The Taylor principle is valid under wage stickiness: an analytical proof

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Consumption, interest rates and expectations: a reconciliation

Despite some strong cases built against it, the Euler equation relating consump-tion, interest rates and expectations remains a cornerstone of monetary policy models.
Under rational expectations, a baseline test of this equation based on the monetary policy rate has a very low fit to the data (Radj2 ∼ 0). Compared to this baseline, tests using interest rates specific to households’ savings and borrowing markedly improve the model’s fit (Radj2 ∼ 30%). A model of limited participation to financial markets, where households are inattentive, and where precaution-ary savings are linked to the risk of unemployment performs even better. It replicates cyclical co-fluctuation and generates similar levels and volatilities of the interest rate to which households react and their marginal rate of intertem-poral substitution since 1990.

Introduction

The Euler equation on consumption is a cornerstone of monetary policy models. In the permanent income hypothesis, when households are assumed to be rational, their con-sumption scheme should verify the following equation:
(c + 1 ) P+t 1 1 = βEt Œ Uc t 1 it ‘ (3.1)
where Uc(t) is the marginal utility of consumption at time t, i is the nominal interest rate at which households can lend and borrow and P the inflation rate of consumption prices. Et is the expectation operator describing how households form their expectations over future variables at time t.
Relaxing the constant interest rate hypothesis Canzoneri et al. (2007) test this first order condition of the representative household maximization program under different utility specifications against US data. Their main result is that when the Euler equa-tion is assumed to hold and the interest rate is treated as the unknown variable, the counter-factual interest rate to which the representative household seems to respond is quite different from the monetary policy rate. In most cases it is negatively correlated with it, a result they pose as a challenge for monetary policy models, which are largely built on the Euler equation.1
This rejection of the Euler equation is too restrictive in two respects.
First, if households were reacting to another interest rate than those of the money market (mortgage, deposit, private loans…), the spread between this rate and the mon-etary policy instrument could account for the negative or weak correlation they observe. Such a result would have strong implications for monetary policy analysis and the way it is modelled (in the new neoclassical synthesis for instance). It would rehabilitate the Euler equation but dampen the transmission mechanism between the policy maker and households.
Second, Canzoneri et al. measure households’ expectations through a VARX model estimated ex ante, i.e. households use optimally all the information publicly available to adjust their consumption-savings trade-off. This assumption can be relaxed in many ways to try not to reject the Euler equation. I inquire one possibility. Contrary to estimating the expectation VARX ex ante, which puts an infinite weight on the minimisation of ex-pectation errors relatively to the Euler equation residual, I estimate the Euler equation and expectations altogether, putting the same weight on both types of residuals.
I build on Canzoneri et al.’s analysis and test household specific interest rates. I find that households’ behaviour is better described by interest rates other than the monetary policy one (specifically a combination of the interest rate on personal loans and the Trea-sury bill); indicating that households encounter sizeable frictions on the credit and loans market which shall not be relegated to a residual.
However, there remain sizeable discrepancies between the Euler equation and the data. In particular, rational expectations as approximated by a VARX model feed too much volatility into the Euler equation and between 1972 and 2013 there has been some struc-tural changes either in households expectation formation process or their behaviour.
Building on these intuitions, I test a model of limited participation to financial markets (i.e. where households react to a combination of the personal loans and the Treasury bill rate but also to their own perception of interest rate as reported in the Michigan survey), where households are inattentive (Reis, 2006), in particular as regards inflation, and where precautionary savings are linked to the risk of unemployment. This set of modelling as-sumptions yield a close link between the MRS and the interest rate to which households are subject (Radj2 = 35%), replicates cyclical co-fluctuation and generates similar levels and volatilities of the two rates; a result in contrast with Canzoneri et al.’s rejection.
Since Hall (1978), macroeconomic data have been confronted to the permanent income-life cycle theory repeatedly leading to successive refutations, validations and improvements in households’ consumption models. Early papers, surveyed by Attanasio (1999), have in-vestigated the properties of consumption within this framework under the assumption of constant real interest rate and often certainty equivalence. Many alternative utility func-tions have been used to reconcile consumption data with the permanent-income hypothesis and understand some major stylized facts. Mankiw (1982), and Bernanke (1985) study the specificities of durables consumption. Abel (1990), and Gali (1994) investigate the equity premium puzzle by modifying the utility function. Campbell and Mankiw (1990) consider a population with only a fraction of households following the model, while oth-ers, financially constrained, consume their current income in each period. Flavin (1981) investigates the excess sensitivity puzzle, that is consumption reacting to lagged changes in income. Campbell and Deaton (1989) investigate the excess smoothness puzzle, that is consumption not responding one-to-one to shocks to permanent income.
The literature on the Euler equation is split between the maximum likelihood estima-tions and the GMM approach pioneered by (Hansen and Singleton, 1982; Hansen, 1982). Because in this paper I test expectations departing from optimality, I use maximum like-lihood estimation. Indeed, the GMM approach relies on the assumption that households expectations are identical to the econometricians, so that conditional on the proper set of instruments an orthogonality condition is built.
Part of the literature based on GMM estimations investigate the asset pricing puzzle. If households arbitrate between consumption and savings on several markets, the Euler equation should be verified for the return on these assets at the same time. Although I consider several households specific interest rate, I work on a weaker assumption of lim-ited participation. While any combination of returns should verify the Euler equation under the CAPM hypothesis, I investigate whether a particular combination of returns does, that is whether the population can be split between fractions arbitrating on one or another market.
Carroll (2000) in his requiem for the representative consumer model interprets the rejection of the Euler equation as an incompatibility between macro data and a micro-founded model due to the differences in the marginal behaviour of individuals. The Euler equation on consumption could be wrong, not because of the form of the utility function but because the representative agent assumption is heroic. Using macroeconomic data to capture the value of structural parameters is not the purpose of this paper: Attanasio and Weber (1993) show that estimates of the intertemporal elasticity of substitution on aggre-gate data are systematically lower than estimates on average cohort data. Attanasio and Weber (1993, 1995) show that tests on aggregate data tend to reject micro-founded models specification too often. However, microeconomics comes against its own difficulties (Car-roll, 2001; Ludvigson and Paxson, 2001) while macroeconomic models remain widespread and have crucial implications for the policy maker, both monetary and fiscal. Thus, the challenge posed by the aggregate Euler equation on consumption is worth investigating.
The remainder of this paper is organized as follows: in section 3.1 I expose the frame-work of the present analysis; in section 3.2 I test the Euler equation against interest rates specific to households borrowing and saving; in section 3.3 I focus on expectations; in section 3.4 I combine results from the previous sections an propose a model which con-vincingly reconciles the Euler equation and rational expectations.

