The explanatory variables don't need to be normal distributed. Only the distribution of the dependent variable should be (approximately) normal conditional on the explanatory variables.

It is more important that the (conditional) relationship of the explanatory variables and the dependent variable is linear.

The explanatory variables don't need to be normal distributed. Only the distribution of the dependent variable should be (approximately) normal conditional on the explanatory variables.

It is more important that the (conditional) relationship of the explanatory variables and the dependent variable is linear.

You will get a bias estimation!! The regression model will not be able to distinguish between your 8 explanatory variables and find out which one has been transformed and which one has not!! You can make sure by analyzing your data in both cases, transformed data and without it, and see how different outputs you will get!

I agree with Jochen. I wouldn't transform anything until I had viewed the scatter plots of the DV against each IV in turn. These should be approximately "cigar shaped". This requirement summarises three assumptions. The two most important ones are linearity and homoscedasticity (equal variance of the DV for different values of the IV).

Once you have run the model you should save the residuals and the predicted values and plot one against the other, looking for the same assumptions to be satisfied.

Thank you so much! I really appreciate your responses, Jochen, Ayoob and Peter. I checked linearity of each of the IVs with the DV by plotting in the scatter-plot. Except two explanatory variables, all others are approximately linearly related with DV (all untransformed). These two variables are not linear with DV even after log transformation. Should I omit these two variables from the model that are not linear with DV? The only problem that I am concerned is one of them was reported in the previous research as significantly related with the same DV in my dataset. My second question is: if normal distribution is not an assumption for explanatory variables, what is the rationale for log transformation? Is this only for meeting linearity assumption for all explanatory variables with DV? My third question is: can you guys suggest me a highly cited book that deals with regression diagnostics and how to satisfy regression assumptions?

If there is previous research on the relationship between that "problematic" predictor and your response, then you can see how these authors established linearity. If they didn't, I would be scpetical about the rest of their model and conclusion anyway.

Instead of looking at the scatterplots of each predictor vs. the response I would look for added variable plots (a.k.a. partial regression plots; https://en.wikipedia.org/wiki/Partial_regression_plot). It may be that the unadjusted relationship is non-linear, but when the other predictors are considered the relation is linear (or, if one is unluky, the other way around).

Predictors are transformed to linearize the relationship with the response (so that linear models can be used). This is the reason, not the distribution of these variables. For instance, consider masuering a calibration curve: you use different concentrations of the analyte and measure a signal. The aim is to find a regression line (tha calibration line or curve) that predicts the concentrations of (new and unknown) samples given their measured signals. As you select the concentrations, those are usually not normal distributed. Only the conditional signal is normal distributed (for each given concentration, the measured signal will scatter around a center, and that distribution can be assumed normal).

The log-transformation turns multiplicative changes into additive changes. Log-tansforming a predictor gives you a model describing a unit change in the response per proportional change of the predictor (like 1 unit increase per doubeling of the predictor value). If such a relationship makes sense, the predictor should be log-transformed.

Literature:

Gelman and Hill: Data Analysis Using Regression and Multilevel/Hierarchical Models.

Draper and Smith: Applied Regression Analysis (Wiley Series in Probability and Statistics):

Harrell: Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis (Springer Series in Statistics)

Chatterjee: Regression Analysis by Example (Wiley Series in Probability and Statistics)

Fox: Applied Regression Analysis and Generalized Linear Models

Thank you so much, Jochen.

You should remember about different interpretation of regression coefficients in lin-log and lin-lin models. In lin-log model coefficient of regression informs about relative change of explanatory variables.

In regression, it is the residuals (of your final model) that should have approx normality, you don't need normality in each IV. Check out the normality plot of your final model (see pic)

In the NORMAL P-P plot, you are hoping that your points will lie in a reasonably straight diagonal line from bottom left to top right. In the SCATTER plot (see enclosed), you are hoping that the residuals will be roughly rectangularly distributed, with most scored consentrated in the center (along the O point). You can also see outliers, i.e., cases that have a std residual >3 or

Yes you can estimate such a model but then the coefficients of the log transformed explanatory variables should be treated as elasticities (i.e log-log model) while the coefficient of the untransformed explanatory variable should be interpreted in line with log-linear model.

There are statistical and substantive consideration.

I'll focus on the former, which have been the object of less answers here, although Hanna's and Paul's answers are great in that respect.

I'd just add that taking logs is used when we are interested in relative changes (such as growth) instead of levels of a variable. That's why, following Paul, a log-log model is ideal to capture elasticities, while, following Hanna, a lin-log is ideal when you believe that relative changes in X give place to absolute changes in Y.

Regards from Madrid,

Andrés

I presume that you are looking at the normality of the residuals in your regressions when you are talking about normality. There is no requirements that the dependent or independent variables follow a normal distribution.

In deciding which variables need to be transformed I would first think about the model that I was estimating and the behabiour of the variables in the model. If one regarded the variable as changing by percentages rather than shifts in levels I would log transform the variable. I would not be concerned about mixing log transformed variables and variables in levels. For example in a money demand equation one would log transform money and the transat ion variable and put the interest rates in levels.

I would regard the normality test of the residuals as a specification test. Most failures of such a test are due to some extreme observations which can often be explained by some data problem or policy intervention which can be modelled ny some kind of dummy or intervention variable. There are many other regression diagnostic statistics (heteroskedasticity, auto-correlation, functional form, stability etc.)that you should also examine with a view to validating your model.

I am so grateful to everyone who added a response to my question. It has been tremendously helpful!

The immediate answer is yes, but some other conditions have also be considered as well. The first question is why you need to transform and leave the variables untransformed. When do we transform variables? log transformation helps to arrest or minimize the effect outlier variable on our dependent variable. the second reason is the theoretical underpinning of the variable; if elasticity interpretation is more theoretically sound than point change. otherwise, provided that if there is a theoretical background in the functional form of the variable it is possible to have such specification:

logy=bo+logx1+x2

yb=bo+logx1+ x2

logy=bo+x1+x2

In linear regression the only thing that should be normally distributed are the residuals - the explanatory variables and the response variable do *not* need to be normally distributed. The word *linear* in liner regression refers to the relationship between the response variable and the unknown (to be estimated) parameters in a model, *not* the relationship between the response variable and the explanatory variables. For instance, y = a + bx +cx**2 can be handled quite easily with *linear* regression.