For my research, I am unsure whether I can convert these variables to LN or not as different papers suggest different things.
Natural Log converts a linear series from linear to non-linear. A linear series in an OLS regression will be 'explained' by the constant term. While adding a non-linear term will need to be explained by other non-linear IVs. The goal of regression design is to remove as much variation (Seasonality, Trend, Cyclicality, etc) as possible from the model a priori, so your explanatory variables (IVs) are only explaining the 'economic' or 'irregular' part of the DV. Adding natural log variables will probably undermine that design and make specification harder, not easier. As an aside: Some macro-level income variables are log-linear by nature. And taking the natural log of those variables converts them to linear, thus removing the non-linear parts of the series. This reduces the variation in the DV that you need to explain with IVs. So in that rare situation, taking the NL will be beneficial to your model design. Otherwise, it just makes specification harder, because you need to explain more variation.
I am not sure about taking the log of a rate, as it is ratio and generally the unemployment and inflation rates are double digit numbers, and talking its logarithm to reduce its magnitude doesn't make much sense, for income it is okay but for unemployment and inflation. I am not sure
The 3 rates are very different, in principle. Inflation (rate) and GDP growth are percentage changes compared to a past period (mostly one year before). These changes can become negative, too, and one may have problems with logs even for positive values, if they are very low. Unless one takes e.g. 2% as 2 (instead of correctly: 2/100=.02), all logs will be highly negative and practically senseless. By the way, percentage changes of X below 10% are very near to dlnX.
The unemployment rate is not a change, but a relation of positive number to another (hopefully much) higher number. Therefore, it is possible to take logs, but (apart from the high negative logs) I think one cannot find a model/equation specification where this makes sense.
In principle, the transformation of the variables is dependent on the specification. If the model is linear, it is not feasible to take logs. If the relation is multiplicative-exponential one must take logs for a linear regression. One can, of course, estimate in differences and take %X instead of dlnX (both are likely to bring the same result). For the linear equation, one can also take differences (dX instead of X). To estimate in differences lowers the danger to get (seemingly) good statistical estimates, which are inm in fact, spurious.
but the question why using Log?
Why not 1/1-Log
Why not square (log(X))
Why note other transformation?
@T.bh the essence of taking the log form is to reduce the magnitudes and bring to par with other variables in the model.
For the variables mentioned, they're already rates and therefore similar in form. You therefore do not need to transform them. If you have other variables in your model that are not in rates, you can only transform such to maintain similarities with the aforementioned.
Logs are not for reducing the magnitudes. For a linear relation between two varaibles with very different magnitudes, one can simply divide (multiply) the higher (lower) one by a round number (e.g. by taking kg instead of g or billions instead of millions). One could also calculate indices. It is not allowed to take logs in this case. On the other hand, if you want to apply a linear statistical method on exponential relations, you must take logs (or their differences or percentages as approximations for dln).
I believe a cautious approach has to be applied. When considering GDP growth rate, Inflation rate, or Unemployment rate all are treated data. They are already the outcome of the raw data. So applying Log would not be suggested unless you have some specific assumption or model applied for.
The feasibility of transforming data into logs or not does not depend on the qualtity (treated or not) of the data, but only on the specification of the model/equationsm one wants to test by econometrics. This may be difficult, because causal effects between GDP development and inflation are likely to go in both directions (simultanuous equation bias). For unemployment, it may be simpler, because I think, inflation has no direct effect on UE, but it likely has positive and negative effects on demand and therefore GDP. BTW, GDP growth rate and inflation (=change of price index CPI) are very near to dlnGDP and dlnCPI, respectively. Therefore, it is unlikely that you find a sensible function, where you have to take logs (apart from the problem, that negative rates could happen).
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