Time Series Forecasting: ARIMA, Exponential Smoothing, ML and What a Forecast Can't Tell You

Time series forecasting is the practice of predicting future values of a variable from its own history, recorded at regular intervals such as days, weeks or months. A forecasting model learns the trend, seasonality and short-run dependence in past observations, then projects them forward, usually as a point forecast with a prediction interval around it.

The methods for time series forecasting run from seasonal naive to ETS, ARIMA, Prophet, gradient boosting and Transformers, and the M4 and M5 forecasting competitions show which kinds of method win on real series and why the answer changes with the data. Even the most accurate model can forecast sales closely and still say nothing about which marketing caused them, which is why marketing mix modeling answers a different question.

What is time series forecasting?

A time series is a sequence of observations of one quantity at equal intervals: weekly sales, daily site sessions, monthly sign-ups. Time series forecasting treats the order of those observations as information. Last December tells you something about this December, which is why rows cannot be shuffled the way they can in cross-sectional prediction.

Each forecast has a frequency, which fixes the seasonal period (about 52.18 for weekly data with an annual pattern, often rounded to 52). It also has a horizon, and accuracy almost always falls as the horizon lengthens. Planning needs the prediction interval, not just the point forecast, because stock, staffing and budgets are sized for the bad weeks.

Univariate, multivariate and dynamic regression models

Forecasting: Principles and Practice by Rob Hyndman and George Athanasopoulos, the free online textbook most of this guide draws on, sorts models into three types. Time series models, such as exponential smoothing and ARIMA, use only the series' own past. Explanatory models use other variables, such as price or ad spend. Mixed models, often called dynamic regression models, combine the two. Strictly, multivariate time series forecasting means modelling several series jointly, as a vector autoregression does. Adding external drivers to one series is dynamic regression, though many tools call that multivariate too.

Time series forecasting is not the same as explaining

Time series forecasting needs patterns that repeat. Explanation needs to know which inputs would change the outcome if you changed them. A model can predict next quarter's sales closely and still credit them to the wrong causes.

The four components of a time series: trend, seasonality, cycles and noise

Every time series forecasting method models some mix of four patterns.

• Trend: the long-run direction, which can change slope.
• Seasonality: a pattern that repeats over a fixed, known period, such as the Q4 retail peak.
• Cycles: rises and falls with no fixed period, often tied to the economy.
• Noise, or remainder: whatever is left once the systematic patterns are removed.

On a few years of data, trend and cycle are hard to tell apart, so most decomposition methods estimate them together as one trend-cycle component.

Additive vs multiplicative decomposition

An additive decomposition treats the series as trend plus seasonal plus remainder, which fits when the seasonal swing stays a similar size as the level changes. A multiplicative one fits when the swing grows with the level, as when a December lift stays at a similar percentage while revenue doubles. Logs turn a multiplicative series into an additive one.

Classical and STL decomposition

Classical decomposition assumes an identical seasonal pattern every year. STL (Seasonal and Trend decomposition using Loess) lets the seasonal pattern change over time and, with outlier downweighting switched on, keeps outliers out of the trend and seasonal estimates, so it is the better default for weekly marketing data.

Promotions and stock-outs land in the remainder, like the spike in Figure 1. A known, recurring event should become a regressor, so the model can forecast the next one. This decomposition by pattern differs from an MMM's base and incremental decomposition, which splits sales by cause.

Stationarity and differencing

A stationary series has statistical properties that do not depend on when you observe it, so trended and seasonal series are not stationary. This matters for some time series forecasting methods and not others. ARIMA needs trend and seasonality removed first, while exponential smoothing models them directly.

How to check: plots, ACF, ADF and KPSS tests

Start with a plot. A visible trend, level shift or growing variance settles it, and the autocorrelation function (ACF) of a non-stationary series decays slowly. The two formal tests have opposite null hypotheses. The augmented Dickey-Fuller (ADF) test assumes a unit root, so a small p-value is evidence against one (stationarity around a constant, or around a trend if the test includes one). The KPSS test assumes stationarity, so a small p-value is evidence against it.

