Chapter 3 of 5

The OLS Solution

Hand calculation, step by step

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

Step 1: Find the averages. x̄ (mean size) and ȳ (mean price).

x̄ (mean size) 0 sqft

ȳ (mean price) $0

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	-246	×	-77,183	=	18,961,323
2	-226	×	-142,551	=	32,169,039
3	-217	×	-56,882	=	12,324,462
4	-144	×	-68,300	=	9,812,452
5	-122	×	-47,959	=	5,835,028
6	-88	×	-8,850	=	775,862
7	-84	×	-14,406	=	1,205,313
8	35	×	46,272	=	1,634,939
9	52	×	5,486	=	287,094
10	110	×	19,034	=	2,100,070
11	113	×	64,752	=	7,338,545
12	154	×	15,569	=	2,402,795
13	199	×	91,526	=	18,244,156
14	222	×	115,675	=	25,718,379
15	237	×	57,819	=	13,722,344
Sum / (n - 1) =					10,895,128.7

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}

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) 10,895,128.7

Var(x) 29,047.7

β₁ (slope) 0.00

For every additional square foot, the predicted price increases by $375.

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

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

ȳ (mean price) 174,664 from Step 1

−

β₁ (slope) 375.08 from Step 5

x̄ (mean size) 348 from Step 1

β₀ (intercept) $0

A hypothetical 0 sqft house would cost $44,262. (It's the baseline, not a real prediction.)