Time Series Analysis

The efficient market hypothesis says prices reflect all available information, so returns are essentially unpredictable. Decades of research say: mostly true, with well documented exceptions. Those exceptions are what factor investing is built on.

Here are the five statistical properties of equity return series that systematic strategies exploit, hedge or must allow for. I will show each one in real Indian data rather than describe it in the abstract, because the gap between what returns look like and what they do is easier to see than to explain.

1. Stationarity checkPrices are not stationary; returns roughly are.
2. TransformPrice to log return.
3. FitA mean model, then a variance model such as GARCH.
4. Check residualsLjung Box and ARCH LM tests.
5. Walk forwardRefit on a rolling window, test on unseen data.

Most retail back tests skip steps 1 and 4, which is how a spurious pattern ends up looking like an edge.

The five properties

Stationarity in plain words

A series is stationary when its average and its spread do not drift over time. Prices fail that test: the Nifty 50 at 22,520 on 9 Oct 2026 has a completely different level from the Nifty of a decade ago, so an average of past prices tells you little. Daily returns come much closer: they hover around a small average with a spread that changes, but does not trend away.

The standard check is the augmented Dickey Fuller test. In plain words, it asks whether the series tends to pull back towards a stable level or wanders off with no anchor. A p value below 0.05 suggests an anchor. Price series almost always fail it; return series almost always pass. If you ever see a back test or a model built on raw price levels, that is the first question to ask.

GARCH: modelling volatility clustering

GARCH (generalised autoregressive conditional heteroskedasticity) models are the standard tool for time varying volatility. The idea: today's variance depends on yesterday's variance and yesterday's surprise.

Worked example: fitting GARCH to the Nifty 50

I fitted a GARCH(1,1) to the same 498 days of Nifty 50 returns, by maximum likelihood with the long run variance fixed at the sample variance. The result: α = 0.08, β = 0.90, so persistence α + β = 0.98. That number has a practical meaning. A volatility shock decays by 2% a day, so half of it is gone after about 34 trading days, roughly a month and a half.

Put that in rupees. Suppose you size a ₹10 lakh equity position to a risk budget. On a calm day the model says annualised volatility is 10%, and you hold the full ₹10 lakh. After a week of 2% to 3% daily moves the model's estimate rises to 20%. To keep the same risk you would hold about ₹5 lakh. A month and a half later, if markets calm down, the estimate is back halfway and so is your position. That is volatility targeting, and it is not forecasting returns. It is forecasting the size of moves, which the data say is genuinely predictable.

Fit by RupeeCase on NSE daily closes, 7 Oct 2024 to 9 Oct 2026. Two years is a short sample for GARCH; treat the parameters as illustrative of the typical shape, which is similar in most equity indices.

Mean reversion vs trend following: the regime question

Mean reversion strategies (contrarian, value) and trend following strategies (momentum) both have empirical support. They do not work at the same time; each has stretches where it dominates.

• Mean reversion tends to dominate in choppy, range bound markets. Momentum strategies get whipsawed, buying breakouts that fail.
• Trend following tends to dominate in strong directional markets. Contrarian strategies keep buying cheap stocks that keep getting cheaper.
• The workable answer is diversification across signal types, not switching between them. Identifying the regime in real time is very hard; holding both is easy.

The monthly data show how noisy this is over short windows. Of the 24 month end to month end changes from October 2024 to October 2026, the Nifty 50 rose in only 10, and the lag 1 autocorrelation of its monthly returns was −0.2, a hint of reversal at the index level over that stretch. Two years later the sign could flip. Do not build a strategy on a pattern you saw in 24 numbers.

Common mistake: modelling prices instead of returns

The common beginner mistake is fitting a model to price levels. Prices trend and wander, so almost anything fitted to them shows a high R² and a strong relationship. Two unrelated rising series will appear correlated simply because both went up. That is called a spurious regression, and it fooled economists long before it fooled traders.

Work with returns, check they are roughly stationary, and test what remains in the residuals. If your model only works on prices, it does not work.

How this shows up in RupeeCase

Questions readers ask