1. Simple Linear Regression – Matrix Approach
Recall our example:
Hypothesize the relationship,
Y = α + βx + ε
and calculate the estimate,
Attendance, x
Amount Bet ($000), Y
117
2.07
128
2.80
122
3.14
119
2.26
131
3.40
135
3.89
125
2.93
120
2.66
130
3.33
127
3.54

2. Matrix Form of the Equation
Define the matrices:
Note: the column of 1s in the X matrix is referred to as a “dummy variable” used to allow us to calculate a in the regression equation.

3. General Matrix Form
We obtain the least squares estimates (a, b) of (α, β) by solving the matrix equation:
for b, or

4. Recall Basic Matrix Operations

5. For Our Example, Step 1: XTX
(note: n=10, Σx = 1254, Σx2 = 157,558)
XTX= ,558

6. For Our Example, Step 2: XTY 30.02
XTY= (note: n=10, Σy = 30.02, Σxy = 3791)

7. Inverting Matrices Calculate the determinant.
2 x 2 matrix (M) , D = ad – bc
3 x 3 matrix (Q), Z = a(ek – fh) – b(dk – fg) + c(dh – eg)
Calculate the elements of the matrix
2 x 2 matrix (M) ,

8. Inverting Matrices (cont.)
(cont.) Calculate the elements
3 x 3 matrix (Q),

9. For Our Example
Step 3: (XTX)-1
D = _______________

10. Our Example
Step 4: Determine b
= ________________________

11. Your Turn (In-Class / Homework)