Neil Arnwine

Taner Yigit

English

Research Papers in Economics

 
 
 
 
 
 
 
  What Fisher Knew About His Relation, We Sometimes Forget
* 
 
 
 
 
Neil Arnwine
a
and 
Taner M. Yigit
b
 
 
 
 
 
A B S T R A C T 
 
 
 
Expected consumption growth increases the real interest rate as the typical consumer tries to 
smooth consumption over time. Thus, while consumption growth should be an integral factor 
in studying the Fisher relation, it is typically excluded from empirical analyses of the 
relation. This paper demonstrates that i) Fisher’s view of the role of consumption growth is 
consistent with a modern understanding of the representative agent’s optimization problem 
via the Euler equation governing the purchase of nominal bonds, and ii) the inclusion of 
consumption growth in the Fisher relation is supported empirically. Finally, our modified 
Fisher relation provides an alternative method for estimating the consumer’s degree of 
relative risk aversion. 
 
JEL Codes: E43 
Keywords:  Fisher relation, interest rate, consumption growth 
 
 
 
 
a Assistant Professor, Department of Economics, Bilkent University, Bilkent, Ankara 06800, Turkey; Email: 
neil@bilkent.edu.tr; Tel: +90-312-290-1890; Fax: +90-312-266-5140   
b Corresponding Author: Assistant Professor, Department of Economics, Bilkent University, Bilkent, Ankara 
06800, Turkey; Email: tyigit@bilkent.edu.tr;  Phone: +90-312-290-1643; Fax: +90-312-266-5140  
* We would like to thank E. Basci, R. Gurkaynak, K. Hasker, I. Pastine, R. Rasche and S. Sayek for their 
comments.  All remaining errors are our own. 
 
   
1.  Introduction 
Fisher recognized that the time shape of income is a determinative factor of the real interest 
rate.   
“The fact that a person’s income is increasing tends to make his preference 
for present over future income high, as compared with what it would be if 
his income were flowing uniformly or at a slackening rate; for an increasing 
income means that the present income is relatively scarce and future income 
relatively abundant (Fisher, 1930, p. 73-74).”   
 
By ‘income’ Fisher meant what we now refer to as ‘consumption’, as he explained in the 
preliminaries of his book, The Theory of Interest, (Fisher, 1930, I.I.61).  Most empirical 
studies of the Fisher relation have generally neglected to include expected consumption 
growth as an explanatory variable. Although some studies have included income growth, e.g., 
Levi and Makin (1978), VanderHoff (1984), and Dotsey et al. (2003), the motive for its 
inclusion was not based upon the idea of income smoothing. In this paper, we derive an 
augmented Fisher relation from the standard representative agent’s Euler equation governing 
the demand for bonds, which differs from the standard Fisher relation by the inclusion of an 
expected consumption growth term. We demonstrate that this inclusion is i) empirically 
significant, and ii) provides an alternative way to estimate the consumer’s degree of relative 
risk aversion. 
The paper is organized as follows:  Section 2 elaborates on why and how we should add 
the consumption growth rate into the Fisher relation, Section 3 empirically examines the 
implications and interprets the results, and Section 4 concludes. 
 
