Interpretation of regression coefficients when dependent variable is standardized

Question

When the dependent variable is standardized, how does one interpret the regression coefficients of continuous or categorical independent variables?

For instance, if we have $K$ groups in the data and the dependent variable is standardized using the mean and standard deviation of within each group. Now we regress this standardized dependent variable on the simple non standardized indepedent variables in a regression as below for each group separately:

$$Y_{it}^{g} = \alpha_{i} + \delta_{t} + \beta^{g} X_{it} + \varepsilon_{it}$$

Where $g$ represents the group for which we are running the regression. $i$ is the id and $t$ is say the year. c

Three questions on this:

1. Can I then interpret the coefficient $\beta^{g}$ as the increase in the dependent variable by $\beta^{g}$ standardized score unit as independent variable increases by one unit?
2. Does this kind of standardization facilitate a comparison of coefficients on $X_{it}$ between the different groups directly? So i mean comparing $\beta^{1}$ with $\beta^{2}$ ?
3. Can you think of any problems with such a regression?

Answer 1

1. If $X_i$ is continuous, you can easily interpret $\beta^g$ as the effect of a change of one unit of $X_i$ on the standard deviation of $Y$ (or on the standardized score). This is a valid approach which allows you to directly assess the magnitude of the relationship between $X_{it}$ on $Y_i$. This same applies to the case in which $X_i$ is categorical.

2. Does this kind of standardization facilitate a comparison of coefficients on $X_{it}$ directly? No, if you want to directly compare different independent variables (e.g. measured in different units), you have to standardize these variables too (if $X_i$ is continuous, if $X_i$ is categorical this is no problem since it is just the effect of moving from one category to another). You can directly compare the effect on $Y_{it}^k$ compared to say $Y_{it}^l$ if $Y^k$, $Y^l$ differ for example in the unit measured.

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Added in response to the comment / updated 2.): Regarding your updated question: You do not take the group-specific mean or variance into account if you standardize the dependent variable by subtracting it's mean ($\bar{Y} = \frac{1}{N\cdot T}\sum_i\sum_t Y_{it}$) and dividing by it's standard deviation. The group-specific means will still differ. If all group-specific means should be zero (and all standard deviations to be 1), you have to take into account the group-specific mean and variance ($\bar{Y}_i = \frac{1}{T}\sum_t Y_{it}$).

However, you assume that all effects are linear. This means that the effect does not depend on the level of the $Y$ (remember standardization should ease the interpretation of your results but the basic assumptions are the same). If you think that this might be violated, you should consider re-thinking your model (e.g. checking this assumption with quantile regression etc.) and looking and non-linear models. Standardization will not help you in this case.

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3. There may be of course all the problems related to linear regression models or longitudinal data in general. But this is a valid approach. You may think of standardizing $X_{it}$ too to ease interpretation (see 2.).