A framework for testing the Euler equation under limited participation and near rationality

The Euler equation relates households’ expectation for consumption growth and inflation to the interest rate at which they can borrow or save. I first consider the standard CES utility function. Under a log-normality hypothesis of forecasting errors, or a first order approximation on these error, equation (3.1) can be linearised (see Appendix 3.B). it = δ + σEtDct+1 + Etπt+1 (3.2)
with σ the relative risk aversion parameter or 1~σ the intertemporal elasticity of substi-tution parameter, EtDct+1 consumption growth expectation (Dlog) and Etπt+1 inflation expectations.2
Such an equation is commonly introduced in monetary policy models but is not as-sumed to holds exactly. A residual to this equation is often interpreted as a preference shock, a demand shock from households part introduced through a time varying discount factor. There are however more reasons for the Euler equation not to hold on aggregate. First, even if each individual behaves in accordance to this equation, on aggregate it should not be exactly verified because households wealth vary and so does their marginal utility of consumption (Carroll, 2000). Second, under its linear form, the constant captures the risk associated to the forecast error. It is often assumed to be constant but changes in the volatility of the economic environment can imply through this channel a precautionary savings behaviour (?). This volatility can account for a significant share of fluctuations (Bloom, 2009). Third, equation (3.2) is based on a separable utility function, but more general utility functions imply non separability in with labour (Mankiw et al., 1985) or in time through durable consumption (Mankiw, 1982; Bernanke, 1985; Ferson and Con-stantinides, 1991; Grossman and Laroque, 1990) or through habit formation (Abel, 1990; Gali, 1994; Constantinides, 1990; Campbell and Cochrane, 1999; Ravn et al., 2006).
The latter has received a lot of attention because it provides a solution to the equity premium/risk free rate puzzles (Abel, 1990; Gali, 1994; Constantinides, 1990; Abel, 1999; Campbell and Cochrane, 1999). In the context of monetary policy models, habit formation also implies a conveniently gradual response of private consumption to monetary policy shocks (Fuhrer, 2000). Habit formation can also explain why savings may be high in a growing economy (Carroll et al., 2000).
Assuming external habit formation, equation (3.2) becomes: it = δ + σEtDct+1 + αDc + Etπt+1 (3.3)
with α = −γ(1 − σ) and 0 < γ < 1 the habit formation parameter.
The Euler equation assumes that households can arbitrate between consumption today and tomorrow. Another popular modelling assumption for households has been proposed by Campbell and Mankiw (1990): a fraction of the population is financially constrained and consumes its income within the period. Introducing this limited participation to financial market in an heterogeneous agent model modifies in particular the aggregate response to fiscal shocks (Mankiw, 2000; Galí et al., 2007). Being financially constrained, these hand-to-mouth consumers are considered as the poor, however Kaplan et al. (2014) show that such a behaviour can also describe rich households whose financial wealth is not liquid.
Assuming that hand-to-mouth households earn a constant share of total income µ, equa-tion (3.2) becomes it δ σ EtDct 1 µ SRt SRt 1 Etπt 1 (3.4) = + Œ − t)( − − −t−1 − )‘+ + (1 − SR + 1 SR µ with SR the aggregate savings ratio of households.
In this paper I consider a less restrictive form of limited participation. The population may be split between households engaged in different financial market (mortgage, savings, consumption loans …) so that on aggregate, the interest rate to which the representative consumer reacts is a combination of household specific interest rates. Although I consider returns on various financial instruments, this paper departs from the CCAPM literature (Campbell, 2003): all households do not arbitrate on all financial instrument returns, i.e. the Euler equation does not hold simultaneously for all interest rates but for aggregate consumption and an aggregate asset. Is this respect I test a more restrictive model than Hansen (1982); Hansen and Singleton (1982); Campbell and Cochrane (1999); Ferson and Constantinides (1991); Hall (1988) who consider simultaneously several rates of returns and assume households arbitrate between them.
