Financial Risk Manager Part 1
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Roderick Jaynes, FRM, analyzed historical sales (S) for over 20 years and found that sales are increasing but its growth rate over the period is relatively constant. Which model is most suitable to forecast out-of-sample sales?
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NNikitesh
A
Sₜ = β₀ + β₁ × Sₜ₋₁
B
Sₜ = β₀ + β₁ × t
C
lnSₜ = ln(β₀ + β₁ × t)
D
Sₜ = β₀ + β₁ × t + β₂ × t²
Explanation:
The correct answer is C because when sales have a relatively constant growth rate over time, the log-linear model (lnSₜ = ln(β₀ + β₁ × t)) is most appropriate. This model transforms the data to linearize exponential growth patterns, where the coefficient β₁ represents the constant growth rate.
Key points:
- Option A (Sₜ = β₀ + β₁ × Sₜ₋₁): This is an autoregressive model that assumes current sales depend on previous period's sales, not suitable for constant growth rates.
- Option B (Sₜ = β₀ + β₁ × t): This is a simple linear trend model that assumes constant absolute increases, not constant percentage growth rates.
- Option C (lnSₜ = ln(β₀ + β₁ × t)): This log-linear model captures constant growth rates as the coefficient β₁ represents the growth rate when the dependent variable is logged.
- Option D (Sₜ = β₀ + β₁ × t + β₂ × t²): This quadratic trend model allows for accelerating or decelerating growth, not constant growth rates.
Mathematical reasoning: When growth rate is constant, sales follow: Sₜ = S₀ × (1 + g)ᵗ, where g is the constant growth rate. Taking natural logs: ln(Sₜ) = ln(S₀) + t × ln(1 + g), which is linear in t with slope ln(1 + g) ≈ g for small g.
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