Why do we study the noise sequence in time series analysis? I am studying time series analysis for the very first time and I cannot seem to get my head around the fact that the entire time series analysis is about studying and modelling the behaviour of the noise component of the time series model. But why is it that we are so concerned about the noise?
From what I could gather is that it's the only component of the time series model (with reference to the classical decomposition model) which is random and thus we should care and try to understand its behaviour. Also, noise is not synonymous to error.
But then what does this exactly imply and how exactly is it useful for understanding the time series at hand?
Any help is much appreciated!
 A: A time-series breaks down to the components seasonality, trend and noise.
Let's take oil prices.
Trend component tells you whether oil prices are going up or down and by approximately how much (steep upward slope, plateauing downward slope etc.)
Seasonality says that given it is winter, what effect does that have on oil prices.
Both the above are derived from the data itself. Now let's take a moment to understand the beauty of the third component- noise.
The most common white noise time series analysis assumes (among other things) that the noise mean zero, some variance and each point of decomposed noise is uncorrelated the signal. The uncorrelated part simply means it is independent of the signal i.e. the trend + seasonality.
Consider the following equations:
y = ax + bz

b is a constant and z is independent of x but contributes to y. 
Now what if the true distribution of z actually looked like
z = mp + nq

Now you cannot directly observe or measure p and q but they contribute to z!
If z is your time-series noise, what seems random is actually controlled by a lot of latent variables. Example- changes in foreign policy that make oil import harder, or massive failure of equipment at major offshore oil plant.
All these effects can then be observed in Noise, but cannot be directly attributed or linked to their root cause. So, it becomes essential to model noise in time series analysis.
(Update: for the sake of an interpretable example I imply that foreign policy changes and equipment failure would not impact trend and seasonality. That's not 100% accurate but it is really hard to come up with intuitive latent variables :D)
A: Why to be concerned about the noise? If you are not concerned about, than you can have troubles with overfitting (you are modelling noise instead of signal). Also good knowledge about noise allows you to filter that noise, so you can get pure signal. Imagine, that you want to study signal y(t)=sin(t)+sin(3t), where sin(3t) is the noise. If you obtain Fourier spectrum of signal y(t), then you can substract noise and you get pure data signal without noise only because you know information about noise. Also if you want to use for example Kalman filter to predict data, you need to have apriori information about noise, so you can make succesful data fusion and get valid results.
