Regression assumptions explained!

This paragraph introduces the topic of regression assumptions, highlighting the importance of understanding the assumptions that underpin regression analysis. It mentions that there are varying numbers of assumptions discussed in different sources, but the video will focus on six key assumptions. The speaker, Justin Zeltser, emphasizes his approach of explaining concepts intuitively without unnecessary formulas, aiming to make the material highly intelligible. The paragraph also briefly describes the graphical representation of regression, including the scatterplot of observations and the line of best fit.

The second paragraph delves into the first assumption of linearity, explaining that it means the relationship between the variables should be additive and linear. It clarifies that including transformations like X squared or log X in the regression equation does not violate the linearity assumption. The speaker uses the example of assessing lung function based on age and discusses how an incorrect functional form can lead to unreliable coefficients and standard errors. The paragraph also introduces residual plots as a tool for detecting issues with the functional form and mentions the use of likelihood ratio tests for comparing models.

This paragraph discusses the assumption of constant error variance, also known as homoscedasticity, which implies that the variance of the error terms should be constant across different levels of the independent variable. The speaker provides an example of household expenditure as a function of income and explains how heteroscedasticity, the violation of this assumption, can occur. The paragraph describes the impact of heteroscedasticity on the reliability of standard errors and hypothesis testing. It also mentions methods for detecting heteroscedasticity, such as the Goldfield-Quandt test and the Breusch-Pagan test, and suggests remedies like using heteroscedasticity-corrected standard errors or logging the variables.

The fourth paragraph addresses the assumption of independent error terms, explaining that each error should be independent of the previous one. Autocorrelation, the violation of this assumption, can occur in time series data where there is a natural order to the observations. The speaker uses the example of a stock index to illustrate how autocorrelation can manifest in a residual plot. The paragraph discusses the implications of autocorrelation on the reliability of standard errors and suggests tests like the Durbin-Watson test and the Breusch-Godfrey test for detection. Remedies include incorporating additional variables that may explain the autocorrelation or using generalized least squares methods.

The fifth paragraph focuses on the assumption of normality of error terms, which means that the distribution of the error terms should be normally distributed. The speaker uses the example of medical insurance payouts to illustrate how violations of this assumption can occur. The paragraph explains that while violations of normality are often not a significant issue with large sample sizes due to the central limit theorem, they can affect the standard errors and hypothesis testing with smaller samples. It suggests using histograms and QQ plots for detecting normality violations and mentions that remedies could include changing the functional form or increasing the sample size.

This paragraph discusses the assumption of no multicollinearity, which means that the independent variables should not be highly correlated with each other. The speaker uses the example of motor accidents as a function of the number of cars and residents in a suburb to illustrate how multicollinearity can occur. The paragraph explains the impact of multicollinearity on the reliability of coefficients and standard errors and suggests detecting it by examining the correlation between variables and using the variance inflation factor. Remedies include removing one of the correlated variables or adding another variable that is not collinear with the others.

The final paragraph addresses the assumption of exogeneity, which means that the independent variables should not be related to the error term. The speaker uses the example of salary as a function of years of education to illustrate how omitted variable bias can occur when a variable like socioeconomic status affects both the independent and dependent variables. The paragraph explains the implications of omitted variable bias on the validity of causative interpretations and suggests using intuition and correlation analysis to detect it. It also introduces the concept of instrumental variables as a remedy, using a historical example of the draft during the Vietnam War as an instrument for education.