First differences, seasonal differences and log transforms

A first difference, the change from the previous period, removes a trend. A seasonal difference, the change from the same period a year earlier, removes stable seasonality. Logs before differencing stabilise a variance that grows with the level. Stop once the series looks stationary. Over-differencing adds spurious autocorrelation and widens intervals.

Baseline time series forecasting methods: naive, seasonal naive, mean and drift

Every forecasting project should start with the four simplest methods. They take minutes and set the bar every other model has to clear.

Method	Forecast for every future period	Suits
Mean	Average of all past observations	No trend or seasonality
Naive	Last observed value	Random walks, such as many prices
Seasonal naive	Value from the same season last year	Strongly seasonal series
Drift	Last value plus the average historical change	A steady trend

Why every model must beat them

Hyndman and Athanasopoulos use these four as benchmarks: a new method that cannot beat them is not worth considering. The M4 competition applied the same discipline, scoring entries against Naive2, a seasonally adjusted naive forecast, and a combination benchmark. Only one of its six pure machine learning methods beat Naive2.

Exponential smoothing, Holt-Winters and ETS

Exponential smoothing produces forecasts that are weighted averages of past observations, with weights that decay exponentially as observations age. The method dates from the late 1950s and the work of Brown, Holt and Winters, as chapter 8 of Forecasting: Principles and Practice records.

Simple, Holt and damped trend

Simple exponential smoothing has one parameter, alpha, between 0 and 1. Near 1, the forecast follows the latest value, like naive. Near 0, it averages a long history, like the mean. Holt added a trend equation with parameter beta, and the damped variant adds phi to flatten the slope over the horizon, a sensible default for growth that will not last indefinitely.

The Holt-Winters seasonal method

Winters added a seasonal equation with parameter gamma. The additive version suits seasonal swings of constant size, and the multiplicative version suits swings that scale with the level, such as a Q4 lift that grows with the business.

The ETS framework: error, trend, seasonal

ETS recasts these methods as statistical models named by three letters. Error is A (additive) or M (multiplicative), trend is N (none), A or Ad (additive damped), and seasonality is N, A or M. ETS(A,A,A) is additive Holt-Winters, and ETS(M,Ad,M) has multiplicative errors, a damped trend and multiplicative seasonality. Because each model has a likelihood, software can fit them all, choose by AICc and produce prediction intervals.

ARIMA models explained: AR, I and MA

An ARIMA model forecasts a series from its own past values and past forecast errors, after differencing it until it is stationary. ARIMA stands for AutoRegressive Integrated Moving Average, and the three parts map to the three numbers in ARIMA(p,d,q). Where exponential smoothing describes trend and seasonality, ARIMA describes autocorrelation, which makes it the other classical family.

AR(p): regression on the series' own past

The autoregressive part regresses the current value on its previous p values. A positive AR(1) coefficient means a high week tends to follow a high week, which is how momentum in demand enters the model.

I(d): differencing to remove trend

The d is the number of differences taken before fitting. "Integrated" names the reverse step: the model forecasts the differenced series, then adds the changes back up to the original level. In practice, d is almost never more than 2, and the advice is to take as few differences as the series needs.

MA(q): regression on past forecast errors

The moving average part uses the previous q forecast errors as predictors. It is unrelated to the moving average that smooths a chart. If the model under-forecast last week, part of that miss carries into this week's forecast.

ARIMA(1,1,1) combines all three. Take an illustrative case: sales rose by 40 units last week, the AR coefficient is 0.5, last week's forecast error was 10 units and the MA coefficient is 0.3. The forecast change is 0.5 × 40 + 0.3 × 10 = 23 units, so from a level of 1,000 units, next week's forecast is 1,023.

Choosing p, d and q: ACF, PACF and automatic selection

The Box-Jenkins approach reads the orders from plots of the differenced series: a partial autocorrelation function (PACF) that cuts off after lag p points to AR(p), and an ACF that cuts off after lag q points to MA(q). Real data rarely looks that clean, so software such as R's auto.arima() picks d with unit-root tests and searches the other orders by AICc. Leftover autocorrelation in the residuals means structure the model missed.

