Describe time trends that are linear and those that are nonlinear.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.a

The idea

A trend model describes how the expected value of a series moves with time. A linear trend is y(t) = β₀ + β₁ × t + ε(t). Here β₁ is a constant change in the level per period, so the series climbs or falls along a straight line. Many financial series, such as asset values or volumes, grow by a roughly constant percentage instead. The log-linear trend fits them: ln y(t) = β₀ + β₁ × t + ε(t). Its slope β₁ is the continuously compounded growth rate. Without the error term, the fitted level of the series is therefore Level = exp(β₀ + β₁ × t). Quadratic and other polynomial trends add powers of t so the slope itself changes. A quadratic with a negative coefficient on t² peaks at Peak = −β₁ ÷ (2 × β₂). The usual mistake is to read the log-linear slope as a unit change in y. On the exam, choose the form that matches the pattern of growth.

Formulas

Fitted level of a log-linear trend

ŷₜ = e^(b₀ + b₁ × t)

b₀: the intercept of the regression of ln(y) on time
b₁: the slope, the continuously compounded growth rate per period
t: the time index

For a series growing at a roughly constant percentage rate. The trap is reading b₁ as a unit change in y: over h periods the level is multiplied by e^(b₁ × h).

Peak of a quadratic trend

peak time = −b₁ ÷ (2 × b₂)

b₁: the coefficient on t
b₂: the coefficient on t², negative for a peak

Only a maximum when b₂ is negative; with a positive b₂ the same point is a minimum. Set the slope b₁ + 2 × b₂ × t to zero, do not forget the 2.

A practice question

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

A risk analyst wants a trend model for assets under management that grows at a roughly constant percentage rate. Which specification is the natural choice?

y_t = b0 + b1 t + e_t, a linear trendWhy not: A linear trend implies a constant absolute increase, so the percentage growth falls over time.
ln(y_t) = b0 + b1 t + e_t, a log-linear trend - correct
y_t = b0 + b1 y_(t-1) + e_t, an autoregressionWhy not: This is a dynamic model of persistence, not a trend in time.
y_t = b0 + e_t, a constant-mean modelWhy not: Has no trend component at all, so it cannot represent sustained growth.

Constant percentage growth means the log of the series rises linearly with time, so ln(y) regressed on t is appropriate; b1 is the continuously compounded growth rate.

A harder one

A log-linear trend fitted to quarterly sales gives ln(y_t) = 2.40 + 0.03 t. By what percentage does the fitted trend value rise over 8 quarters?

Closest to 24.0%Why not: Treats the change in the log (0.24) as a percentage change; exponentiation is needed to convert it.
Closest to 127.1%Why not: Reports the ratio of the final to initial trend value (1.271) as the increase, forgetting to subtract 1.
Closest to 27.1% - correct
Closest to 3.0%Why not: Reads the one-quarter log growth rate as the growth over the whole eight-quarter horizon.

The log of the trend rises by 0.03 x 8 = 0.24, so the level is multiplied by exp(0.24) = 1.2712, a rise of 27.12%.

The hardest tier

A log-linear trend fitted to quarterly data is ln(y_t) = 1.80 + 0.0085 t. Assuming the trend continues, how many quarters does the fitted trend take to double?

Closest to 117.6 quartersWhy not: Divides 1 by the slope, which gives the time for the log to rise by 1 (a factor e), not 2.
Closest to 81.5 quarters - correct
Closest to 235.3 quartersWhy not: Divides 2 by the slope, treating the doubling as a rise of 2 in the log.
Closest to 35.4 quartersWhy not: Uses the base-10 logarithm of 2 although the trend is in natural logs.

Doubling requires the log to rise by ln 2 = 0.6931, so the time is 0.6931 / 0.0085 = 81.5 quarters.

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