P McDTo understand the serial correlation in a time series it is helpful to look at a simple random walk model such as:

p(t) = p(t-1) + u

where p(t) is the price level at time t and u is a random variable, the net change for the time period.

Suppose the price started at 100 years ago. Then the current price level will be the sum of all the previous daily changes plus the original 100. Over time the dispersion of the sum of these random changes will increase with the square root of elapsed time. This growing dispersion can be interpreted as a trend by one reading a chart even for a truly random price series.

As far as real markets go, we know that the daily serial correlations between changes is small and generally has tended to be negative in recent years. But it is also true that the variance of the market movement is serially correlated. In other words volatility persists. So we wind up with a slightly more complicated model where the u's in the above formula have a wandering variability.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008

Jim Sogi comments:

Phil has written often that price levels are serially correlated. This is a characteristic of a time series. Why is this? For some mechanical or structural reason levels are limited in the amount of their change over a given time. In markets in general, the market forces and competition of bid and ask tend to compress prices. In Globex, the 1/4 point tick and the depth prevents large jumps or moves in between trades. I posit that this is one of the basic market forces, and can be used to the trader's advantage. Globex itself as a market making mechanism has incentive to dampen wild price swings. Market makers, and stat arb traders have incentives to prevent wild price swings and big trends.

The second main force in markets is the force that overcomes the gravity, so to speak, of the serial correlation, and overcomes the structural dampening mechanisms inherent in a time series. This is described in part as volatility, or also as trending. This is the force that moves the market away from any given price. What is this force? It might be market imbalance of supply and demand much larger than the inside depth or of one day or week's trade volume as we saw this year and last. It could be underlying financial movements, changes in investor perceptions that outweigh the tendency of correlation. It might be government risk. This appears to be the balance of force between mean reversion and trending.

Another aspect of trending is the appearance of trends in a random time sequence as an artifact of the correlation and the increase or variation in volatility. This calls into question the fundamental nature of the force or explanation of the market forces as an explanation for trends.

These two forces might be quantified, and a metric used to define a trend. The serial correlation appears to be the basis for mean reversion trading for which trending is an anathema. Some trading methods utilize trending but it is problematic defining a trend. One of the problems is the existence of trends as an artifact of random time series and how to distinguish such artifacts from a fundamental move, in advance. The answer will also change with cycles.

The worst thing for a range trader is for a break out. The worst thing for a trend follower is a range. A single big move can wipe the reverter, and multiple range entries can ruin the trend follower before he catches the train. The correlation continues during a trend and results in the channel phenomenon, so this can be used to the traders advantage both contra and following.


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