Seasonal ARIMA (SARIMA)

A seasonal ARIMA model adds terms at the seasonal lag, written ARIMA(p,d,q)(P,D,Q)m, where m is the seasonal period. ARIMA(0,1,1)(0,1,1)12 takes a first and a seasonal difference of monthly data and adds one regular and one seasonal MA term. Weekly data is awkward because a year is not a whole number of weeks, so Fourier terms as regressors are the usual workaround.

ARIMAX and regression with ARIMA errors

Adding exogenous regressors, meaning external drivers such as price, holidays or ad spend, gives a model usually called ARIMAX, where the X stands for exogenous. When regressors are supplied, statsmodels' SARIMAX actually fits a related form, a regression with SARIMA errors, an ordinary regression on the drivers whose errors follow a seasonal ARIMA process. That form keeps the coefficients easier to read. Either way, a coefficient on TV spend is a forecasting weight, and it can sit far from what TV actually caused.

ARIMA vs exponential smoothing

Every linear exponential smoothing model has an ARIMA equivalent, but multiplicative ETS models have none, and many ARIMA models have no ETS counterpart. Neither family contains the other, and neither dominates on accuracy. ETS handles changing and multiplicative seasonality naturally. ARIMA handles short-run autocorrelation and regressors. Backtest both and keep the winner, or average them.

Prophet: fast, automatic and rarely the most accurate

Prophet is Meta's open-source tool for time series forecasting, released when the company was still called Facebook. It adds up a piecewise trend that bends at detected changepoints, Fourier-based seasonality and holiday effects, and it fits quickly with default settings.

When Prophet is worth using

Hyndman and Athanasopoulos, reviewing the model Taylor and Letham (2018) describe, find it fast and fully automated, and add that it rarely gives better forecast accuracy than the alternatives. Use it for daily data with several seasonal periods and many holidays, and backtest it against ETS, ARIMA and seasonal naive on anything else. Meta's open-source MMM, Robyn, uses its decomposition.

Machine learning and deep learning for time series forecasting

Machine learning reframes time series forecasting as supervised learning. Each row is a time point, and the features are lagged values (sales one and 52 weeks ago), rolling means, calendar fields and known drivers such as price and promotion.

Gradient boosting for time series forecasting: LightGBM and XGBoost

Gradient-boosted trees fed with lag features capture non-linear effects and interactions, such as a promotion that lifts sales more in December, without being told the functional form. A tree cannot predict outside the range it saw in training, so a trending series needs detrending or a target defined as change.

Global models trained across many series

A global model is one model trained on thousands of related series at once, such as every product in every store, so patterns learned on data-rich series carry over to sparse ones. That is where machine learning gains most of its edge. For multi-step forecasts, the recursive approach feeds each forecast back in as a lag, so errors compound, while the direct approach trains one model per horizon.

LSTMs and recurrent networks

LSTMs (long short-term memory networks) carry a memory state through the sequence, so they learn long dependencies without hand-built lags, but they need a lot of data and tuning and are hard to interpret. On a single business series of a few hundred points, LSTMs rarely repay the effort over ETS. In the M4, the pure machine learning methods did poorly, while the winner paired exponential smoothing with LSTM networks trained across all the series at once.

Transformers and the linear-model surprise

In "Are Transformers Effective for Time Series Forecasting?" (AAAI 2023), Zeng, Chen, Zhang and Xu tested Transformer models built for long-horizon forecasting against one-layer linear models, which they called LTSF-Linear. On all nine real-life datasets, the linear models beat the Transformer-based ones, often by a wide margin. A later design, PatchTST (ICLR 2023), reported beating DLinear, one of those linear models, in general and especially on the larger datasets.

Foundation models and zero-shot forecasting

Pretrained foundation models for time series forecasting, such as Amazon's Chronos and Google's TimesFM, forecast new series without training on them. Backtest them like any other candidate, especially when history is short.

Which forecasting methods actually win? Evidence from the M competitions

The Makridakis, or M, competitions are open tests of time series forecasting methods on large sets of real series, scored on data the entrants never saw.

