In Stata, you can calculate this like this:
. sysuse auto, clear
(1978 Automobile Data)
. gen lnx = ln(mpg)
. qui probit foreign lnx weight, nolog
. margins, expression(normalden(xb())*_b[lnx]/100)
Predictive margins Number of obs = 74
Model VCE : OIM
Expression : normalden(xb())*_b[lnx]/100
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | -.0070446 .0022689 -3.10 0.002 -.0114915 -.0025977
------------------------------------------------------------------------------
This can be interpreted as saying that a 1% increase in miles per gallon is associated with a 0.007 reduction in the probability of the car being imported (on a [0,1] scale), holding the weights of the cars constant.
In general, you cannot use plain OLS on market data to estimate demand elasticity. The one, unusual, exception would be if supply was not price sensitive (the supply curve was a straight up and down line with the same price for every quantity). The same goes for supply elasticity. See this link which explains it well. You can also see Greene's Econometric Analysis textbook (edition 5 or 7, 5 is available for free in pdf online) which covers this topic thoroughly. I will make an informal attempt at explaining below.
Suppose we want to estimate a linear supply and demand curve, something like;
$$
\begin{array}{llc}
q^s_t &= \alpha_0 + \alpha_1 p_t + \alpha_2 w_t+ u_t&(Supply) \\
q^d_t &= \beta_0 + \beta_1 p_t +\beta_2 y_t+ v_t&(Demand)
\end{array}
$$
Where $q^s_t=q^s_t$ is log quantity, $p_t$ is log price, $w_t$ are the log input costs, and $y_t$ is log consumer income. (you can have other explanatory variables too if you like, but this is the basic case).
With the above specification, the demand and supply price elasticities are
$$
\frac{\partial q^s_t}{\partial p_t}= \alpha_1\;\;(Supply)\;\;\;\;
\frac{\partial q^d_t}{\partial p_t} = \beta_1\;\;(Demand)
$$
BUT THE ABOVE CANNOT BE ESTIMATED WITH OLS because price is endogenous. Why? because, for the market to clear, price and quantity are decided simultaneously with both demand and supply side mechanisms. For example, a firm may change it's prices in response to increases or decreases in it's competition. Such variation in prices is not exogenous but rather caused by shifts in $q^s_t$. This reverses the order of causality that would be implied by an OLS regression model of the above supply and demand functions.
Demand elasticity is generally upward bias when estimated with OLS and I am pretty sure supply elasticity is downward bias (but check the references above too, I think the bias is usually attenuating (toward 0) in all cases).
There are two prominent econometric methodologies for estimating demand/supply curves. These are
Structural equation modeling: $$ \begin{bmatrix}q_t \\ p_t \end{bmatrix} = \begin{bmatrix}\pi_0 \\ \omega_0 \end{bmatrix} + \begin{bmatrix}\pi_1 & \pi_2 \\ \omega_1&\omega_2 \end{bmatrix}\begin{bmatrix}w_t \\ y_t \end{bmatrix}+\begin{bmatrix}e_{1,t} \\ e_{2,t} \end{bmatrix}$$ which is a multivariate likelihood.
Instrumental Variables (as suggested in your question). In the case of demand this is: $$\begin{array}{llc}
p_t &= \omega_0 + \omega_1 w_t + \omega_2 y_t + e_t&(First\;Stage) \\
q^d_t &= \beta_0 + \beta_1 \hat p_t +\beta_2 y_t+ v_t&(Second\;Stage)
\end{array} $$
$w_t$ works as an instrument because it acts like an exogenous “shifter” in supply. The basic idea is that by moving the supply curve up and down and recording the equilibrium quantities and prices, we can trace out the demand curve. The assumption here is that the suppliers cannot control their input costs (this is not always true when suppliers can negotiate their input costs as in oligopsony but we assume it is here). For supply we would use the same methodology but with $y_t$ as the instrument.
Best Answer
I guess Promo_1 and Promo_2 are categorical variables, in this case binary, representing the presence of the promotion.
The elasticity of interest here is the price elasticity of demand, in the presence of Promo_1. So you need to compute:
$$ dD/dPA * PA/Q = dlogD/dlogPA $$ for Promo_1 = 1. Here you have a log-log specification, so the parameters are elasticities. Differentiate the model equation with respect to logPA, you obtain b1 + b5*Promo_1.
Hence, in the presence of the promotion, i.e. Promo_1 = 1, the estimated elasticity is b1+b5.
Edit: For the record, estimating demand function has a long history in econometrics because of the potential endogeneity issues. You should probably look up for 2SLS and structural models.