I. INTRODUCTION

In a world with asymmetric information and other market imperfections, financial intermediaries provide credit to otherwise liquidity constrained agents. If lending without screening and monitoring entails large deadweight losses, and if market financing is prevented by free-rider problems, banks emerge as the only source of external financing for potentially productive agents (Diamond, 1984).¹ As a result, the relationship between monetary perturbations and aggregate economic activity is necessarily linked to bank lending behavior.² However, the response of bank lending to positive and negative interest rate changes may be inherently different, and potentially asymmetric. Even though several papers have studied the asymmetric effects of monetary policy on real economic activity, little attention has been paid to the asymmetric response of bank lending to interest rate changes.³

This paper has two aims. First, it proposes and solves a dynamic matching model, where credit expansion and contraction are time consuming, and shows that bank lending is likely to respond asymmetrically to interest rate changes.⁴ Second, it provides empirical evidence that bank lending in Mexico and the US responds asymmetrically to positive and negative innovations in money market rates.

The paper argues that lending expansion and contraction are time consuming activities, whereas buying and selling money market funds takes place without time delays. In reality, there are several reasons why lending expansion may be a time consuming process: investment opportunities might be difficult to find, or screening potential applicants might impose a time constraint on the banking system’s ability to issue new loans. In the case of existing bank-client relationships, these problems are potentially less severe. However, financial institutions still need to evaluate the profitability of expanding an existing loan. Similarly, recovering non-performing loans may require a time consuming legal procedure.⁵ As a result, the speed at which lending opportunities become available, and the speed at which banks can recall existing loans are important determinants of the dynamic response of bank lending to shocks in the money market. In particular, when banks can rapidly recall non-performing loans, but experience technological delays in expanding credit, the response of bank lending to interest rate changes is likely to be asymmetric: positive changes result in the immediate contraction of bank loans, whereas negative changes produce only a gradual expansion of bank lending.

A large literature in theoretical macroeconomics has emphasized the role of search frictions in the labor market, and the existence of a matching problem between vacant jobs and unemployed workers. Similar frictions are also relevant in intermediated capital markets, where banks are the main source of productive capital. As vacancies and workers search for each other in a world with imperfect information, so do bank’s capital and idle projects. Theoretically, we model the market for lending as a matching environment, where banks and entrepreneurs search for each other with a view toward establishing profitable relationships. A large microeconomic literature has shown that asymmetric information may lead to equilibrium credit rationing in the banking system.⁶ In this paper, even though we do not directly deal with informational asymmetries in bank-client relationships, we do model an aggregate form of credit-rationing. Throughout the analysis we assume that there is a positive probability that bank funds and idle projects do not succeed in finding each other in a given period. In other words, we assume that new loan contracts can profitably take place only after a bank and an entrepreneur have been randomly matched. This over-simplification, while extreme from the perspective of microeconomic theory, is meant to capture in an aggregate model the time consuming process of credit formation.

Banks are endowed with a given quantity of capital to be invested in two assets: money market funds and entrepreneurial projects. Technologically, the two assets differ in various respects. First, investments in the money market are risk free, whereas loans have an idiosyncratic probability of default, and are heterogeneous in terms of risk and return. Second, the technology to issue and recall loans is time consuming, while investing in the money market is not. From the banks’ perspective, the probability of forming a credit relationship, and the probability of recovering non-performing loans are two exogenous and independent stochastic processes⁷. Conversely, the timing for investing and disinvensting in the money market is deterministic and immediate. Hence, banks’ evaluation of entrepreneurial projects will reflect not only the immediate return generated by the associated loan, but also their value as an asset that might be difficult to replace.⁸

We show that banks monotonically rank individual projects, and optimally choose a reservation project below which buying money market funds is strictly preferred to project financing. Since the value of the marginal project depends on the return on the banks’ alternative asset, in equilibrium, any shock to the money market (securities) rates affects the optimal allocation of bank capital. The paper shows that the dynamic response of bank lending to interest rate shocks depends crucially on two structural parameters: the speed at which lending opportunities become available, and the speed at which banks can recall existing loans. From this perspective, econometric estimates that force lending to respond symmetrically to interest rate changes are likely to impose undue restrictions.

Empirically, we investigate whether the response of bank lending to interest rate changes is indeed asymmetric, as most of the parameterization of the model would suggest. Empirical evidence for the US and Mexico confirms our intuition: bank lending reacts asymmetrically to exogenous money market perturbations. In particular, we find that banks react more rapidly to market interest rate increases in both countries.⁹

The paper proceeds as follows. Section II introduces concepts and notations, while section III presents and solves the steady-state model, where returns to the money market are fixed and time invariant. Section IV solves a stochastic dynamic version of the model with shocks to market returns, provides numerical simulations, and briefly highlights the model’s empirical implications. Section V develops the empirical analysis and provides evidence of asymmetry in the response of aggregate lending to money market shocks in Mexico and in the United States. Section VI discusses policy implications and alternative theoretical interpretations. Section VII briefly summarizes and concludes the paper.