M4: combinations and a hybrid beat pure machine learning

Makridakis, Spiliotis and Assimakopoulos (2018) report the M4 findings in the International Journal of Forecasting. Of the 17 most accurate methods, 12 were combinations of mostly statistical approaches. The winner was a hybrid of statistical and machine learning features, with an average sMAPE close to 10% better than the combination benchmark. The six pure machine learning methods performed poorly, and none beat the combination benchmark.

M5: gradient-boosted trees took the top ranks

The M5 used 42,840 hierarchical Walmart sales series, which Makridakis, Spiliotis and Assimakopoulos (2022) describe in the International Journal of Forecasting. Most of them are intermittent, and the competition supplied explanatory variables alongside them: price, promotions, day of the week and special events. Januschowski and colleagues report that tree-based methods, mainly gradient-boosted trees in LightGBM, outperformed other solutions, deep learning included, and that the winners of both M5 tracks relied heavily on them.

Why the two results differ

The M4 was mostly unrelated single series with no explanatory variables, which suits statistical methods. The M5 was thousands of related series with prices and promotions attached, so a global gradient-boosted model could learn how those drivers move sales across every item.

What this means for your data

Match the time series forecasting method to the data you actually have.

Your situation	Starting point
One or a few series, no reliable drivers	ETS and ARIMA, then a combination
Daily data, several seasonal cycles, holidays	ARIMA with Fourier and holiday terms, or Prophet
Many related series with price and promotion data	A global gradient-boosted model
Intermittent series with many zeros	Intermittent-demand methods, scored with MASE
You need to know what drove the outcome	A causal model, covered below

In every forecasting row, the first step is to beat seasonal naive.

Forecast accuracy metrics: MAE, RMSE, MAPE and MASE

Each of these metrics summarises the forecast errors, actual minus forecast, on data the model did not see.

Metric	What it measures	When it misleads
MAE	Average absolute error, in units	Across series of different scales
RMSE	Like MAE, but large misses count more	When a few big misses dominate
MAPE	Average error as a percentage of the actual	At or near zero actuals
MASE	MAE relative to an in-sample naive forecast	Harder to explain to non-specialists

Scale-dependent errors: MAE and RMSE

MAE and RMSE are in the data's units, so they compare models on one series but not across series of different sizes. Minimising MAE targets the median of the future distribution and minimising RMSE targets the mean, which differ for skewed sales.

Percentage errors: MAPE and its traps

Percentage errors are infinite or undefined when any actual value is zero, and extreme when actuals are near zero, which rules MAPE out for intermittent sales and makes it unstable for small channels. The symmetric sMAPE has its own instability near zero. A worked example with illustrative numbers:

Week	Actual	Forecast	Absolute error	Percentage error
1	120	110	10	8.3%
2	80	90	10	12.5%
3	100	95	5	5.0%
4	2	12	10	500.0%
5	90	85	5	5.6%
6	110	100	10	9.1%

MAE is 8.3 units and RMSE 8.7. MAPE is 90.1%. Drop week 4, which sold 2 units, and MAPE falls to 8.1%. The absolute error stayed between 5 and 10 units every week, so the jump comes entirely from the small denominator in week 4.

Scaled errors: MASE

MASE, proposed by Hyndman and Koehler, divides the forecast's MAE by the in-sample MAE of a naive forecast, or seasonal naive for seasonal data. A value below 1 means the forecast beats the average one-step naive forecast. If the naive method's in-sample MAE in the example were 25 units, MASE would be 8.3 / 25 = 0.33. It works across scales and builds the benchmark rule into the number.

Which metric to report

Report MASE across series, MAE or RMSE in units for planners, and MAPE only for series well away from zero. Add prediction interval coverage, which the M4 scored alongside point forecasts: if 80% intervals contain the actual value far less often than that, the model is overconfident. The MAPE an MMM tool reports validates prediction only. It says nothing about how the model split sales between channels.

How to backtest a time series forecasting model

Backtesting is the standard accuracy check. A single train/test split gives one accuracy number from one stretch of time, which may include a one-off promotion.

