3. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
R² is called coefficient of determination
This gives the contribution made by regression in explaining the variations in dependent variable
This is worked out as a ratio between the regression sum of square of squares and total sum of
square.
TSS = ESS + RSS
We can show the goodness of fit of a regression line through the graph (Figure 1) and from that we can
calculate the value of r2.
Basic Econometrics
4. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
Figure 1 Goodness of fit of estimated regression line
There are two lines in Figure 1, a horizontal line placed at
the average response, , and a shallow-sloped estimated regression line, . From Figure 1, the calculation of
sum of squares are;
Basic Econometrics
5. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
There are two lines in Figure 1, a horizontal line placed at
^
the average response,y and a shallow-sloped estimated regression line,y . From Figure 1, the calculation of
sum of squares are;
Explained Sum of Squares (ESS) quantifies how far the estimated sloped regression line, , is from the horizontal
"no relationship line," the sample
mean ESS =
Residual sum of Squares (RSS) quantifies how much the data points, yi, vary around the estimated regression
line,
Basic Econometrics
6. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
Total Sum of Squares (TSS) quantifies how much the data points, yi, vary around their mean,
Basic Econometrics
7. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
Basic Econometrics
TSS =Total Sum of Squares
ESS =
Explained Sum of Squares
RSS = u^2
I = Residual Sum of
Squares
ESS RSS
1 = -------- + -------- ; or
TSS TSS
RSS RSS
1 = r2 + ------- ; or r2 = 1 - -------
TSS TSS
8. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
Basic Econometrics
r²= 𝐸𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
So the above can be written as
𝑅2 = 1- 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 − 𝑈𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
or
1- ∑ 𝑒𝟐
∑(𝑌 − ^𝑌)2
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
9. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
Basic Econometrics
PROPERTIES OF r²
r2 = ESS/TSS
is coefficient of determination, it measures the proportion or
percentage of the total variation in Y explained by the regression
Model
0 r2 1;
r = r2 is sample correlation coefficient
Some properties of r
10. BASIC ECONOMETRICS
Simple Linear Regression
COEFFICIENT OF DETERMINATION-GOODNESS OF FIT
FORMULA
r² =
1- 𝑵 ∑ 𝑒𝟐
𝑁 ∑ 𝑌2 − (∑ 𝑌)2
Basic Econometrics
