Time series analysis is about learning the fundamental features of the given series in order to make predictions in the future. One of the most important facts about financial time series is autocorrelation (serial correlation). So we want to end up with a model capable of expalining why there is serial correlation.
As you can see from the formulas above, serial correlation is the correlation of the time series with itself. So this is the fundamental feature of financial data: observations far away from each other correlate.
If we fetch for example AAPL stcok prices from Yahoo Finance and plot the histogram, we can make sure daily returns has approximately normal distribution.
What does it mean? That we can build models to explain this distribution. Thats why random walk (Wiener-process) is a good start. We assume stock prices and portfolios follow a random walk. Random walk time series is something like x(t)=x(t-1)+w(t) where w(t) is a white noise term so N(0,σ2). So what do we have to do? If we have a time series model (random walk, AR, MA, ARMA …) we have to check the autocorrelation-plot (ACF). It tells us whether out model is able to explain serial correlation or not.
This is the aim when constructing models for the FOREX: we would like to explain serial correlation and end up with a ACF plot like this. If we use random walk for S&P500 we can come to the conclusion that random walk is not the best model possible for financial assets!
Autoregressive Model (AR)
The autoregressive model is the generalization of random walk: instead of considering just a single observation in the past w(t-1), we keep including more and more past values. An AR(p) autoregressive model order p considers p past observations.
As you can see an AR(1) is the random walk itself. If we try to model financial assets (stocks and portfolios) with autoregressive model, we are not able to explain all serial correlation. What does it mean? It means we need more complex models to grasp the complexity of the market
Moving Average Model (MA)
Moving average model is about including more and more past white noise terms in our model. Sometimes it is working fine, sometimes it is not.
In my opinion, moving average model is still not complex enough to grasp the fundamentals as far as the market is concerned. The main problem with these models is that they do not take volatility clustering into account. What is volatility clustering? It means that financial series are not stationary: the variance (volatility) is not constant. We have to find models capable of explaining volatility clustering as well. Combined ARIMA and GARCH model is able to explain both serial correlation as well as volatility clustering!