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Proof of Sum of Squares TSS = ESS + RSS in Econometrics Mathematical Explanation



TSS (total sum of squares) is equal to ESS (explained sum of squares) plus RSS (residual sum of squares), we need to start with the definitions of these terms and then use some algebraic manipulations to arrive at the desired result.


Let us begin by defining the three terms:


TSS = ∑(Yi - Ȳ)², where Yi is the actual value of the response variable for observation i, and Ȳ is the mean of the response variable.


ESS = ∑(Ŷi - Ȳ)², where Ŷi is the predicted value of the response variable for observation i.


RSS = ∑(Yi - Ŷi)², which is the sum of squared differences between the actual and predicted values of the response variable.


Now, we can expand the terms in the definition of TSS using the definition of Ŷi:


TSS = ∑(Yi - Ȳ)² = ∑[(Yi - Ŷi) + (Ŷi - Ȳ)]² = ∑(Yi - Ŷi)² + ∑(Ŷi - Ȳ)² + 2∑(Yi - Ŷi)(Ŷi - Ȳ)


Next, we can use the definition of RSS to simplify the first term on the right-hand side:


∑(Yi - Ŷi)² = RSS


Similarly, we can use the definition of ESS to simplify the second term:


∑(Ŷi - Ȳ)² = ESS


Now, we need to simplify the third term using some algebraic manipulations. We can start by expanding the product:


2∑(Yi - Ŷi)(Ŷi - Ȳ) = 2∑(YiŶi - YiȲ - ŶiȲ + Ŷi²)


Then, we can use the fact that the sum of the residuals (Yi - Ŷi) is zero, which can be shown as follows:


∑(Yi - Ŷi) = ∑Yi - ∑Ŷi = nȲ - nȲ = 0


Using this fact, we can simplify the third term:


2∑(Yi - Ŷi)(Ŷi - Ȳ) = 2∑(YiŶi - ŶiȲ) = 2(∑YiŶi - ∑ŶiȲ) = 2(∑ŶiYi - nȲ²) = 2(ESS - n(Ȳ - Ŷ)²)


where Ŷ is the sample mean of the predicted values, which is equal to Ȳ.


Substituting these results back into the equation for TSS, we get:


TSS = RSS + ESS + 2(ESS - n(Ȳ - Ŷ)²) = RSS + 2ESS - 2n(Ȳ - Ŷ)² = RSS + 2ESS - 2n(Ȳ - Ȳ)² = RSS + 2ESS - 0 = RSS + ESS


Therefore, we have shown that TSS is equal to ESS plus RSS.


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