Interpreting log-difference in log-log model

[Luna Schmidt]

Dear all,

I read various similar posts but I'm still confused and new to econometrics, Stata, and Statalist. I would highly appreciate any help and hope this question is formatted and phrased appropriately.
In short: I am unsure about the interpretation of the coefficients in my regression analysis – is it percentage, is it percentage points, etc. – when using growth rates calculated as log-difference as a dependent variable.

I am running regressions (employing OLS) with the dependent variable as the growth in real GDP per worker over a certain period, calculated as log-difference, so that: growth over the period = ln(real GDP per worker in the final year of the period) - ln(real GDP per worker in the initial year of the period).

Some major independent variables are: log real GDP per worker in the initial year, and an index of economic freedom (measured on a scale of 0-1 and not log-transformed, called EF).

I estimate the equation:
ln(real GDP per worker in the final year of the period) - ln(real GDP per worker in the initial year of the period) = beta0 + beta1*(log real GDP per worker in the initial year) + beta2*EF + e

After running the regression in Stata, I obtain the following estimates:
The coefficient for initial real GDP per worker: - 0.321, p < 0.01 and the coefficient on the index of economic freedom: 1.239, p < 0.01

Is the following correct?
- a 1% higher initial income per worker is associated with about a 0.321 percentage point reduction in the growth rate, holding all else constant.
- an increase of the index of economic freedom by 0.1 is associated with about a 0.1239 percentage point increase in the growth rate, holding all else constant. (I think that interpreting the change in the index by one "full unit" (1) does not appear sensible for the index measured on a scale from 0-1).

So far, I am assuming that I should express the effects in percentage points, as I investigate the growth rate. However, I am unsure, as I usually see similar interpretations of a log-log model in percentage changes and I'm not familiar with the interpretation of log-differences in the context of log-log models. Or do I need to adjust the coefficients in any way (e.g. multiplying/dividing by 100) to sensibly interpret the results?

I highly appreciate any helpful comments.

Kind regards,
Luna Schmidt

[Clyde Schechter]

Because your DV is a log-difference, this is not the same as a simpler log-log or log-linear model. I don't think I've seen this come up previously on Statalist. The way to handle problems like this is to go back to the underlying algebra.

The first step is to notice that the dependent variable, a difference in logarithms is the logarithm of the ratio. So the dependent variable here is best thought of as log(GDP per worker final / GDP per worker initial). This log of the ratio is also an approximation to the actual period rate of growth in GDP per worker in percents divided by 100. (And at least for reasonable short periods, the rate of growth for GDP per worker will be small enough that the approximation is decent. For long periods or large growth rates, it would be better to use the actual growth rate rather than the approximation through logarithms as the two start to diverge sharply. I will assume that the growth rate is what you are actually interested in, and that the circumstances are such that your difference in logarithms remains a useful approximation.

So with that in mind, your equation

ln(real GDP per worker in the final year of the period) - ln(real GDP per worker in the initial year of the period) = beta0 + beta1*(log real GDP per worker in the initial year) + beta2*EF + e

can be rewritten as:

GDP per worker growth rate = beta0 + beta1*(log real GDP per worker in the initial year) + beta2*EF + e

And at this juncture it is good to make note of the fact that a growth rate is measured in percent, not percentage points. That's because a (non-transformed) growth rate is basically a ratio, not a difference. Ratio changes can be expressed as percentages, differences must be expressed as percentage points. (This is a generic principle: it applies in all contexts.)

Now, let's deal with EF first, because it's simpler. If EF increases by 0.1, the EF term in the model increases by beta2*.1, which in this case is 1.239*.1 = 0.1239. From this it follows that the dependent variable, growth rate/100, grows by 0.1239. Note that this is an additive change, not multiplicative. So, if the dependent variable were originally .05 (corresponding to a 5% growth rate) the DV would change to .1739, which corresponds to a growth rate of 17.39%, which is an increase in the growth rate of 12.39 percentage points over the original 5%.

