When you include dummy variables for categories, you generally leave one dummy out so that the design matrix $X$ is not rank deficient. (This is an extremely important, somewhat abstract concept.)
- Let's say you have dummies for Jan, Feb, ..., Nov, that is, you leave out the dummy for December.
- The coefficient on Jan is then the effect of being in Jan relative to December.
- Eg. if $b_2$ were .02, it would imply that sales are about 2% higher in January than December.
Changes in logs is conceptually similar to percent changes in levels.
Side note:
For $y_2$ near $y_1$, you have:
$$\log y_2 - \log y_1 \approx \frac{y_2 - y_1} {y_1}$$
That is, the log difference is close to the percent change (you can prove this using first order Taylor expand to linearize the log near 1). Eg. $\log(2.02) - \log(2) = .01$. As $y_2$ and $y_1$ get farther apart though, that approximation breaks down. Eg. $\log(1.4) - \log(1) = .3365$. 1.4 is 40% higher than 1, but the log difference is only .3365.
The simplest thing is to simplify and look at the math.
If $y=\beta{x}+\gamma\delta$, then $\exp(y)=\exp(\beta{x})\times\exp(\gamma\delta),$ where $\delta$ is your dummy and $\gamma$ is your coefficient.
When $\delta=0,\exp(\gamma\delta)=1,$ so your problem reduces down to $\exp(y)=\exp(\beta{x})$. When $\delta=1$, $\exp(\gamma\delta)$ is a positive contant. In this case it is $1.012174$. This is a multiplier that rescales the product of your other variables. Because it is greater than unity, it multiplies the consumer price index by an increasing amount over the factors.
When zero, the consumer price index is predictable from the factors alone. When non-zero, the scale of the factors is $1.012174$ greater than without the intervention.
An alternative way to think of it would be that in the presence of intervention, the effect of the $\beta$ terms is $1.012174$ larger.
$$CPI = \frac{1,2922}{GDP^{0,3525}}\times{Import^{0,4239}}\times{NEER^{0,2115}}\times{WAGE^{0,6291}}\times{1.012174},\text{if }\delta=1$$ and $$CPI = \frac{1,2922}{GDP^{0,3525}}\times{Import^{0,4239}}\times{NEER^{0,2115}}\times{WAGE^{0,6291}}\times{1},\text{if }\delta=0$$
Best Answer
You go right ahead and include that index (in levels) in the model you wish to estimate. The interpretation is the same always, when the x increase by 1 y increase by $\beta$.
You can log the index, provided it is never 0. Then you have a standard level-log model, with the semi elastic interpretation.
Also note (if you log the index), you can rescale the index to any base year you wish. It does not make any difference for the estimate. It will however change the intercept, but very often one does not care about the intercept.