Khwaja-Mian (2008) Estimator

DateJuly 2020
KeywordsCredit ChannelFixed Effects

In this post, I explain the Khwaja-Mian (2008) estimator, which is a technique used to control for credit demand shocks in order to identify credit supply shocks i.e. the "bank lending channel."


Authors introduce time-variant firm fixed effects to fully capture the demand for credit by focusing on firms with multiple banking relationships. They actually do more than this, but subsequent papers citing this paper only refer to the inclusion of fixed effects.


Khwaja and Mian (2008) examines the credit supply consequences of the run on bank deposits.

They exploit differential shocks to the liquidity of banks due to unanticipated nuclear tests in Pakistan and find that a bank that experienced a one percent larger fall in liquidity reduces lending by an additional 0.6 percent.

What they do in the original paper

In an attempt to isolate the credit supply channel, they need to control for loan demand.

However, loan demand is not fixed over time — it is, though, for a borrower within each single time period. Therefore, if we can have more than one observation per borrower for each time period, we can then control for loan demand.

Loan-level Data

The first key step is to obtain a panel of bank-firm specific loans. This step is critical because you need to observe credit lines of multiple banks for each firm.

Empirical Problem

The banks with a lot of dollar-denominated deposits experienced a sharper deposit run. It could be that investment (loan) demand may be dropping more for firms that traditionally borrow from dollar-denominated banks.

To see this more clearly, consider the change in loans for firm jj made from bank ii (ΔLij)(\Delta L_{ij}). A simple OLS would regress this on the change in deposits at bank ii (ΔDi)(\Delta D_i)

(1):ΔLij=β0+β1ΔDi+ηj+ϵij(1): \Delta L_{ij} = \beta_0 + \beta_1 \Delta D_i + \eta_j +\epsilon_{ij}

where ηj\eta_j is the firm-specific characteristic that is unobserved.

Then the concern is Corr(ηj,ΔDi)>0Corr(\eta_j, \Delta D_i)>0 which implies β1OLS>β1\beta_1^{OLS} > \beta_1.

Restricting Analysis to Firms with Multiple Banking Relationships: Bank Lending Chanel

To confront the above empirical challenge, authors introduce firm-fixed effects (FE) in the first-differenced data. Therefore, the authors compare how the same firm's loan growth from one bank changes relative to another more affected bank.

Therefore the equation that they estimate is:

(2):ΔLij=γj+γ1ΔDi+ϵij(2): \Delta L_{ij} = \gamma_j + \gamma_1 \Delta D_i +\epsilon_{ij}

where γj\gamma_js represent the firm fixed effects.

To the extent this within firm comparison fully absorbs firm-specific changes in credit demand, the estimated difference in loan growth can be plausibly attributed to differences in bank liquidity shocks.

Comparing FE vs. OLS Estimates for Additional Insight: Firm Borrowing Channel

Authors also use the firm fixed effects to provide a conservative estimate of the impact of the liquidity shock on firm's total borrowing.

Specifically, the equation that they estimate is:

(3):Yj=β0F+β1FΔDˉj+ηj(3): Y_j =\beta_0^F + \beta_1^F \Delta \bar{D}_j + \eta_j

where Dˉj\bar{D}_j is the average liquidity shock faced by firm jj's pre-shock banks. The idea is that if the liquidity shocks have no impact on firm borrowing, then β1F\beta_1^F should be zero.

Here's where the authors really shine. Equation (3) is subject to the same endogeneity concern as the equation (1): Corr(ηj,ΔDi)0!Corr(\eta_j, \Delta D_i)\neq 0!

Unlike before, however, we can no longer focus on firms with multiple banking relationships and use firm fixed effects. Thus the second-best option is to hope that the sign of the correlation is negative so that the OLS estimate can be interpreted as a conservative estimate.

And it indeed turns out that the sign is negative!

Comparing columns (3) and (4), we see that the OLS estimate drops from 0.60 to 0.46. The drop in the OLS coefficient implies that a bank’s liquidity supply and its client firms’ loan demand shocks are cross-sectionally negatively correlated. Consequently, OLS provides an underestimate of the true effect

Some Comments

  1. Clearly, Khwaja and Mian (2008) does more than just to introduce time-variant firm fixed effects by looking at firms with multiple banking relationships. The clever part of the paper is to use the results to put a sign on a particular covariance of interest and use it to justify the interpretation of an alternate regression.
  1. When people cite the methodology in Khwaja and Mian (2008), they typically refer to the introduction of fixed effects rather than the putting a sign of the covariance.

Follow-up work using this method