Looking for non causal time series in real life

Question

Is there examples of real life(e.g. financial/economic) non-causal time series? It seems that most models are assuming causality(i.e. Furture data depend upon past and present only).

Answer 1

Causality means "dependent on information only up to the present", in the context of time series and stochastic processes.

There are no non-causal data generating processes in finance, or economics in general. (There are very basic economic reasons for this.)

To say that $dr_t=\mu_tdt+\sigma_t dW_t$ is "non-causal" is fundamentally incorrect. Brownian motion $W_t$ does not depend on its future value. Indeed, mathematically it's a Markov process, and a martingale. So is its discrete-time version, the random walk. A quick casual browsing will tell you that such a process by design depends only on information up to present.

In less elementary settings, in order to be economically and mathematically correct, in economic/finance models such processes are always considered together with a filtration. Standard empirical interpretation of a filtration is "increasing flow of information". In layman's terms, asset prices evolve as information, up to the present, becomes incorporated into prices. No future information. (Note a forecast is not information from the future.)

Further Comments

1. Causality and stationarity should not be conflated. In time series, one usually first encounters the notion of causality in the context of stationary linear processes---a linear time series is non-causal if it depends on future innovations. However, causality, or lack thereof, can be considered regardless of stationarity, as well as for continuous-time processes. Causality is about direction of information flow---from the past to future or vice versa.

2. In the stationary time series setting, for every non-causal model there is a causal model that is observationally equivalent---e.g. consider the AR(1) case. Statistically, you cannot distinguish these two cases. Therefore one chooses to consider only causal models for both statistical and non-statistical reasons.

3. To claim asset prices in an asset bubble is "non-causal" is an incorrect statement building on an already incorrect statement. Asset bubbles occur when prices deviate from its fundamental value, without getting into the precise economic notion. The first such historical episode is the South Sea Bubble of 1720. More recently, there is the US housing bubble. Bubbles occur not because market participants can see into the future. That would be absurd. Indeed, bubbles always burst precisely because people cannot see into the future. They tend to be driven by irrational beliefs and/or excessive risk-taking. A "live" example of a bubble occurring at the present moment is here. The reason people buy the stock of a bankrupt company, thereby drive up its price by 400%, is not because they have information about the future.

Answer 2

The tongue in cheek type of example is a random walk model. The asset prices in finance, or more precisely their returns $r_t$, are often modeled as a random walk: $dr_t= \sigma^2\xi_t$ where $\xi_t\sim\mathcal N(0,1)$. In discrete time you would say $r_{t+1}=r_t+\xi_t$.

However, this doesn't directly address your question. The reason is that causal/noncausal in time series are discussed in the context of stationary models, see e.g. these notes p.9. Here, AR(1) model such as: $x_t=\phi_1 x_{t-1}+w_t$ will be causal when $|\phi_1|<1$ and noncausal when $|\phi_1|>1$ and both cases are stationary, despite the latter being explosive. Random walk is AR(1) with $\phi_1=1$ is non stationary, so causality in this sense is meaningless to talk about, although it is not causal, neither it is noncausal.

Hence, the better answer would be to point you to "bubble studies" in asset pricing literature. Frankly, I haven't seen anything that would convince me in the existence of them, but some folks look at them from different angles. The one that might be inline with what you're looking for is "Rational Bubbles in Stock Prices?" at NBER by Diba and Grossman (1988), see section 1. They start with a typical consumption model, and come to a special case where $B_t=(1+r)B_{t-1}+xi_t$ in Eq(5), where $B_t$ is the explosive component of asset prices that is usually assumed to be zero. You can also see how they deal with nonconvergence of the sum that I mentioned above: they make past finite! There's time zero of inception of stock market, see Eq(7)

If you look up in Google Scholar the paper that cite this one, you'll find pretty much all bubble studies to date, e.g. Jarrow's "How to detect a bubble" gives you DIY instructions, and also claims that post IPO LinkedIn price was a bubble.