Please take note, before we begin, the term variation and variance are different. Variation
means the sum of squares of the deviations of a variable from its mean value. Variance is this
sum of squares divided by the appropriate degrees of freedom. In short, variance =
variation/df
Variance of the estimated residuals and standard error:
𝜎̂ 2 =
̂𝑖2
𝑅𝑆𝑆 𝑜𝑟 ∑ 𝜎
𝑛−2

Variance of an equation is, equal to the residual sum of squares, divided by the
degree of freedom, number of observations, minus 2.
𝜎̂ = √

𝑅𝑆𝑆
𝑁−2
Standard error of an equation, you simply square root the answer for variance.
̂ 𝟐)
Variance and standard error for (𝜷
2
̂
𝜎
Var(𝛽̂2 ) = ∑ 𝑥 2
𝑖

Variance divided by the summation of “small x”
2
̂
𝜎
se(𝛽̂2 ) = √∑ 𝑥 2
𝑖

Standard error of (𝛽̂2 ), is the square root of variance.
̂ 𝟏)
Variance and standard error for (𝜷
2
∑𝑋
Var(𝛽̂1 ) = 𝑛 ∑ 𝑥𝑖 2
𝑖

you divide the summation of the big 𝑋 2 by the number observations × the
summation of the small 𝑥 2 .
2
∑𝑋
se(𝛽̂1 ) = √𝑛 ∑ 𝑥𝑖 2
𝑖

Standard error of (𝛽̂1 ), is the square root of variance.
RSS or Residual error (NB, the word error here should not be associated with its intrinsic
meaning, mistake)

Depicted as ∑ 𝑢̂𝑖2 , residual or unexplained variation of the Y values about the
regression line.

It is the vertical difference between the actual value of Y and the predicted value of Y.

(𝑌𝑖 − 𝑌̂𝑖 )2

The divination of the Y value.

How the predicated Y value, differs from the mean value of Y.

(𝑌̂𝑖 − 𝑌̅𝑖 )2

The difference between the actual value of Y and the mean value of Y

(𝑌𝑖 − 𝑌̅𝑖 )2
ESS
TSS
Note: the smaller the RSS is relative to the TSS, the better the estimated regression line fits the data.
OLS is the estimating technique that minimizes the RSS and therefore maximizes the ESS for a given
TSS.
- the “squared” variations of Y around its mean as a measure of the amount of variation to be
explained by the regression.
- For Ordinary Least Squares, the total sum of squares has two components, variation that can be
explained by the regression and variation that cannot.
At Eben, please enlighten us which squared divination is explained by the regression line, and which
is not.