Log-linear trend model

A trend model regresses a series on calendar time. The linear trend is

y_t = b0 + b1 t + e_t

and its slope b₁ is a claim about the absolute change per period: rupees per quarter, percentage points per year. Fit it exactly as you would fit any simple regression, with t = 1, 2, 3, ... as the sole explanatory variable. The forecast for period T + k is b₀ + b₁(T + k), and nothing about the series' own past values enters the arithmetic.

How does a log-linear trend differ from a linear one?

The log-linear trend regresses the natural logarithm of the series on time:

ln(y_t) = b0 + b1 t + e_t

Here b₁ is a claim about the proportional change per period. The exact per-period growth rate is e^b₁ − 1, not b₁ itself, though the two agree to about a tenth of a percentage point for small b₁. In the revenue example in Worked Example 3, b₁ = 0.062567 and the true implied quarterly growth is 6.4565%, so quoting 6.2567% would be wrong by 20 basis points a quarter, which compounds to 0.96 percentage points a year (28.4367% against 27.4751%).

Choosing between them is not a matter of taste, and there are two ways to decide. The first is the shape of the underlying process: a series generated by a constant growth rate is exponential in levels and linear in logs, and revenue, dividends, index levels and populations are usually of that kind, while a series generated by a constant absolute increment is linear in levels. The second is empirical, and it is the whole reason Durbin-Watson comes up when you choose a trend model.

How do you forecast from a log-linear trend?

Back-transforming a log-linear forecast has a trap in it. Exponentiating the fitted log value gives the median of the forecast distribution, not the mean, because the exponential of a normal variable is lognormal and the lognormal's mean sits above its median by a factor of exp(s²/2). Duan's smearing estimator is the standard correction: multiply the naive back-transform by exp(s²/2), where s² is the residual variance of the log regression. On the revenue series in Worked Example 3 s = 0.03161349, the correction factor is 1.00049983, and the twenty-first quarter's forecast moves from ₹142.3819 crore to ₹142.4531 crore. Five hundredths of a percent is negligible here and is not negligible everywhere: at a residual standard deviation of 0.30 the correction is 4.6%, which is the difference between a beat and a miss.