II. DESCRIPTION OF THE MODEL

We consider an economy populated by a fixed number of risk neutral banks and risk neutral entrepreneurs. Entrepreneurs are endowed with projects of different qualities and seek project financing. Banks are endowed with liquid funds and seek investment opportunities. Entrepreneurs have no private source of funds, and have no access to market financing, so that bank loans are their only source of external capital. Banks, however, may invest their capital in two alternative assets: money market bonds and project-loans. For simplicity, we assume that each entrepreneur is endowed with an indivisible project requiring an initial investment of $1 that is productive only when it is matched to a unit of bank capital. Since our focus is on loans, rather than on banks, we assume that each bank has a single unit of funding capital. As a result, we abstract from issues related to market structure in the banking system.

Throughout the analysis, the aggregate capital available to the banking sector, and the aggregate number of entrepreneurs/projects are constant and time invariant. However, we assume that there are n different types of projects, and k projects of each type, so that the aggregate number of projects is nk. For analytical convenience, we let nk be also the fixed number of banks or, alternatively, the stock of capital in the banking system.

Each bank can invest its indivisible unit of capital in two different assets: money market funds or project loans. We let the return on the money-market investment be risk-free, and we let r_d indicate its instantaneous return. Conversely, a project of type i, with $i\in (1,2,\dots ..,n)$ is characterized by a pair (y_i, λ_i), where y_i is a time-invariant return, and λ_i is an instantaneous probability of destruction, a Poisson process that measures the project’s idiosyncratic risk. Since most models predict that riskier projects must have higher returns, we assume that the elements of the pair (y_i, λ_i) are positively correlated, and strictly increasing in i. Furthermore, in order to obtain a monotonic ranking of the projects, and reasonable aggregate results, we need to assume that $\frac{y_i}{{{\lambda }}_i}>\frac{y_j}{{{\lambda }}_j},\forall i>j$. Thus, dividends grow faster than destruction rates, and the index i is a proxy of a project’s quality. Finally, each project can be in two different states, depending on whether or not it is matched to a bank. A financed project is active and it produces an idiosyncratic dividend y_i, while an unfinanced project is idle and does not yield any dividend.

We model credit expansion and contraction as time-consuming processes. When time elapses in a continuous way, as we assume in the rest of the analysis, money market investment and disinvestment can be undertaken immediately, whereas loan expansions and contractions are time consuming. Formally, an analytically convenient way to model credit formation as a time-consuming process can be borrowed from the traditional matching literature (Diamond, 1982, and Mortensen and Pissarides, 1994, 1998). In what follows, we assume that the number of credit applications that are fully screened and evaluated in a given interval of time is described by a unique function of few aggregate variables: the stock of capital in the money market, and the number of idle projects.¹⁰ Thus, a function x(v, m), where m is the number of idle projects, and m is the stock of capital invested in the money market, records the number of loan applications completely screened and evaluated in a given period. In terms of the function x(.), the assumption that screening and evaluating projects are time consuming activities is equivalent to assuming that

Furthermore, since every credit relationship involves one unit of funds and a single project, in our simple set-up, υ = m. If we also assume that credit formation can be described by a constant return technology, as we do in the rest of the paper, x can be written simply as

where α ≡ x (1, 1) < 1, is the instantaneous probability that a bank screens a project in a infinitesimal time interval. Alternatively, given our symmetric structure, α is the probability that a credit application is completely screened.¹¹ In other words, we are assuming that in the economy, during a short period of time δt there is a positive probability 1 − αδt that a unit of capital and an idle project do not succeed in finding each-other. As a result of equation (1), the economy is characterized by aggregate (and stochastic) credit rationing. Formally, we do not need to specify whether banks meet entrepreneurs randomly over time, or whether banks find new projects at an infinite speed but their screening technology is intrinsically time-consuming. Nevertheless, aggregate credit formation is time consuming, and the parameter α captures this property in a simple way.