Now suppose initial GDP per worker increases by 1%. Then log(increased initial GDP per worker) = log(1.01*original initial GDP per worker) = log(1.01) + log(original initial GDP per worker) = .00995 + log(orginal inital GDP per year). Therefore, this log-transformed independent variable has increased (additively) by 0.00995. It follows that the expected value of the dependent variable grows by beta1*0.00995 = -0.321*0.00995 = -0.00319. Again this is an additive change, so the result will be percentage points, but we must account for the fact that the DV is scaled down by a factor of 100. So again, suppose the original growth rate were 5%, corresponding to an original DV of 0.05. Then the new DV is lower by 0.00319; 0.05 - 0.00319 = 0.04681. And a DV of 0.04681 corresponds to a growth rate of 4.681%. The change in the growth rate, then, is the difference between 5% and 4.681%, which is a decrease of 0.319 percentage points.

[Luna Schmidt]

Dear Clyde,

I thank you tremendously for your time and this thorough and helpful response.

If you (or someone else, I do not want to ask for too much) don't (does not) mind, I'd have some follow-up questions to clarify:

Sorry in advance for any questions appearing mundane – I have the feeling that the more I try to understand/read about it the less I know.

1. Regarding the growth rate approximated by log-difference: I am replicating another study which calculates the log-difference for an observational period of 20 years. In this case, I assume it could be more sensible to "ex-post" interpret the coefficient in terms of the average annual growth rate, would you agree? I elaborate to ensure any mistakes in my thoughts are transparent:
For instance, if the log-approximated GDP per worker growth rate = 0.5 and following your response, the growth rate over the 20-year period is 50%. In the case of a 0.1 increase in the EF index (coefficient 0.1239) this leads to a change of the DV to 0.6239, so that the growth rate is 62.39%, thus an increase of 12.39 percentage points, as in your example. This, however, does not really seem like a sensible measure. If I want to express it as the average annual growth rate, how do I adjust the interpretation? Can I just divide the "example growth rate" by the years in the period? Is that allowed or does this change the estimates? Or is it the same, as, e.g., expressing the estimates for the EF index in 0.1 units instead of 1?

2. I have frequently read that the log-log interpretation is just an approximation for small coefficients. So in the examples above, should I somehow adjust the interpretation of the coefficients to account for that (e.g. exponentiation etc.)? Or is this, again, different in the case of log-differences?

3. Suppose that I have another independent variable in the estimated equation above: share of exports in real GDP (EX/GDP). It is calculated as the share of Exports in real GDP (e.g., EX/GDP = 0.1). I introduce this variable into the specification in log-form, so that the new equation becomes:
GDP per worker growth rate = beta0 + beta1*(log real GDP per worker in the initial year) + beta2*EF + beta3*(log EX/GDP) + e Am I correct in the assumption that I can express the change in the independent variable as a percentage change? Or is it, again, a matter of percentage points? So that a coefficient on EX/GDP of 0.45 means that a 1% increase in the share of exports in real GDP leads to, c.p., an average increase in the growth rate by 0.45 percentage points (as in the example of initial GDP per worker)? And how would that change if the dependent variable was not growth over the period (as log-difference) but log(real GDP in a year)? Is it then just the typical "a 1% increase in the IV is associated... with x% increase in the DV" interpretation?

Again, thank you a lot.

Kind regards,
Luna Schmidt

[Clyde Schechter]

When you are dealing with period growth rates like 50%, you are way beyond the territory where the difference in logarithms is a good approximation. For a 50% growth rate, we are talking about a final/initial ratio of 1.5, and the log of 1.5 is 0.405--which is way off from 0.5. So I wouldn't use the difference of logarithms in this data at all. I would go back and calculate the actual period growth rates, namely (final/initial)-1, or 100 times that if you prefer to have it in percents. And I would use that as the dependent variable. That also greatly simplifies the interpretation of all the coefficients because now you only have to worry about the logs on the independent variables, so your life is easier.

As for converting from 20 year growth rates to annualized growth rate, dividing by the number of years would just be an approximation, and it would only be a good approximation for a small growth rate and a small number of years. The correct derivation is as follows. If the annual growth rate is x%, then the 1 year later to present ratio is 1+(x/100). Over N (think 20) years, this compounds to an N-year final:initial ratio of [1+(x/100)]^N, or an N-year growth rate of 100 times that. So if you already have an N year growth rate value of X%, we can solve for the annualized growth rate x as x = [(X/100)^(1/N) - 1]*100.