2.  Model 
Fisher’s quote above is consistent with the analysis of the consumer’s Euler equation for bond 
purchases, which is derived from a standard representative agent problem.  
  () () ( )
1
c uc uc
iE
pp
β
c ′ ⎡ ⎤
=+⎢ ⎥ ′ ⎣ ⎦
  (1) 
Here a prime denotes variables measured at time t + 1, a tilde distinguishes the interest rate in 
levels from the log of one plus the interest rate used in some of the equations below and E is 
the expectations operator at time t. The optimizing consumer equates the marginal utility 
value of current consumption with the discounted expected future utility value consumption in 
the next period. Assuming a constant degree of relative risk aversion utility function, 
1     ()
1 1
1
1
ln 1
c
uc
c
γ γ
γ
γ
− ⎧ ≠ ⎪ − = ⎨
⎪ = ⎩
 (2) 
Equation (1) becomes 
  ()
1
1 t
pc
iE
pc
γ
β
− ⎡ ⎤ ⎛⎞ += ⋅ ⎢ ⎥ ⎜⎟ ′ ′ ⎝⎠ ⎢ ⎥ ⎣ ⎦
  (3) 
where γ  represents the consumer’s degree of relative risk aversion. Expanding the 
expectation term we obtain  
  ()
1
1 cov , tt
pc p c
iE E
pc p c
γγ
β
− ⎧⎫ ⎡ ⎤ ⎛⎞ ⎪ ⎛⎞ ⎛⎞ += + ⎪
⎢ ⎥ ⎨ ⎜⎟⎜⎟ ⎜⎟ ′′ ′ ′ ⎝⎠ ⎝⎠ ⎝⎠
⎬
⎢ ⎥ ⎪⎪ ⎣ ⎦ ⎩⎭
  (4) 
To simplify this expression we define the following terms  
  ()
1 ln , ln 1 , ln , ln
ee p c
ii E g E
p c
ρβ π
− ⎛⎞ ⎛⎞ == + = − = − ⎜⎟ ⎜⎟ ′ ′ ⎝⎠ ⎝⎠
  (5) 
where ρ represents the consumer’s rate of time preference, i represents the nominal interest 
rate, π
e is the expected inflation rate, and g
e represents the expected consumption growth rate.
1   
Assuming that the covariance term in Equation (4) is negligible, as expected in a low 
inflation risk economy (Sarte, 1998), we obtain our augmented Fisher equation by taking the 
natural log of both sides of Equation (4) and substituting in the terms from Equation (5): 
 
e i
e g ρ πγ =+ +  (6) 
The nominal interest rate must compensate the consumer for i) the rate of time preference, ii) 
the loss in purchasing power due to inflation, and iii) the utility cost of expected consumption 
fluctuations. Note that the coefficient on the consumption growth rate is the consumer’s 
degree of relative risk aversion. An estimation of Equation (6) provides us with an alternative 
way to measure this parameter. 
 
3.  Data and Estimation 
In testing the implications of our model, we use US quarterly data for 3-month Treasury 
constant-maturity bonds (Board of Governors), seasonally-adjusted real personal consumption 
expenditures (Bureau of Economic Analysis), and the consumer price index for urban 
                                                 
1 The proper inflation measure is the ‘inverse of the expected rate of decrease in purchasing power due to 
inflation’ rather than the ‘rate of inflation’ itself. According to Jensen’s inequality these are not the same in the 
presence of risk.  The same also applies to the consumption growth rate. In this study we proxy for the measure 
of inflation and consumption expectations by a single point, namely the actual inflation or growth, so there is no 
difference between  ( ) ln Epp ′ −  and  ( ) ln Epp ′  or  ( ) ln Ecc ′ −  and  ( ) ln Ecc ′ . 
2   consumers (Department of Labor) for the sample period of 1960Q4 to 2005Q4. Inflation and 
consumption growth expectations are proxied for by the actual 3-month growth rates of CPI 
and consumption data, respectively. Instrumental variable estimation is used to overcome 
potential endogeneity problems caused by the proxies. Also, to avoid artificial moving 
average problems, we select non-overlapping data points for (3-month) inflation and 
consumption growth and also divide the T-Bill rates by four (Sun and Phillips, 2004). Finally, 
since the Treasury rate reflects the yield to be collected one quarter after purchase, we align 
next quarter’s inflation and consumption growth rates with the current quarter’s interest rate 
(Sun and Phillips, 2004). 
The standard cointegration techniques by Crowder and Hoffman (1996), Ng and Perron 
(1997), Dotsey et al. (2003), Sun and Phillips (2004), and Caporale and Pittis (2004) do not 
perfectly fit the purposes of our study since the consumption growth component is found to be 
stationary,
2 and is used to explain the short run fluctuations around the long-run Fisher 
relation. We follow Mehra (1993) who uses an error correction method to tackle a similar 
problem for the joint estimation of the long- and short-run money demand function. While our 
long-run Fisher equation is 
  01 2 tt i t t g u ρ ρπ ρ = +++  (7) 
its short run adjustment process incorporates the dynamic error correction changes: 
 