When a unit of bank capital and an idle project match, all uncertainty is resolved, and the bank immediately knows the type of the project. An active loan is subject to idiosyncratic risk of destruction at rate λ_i. For the entrepreneur, the realization of the shock represents immediate bankruptcy, and its life-time utility immediately drops to zero. For the bank, the bankruptcy of a project brings to an end the income generated by the associated loan. Immediately thereafter, the bank initiates a costly and time consuming bankruptcy procedure. For analytical simplicity, we assume that recovering capital out of a bankrupt project involves a flow cost f, and an instantaneous probability of success equal to σ, where σ is the arrival rate of a simple Poisson process.¹² This assumption captures the idea that bankrupt firms have assets that can potentially be liquidated, but only via a time consuming and (stochastic) device. Furthermore, to keep symmetry between banks’ aggregate capital and the entrepreneurial population, we assume that a new type i project appears idle only when the bank has successfully completed the bankruptcy procedure.

The existence of credit rationing and a finite α generate a pure economic rent to be split between entrepreneurs and banks that successfully match. As a result, to formally close the model, we need a sharing rule that determines the interest rate charged to different projects. We follow the standard matching literature and assume that the total surplus generated by an active project is continuously shared in fixed proportions, and we let β represents the bank’s share.

Banks choose a search strategy that maximizes the expected value of their capital; they select a decision rule that describes whether to finance a specific project, whenever it becomes available. Since the present value of financing each loan is monotonic in i, the bank’s decision rule satisfies a “reservation” property. We show that in equilibrium each bank selects a cut-off quality i^*, such that for projects of quality lower than i^* banks prefer to invest in the money market. In the model, an equilibrium is a reservation rule i^*, or alternatively, a stationary allocation of capital (a distribution of bank capital among project financing, recovery loans, and money market investments) that is consistent with the optimal reservation rule.

III. THE MODEL: STEADY STATE

This section presents and solves the steady state model, with a fixed and time invariant money market rate (r_d). In what follows, we shall indicate with V_i and J_i, the present discounted values for a type i entrepreneur of, respectively, an idle project and an active project, while ρ < 1 shall be the discount rate, which is assumed to be the same for the banks and the entrepreneurs. Even though idle projects do not yield any dividend, their present discounted value may still be positive, by virtue of an expected capital gain associated with successful matching.¹³ More formally, if α is the entrepreneur’s probability of having his or her project screened, the valuation of an idle type i project is

where the max operator in equation (2) indicates that an entrepreneurs has always the option to leave his or her project inactive. An active, or financed, type i project yields an instantaneous dividend y_i, and is characterized by an instantaneous destruction probability λ_i. If a destroyed project yields a zero value to the entrepreneur, the present discounted value of an active project reads

where r_i is the interest that a type i project pays to the bank.

Similarly, we shall indicate with C_i the bank’s value of a unit of capital invested in a type i loan, and with D the value of investing in the money market. Finally, B shall indicate the present discounted value of a bad loan (a unit of capital in the recovery state). If the interest rate on the money market is r_d, and α is the probability of completely screening a project, the value of investing a unit of capital in the money market reads

Equation (4) is one of the key equations of the model, and the max operator in the capital gain term encodes the bank’s choice between investing in project loans or in money market bonds. Once a bank has successfully (or luckily) screened a project j, it will convert its funds into a type-j loan as long as C_j is greater than D. Ex-ante, however, the bank does not know the quality of the project, but only its overall distribution, which we have assumed uniform for analytical convenience. Hence, the summation term in equation (4) represents the conditional value of a loan.

The asset valuation of a type i loan depends on the interest rate charged, r_i, and the capital loss suffered by the bank in case of bankruptcy. Since the bank will start the bankruptcy procedure when the project is hit by an idiosyncratic shock¹⁴, the value of a type i loan reads

where B is the present discounted value of a bad loan. In the recovery state the bank pays a flow cost f, and expects to successfully recover its capital with instantaneous probability σ. Since the bank will immediately invest the capital that it has successfully recovered in the money market, B solves

For the bank, establishing a credit relationship is optimal as long as C_i > D. Similarly, for the entrepreneur, having an active project is optimal as long as J_i > V_i. As a result, the surplus from a credit-match can be formally represented as the net value of establishing a credit relationship, and reads

where W_i, the surplus of a type i loan, is the measure of the quasi-rent generated by the match. To formally close the model, we need a rule that determines how the bank and the entrepreneur will divide the surplus generated by a match. Since we assume that the bank gets a fixed share β of the total surplus, it must be true that

Equation (8) reflects two characteristics of the model’s equilibrium: first, it is profitable for the bank and the entrepreneur to establish a credit relationship as long as the total surplus is positive; second, there is agreement between the bank and the entrepreneurs on which projects should be financed. Making use of equations (5), (3), (2) and (4), the surplus from the credit relationship (7) can be conveniently written as

where

After dividing both sides by (ρ + λ_i), the right hand side of equation (9) is a mapping. Unfortunately, equation (9) does not satisfy the Blackwell sufficient conditions for a contraction. Nevertheless, Appendix I shows that a simple restriction on the parameters provides us with a sufficient condition for the mapping of equation (9) to be a contraction, and thus, to guarantee the existence of a fixed point.¹⁵

