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Sum of Squared Residuals: Looking at the Model’s Mistakes
Whenever a regression model is estimated, it produces a predicted value for each observation. But predictions are rarely exact. The predicted value and the actual observed value are usually a little different.
That difference is called a residual.
A residual simply shows how far the model’s estimate is from what actually happened. Some residuals are small, some are larger. Some are positive, others negative.
Looking at one residual by itself does not reveal much. What matters is the pattern across the entire dataset. This is why analysts calculate the Sum of Squared Residuals, often abbreviated as SSR.
Why Residuals Are Squared
Residuals can be both positive and negative. If we added them together directly, positive values could offset negative ones. The total might look small even when individual errors are large.
To avoid this problem, each residual is squared before being added. Squaring removes the negative sign and ensures that larger deviations from the model receive greater weight.
Once all the squared residuals are added together, we obtain the Sum of Squared Residuals.
What the Measure Shows
The Sum of Squared Residuals captures how far the regression model’s predictions are from the actual observations.
If this value is large, the regression line is missing the data points by a wider margin. That usually suggests the model is not explaining the relationship particularly well.
If the value is smaller, the predicted values sit closer to the observed values. The model is doing a better job of describing the pattern in the data.
The Link with Ordinary Least Squares
Most regression models used in finance rely on the ordinary least squares (OLS) method.
OLS works by choosing the regression coefficients that make the Sum of Squared Residuals as small as possible. In simple terms, it searches for the line that keeps the total squared prediction errors to a minimum.
This idea is central to how regression models are estimated.
A Broader Perspective
In regression analysis, the total variation in the dependent variable can be divided into two parts. One part is explained by the regression model, and the other part remains unexplained.
The Sum of Squared Residuals represents this unexplained portion of the variation.
Because of this role, it is closely related to other statistics used in regression, including R² and the standard error of the regression.
Final Thought
The Sum of Squared Residuals is essentially a measure of the model’s prediction mistakes. By squaring and adding the residuals, analysts obtain a single number that reflects how far the model’s estimates are from the observed data. Regression techniques such as ordinary least squares aim to minimise this quantity so that the fitted model follows the data as closely as possible.
