# Linear and semi-log regression model

Is this equation:

$$\log{(y)} = a + bx$$

semi-log or log-linear mode (or it is the same thing)?

I have two models: linear (1) and semi-log (2). The values of $R^{2}$, adjusted $R^{2}$, and Standard Error are:

• Linear: $R^{2} = 0.6780,~\mathrm{adj.}~R^{2} = 0.6513,~~\mathrm{SE}=94.101$
• Semi-log: $R^{2} = 0.5803,~\mathrm{adj.}~R^{2} = 0.5455,~~\mathrm{SE}=0.5493$

How to interpret this values especially from the second model?

This is an answer to the first part of the question regarding the description of the model: $$\log{(y)} = a + bx.......(1)$$ It is important to distinguish: i) whether a model is linear in the sense of the Classical Linear Regression Model (CLRM), and ii) whether a model has linear functional form. Model (1) is linear in the first sense because it is linear in the parameters $a$ and $b$, and this is not affected by the log of $y$. Similarly, models (2), (3) and (4) below are all linear in the CLRM sense: $$y = a + bx.......(2)$$ $$y = a + b*log(x).......(3)$$ $$log(y) = a + b*log(x).......(4)$$ However, of the above models only Model (2) has linear functional form. Models (1) and (3) could both be said to have semi-log functional form, although it is better I suggest to be more precise and indicate which variable is logged by describing (1) as semi-log (dependent) and (3) as semi-log (independent). The functional form of Model (4) is sometimes described as log-linear and sometimes as double log.

I have never heard the term "semi-log regression" in 20 years. It may be in use in some substantive areas. Log linear analysis is something else - it is used when you have multiple categorical variables.

Both of your models are linear regressions. It's just that the second uses the log of y rather than y. $R^2$ has the same meaning as usual - it is the proportion of variance in y explained by the model. Adjusted $R^2$ is one way of penalizing for complexity. Since your model has only one independent variable, it is very close to the unadjusted $R^2$

EDIT answer to comment For SE of the regression see this article. It is entirely reasonable that this changes a lot when you change the scale of the dependent variable.

• @ Peter Flom Thanks for you answer. How to interpret SE values i.e. such a great difference between this two? Can SE be 94.101 when R2 = 0.6780 is much lower? Jul 17, 2014 at 9:41
• What is the SE value? What is it the standard error of? Jul 17, 2014 at 15:00
• @Navi to clarify Peter's last comment: in your equation I can off the top of my head think of four standard errors: (1) the SE of $\hat{a}$, (2) the SE of $\hat{b}$ the SE of $\hat{\log(y)}$ (i.e. the predicted value of the regression line), and the SE of $\tilde{\log(y)}$ (i.e. of the predicted value of $\log(y)$ for a given value of $x$. Which SE is your question about? Jul 17, 2014 at 16:39
• @Peter Flom Standard error of regression Jul 17, 2014 at 16:43
• OK, I will edit my answer. Jul 17, 2014 at 17:37

Semi-log and log-linear models are terminologies used in Econometrics. They are not the same.

Log-linear models $$Y = e^{\beta _1 }X_2 ^{\beta _2 }X_3 ^{\beta _3 }e^\epsilon$$ it is linearized as $$\log Y = \beta_1 + \beta_2 \log X_2 +\beta_3 \log X_3 +\epsilon$$ Here $$\beta$$-coefficients are elasticities. For this reason, the log-linear model is also known as the "constant elasticity model".

Semilog-linear models

these are useful when coefficients are interpreted as relative variations (rate of change). The original model is $$Y = \exp\left(\beta _1 +\beta _2 X_2 + \beta _3 X_3 + \epsilon\right)$$ and it is linearized as $$\log Y = \beta _1 +\beta_2 X_2 + \beta_3 X_3 + \epsilon$$ In this case $$100 \times (\exp(\beta_k) - 1)$$ measures approximately the expected percentage change in $$Y$$ for a unit change in $$X_k$$ holding all else equal.

R squared interpretation

If you fit a linear model that is a linearized version of a non-linear model the $$R^2$$ must be carefully handled.

The $$R^2$$ is the fraction of variability of the outcome variable (or dependent variable) captured by the regression function. But the previous is referred to the linear model.

Now, for the previous log-linear model the outcome variable is $$\log Y$$ while the regression function is $$\beta_1 + \beta_2 \log X_2 +\beta_3 \log X_3$$

In the case of the semilog-linear model example, the outcome is again $$\log Y$$, while the regression function is $$\beta _1 +\beta_2 X_2 + \beta_3 X_3$$.

Side comment on the use of the $$R^2$$

The $$R^2$$ is one of the most dangerous tools. It's almost worthless and should be removed from any introductory-intermediate statistics book. People use it following silly rules of thumb, but there are serious technical issues why its misuse can be dangerous.

Intuition: suppose you run a regression and you find a spectacular $$R^2 = 0.999999999999....$$ approaching its "ideal maximum value". You are happy, but this means that the estimated variance of error term in the regression is close to 0... this is a violation of the assumption that your observed outcome $$Y$$ is a random fluctuation around the conditional mean of $$Y|X$$ represented by the (linear) regression function.

In the best case, a too good $$R^2$$ is an indication that you are fitting noise, then the question is: how large should be $$R^2$$ before to worry about it? Nobody knows the answer, because this depends on the bias-variance trade-off. Since of this I strongly suggest looking at other better measures of fitting.