The surplus function W_i is a monotonic increasing function of i. Intuitively, as long as the dividends of higher quality projects grow faster than the associated bankruptcy probability, the total rent generated by a project should be monotonic in i. This result, albeit analytically intuitive, is formally derived in Appendix II. By virtue of (8), monotonicity of projects’ surplus implies monotonicity of the bank’s surplus, which allows us to characterize banks’ behavior through the reservation property. In other words, banks select a marginal type i^*, or a cut-off value $W_{i^{*}}$, below which investment in the money market is the optimal policy. More formally, i^* solves

In the case in which $W_{i^{*}}\equiv 0$, from equation (9), the marginal returns of the project can be conveniently written as

Equation (11) describes in a simple way the return on the marginal project. In order to be worthwhile to grant a loan, the marginal project i^* must at least compensate the bank for three different elements: the per period return on the money market $(a_{i*}{r}_d)$, the expected value of an alternative project (the expression in square brackets), and the expected cost of recalling the loan if conditions turn bad $\left(\frac{{{\lambda }}_{i*}f}{\sigma +\rho }\right)$.

Before analyzing several comparative static results, we solve for the equilibrium interest rate r_i. Rearranging equation (3), and making use of equation (8), the interest rate on a type i loan reads

Finally, making use of equations (5) and (4), the interest rate can be simply written as

Equation (13) shows that the interest rate on a type i project is equal to the return on the alternative banks’ asset (ρD), augmented by a strictly positive spread. Furthermore it is possible to show that the spread component of equation (13) is monotonic in the quality i, so that banks earn higher returns on riskier projects. For this purpose, it is sufficient to show that the interest rate is increasing in i. Making use of equation (13), the spread between two different quality projects can be conveniently written as

Since W_i > W_i−j and λ_i > λ_i−j, it immediately follows that r_i − r_i−j > 0 for any positive integer j. As a result, the interest rate spread between a type i loan and an alternative investment grows monotonically with the quality i.

We are now in the position to analyze the most important comparative static results of this section, namely the relationships between the money market rate, r_d, the interest rates on loans, r_i, and the banks’ optimal portfolio allocation, as described by the reservation quality i^*. In general, changes in r_d will affect both the amount of lending and the interest rate charged on loans. First, following an increase (decrease) in r_d, it is quite likely that some project becomes immediately unprofitable (profitable) relative to the money market investment. Since the surplus of a type i match is decreasing in the money market rate,¹⁶

an increase in r_d may affect the bank’s equilibrium allocation between project loans and money market investments. If $W_{i^{*}}\approx 0$, equation (15), by virtue of (10) produces an immediate change in i^*. In general, for any marginal increase (decrease) in r_d, if n is sufficiently large, the bank will change its optimal asset allocation and it will decrease (increase) its share of loans. Similarly, making use of equation (13), the effect of a change of the money market rate on the interest rate charged to a type i project is

Appendix III. shows that both terms in the brackets of equation (16) are positive. Thus, there is a positive pass-through effect between money market rates and interest rate changes. Equation (16) shows that the magnitude of the pass-through, albeit strictly positive, crucially depends on several structural parameters. First, $\frac{\partial r_i}{\partial r_d}$ is a decreasing function of β. If we take β as a proxy for the bank’s market power, this result implies that the pass through is larger in less concentrated markets, a result familiar in the industrial organization literature (Tirole, 1988). Second, when β is zero and σ tends to infinity, the pass-through is exactly one. In this case, bank lending to private entrepreneurs does not entail any recovery risk, and banks have no market power. Hence, all projects pay to the bank the money market return.

Now we can specify the equilibrium allocation between alternative investments. In equilibrium, banks will finance only projects of type i ≥ i^*, leaving unfunded the remaining types. Nevertheless, the economy experiences continuous turnover of projects, and to complete the model we need to specify the distribution of banks’ asset among active loans, bad loans and money market investment. If we indicate with υ_i, b_i, and c_i the steady state quantity of type i projects which are, respectively, idle, bankrupt, and fully active, the flow balance conditions are

Solving the system of equations (17) yields, for i ≥ i^*, the following equilibrium quantities

while for i < i^*, all projects are idle and υ_i = k. The comparative static properties of the quantities in equations (18) show that υ_i is a decreasing function of α. The easier the screening process, the larger the proportion of funds kept in the money market. Similarly, higher σ leads to a lower quantity of projects in the bankruptcy state. Finally we aggregate equation (18) over different quality indices to obtain

From equations (19), it is clear that an increase in market rates, by raising i^*, reduces aggregate lending c and the quantity of capital in the bankruptcy state. As a result, the capital invested in the money market rises. The next section looks at interest rate changes in a stochastic setting.

