Transform regression coefficients in semi-elasticities for different functional forms

Question

I have a number of studies examining the impact of corporate diversification ($D$) on firm value ($Q$) using standard regression models like:

$$Q_{it} = \beta_1 + \beta_2D+\gamma X'+u_{it} \tag 1$$

The subscripts indicate the time ($t$) and the firm ($i$). $X'$ is a vector of control variables. I am interested in the value of the coefficient $\beta_2$, which shows the "diversification premium" ($DP$).

The functional forms of the regression models above are somewhat different across studies. In detail, four different types are observable:

1. Linear model as above
2. Log-linear model, i.e. the dependent variable is $log(Q_{it})$
3. $D$ is a dummy variable with 1 indicating a diversified firm and 0 indicating a non-diversified firm
4. $D$ is a continuous variable measuring the degree of diversification

My aim is to make the estimates for $\beta_2$ comparable across the four types of cases mentioned above. Thus I want to convert $\beta_2$ such that it shows the value premium of diversification, i.e. the percentage change of firm value for a diversified firm (if $D$ is a dummy) or for an average diversification level ($D$ is a continuous variable). I want to use the following transformations:

1. If $D$ is a dummy & Linear model (as in Equation (1)) -> $DP = \beta_2(1/\bar{Q})$
2. If $D$ is a dummy & Log-linear model with $log(Q_{it})$ -> $DP = EXP(\beta_2)-1$
3. If $D$ is a continuous measure & Linear model (as in Equation (1) -> $DP = \beta_2(\bar{D}/\bar{Q})$
4. If $D$ is a continuous measure & Log-linear model with $log(Q_{it})$ -> $DP = EXP(\beta_2\bar{D})-1$

As I am not totally familiar with transformations of regression coefficients from models with different functional forms, I want to ask if these transformations are correct?

Answer 1

"Thus I want to convert $\beta_2$ such that it shows the value premium of diversification, i.e. the percentage change of firm value for a diversified firm (if $D$ is a dummy)..."

"Percentage change" is essentially a mark-up. This is expressed mathematically as

$$\text{%}DP \equiv \frac {\text {(average)Change in firm value}}{\text{(average) Firm value without the diversification premium}}$$

$$=\frac {DP}{\bar Q_d-DP}$$

So
1. If $D$ is a dummy & Linear model (as in Equation (1)) : $\hat \beta_2$ estimates the level of the premium ($DP$), but for those firms that are characterized as "diversified" (i.e as a discrete marginal change). So to obtain a percentage expression for the denominator you should consider the average firm value of only those firms that are diversified:

$$\text{%}DP = \frac {\hat \beta_2}{\bar Q_d - \hat \beta_2}$$

This is what a firm's value gains in percentage term due to diversification on average.

2. If $D$ is a dummy & Log-linear model with $\log(Q_{it})$ as the dependent variable: $$Q_{it} = \exp\{\beta_1 + \beta_2D+\gamma X'+u_{it}\}$$

and indeed here

$$\text{%}DP = \exp\{\hat \beta_2\}-1$$

Note that due to the multiplicative nature of the original non-linear expression, $\hat \beta_2$ inherently represents and is estimated as an average mark-up factor (i.e. level-free), and this is why the average value of the firms does not appear in the calculation.

"...or for an average diversification level ($D$ is a continuous variable).

I guess you can take it from here.