1. An increase in 1 percent point of health expenditure as % of the GDP reduces infant mortality rate by 0.39 deaths per 1000 live births.

2. An increase in 1 percent in urbanization reduces infant mortality rate in 0.42%.

José's interpretation is correct.

In addition, you might want to standardise your independent variables, so that instead of reading coefficients as 1 percent point increases you would be able to interpret them as 1 standard-deviation (SD) increases. It depends what you think might be easier to interpret in your case (1 percent point or 1 SD, the latter being a "plausible" increase based on the sample you observe and perhaps more comparable to other results in the literature).

I agree with other colleagues José-Ignacio Antón and Giorgia Menta In addition, keep in mind about the t-statistic of coefficient whether it is statistically significant or not. Interpretation of coefficient makes more sense when it is significant.

José-Ignacio Antón Giorgia Menta Proloy Barua Thank you so much, Dear colleagues.

I would be very surprised if in case 1 you can regard health expenditure as % of the GDP as a sole determinant of infant mortality or in case 2 increase in urbanization as the sole determinant.

Both regressions suffer from omitted variable bias. The coefficients do not reflect the causal effect of changes in the explanatory variables on the dependent variable.

For example, an increase in average incomes might give rise to an improvement in infant mortality. The increase in income might be correlated with urbanization. In your measure, the improvement due to the increase in incomes would be reflected to some extent as due to urbanization. This would inflate wrongly the effect of urbanization.

All answers/comments are correct. In addition to John, may I remark, that it is important to look at possible correlations between independent variables. Let us assume, that health expenditures and urbanization were really the main variables affecting infant mortality (and other variables "produce" white noise). As these variables are likely to be correlated, the results of the estimation may be spurious. Anyway, for a good interpretation, one should know what data are used: cross-section of (which) countries or regions - time series - combination of both?

By the way, the interpretation of changes of health expenditures/GDP is highly problematic, because e.g. an increase may be due to higher expenditures or a lower GDP. And the influence of urbanization is so low, that it is not worth to estimate. Try e.g. an equation like ln(InfMort) = -.42*Urb + Constant (between 1 and 4), and you will see not much difference in the mortality rates for very different urbanization percentages. I suspect that both estimated coefficients are not significant.

When José-Ignacio Antón osé-Ignacio Antón says

"An increase in 1 percent point of health expenditure as % of the GDP reduces infant mortality rate by 0.39 deaths per 1000 live births."

I would prefer if he said

"In a bivariate regression, an increase in 1 percent point of health expenditure as % of the GDP has been associated with a reduction in the infant mortality rate by 0.39 deaths per 1000 live births."

To me, the first statement implies causation. If read by a non-expert it will be taken as causation. in the bivariate regression with omitted variable(s), you have correlation and not causation. The explanation should make this clear.

@ sajjad ,I think the 2nd case is log-lin model of the type of semi log model in which cofficient measures relative change in infant mortality for an absolute change in urbanization

Dear John C Frain, thank you for pointing critical differences between two statements: causality and association. This is a good learning for me.

I thought point 2 should be interpreted as 42% reduction for 1% urb because the unit measurement in independent var is already in %??? Please check again.

A 42% reduction per percent urb is much too high, 0,42 is too low to be relevant. One remark to the question in general: If one wants to estimate coefficients of an equation, one should know before the estimation how the results shall be interpreted. That is part of a careful specification.

Carlos Mangunsong Thinking about these numbers. Suppose that the infant mortality rate is 10 per 1000. (order of magnitude).

log(.01) = -4.6053

An increase of 1 in the urbanization rate reduces this by 0.42. and now

log(infant mortality rate) = -5.0253

infant mortality rate = 0.0066 or 6.6 per 1000.

If your urbanization rate is less than 20% then an increase of 1% in the rate is a large change. If your urbanization rate is more than 80% a 1% change is a small change. The size of the coefficient depends on the urbanization rate. The semi elasticities then will differ from country to country and the value found may be reasonable if urbanization is low in the country under examination. As I have stated the semi-elasticities are subject to omitted variable bias.

The calculation above illustrates the differences between continuous growth rates and % changes. It is correct to use continuous growth rates in this analysis.

Anton Rainer's last comment is essential.

John, it is not the difference between discrete and continuous growth rates that is relevant: 1) If one estimates on the basis of cross section data (e.g. the values of a certain period or point of time for different regions or countries), growth does not play any role, it simply has no meaning. 2) If one has to do with time series, the percentage growth and growth measured as dln is, in general, so near to each other, that one will not find a significant difference. For a percentage growth of 10%, dln is 9.53% (which is not very far away).

I think, the problem is that most of us have been taught percentage(s) (changes) in a rather strange manner. We have learned, that percentage (change) p of X is the difference to a base value or the value of the preceding year Xb divided by Xb multiplied by 100, i.e. p=(X-Xb)/Xb*100.

