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1. Introduction: Three Concepts, One Family

Students of econometrics typically encounter these three concepts at three different points in time and from three different directions:

When studying statistics, they first learn the correlation coefficient r—which measures the strength of the linear association between two variables and takes values between -1 and 1.
Next, they learn simple linear regression and encounter the slope coefficient β̂₁—how much Y changes on average when X changes by one unit.
Later, in multiple regression, the partial regression coefficient appears—the β̂₁ that holds "all else constant."

In most textbooks, these three concepts are each discussed separately, with their interrelationships buried in the crevices of formula derivations. But if you put them side by side, you will find that they share a close mathematical kinship. Moreover, once you see this relationship clearly, many previously isolated pieces of knowledge—R², standard errors, multicollinearity—will connect on their own.

Core message: The simple regression coefficient is a "scale-adjusted correlation coefficient." The partial regression coefficient in multiple regression is a "scale-adjusted partial correlation coefficient." The correlation coefficient answers "how strong is the association," while the regression coefficient answers "how much does it change"—the only difference between them is the ratio of standard deviations.

2. The Simple Regression Coefficient: A Correlation Coefficient in Disguise

2.1 Starting from the OLS Derivation

The simple linear regression model:

Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i

OLS minimizes the sum of squared residuals, yielding the expression for β̂₁:

\hat{\beta}_1 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{n} (X_i - \bar{X})^2} = \frac{\text{Cov}(X,Y)}{\text{Var}(X)}

2.2 The Key Algebraic Step—Factoring Out the Correlation Coefficient

Recall the definition of the correlation coefficient:

r_{XY} = \frac{\text{Cov}(X,Y)}{s_X \cdot s_Y}

where $s_X = \sqrt{\frac{\sum (X_i - \bar{X})^2}{n-1}}$ is the sample standard deviation of X, and $s_Y$ is defined analogously.

Now, apply an algebraic transformation to the expression for β̂₁:

\hat{\beta}_1 = \frac{\text{Cov}(X,Y)}{\text{Var}(X)} = \frac{\text{Cov}(X,Y)}{s_X \cdot s_Y} \cdot \frac{s_Y}{s_X} = r_{XY} \cdot \frac{s_Y}{s_X}

This is the precise mathematical relationship between the simple regression coefficient and the correlation coefficient:

\boxed{\hat{\beta}_1 = r_{XY} \times \frac{s_Y}{s_X}}

The regression coefficient = the correlation coefficient × (standard deviation of Y / standard deviation of X).

2.3 What Does This Formula Tell Us?

It decomposes β̂₁ into the product of two factors:

rXY: Pure correlation—the degree of "co-movement" between X and Y, dimensionless, taking values in [-1, 1].
sY / sX: Pure scale adjustment—converting the rate of variation measured in "units of X" into "how much Y changes when X changes by one unit."

Interpretation:

If rXY = 0.8, sX = 5, and sY = 10, then β̂₁ = 0.8 × (10/5) = 1.6. This means: X and Y are strongly associated (r = 0.8), and Y's range of variation is twice that of X (sY/sX = 2), so when X changes by 1 unit, Y must change by 1.6 units to "keep up" with such a strong and "larger-scale" co-movement.

An intuitive conclusion: The regression coefficient is a correlation coefficient that has been "rescaled." The scaling factor depends on the relative dispersion of X and Y.

2.4 Verifying a Special Case: Standardized Regression

If you first standardize both X and Y (subtract the mean and divide by the standard deviation) to obtain Z_X and Z_Y (with mean 0 and standard deviation 1), then:

$s_{Z_X} = 1$ , $s_{Z_Y} = 1$
Substituting into the formula: $\hat{\beta}_1^{standardized} = r_{XY} \times \frac{1}{1} = r_{XY}$

The standardized regression coefficient = the correlation coefficient. In standardized regression, the two are exactly equivalent. This is no coincidence—it is a direct consequence of the definition of the correlation coefficient.

3. Geometric Intuition—Two Lines in the Scatter Plot

There is another more intuitive way to understand the relationship between β̂₁ and rXY: by looking at the two lines in a scatter plot.

3.1 The SD Line

For all observed points, if X deviates from its mean by a distance of $k \cdot s_X$ , then Y deviates from its mean by a distance of $k \cdot s_Y$ . Connecting all such points forms a line passing through (X̄, Ȳ)—the SD line—whose slope is:

\text{Slope of SD line} = \frac{s_Y}{s_X}

This line does not care about how strongly X and Y are "actually" associated—it simply "compares" the dispersion of the two variables in a pure ratio.

