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Identify which models linear regression can estimate and set them apart from those it cannot.

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

The idea

Ordinary least squares can estimate any model that is linear in its parameters, even when it is not linear in the variables. A quadratic model is fine, because the coefficients a, b and c enter additively and X² is simply another regressor: Y = a + b × X + c × X² + error. A multiplicative model is not additive as written, so it needs one more step: Y = A × X^b × error. Taking logarithms makes it linear, so it also qualifies after transformation: lnY = ln A + b × ln X + ln error. OLS cannot handle models where a parameter sits in an exponent or in a ratio that no transformation removes, such as this one: Y = a + X^b + error. It also fails where parameters multiply each other. Those models need nonlinear least squares. The usual mistake is to read linear as a straight line. For the exam, test the parameters, not the variables.

A practice question

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

In the term linear regression, the word linear refers to the requirement that the model be:

  1. linear in the variables, so that only a straight-line relationship between Y and X can be estimated, with no curvature allowedWhy not: Confuses linearity in parameters with linearity in variables; a model with X² is still a linear regression.
  2. linear in the parameters, although the regressors may be transformed variables such as squares or logarithms - correct
  3. linear in the error term, so that residuals must be normally distributed around the fitted lineWhy not: Mixes up linearity with the error distribution; normality is a separate, mostly small-sample, assumption.
  4. linear in time, so that the data must be a time series with a steady trendWhy not: Invents a time-series requirement; linear regression applies equally to cross-sectional data.

Linearity means the dependent variable is a linear combination of the parameters. Transforming the variables (squares, logs, reciprocals) is allowed because the transformed variable simply becomes a regressor.

A harder one

A risk manager believes trading revenue follows R = A x S^b x u, where S is trading volume, A and b are parameters and u is a positive multiplicative error. Which approach allows OLS to be used?

  1. Take natural logarithms of both sides: ln R = ln A + b ln S + ln u, then regress ln R on ln S - correct
  2. Regress R on S directly, because the model is already linear in the variablesWhy not: The model is not linear in S, and it is nonlinear in the parameter b, so a direct regression of R on S misspecifies it.
  3. Square both sides so that the exponent b disappears, then regress R² on SWhy not: Squaring does not remove b; it doubles the exponent, and the model stays nonlinear in the parameters.
  4. OLS cannot be used under any transformation of the data, so nonlinear least squares is the only option hereWhy not: Overlooks that a multiplicative error makes the model log-linear; the log transform gives a linear-in-parameters model.

Taking logs turns the product into a sum, giving an intercept (ln A) and a slope (b) that are linear parameters, and the error ln u is additive. This is the log-log specification.

The hardest tier

Which of the following models can be estimated by OLS without any reparametrisation?

  1. Y = b0 + b1 x X1 + (b1 x b2) x X2 + errorWhy not: The coefficient on X2 is a product of two parameters, so the model is nonlinear in the parameters.
  2. Y = b0 + X1^b1 + error, where b1 is the unknown exponentWhy not: The parameter b1 is an exponent; this is nonlinear in the parameter.
  3. Y = 1 / (b0 + b1 x X1) + error, where the error is added at the endWhy not: With an additive error, the reciprocal of the linear index cannot be turned into a linear-in-parameters equation.
  4. Y = b0 + b1 x X1 + b2 x X2 + b3 x (X1 x X2) + error - correct

The interaction term X1 x X2 is just another regressor with its own coefficient b3, so the model is linear in b0 to b3. The other models have parameters multiplied together, in an exponent, or inside a reciprocal.

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

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