Borrowing constraints or non-separability? Seeking an answer from the data

The relationship between consumption and income variability has been an integral part of the research on household consumption behaviour for at least most of the time since systematic modelling of optimal life-cycle consumption behaviour was first introduced into economics more than fifty years ago. Recent vintages of models seeking to account for the relationship between consumption and income variability have started to employ ideas from the class of private information models with asset accumulation to derive empirical implications to be tested on data. Interest in these models is to some extent motivated by the empirical failure of simpler approaches, including the hypothesis of complete insurance markets and models where the only insurance available to households is self-insurance, such as simpler versions of the life-cycle and permanent income hypothesis.

As the complete insurance hypothesis is rejected by the data, many researchers have followed the alternative approach of assuming exogenously incomplete markets. To take a concrete example, the Bewley model embeds a version of the permanent income model in a market structure where the only mechanism available to households to smooth consumption over time is through personal savings, possibly with a single asset. Inter-temporal trades can be further limited by the impossibility of borrowing beyond a certain level. In between the two extremes of complete markets and highly limited and exogenously given inter-temporal trade opportunities, research features other possibilities where individuals have access to some state-contingent mechanisms providing insurance over and above that considered in the Bewley model.

The robustness of the Euler equation, one of the key conditions pinning down consumption levels in life-cycle models, is a big advantage from an empirical point of view. Since Hall’s (1978) seminal contribution[1], many authors have focused on the orthogonality restrictions implied by the Euler equation for consumption that can be derived from the consumer’s inter-temporal optimization problem. Under this approach, we do not need to be informed about many aspects of the environment in which the consumer operates or even about the information sets available to consumers. Whereas the level of consumption could depend in an unknown way on income expectations and other unobservable quantities, the Euler equation does not need a closed-form solution and basically exploits the fact that, approximately, changes in log consumption are unpredictable given currently available information. In particular, these changes in consumption should not be related to predictable changes in income. Violations of the orthogonality conditions are all too familiar from the existing research. Many of the reported violations take the form of excess sensitivity of consumption growth to expected changes in income, which evidence has been interpreted as reflecting restrictions on inter-temporal trade, ie as evidence of the existence of (binding) borrowing constraints. However, there are also authors who argue that excess sensitivity does not necessarily reflect the existence of binding borrowing constraints faced by consumers. Instead, excess sensitivity can be due to non-separable preferences between consumption and leisure (labour supply), demographic effects or aggregation problems.

A reliable empirical test which would enable us to differentiate the effects of borrowing constraints from those of non-separabilities on consumption in different times and states of the world would of course be most welcome. Designing such a test has proven to be difficult, however. On the other hand, we do not lack efforts to tackle this issue. In a recently published Bank of Finland discussion paper, Consumption Euler equation with non-separable preference over consumption and leisure and collateral constraints (BoF 09/2009), Juha Kilponen sets up an inter-temporal consumption problem for a household displaying non-separable preferences where the household faces (binding) collateral constraints. The particular model that Kilponen employs is an extension of Iacoviello’s (2004) model,[2] which displays an economy consisting of two types of households, unconstrained and constrained ones. Both types have preferences defined over consumption, leisure and housing. As is typical in these models, the unconstrained households are more patient, valuing future consumption relatively more than the constrained households. Furthermore, housing is separable from consumption and leisure, and households can trade houses, the consumption good and a riskless real bond. Consumption and leisure enter non-separably in the per period utility function, the form of which imposes cancellation of income and substitution effects between consumption and leisure. This functional form is used in eg neoclassical models of business cycle and growth[3] and also in modelling optimal labour supply decisions by households.[4]

Kilponen derives the consumption Euler equations under the specified conditions for both the constrained and unconstrained households, which, after linearization, he aggregates using the ‘λ aggregator’, ie assuming that the share of constrained households equals λ, to arrive at the aggregate Euler equation for the economy. The resulting Euler equation for aggregate consumption incorporates the effects of non-separability and collateral constraints and displays the interdependence between changes in aggregate consumption, changes in aggregate hours, changes in housing demand and real interest rates along the aggregate consumption path. Real interest rates are measured in terms of both consumer prices and house prices. To estimate the Euler equation, Kilponen collects quarterly data on aggregate consumption, aggregate hours, consumer prices, house prices and interest rates in Finland over the period 1987Q1 – 2008Q2. As a proxy for housing demand, Kilponen uses (de-trended) per capita total residential investment over the same sample period. Given this data, the aggregate Euler equation is then estimated by GMM using three to four lags of each ‘right-hand side variable’ as instruments together with the propensity to consume (consumption-to-income ratio), world output and the debt-to-income ratio as additional instruments. To account for the possibility of moving average errors, only lags greater than or equal to two for the ‘right-hand side variables’ are included in the instrument set.

