Chapter 3 of 5

The OLS Solution

Hand calculation, step by step

OLS finds the one line with the smallest sum of squared error. Let's calculate it by hand.

Step 1: Find the averages. x̄ (mean salary) and ȳ (mean value).

x̄ (mean salary) 0.0 M€

ȳ (mean market value) 0.0 M€

Step 2: How far is each point from the average? These are deviations.

Step 3: Multiply the deviations together. Sum them up, then divide by n − 1 — that's the covariance.

\text{Cov}(x, y) = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{n - 1}

#	x_i - x̄	×	y_i - ȳ	=	product
1	-4.21	×	-13.6	=	57.2
2	-3.21	×	-11.6	=	37.2
3	-2.61	×	-3.6	=	9.4
4	-1.56	×	-9.6	=	15.0
5	-1.14	×	-1.6	=	1.8
6	-0.41	×	3.4	=	-1.4
7	-0.01	×	0.4	=	-0.0
8	0.19	×	-6.6	=	-1.3
9	0.49	×	0.4	=	0.2
10	0.99	×	-1.6	=	-1.6
11	1.39	×	5.4	=	7.5
12	1.69	×	8.4	=	14.2
13	2.29	×	6.4	=	14.7
14	2.69	×	13.4	=	36.1
15	3.39	×	10.4	=	35.3
Sum / (n - 1) =					16.0

Cov(x, y) 0.0

Step 4: Square the x-deviations and average them. That's the variance of x.

\text{Var}(x) = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}

#	x_i - x̄	(x_i - x̄)²
1	-4.21	17.71
2	-3.21	10.29
3	-2.61	6.80
4	-1.56	2.43
5	-1.14	1.30
6	-0.41	0.17
7	-0.01	0.00
8	0.19	0.04
9	0.49	0.24
10	0.99	0.98
11	1.39	1.94
12	1.69	2.86
13	2.29	5.25
14	2.69	7.25
15	3.39	11.51
Sum / (n - 1) =		4.9

Var(x) 0.0

Step 5: Divide covariance by variance. That's β₁, the slope.

\beta_1 = \frac{\text{Cov}(x, y)}{\text{Var}(x)}

Cov(x,y) 16.0

Var(x) 4.9

β₁ (slope) 0.00

For every additional €1M in annual salary, the predicted market value increases by €3.3M.

Step 6: Calculating the intercept. β₀ = ȳ − β₁ · x̄

\beta_0 = \bar{y} - \beta_1 \cdot \bar{x}

ȳ (mean market value) 31.6 from Step 1

−

β₁ (slope) 3.26 from Step 5

x̄ (mean salary) 7.8 from Step 1

β₀ (intercept) 0.0 M€

A player with €0M salary would have a predicted market value of €6.1M. (It's the baseline, not a real prediction.)

The intuition behind this formula is: we need to select an intercept, so that when we plug in the mean value of X in the model, we would get the mean value of Y.