Log-Linear Regression Analysis

00:00 8 marks

Consider a dataset that follows an exponential trend modeled by \(y = ab^x\).

(a) Explain how taking the logarithm of the y-values transforms the exponential relationship into a linear one. State the resulting linear equation in terms of \(x\).

(b) Using the transformed data, fit a linear regression model of the form \(Y = mx + c\) to estimate \(\ln(b)\) and \(\ln(a)\). Explain how these estimates relate back to the original exponential parameters.

Part a

6 minutes 4 marks

Question

Explain how taking the logarithm of the y-values from an exponential relationship of the form \(y = ab^x\) can lead to a linear relationship. Perform the transformation and state the resulting linear equation.

Hint Hint

Hint

Recall that taking natural logarithms gives \(\ln(y) = \ln(a) + x\ln(b)\).

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Complete Answer

After the transformation, the relationship becomes \(\ln(y) = \ln(a) + x\ln(b)\), which is linear with slope \(\ln(b)\) and intercept \(\ln(a)\).

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Part b

6 minutes 4 marks

Question

Given the transformed data from part (a), fit a linear regression model of the form \(Y = mx + c\) to estimate the values of \(\ln(b)\) and \(\ln(a)\). Explain how these estimates relate to the original exponential model \(y = ab^x\).

Hint Hint

Hint

Remember that in the regression model \(Y = mx + c\), the slope \(m\) corresponds to \(\ln(b)\) and the intercept \(c\) corresponds to \(\ln(a)\).

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Complete Answer

The slope \(m\) from the linear regression model provides an estimate for \(\ln(b)\), and the intercept \(c\) estimates \(\ln(a)\). To revert to the original exponential parameters, we calculate \(a = e^c\) and \(b = e^m\).

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