Return forecasting models: ARIMA, GARCH, LSTM and machine learning

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Forecasting models aim to estimate the future from the past and the present: next month's return, next week's volatility or the direction of a stock tomorrow. Research draws a clear line here: volatility can be predicted fairly well, returns hardly at all. This article explains ARIMA, GARCH, regressions with valuation variables and machine learning up to LSTM networks. Above all, it shows how to test whether a forecast is really any good.

What can and cannot be predicted

Returns in liquid markets are barely autocorrelated; Cont (2001) counts this among the stylized facts. This fits the efficient market hypothesis: an easily predictable pattern would be traded until it disappears. Fluctuations, by contrast, cluster: turbulent days tend to be followed by turbulent days (volatility clustering). Over many years, valuations such as the Shiller P/E say something about returns, albeit with a wide dispersion.

The ratings summarize the research cited below and are an assessment, not a measured quantity.

ARIMA: autoregression and moving averages

Box and Jenkins (1970) made ARIMA models the standard. AR(p) means that today's value depends on the last p values. MA(q): it depends on the last q shocks. I(d): the series is differenced d times until it is stationary. Prices have a unit root, which the test of Dickey and Fuller (1979) checks for; returns are approximately stationary, i.e. d = 0.

The order is identified via the autocorrelation function (ACF) and the partial autocorrelation function (PACF): for an AR(p), the PACF cuts off after p lags while the ACF decays slowly; for an MA(q), it is the other way round. After estimation, the residuals should no longer show any autocorrelation, which the Ljung-Box test of Ljung and Box (1978) checks.

GARCH: forecasting volatility

For fluctuations, there are models that really deliver. Engle (1982) introduced ARCH, and Bollerslev (1986) generalized it to GARCH. In GARCH(1,1), tomorrow's variance is made up of a base level, today's squared shock and today's variance. If α + β is less than 1, the forecast reverts to the long-run variance; α + β measures how persistently a shock lingers. More on the basics under volatility.

According to Andersen and Bollerslev (1998), the reason GARCH forecasts often look weak in tests lies in the yardstick: squared daily returns measure actual volatility only with a great deal of noise. Measured against the hidden volatility itself (better estimated from intraday prices), the forecasts are remarkably accurate. Hansen and Lunde (2005) compared 330 models of this family out of sample: for exchange rates, none clearly beat GARCH(1,1); for a stock, models with a leverage effect (a stronger reaction to price falls) were clearly better.

Regressions with valuation and interest rates

The classic approach for returns is a regression of the next period on variables known today: dividend yield, earnings yield, term structure, credit spreads, inflation. In sample, many of them are significant. Welch and Goyal (2008) re-examined the well-known variables comprehensively: by and large, they had forecast poorly for 30 years, both in and out of sample, were unstable and would not have helped an investor with market timing using the data available at the time.

Campbell and Thompson (2008) counter that with weak restrictions (such as no negative forecast of the equity premium and coefficient signs as expected by theory), many regressions beat the historical mean. The explanatory power out of sample is small, but economically significant for mean-variance investors. Rapach et al. (2010) show that an average of many individual forecasts beats the historical mean more consistently than any single model.

Machine learning: from lasso to LSTM

Machine learning helps above all when there are many similar signals. Ridge and lasso shrink regression coefficients, and lasso sets some of them to exactly zero. Random forests average many decision trees grown on random samples; gradient boosting builds trees one after another, each correcting the errors of the previous ones. Neural networks chain together many non-linear layers. Gu et al. (2020) compared these methods for the risk premiums of individual US stocks: the forecasts brought investors large economic gains, in some cases doubling the performance of leading regression-based strategies. Trees and neural networks performed best because they capture non-linear interactions. All methods relied on similar signals, mainly variants of momentum, liquidity and volatility.

For sequences such as price series, there are recurrent networks. The LSTM of Hochreiter and Schmidhuber (1997) has memory cells with gates that control what is kept and what is forgotten. Fischer and Krauss (2018) used it to predict the daily direction of S&P 500 stocks out of sample from 1992 to 2015: before transaction costs, a return of 0.46% per day and a Sharpe ratio of 5.8, better than a random forest, a deep neural network and logistic regression. From 2010 onwards, however, the excess returns appeared to have been arbitraged away. The selected stocks were often volatile and showed a pattern of short-term reversal. So part of the success was a known effect.

Newer architectures such as transformers process entire sequences with an attention mechanism and are now also offered as pre-trained models for time series. As of October 2026, we are not aware of independently confirmed evidence that they predict returns better after costs. This is an assessment. Kelly et al. (2024) argue that models with more parameters than observations capture the predictability of the market return better than simple ones. This is a case for machine learning that research continues to debate.

