Read the estimates from an ordinary least squares (OLS) regression of a dependent variable on one explanatory variable and say what they mean.

FRM® Part I · Quantitative Analysis · Linear Regression · FRM1.QA.7.b

The idea

In a regression of Y on one explanatory variable X, OLS picks the intercept and slope that minimise the sum of squared residuals. The slope estimate is the covariance of X and Y divided by the variance of X: Slope = Cov(X,Y) ÷ Var(X). It is the average change in Y for a one-unit change in X, measured in units of Y per unit of X. The intercept follows from the sample means, as the mean of Y minus the slope times the mean of X: Intercept = mean(Y) − Slope × mean(X). It is the fitted value of Y when X is zero, and it means little if zero lies far outside the range of the data. The fitted line always passes through the point of sample means, and with an intercept in the model the residuals sum to zero. A fitted value is the intercept plus the slope times X. The usual mistake is dividing the covariance by the variance of Y instead of X, or ignoring the units of X when reading the slope.

Formulas

OLS slope with one explanatory variable

b = Cov(X,Y) ÷ Var(X)

Cov(X,Y): the sample covariance of X and Y
Var(X): the sample variance of the explanatory variable X

Single regressor with an intercept. The trap is dividing by Var(Y), which gives a different number; dividing by both standard deviations gives the correlation.

OLS intercept with one explanatory variable

a = Ȳ − b × X̄

Ȳ: the sample mean of Y
X̄: the sample mean of X
b: the OLS slope

Follows because the fitted line passes through the point of means. It is the fitted Y at X = 0, which means little when zero lies outside the data.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the fitted regression Y = a + b X, the OLS intercept a is best described as:

the expected change in Y for a one-unit change in X, measured in units of Y per unit of XWhy not: That is the slope, not the intercept.
the correlation between X and YWhy not: The correlation is unit-free and bounded by -1 and +1; the intercept is in units of Y.
the fitted value of Y at X = 0, which equals mean Y minus b x mean X - correct
the mean of Y, regardless of the slopeWhy not: The intercept equals mean of Y only when the slope times the mean of X is zero.

Because the line passes through the point of sample means, a = mean(Y) - b x mean(X), which is the predicted Y at X = 0.

A harder one

A fund's returns Y are regressed on market returns X. The sample covariance of X and Y is 0.0042, the sample variance of X is 0.003 and the sample variance of Y is 0.0071. The OLS slope estimate is closest to:

0.59Why not: Divides the covariance by the variance of Y rather than the variance of X.
0.91Why not: Divides the covariance by the product of the standard deviations, which gives the correlation, not the slope.
4.2Why not: Reports the covariance scaled by 1,000; the covariance alone is not the slope until divided by Var(X).
1.40 - correct

Slope = Cov(X,Y) / Var(X) = 0.0042 / 0.003 = 1.40. Dividing by Var(Y) gives 0.59; the correlation is 0.91.

The hardest tier

A risk analyst fits Y on X using n = 8 observations and records: sum of X = 92, sum of Y = 19.6, sum of XY = 285.8, sum of X² = 1100, sum of Y² = 137.64. The OLS slope estimate is closest to:

0.26Why not: Uses sum of XY divided by sum of X², skipping the centring on the sample means.
0.67Why not: Divides by the corrected sum of squares of Y; this is the slope from regressing X on Y.
0.98Why not: Computes the correlation coefficient (Sxy over the square root of Sxx x Syy), not the slope.
1.44 - correct

Sxy = 285.8 - 92 x 19.6 / 8 = 60.40; Sxx = 1100 - 92² / 8 = 42.0; slope = 60.40 / 42.0 = 1.438. The correlation is 0.98 and the reverse slope 0.67.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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