The decision to use levels of a variable should be based on the underlying economic model that you are estimating.

Consider a variable such as the Consumer Price index. When a shock hits the consumer price index (CPI) it will increase by a percentage. I.e. the effect is multiplicative. If the index is at 100 a shock may raise it to 102. If the CPI is at 200 some years later the same shock will raise it to 204. Taking logs

ln(100) = 4.6052, ln(102) = 4.6250 = ln(100)+0.0198

ln(200) - 5.2983, ln(204) = 5.3181 = ln(200) +0.0198

When you transform to logs the shock becomes additive and has the same value.

Consider something like an interest rate. One considers that a shock adds a certain number of basis points to the interest rate. It is often the case that the effect of the shock is the same regardless of the actual level of the interest rate.

Most of the statistical tests mentioned are simply confirmation of the model that you are estimating. If you fail various specification tests (heteroskedasticity, serial correlation, functional form, stability, etc.) it is more a problem with your underlying model. Perhaps you have not chosen the right transformation but most times there is more wrong than that.

Using the above reasoning variables such as GDP, Consumption, investment, money demand, prices and similar should be log-transformed. Variables such as interest rates, asset returns, and similar should not be transformed.

GDP and log(GDP) are both non-stationary. (A well-behaved function of a non-stationary variable is non-stationary). However, log(GDP) is usually found to be I(1). If this is so GDP is not I(1). Choosing the correct version when working in a VAR or VECM is important.

Nana Musa There is no normality requirement for the variables in a VAR or VECM to be normally distributed

It depends on the features of your data.

To check if your transformation works, let me recommend X'_vs_lag(X') scatterplot, where X' is transformed X.

This poster shows a framework for visual analysis involving time series transformations:

Poster A Visual Framework for Longitudinal and Panel Studies (with ...

More examples in this article: Article Informational Value of Dividend Initiations: Impact of Cash ...

Apart from the log-transformation you can use different alternative methods. You can use this quick guide to try to select a good transformation for your data:

https://xdplot.blogspot.com/p/data-transformations-guidelines.html

Thank you so much dear Andrey Davydenko .

HTH! Please let me note:

Generally, if you transform your time series and calculate forecasts and then back-transform forecasts to the original scale, then you need to use MAE to measure accuracy, more details here (page 3):

Chapter Forecast Error Measures: Critical Review and Practical Recommendations

If you have many time series, please let me recommend the AvgRelMAE metric proposed in (Davydenko and Fildes, 2013):

Article Measuring Forecasting Accuracy: The Case Of Judgmental Adjus...

A more general workflow for measuring forecasting accuracy is proposed here: Conference Paper Data Formats and Visual Tools for Forecast Evaluation in Cyb...

It depends on data and model used. You should read through several papers related to your scope, hope that you will know whether they take logarithm or not.

All right, but it is not just a question of forecasting accuracy. If a log transform is needed, the forecast intervals will be dissymmetric, going up more than down. Another way to assess the need of a transform is to look at residuals. If their magnitude grows with the observations, it is indication that a transform is needed.

Great response dear professor Guy Mélard . In fact I run a SVAR model with and without log transform, the results were so contradictory with Log transformation !!.

Thank you again dear.

Log transformation becomes necessary when the series in your model are not normally distributed and burdened with outliers. Secondly, when you are estimating a non-linear relationship or when the model generated from the series is not stable as may be shown by cumulative sum (CUSUM) plot, it is advisable to transform such series. Log transformation enhances additivity, linearity and validity.

The decision to use levels of a variable should be based on the underlying economic model that you are estimating.

Consider a variable such as the Consumer Price index. When a shock hits the consumer price index (CPI) it will increase by a percentage. I.e. the effect is multiplicative. If the index is at 100 a shock may raise it to 102. If the CPI is at 200 some years later the same shock will raise it to 204. Taking logs

ln(100) = 4.6052, ln(102) = 4.6250 = ln(100)+0.0198

ln(200) - 5.2983, ln(204) = 5.3181 = ln(200) +0.0198

When you transform to logs the shock becomes additive and has the same value.

Consider something like an interest rate. One considers that a shock adds a certain number of basis points to the interest rate. It is often the case that the effect of the shock is the same regardless of the actual level of the interest rate.

Most of the statistical tests mentioned are simply confirmation of the model that you are estimating. If you fail various specification tests (heteroskedasticity, serial correlation, functional form, stability, etc.) it is more a problem with your underlying model. Perhaps you have not chosen the right transformation but most times there is more wrong than that.

Using the above reasoning variables such as GDP, Consumption, investment, money demand, prices and similar should be log-transformed. Variables such as interest rates, asset returns, and similar should not be transformed.

GDP and log(GDP) are both non-stationary. (A well-behaved function of a non-stationary variable is non-stationary). However, log(GDP) is usually found to be I(1). If this is so GDP is not I(1). Choosing the correct version when working in a VAR or VECM is important.

Nana Musa There is no normality requirement for the variables in a VAR or VECM to be normally distributed

Thank for your answer Prof. John C Frain. It is good enough I think.

