Is there a way to tell if a time series price data is reversed?

Assuming you are given an array of values representing a stock's historical price, without timestamps, is there a way to tell if this array of prices has been reversed?

This is a great question!

The simple, boring answer is clearly "no" - given a time series of stock prices, there is no way to tell for certain whether the price series has been reversed or not. For any time series of stock prices, the time-reversal is also a possible time series of stock prices.

However, there can be indications that a time series of price data has been reversed. In general, the longer the time series, the easier it will be. For example,

1. Stock splits. Companies that do stock splits tend to have high stock prices (e.g. above \$100) and split their stock to lower ranges (e.g. \$10-40) so if you see the reverse happening (a lower range jumping to a higher range) it could be an indication of a reversed series. Obviously this only applies if you have unadjusted prices.
2. Reverse stock splits - the opposite applies, as companies only tend to do reverse stock splits when they are trading at distressed prices, e.g. \\$1-5, so a number of falls from a reasonable range to a distressed range could indicate a reversed series.
3. For various reasons stock splits are more common than reverse stock splits, so if you see a lot of near-integer falls in the stock price and no near-integer gains, it could indicate a reversed series.
4. Dividend payments. When a stock makes a dividend payment, all else being equal, you expect the stock price to drop by the amount of the dividend. This means that drops of 1%-5% in the stock price are more common than you would otherwise expect. If you see more jumps of this magnitude than expected, it could indicate a reversed series. Again, only works if you have unadjusted prices.
5. Stock prices tend to appreciate slowly and have sharp falls i.e. "up by the stairs and down by the elevator". The sharp falls can often be followed by sharp rallies. If you see the opposite, i.e. the stock price falling slowly with occasional sharp rallies (where the rallies are followed by sharp falls) it could be indicative of a reversed series.

I illustrate point 5 below, to show what I mean. The blue line is the log of the S&P 500 index from 1990 to today, whereas the red line is a synthetic index generated by reversing the time order of the S&P 500 returns, and multiplying them back up. To an experienced eye, the blue line looks much more plausible as a time series of the stock prices than the red line does.

• Thanks, I agree 4 and 5 would be especially useful. Do you think mathematically, there would be any method to evaluate as well?
– Will
Commented Apr 20, 2020 at 22:07

This question about reversibility of time attracted a lot of attention of econophysicists. Laws of physics guarantee no time reversion in real life (because of entropic principle): is it the same on financial markets?

Somehow, the returns of stocks should reflect the growth of economy and the risk of projects the companies are betting on. All that taking its ground in the physical world, one could expect no reversion of time on financial markets.

One empirical evidence is that recent trends are predicting bursts of volatility only in one direction: "future trends are not following volatile periods" with the same intensity.

To test that

• build a very simple short term trend indicator $${\cal T}(t,t-\delta t)$$
• estimate the short term volatility $$\sigma(t,t-\delta t)$$
• add any indicators/analytics you like during the same interval $$[t,t-\delta t]$$.

So that you can build a feature space $$X_t$$ by collecting in a vector all these features.

Then for any stopping time $$\tau$$ you need to store the volatility from $$\tau+1$$ to $$\tau+\delta t$$ in a $$Y_\tau$$ variable.

Use your favourite model (linear regression, random forest, etc) to predict $$Y$$ as a function of $$X$$. Keep the residuals and the $$R^2$$ of this regression.

Do the same on the time-reversed time series: you will see that the $$R^2$$ should be worst.

If it is time serie object (whatever programming language you use) it should at least consist of column with time stamps and column with prices.

If it is just vector of prices there is no way to determine wheter is is reversed or not.

However if you plot this vector and you see for instance decreasing trend and you know that in this time period underlying stock performed well, you can assume there could be potentially reversed order.