REGRESSION: Non-Linear relationships & Logarithms

This paragraph introduces the regression series by Zed Statistics, hosted by Justin Seltzer. It's the fourth video of a five-part series focusing on functional form and transformations. The video is split into three segments due to the extensive content. The first segment discusses numerical variables and non-linear relationships, such as squared relationships and logarithms, which are crucial in regression studies. The second segment addresses categorical X variables, like yes/no types, and how to model these in regression. The third segment covers the scenario where the Y variable, the dependent variable, is categorical, with a focus on discrete choice modeling and logit models. The data set used is 'J Bob's used car sales,' created by the host for illustrative purposes, and is available for download. The video also provides a link to an Excel file with all the models used in the video.

This paragraph explains how to run a regression analysis using Microsoft Excel. It guides the viewers through the process, starting from the 'Data' tab and selecting 'Data Analysis' to access the 'Regression' tool. The paragraph details the input requirements for Y and X data ranges, including the inclusion of headings, and the option to output the results in a new worksheet. The output includes the dependent variable (price), coefficients, standard errors, T-stats, and p-values. A sample regression equation is provided, along with an interpretation of coefficients for age and odometer. The paragraph also discusses the limitations of the model, such as the unexpected increase in car price with age, and the need for scatter plots to better understand the relationships between variables.

In this paragraph, the host addresses the non-linear relationships between the variables by introducing transformations. The discussion begins with the realization that the relationship between car price and age, as well as odometer reading, may not be linear. The host suggests using scatter plots to visualize these relationships better. The video then introduces the concept of squaring the age variable and taking the inverse of the odometer to create non-linear models. The host explains that despite these transformations, the model is still considered linear regression because it refers to the coefficients. The paragraph emphasizes the importance of including both the linear and squared versions of age for flexibility in modeling. The output of the transformed model shows an improved R-squared value and significant p-values, indicating a strong relationship between the variables.

This paragraph delves into the concept of logarithms and their role in handling non-linearity in regression analysis. The host explains that logarithms are used to scale skewed data, converting numbers from a large range into a more manageable scale. The paragraph describes the process of taking the natural log of the odometer variable to create a more symmetrical and bell-shaped distribution. The host also introduces the idea of taking the log of the dependent variable (price) to improve the model's distribution. The video explains how to interpret the coefficients of logged variables, emphasizing that the change in price is now expressed as a percentage change for both X and Y variables. The paragraph concludes by highlighting the benefits of using logarithms, such as improving the model's interpretability and meeting regression assumptions.

The final paragraph of Part A wraps up the discussion on regression analysis and model interpretation. The host reiterates the importance of correcting skewed variables like price and odometer to improve the model's assumptions. The paragraph explains how the interpretation of the log of odometer coefficient has changed, now expressing the effect as a percentage decrease in price for a 1% increase in odometer reading. The host also addresses the decrease in R-squared value when the dependent variable is transformed, clarifying that R-squared values are not directly comparable between models with different Y variables. The paragraph ends with a teaser for Part B, promising further insights after a break.