Deterministic Trends: Linear, Quadratic and Log-Linear Trend Models

Understand Deterministic Trends and Trend Models

Key formulas to remember

Linear trend

y_t = β0 + β1·t + ε_t

β1 is the constant change in y per period. t = 1, 2, ..., T.

Quadratic trend

y_t = β0 + β1·t + β2·t² + ε_t

Slope at time t = β1 + 2β2·t. A turning point exists at t = −β1 ÷ (2β2) when β2 ≠ 0.

Log-linear (exponential) trend

ln y_t = β0 + β1·t + ε_t, equivalently y_t = exp(β0 + β1·t + ε_t)

Requires y_t > 0. β1 is the continuously compounded growth rate per period.

Exact growth rate from log-linear model

Growth per period = exp(β1) − 1

For small β1 this is close to β1. Over n periods, growth = exp(n·β1) − 1.

Trend forecast, h periods ahead

Linear: ŷ_{T+h} = β̂0 + β̂1·(T+h). Log-linear: ŷ_{T+h} = exp(β̂0 + β̂1·(T+h))

Plug the future time index into the fitted equation. Ignore the error term, whose expected value is zero.

Detrended series (trend stationary)

Residual_t = y_t − fitted trend_t

If the series is trend stationary, the residuals are covariance stationary and can be modelled with AR/MA terms.

How to solve Deterministic Trends and Trend Models questions

1. 1Identify the model form from the question: y on t (linear), y on t and t² (quadratic), or ln y on t (log-linear).
2. 2Check the time index. Find out whether t starts at 1 and which date the question is asking about. Many errors come from a shifted t.
3. 3Decide whether the dependent variable is y or ln y. This decides how you read β1 and whether you must exponentiate at the end.
4. 4Interpret the coefficients. Linear: β1 is a change in units per period. Log-linear: β1 is a percentage growth rate per period. Quadratic: slope = β1 + 2β2t.
5. 5For a forecast, substitute the future t into the fitted equation. Do the arithmetic in the model's own scale first.
6. 6For a log-linear model, convert back with exp() only at the end. Use exp(β1) − 1 if the question wants the exact growth rate.
7. 7Sanity-check the result. A log-linear forecast must be positive. A quadratic forecast far out can behave oddly, so check the sign of β2.
8. 8If asked about model choice, prefer the lower AIC or SIC and a residual plot with no pattern. Do not choose on R-squared alone.

Quickest way: Read the dependent variable, then plug in t

Common mistakes in Deterministic Trends and Trend Models

• Forgetting to exponentiate a log-linear forecast

  The fitted equation returns ln y, and the number looks like a plausible answer.

  Fix: If the left-hand side is ln y, press e^x before reporting a level. Ask whether the answer is a log or a level.

• Reading β1 in a log-linear model as a change in units

  Students carry over the linear-trend interpretation.

  Fix: In ln y = β0 + β1t, β1 is the proportional growth per period. 0.012 means about 1.2% per period, not 0.012 units.

• Using the wrong value of t for the forecast

  The time index is assumed to start at 0, or the forecast date is counted from the wrong origin.

  Fix: Read how t is defined. If t = 1 is the first observation and T is the last, the h-step-ahead forecast uses t = T + h.

• Treating the quadratic slope as β1

  In a linear model the slope is constant and equals β1, so students assume the same here.

  Fix: The slope of a quadratic trend is β1 + 2β2t, so it depends on t. β1 is only the slope at t = 0.

• Choosing the model with the highest R-squared

  R-squared always rises when t² is added, so the larger model looks better.

  Fix: Compare AIC or SIC, which penalise extra parameters, and check that the residuals show no leftover pattern.

• Confusing a deterministic trend with a unit root

  Both make a series drift upward, so they look alike on a chart.

  Fix: A deterministic trend is a fixed function of time with stationary deviations. A unit root has shocks that permanently shift the level. Detrending fixes the first. Differencing is used for the second.

Worked examples