 # Non-Linearities and the Euler Equation: Does Uncertainty Have an Effect on the Approximation Quality?

From the journal Review of Economics

## Abstract

Deriving a forward-looking Euler equation, this paper compares two fully identified non-linear versions. The difference (or bias) between them stems from an approximation by extracting parameters from the expectation values (Jensen’s inequality) as it is common practice in the literature. Furthermore, the model is completely identified using Consensus Forecasts data for the expectations, inflation-indexed bonds as a proxy for the long-run real interest rate, and estimates for the elasticity of intertemporal substitution. Regression analyses using data for three major economies reveal that the difference between the two Euler versions can be explained by uncertainty in the data itself and external uncertainty measures. The results confirm a connection between theoretical and empirical higher-order moments in economic models.

JEL Classification: C02; C12; E31; E43

## Acknowledgements

Thanks to Christian Bauer, Matthias Neuenkirch, and participants of the 2019 Workshop on Monetary and Financial Macroeconomics at Deutsche Bundesbank Hamburg for helpful comments on earlier versions of the paper.

## Appendix

### A.1 Euler Equation – Calculation Steps

The optimization problem uses the constraint

(10)CtPt+Bt+1=Wt+(1+it1)Bt,

with Wt as the nominal wage and Bt as the nominal value of bonds. The latter provides the link between two periods. Depending on the definition of the interest rate, the horizon can vary. Since the right-hand side of eq. (10) represents the disposable income in period t, Bt is the investment by households starting in period t - 1 by the interest-bearing condition it - 1.

Dynamic Programming uses the additively separable utility function and the envelope theorem to set up optimality conditions for two consecutive periods. The procedure can be divided into three parts. The first part is to write a value function, the Bellman equation. Under the assumption that the second term of the expanded utility

(11)U(Ct)+Ets=t+1ϱsst1U(Cs)

is maximized in period t, the Bellman equation is

(12)V(Bt)maxCtU(Ct)+ϱtV(Bt+1).

The expected value vanishes since Bt + 1 is determined by variables in period t in the constraint. Differentiating with respect to Ct gives the first-order condition

(13)dU(Ct)dCt+ϱtdV(Bt+1)dBt+1dBt+1dCt=0,

which results, when including eq. (10), in

(14)dU(Ct)dCt=PtϱtdV(Bt+1)dBt+1.

Equation (14) relates the marginal utility to the marginal value in the following period, the time preference, and prices in the same period. Therefore, a higher ϱt and Pt results in a lower Ct.

In the next part, the envelope theorem is used to differentiate the value function (by inserting the optimized Ct) with respect to the costate variable Bt:

(15)V(Bt)=U(Ct)+ϱtV(Bt+1)
(16)dV(Bt)dBt=ϱtdV(Bt+1)dBt+1dBt+1dBt
(17)dV(Bt)dBt=ϱtdV(Bt+1)dBt+1(1+it1).

Equation (17) reveals the relationship of the marginal value functions.

In a third step, the first-order condition eq. (14) can be used to replace the value functions in eq. (17) with the marginal utility in both periods t and t - 1:

(18)dU(Ct1)dCt11Pt1ϱt1=ϱtdU(Ct)dCt1Ptϱt(1+it1)
(19)dU(Ct)dCt=ϱt(1+it)EtPtPt+1dU(Ct+1)dCt+1.

The time shift yields the Euler condition.

### A.2 Constant Relative Risk Aversion – Utility Function

The CRRA-function,

(20)U(C)=C1γ11γ,   γ>0,

transforms into a root-, log-, or inverse hyperbolic-form, depending on the value of γ. For γ∈ ]0,1[, a root-function appears, for example the square root:

(21)U(C | γ=0.5)=2(C1).

Log-utility (γ = 1) can be considered as a special case since l’Hôpital’s rule is needed to reveal the ratio between the limit values in numerator and denominator:

(22)U(C | γ=1)=limγ1C1γ11γ=limγ1log(C)C1γ(1)1=log(C).

Finally, a hyperbolic-form, mirrored at the abscissa, emerges for γ > 1:

(23)U(C | γ=2)=C111=1C+1.

Note that for all cases, a positive utility is only assured if more than one unit is consumed.

### A.3 Covariances of Transformed Random Variables

The idea is to show the substantial difference in the correlation after transforming two random variables (RV’s). Figure 6 compares the correlation of repeated draws between two (i) normally-distributed and (ii) the inverse of normal-distributed RV’s. Inverting the variables as transformation is chosen because it is relatively simple and basically matches the non-linearities in eq. (5). Per repetition, 30 RV’s are drawn since this roughly corresponds to the number of observations per cross-section, presented in Section 3. We calculate the correlation as standardized covariance, otherwise outliers would distort the density function. Figure 6:

Resulting correlation with 105 repetitions, each with 30 draws. Distributions: Standard-Normal (left) and inverted Standard-Normal (right).