Under the assumption that each households arbitrates between consumption and savings on one interest rate, equation (3.2) becomes it 1 γ rate it γ it δ σEtDct 1 Etπt 1 (3.5) ¯ =(− Q ) fed + Q rate rate = + + + + rates fed rates fed ≠ ≠ with ¯ an interest rate combination, with weights assumed constant. As it is written, the i interest rate combination imposes that the weights sum to one. In addition, one may impose, as these weights should reflect consumption shares, that these weights are con-strained between 0 and 1.
From an econometric point of view, it is noteworthy that while the residual of the Euler equation due to the aggregation of individual consumption growth and more gen-erally the demand shock is naturally correlated to consumption growth, the aggregation approximation for interest rates or missing components in this combination (taxation, management fees, differences in liquidity) imply a residual which is correlated to the in-terest rates. Either interpretation of the residual imply opposite estimations in terms of explained variable (it Etπt 1 or EtDct 1). The approach usually favoured regresses ¯ the Fedfunds and as such considers the residual as orthogonal to consumption growth on − + + this interest rate. Nevertheless as this residual should in particular capture the spread between the Fedfunds and the interest rate to which households are subject, this orthog-onality assumption is quite strong. For this reason, I favour the approach regressing an optimal interest rate combination on expected consumption.
More problematic for the estimation is the expectation operator. Indeed, expectations are not directly measured which calls for elaborate estimation strategies. Hansen (1982); Hansen and Singleton (1982) proposed a GMM estimator built on the orthogonality of the Euler equation with the information set of the representative household. Despite its statistical properties this estimator has a major drawback for the present exercise: with a GMM estimation, I could neither extract the expectations nor allow them to depart from optimality. For this reason, I use a maximum likelihood estimation and as in (Fuhrer, 2000; Canzoneri et al., 2007), I assume that households are forming their expectations of consumption growth and inflation through a VARX model: Yt+1 = L(L)Yt + GXt + t+1 (3.6)
with Y = [Dc, π]3 = [ c, π] is the error of the VARX model and is by assumption identi-fied to the expectation errors of consumption and inflation (EtYt+1 = L (L)Yt + GXt).
Plugging expectations in the Euler equation is a compatibility check of two key ele-ments of monetary policy models, namely the Euler equation and rational expectations. In addition to the CES function considered here, Canzoneri et al. (2007) have tested the Eu-ler equation implied by other forms of utility functions in a similar approach, but conclude that these functions also invalidate the Euler equation on aggregate data. In (Poissonnier, 2015b) I confirm their results with other specifications of the VAR, with other definitions of the consumption bundle and on French data. To further analyse the compatibility of the Euler equation (3.5) (possibly with habit formation and hand-to-mouth consumers) and the VARX model (3.6) I thus focus on the household specific interest rate combination ¯ in the full model:
In section 3.2 I focus on the interest rate combination ¯ and consider expectations (3.8) i estimated ex ante. However, even if I assume the shocks to be independent, the fact that the Euler equation includes expectations derived from the other equations should prevent me from treating the shocks sequentially. Estimating expectations ex ante is then equiv-alent to putting an infinite weight on the minimisation of relative to the minimization of ζ and bias the test of the Euler equation towards its rejection. Indeed, small devia-tions from the estimation ex ante of the VARX model, within its confidence interval, may improve the fit to the Euler equation. To investigate this possibility of near rationality, I estimate the system (3.9) altogether in section 3.3. Near rationality is here the other side of the coin considered by Cochrane (1989): I do not consider near rationality as small deviations from the optimal arbitrage, i.e. the Euler equation, but small deviations from the optimal forecast of consumption and inflation required to make such an arbitrage.

