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Explain how the coefficients of a multiple regression should be read.

FRM® Part I · Quantitative Analysis · Regression with Multiple Explanatory Variables · FRM1.QA.8.b

The idea

In a multiple regression each slope is a partial effect: the expected change in Y for a one-unit change in that regressor with all the other included regressors held constant. The intercept is the expected Y when every regressor is zero, which may not be meaningful. Because other variables are held fixed, a slope usually differs from the slope in a simple regression on the same variable. The simple slope also picks up the effect of correlated omitted variables. A dummy variable coefficient is a shift in the conditional mean relative to the base group. With an interaction term, the effect of X for the dummy group is the sum of the base slope and the interaction coefficient. Rescaling a regressor rescales its coefficient inversely and leaves t-statistics and R² unchanged.

A practice question

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

In a multiple regression, how should the estimated slope coefficient on explanatory variable Xj be read?

  1. The expected change in Y for a one-unit change in Xj, ignoring the other regressorsWhy not: That describes the total association in a simple regression. In multiple regression the other variables are held fixed.
  2. The expected change in Y per unit change in Xj, other regressors held fixed - correct
  3. The share of the variation in Y that is explained by Xj aloneWhy not: Confuses a slope with a contribution to R2; slopes are measured in units of Y per unit of X.
  4. The expected change in Xj for a one-unit change in YWhy not: Reverses the direction of the relationship; the regression of Y on X does not give the effect of Y on X.

A slope in a multiple regression is a partial effect: the change in the conditional mean of Y for a one-unit rise in Xj with all other included regressors fixed. The value will generally differ from the slope in a simple regression on Xj alone.

A harder one

In a simple regression of a stock's return on factor X1 the slope is 0.90. In a multiple regression that adds a correlated factor X2, the slope on X1 falls to 0.40. What is the best explanation?

  1. X1 and X2 are correlated, so the simple slope also captured X2's effect - correct
  2. Adding regressors always shrinks every coefficient, which is a bias in the multiple regressionWhy not: Coefficients can rise, fall or stay the same; the change reflects overlap between regressors, not a general shrinkage bias.
  3. The larger standard errors of the multiple regression shrink the point estimate towards zeroWhy not: Standard errors measure precision and do not move the point estimate.
  4. The two slopes must be equal if both models are unbiased, so one calculation is wrongWhy not: Unbiasedness of different models does not imply equal slopes; they answer different questions about holding X2 fixed.

The simple slope measures the total association of X1 with Y, including whatever X1 proxies for in X2. The multiple slope holds X2 fixed, so the two coincide only if X1 and X2 are uncorrelated or X2 has no effect.

The hardest tier

Credit spread (in bp) is modelled as: spread = 35 + 0.8 x LEV + 60 x HY + 0.5 x (HY x LEV), where LEV is leverage in percentage points and HY equals 1 for high-yield issuers and 0 otherwise. For a high-yield issuer, a one-point rise in leverage changes the expected spread by how many bp?

  1. 0.50Why not: Uses only the interaction coefficient and drops the baseline leverage slope that also applies when HY = 1.
  2. 0.80Why not: Applies the investment-grade slope; ignores the interaction term that adds to the slope when HY = 1.
  3. 0.40Why not: Multiplies the two coefficients instead of adding them.
  4. 1.30 - correct

With HY = 1 the leverage slope is the sum of the base slope and the interaction coefficient: 0.8 + 0.5 = 1.3 bp per point. The dummy coefficient only shifts the level of the spread.

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