While the second moments are equal (Std.Dev. = 0.2), the kurtosis shows a fundamental difference: 2.8 (non-transformed) vs. 8.1 (transformed).

### A.4 Euro Area – Weighting Scheme

Weighting of EA countries (France, Germany, Italy, Netherlands, and Spain) is conducted from January 1999 to November 2002. Seasonal- and calendar-adjusted real GDP data from Eurostat show that these five countries contribute around 86% to the EA’s GDP. Over time, there is a slight downward trend and a minor structural break in 2001 when Greece joined the euro zone. Quarterly GDP data are converted to a monthly basis via linear interpolation. The weighting scheme is as follows:

(24)μEA=i=15κiE[Yi]=E[λY],

where κi are GDP-weights with i=15κi=1 and λ, the firm level weights for each country (depending also on number of firms: λi=(N/Ni)κi), is needed to insert the individual forecasts directly into the expectation value, i. e. E[(1+λY)γ].

Using this method for the period after 11/2002 shows no substantial difference, when comparing with the actual values for the EA.

### A.5 Descriptive Statistics

Table 6:

Consumption and price level measures are represented as one-year ahead expected growth rates.

Euler Equation DataMean (%)Std.Dev. (pp)Min. (%)Max. (%)
Personal Consumption (US)2.520.871.174.49
Household Consumption (UK)1.921.081.953.97
Private Consumption (EA)1.310.840.532.88
Consumer Prices (US)2.460.870.665.2
Retail Prices (UK)3.011.140.447.55
Consumer Prices (EA)1.620.530.023.01
Effective FFR (US)3.072.550.049.52
SONIA (UK)3.022.530.168.61
EONIA (EA)1.71.70.365.16
Inflation-Indexed Security (US)1.070.890.793.14
Inflation-Indexed Bond (UK)1.821.751.685.08
Gov. Benchmark Bond Yield (EA)2.741.9817.02

Table 7:

Consensus Forecasts uncertainty measures are based on the one-year ahead expected growth rates. The last column shows the coefficient of variation.

CF UncertaintyMeanStd.Dev.Min.Max.CV
GDP Std.Dev. (US)0.30.110.130.70.37
GDP Std.Dev. (UK)0.370.120.170.760.32
GDP Std.Dev. (EA)0.270.140.080.70.51
Consumption Std.Dev. (US)0.290.10.110.770.34
Consumption Std.Dev. (UK)0.480.170.181.120.36
Consumption Std.Dev. (EA)0.310.150.120.70.49
Prices Std.Dev. (US)0.280.10.110.920.35
Prices Std.Dev. (UK)0.330.170.11.120.5
Prices Std.Dev. (EA)0.240.180.060.770.76

Table 8:

The last column shows the coefficient of variation, only differing by a small amount across related variables. CLIFS is an index of financial stability by the ECB. EPU and MPU are the Baker, Bloom, and Davis (2016) uncertainty measures. Macroeconomic and Financial Uncertainty are the indices by Jurado, Ludvigon, and Ng (2015). Systemic Stress is an indicator provided by the ECB. Originally, the Forecast Uncertainty data by Rossi and Sekhposyan (2015); Rossi and Sekhposyan (2017) are standardized such that the maximum is 1. Slightly larger values in our data are due to interpolation. Interestingly, there is no or weak negative correlation between these measures. The correlation between all other related variables across regions is as expected: positive and significant (available on request). The World Uncertainty Index comes from the paper by Ahir, Bloom, and Furceri (2018).