3.2 The Regression Line

The slope of the OLS regression line is $\hat{\beta}_1$ , and its relationship to the SD line is obtained by multiplying by rXY:

\text{Slope of regression line} = r_{XY} \times \text{Slope of SD line}

If rXY = 1 (perfect positive correlation), the regression line = the SD line. Every change in X is matched by Y responding precisely at the magnitude of the "standard deviation ratio."

If rXY = 0 (no correlation), the regression line is horizontal—β̂₁ = 0. No matter how X changes, the predicted value of Y equals the mean of Y. The SD line may still be steeply sloped, but that is a pure "scale illusion"—in reality, changes in X do nothing to help predict Y.

If rXY = 0.5 (moderate correlation), the slope of the regression line equals half the slope of the SD line. When X changes, the predicted change in Y is only half of what would be expected "purely from the standard deviation ratio"—because X explains only part of the variation in Y.

This is the intuitive meaning of the regression line: it "compresses" the slope of the SD line back toward zero, and how much it compresses depends on how strongly X and Y are correlated.

4. Partial Regression Coefficients in Multiple Regression: Blood Relatives of Partial Correlation

Now we move to multiple regression. The model has two independent variables:

Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon

The estimated β̂₁ is called the partial regression coefficient—"the effect on Y of a one-unit change in X₁, holding X₂ constant."

4.1 Recap: The Three Steps of the FWL Theorem

From the perspective of the FWL theorem (see the previous article for details), β̂₁ is equivalent to the result of the following "stepwise regression":

Step 1: Regress X₁ on X₂ and take the residuals. Denote these residuals as $\tilde{X}_1$ —the part of X₁ that "cannot be explained by X₂."

Step 2: Regress Y on X₂ and take the residuals. Denote these residuals as $\tilde{Y}$ —the part of Y that "cannot be explained by X₂."

Step 3: Regress $\tilde{Y}$ on $\tilde{X}_1$ in a simple regression. The resulting slope is β̂₁.

Thus, β̂₁ essentially measures: "the strength of the association between the remaining parts of X₁ and Y after each has been purged of the linear influence of X₂."

4.2 The Partial Correlation Coefficient Takes the Stage

In Step 3 above—regressing the "purified" Y on the "purified" X₁—the simple correlation coefficient between these two "purified" variables is the partial correlation coefficient, typically denoted $r_{YX_1 \cdot X_2}$ .

Read as: "The partial correlation between Y and X₁, controlling for (or having removed) X₂."

The partial correlation coefficient measures: the residual "pure" correlation between Y and X₁ after removing the linear influence of X₂.

4.3 The Relationship Between Partial Regression Coefficients and Partial Correlation Coefficients

Since Step 3 of FWL is a simple regression of Y on X (using residual variables), the earlier formula "simple regression coefficient = correlation coefficient × standard deviation ratio" naturally applies:

\hat{\beta}_1 = r_{YX_1 \cdot X_2} \times \frac{s_{\tilde{Y}}}{s_{\tilde{X}_1}}

where:

$s_{\tilde{Y}}$ is the sample standard deviation of the residual variation in Y after removing the linear influence of X₂.
$s_{\tilde{X}_1}$ is the sample standard deviation of the residual variation in X₁ after removing the linear influence of X₂.

The relationship between the partial regression coefficient and the partial correlation coefficient is structurally identical to the relationship between the simple regression coefficient and the simple correlation coefficient. The only difference is that both the partial correlation and the partial regression coefficient are measured using residual variation after "purifying" the effect of the control variables.

5. A Table Unifying All Three Relationships

Concept	Mathematical Definition	Relationship to the Correlation Coefficient	Range of Values
Correlation coefficient $r_{XY}$	$\frac{\text{Cov}(X,Y)}{s_X s_Y}$	It is itself the correlation coefficient	[-1, 1]
Simple regression coefficient $\hat{\beta}_1$	$\frac{\text{Cov}(X,Y)}{\text{Var}(X)}$	$r_{XY} \times \frac{s_Y}{s_X}$	Unbounded—depends on the relative scales of Y and X
Partial correlation coefficient $r_{YX_1 \cdot X_2}$	$\text{Corr}(\tilde{Y}, \tilde{X}_1)$ , where $\tilde{Y}, \tilde{X}_1$ are residuals after removing the influence of X₂	It is itself the (partial) correlation coefficient	[-1, 1]
Partial regression coefficient $\hat{\beta}_1$ (multiple)	Derived from the FWL theorem	$r_{YX_1 \cdot X_2} \times \frac{s_{\tilde{Y}}}{s_{\tilde{X}_1}}$	Unbounded

