What common forecasting models can be seen as special cases of ARIMA models?

Question

This morning I woke up wondering (this could be due to the fact that last night I didn't get much sleep): Since cross-validation seems to be the cornerstone of proper time-series forecasting, what are the models I should "normally" cross-validate against?

I came up with a few (easy) ones, but I soon realized they were all but special cases of ARIMA models. So I'm now wondering, and this is the actual question: What forecasting models does the Box-Jenkins approach already incorporate?

Let me put it this way:

1. Mean = ARIMA(0,0,0) with constant
2. Naive = ARIMA(0,1,0)
3. Drift = ARIMA(0,1,0) with constant
4. Simple Exponential Smoothing = ARIMA(0,1,1)
5. Holt's Exponential Smoothing = ARIMA(0,2,2)
6. Damped Holt's = ARIMA(0,1,2)
7. Additive Holt-Winters: SARIMA(0,1,m+1)(0,1,0)m

What else can be added to the previous list? Is there a way to do moving average or least squares regression "the ARIMA way"? Also how do other simple models (say ARIMA(0,0,1), ARIMA(1,0,0), ARIMA(1,1,1), ARIMA(1,0,1), etc.) translate?

Please note that, at least for starters, I'm not interested in what ARIMA models cannot do. Right now I only want to focus on what they can do.

I know that understanding what each "building block" in an ARIMA model does should answer all of the above questions, but for some reason I have difficulties figuring that out. So I decided to try a "reverse engineering" kind of approach.

Answer 1

The Box-Jenkins approach incorporates all well-known forecasting models except multiplicative models like the Holt-Winston Multiplicative Seasonal Model where the expected value is based upon a multiplicand.

The multiplicative seasonal model can be used to model time series where one has the following (in my opinion a very unusual) case: If the amplitude of the seasonal component/pattern is proportional to the average level of the series, the series can be referred to as having multiplicative seasonality. Even in the case of multiplicative models, one can often represent these as ARIMA models thus completing the "umbrella."

Furthermore since a Transfer Function is a Generalized Least Squares Model it can reduce to a standard regression model by omitting the ARIMA component and assuming a set of weights needed to homogenize the error structure.

Answer 2

You can add

Drift: ARIMA(0,1,0) with constant.

Damped Holt's: ARIMA(0,1,2)

Additive Holt-Winters: SARIMA(0,1,$m+1$)(0,1,0)$_m$.

However, HW uses only three parameters and that (rather strange) ARIMA model has $m+1$ parameters. So there are a lot of parameter constraints.

The ETS (exponential smoothing) and ARIMA classes of models overlap, but neither is contained within the other. There are a lot of non-linear ETS models that have no ARIMA equivalent, and a lot of ARIMA models that have no ETS equivalent. For example, all ETS models are non-stationary.

Answer 3

• The exponentially weighted moving average (EWMA) is algebraically equivalent to an ARIMA(0,1,1) model.

To put it another way, the EWMA is a particular model within the class of ARIMA models. In fact, there are various types of EWMA models and these happen to be included in the class of ARIMA(0,d,q) models - see Cogger (1974):

The Optimality of General-Order Exponential Smoothing by K. O. Cogger. Operations Research. Vol. 22, No. 4 (Jul. - Aug., 1974), pp. 858-867.

The abstract for the paper is as follows:

This paper derives the class of nonstationary time-series representations for which exponential smoothing of arbitrary order minimizes mean-square forecast error. It points out that these representations are included in the class of integrated moving averages developed by Box and Jenkins, permitting various procedures to be applied to estimating the smoothing constant and determining the appropriate order of smoothing. These results further permit the principle of parsimony in parameterization to be applied to any choice between exponential smoothing and alternative forecasting procedures.

Answer 4

"The Gauss-Markov plus white noise model of the first difference is a special case of an ARIMA (1,1,1)

and the damped cosine plus white noise model is a special case of an ARIMA (2,1,2)."

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=9290&context=rtd