Dear Chellai

You have been provided with two good answers from Guy and John Frain. It does heavily depend on the nature of the underlying nature of the model and the data.

Specification is an important feature of the literature on Demand equations for which the specification has implications for the theory. I have some discussion of this in the introduction to Burke and Hunter (2005), which you can find in RG. There are particular models such as Almost Ideal Demand Systems (AIDS) that aggregate perfectly and exhibit certain nice features. In the levels form the dependent variable is a proportion the budget share and the right hand side is in logarithms. I estimated for JNJ Muellbauer various Stone Demand Equations in dynamic error correction form and the DYNAIDS model in error correction form can be estimated by regression without imposing the full non-linear restrictions implicit in the model. However, the budget share is bounded between zero and one implying that it will not usually be conventionally non-stationary not that that greatly affects estimation in difference and levels as a VAR (I set this as an exercise for various of my students in Applied Econometrics and Microeconomics at Southampton, Surrey, Cardiff and Brunel. I now teach Finance students so generally different models apply though I believe David Blake has applied DYNAIDS in Money/Finance context. These types of equations also aggregate to a characteristics adjusted version of the consumption function that also would have a dynamic form. The log/log model does not aggregate perfectly and more fundamentally it violates adding up. Similar discussions might also be considered in the context of production functions.

At the macro or aggregate level then logarithmic models are popular, because the dependent variable in difference form defines a growth or an inflation rate which is a proportion while an equivalent dependent variable specifications in pure differences arise as a result of dividing through by the lagged level to create a growth rate and conceptually that implies everything on the right hand side divided by this value so it has implications for either the original specification or the one which is left following this rescaling. Hence, the same transform in relation to the model in logs is much more convenient.

This convenience is bought at a cost that these relations arise differently and thus do not generally follow from the above discussion of microeconomic theory. This is probably most difficult for quantities that arise from demand relations.

In Finance the logarithmic form is convenient and has a logic that complies with Finance theory. The change in the logarithmic price is a continuously compounded return and this is also consistent with the observation that some financial series are bounded at zero and may often be well approximated by a log normal distribution. Hence, the log transformation implies that the resulting series are normal and their difference, which is the return (see the chapters on Stochastic Processes and Black Scholes in John Hull - Options and Futures, Person various editions). Again this is not without implications as daily and weekly returns will not usually be normal. However, there are some useful results in Finance that suggest that under fairly broad assumptions as to the behaviour of the underlying price series that in the aggregate a central limit theorem applies and they tend to normality. In simple terms, the quarterly difference in the log of share prices is equivalent to the sum of their daily changes and it is that type of sum that may be normal.

The natural log data will also be better behaved meaning that in first difference they are more likely to be homoscedastic while over time the conventional random walk model that although being consistent with market efficiency in the sense that tomorrows price is not impacted by the past history of prices, but when the series evolves in this nominal form the process cannot have constant variance and drift rate (the random walk or AR model with unit coefficient and non-zero intercept) will not be coherent. As time evolves and the dependent variable increases then the rate of increase diminishes pro-rata. In the limit the drift rate tends to zero and more worrying the same applies to the variance. So as the value of a portfolio as described by the nominal value of an index becomes less risky. Possibly explains why buy hold works so well for the ordinary investor (joke). The same does not apply to the discrete version of the Ito process, in which drift and the volatility evolves over time in proportion to the time stock price or index value. Put simply, the Ito process implies the change in the stock price depends on the drift rate times the stock and the error with a volatility being a unit variance times the stock price. In discrete form this implies that the ration of the difference to the stock which measure the return has a constant mean and variance. Or equivalently given the precise selection of divisor is not precisely defined or as the time frame tends to zero in the limit this approaches the derivative, then for small differences any measure of the rate of change will tend to the same and thus the continuous rate of return given by the change in the log is equivalent as is the log form of this model from which the stock price in logs evolves as a random walk which has constant drift and constant variance.

If prices or other economic series follow similar diffusion processes, then the log form of consumption and income may be reasonably well approximated by random walks.

Another less easy way would be to use a Box-Cox transformation of the variables, but this specification would no-longer be linear.

Many thanks dear professors John C Frain John C Frain and John Hunter .

I appreciate your responses.

Best wishes

Like usual, John's comment is almost perfect. Theory determines the specification of the model (equation) one wants to estimate. But that means, that it may not be feasible to take the logs of GDP, consumption etc., if you think that these variables are linearly dependent on explaining variables. Another problem with logs for a behavioural equation is, that people do not think in logs (most do not even know what a log is). Therefore, for multiplicative relations, I recommend to formulate the specification in percentage changes (which - by the way - are near to dln as long as changes are not too high). If the relations between the exogenous variables are fairly constant over time, one could get (seemingly) good estimates with logs (or percentages) even for a principally linear relation. But one might have problems to interpret the coefficients.

Dear Chellai.

It depends on the type of data and the economic model you are looking at for your analysis.

But when you series are not normally distributed, log transformation becomes necessary.

Regards