[Hassam]

Predicting financial time series is a challenging task. If predicting financial time series was easy, it would have been very easy to make fortunes in the financial markets. Financial time series modeling is of the most actively researched areas now a days. Time series are important in many areas that we come across. Most important areas where time series matters is the health services sector, weather forecasting, earthquake forecasting, marketing, inventory and so many other areas. Price data is a perfect example of time series data. You can open your trading platform and find price being plotted at specific time periods also known as timeframes.

Autoregressive Financial Time Series Models

Most time series have some relation with the past values meaning present value is being influenced by past values to some degree. This is precisely what we believe when we do technical analysis. We look at the chart past price values and try to predict the future price. Now I have said this dependence is not 100%. Technicians also known as chartist believe that past price can be used to predict future price. So the first financial time series model that comes to mind is the Autoregressive Models. I have used autoregressive (AR) models in trading and find that AR models do have some predictive power.

Mean Reversion Trading Strategies

The problem with the standard AR models is that they assume the time series to be stationary. Stationary is a concept in statistics that means the expected mean of the time series is constant or level. So a stationary time series will most of the time fluctuate around that level. If it strays far from that level, it will try to return to the mean. This is also known as the Reversion to Mean. Mean Reversion Trading Strategies are popular in the quant world. On mean reversion trading strategy that is very popular is the pair trading. We find two instruments like two stocks in the same sector or two currency pairs that have some correlation. We test for mean reversion and if the test is statistically significant, we wait for the two instrument to diverge from the mean. When it happens, we short the stock what is above the mean and go long the stock that is below the mean.

Autoregressive Integrated Moving Average Models
But there are many times when the price time series is not stationary at all meaning its long term mean is changing. So we have the mean level moving steadily either up or down.As I said most of the AR time series models use stationary assumption in making the forecasting. Another model that is popular is the ARIMA (Autoregressive Integrated Moving Average Model). In an ARIMA model we make the time series stationary by differencing. Differencing we detrend the time series of its trend component. I have used ARIMA models and they produce very poor financial time series forecasts. What to do? I have started this thread in which I want to discuss how we can use Bayesian Models to improve the financial time series forecasting.

[Hassam]

Most of the time we will observe stochastic dependence between consecutive values of a time series. If you are a trader, you should be knowing that price depends on its past values. Sometimes this dependence is strong and sometimes it is weak but this dependence is always there. Structural time series models break the time series into a trend component, a seasonal component, a cyclical component and pure while noise. We can model the trend with a polynomial, seasonal/cyclical components with a periodic functions like sines and cosines and so on.

The simplest are the autoregressive models that model structural dependence conditional on past outcomes. If you use R and let it choose the AR order automatically, most of the time you will find it choosing the first order AR model as the most appropriate model. AR(1) is the simplest model that we can use in building a trading model: y(t)=b0+b1y(t-1)+u(t). u(t) is white noise and has the distribution u(t)~N(0,a2). u(t) is considered as a random shock and b1y(t-1) is the prediction part. As a trader we are interested in this prediction part b1y(t-1).

Financial time series are non stationary. Stationary means the probability distribution generating the time series data is time invariant. Stationary condition is not possible in financial time series and the probability distribution that is generating the time series is continuously changing with time. As said above, our goal is out of sample prediction that we can use as a trading signal. When we try to recursively make the prediction, each new prediction becomes a source of error. In traditional ARIMA modeling, first we need to make the time series stationary. This is done by differencing.

In Bayesian time series modeling, we don't need to do differencing. In Bayesian statistics, parameters are treated as random variables while data is considered as fixed unlike traditional statistics where model parameters are considered to be fixed and data is considered to be random. Another thing good in Bayesian modeling is that we can model explosive non stationary time series also. Many times we find financial time series exhibiting explosive non stationary behavior.