12
01 , 2 , 3 1
00
nn
ts t s s t s t
ss
ig t u θ θπ θ θ ε −−
==
− ∆ =+ ∆ + ∆ + + ∑∑   .  (8) 
where   and  t u t ε  are deviations from long and short run equlibriums, respectively. Estimating 
both equations simultaneously in a reduced form equation yields: 
123
01 , 2 , 3 , 4 1 5 1 6 1
000
nn n
ts t s s t s s t s t t t
sss
ig i g t i ψ ψπ ψ ψ ψ π ψ ψ ε −− − − −
===
∆= + ∆ + ∆ + ∆ − − + + ∑∑∑ −  (9) 
which provides us with consistent estimates of the structural parameters of interest  36 θ ψ = , 
00 6 ρ ψψ ≅−  (since  00 3 0 ψ θθ ρ =− ),  14 6 ρ ψψ = −  and  25 6 ρ ψψ = − . Due to possible 
endogeneity resulting from proxying for expectations, we estimate the model with both OLS 
and IV methodologies.
3 The results are displayed in Table 1.
4  
                                                 
2 ADF tests, using the modified Akaike information criteria for lag selection, find that the T-Bill and inflation 
rates have a unit root, while rejecting the non-stationarity of consumption growth; the corresponding test 
statistics are -1.87, -2.38, and -5.92, respectively. 
3 In OLS, n1, n2 and n3 are all chosen as 1 while they are 2 in the IV estimation. They are determined by using 
the Schwartz information criterion.  The instruments used in the IV model are 2 non-coincident lagged levels of 
the T-Bill rate, inflation and consumption growth with 2 coincident lagged differences of the interest rate and 4 
lagged differences of inflation and consumption growth. 
3   (Insert Table 1 here) 
The first two columns of Table 1 show the reduced form estimates while the latter two 
display the structural parameters of Equation (7) derived from the reduced form estimates.  
While we use standard t-statistics to calculate the significance levels of reduced form 
coefficients, we use a combination of Wald and t-statistics to test for null hypotheses of the 
long-run Fisher relation, namely  1 1 ρ =  ( 64 0 ψ ψ + = ) and  02 ,0 ρ ρ =  ( 50 0, 0 ψ ψ == ). The 
findings support our theory that consumption growth is an important element in the Fisher 
relation explaining the short-run fluctuations around the long-run Fisher relation. While the 
long-run inflation coefficient is not significantly different than one and the time preference 
rate is insignificantly different than zero, consumption growth is marginally significant. Its 
coefficient, the degree of relative risk aversion is within the acceptable range of values in the 
literature.
5 This last finding is not surprising considering the expected value of consumption 
growth is a constant part of the real rate in the limit.  In addition to the marginal significance 
in the long-run Fisher relation, the dynamic error correction estimates show that consumption 
growth changes are important in explaining the adjustments in the interest rate as well as their 
level.  
 
4.  Conclusion 
The effect of expected consumption growth on the interest rate has been overlooked in studies 
of the Fisher relation.  We find that including a consumption growth term in the estimation of 
the Fisher equation yields a significant coefficient for consumption growth and allows us to 
estimate the consumer’s degree of relative risk aversion; therefore, we conclude that the future 
studies of the Fisher relation would benefit from including expected consumption as an 
explanatory variable. 
                                                                                                                                                         
