US household deleveraging following the Great Recession – a model-based estimate of equilibrium debt

A Deriving the Common Correlated Effects Pooled Mean Group (CCEPMG) equation

We estimate a dynamic panel error correction model based on quarterly data for 50 US states (plus the District of Columbia) over the period 1999Q1–2012Q4, with subindices t=1, 2,..., T (T=56) for quarters and i=1, 2,..., N (N=51) for states. Following Pesaran, Shin, and Smith (1999), we assume an autoregressive distributed lag (ARDL) (1,1,1,...,1) dynamic panel specification of the form:

(1)dit=μi+ρi1di,t−1+δ′i0Xit+δ′i1Xi,t−1+uit (1)

where  Xit=[hpithownitiituritdemitltvitfrclit], δi0=[δi01.....δi07] and δi1=[δi11.....δi17]

If the variables are I(1) and cointegrated, then the error term is I(0) for all i.

Using d_it=d_{i, t–1}+Δd_it, (1) can be written as:

(2)Δdit=μi+(ρi1−1)di, t−1+δ′i0Xit+δ′i1Xi, t−1+uit (2)

Furthermore, since X_{i, t–1}=X_it–ΔX_it, we can write the above equation as:

(3)Δdit=μi+(ρi1−1)di, t−1+(δ′i0+δ′i1)Xit−δ′i1ΔXit+uit (3)

To highlight the long-run relationship, we can write (3) in error-correction form:

Δdit=μi−(1−ρi1)[di, t−1−δ′i0+δ′i11−ρi1Xit]−δ′i1ΔXit+uit

(4)Δdit=μi+ϕi(di, t−1−θ′i1Xit)+δi1∗′ΔXit+uit (4)

where ϕi=−(1−ρi1), θi1=[θi11.....θi17]=[δi01+δi111−ρi1.....δi07+δi171−ρi1]=−(δi0+δi1ϕi) and δi1∗=−δi1

To estimate the parameters of the model, we apply the Pooled Mean Group (PMG) estimator as described in Pesaran, Shin, and Smith (1999). Under the PMG estimator assumption of long-run homogeneity, the long-run coefficients are assumed to be the same across states, i.e., θ_i1=θ₁. By contrast, the short-run coefficients and the group-specific error correction coefficients (the speed of adjustment) are allowed to differ across states, so that δ_ij=δ_ij, and ϕ_i=ϕ_i. In this case, the reported coefficient values are given by the means of the respective estimates for individual states. The PMG also assumes that the disturbances u_it are independently distributed across states i and time t with zero mean and state-specific variances var(uit)=σi2.

With the PMG assumption of long-run homogeneity, (4) can be written as:

(5)Δdit=μi+ϕi(di, t−1−θ′1Xit)+δi1∗′ΔXit+uit (5)

where θ1=[θ11.....θ17]

We assume that the foreclosure rate affects debt only in the short run but not in the long run (i.e., θ₁₇=0), so (5) can be written in its extended form as:

(6)Δdit=μi+ϕi(di, t−1−θ1hpit−θ2hownit−θ3iit−θ4urit−θ5demit−θ6ltvit)+δi11∗Δhpit+δi12∗Δhownit+δi13∗Δiit+δi14∗Δurit+δi15∗Δdemit+δi16∗Δltvit+δi17∗Δfrclit+uit (6)

The term in brackets is the long-run relationship between debt and the explanatory variables, while θ₁, …, θ₆ are the long-run coefficients, which are typically the object of primary interest (see Blackburne III and Frank, 2007). ϕ_i is the speed of adjustment, which shows what percentage of the gap is being closed in each period and is expected to be negative and significant, if the variables are cointegrated and exhibit a return to long-run equilibrium.

In the long-run, debt at the state level will be determined by:

(7)dit=θio+θ11hpit+θ12hownit+θ13iit+θ14urit+θ15demit+θ16ltvit=θi0+θ′1Xit (7)

where θi0=μi1−ρi1=μi−ϕi is the state-specific long-run constant.

As an alternative to our PMG estimator, we also apply the Mean Group (MG) estimator, which estimates independent error-correction equations for each state without imposing homogeneity restrictions on long-run effects, but rather computes the mean of estimated state-specific long-run coefficients. We provide formal statistical evidence for choosing whether our PMG estimator is preferred to the MG estimator by applying a Hausman test on the homogeneity restriction that the long-run coefficient is the same for all states (see Pesaran, Shin, and Smith 1999).

To compute debt in the long run at the aggregate level, we apply the estimated long-run coefficients from the PMG, assumed to be homogeneous, to the US aggregate data:

(8)dtagg=θ¯0+θ11hptagg+θ12howntagg+θ13itagg+θ14urtagg+θ15demtagg+θ16ltvtagg=θ¯0+θ′1Xtagg (8)

The long-run aggregate constant term θ¯0 is computed using the state-specific constants and the autoregressive coefficients on the lagged dependent variable as follows:

(9)θ¯0=∑i=1Nμiρi10N+∑i=1Nμiρi11N+∑i=1Nμiρi12N+...+∑i=1Nμiρi1∞N (9)

which makes use of μ−ϕ=μ11−ρ1=∑k=0∞μρ1k⇔|ρ|<1

The reason why θ¯0 is not derived from the simple averages across states is because we cannot assume that the constant terms and the coefficients on the lagged dependent variables are independently distributed across panel members.

