The Bank Underground blog A question of interest: Is UK household debt unsustainable? presents some of the key insights from a VAR model linking household debt, house prices and interest rates. Since we have not released details of the model in any other publication, this technical annex explains the estimation and identification of our model in more detail.
Our baseline VAR is similar to the US model estimated by Juselius, Borio, Disyatat and Drehmann. We make some minor changes. First, since our primary objective is to study the drivers of debt rather than monetary policy, we drop inflation from our set of variables. We also use GDP in our system rather than including private demand and other components of GDP separately. Finally, we extend our estimation back to 1975 rather than to 1985, because we want to include another financial cycle and check that the cointegrating relationships we identify still hold. We found that the cointegration properties did change with an extra decade of data, something we suspected was driven by financial liberalisation in the 1970s and 1980s. This meant that households were able to take on more debt at a given level of interest rates and house prices. We add an additional exogenous variable into the system to proxy this effect.
Table 1 below shows the data definitions and sources used in our model. (The variable codes are not used in the main blog post but they are referred to elsewhere in this annex.)
Table 1: Data definitions for our VAR model
Estimation of cointegrating vectors
We make judgements in the main blog about the sustainability of UK debt by comparing current debt levels with “equilibrium” levels. Those equilibria are derived by fixing a particular value of Bank Rate, and calculating the level of debt or house prices implied by the cointegrating relationships in our model. Standard tests for cointegration suggest that there are two or three cointegrating relationships. Following the US analysis, we hypothesise that there are three, and impose restrictions so that these can be interpreted as a leverage constraint for households, a debt service constraint, and a pass-through equation linking policy rates to the effective interest rates facing households. These are shown in Table 2. All of the unrestricted coefficients are highly significant and the zero restrictions imposed pass standard statistical tests.
Table 2: Long-run equilibrium relationships
Chart A shows the deviations from equilibrium for each of the three cointegrating vectors. Group unit root tests indicate that the cointegrating vectors are stationary (at the 5% significance level). The cointegrating vectors broadly tell an intuitive story. For example, one thing that stands out clearly is the elevated level of the debt service burden in the late 1980s and early 1990s, and again in the mid-2000s, consistent with the raw debt service ratio measure shown in Chart 2 of the main blog.
Chart A: Residuals from the three cointegrating relationships shown in Table 2
Sources: ONS, Bank of England, Nationwide, Halifax, IMF and Bank calculations.
Estimating the short run dynamics of our VAR
Following Juselius et al., we embed our long-run estimates in a simple VAR model. This allows us to understand dynamic correlations between the variables, and model feedback from the deviations in Chart A to other variables, including GDP growth. We applied standard tests for lag length and found that a two-lag specification was acceptable. Table 3 shows the full set of coefficients. Note that for this step we replace Bank Rate with its real rate counterpart in order to try and control for long-run trends in inflation. Changes in trend inflation and financial liberalisation are treated as exogenous and so are not modelled explicitly.
Table 3: Reduced form VAR model
The VAR system has several intuitive features, such as:
- When leverage is low (third row from the bottom), credit growth picks up (first column). This could partly reflect equity withdrawal as house prices rise.
- When debt service costs are high (second row from the bottom) credit and house price growth fall back. GDP growth falls too, although only modestly. In other words, high debt service costs can be a drag on economic growth.
- Real policy rates respond negatively to ‘excess’ credit, particularly high debt service costs (the right hand column). This may reflect monetary policy loosening in response to lower household spending.
We deliberately exclude ‘drift’ from the VAR, to ensure that interpretable shocks, rather than deterministic trends, explain the drift in variables like credit-to-GDP over time. We also feel this is justified on statistical grounds, as most constants are found to be insignificant when we estimate the VAR in first differences.
Identification of structural shocks
A key innovation in our work is that we attempt to isolate the underlying forces driving the variables in our system over time. In the blog we highlight the contribution falling equilibrium real interest rates have made to increases in household debt in the UK over the past 30 years. Since interest rates respond endogenously to all the other variables in our model, we can only isolate properly the contribution of lower equilibrium interest rates if we “identify” the VAR under certain assumptions.
Following King et al (1991), we use an identification scheme which distinguishes between transitory and permanent shocks, where only the latter have an impact on the levels of variables in the system, like real interest rates, in the long run. Since the system has five endogenous variables and three cointegrating vectors, under our identification scheme there are only two permanent shocks, and three transitory shocks. We assume that only one permanent shock has a long-lasting effect on real policy rates: we term this a ‘long-run equilibrium rate’ shock. This is a practical approach to investigate the dynamics of the system, but it is theoretically contentious – more so than the rest of our work.
We interpret the other permanent shockas a long-run aggregate supply shock, which could capture permanent changes in productivity or labour supply. We do not impose further restrictions to identify the remaining, transitory, shocks in the system as our focus is on long-run trends in the data. These transitory shocks, along with the gradual fading of pre-sample shocks, are grouped together in Chart 5. As before, changes in trend inflation and financial liberalisation are treated as exogenous shocks, the effects of which show up separately in Chart 5.
Impulse response analysis
In the main blog we show a decomposition of debt-to-income including the contributions of different exogenous shocks. Here we focus on the impact of the “long-run equilibrium rate shock”, which is assumed to move the real policy rate permanently. We trace through the full impact of that shock on other variables (Chart B).
We introduce a shock to long-run equilibrium rate which is sufficient to lower real policy rates by 1pp in the long run – though note that it takes several years for this to happen. When this shock first hits, real policy rates fall (1) – perhaps because the shock has initially depressed demand and that requires a monetary policy response. This reduces debt service costs (2), and drives up house prices (3), sending households’ leverage below equilibrium (4).
Chart B: Impact of a shock to the equilibrium interest rate
This lower level of leverage prompts higher credit growth (5), a modest pick-up in GDP (6) and, over time, a gradual rise in policy rates (7) – perhaps to mitigate the effect of higher spending on inflation. But since house prices initially respond much faster than credit volumes, leverage remains below equilibrium (8), continuing to spur credit growth (9). That leads to an ‘overshoot’ of debt service costs (10), and a further fall in real rates in response (11), before the system stabilises after around 10 years.