Language models as a forecasting tool

Large language models (LLMs) are mainly used to assess texts. According to Lopez-Lira and Tang (2026), ChatGPT's ratings of headlines predicted daily returns out of sample, more strongly for small stocks and after bad news; the authors expect broad adoption to make markets more efficient. Two caveats, as an assessment: a language model may have seen texts during training that were written after the test period. That would be a hidden look-ahead bias. And there is no evidence that a language model predicts a return figure well directly. More under alternative data.

Testing forecasts honestly

• Out of sample: every forecast may only use data up to the previous day, including for parameters, normalization and variable selection. A rolling walk-forward test with the historical mean as the opponent is standard (out-of-sample R²).
• Significance: the Diebold-Mariano test of Diebold and Mariano (1995) tests whether two forecasts are equally accurate, with any loss function and even with autocorrelated, non-normally distributed errors.
• Multiple testing: if you compare many models, you will find a good one by chance. The reality check of White (2000) and the SPA test of Hansen (2005) test whether the best model really beats the benchmark; Harvey et al. (2016) require t-statistics above 3.0 for new factors.
• Data leakage: revised macro data instead of the data published at the time (Croushore and Stark 2001), variables normalized over the whole sample, or random cross-validation that mixes future and past. For this, López de Prado (2018) describes purged cross-validation for time series.
• Costs: accuracy is not profit. Strategies with daily turnover, as in the LSTM example, lose a lot after costs (see backtest overfitting).

Advanced: why good models forecast poorly

• Signal and noise: if a model explains 1% of the variance of monthly returns with a volatility of 4.47%, the predictable part has a volatility of only 0.45 percentage points. Even the mean itself is known only to within about 0.29 percentage points after 240 months (4.47 / √240).
• Bias-variance tradeoff: a flexible model fits the sample better but varies more with the data. Shrinkage as in ridge and lasso, the restrictions of Campbell and Thompson (2008) or forecast combinations following Rapach et al. (2010) reduce the variance at the cost of a small bias.
• Instability: relationships change with monetary policy and market structure. Welch and Goyal (2008) found unstable models; regimes and structural breaks are covered in the article regime models.
• Overlapping returns: multi-year forecasts with windows shifted monthly produce strongly dependent observations. Without a correction following Newey and West (1987), t-statistics look far too large.
• Distribution instead of a point: for risk, what often counts is not the expected value but the range. Volatility models deliver exactly that, for example for value at risk.

What the research shows

• Market return: classic predictor variables mostly fail to beat the historical mean out of sample (Welch and Goyal 2008); with restrictions and combinations, they narrowly succeed, with a small but economically relevant R² (Campbell and Thompson 2008; Rapach et al. 2010).
• Volatility: GARCH models forecast actual volatility well (Andersen and Bollerslev 1998); GARCH(1,1) is hard to beat, and for stocks, asymmetric variants help (Hansen and Lunde 2005).
• Cross-section of individual stocks: trees and neural networks outperform linear models, relying on momentum, liquidity and volatility (Gu et al. 2020).
• Deep learning for daily directions: high returns before costs until about 2009, hardly any edge thereafter (Fischer and Krauss 2018).
• Language models: ratings of news predict short-term returns, especially for small stocks (Lopez-Lira and Tang 2026).

Limits and common mistakes

• Hit rate instead of return: getting the direction right 55% of the time means nothing if the days you get wrong are the big ones.
• Tests that are too short: a few years out of sample cannot separate small advantages from chance.
• Hidden future: variable selection, hyperparameters and model architecture are often chosen after the test period is already known.
• Costs, capacity, taxes: short-term signals require high turnover; what works with small stocks in a backtest often cannot be traded at scale.
• Publication: known patterns get traded and weaken (alpha decay).

This article explains methods and is not investment advice.

How Portfolio Lab calculates it

Portfolio Lab does not make return forecasts; there is no ARIMA, no GARCH and no machine learning in the app. Instead, the Assets analysis tests whether there is anything to forecast in a series at all: autocorrelation and partial autocorrelation for 4 to 60 lags with the band ±1.96 / √n, optionally on squared deviations from the mean, in which volatility clustering shows up; the Ljung-Box test for horizons 1, 3, 6, 12 and 24; and the variance ratio following Lo and MacKinlay (1988) for 2, 3, 6 and 12 months or 2, 5, 21 and 63 days (methodology).

The Monte Carlo simulation draws monthly returns from the portfolio's own history, individually or in blocks of 12 months, or from a normal or Student's t-distribution with 5 degrees of freedom. It shows a range of possible paths, not a forecast (methodology, Monte Carlo simulation). The Volatility dashboard sets the volatility priced in by the options market (the VIX for maturities from 9 days to 1 year) against the realized volatility of the last 21 trading days. The difference is a snapshot, not a forecasting model (methodology).

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.