IV. STOCHASTIC SHOCKS TO THE MONEY MARKET

We now extend the analysis of the previous section and consider a world in which money market returns are stochastic and time variant. For analytical simplicity, we assume that the money market rate jumps stochastically between two different values, $r_d^T$ and $r_d^E$, with $r_d^T>r_d^E$. Since the money market can be in two different states, in what follows we shall call $r_d^T$ the tight state, and $r_d^E$ the easy state. We also assume that the state of the money market moves according to a symmetric Markov process, and we shall indicate with μ the instantaneous probability of a state switch. To solve the model, we need to specify a full set of state contingent value functions, which makes the analytic of the model particularly cumbersome. Hence, we do not provide a close form solution of the model, or comparative static results. However, as a way to maintain the discussion at an intuitive level, we describe the dynamics of the model through numerical simulations.

B. The Aggregate Dynamics: Simulations and Discussions

This section simulates the aggregate dynamic response of bank lending to changes in market conditions. In what follows we assume that time is discrete and that the state of the money market (tight, $r_d^T$ or easy, $r_d^E$) is realized at the beginning of the period, and constant throughout. The timing of the decision is as follows. Banks observe the money market realization, and immediately select a reservation i^*T or i^*E. Thereafter, each entrepreneur learns whether his or her project has been screened, the credit allocation (c, r, υ) is determined, interest rates are paid, and the period is completed.

In a well behaved equilibrium, given the n project qualities, there is always a subset of project types whose quality at time t is above the highest reservation quality (i ≥ i^*T), and another subset of projects whose quality is below the lowest one (i < i^*E). For these projects, the allocation of capital is not affected by regime switches, even though the interest rate charged on the associated loans will be conditional on the state of the market. More formally, for i ≥ i^*T, the shares of projects that are financed, not-financed, or are in the recovery state are those described by equations (18).

Conversely, for types i < i^*E no project is ever financed, and we just have υ_i = k. However, for those projects whose quality lies between the two reservation thresholds (i^*E ≤ i < i^*T), the composition of projects that are financed, not-financed, or are in the recovery state is state dependent. If we let x(t) be an indicator function

$x(t)=\{\begin{array}{l}{i}^{T*}\text{if}{r}_d=r_d^T\\ i^{E*}\text{if}{r}_d=r_d^E\end{array}$

that records the state of the money market at time t, the dynamics of a type i project is

$\begin{array}{ccc}{c}_{i,t+1}=(1-{{\Phi }}_1)(1-{{\lambda }}_i\mathit{\text{dt}})c_{i,t}+{{\Phi }}_2\alpha {\upsilon }_{i,t}\mathit{\text{dt}}& & (29)\end{array}$

$\begin{array}{ccc}{b}_{i,t+1}=(1-\sigma \mathit{\text{dt}})b_{i,t}+(1-{{\Phi }}_1){{\lambda }}_i{c}_i\mathit{\text{dt}}+{{\Phi }}_1{c}_{i,t}& & (30)\\ c_{i,t}+b_{i,t}+{\upsilon }_{i,t}=k\forall i,t,\end{array}$

where

$\begin{array}{ccc}{{\Phi }}_1=\{\begin{array}{l}1\text{if}x(t+1)=i^{*T},x(t)=i^{*E},W_i^T<-\frac{\rho D^t}{\rho +\sigma }\\ 0\text{otherwise}\end{array}& & (31)\end{array}$

and

$\begin{array}{ccc}{{\Phi }}_2=\{\begin{array}{l}1\text{if}x(t+1)=i^{*E}\\ 0\text{otherwise}\end{array}& & (32)\end{array}$

Consider first the dynamics of equations (29) and (30). If the state switches from easy to tight (Φ₁ = 1), all active projects enter the bankruptcy state b_i, while when there is no state switch the flow from active to recovery is governed by the natural turnover λ_i. Conversely, successfully screened projects are converted into active loans only when the market is easy, and the indicator function Φ₂ keeps track of this condition in equation (29).

In what follows, we simulate the dynamic response of the aggregate economy using the baseline parameter values specified in Table 1. We focus on two aggregate statistics, the aggregate credit (c_t) and the average interest rate on active loans, which we formally indicate with

${\overline{r}}_t=\frac{{\displaystyle {\sum }_{i>x(t)}^n{r}_{\mathit{\text{it}}}{c}_{\mathit{\text{it}}}}}{{\displaystyle {\sum }_{i>x(t)}^n{c}_{\mathit{\text{it}}}}},$

Table 1.