As discussed in the main text, the econometric estimation of Equation (5) may be affected by the presence of cross-section dependence across panel members. In what follows, we relax the assumption of cross-section independence of the error term of the standard PMG model by expressing the error term u_it as:

(10)uit=λ′ift+εit (10)

where an unspecified number of unobserved common factors f_t with idiosyncratic factor loadings λ_i are allowed to capture time-variant heterogeneity and cross-section dependence, while ε_it are now idiosyncratic errors independently distributed across i and t. In this set-up, the factors f_t can be non-linear and also non-stationary. The regressors X_it are allowed to be driven by some of the same common factors as the dependent variable. We employ two alternative estimators that have been developed to allow for correlation across panel members due to unobserved common time-specific effects as described above: the Augmented Mean Group (AMG) estimator introduced in Eberhardt and Teal (2010) and the Common Correlated Effects Pooled Mean Group (CCEPMG) estimator (see Pesaran 2006; Binder and Offermanns 2007; Chudik and Pesaran 2013).

The basic procedure behind the AMG estimator by Eberhardt and Teal (2010) is described in the main text. The other method used to correct for cross-section dependence in the disturbances is the CCEPMG estimator, which is chosen as our preferred specification when reporting the main results. It is based on the Common Correlated Effects Mean Group (CCEMG) estimator developed by Pesaran (2006). The basic idea is to filter the individual-specific regressors in a way that the differential effects of unobserved common factors are eliminated. The CCEMG solves the problem by augmenting the regressors in the group-specific regression equation with cross-section averages of the dependent variable and the individual-specific regressors.

The focus of the CCEMG estimator is on obtaining consistent estimates of the parameters related to the observable variables, while the estimated coefficients on the cross-section averaged variables are not interpretable in a meaningful way: they are merely present to filter out the biasing impact of the unobservable common factor (see Eberhardt 2012). The CCEMG estimator is robust to the presence of a limited number of “strong” factors (which can represent global shocks, as well as an infinite number of “weak” factors possibly associated with local spillover effects. Moreover, the CCEMG estimator is robust to nonstationary common factors (Kapetanios, Pesaran, and Yamagata 2011) and it continues to hold under slope homogeneity and in the presence of any fixed number of unobserved factors, (see Pesaran 2006).

Augmenting the standard PMG model in Equation (5) with cross-section averages as discussed above, our equation for the CCEPMG estimator becomes:

(11)Δdit=μi+ϕi(di, t−1−θ′1Xit)+δi1∗′ΔXit+αid¯t+βiX¯t+γiΔd¯t+ηiΔX¯t+εit (11)

where d¯t and X¯t are averages of the dependent variable and the regressors across states, computed at every time period t.

Following (11), the long-run relationship between d and X at the state level is given by:

(12)dit=θi0+θ′1Xit+ϑid¯t+ψ′iX¯t (12)

where ϑi=αi−ϕi and ψi=[ψi1.....ψi7]=[βi1−ϕi.....βi7−ϕi]=−(βiϕi)

In our final CCEPMG specification, the CCE augmentation of the standard PMG model employs the following cross-section averages (included both in levels and in differences): log of house price to income ratio (hp), unemployment rate (ur), and 35–54 age group (dem).^[27]

Based on our final CCEPMG specification, long-run debt at the US aggregate level will be given by:

(13)dtagg=θ¯0+θ11hptagg+θ12howntagg+θ13itagg+θ14urtagg+θ15demtagg+θ16ltvtagg+∑j=0∞ψjX¯t−j=θ¯0+θ′1Xtagg+∑j=0∞ψjX¯t−j (13)

where θ¯0 is computed as described in (9), while the contribution from the cross-section averages ∑j=0∞ψjX¯t−j uses ψj=∑i=1Nβiρi1jN based on the approach used for the aggregation of the long-run constant term.

For practical reasons, in the main text we report the estimates for equilibrium debt and debt gaps using the term ∑j=0∞ψjX¯t(without lagged values of the cross-section averages), so as to have estimated values for the whole sample period. We do this after having checked that the results are rather similar to the ones using the term ∑j=0∞ψjX¯t−j, computed with up to 30 lags for the term X¯t−j (results are available upon request).

B Descriptive statistics and econometric tests

C Additional tables and figures

Figure C.1

Decomposing the decline in CCEPMG equilibrium debt from 2007Q2.

Source: Authors’ calculations.

Figure C.2

Equilibrium debt and implied gap using a bottom-up approach.

Source: FRBNY/Equifax Consumer Credit Panel and authors’ calculations.

Notes: Equilibrium debt in the bottom-up approach is computed from the aggregation of state-level equilibrium estimates, derived by taking state-weights for income. Last observation refers to 2012Q4.

Figure C.3

Equilibrium debt and implied gap using the ltv at the state level and weighted cross-sectional averages.

Source: FRBNY/Equifax Consumer Credit Panel and authors’ calculations.

Notes: The “Avg_income weights” specification applies state-income weights to the cross-sectional averages in the CCEPMG. Last observation refers to 2012Q4.

Figure C.4

Recursive CCEPMG estimates using different sample periods.

Note: The CCEPMG is estimated recursively over different sample periods: the first estimate considers the 1999–2005 period, and in the subsequent estimates it adds 1 year at a time. The speed of adjustment corresponds to the error correction term in the CCEPMG model. The Hausman test reports p-value under the null hypothesis that the CCEPMG estimator is both efficient and consistent, i.e., that the long-run homogeneity restriction is valid. The remaining solid lines refer to the long-run coefficients on the variables included in the CCEPMG model. Bands around the point estimates consider ± 2 standard errors.

Figure C.5

Average developments in economic indicators for high vs. low deleveraging states.

Note: “High deleveraging states” are those states that featured the largest declines in their household debt-to-income ratios between the peak for each state and 2012Q4, defined by the 90th percentile. These include Arizona, California, Florida, Hawaii, Nevada and South Dakota. The “low deleveraging states” are those that featured the smallest declines, defined as the 10th percentile and include Arkansas, Iowa, Kansas, Mississippi, North Dakota and West Virginia.