BLP Instruments Are a Mundlak Device, Used Backwards
BLP Instruments Are a Mundlak Device, Used Backwards
I recently noticed a connection between BLP-style demand instruments and the Mundlak device from panel econometrics that I think is underappreciated. The algebra is not hidden; the two ideas just live in literatures that rarely cite each other and describe the same object in opposite terms.
The claim is this: BLP instruments are built from the same object as a correlated-random-effects (CRE) control, but they play the opposite role. The Mundlak device puts a group mean into the regression to absorb variation; BLP puts a close relative of that group mean on the excluded list to supply variation. Same object, opposite job. Most of the rest follows from that one reversal.
The setup
Let’s focus on the linear Berry (1994) logit specificatio, so we can reason about ordinary linear IV rather than the nonlinear BLP (1995) inversion. After inverting shares,
\[ \delta_{jt} \;=\; \ln s_{jt} - \ln s_{0t} \;=\; x_{jt}\beta \;-\; \alpha\, p_{jt} \;+\; \xi_{jt}. \]
Let \(\xi_{jt}\) be either some sort of unobserved demand shock or an unobserved quality index. Because firms price knowing the demand shock \(\xi_{jt}\), there is correlation between prices and the unobservable: \(\operatorname{corr}(p_{jt},\xi_{jt})\neq 0\). We need an instrument that moves price for reasons unrelated to \(\xi_{jt}\).
The classic BLP instrument is the sum of rival characteristics:
\[ z_{jt} \;=\; \sum_{k\neq j} x_{kt} \;=\; \underbrace{\sum_k x_{kt}}_{S_t} \;-\; x_{jt} \;=\; J_t\,\bar{x}_t \;-\; x_{jt}. \]
Clearly, \(S_t = J_t \bar{x}_t\). The market aggregate of characteristics is close what a Mundlak device is built from, averages of \(X\) variables. The rival sum is that aggregate with two adjustments: leave-one-out (the focal product is subtracted, since \(x_{jt}\) is already a regressor, so what is excluded is the rest of the market), and the own/rival split (with multiproduct firms, BLP separates the own-firm aggregate from the rival aggregate so the markup function can carry the oligopoly asymmetry). So a BLP instrument is a leave-one-out, own/rival-split Mundlak device built on characteristics.
Same object, opposite job
For a while, I thought that BLP and CRE are the same idea, but that’s not right. They use the same object under opposite identifying assumptions. The Mundlak route treats group-mean variation as a nuisance: it presumes \(\bar{x}_t\) correlates with the error and soaks it up. Note, this is also the fixed-effects route by Mundlak’s own result: including group means of the regressors reproduces the fixed-effects estimator, so “control for \(\bar{x}_t\)” and “absorb a market fixed effect” are two descriptions of the same thing.
The BLP route treats the leave-one-out version of that same variation as the instrument: it presumes rival characteristics are excludable from utility and uncorrelated with \(\xi_{jt}\). The validity of the BLP instrument is the assumption the FE approach treats as false. Given this, we can see that the two approaches are different.
If one fears endogenous product positioning or entry (e.g., characteristics chosen in response to demand shocks, so that \(\bar{x}_t\) is correlated with \(\xi_{jt}\)), then one wants to absorb \(\bar{x}_t\) as a control; however, the same correlation that motivates using \(\bar{x}_t\) as a control would disqualify it as an instrument. If instead one is willing to assume rival characteristics are placed without regard to a firm’s own demand shock, \(\bar{x}_t\) becomes a valid excluded instrument, and absorbing it would throw away exactly the variation used to identify \(\alpha\).
So \(\bar{x}_t\) is either the thing to purge or the thing to exploit. Same decomposition, opposite bin.
That reversal is the bridge between the two literatures. To an empirical IO economist, “BLP instruments are leave-one-out rival sums” is a definition. To a panel econometrician fluent in Mundlak and Chamberlain, the reversal is the news: the rival aggregate is the object their machinery exists to purge, and the BLP exclusion restriction is the negation of the CRE motivation.
Three shortcuts that fail
If BLP instruments are “just means,” three shortcuts suggest themselves. Each one fails in an instructive way.
Means as controls
Putting market FEs or a Mundlak device into the regression as controls, with price left as a raw regressor, does not recover \(\alpha\). An instrument and a control do opposite jobs: IV creates exogenous price variation; a control removes variation and leaves price still endogenous. Price endogeneity is within-market — even within one market, the high-\(\xi_{jt}\) product carries a high price — and market FEs do nothing to within-market correlation. Thus, this approach is essentially running OLS on price.