Baseline Parameter Values

Source: Authors’ calculation

Table 1.

Baseline Parameter Values

Variables	Notation	Value
Screening probability	α	0.050
Recall probability (low)	σ	0.050
Recall probability (high)	σ	1.000
Tight Money market rate	$r_d^T$	0.050
Easy Money market rate	$r_d^E$	0.020
discount rate	ρ	0.100
quality index	n	12.000
bankruptcy rate (min)	λ1	0.070
bankruptcy rate (max)	λn	0.020
idiosyincratic return (min)	y 1	0.100
idiosyincratic return (min)	y n	0.000
banks’ surplus share	β	1.000

Source: Authors’ calculation

View Table

Table 1.

Baseline Parameter Values

Variables	Notation	Value
Screening probability	α	0.050
Recall probability (low)	σ	0.050
Recall probability (high)	σ	1.000
Tight Money market rate	$r_d^T$	0.050
Easy Money market rate	$r_d^E$	0.020
discount rate	ρ	0.100
quality index	n	12.000
bankruptcy rate (min)	λ1	0.070
bankruptcy rate (max)	λn	0.020
idiosyincratic return (min)	y 1	0.100
idiosyincratic return (min)	y n	0.000
banks’ surplus share	β	1.000

Source: Authors’ calculation

where c_it is described by equation (29) for i^*E ≤ i < i^*T, and by equation (18) for i ≥ i^*T.

Figures 1 to 4 plot the dynamics response of c_t and ${\overline{r}}_t$ for different values of α and σ. Figure 1 plots the dynamics response of c_t for σ > α along a full easy-tight-easy cycle, σ is the institutional parameter that describes the average speed of recovering a loan, while α is the technological parameter that describes the average screening period of a credit application. In Figure 1, when the market state turns tight (approximately at period 9 in Figure 1) i^*E jumps to i^*T, and all projects whose idiosyncratic quality lies between the two reservation qualities i^*E ≤ i < i^*T are rapidly recalled. Since σ is relatively high, the average duration of the recovery state is very short, and the bank rapidly recovers the invested capital. When the market state switches to tight (approximately at period 30 in Figure 1), i^*T jumps immediately to i^*E, but the dynamic response of aggregate credit is time consuming. As the interest rate falls, banks immediately change the reservation quality, but they still have to process the additional loan applications, and the associated credit expansion takes place only at rate α. Thus, when σ > α, there is a short-run asymmetric response of aggregate lending to interest rate changes. However, as the projects are successfully screened, credit returns to its original level, without any long-run asymmetry. These results represent a first set of empirical implications of our model. Figure 2 plots the dynamic response of the average interest rate when σ > α. In general, the dynamic response of ${\overline{r}}_t$ to changes in the money market rate is affected by two different effects. First, there is an instantaneous pass-through effect, linked to the translation of the new r_d into the existing r_i, as was formally described by equation (16). Second, there is a portfolio effect, induced by the modification of the bank’s optimal portfolio towards more profitable projects. When r_d switches to tight in Figure 2, the average interest rate immediately jumps to a new level, as a result of the portfolio effect and the pass-through effect. However, when policy switches to easy, on impact, the change in r_d is only related to the pass-through effect, since the portfolio effect is time-consuming. From Figure 2, is clear that the asymmetric response of the average interest rate depends entirely on the portfolio effect.

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Banks’ Capital Allocation over a easy-tight-easy cycle; σ > α

Citation: IMF Working Papers 1998, 093; 10.5089/9781451951318.001.A001

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Average Interest Rate over an easy-tight-easy cycle; σ > α

Citation: IMF Working Papers 1998, 093; 10.5089/9781451951318.001.A001

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Banks’ Capital Allocation over a easy-tight-easy cycle; σ = α

Citation: IMF Working Papers 1998, 093; 10.5089/9781451951318.001.A001

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Average Interest Rate over an easy-tight-easy cycle; σ = α

