Forecasting methods

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Comparison

Name What it assumes the future looks like	What it needs	Seasonality	Other variables	Prediction interval
Naive / seasonal naive Tomorrow equals today, or equals the same weekday last week. No parameters, no fitting, and the benchmark every other method on this page has to beat.	one observation	seasonal variant only	none	from residuals
Moving average Tomorrow equals the mean of the last k days. Kills noise, lags the trend by about half the window, and cannot see a cycle shorter than it. Full page →	k observations	no	none	from residuals
Exponential smoothing (ETS) A weighted average of the whole past with weights decaying exponentially. The Holt-Winters form carries level, trend and season as separate states.	a few cycles	yes, additive or multiplicative	none	yes, via state space
ARIMA / SARIMA Regress the differenced series on its own lags and on past shocks. The d is the number of differences taken, which is the stationarity decision made explicit. Full page →	~50+ observations	yes, via seasonal terms	yes, as ARIMAX	yes, closed form
Regression on lags and calendar An ordinary linear model whose features are lagged values, day of week, holidays and anything else known in advance. Interpretable, and it takes exogenous inputs for free. Full page →	a few cycles	yes, as dummies	yes	yes, closed form
Gradient boosting on features The same feature table, fitted by trees instead. Handles non-linearity and interactions, cannot extrapolate a trend beyond the range it was trained on. Full page →	thousands of rows	yes, as features	yes	quantile loss, or bootstrap
Neural networks Learn the mapping from a window of history to a horizon. Strongest with many related series to share structure across, and hungry for both data and tuning. Full page →	many series, long history	learned, if shown enough	yes	quantile or probabilistic heads
Forecast combination Average several forecasts instead of choosing one. Generally more accurate than the forecasts it is made of, which makes it the cheapest improvement available.	two or more forecasts	inherited	inherited	inherited or pooled

Forecasting has an unusual property for a modelling problem: the answer arrives on its own. Wait, and you find out. That makes it one of the few areas where methods can be ranked honestly — and the ranking is less flattering to complexity than most people expect.

Before any of this, note what a forecast is. "Among the many options for a forecast of a single time series at a future time period are an expected value (known as a point forecast), a prediction interval, a percentile and an entire prediction distribution" [1, §2.1]. A single number is the least of these, and the one most often delivered.

Start at the bottom of the table, not the top

The naive forecast is a real competitor. It is not a straw man included for completeness. Any method that cannot beat "the same value as last week" on your data has not earned its maintenance cost, and on noisy, short or intermittent series many do not.

Exponential smoothing is the next hardest thing to beat. The review is blunt about a method from 1944: "exponential smoothing is one of the workhorses of business forecasting. Despite the many advances in the field, it is always a tough benchmark to bear in mind" [1, §2.3.1]. Its simplest form is one line — "f_t+1 = α y_t + (1 − α) f_t" [1, §2.3.1] — and it is also known as the exponentially weighted moving average [1, §2.3.1].

Modern practice does not stop at the point forecast: state-space formulations "equip them with a statistical framework capable of providing future probability distributions" [1, §2.3.1], which is what turns a smoother into a model.

The statistical middle

AR and MA are two different ideas about where the pattern lives. An autoregressive model "is simply a linear regression of the current value of the series against one or more prior values of the series" [2, §6.4.4.4]. A moving-average model is "conceptually a linear regression of the current value of the series against the white noise or random shocks of one or more prior values" [2, §6.4.4.4] — and because those shocks are not observed, "fitting the MA estimates is more complicated than with AR models ... iterative non-linear fitting procedures need to be used in place of linear least squares" [2, §6.4.4.4].

Box and Jenkins' contribution was not either model but "developing a systematic methodology for identifying and estimating models that could incorporate both approaches" [2, §6.4.4.4], which is what ARIMA is.

The d in ARIMA is a stationarity decision. "An important decision when using an ARIMA model is the choice for the value of d" [1, §2.3.4] — see stationarity for what that choice is about.

The two families overlap more than their names suggest. "Some exponential smoothing models can also be written in ARIMA format ... For example, SES is equivalent to ARIMA(0,1,1) when θ₁ = α − 1" [1, §2.3.4].

Order selection is now usually automatic. The ACF and partial ACF identify small p and q; beyond that "in practice one relies on the well-known information criteria like AIC and BIC", and auto.arima "has been found to be very effective and increasingly being used in ARIMA modelling" [1, §2.3.4].

Machine learning, and what it is actually for

Neural networks are characterised by their flexibility, not their accuracy on any one series. They are distinguished by "their ability to model non-stationary, nonlinear and high complex datasets" [1, §2.7.8], and they are trained the same way every other network is — "the most common NN procedure, is the back-propagation" [1, §2.7.8]. See backpropagation.

The practical rule of thumb the table encodes: gradient boosting and neural methods earn their place when you have many series rather than one long one, or when exogenous variables carry most of the signal. On a single series of a few hundred points, the statistical rows are usually better and always cheaper.

One structural limitation is worth stating plainly, since it is arithmetic rather than a tuning problem: a tree's prediction is an average of training targets in a leaf, so it can never fall outside the range of targets it was trained on. A series that keeps rising is therefore forecast flat as soon as it leaves that range. The usual fix is to de-trend first, which is decomposition again.

The row most people skip

Combination. "Given N forecasts of the same event, forecast combination involves estimation of so called combination weights assigned to each forecast, such that the accuracy of the combined forecast generally outperforms the accuracy of the forecasts included" [1, §2.6.1].

That is the whole argument, and it has held since the late 1960s [1, §2.6.1]. It costs nothing but the average, needs no new theory, and is the first thing to try after a second method exists.

Choosing

Start with seasonal naive and exponential smoothing. Establish what "good" means on your data before choosing anything you will have to maintain.

Move to ARIMA or a lag regression when you need intervals and interpretation. Both give closed-form prediction intervals, and the lag regression takes exogenous variables without ceremony.

Reach for boosting or a neural model when the problem is wide, not long — many series, many features, or a relationship that no linear lag structure captures.

Then combine. Two mediocre forecasts averaged usually beat either [1, §2.6.1].

And check the series first. Every row here assumes something about trend, seasonality or stationarity, and those are questions for decomposition and the ACF rather than for the model.