Mean of characteristics as an instrument
This works, but it is likely to be weaker than the BLP instruments. The exclusion restriction \(E[\bar{x}_t\,\xi_{jt}]=0\) is the standard BLP assumption. After own \(x_{jt}\) is partialled out, the excluded content of \(\bar{x}_t\) is rival characteristics, which shift markups. This approach treats own-firm and rival products symmetrically, and so it discards structure that give BLP IVs additional strength.
Market FE as an instrument
An off-the-wall idea is to instrument price with market dummies. There is a kinda-connection between BLP IVs and Mundlak device, and between Mundlak and Market FEs, so what about this? Can market FEs be like a nonparametric BLP instrument? Sadly, no.
Using market FEs as instruments makes the fitted price the (\(X\) adjusted) market average \(\bar{p}_t\), so \(\alpha\) is identified off between-market variation. The implied exclusion restriction is that market identity is orthogonal to \(\xi_{jt}\). But the market dummies (our FEs) span contains \(\bar\xi_t\), so the instrument is valid only if there is no market-level demand shock at all. Once high-demand markets also have high average prices, the instrument is correlated with the error. Note the mirror image of the BLP IV case: FE as a control needs the within piece clean; FE as an instrument needs the between piece clean.
Why characteristics work and but market FEs do not
Price moves for two distinct reasons. Cost and competition (e.g., nearby rivals, many firms, a rival cost shock) move markups and have nothing to do with own \(\xi_{jt}\). Demand (e.g., high \(\xi\), so the firm charges more because people want the product) is the endogenous channel. A valid instrument isolates the price variation from the first channel only.
BLP instruments price with rival characteristics, which move price through the cost-and-competition channel, and, under the exclusion, are unrelated to own \(\xi_{jt}\). Within-market variation survives to identify parameters. Rival characteristics are an input that exists (somewhat) independently of any one product’s demand shock, so they can shift the markup without touching \(\xi\). Mundlak mean-\(x\) uses market-average characteristics, so, under the exclusion restriction, are like BLP IVs, but there is less variation within market.
Market FE uses market identity, so the fitted price \(\bar{p}_t\) mixes both channels, and this becomes the main issue with it. The next section clarifies the issue.
Market FE includes unobserved correlation
This will get a little loose with notation, but I think it brings the point home.
The exclusion restriction is: \[ E[\,\xi_{jt}\mid x_{jt},\, X_t\,]=0, \] which is that unobservables are conditionally independent of observed covariates.
Thus, the empirical target is to capture variation tied to \(x\):
\[ E[\,p_{jt}\mid x_{jt},\, X_t\,], \]
which is only a function of characteristics.
However, if we used market FEs, then we get:
\[ E[\,p_{jt}\mid x_{jt},\, \text{market}_t\,] \;=\; E[\,p_{jt}\mid x_{jt},\, X_t,\, \text{Unobserved}_t\,], \] which necessarily includes and market level trends, \(\bar \xi_t\). That is, the market FE includes market level unobservable factors that are likely correlated with \(\xi_{jt}\). Additionally, this is why the Mundlak device works: it only uses \(X\) variables when making the market averages.
What if all firms were single-product
Suppose every firm has only one product per market, so the own/rival split has nothing to split. Do Mundlak-IV and BLP-IV coincide? Yes. BLP IVs are just leave-out sums; Mundak is (or can be) leave out averages (if we are not explicitly targting an FE equivalence). So with single-product firms, Mundlak-IV and BLP-IV become near-twins, both valid and differing only by count-weighting. However, if one includes market FEs as controls because one wants to control for market factors, then that will almost completely absorb all variation in the IVs.
Recently, there is another set of rival-characteristic IVs one can use that would survive adding market FEs as controls. BLP IVs are really just one approximation to \(E[p\mid x, X]\) that one could form, and there are technically infinite versions one could use. Differentiation instruments such as \(\sum_{k\neq j}(x_{jt}-x_{kt})^2\) still have variation. These IVs are still built as functions of rival characteristics (i.e., measure local competition in characteristic space), but there is stronger variation across products within a market. Their nonlinearity keeps them out of the span of \(\{\)market FE, own \(x\}\).
Takeaway
Every method here builds a fitted price from some conditioning set and identifies the price coefficient off that fitted price. The methods differ only in the conditioning set:
\[ \underbrace{x_{jt},\, X_t}_{\text{BLP and Mundlak-}x:\ \text{clean}} \qquad\text{versus}\qquad \underbrace{x_{jt},\, X_t,\, \text{Unobserved}_t}_{\text{market FE:\ dirty}}. \]
“Being in market \(t\)” conditions on a set that contains the unobserved demand shock; “\(x_{jt}, X_t\)” does not. BLP instruments are a Mundlak device run backwards, and they work for the same reason the Mundlak device, run forwards, would have removed the wrong thing.