series with an Inverse Gaussian error distribution, the 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 · Simulation: E2 (T1), artefacts A3 · Assumes: C12, C13 · Limitations: L2 · Contradicts: 978-3-540-71918-2

Evidence

E1 · formal

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).

Foundation: classical_probability · Assumes: C12, C13

Reformulate pure multiplicative ETS in logarithmic state space form: $\log y_t = w' \log(v_{t-l}) + \log(1 + \epsilon_t)$ and $\log v_t = F \log v_{t-l} + \log(1_k + g \epsilon_t)$ . (logarithmic_transformation; from S, e, c, t, i, o, n, , 2, ,, , E, q, ., , (, 2, ., 1, ))
Expand $h$ -steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation $E(\log y_{t+h}|t)$ . (recursive_substitution; from S, e, c, t, i, o, n, , 2, ,, , E, q, s, ., , (, 2, ., 5, ), -, (, 2, ., 6, ))
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 $ETS(M,N,N)$ . (jensens_inequality; from S, e, c, t, i, o, n, , 2, ,, , E, q, s, ., , (, 2, ., 8, ), -, (, 2, ., 1, 1, ))

E2 · simulation · cited by C3

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 ETS(M,Md,N), Inverse Gaussian error distribution · cases Simulation experiment 1 synthetic series · procedure Box-Cox odds regression procedure · lens percentage difference (PD)

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 · reduces transferability · affects C3 · mitigation: none

Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters.

Assumptions

C12 · statistical · unchecked

The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$ . Rested on by C3, E1.

C13 · methodological · tested

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 $1 + \epsilon_t > 0$ . Rested on by C3, E1.

Definitions

Methods 2

D1 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: Svetunkov (2023).

D13 · a kind of D1 ETS(M,Md,N): Pure multiplicative exponential smoothing model with damped multiplicative trend and no seasonality. Source: 978-3-540-71918-2.

Data 1

D14 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 $h \in \{10, 50, 100\}$ .

Variables and measures 1

D8 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 $[0, 1)$ .

Study design 1

D19 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').

Concepts 1

D6 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 $V(1+\epsilon_t) = \sigma^2$ .

Context

Contradicts 978-3-540-71918-2 · from C3: Shows that the difference between point forecasts and conditional expectations is non-negligible and grows substantially with horizon and parameter values.

References

978-3-540-71918-2 978-3-540-71918-2 Used by: D13
Svetunkov (2023) Svetunkov, I. (2023). Forecasting and Analytics with the Augmented Dynamic Adaptive Model (ADAM). Chapman and Hall/CRC. doi:10.1201/9781003452652 Used by: D1

Artefacts

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

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.4): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C3: 36, Weakly supported

Novelty: new, by the authors

−45 Checkable Verifiability: unverifiable. To remove it: Make the data, code and models the claim rests on available, or state why they cannot be.
−10 Evidence Rests on an unchecked proof. To remove it: Support the claim with evidence the claimset records, citing the result entries or table rows it reports, or have its proof checked.
−5 Plan A principal claim reported without a plan recorded beforehand. To remove it: Plan the measurement in the lab record or a registration before running it, or publish the lab record that shows every run.
−3 Open problems Assumption C12 is not checked: The multiplicative error term $(1+\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\epsilon_t) = 1$ . To remove it: Test the assumption, or analyse how sensitive the claim is to it.
−1 Open problems Limitation L2 (reduces transferability, not mitigated): Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models · C3