RDP 2013-05: Liquidity Shocks and the US Housing Credit Crisis of 2007–2008 Appendix A: Identifying the Effect of Credit Supply Shocks

In this Appendix I outline a model that describes how credit demand and supply shocks affect mortgage lending. The model is closely related to that of Khwaja and Mian (2008). The purpose of the model is to highlight the identification problem and explain the construction of the estimator that controls for credit demand shocks. In this simple model I assume that each bank i lends to region j at time t the amount L_ijt. There are two time periods. I assume that each bank can lend to only one region, but a region can borrow from multiple banks. On the credit supply side, I assume that each bank loan to a particular region is financed through a combination of securitisation S_it and other forms of external financing W_it. These assumptions generate the following flow of funds constraint:

I assume that the bank can securitise loans at no cost, but, importantly, there is a quantitative limit on how much the bank can securitise (i.e. S_it < ). Beyond this limit, the bank must turn to costly external finance to support higher levels of lending. These assumptions ensure that the level of securitisation matters to the lending decision of each bank. If there was no limit on securitisation and/or no cost of accessing external finance, then the funding structure of the bank would not affect lending.^[15] Under this set of assumptions, the marginal cost of lending for the bank is solely a function of the volume of wholesale debt (i.e. external finance):

The cost parameter (γ > 0) denotes the slope of the marginal cost curve. On the credit demand side, I assume the marginal loan return is given by the following equation:

The borrower quality parameter allows for variation in loan returns across regions. Given the slope parameter (α) is a positive constant, the formulation assumes that there are diminishing marginal returns to borrowing. I solve for the first-period equilibrium by equating marginal revenue with marginal cost and substituting the flow of funds identity:

where the superscript ‘*’ denotes equilibrium. At the end of the first period, the credit market experiences two types of shocks:

1. Credit demand shock:
2. Credit supply shock: .

The credit demand shock consists of two terms – an aggregate shock that is common to all regions and an idiosyncratic shock that is specific to each region (η_j). In terms of the econometric framework, the aggregate credit demand shock might be an unexpected change in US monetary policy while the region-specific demand shock might be a shock to regional house prices. The credit supply shock also consists of two terms – an aggregate shock that is common to all banks and a bank-specific shock (δ_i). The aggregate credit supply shock might reflect some change in financial regulation that affects the ability of banks to securitise loans while the bank-specific credit supply shock could reflect each bank's ability to securitise assets.

Following the same approach as before, I solve for the second-period equilibrium:

As the two solutions are linear, I can then take the difference in (equilibrium) lending over time to obtain:

The change in the amount of each loan consists of two terms. The first term on the right-hand side denotes the impact of the region-specific credit demand shocks. The second term denotes the impact of the bank-specific supply shocks. If there is no cost of external finance (γ = 0), the credit supply shocks will not affect the equilibrium growth rate of lending; lending growth will only be a function of demand shocks. In other words, credit supply shocks only matter if there are financing frictions on the lender side.

Now suppose I re-arrange the equation to combine all the aggregate shocks in a single term and have two separate terms for the bank-specific and region-specific shocks:

If I assume that the share of loans that are securitised (SALESHARE_i) by each bank is a suitable proxy for the bank-specific credit supply shock (δ_i) then I could run the OLS regression:

where there is an intercept that captures all the aggregate effects , a slope coefficient that captures the relationship of interest and a composite error term (ν_ij) which consists of a region-specific component (η_j) and a bank-region specific component (ε_ij). If the share of loans that are securitised (SALESHARE_i) is correlated with the unobservable credit demand shocks (η_j) then the OLS estimate of β₁ will be biased. But suppose the region borrows from both an OTD lender and an non-OTD lender. Denote the OTD lender with subscript O and the non-OTD lender with subscript N. For a region j that has a loan from each type of bank, the within-region difference in lending growth is:

This equation eliminates both the effect of the aggregate shocks and the unobservable region-specific credit demand shocks. Equivalently, the OLS regression with the inclusion of borrower-specific fixed effects controls for the unobservable credit demand shocks (η_j). An unbiased estimate of the causal effect of the credit supply shock can be obtained under the identifying assumption that each bank's loan securitisation share is uncorrelated with the bank-region specific errors (corr(SALESHARE_i,ε_ij) = 0)).

Footnote

I further assume that the bank cannot fund loans through internal finance (e.g. retained earnings or deposits). While this appears to be a strong assumption, it is only made to simplify the algebra – the results still hold if I instead assume that banks finance lending through deposits (D_it), where deposits are the cheapest form of funding. In that case, the key assumptions are that there is also a quantitative limit on internal finance (i.e. D_it < ) and that securitisation is a cheaper form of funding than other forms of external finance. [15]