Functional Forms for Simple Linear Regression | CFA Level I Quantitative Methods

In this lesson, we’ll explore the functional forms for simple linear regression, including time-series regressions, log-lin models, lin-log models, and log-log models.

Time-Series Regressions

A time series is a set of observations for a variable over successive periods of time. The x-axis represents time, while the y-axis represents the dependent variable. A series has a trend if a consistent pattern is visible when plotting the observations on a graph.

A linear trend can be represented by a linear trend model, similar to a simple linear regression, but with two key differences:

• Time is the independent variable
• Observations are sequential and in fixed time periods

We can apply most techniques from simple linear regression, such as the least squares method, measures of fit (SEE and R-squared), and T-tests and F-tests for the significance of the slope coefficient.

Log-Lin, Lin-Log, and Log-Log Models

If a relationship exhibits exponential patterns, we can use a log-linear trend model. This model allows us to use linear regression techniques while modeling exponential relationships. The slope coefficient represents the relative change in the dependent variable for an absolute change in the independent variable.

In cases where the dependent variable increases at a slower rate over the independent variable, a lin-log model may be more suitable. The slope coefficient represents the absolute change in the dependent variable for a relative change in the independent variable.

For relationships that appear random but reveal a linear pattern when applying natural logs to both dependent and independent variables, we can use a log-log model. The slope coefficient represents the relative change in the dependent variable for a relative change in the independent variable.

Choosing the Appropriate Functional Form

Choosing the right functional form can be challenging. One objective way is to evaluate the goodness of fit by calculating the R-squared, SEE, and F-statistic for each model and comparing their results.