Understanding Random Residuals in Linear Regression: A Visual Analysis

Target Audience: New Graduates/Analysts

In linear regression analysis, residuals are the difference between the observed values of the dependent variable and the values predicted by the regression model.

The concept of randomness of residuals can be understood as the absence of any discernible patterns, trends, or correlations in the residuals. The residuals should be scattered around the regression line with no systematic deviations from zero. When the residuals exhibit randomness, it indicates that the model has captured all the available information, and any unexplained variability is due to random chance, making the model more reliable.

Having random residuals is essential for several reasons:

1. If the residuals show patterns or trends, the model is not capturing all the essential underlying relationships in the data. Non-random residuals suggest that the model is missing some critical explanatory variables or that the relationship between the variables is not linear.

2. Random residuals are a vital assumption in linear regression that enables valid statistical inference. If the residuals are not random, then the estimates of the coefficients, standard errors, and hypothesis tests can be biased or invalid.

3. A regression model with random residuals is more likely to provide accurate predictions for new data, as it does not make systematic errors in predicting the dependent variable.

4. Checking for randomness of residuals is an essential diagnostic tool in regression analysis. By examining the residuals for patterns or trends, analysts can identify potential problems with the model and make necessary adjustments.

In summary, ensuring that the residuals in a linear regression model are random is crucial for its accuracy, reliability, and validity. It allows analysts to make sound inferences, create robust predictions, and ensure that the model appropriately captures the relationships within the data.

Homoscedasticity

The concept of random residuals in linear regression is related to the idea of homoscedasticity. Homoscedasticity refers to the assumption that the variance of the residuals is constant across all levels of the independent variables. In other words, there should be no systematic patterns in the residuals' variability as the independent variables' values change.

The residuals are considered homoscedastic when they exhibit constant variance and show no patterns or trends. This means that the variability in the residuals is random and does not depend on the values of the independent variables, which is a desirable property in regression analysis.

Having homoscedastic residuals is essential for the validity and reliability of the regression model. If the residuals display a heteroscedasticity pattern (where the residuals' variance is not constant), it can lead to biased parameter estimates, incorrect standard errors, and invalid hypothesis tests. Consequently, ensuring that the residuals are homoscedastic is crucial for making accurate inferences and predictions using the regression model.

Testing the Randomness of Residuals

Analysts can create a scatter plot with the residuals on the y-axis and the predicted values from the regression model on the x-axis to test the randomness of residuals in linear regression. Each point on the plot represents a data point in the dataset, with the x-coordinate being the predicted value and the y-coordinate being the corresponding residual.

Analysts can visually inspect whether the residuals exhibit any systematic pattern or trend by examining this scatter plot. A random scattering of points around the horizontal line at y=0 would indicate that the assumption is likely satisfied in the regression model. Conversely, if a clear pattern or structure is observed in the plot, it suggests a violation of the assumption.

Analysis of the Plot

The random scatter of the residuals from the regression model around the horizontal axis (predicted prices) suggests that the errors are independent and identically distributed (homoscedastic), meeting one of the assumptions of linear regression.

An R-squared of 0.000 further strengthens this point, indicating that there is no correlation between the residuals and the predicted values. This is ideal, as it suggests that the model has captured all the linear relationships in the data, and the residuals are just random noise. The fact that the residuals are scattered around zero indicates that there is no systematic bias in the model. In other words, the model is not consistently over- or underestimating the actual home values.

The residual plot suggests that the model is on the right track. However, it is important to remember that this is just one step in the model development process. To assess its generalizability, it is important to evaluate the model's performance on a hold-out set (data that was not used to build the model or train the ML model).

Attention Mass Appraisal Analysts: A few outliers exist, especially at the higher end of the predicted prices. Therefore, while the residual plot passes the test, it would be prudent to have the in-house data team investigate these outliers and document their findings. The findings should be included in the model’s final documentation as an appendix.

Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a crucial statistical concept that explains large samples' average (mean) behavior. This regression analysis was conducted using a large sample of 11,224 sales. Due to the large sample size, CLT ensures that even with inherent variability in real estate prices, the residuals will tend toward a normal distribution, indicating randomness and implying that deviations from the predicted prices (residuals) occur by chance and are not due to any systematic pattern or bias in the model. The minor deviations often observed in residuals (e.g., skewness and kurtosis) are likely because real-world data is not perfectly normal. However, with such a large modeling dataset, CLT suggests that these deviations are likely insignificant, and the residuals can be considered random.

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