Unfortunately, the problem you will run into here is one of endogeneity. The observed p and q are market equilibrium prices and quantities, not "samples" of points along a demand curve. While occasionally you may get lucky, usually if you naively regress quantity on price, you will end up with a positive coefficient. According to economic theory, this doesn't make sense (unless you've actually stumbled upon a Giffen good), but what one is typically observes are different levels of demand, and prices adjusting accordingly.
Therefore, to estimate accurate demand coefficients on price, it is typical to either estimate a system of equations (much as one solves a simple supply-demand problem in intro-micro), or to use instrumental variables. In either case, for identification, it requires some variables that affect only the supply side of the equation.
And finally, if you've solved all that such that you can overcome the endogeneity issues with your regression, remember that while taking logarithms is often a convenient way to do regressions, it implies a fundamental distinction about the model and the error terms, and if that doesn't fit the actual process generating the data, the "elasticity" you get will be the right units, but won't be the right number.
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
Well your problem, is the
Volume ~ Function(Price)and that Beta can give you the elasticity, but in your caseVolumeis not only a function ofPricebut also demand. I would solve the problem two ways:Volume ~ F(Price, Some Quantifier which is correlated to Demand that impacts demand irrespective of Price)For example Time, as time goes up Demand goes up