Do two things really move together?
Real estate analysts constantly ask questions shaped like 'does X move with Y?' Does a submarket's cap rate move with the 10-year Treasury yield? Does a unit's monthly rent move with its square footage? Does vacancy move with new supply? Correlation is the statistical tool that answers the first half of that question — it measures the strength and direction of the straight-line (linear) relationship between two numeric variables and expresses it as a single number called the correlation coefficient, written r, which always falls between −1 and +1. Regression answers a related but different question: given that relationship, can you fit a line through the data that lets you estimate or predict one variable from the other? Together, correlation and regression turn a scatter of dots on a chart into two things a beginner can actually use: a number describing how tightly two variables move together, and an equation for estimating one from the other.
Reading the correlation coefficient
Picture a scatter plot — a chart with one variable on the horizontal (X) axis and the other on the vertical (Y) axis, with one dot per data point (per property, per lease, per month). If the dots trend upward from lower-left to upper-right, that's a positive correlation — as X increases, Y tends to increase too. If the dots trend downward from upper-left to lower-right, that's a negative correlation — as X increases, Y tends to decrease. If the dots are scattered with no visible pattern, correlation is near zero.
The correlation coefficient r quantifies this: r = +1 means a perfect positive linear relationship (every dot falls exactly on an upward-sloping line); r = −1 means a perfect negative linear relationship; r = 0 means no linear relationship at all. In practice you rarely see exactly +1, −1, or 0 — real-world data lands somewhere in between, and the closer |r| (the absolute value of r, ignoring its sign) is to 1, the stronger the linear relationship.
Worked example 1. Four comparable office suites have the following (size index, rent index) pairs, both measured on a simplified, unitless index scale for ease of calculation: (1, 2), (2, 3), (3, 5), (4, 6).
To compute r by hand, first calculate five sums: - ΣX (sum of X values) = 1 + 2 + 3 + 4 = 10 - ΣY (sum of Y values) = 2 + 3 + 5 + 6 = 16 - ΣXY (sum of each X times its paired Y) = (1×2) + (2×3) + (3×5) + (4×6) = 2 + 6 + 15 + 24 = 47 - ΣX² (sum of each X squared) = 1 + 4 + 9 + 16 = 30 - ΣY² (sum of each Y squared) = 4 + 9 + 25 + 36 = 74
With n = 4 pairs, plug these into the correlation formula:
r = [n·ΣXY − ΣX·ΣY] ÷ √{[n·ΣX² − (ΣX)²] × [n·ΣY² − (ΣY)²]} = [4(47) − (10)(16)] ÷ √{[4(30) − 100] × [4(74) − 256]} = [188 − 160] ÷ √{[20] × [40]} = 28 ÷ √800 = 28 ÷ 28.28 ≈ 0.99
An r of 0.99 is an extremely strong positive correlation — in this small sample, suite size and rent move together almost perfectly.
Pearson Correlation Coefficient
r = [n·ΣXY − ΣX·ΣY] ÷ √{[n·ΣX² − (ΣX)²][n·ΣY² − (ΣY)²]}
- r
- — The correlation coefficient, always between −1 and +1
- n
- — The number of paired data points
- ΣXY
- — The sum of each X value multiplied by its paired Y value
- ΣX
- — The sum of all X values
- ΣY
- — The sum of all Y values
- ΣX²
- — The sum of the squares of the X values
- ΣY²
- — The sum of the squares of the Y values
This formula compares how X and Y move together (the ΣXY term) against how much each varies on its own (the ΣX² and ΣY² terms). Plugging in the sums from Worked Example 1 produces r ≈ 0.99, indicating an almost perfectly straight-line positive relationship between suite size and rent in that sample.
Fitting a line: simple linear regression
Correlation tells you how strong a relationship is; regression goes a step further and fits an actual straight line through the data so you can estimate Y for any given X. The line takes the form Y = a + bX, where b is the slope (how much Y changes for every one-unit increase in X) and a is the intercept (the predicted value of Y when X = 0). The variable being predicted, Y, is called the dependent variable; the variable used to make the prediction, X, is called the independent variable.
Worked example 2. Using the same four (size index, rent index) pairs from Worked Example 1 — and the same sums (ΣX = 10, ΣY = 16, ΣXY = 47, ΣX² = 30, n = 4) — the slope and intercept are:
- Slope: b = [n·ΣXY − ΣX·ΣY] ÷ [n·ΣX² − (ΣX)²] = [4(47) − (10)(16)] ÷ [4(30) − 100] = [188 − 160] ÷ [120 − 100] = 28 ÷ 20 = 1.4 - Intercept: first find the means: X̄ = ΣX ÷ n = 10 ÷ 4 = 2.5, and Ȳ = ΣY ÷ n = 16 ÷ 4 = 4. Then: a = Ȳ − b·X̄ = 4 − (1.4 × 2.5) = 4 − 3.5 = 0.5
The fitted regression equation is: Rent Index = 0.5 + 1.4 × Size Index.