Uncertainty MeasuresMeanStd.Dev.Min.Max.CV
S&P 500 Std.Dev. (US)18.8514.411.42111.410.76
FTSE 100 Std.Dev. (UK)79.7649.3511.09335.780.62
Euro Stoxx 50 Std.Dev. (EA)55.3740.314.75245.330.73
S&P Avg. Daily Range (US)1.240.690.376.620.55
FTSE Avg. Daily Range (UK)1.280.670.535.950.52
EStoxx Avg. Daily Range (EA)1.690.840.65.880.5
VIX (US)19.297.429.5159.890.38
VFTSE (UK)19.197.689.5554.150.4
VSTOXX (EA)23.938.8111.9961.340.37
WTI Std.Dev. (US)1.641.350.1511.290.82
Brent Std.Dev. (UK/EA)1.641.310.1610.790.8
St.Louis FSI (US)011.534.71
CLIFS (UK)0.120.090.010.560.78
Weighted CLIFS (EA)0.120.080.030.420.66
EPU (US)113.7943.0544.78284.250.38
EPU (UK)119.6668.7724.04558.220.57
EPU (EA)131.3561.4845.3433.280.47
MPU (US)106.1450.8412.8362.440.48
Macro Uncertainty (US)0.910.050.851.150.05
Financial Uncertainty (US)0.980.050.911.130.05
Systemic Stress (EA)0.180.160.020.80.92
Forecast Uncertainty (US)0.730.130.491.010.17
Forecast Inf. Uncertainty (EA)0.750.140.451.020.18
Forecast GDP Uncertainty (EA)0.750.140.5110.19
World Uncertainty Index122.6337.6763.76250.520.31

### A.6 Correlation – Uncertainty Measures

Table 9:

S&P and WTI are the monthly standard deviations. S&PR are monthly averages of the daily range in percent (S&P 500). FSI is the St. Louis Financial Stress Index. EPU and MPU are the Baker, Bloom, and Davis (2016) uncertainty measures. Macro (Macroeconomic Uncertainty) and Fin. (Financial Uncertainty) are the uncertainty measures by Jurado, Ludvigon, and Ng (2015). Rossi is the forecast uncertainty by Rossi and Sekhposyan (2015). WUI is the World Uncertainty Index by Ahir, Bloom, and Furceri (2018). Numbers in bold denote significance at the 5% level.

USS&PS&PRVIXWTIFSIEPUMPUMacroFin.Rossi
S&P1
S&PR0.61
VIX0.490.91
WTI0.450.450.371
FSI0.110.680.670.081
EPU0.450.390.390.320.041
MPU0.230.070.060.040.080.481
Macro0.360.630.60.590.550.230.061
Fin.0.460.760.830.310.640.30.020.651
Rossi0.030.140.190.080.170.160.050.10.181
WUI0.13–0.28–0.290.14–0.620.440.37–0.21–0.360.08

Table 10:

FTSE and Brent are the monthly standard deviations. FSTER are monthly averages of the daily range in percent (FTSE 100). CLIFS is an index of financial stability by the ECB. EPU is the Baker, Bloom, and Davis (2016) uncertainty measure. WUI is the World Uncertainty Index by Ahir, Bloom, and Furceri (2018). Numbers in bold denote significance at the 5% level.

UKFSTEFSTERVFTSEBrentCLIFSEPU
FSTE1
FSTER0.761
VFTSE0.650.931
Brent0.390.430.281
CLIFS0.380.560.60.321
EPU0.20.210.130.170.181
WUI00.11–0.260.18–0.180.59

Table 11:

EStoxx and Brent are the monthly standard deviations. EStoxxR are monthly averages of the daily range in percent (ESTOXX 50). CLIFS is an index of financial stability by the ECB. EPU is the Baker, Bloom, and Davis (2016) uncertainty measure. INF and GDP are forecast error measures by Rossi and Sekhposyan (2017). WUI is the World Uncertainty Index by Ahir, Bloom, and Furceri (2018). Numbers in bold denote significance at the 5% level.

EAEStoxxEStoxxRVStoxxBrentCLIFSEPUCISSINFGDP
EStoxx1
EStoxxR0.71
VStoxx0.620.921
Brent0.320.20.181
CLIFS0.40.720.750.311
EPU0.150.150.130.290.141
CISS0.370.590.610.480.790.151
INF0.12–0.21–0.210.310.050.110.061
GDP0.270.240.250.270.280.120.360.11
WUI0.1–0.13–0.140.18–0.160.780.120.18–0.38

Table 12:

Symbols.

LetterDescription
βiParameters (“bias” equation, i∈{0,1,2})
γτReciprocal value of the EIS (τ∈{1,2})
ΔiResiduals (Jensen’s inequality, i∈{c,π})
εt(i)Error terms (i∈{J,bias})
κi(Aggregated) GDP weights (i{1,,5})
λiGDP weights on individual level (i∈{1,…,5})
µMean
πtInflation
ϱtDiscount factor (time preference)
σStandard deviation
BtBonds
ctConsumption growth
CtConsumption
itNominal interest rate
mIndex of months
NNumber of forecasters
NiNumber of forecasters in a specific country (i{1,,5})
PtPrice; Price level
rtLong-run real interest rate
U(.)Utility function
V(.)Value function (Bellman equation)
WtWage

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Published Online: 2020-01-09
Published in Print: 2020-01-28