Hi,

I have general questions regarding fpp3/tidyverts:

[1] For the 4 simple methods: mean, naive, snaive and rw_drift in 5.2 Some simple forecasting methods | Forecasting: Principles and Practice (3rd ed)

(a) Can all 4 simple methods be estimated/used with ANY series, e.g. rawdata as is: no transformations, no differencing?

(b) For all 4 simple methods are bootstrap PI's applicable to ANY series? 5.5 Distributional forecasts and prediction intervals | Forecasting: Principles and Practice (3rd ed)

[2] Are bootstrap forecasts valid for ARIMA models including MA terms, as in 9.8 Forecasting | Forecasting: Principles and Practice (3rd ed) ? I have read elsewhere, that simple shuffling of residuals is NOT completely appropriate for a model with MA terms for an entire sample.

[3] Non-Gaussian forecasting using fable is described in: Rob J Hyndman - Non-Gaussian forecasting using fable.

Non-normality within the modelling framework arises due to,

(a) transformations: normal if no transformation used, non-normal if transformation applied i.e. box-cox.

(b) non-normality of residuals.

Are there any other sources of non-normality?

For forecasting in COMBINATION and ENSEMBLE models:

(i) COMBINATION: if both (a) and (b) are normal, use either simulate or bootstrap, or none i.e. simulate = FALSE, bootstrap = FALSE.

(ii) COMBINATION : if either (a) or (b) are non-normal, NOT use simulate as that draws from independent N(0,sigmasq) shocks over the forecast period i.e. NOT use fabletools:::forecast.model_combination() as this applies the formula Var(aX + bY), based on the normality assumption. Likewise, NOT use bootstrap.

(iii) ENSEMBLE: if both (a) and (b) are normal, use either simulate or bootstrap, or none i.e. simulate = FALSE, bootstrap = FALSE.

(iv) ENSEMBLE: if either (a) or (b) are non-normal, use bootstrap as these are more representative of the series, NOT use simulate as that draws from independent N(0,sigmasq).

Correct?

Further, if [2] is true i.e. I cannot bootstrap forecasts from an ARIMA model, can I mix an ETS with bootstrap forecasts and an ARIMA with no shocks, in an ENSEMBLE? If so how?

[4] Formal normality tests, used for residuals, are NOT included in tidyverts. Why? However, in R there are:

stats::shapiro.test

moments::jarque.test

DescTools::AndersonDarlingTest

nortest::ad.test

etc, etc

Importantly, which test is the most appropriate/recommended for use with fable (e.g. Jarque-Bera tends to be sensitive) as this will determine whether to bootstrap PI's or not?

[5] Seasonal Characteristic Roots:

I understand non-seasonal characteristic roots, but am confused with seasonal characteristic roots.

In Forecasting Methods,3e, by Makridakis, Wheelwright, and Hyndman p.346, specification ARIMA(1,1,1)(1,1,1)[4], quarterly data, it's simple to multiply out the factors, but the general model has 8 AR terms at lags 1,2,4,5,6,8,9 and 10 (implying 8 roots) and 3 MA lagged terms at 1,4,5 (implying 3 roots).

Based on, Rob J Hyndman - Plotting the characteristic roots for ARIMA models plotting an ARIMA(1,1,1)(1,1,1)[4] should have 5 AR roots and 5 MA roots. Yet, for an ARIMA(1,1,1)(1,1,1)[4] there are only 2 AR coeffs (1 non-seasonal, 1 seasonal) and 2 MA coeffs (1 non-seasonal, 1 seasonal), implying 2 AR roots and 2 MA roots. Please can a BY-HAND example be provided, algebraically and in fable, as to why?

many thanks,

Amarjit