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.

Using multiple discriminate analysis, Edward Altman created a model that provides guidance regarding a company’s credit health. The model is named The Z-Score Model. Upon providing measures from a company’s financial statements, the model would yield a single variable that is designed to be compared to a pre-specified scale. The placement of the Z-score on the scale would indicate whether a company was likely to head toward bankruptcy.

I implemented the model in Excel, and it can now be accessed through the Models page of the foquant.com website.

This model was also covered in the Fixed Income curriculum for the CFA Level 2 exam.

Two new models have been added to the Models page at foquant.com.

The first is the Gordon Growth Model, which is a very simple equity valuation tool. This model features a chart that shows the difference in price dependent upon the dividend growth rate. This is important, because as the growth rate approaches the required rate of return, the value of the equity approaches infinity.

The second is a C++ calculator for a standard coupon bond. The user is asked for inputs concerning the bond, and the value is printed to the screen. Similar to the GGM, this file will also show a range of prices depending upon discount rates, providing almost a high to duration and convexity.

On the topic of the Models page - this page will soon be redesigned. It will be laid out by topic model type, making it easier to view. While this may not seem necessary now as there are only a handful of models, this will be useful as the page grows and the models increase in complexity.