Hello, Hubert. It seems you are working hard and successfully.

As for taking logarithm of rates, there seems to be another reason. If you are led to take rates, it indicates that the time series you are observing is a multiplicative series. I say this in comparison to additive series.

Here is a rough idea. Let X(t) be a series of random variables, mutually independent and having two first moments (i.e. averages and variations). Then

Y(t) = Y(t-1)*X(t) ∀ t ∈ Z

is a multiplicative stochastic series and

U(t) = U(t-1) + X(t) ∀ t ∈ Z

is an additive stochastic series. For example, the amount of IOU, GDP of a nation, the price index can be considered as multiplicative series.

It is a well known theorem that a multiplicative series forms a normal distribution, when it is measured by logarithms and by subtracting trends. This theorem is easily derived form the theorem on additive series.

As you see it, taking logarithms is to transform a multiplicative series into an additive series. Many theorems are expressed for additive series and we are more accustomed to additive series than multiplicative series. For this reason, for example, we are more accustomed to take arithmetic average than geometric average. I do not why but it is possible that physical processes are additive rather than multiplicative.

In a multiplicative series, it is more natural to take rates in place of differences, but if you want to use theorems of stochastic process, you are led to take logarithm as it transforms multiplicative series into additive series.

I hope I am clear.

in my humble opinion there are two aspects here: 

1) pure technical aspect, that means that logarithms are not defined for negative values and for values between 0 and 1 they are negative. That means that if your growth rate is -0.05 which is -5%, then you will get a missing value. On the other hand an unemployment rate of say 0.08 or 8% will give you a negative logarithm. But one could just calculate 0.08*100 and get 8 to resolve this issue and avoid negative values. 

2) the interpretation, that means that if we consider marginal effects, a 1% change of unemployment rate (which is already a %) leads to a 1% change in ... 

Whereas 1) is crucial for the analysis, 2) is somewhat vague. I would say, that if you bear in mind the mathematical properties of a logarithmic transformation, it should not be wrong. 

i thought as much. Similar question troubled me. i had percent change in PPI data. I did not log because they were too small in magnitude and also the negatives are impossible to log.

Good morning.

The main question is "why"? I see two reasons (but I am sure there must be others).

One is based on the theoretical model: you have a non-linear model (e.g., an elasticity appearing as exponent) and you wish to estimate it using linear regression: logs make the trick.

Another is statistical: your observations are badly distributed (from a "normal" perspective: asymmetric, with most observations on one side of the mean and a few outliers on the other side). Taking the log of the variable gives you a better  shape in terms of normal distribution (symmetry as coincidence of mean and median, variance, etc.)

If the rate of growth is in the explanatory variable, the right way is to move to a time serie model by moving the lagged variable ( Y(t-1) ) to the right-hand side of the regression equation. If it is on the right hand side, well,... I cannot think right now about theoretical models having percentage rates as exponents in a non-linear model (but why not).

So, the empirical reason for using logs on rates (have a transformed variable behaving "normally") is probably the main reason. But you need first to check if the distribution of your original series is really so bad and if using logs improves it.

Negative is not an issue: a rate of change is the ratio X(t)/X(t-1) and should not be negative if both Xs are of the same sign. If X(t) is >< 0 for all (t), the ratio is always strictly positive. You may even use log(X(t) - log(X(t-1)) instead of X(t)/Xt-1), because it behaves better from a mathematical perspective (it is symmetric: the growth from (t) to t(+1)) equals the de-growth from (t+1) to (t), which is not the case with rates computed using  X(t)/Xt-1)).

What is usually referred as "negative" rates are those that give a result lower than 1 [ X(t) < X(t-1) ], and comes from the arithmetic way of presenting the variation in percentages: [X(t)- X(t-1)/X(t-1)] or [ X(t-1)/X(t-1) -1 ]. But using log(X(t) - log(X(t-1)) instead clears the way.

Hope it helps,

Hubert

Hello, Hubert. It seems you are working hard and successfully.

As for taking logarithm of rates, there seems to be another reason. If you are led to take rates, it indicates that the time series you are observing is a multiplicative series. I say this in comparison to additive series.

Here is a rough idea. Let X(t) be a series of random variables, mutually independent and having two first moments (i.e. averages and variations). Then

Y(t) = Y(t-1)*X(t) ∀ t ∈ Z

is a multiplicative stochastic series and

U(t) = U(t-1) + X(t) ∀ t ∈ Z

is an additive stochastic series. For example, the amount of IOU, GDP of a nation, the price index can be considered as multiplicative series.

It is a well known theorem that a multiplicative series forms a normal distribution, when it is measured by logarithms and by subtracting trends. This theorem is easily derived form the theorem on additive series.

As you see it, taking logarithms is to transform a multiplicative series into an additive series. Many theorems are expressed for additive series and we are more accustomed to additive series than multiplicative series. For this reason, for example, we are more accustomed to take arithmetic average than geometric average. I do not why but it is possible that physical processes are additive rather than multiplicative.

In a multiplicative series, it is more natural to take rates in place of differences, but if you want to use theorems of stochastic process, you are led to take logarithm as it transforms multiplicative series into additive series.

I hope I am clear.

My answer is that you should not given that growth rate may be negative. Also, if you choose to use the level (rather than the logarithm), it will change your interpretation of the estimated coefficient (semi-elasticity vs elasticity).

Mankan,

Hello, 

I agree with all what is presented by all the colleagues. Also what I learned from my experience while working on my PHD is that you need to check the other variables as well. If for example for you have 4 independent variables that don't have the same measurement as such : X1=number of people, X2= index , X3=values in billions of USD, x4=rate . Then yes it is better to take the log of all of your independent variables.

But here be careful while interpreting the coefficients. The analysis should be in terms of elasticity or % nit units.

2- Another reason to take a log of a variable is that if it has a trend.

Good Luck