Expanding vs sliding windows

Time series cross-validation, or evaluation on a rolling forecasting origin, repeats the test. Fit on data up to an origin, forecast the next few periods, record the errors, move the origin forward and refit. In an illustrative setup, fold 1 trains on weeks 1 to 104 and forecasts 105 to 108, and fold 2 adds four weeks and repeats. That is an expanding window. A sliding window keeps the training length fixed and drops the oldest data, which suits series whose behaviour drifts.

Backtesting multi-step horizons

Score each horizon separately, never shuffle rows, and never use a feature that would not be known at the forecast origin. Regressors must be known in advance, such as planned prices, or forecast themselves. Plugging in their actual future values flatters the backtest. The same discipline belongs in MMM validation, where it shows that a model predicts, though not that it attributes correctly.

Time series forecasting models: how to choose one, and the tools to run it

A decision checklist

Six questions narrow the choice of time series forecasting model.

1. How many series? A few point to ETS and ARIMA, while thousands of related series point to a global model.
2. How much history? Seasonal models need several full seasonal cycles.
3. Is the data intermittent? Use intermittent-demand methods and MASE.
4. Which drivers are known in advance? Planned prices and holidays can enter the model. Unknown drivers must be forecast too.
5. Do you need intervals? ETS and ARIMA produce them directly, while many machine learning pipelines need extra work.
6. Do you need to explain the outcome? Then forecasting is the wrong tool, because accuracy cannot separate correlated drivers.

Time series forecasting in Python

For time series forecasting in Python, statsmodels covers ETS and SARIMAX, statsforecast offers fast automatic ETS and ARIMA, Prophet has its own package, and LightGBM or XGBoost with lag features covers machine learning.

R, Excel and the standard reference

In R, the fable ecosystem accompanies Forecasting: Principles and Practice, which works through most methods here with code. If your team reads one book on time series forecasting, make it that one. Time series forecasting in Excel starts with FORECAST.ETS, a fair single-series baseline.

A forecast is not an explanation

Hyndman and Athanasopoulos warn readers "not to confuse correlation with causation, or causation with forecasting". A variable can help forecast another without causing it. For marketing, it means a time series forecasting model can predict sales well while crediting the wrong channels.

Correlation is useful for forecasting, even without causation

Their example is a beach resort, where monthly ice-cream sales help forecast drownings because hot weather drives both. Marketing data has the same structure: spend rises before Christmas because demand does, and brand search rises when TV airs. A time series forecasting model can use these patterns without knowing which way the causation runs.

Correlated drivers: rarely a problem for forecasts, a real one for contribution

The same section explains that correlated predictors are not really a problem for forecasting, because forecasts can be computed without separating their effects. They become a problem for scenario forecasting and for any historical analysis of each predictor's contribution. Those are the two questions marketers ask of spend data.

Consider a brand that flights TV and paid search together every autumn, into the Q4 peak. An ARIMAX model with TV, search and seasonal terms will forecast Q4 well, but because the three move together, the data cannot say how to split the lift. In the illustrative example in Figure 3, a 60/25/15 split and a 20/35/45 split across TV, search and seasonality fit the history equally well, yet they point the budget in opposite directions.

Why an ARIMA sales forecast cannot tell you which campaign worked

Autoregressive terms make it worse. If last week's advertising lifted last week's sales and part of that effect carries over, the AR term absorbs it: lagged sales stand in for the advertising that drove them. The model forecasts well and gives advertising a small coefficient, which shows only that past sales predict current sales.

Scenario questions need a causal model

"What if we cut TV by half?" changes one input in a way the history may never have shown, and a forecast extrapolates exactly the correlations that would break. Answering it needs a causal model with carryover and saturation specified, controls for season and price, and calibration against experiments. Our guides to what-if analysis and incrementality testing cover the scenarios and the experiments that anchor them.

How marketing mix modeling uses time series structure

A marketing mix model is a regression on weekly or daily series, so it borrows from time series forecasting while asking a different question: how much of the outcome each driver caused, and what the next dollar will return. Our guide to ROAS works through that marginal return.

Trend and seasonality as control variables

In MMM, trend, seasonality and holidays absorb demand that would have occurred anyway, so it is not credited to media that ran alongside it. Robyn, Meta's open-source MMM, decomposes trend, season, holiday and weekday effects out of the box using Prophet.