6. What Phenomena Can These Two Relationships Help You Understand Directly?

6.1 R² in Simple Regression Is the Square of the Correlation Coefficient

R² in simple regression is defined as "the proportion of the variation in Y explained by X." Starting from the relationship β̂₁ = rXY × (sY / sX) above, we can derive:

R^2 = r_{XY}^2

In simple regression, R² is simply the square of the simple correlation coefficient. This means—if the correlation between X and Y is 0.6, then no matter what software you use to run the regression, R² will always be 0.36.

6.2 When Does β̂₁ = 0?—Two Completely Different Reasons

From β̂₁ = rXY × (sY / sX), we can see that β̂₁ = 0 can occur in only two cases:

rXY = 0: X and Y are completely uncorrelated. OLS correctly detects this fact.
sY = 0: Y has no variation at all—all observed values of Y are equal. In this case, you do not need a regression to analyze "changes in Y" either.

In any other case—regardless of how large sX is or how small sY is—as long as rXY ≠ 0, β̂₁ will not be zero.

6.3 Why Can the Sign of β̂₁ in Multiple Regression Differ from That of r?

Consider an example:

Y = wage, X₁ = years of education, X₂ = years of work experience.
In a simple regression, education and wages are positively correlated—r > 0.
But after adding work experience, the partial regression coefficient may shrink or even reverse sign (in extreme cases)—why?

From the perspective of the partial correlation formula: the sign of the partial regression coefficient = the sign of the partial correlation coefficient. If, within the subgroup of "people with the same work experience," education and wages are actually negatively partially correlated (a more extreme Simpson's paradox scenario), then the partial regression coefficient will be negative—even though, overall, the simple correlation between education and wages is positive.

This is the core magic of "holding all else constant"—it allows you to see the correlation "after controlling for X₂," rather than the raw correlation of "the full sample mixed together."

7. Summary

Three layers of relationships to help you connect these three concepts in one go:

Simple regression coefficient = correlation coefficient × standard deviation ratio. After standardizing the variables, the regression coefficient = the correlation coefficient. The two are not independent concepts; they are the same strength of association expressed on the "original scale" versus the "standardized scale."
Partial regression coefficient in multiple regression = partial correlation coefficient × residual standard deviation ratio. This structure is identical to that of simple regression. This is because the FWL theorem tells us that β̂₁ in multiple regression is essentially a simple regression between "purified" variables.
The correlation coefficient answers "how strong," while the regression coefficient answers "how much." The former removes scale; the latter retains it. In empirical research, both are worth reporting—the correlation coefficient allows you to compare the strength of associations across different models, while the regression coefficient allows you to interpret the real-world meaning directly.

One sentence to close:

"Every time you run a regression and obtain a coefficient, you are actually saying two things: one about 'correlation' (sign and significance), and one about 'magnitude of change' (the size of the coefficient). Between these two things there is only one wall—the ratio of standard deviations. Tear down that wall, and the regression coefficient and the correlation coefficient are one and the same."

8. Presentation Suggestions for Bilibili/WeChat

Bilibili video: The core animation is suggested in three steps—① Display the SD line and the regression line on a scatter plot, with the red SD line at a fixed angle and the blue regression line "collapsing" toward the horizontal as r changes; ② Show the "black-box purification" process of multiple regression—after both variables have the influence of X₂ removed, plot the remaining residuals in a scatter plot; ③ Display the three formulas side by side, using connecting lines to emphasize the structural consistency among them.
WeChat article: For the derivation of β̂₁ = r × (sY/sX) in simple regression, use a step-by-step formula box. For the connection between partial correlation and FWL, use a flowchart to represent the three steps of "purify → regress again."
Recommended titles:
- Main title: 《What Exactly Is the Relationship Between Regression Coefficients and Correlation Coefficients?》
- Alternative title: 《The Regression Coefficient Is Just a Correlation Coefficient with a Different Ruler—and the Partial Regression Coefficient Is No Different》
Key quotes:

"The correlation coefficient answers 'how strong,' and the regression coefficient answers 'how much.' Between the two lies only one wall—the ratio of standard deviations."