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The Euler equation with households-specific interest rates

In this section, I consider households-specific interest rates in testing the Euler equation. The interest rates I consider are depicted on Figure 3.1. The correlation of real mone-tary policy rate and the spreads on households-specific rates are reported in Table 3.1. It is noteworthy that households-specific spreads are negatively correlated to real mone-tary policy rate, which could a priori account for the results exposed by Canzoneri et al. (2007) and also motivate this analysis where this spread is not assumed orthogonal to the monetary policy stance.
US interest rates (Figure 3.1) I use the Effective Federal Funds Rate (Fedfunds), a mortgage rate (30-Year conventional mortgage rate), a car loans rate (Finance Rate on Consumer Instalment Loans at Commercial Banks, New Autos 48 Month Loan), a personal loans rate (Finance Rate on Personal Loans at Commercial Banks, 24 Month Loan), a deposit rate (3-Month Certificate of Deposit: Secondary Market Rate) and the 3 month treasury bill (3-Month Treasury Bill: Secondary Market Rate). I add to this list the null return on cash holdings.4 Apart from the mortgage rate, maturities of these interest rates are short. The mortgage rate is included because wealth effects are empirically stronger in the US than in France, a stylized fact sometimes attributed to the higher flexibility of mortgages in the US where real estate can be used as a collateral for consumption. Some of these rates are closely related (treasury bill, fedfunds and certificates of deposits). Appart from the rate on the Certificate of Deposit which is superimposed to the Fedfunds, I keep all these rates to remain as exhaustive as possible.5
The estimates for the different cases (CES utility function, with habit, with constrained households à la (Campbell and Mankiw, 1990) (CM), and both habit and CM households) are reported in Table 3.2. The model’s constant allows to compute the discount factor.6 It is equal to 98%. The estimated inverse elasticity of substitution or CRRA parameter is below one as in (Hansen and Singleton, 1982) and in constrast with (Hall, 1988). Habit formation is found either to be at its upper bound (1), which contradicts the model’s as-sumption, or 0.36 when estimated together with a constrained households à la Campbell and Mankiw hypothesis. However the share of these households is very small (4% or null) compared to the original paper’s result (between 40 and 50%), making a small case in favour of either assumption. As for the optimal interest rate combination, the weights are constrained between 0 and 1. I find a combination of 80% personal loans and 20% Treasury bill throughout the specifications with in particular a null weight on the Fedfunds.
As I estimate the Euler equation under the non standard assumption of the residuals being correlated to the Fedfunds, for comparison, I run the same estimations under the reverse assumption, as in (Hall, 1988). First, these estimations reject the habit formation or constrained households assumptions, the habit parameter is found to be null and the share of constrained households equal to 1%. The CRRA parameter is estimated to be large (around 10). Such a value is sometimes encountered in the equity premium puzzle literature as it increases the MRS closer to the average return of risky assets but at the same time it raises another issue as it increases the volatility of the MRS and thus of the implied risk free rate (Abel, 1990). Here, it can be seen as the result of an estima-tion bias as this strategy minimizes the variance of ζt~σ so that increasing the value of σ indeed improves the fit of the regression but may well deteriorate the fit of the Euler equation. This may explain (Hall, 1988) findings of a non significant intertemporal elastic-ity of substitution (i.e. a high risk aversion σ). In line with this high point estimate, the discount factor is found larger than one: with long term growth, this does not jeopardize the model’s stability (see (Kocherlakota, 1990) in the RBC case and (Poissonnier, 2015a) in the neo Keynesian case). As for the interest rate, this estimation yields a combination of 20% personal loans rate and 80% car loans rate but again a null weight on the Fedfunds.
Comparisons of the MRS with the optimal rate combination are depicted in Fig-ure 3.2.7 Table 3.3 displays some comparative statistics for actual and Euler interest rates in real terms.
The estimated MRS is not correlated with the Fedfunds (< 11%). The correlation with the optimal rate combination is larger (between 20% and 30%) whether deflation is based on expected future inflation (correlations 2) on contemporaneous inflation (correlations 1) and regardless of the model considered. In real terms, the MRS is less volatile than the corresponding optimal rate combinations (Table 3.3) but in nominal terms the former exhibits small but rapid fluctuations, while the optimal rate combination and actual rates are much smoother (Figure 3.2).
Finally, the optimal rate combination markedly differs from the monetary policy rate although it replicates similar fluctuations. Considering a weighted average of the rate on personal loans and the treasury bill explains a large share of the spread between the MRS and the monetary policy rate but there remains a sizeable discrepancy in the short term volatility which is largely due to the volatility of inflation expectations.
As a partial conclusion: overall fit to the data To measure the fit to the data I compute both the log-likelihood of the full model (3.9) (even though it is estimated in two steps) and the adjusted R2 of the Euler equation (3.7) (Table 3.4).8

