_a_ 15 _/a_ 35, and _a_ 45 _/a_ 35 are less than the upper bound. Since the analysis is at quarterly frequency, I choose 0.077 as an upper bound for the one-quarter oil supply elasticity. This value is a conservative estimate of the upper bound given that it is three times the one-month oil supply elasticity estimate used in Kilian & Murphy (2012). Moreover, in comparison to a small one-month price elasticity of oil supply, evidence from other studies suggest a small price elasticity of oil supply at quarterly frequency as well. For example, Newell & Prest (2019) use microeconomic data to estimate a onequarter oil supply elasticity of 0.017 for conventional crude oil.

However, it can be argued that short-run price elasticity of oil supply would be higher when incorporating shale oil production. Empirical findings suggest a one-quarter supply elasticity that is close to zero even for shale oil production (see Newell & Prest, 2019; Kilian, 2020). This result does not invalidate the common view that, relative to conventional producers, shale oil producers are more agile in responding to movements in oil prices. It only implies that it takes more than three months for both conventional and shale oil producers to respond to changes in oil market conditions (Kilian, 2020). Moreover, based on survey evidence from U.S. shale firms, Golding (2019) states that “the average horizontal well pad in the Permian Basin takes four to six months from the commencement of drilling to production coming online,” and for wells that are drilled but not hydraulically fractured “it may take one to three months to go into production.”

**_Estimation._** The sign restrictions are imposed using the procedure developed by Rubio-Ramirez, Waggoner, & Zha (2010). The procedure involves generating a random orthonormal matrix _Q_ such that _QQ_<sup>_′_</sup> = _I_ by obtaining the _QR_ decomposition of a _K × K_ matrix _X_ of independent _N_ (0 _,_ 1) values. I then let _B_ be a lower triangular matrix corresponding to the Cholesky decomposition of the variance-covariance matrix of the reduced-form residuals, Σ _u_ = _PP_<sup>_′_</sup> , obtained from estimating the model (6). Thus, the following equality holds: Σ _u_ = _PP_<sup>_′_</sup> = _PQQ_<sup>_′_</sup> _P_ = ( _PQ_ )( _PQ_ )<sup>_′_</sup> , where _PQ_ = _B_ .<sup>17</sup> The matrix _B_ is a candidate structural impact multiplier matrix, and we can verify whether its elements satisfy the maintained sign restrictions discussed in Section IV.2.2.

The procedure can, thus, be summarized by the following steps:

1. Draw an independent standard normal _K × K_ matrix _X_ . Obtain the QR decomposition of _X_ such that _X_ = _QR_ and _QQ_<sup>_′_</sup> = _I_ .

2. Compute the the candidate solutions _B_ such that _B_ = _PQ_ and _P_ is the Choleksy decomposition of the reduced form residuals Σ _u_ .

3. Use the candidate solutions to compute the impact effects of structural shocks.

4. Repeat step 1-2 until the desired number of iterations. Record each candidate solution and the corresponding impulse response that satisfies the restrictions.

> 17Note that unlike _P_ , the matrix _B_ = _PQ_ is not a lower triangular matrix and, thus, does not depend on the ordering of the endogenous variables in the model (5).

16