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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:
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.
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.
