Slopes of y-on-x vs x-on-y Regression: Relationship and When They Are Equal

You run two simple ordinary least squares (OLS) regressions, each with an intercept, on the same paired observations $(x_i, y_i)$ for $i = 1, \dots, n$:

1. Fit $y$ on $x$: $y = \alpha_1 + \beta_{y|x}\, x + \varepsilon$
2. Fit $x$ on $y$: $x = \alpha_2 + \beta_{x|y}\, y + \eta$ The interviewer asks you to reason through how the two fitted slopes relate to each other, and what it takes for them to be equal.

Constraints & Assumptions

• Both fits are plain OLS with an intercept and a single predictor.
• The data are non-degenerate: $n \ge 3$, and the sample variances of $x$ and $y$ are strictly positive.
• "Coefficients" means the estimated slopes $\hat\beta_{y|x}$ and $\hat\beta_{x|y}$.
• When asked how to make the slopes equal, you may transform the data (for example, rescale variables).

Clarifying Questions to Ask

• Are both fits ordinary least squares, or is a symmetric method such as orthogonal / total least squares intended?
• Should I treat the data as raw, or may I assume it has already been centered or standardized?
• Do the intercepts matter as well, or only the slopes?
• When you ask how to make the coefficients equal, can I transform the variables, or must a condition hold on the raw data?

Part 1

Derive both slope estimates using sample moments. How are $\hat\beta_{y|x}$ and $\hat\beta_{x|y}$ related? Specifically, what is their product, and why is $\hat\beta_{x|y}$ generally not equal to $1/\hat\beta_{y|x}$, even though the second regression looks like the first one "solved for $x$"?

Hint — Write both slopes in terms of sample moments: In simple OLS, the slope equals the covariance divided by the variance of the predictor: $\hat\beta_{y|x} = \widehat{\mathrm{Cov}}(x,y)/\widehat{\mathrm{Var}}(x)$. Write the corresponding expression for the reverse regression and compare the two.

Hint — Bring in the correlation: Write each slope as the sample correlation $r$ multiplied by a ratio of standard deviations. Multiplying the two forms reveals the relationship — and ties it to a familiar goodness-of-fit quantity.

What This Part Should Cover

Part 2

Under what data condition are the two slopes exactly equal? How would you transform the data so that the two regressions are guaranteed to yield the same slope, and what is the value of that common slope?

Hint — Set the two expressions equal: Using the moment forms from Part 1, set them equal and see what must hold for the sample variances of $x$ and $y$ (watch for the degenerate zero-covariance case).

Hint — A standard preprocessing step works: Consider what happens to both slopes if you z-score each variable. What do the standard-deviation ratios become, and what quantity does each slope then equal?

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

• Suppose both $x$ and $y$ are noisy measurements of the same underlying quantity. Is either OLS slope an unbiased estimate of the true relationship? What symmetric alternatives exist, and what assumptions do they make?
• In the $(x, y)$ plane, how do the two fitted lines (not just slopes) compare geometrically? When do they coincide?
• What happens to each slope if you add independent noise to $x$ only, leaving $y$ unchanged?

Overview: This question tests understanding of simple OLS linear regression estimators, moment-based slope expressions, the relationship between y-on-x and x-on-y fitted slopes, and how correlation and variable scaling affect coefficient values.