Citation: IMF Working Papers 1998, 093; 10.5089/9781451951318.001.A001

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Figures 3 and 4 plot the dynamic response of c_t and ${\overline{r}}_t$ along a full easy-tight-easy cycle when σ = α. Qualitatively, the dynamic profile of Figure 3 is clearly different from the dynamic profile of Figure 1. In Figure 3, when the market turns tight (approximately at period 9) i^*E jumps to i^*T, but nothing happens on impact, neither to aggregate credit nor to the recovery state. To properly understand Figure 3 it is necessary to go back to equation (22). When r_d switches to tight, and σ is very low, banks expect a very long and costly recovery process, and may well prefer to keep active some of the projects with i < i^*T. When the latter is true, banks prefer to wait for the natural bankruptcy of the project, rather than go through a costly separation. Nevertheless, during the tight phase banks choose not to finance projects that are successfully screened but have an idiosyncratic quality below i^*T. Basically, projects of quality i, with i^*E ≤ i < i^*T, are destroyed at rate λ_i, but are not replaced at rate α. As a result, aggregate credit gradually falls. When the policy switches to easy (approximately at period 30 of Figure 3), banks begin to finance all projects with reservation quality greater than i^*E, and aggregate credit rises smoothly. Overall, there is no asymmetry over the full cycle. Comparing Figures 3 and Figure 1, it is clear that the dynamic response of aggregate credit to interest rate changes depends crucially on the relative values of σ and α. Finally, Figure 4 plots the dynamic behavior of the average interest rate. When σ = α, there is no portfolio reallocation and, as a result, there is almost no asymmetry in the dynamic response of the average interest rate.

V. EMPIRICAL EVIDENCE

In this section we provide some empirical evidence on the effects of interest rate changes on bank lending, focusing on the difference between positive and negative shocks. We consider two countries, in representation of mature and emerging markets: the United States and Mexico. The data for the US are quarterly, and run from 1960 to 1997. For Mexico we have monthly data, from 1982 to 1997. The source of all the data is the International Financial Statistics. Market rates are taken from line 60b, which records the most representative short term rates.¹⁹ We use “claims on the private sector” (line 22d) as a proxy for loans. This choice, as was the use of IFS data, was mainly motivated by the need of international comparability. For the US we use the Consumer Price Index (CPI) and GDP growth as a way to construct exogenous interest rate shocks. For Mexico we use Industrial Production in the absence of monthly GDP data.

A. Methodology

We employ a two-step procedure similar to that applied by Cover (1992) and Garibaldi (1997), but adapted to consider the stationarity problems of aggregate credit. First, we estimate the money market rate processes, and we use the associated residuals for constructing positive and negative interest rate shocks. Second, we estimate the effect of those shocks on a properly defined specification of aggregate credits.²⁰

In the first step we estimate an autoregressive-distributed lag model (ARDL) of the money market rate, CPI inflation, and GDP (or industrial production). If MM is the money market rate, the ARDL regression reads

$\begin{array}{ccc}{\mathit{\text{MM}}}_t=a+{\displaystyle \sum _{i=1}^M{\alpha }_i{\mathit{\text{MM}}}_{t-i}+{\displaystyle \sum _{i=0}^N{\beta }_i{G}_{t-i}+{\displaystyle \sum _{i=0}^Q{\gamma }_i{P}_{t-i}+{\epsilon }_t,}}}& & (33)\end{array}$

where G is the real output growth, and P is the CPI inflation. When the lags are properly chosen, ε_t mimics a simple white noise process, and we take its estimated values as exogenous shocks to the money market rate. As a way to test the robustness of our generated shocks, we use the Akaike and Schwartz criteria to optimally select M, N, and Q (the order of the ARDL)²¹. Results of regression (33) are reported in Table (2) for the US and Table (3) for Mexico. Next, we divide the residuals generated in regression (33) into positive and negative values, and we introduce two different variables. More specifically, we define a positive shock to the money market interest rate as

${\mathit{\text{tight}}}_t=\max ({\epsilon }_t,0),$

Table 2.

U.S Federal Fund Rate

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

Table 2.

U.S Federal Fund Rate

Dependent Variable: Federal Fund Rate
Quarterly Data: 1960(1) to 1997(4)
(Ordinary Least Square, First stage regression
Regressor a	Notation a	Akaike b	Schwartzc
M M t−1	α 1	0.99***	0.91***
		(0.087)	(0.034)
M M t−1	α 2	-0.32***	-
		(0.124)
MM t -3	α 3	0.27 ***	-
		(0.090)
G t	β 0	0.23**	0.31***
		(0.102)	(0.100)
G t−1	β 1	0.25	0.31
		(0.102)	(0.100)
G t−2	β 2	0.14	-
		(0.106)
G t−3	β 3	0.15	-
		(0.095)
P t	γ 0	0.91 ***	0.81***
		(0.19)	(0.174)
P t−1	γ 1	-0.63 ***	-0.67 ***
		(0.226)	(0.217)
P t−2	γ 2	0.73 ***	0.51***
		(0.222)	(0.182)
P t−3	γ 3	-0.39 *	-
		(0.211)
Cost	a	-0.83 ***	-0.58**
		(0.304)	(0.259)

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

View Table

Table 2.

