Dear Tomas,

Thank you very much! If your time allows, could you please explain a bit more about why the exponential trend formula is not appropriate for large growth rates?

I find from my data that the R-square of exponential trend function tends to increase in the estimated annual growth rate (the coefficient b in y=ae^bx). Put it differently, if data show an obvious and steady increasing trend, then R-square tends to be really high; but if the trend is not obvious or highly fluctuant, then R-square tends to be small.

If annual growth is higher than 10%, then the R-square tends to be above 0.9. But if annual growth rate is only 0.4%, for example, R-square is only around 0.02.

The correct formula for calculating annual growth is given below:

Assume that average exports in 1997 is $40.28,  and it is $42.51 in 1998, then the growth rate is as follows. First, u= {log base 10(Xt /Xt-1)} then u /diff where dff=1998-1997.

Next, jj= uu/diff minus 1, then 

(Antilog of jj) * 100 gives you the annual growth of 5.5363%

** [Antilog{.023402/1) -1 ] * 100 ==5.5336% Note that this how to calculate annual growth.  If you want overall growth, tell me. If you stayed with this approach you cannot make mistake!!

Exponential models can not be used in the case of large disturbance term, but they alow to see the total tendency. It can be easily prooved that AAGR is based only on first and last observation, so it needs some improvement. See for instance that proposed by Prof. Timofiejuk in:

Timofiejuk I.: O liczeniu średniego tempa wzrostu. Wiadomości Statystyczne 12/1994, s. 6-9 (this is Polish journal).

I do not agree with your point --that it is based on two points. I can show you that my method is optimal and used by the "most respected" in all their calculations. I am in bed now. I have never heard of who you have cited but you have heard of the authorities I will cite when I wake up in the morning. Does it mean that every minor stuff in a journal makes sense. By the way, what do mean by large disturbance terms? 

Thank you all. I have run the formula by Chuck and reach the same result. It seems to be equivalent to (Xt/Xt-1)^(1/diff)-1.

But I think a problem is that such a "compound growth rate" method neglects observations between the starting time point and the ending time point.  By contrast, exponential trend function can solve this problem, but it performs poorly (low R-square) when the growth rate is small or there is strong fluctuation. But I find that the annual growth rates calculated from the "compound growth rate" method you suggested and from exponential trend function are very similar.

Jacek,

The question is, "What is the best method to calculate an annual growth rate of trade?" Bernanke, former federal reserve chairman and Abel textbook; Dornbusch and Fischer including Kenneth Stewert Econometrics textbook all recommend the approach I have suggested. Given aggregated data, working with continuous growth rates data analyzes is the best method. Best in the sense that all good scholars use a common easy to understand measure. Indeed, the perfect is always the enemy of the good. What obtains NBER calculations and the trade data manuals should be seen as the best. Yes, it is not every minor perturbation that is relevant in calculations and one does not have to think whether there is large growth or small growth in writing their work. By the way, when was the last time we've had large growth in the world?

Yue

Yes, I have read your comments and they are very good. As you can see there is no perfect measure. However there is the best measure and it is the one used in textbooks and journals of the development of economics.  My suggestion, as an editor, is that you do what readers can easily understand in spite of minor perturbations. Thank you for giving us the opportunity for us add our two-cent opinions. 

Dear Chuck, many thanks again to your attentions to my questions! Your answers always help so much.

Dear Tomas, Thank you very much for the mathematical derivation!

Dear Yue Teng,

You can use Kinked exponential model.

The advantage of using this method for the two sub-periods is that the assumption of continuity in time series (i.e. there is no structural break in the time series used) is methodologically relaxed.

You may use semi-log or log linear method in which all the observations are taken in to account which is called Annual Compound Growth Rate (ACGR or CAGR). But major limitation of this method is that if you have negative observation(s) then you can not use it. For longer period to me it is better one. For annual compound growth rate take anti-log of regression coefficient, subtract one from that and multiply by hundred then you will get per cent annual compound growth rate.

@ L.G : 

your semi-log may not be good for (a) if the growth is non-linear (b) if there is non-spherical errors and (c) in the presence of weekly or even quarterly data

It is log-linear, one can estimate ACGR daily / weekly / monthly / quarterly / yearly / etc depends on frequency and length of the data. If structural breaks are there then one can use for major breaks, for fairly long series kinked exponential growth rates. Of course, there are some limitations too. If non-linearity is to be taken into account then one may go for Y-o-Y growth rate for a short time period. However, each and every growth rate technique has its own merits and demerits. It depends on the purpose of growth rate estimation. If there is any growth rate technique without any limitations which can be used universally, it would nice know it.

@L. G. I wanted you to acknowledge that there are huge limitations to your what you posted. As an econometrician, I dislike the simplicity implied in your original post --just run log(y) on c time and take anti-log. That is a deterministic trend. It is not true even r structural break as indicated above. The legitimacy of any conclusion approach depends on the proper specification and this has implications on the how the error term behaves. In fact, I would argue that a stochastic trend is less evil than your deterministic trend given stationarity. I do not think you addressed my non-spherical errors, and I wished you had flavored your message as a quick way. I do not know what you refer to Y-0-Y, however, it is not important since you have addressed some limitations for our audience. 

The technique which I mentioned was  common and meant for general researchers and not for specialists in econometrics, as you rightly mentioned

A decision concerning annual growth rate should be made based on a better model (not estimator). Economic processes evolve in a continuous fashion and tend to grow largely at a constant rate so that one standard deviation of economic time series  is proportional to its level so that a stochastic trend obtains and hence stationarity obtains. That is, percentage changes do not become systematically larger or smaller over the period.. One can say they dance around a mean growth rate. This nullifies deterministic trend, hence OLS on time will be biased downwards since it assumes an absence of a unit root. Therefore, although in between values may be ignored, my earlier suggestion is the better approach. Being geometric in nature, the results are similar to to the log-difference approach. Consequently, all the near-Nobel laureates use that approach in their textbooks when calculating an annual growth rate. 

https://www.wto.org/english/res_e/statis_e/wts2016_e/wts2016_e.pdf

(i) if we use the Log values (to base 10), then we have to necessarily continue with the three additional steps [i.e. (i) taking the antilog of slope coefficient for the time variable T, (ii) obtaining the difference by subtracting 1 and (iii) multiplying the resultant value by 100 to obtain percent per annum growth rate] to arrive at the compound growth rate;

(ii) if we use the natural log values but don’t continue with the above three additional steps, then the coefficient of time variable ‘T’ multiplied by 100 gives the instantaneous growth rate of the dependent variable. This will be near about to the growth rate obtained by taking the log values; and

(iii) if we take the natural log values and continue with the above three steps, then the growth rate obtained would not only be the compound growth rate but it will exactly be the same as we obtain when we proceed by taking the logarithm to base 10 values.