4 Since the estimates of the standard Fisher relation in the literature are not too different than our estimates, we 
refrain from including them in Table 1 in the interest of brevity. 
5 The range of CRRA estimates range from 0.5 to 2.5. 
4   References 
Caporale, G.M., and N. Pittis, 2004, Estimator Choice and Fisher’s Paradox:  A Monte Carlo 
Study, Econometric Reviews 23, 25-52. 
Crowder, W.J. and D.L. Hoffman, 1996, The Long-Run Relationship between Nominal and 
Inflation:  The Fisher Equation Revisited. Journal of Money, Credit, and Banking 28, 
102-118. 
Dotsey, M., C. Lantz, and B. Scholl, 2003, The Behavior of the Real Interest Rate, Journal of 
Money, Credit, and Banking 35, 91-110. 
Fisher, I., 1930, The Theory of Interest. New York, MacMillan. 
Levi, M.D. and J.H. Makin, 1978, Anticipated Inflation and Interest Rates, Further 
Interpretation of Findings on the Fisher Equation, American Economic Review 68, 801-
812. 
Mehra, Y.P., 1993, The Stability of the M2 Demand Function:  Evidence from an Error-
Correction Model, Journal of Money, Credit, and Banking 25, 455-460. 
Ng, S., and P. Perron, 1997, Estimation and Inference in Nearly Unbalanced Nearly 
Cointegrated Systems, Journal of Econometrics 79, 53-81. 
Sarte, P.D.G., 1998, Fisher’s Equation and the Inflation Risk Premium in a Simple 
Endowment Economy, Federal Reserve Bank of Richmond Economic Quarterly 84, 53-
72. 
Sun, Y., and P.C.B. Phillips, 2004, Understanding the Fisher Equation, Journal of Applied 
Econometrics 19, 869–886. 
VanderHoff, J., 1984, Evidence on the Varying Effect of Expected Inflation on Interest Rates, 
Review of Economics and Statistics 66, 477-481. 
5    
Table 1: Reduced Form and Structural Parameters from the Joint Test of the Long- and Short-
Run Fisher Relation  
  OLS (reduced)  IV (reduced)  OLS (structural)  IV (structural) 
0 ψ   -0.04 (0.06)  -0.08 (0.06)  -0.37  -0.82 
1 t π −   0.14
*** (0.03)  0.14
*** (0.03)  1.23
† 1.46
†
1 t g −   0.06
* (0.03)  0.07
* (0.04)  0.51
*  0.77
*
1 t i −   -0.11
*** (0.03)  -0.09
*** (0.03)     
t π ∆    0.10 (0.03)  0.11
*** (0.04)     
1 t π − ∆    0.07
*** (0.03)  0.07 (0.04)     
2 t π − ∆    0.004  (0.03)     
t g ∆   -0.01 (0.02)  -0.01 (0.02)     
1 t g − ∆   -0.06
*** (0.02)  -0.09
*** (0.03)     
2 t g − ∆    -0.02  (0.01)     
1 t i − ∆   -0.03 (0.07)  -0.12 (0.08)     
2 t i − ∆    -0.09  (0.09)     
2 R   0.27 0.26     
Prob. of J stat   0.99     
DW  2.11 1.96     
Notes: Standard errors are reported in the parentheses. 
*** indicates 99% significance while 
**(
*) indicates 
95%(90%). The structural parameters are estimated using the reduced form estimates, and their significance 
levels are tested using Wald ( ) and t-tests (
64 0 ψψ +=
50 0, 0 ψψ = = ). 
† indicates failure to reject the null of the 
coefficient equaling 1. 
  
 
6   

Anticipated Inflation and Interest Rates,

Estimation and Inference in Nearly Unbalanced Nearly Cointegrated Systems,

Estimator Choice and Fisher’s Paradox: A Monte Carlo Study,

Evidence on the Varying Effect of Expected Inflation on Interest Rates,

Fisher’s Equation and the Inflation Risk Premium in a Simple Endowment Economy, Federal Reserve Bank of

The Behavior of the Real Interest Rate,

The Long-Run Relationship between Nominal and Inflation: The Fisher Equation Revisited.

The Stability of the M2 Demand Function: Evidence from an ErrorCorrection Model,

The Theory of Interest.

Understanding the Fisher Equation,

What Fisher Knew About His Relation, We Sometimes Forget

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What Fisher Knew About His Relation, We Sometimes Forget

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