Adstock and saturation

Adstock spreads each week's advertising effect across later weeks at a decay rate estimated per channel, a distributed-lag structure that the MMM guide illustrates. Saturation, the flattening of response as spend rises, lives on the spend axis rather than the time axis. Our guide to the law of diminishing returns covers it.

Validation borrowed from forecasting, calibration borrowed from experiments

Rolling-origin holdouts show whether an MMM predicts sales it has not seen, which is necessary and far from sufficient, because several contribution splits can predict equally well (Figure 3). Experiments decide between them: a geo lift test measures one channel's incremental effect, and the model is calibrated to agree.

	ARIMAX forecast	Marketing mix model
Objective	Predict future values	Estimate each driver's contribution and return
Output	Forecasts with intervals	Contribution, ROI and response curves
Validated by	Rolling-origin backtests	Holdouts plus calibration to experiments
Correlated drivers	Rarely a problem, if their historical relationship holds	The central problem

Frequently asked questions

What is the difference between ARIMA and SARIMA?

ARIMA models a series from its own past values (AR), differencing (I) and past forecast errors (MA), written ARIMA(p,d,q). SARIMA adds the same terms at the seasonal lag, written ARIMA(p,d,q)(P,D,Q)m, where m is the seasonal period, such as 12 for monthly data. Use it when a regular ARIMA leaves seasonal autocorrelation in the residuals.

Is ARIMA a machine learning model?

ARIMA is a classical statistical model from the Box-Jenkins tradition, with a few parameters estimated by maximum likelihood. Loose definitions of machine learning include it, since it learns from data, but it lacks the flexibility of gradient boosting or neural networks. On short single series, that simplicity makes it a hard baseline in time series forecasting.

What is the best model for time series forecasting?

No single model wins everywhere. In the M4, on mostly unrelated series, 12 of the 17 most accurate methods were combinations of mostly statistical approaches. In the M5, with related retail series and price data, gradient-boosted trees led. Start with seasonal naive, ETS and ARIMA, and keep whatever wins a rolling-origin backtest.

Are Transformers effective for time series forecasting?

The evidence is mixed. Zeng and colleagues (AAAI 2023) found that a one-layer linear model beat the Transformer-based forecasters they tested on all nine real-life datasets, often by a large margin. A later design, PatchTST (ICLR 2023), reported outperforming one of those linear models in general, especially on the larger datasets. Backtest any Transformer against a linear model and seasonal naive before trusting it.

What is a good MAPE?

There is no universal MAPE threshold in time series forecasting. A good figure depends on volume, volatility and horizon, and MAPE breaks entirely when actual values reach zero. Compare the model's error with seasonal naive on the same series instead, or report MASE, where a value below 1 means the model's errors are smaller than the average one-step naive error on the training data.

Can a time series forecast measure marketing ROI?

No. Time series forecasting can predict sales accurately while crediting the wrong drivers, because correlated spend, seasonality and lagged sales can trade off against each other without changing accuracy. Measuring ROI needs a causal model, such as a marketing mix model with adstock, saturation and seasonal controls, calibrated against experiments. Forecast accuracy cannot confirm its ROI.

How we approach it at Aryma

Time series forecasting and attribution are separate jobs that share one foundation, the exploratory analysis that shows which drivers lead sales, by how many weeks, and which variables move together too tightly to separate. Aryma Intellisheet does that in Excel with no coding, as our piece on bringing EDA into Excel describes. It cleans and reshapes MMM data before modelling and charts the outputs afterwards. Its EDA includes a colour-coded correlation matrix and heatmap, and cross-correlation analysis that finds lagged relationships between variables.

Lags found there feed both time series forecasting and MMM, as lag features in one and starting points for adstock in the other. MMM Singularity is the interpretation layer that connects attribution, incrementality, saturation and prediction into one explainable view. Our guide to AI for data analysis covers how to keep AI tools grounded in validated models.

If your measurement problem needs a design rather than a product, see our custom solutions. To talk through your own data, get in touch with us.