Kilponen estimates three different Euler equations, depending on whether effects from non-separability or collateral constraints and non-separability are present in the estimated equation. According to the results from the whole sample, the estimated elasticity of inter-temporal substitution varies depending on the exact form of the estimated Euler equation, with point estimates spanning the range from 0.3 to 0.6. Apart from the standard Euler equation, ie the one without non-separabilities and collateral constraints, the estimated elasticity of inter-temporal substitution is significantly positive at conventional significance levels. The estimated consumption share of the constrained households, 0.63, is, on the other hand, somewhat high relative to existing international evidence. Kilponen conjectures that this may be due to delayed financial liberalization that started at around the time the sample period starts. This may well be the case, and, while financial liberalization progressed fairly rapidly, resulting in rapid house price inflation potentially relaxing the collateral constraints, the boom was short-lived, as the economy plummeted in the early 1990s as a result of the collapse of the fixed exchange-rate system and the ensuing banking crisis. However, a fair amount of uncertainty remains about the true value of the consumption share of constrained households, as the estimated standard error is relatively high. Thus, the data suggests there is a high probability the true value of λ can be as low as 0.18 and as high as 1.

The estimation results on the whole sample also indicate that the point estimate of the liquidation cost parameter and the inverse (price) elasticity of housing demand is perhaps surprisingly low, 0.11 and 0.03 respectively, implying, in the latter case, that the underlying household preferences are almost linear in housing. That the estimated liquidation cost parameter is low (and imprecisely estimated) suggests either that only a small fraction of the housing wealth is pledgeable, so that the collateral constraint is fairly tight, or that the discount factor of unconstrained households is also very small. Neither of these seems plausible; so the low estimate is slightly puzzling, as Kilponen also seems to acknowledge when he says that the estimate value is unrealistically low and the liquidation cost parameter is typically calibrated in DSGE models to higher values. Kilponen re-estimated his model after fixing the liquidation cost parameter at higher values, but the outcome seems to suggest a nasty trade-off with a smaller estimated inter-temporal elasticity of substitution and a negative consumption share of constrained households.

As Kilponen rightly notes, visual inspection of the goodness of fit of the various models clearly suggests that, for the whole sample, the extended model with both non-separable utility and collateral constraints does better than the two alternatives of separable utility with no collateral constraints and non-separable utility with no collateral constraints. Root-mean-squared errors and the correlation coefficient between actual consumption growth and dynamic forecast ranges, which range from -0.46 in the standard model to 0.57 in the model with non-separable utility and collateral constraints, agree with the eye.

Evidence from the post-crisis period, from 1995 onwards, indicates that the elasticity of inter-temporal substitution has increased and the consumption share of constrained households fallen somewhat. The liquidation cost parameter, too, seems to have increased, but continues to be imprecisely estimated. Moreover, the inverse elasticity of housing and the liquidation cost parameter, although of the right sign, are also imprecisely estimated. In terms of the goodness of fit, the two extensions of the standard model seem to perform equally well with the correlation between actual consumption growth and dynamic forecasts at around 0.25, which is thus clearly smaller for the latter model than in the whole sample. The empirical fit of the standard model is also better for the latter part of the sample, although the correlation between actual consumption growth and dynamic forecasts continues to be negative. Towards the end of his estimation exercise, Kilponen runs a number of robustness checks with different subsamples and instrument sets. The picture that emerges from these checks suggests that the elasticity of inter-temporal substitution seems to have risen towards the end of the sample period; the data does not contain much information on the value of the inverse elasticity of housing demand, the liquidation cost parameter or the fact that the consumption share of constrained households remains persistently high across sub-periods and instrument sets.