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Contrast the assumptions made in single regression with those made in multiple regression.

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

The idea

Single regression assumes a linear model, errors with zero mean conditional on X, constant error variance, no correlation between errors, and some variation in X. Normality matters for small-sample inference. Multiple regression keeps all of these, now conditional on every regressor, and adds one. There must be no exact linear relationship among the explanatory variables, so no perfect multicollinearity. If one regressor is a constant times the intercept or an exact combination of others, OLS cannot be computed. High but imperfect correlation is different: estimates stay unbiased, but standard errors rise and t-statistics fall. Bias comes from a failure of zero conditional mean, typically an omitted relevant variable correlated with an included one. Heteroskedasticity and autocorrelation leave OLS unbiased but make the usual standard errors unreliable. The exam favours the distinction between perfect and imperfect multicollinearity.

A practice question

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

Which assumption is required in a multiple regression but is not needed in a regression with a single explanatory variable?

  1. The error term must have a conditional mean of zeroWhy not: Zero conditional mean of the error is already required in the single-variable model, so it is not new.
  2. The explanatory variables must be uncorrelated with one anotherWhy not: Confuses 'no perfect multicollinearity' with zero correlation. Only an exact linear dependence is ruled out.
  3. No exact linear relationship among the explanatory variables - correct
  4. The errors must have constant varianceWhy not: Homoskedasticity is assumed in the single-variable model too, so it is not the added assumption.

In a single regression the only requirement on X is that it varies. With several regressors, none may be an exact linear combination of the others, otherwise the coefficients cannot be separately identified. Linearity, zero conditional mean and constant error variance carry over unchanged.

A harder one

A risk manager regresses a bond fund's return on changes in the 5-year yield and the 10-year yield. The sample correlation between the two yield changes is 0.92. Which statement is correct?

  1. OLS cannot be computed because the no-multicollinearity assumption is violatedWhy not: Treats a high but imperfect correlation as perfect multicollinearity. OLS is still computable.
  2. The estimates are biased because correlated regressors violate the zero conditional mean assumptionWhy not: Correlation among included regressors does not bias OLS; bias arises from an omitted variable correlated with the included ones.
  3. The standard errors are understated, so the null is rejected too oftenWhy not: Direction is reversed: collinearity inflates the variance of the estimators, making t-statistics smaller, not larger.
  4. The estimates stay unbiased, but standard errors rise and t-statistics tend to be low - correct

Imperfect multicollinearity leaves OLS unbiased and consistent but raises the variance of each affected coefficient, because the data cannot separate their effects. The usual symptom is a high R2 or significant F-statistic with insignificant individual t-statistics.

The hardest tier

A risk manager regresses monthly changes in a fund's NAV on an equity index return, a credit spread change and the central bank policy rate level, with an intercept. The policy rate was held at exactly 0.25 percent throughout the 60-month sample. What is the result?

  1. The policy rate column is a multiple of the intercept column, so OLS cannot be computed - correct
  2. The policy rate coefficient is estimated as zero with a very large standard error, as for any irrelevant variableWhy not: A constant regressor is not just irrelevant; it is exactly collinear with the intercept, so no unique estimate exists.
  3. All slope coefficients are biased towards zero because a fixed regressor carries no informationWhy not: A constant column creates non-identification, not bias in the other slopes.
  4. The errors become heteroskedastic because one regressor has no varianceWhy not: Zero variation in a regressor affects identification of coefficients, not the variance pattern of the errors.

A variable that never changes is a constant times the intercept's column of ones. That is an exact linear dependence, so the coefficients on the intercept and the constant regressor cannot be separately identified. A regressor must show sample variation, and the multiple regression requires no perfect multicollinearity.

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