Time Series Models with Multiple Time Series
by theorangedog on Feb.10, 2008, under Skills
Can least squares regression be used if we are creating a model that includes more than one time series? An example of a model with more than one time series would be as simple as a model that includes a dependent variable and a single independent variable that are both time series. Such as:
Working with that example, there are five possible outcomes that dictate whether or not least squares is an appropriate method. Generating these outcomes begins by testing each variable, again the dependent variable and the single independent variable, for a unit root. This can be done by using the Dickey-Fuller test.
Outcome 1:
Neither variable has a unit root. In this case, least squares can be used to estimate the model.
Outcome 2:
The dependent variable has a unit root while the independent variable does not. This occurs by failing to reject the hypothesis of a unit root for the dependent variable. If this is the case, the error term is not covariance stationary and least squares is not appropriate for that specific model.
Outcome 3:
The dependent variable does not have a unit root while the independent variable does. Opposite of Outcome 2, this occurs by failing to reject the hypothesis of a unit root for the independent variable. Similar to Outcome 2, the error term is not covariance stationary and least squares is not an appropriate method.
The next two outcomes occur when both the dependent variable and independent variable have a unit root. In these cases, the next step would be to test for cointegration between the two time series. This can also be performed using a Dickey-Fuller test. The time series are said to be cointegrated if their divergence has some boundary.
Outcome 4:
Both variables have a unit root but they are not cointegrated. As in Outcome 2 and Outcome 3, the error term will not be covariance stationary and thus least squares is not an appropriate method.
Outcome 5:
Both variables have a unit root and they are cointegrated. In this case, the error term is covariance stationary so the model may be valid. However, this relationship models the long term, and short term results may not be as expected. That is primarily the cause of the fall of LTCM. While there may be boundaries to a divergence in the two time series, there are not necessarily natural rules in all models that dictate maximum time frames for desired changes in the divergence. In other words, if a trader is hoping for a wide divergence to narrow with the two time series meeting, there is not necessarily a time frame in which that must happen.
This is a quick overview of testing the quality of a model with two time series variables using least squares. The intent was to build upon my prior post, Test for Random Walk in Time Series. In an upcoming post, I will provide the results (and excel spreadsheets) that result in one of the five outcomes, and will provide additional detail and connection within the method.
Leave a Reply
-
Welcome to theorangedog.net.
This site is my dashboard for links and information on topics related to Skills (Finance), Spirits (Whisky & Beer), and Speed (Races & Training).
-
.
Check these guys out!
February 12th, 2008 on 8:16 am
One of the things with least squares regression is that it takes into account all points to do the regression, unlike L1 regression which gives less weight to points with extreme deviations to all other points.
I guess the question to ask is do we need to treat extreme outliers (if any) like any other point or would it be safe to ignore them in much the same way as L1 regression does?
February 12th, 2008 on 9:07 am
If we can reasonably make the assumption that the data are distributed in a Gaussian manner, then sure, least squares is fine. Off the top of my head, and without providing sources, I would say that one could easily find evidence for or against that assumption. It would likely simply be a function of time frame and market/security.
However, most of what I can come across does not seem to fit that assumption, referring to both security price returns and trading return series. In those cases, I think its important to consider the outliers, especially as they often relate to draw-downs, which is a hot topic.
Data Mining in MATLAB has a discussion on LSR versus L-1 here, and Neural Market Trends brought up the point of outliers in options here. Tom actually did a follow up post today on that, where he trims the outliers here.
The topic of outliers in trading is especially interesting due to its complexity. Often time, shocks to liquidity mean that even in a well-modeled system, the realized exit/entry price will likely largely vary when compared to the modeled exit/entry. Kind of an obvious statement, but it is what I find interesting.
A quick scratch and read test I did a while back pitted the slippage of two securities, one with an absolute price about 1/4th or 1/5th of the other, against each other. What I found was that the higher-priced security had proportional slippage greater than that of the lower-priced security. Maybe just an anomaly due to the small dataset. Not necessarily related to outliers, but the outliers in the slippage series would be interesting as well.