Table of contents :

1 Résumé 
1.1 Consommation, taux d’intérêt et anticipations : vers une réconciliation
1.2 Le principe de Taylor est vérifié lorsque les salaires sont rigides
1.3 Un compte satellite des ménages français
1.4 Réformes structurelles dans les modèles DSGE
Bibliographie
2 Introduction 
Introduction
Main results
Relation to the literature
Connections between chapters
Bibliography
3 Consumption, interest rates and expectations: a reconciliation 
3.1 A framework for testing the Euler equation under limited participation and near rationality
3.2 The Euler equation with households-specific interest rates
3.3 Expectations in the Euler equation
3.4 One way of reconciliation
3.5 Conclusion
Appendices
3.A Data and the VAR of expectations
3.B The conditional log-normality hypothesis
3.C On French data
Bibliography
4 The Taylor principle is valid under wage stickiness: an analytical proof 
4.1 Introduction
4.2 A monetary model with sticky wages and prices
4.3 Outline of the proof
4.4 Conclusion
Appendices
4.A Proof
Bibliography
5 Households Satellite Account for France. Methodological issues on the assessment of domestic production
5.1 Introduction
5.2 Domestic production amounts to 30 to 50% of GDP in most studies
5.3 The accounting and valuation of hours of domestic work
5.4 From TUS to HHSA
5.5 A households satellite account for France in 1998 and 2010
5.6 Conclusion
Appendices
5.A Additional tables for international comparisons
5.B Activity, time and wage
5.C Comparisons on the main issue: the frontier of production
Bibliography
6 Structural reforms in DSGE models: a plead for sensitivity analyses 
6.1 Introduction
6.2 Model
6.3 Understanding the mechanisms
6.4 A sensitivity analysis
6.5 Conclusion
Appendices
6.A Sensitivity analysis
6.B Some calculations
Bibliography

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