To predict the rent index for a new suite with a size index of 5 (a value not in the original data), substitute X = 5: Rent Index = 0.5 + 1.4(5) = 0.5 + 7.0 = 7.5. This is the model's best single estimate, based purely on the linear pattern in the four original suites.
Simple Linear Regression
b = [n·ΣXY − ΣX·ΣY] ÷ [n·ΣX² − (ΣX)²] ; a = Ȳ − b·X̄ ; Ŷ = a + bX
- b
- — Slope of the regression line — the change in Y for each one-unit increase in X
- a
- — Intercept of the regression line — the predicted value of Y when X = 0
- Ŷ
- — The predicted (estimated) value of Y for a given X ('Y-hat')
- X̄
- — The mean of the X values
- Ȳ
- — The mean of the Y values
The slope tells you how much the predicted Y increases for each one-unit increase in X; the intercept anchors the line. In Worked Example 2, the fitted line Rent Index = 0.5 + 1.4 × Size Index predicts a rent index of 7.5 for a size index of 5.
How good is the fit? R² and the causation trap
A regression line can always be drawn through data, even data with almost no real relationship, so you need a way to measure how well the line actually fits. That's the job of R² (R-squared, the coefficient of determination), which tells you what percentage of the variation in Y is 'explained' by its linear relationship with X. R² ranges from 0 (the line explains none of the variation) to 1 (the line explains all of it); for a simple linear regression, R² is simply r² — the correlation coefficient, squared.
Worked example 3. Using the regression equation from Worked Example 2 (Ŷ = 0.5 + 1.4X), compute the predicted Ŷ for each of the four original suites, then the residual (the difference between the actual Y and the predicted Ŷ):
X = 1: Ŷ = 0.5 + 1.4(1) = 1.9; residual = 2 − 1.9 = 0.1; residual² = 0.01 X = 2: Ŷ = 0.5 + 1.4(2) = 3.3; residual = 3 − 3.3 = −0.3; residual² = 0.09 X = 3: Ŷ = 0.5 + 1.4(3) = 4.7; residual = 5 − 4.7 = 0.3; residual² = 0.09 X = 4: Ŷ = 0.5 + 1.4(4) = 6.1; residual = 6 − 6.1 = −0.1; residual² = 0.01
Summing the residual² column gives the sum of squared errors: SSE = 0.01 + 0.09 + 0.09 + 0.01 = 0.20.
Next, compute the total sum of squares (SST), which measures how much Y varies around its own mean (Ȳ = 4), ignoring X entirely: (2−4)² + (3−4)² + (5−4)² + (6−4)² = 4 + 1 + 1 + 4 = 10.
Finally: R² = 1 − (SSE ÷ SST) = 1 − (0.20 ÷ 10) = 1 − 0.02 = 0.98.
This matches r² from Worked Example 1 (0.99² ≈ 0.98), confirming the calculation two independent ways: about 98% of the variation in rent index across these four suites is explained by suite size, leaving only about 2% explained by everything else.
A high R² is not permission to assume causation, though. Correlation (and regression) never prove that one variable causes the other to change — a relationship can exist because X causes Y, because Y causes X, because a third factor drives both, or purely by coincidence. A classic non-real-estate example: monthly ice cream sales and monthly drowning incidents are strongly positively correlated, but ice cream doesn't cause drowning — both rise together in the summer heat, a third factor (hot weather, and the resulting jump in swimming) drives both.
Rule-of-thumb strength of a correlation coefficient (based on |r|)
| |r| range | Typical interpretation |
|---|---|
| 0.90 – 1.00 | Very strong relationship |
| 0.70 – 0.89 | Strong relationship |
| 0.40 – 0.69 | Moderate relationship |
| 0.20 – 0.39 | Weak relationship |
| 0.00 – 0.19 | Negligible / no meaningful linear relationship |
Don't extrapolate, and don't confuse a good fit with a guaranteed future
A regression line is only reliable within (or close to) the range of X values used to build it. **Extrapolation** — stretching the line far beyond that range — is dangerous because the real relationship may bend, flatten, or break down entirely outside the data you actually observed. For example, a regression built from comps within 0.5 to 5 miles of a highway on-ramp says nothing reliable about a property 40 miles away; the model was never shown data out there. Likewise, a high R² describes how well the line fit the historical data used to build it — it is not a guarantee about future data, and it says nothing about whether the relationship will hold going forward. Treat regression output as a well-informed estimate with real uncertainty around it, not a promised outcome.
Module Check
Four comparable office suites have the following (size index, rent index) pairs, both on a simplified unitless scale: (1,2), (2,3), (3,5), (4,6). Using the Pearson correlation formula, what is the correlation coefficient r between size and rent, rounded to two decimals?