U.S Federal Fund Rate

Dependent Variable: Federal Fund Rate
Quarterly Data: 1960(1) to 1997(4)
(Ordinary Least Square, First stage regression
Regressor a	Notation a	Akaike b	Schwartzc
M M t−1	α 1	0.99***	0.91***
		(0.087)	(0.034)
M M t−1	α 2	-0.32***	-
		(0.124)
MM t -3	α 3	0.27 ***	-
		(0.090)
G t	β 0	0.23**	0.31***
		(0.102)	(0.100)
G t−1	β 1	0.25	0.31
		(0.102)	(0.100)
G t−2	β 2	0.14	-
		(0.106)
G t−3	β 3	0.15	-
		(0.095)
P t	γ 0	0.91 ***	0.81***
		(0.19)	(0.174)
P t−1	γ 1	-0.63 ***	-0.67 ***
		(0.226)	(0.217)
P t−2	γ 2	0.73 ***	0.51***
		(0.222)	(0.182)
P t−3	γ 3	-0.39 *	-
		(0.211)
Cost	a	-0.83 ***	-0.58**
		(0.304)	(0.259)

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

Table 3.

Mexico Money-Market Rate

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

Table 3.

Mexico Money-Market Rate

Dependent Variable: Money Market Rate
Monthly Data: 1982(1) to 1997(12)
(Ordinary Least Square, First stage regression
Regressora	Notationa	Akaike-Schwartzb
M M t − 1	α 1	1.24 ***
		(0.076)
M M t − 2	α 2	-0.59 ***
		(0.114)
M M t − 3	α 3	0.29 ***
		(0.070)
G t	β 0	0.1
		(0.118)
P t	γ 0	2.21 ***
		(0.363)
P t − l	γ 1	-3.26 ***
		(0.705)
P t − 2	γ 2	1.5 *
		(0.769)
P t − 3	γ 3	-2.00 ***
		(0.747)
P t − 4	γ 3	1.54 ***
		(0.402)
Cost	a	3.76 **
		(1.766)

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

View Table

Table 3.

Mexico Money-Market Rate

Dependent Variable: Money Market Rate
Monthly Data: 1982(1) to 1997(12)
(Ordinary Least Square, First stage regression
Regressora	Notationa	Akaike-Schwartzb
M M t − 1	α 1	1.24 ***
		(0.076)
M M t − 2	α 2	-0.59 ***
		(0.114)
M M t − 3	α 3	0.29 ***
		(0.070)
G t	β 0	0.1
		(0.118)
P t	γ 0	2.21 ***
		(0.363)
P t − l	γ 1	-3.26 ***
		(0.705)
P t − 2	γ 2	1.5 *
		(0.769)
P t − 3	γ 3	-2.00 ***
		(0.747)
P t − 4	γ 3	1.54 ***
		(0.402)
Cost	a	3.76 **
		(1.766)

Notes: Standard errors are in parentheses One, two and three asterisks indicate significance at the 10, 5, and 1 percent respectively.

^a See equation (33)

^b ARDL order selected with Akaike criterion

^b ARDL order selected with Schwartz criterion Source: Authors’ calculation

and a negative shock as

${\mathit{\text{easy}}}_t=\min ({\epsilon }_t,0).$

In the second regression we use “claims on the private sector over CPI” as a proxy for aggregate real loans, and we regress this variable on its lagged values, the lagged values of real deposits, and the lagged values of positive and negative market rate shocks. Real deposits are included as a control variables to take into account changes in the amount of funds available to the banking system.²² The ARDL model of the second step is

$\begin{array}{ccc}{L}_t=b+{\displaystyle \sum _{i=1}^R{\delta }_i{L}_{t-i}+{\displaystyle \sum _{i=0}^S{\vartheta }_i{D}_{t-i}+{\displaystyle \sum _{i=1}^U{\phi }_i{\mathit{\text{tight}}}_{t-i}+{\displaystyle \sum _{i=1}^V{\psi }_i{\mathit{\text{easy}}}_{t-i}+\xi t+{\eta }_t}}}}& & (34)\end{array}$

where L_t is the log of real claims on the private sector, D_t−i is the log of real deposits, t is a time trend, and η is a white noise error. Since we need to allow credit to react to interest rate changes we do not include the contemporaneous values of easy and tight, and the variables tight and easy in equations (34) are only lagged.

To check the stability of the model, we employ the two-stage procedure proposed in Pesaran et al. (1996). We use the error correction specification of regression (34), and perform a variable deletion test on the coefficients of the lagged levels of L, D, tight, and easy. The null hypothesis is the non-existence of a long-run relationship, or in other words, the instability of the model. This procedure allows us to avoid the pre-testing problems associated with standard cointegration analysis, which requires the classification of variables into I(1) and I(0). Fortunately, we are able to reject the null hypothesis of instability (no-cointegration) for both countries.²³