In fact, percentages are only one of (infinitely) many versions to write a rational number, e.g. 0.625, 625/1000, 6.25/100, 5/8, 625000 ppm, 625 promille and 62.5*1/100 are different writings for the same number. % simply means 1/100. The problem is that some people take the number in front of % and others the rational number itself for a statistical analysis: e.g. 62.5% is taken as 62.5 or 0.625. Of course, an estimated coefficient in the second case will be 100 times higher.

If in EXCEL, similar programs and simple calculators you divide one amount by another and you choose "%", it becomes clear, that there is only a formal difference. Unfortunately, the teaching has not changed since more than 100 or 200 years, which leads to troubles even among economists.

Anton, as I have understood the data the difference between continuous compounding (or growth rates) and percentage change is important. The continuous figure of -0.42 corresponds to a percentage change of -34%. In this instance, the difference can not be disregarded. ( I suspect that Sajjad Sargolzaei is from Iran. Iran is probably highly urbanized and therefore the -0.42 may be high.)

In econometric analysis, I would generally use continuous rates (dln or 100*dln). I would even convert official inflation rates to continuous rates. I prefer this for the following reasons reasons

I have even converted inflation rates to continuous rates. Perhaps, this was necessary for Ireland during the 70's, 80's and 90's when inflation rates (and other nominal data) were much more volatile than they would have been in Germany. The % changes would, on occasion, have lead to problems. Perhaps it is my mathematical training. Anyway, on this point, we will have to agree to differ.

Dear John, both change rates, dln and %, are “continuous”, but mostly they are applied on discrete data. If p1 is the %-change rate and p2 the dln-change rate, the development of Y(t)=Y(0)*(1+p1)t=Y(0)*exp(p2*t) is the same. If we leave the assumption of constant p’s, we have Y(t)=Y(t-1)*(1+p1(t))=Y(t-1)*exp(p2(t)) or 1+p1(t)=exp(p2(t)), where t can, in principle, be any real number. If we take the nlog, we get ln(p1)=p2. Both p’s can be expressed in any form of rational numbers, so in %, too.

As ln(1+p1)=p1-p1²/2+p1³/3-…+…etc. p2 is lower than p1 with the difference relatively higher for negative p’s. For low growth rates (up to about 10%) the difference is statistically negligible, because, in general, the standard deviation of estimated coefficients is considerably higher. The main problem with dln is, that we do not think in logs and differences of logs. Have you ever tried to tell somebody, that e.g. the inflation rate published by a statistical office is not 15%, but 14% (=ln1.15)? What of the two to use depends on the model and its specification. But for behavioural equations, I think, one would mainly use the %-version, because I know nobody, who thinks in logs. Most people even do not know what a log is and even much more people cannot calculate in logs (I can, but I have to translate dln to %, though). Why then should it influence their decisions?

What concerns the urbanization-infant mortality relation: -.42 really means a (not exact) reduction of 0.42% for 1% increase, but for high coefficients this relation is not valid any more. But even -34% is much too a high reduction. The main problem is that this relation cannot be considered as causal (as you rightly wrote in your former answer). I enclose a table with simulations of urb (upwards trend) and infant mortality (decreasing) which are independent of each other. Nevertheless, diagrams and regressions show rather good results. Always when correlating or regressing variables with trend, one can get wonderful results – but without any sensible meaning.

By the way, I am Austrian (not German, but German speaking), but many economic developments are similar in Austria and Germany.

Anton Rainer Sajjad Sargolzaei I attach a pdf with some analysis of corresponding data taken from the World Bank database. My answers are very different from those of Sajjad. Perhaps we are using different data sets. Leaving aside the problems of omitted variables there appear to be interpolated trends in the data. My regression reflects more the interpolation methods used by the compiler of the series. The y are not suitable for any model fitting exercise.

I have also tried to clarify some of my views on the use of dnls, exponentials, and percentages in economics.

John, I think, there is no much difference between our views. You prefer differential equations, I am more for difference equations. My reasons are, that I do not see any continuous process in the economy and (even if there were) all data are discrete. Obviously, economists wanted to copy the physicists and built mathematical models with integrals and differentials to show that they are "scientific", too (inferiority complex?). It may also be that continuous models are easier to handle (but often at the price of less reality). One of the main problems of these models is that they mostly blur the (principal) difference between stock and flow variables. One example is the Domar model (E.D. Domar: The “Burden of the Debt“ and the National Income; in: American Economic Review, Vol.34 (1944), where he used integrals of GDP and public debt to show that the dept-to-GDP ratio tends to the deficit-GDP ratio (d) divided by nominal GDP growth (g). If d=3% (Maastricht limit) and g=5%, one arrives at the 60% debt (EU limit). If one uses a more realistic discrete model (debt as a stock variable measured at the end of the year, GDP as a (not constant!) flow akkumulated over the year), the result is 63% (=d/g*(1+g)).

For a 1 unit increase in Health expenditure, the Infant mortality rate decreases by -0.39.