Software 3.6.1 (A constant growth curve). Transformations are written directly in the model formula:

> subs <- c(38.1, 80.0, 170.4, 354.5, 744.4); T <- -2:2
> g <- lm(log(subs) ~ T)
> summary(g)$coef
            Estimate  Std. Error   t value     Pr(>|t|)
(Intercept) 5.128735 0.002747891 1866.4259 3.391861e-10
T           0.743341 0.001943052  382.5635 3.938668e-08
> b <- coef(g)["T"]; exp(b) - 1
      T
1.10295

Reading the output. The slope \(0.7433\) and intercept \(5.1287 = \overline {\log Y}\) agree with the hand calculation of Example 3.2.2. The growth rate is computed from the coefficient, not printed by lm: \(\hat g = 1.103\), or \(110\%\) a year. The tiny standard errors reflect an almost exact fit with only three residual degrees of freedom, and should not be over-interpreted: five points that lie on a smooth curve say little about how long the curve will continue.

Stata: generate lnsubs = ln(subs), regress lnsubs T, then nlcom exp(_b[T]) - 1, which also gives the delta-method standard error. EViews: ls log(subs) c t; then scalar g = exp(@coefs(2)) - 1.

Software 3.6.2 (Comparing a linear and a linear-log model). For the savings data, the savings rate against income per head, in levels and in logarithms:

> f1 <- lm(sr ~ dpi, data = LifeCycleSavings)
> f2 <- lm(sr ~ log(dpi), data = LifeCycleSavings)
> c(summary(f1)$r.squared, summary(f2)$r.squared)
[1] 0.04855805 0.07636831
> summary(f2)$coef
            Estimate Std. Error   t value   Pr(>|t|)
(Intercept) 1.756107  4.0203461 0.4368048 0.66421166
log(dpi)    1.207970  0.6063571 1.9921764 0.05205621

Reading the output. Both models explain sr, so their \(R^2\) values are comparable, and the linear-log form fits better. Income per head ranges from about $89 to $4 000, and a model in which each extra dollar matters equally across that range is implausible. The coefficient means that a \(10\%\) higher income per head is associated with a savings rate higher by \(1.208 \times \log 1.1 = 0.115\) percentage points, roughly \(1.208/10\). The \(p\)-value of \(0.052\) is borderline, and income on its own explains little of the variation in savings rates.

Stata: generate lndpi = ln(dpi), then regress sr lndpi. EViews: ls sr c log(dpi).

Software 3.6.3 (Box–Cox comparison of a linear and a growth model). For the subscriber data, compare a linear trend in \(Y\) with the growth curve, after scaling by the geometric mean:

> gm <- exp(mean(log(subs))); ys <- subs / gm
> rss_lin <- sum(resid(lm(ys ~ T))^2)
> rss_log <- sum(resid(lm(log(ys) ~ T))^2)
> c(gm, rss_lin, rss_log)
[1] 1.688035e+02 1.652347e+00 1.132636e-04

Reading the output. On the common scale the growth curve’s residual sum of squares is about \(14\,600\) times smaller. By Theorem 3.5.2 the log-likelihood difference is \(\frac 52\log 14\,589 = 24.0\) in favour of the growth curve, an overwhelming margin. The \(R^2\) values (\(0.858\) and \(0.99998\)) happened to point the same way here, but they could not have been relied on to do so.

Stata: boxcox subs T estimates \(\lambda \) by maximum likelihood. EViews has no built-in Box–Cox command; the scaled comparison above is done with two ls commands on the scaled series.