Supplemental Material for ’Testing for Money Illusion Hypothesis in Aggregate Consumption Function: Mixed Data Sampling Approach’ Kaiji Motegi ∗ Akira Sadahiro † First Draft: November 29, 2014 This Draft: November 27, 2015 Abstract Testing for the money illusion hypothesis in aggregate consumption function generally involves a regression model that projects real consumption onto nominal disposable income and a consumer price index. Price data are usually available at a monthly level, but consumption and income data are sampled at a quarterly level in some countries like Japan. This paper takes advantage of mixed data sampling (MIDAS) regressions in order to exploit monthly price data. We show via local power analysis and Monte Carlo simulations that our approach yields deeper economic insights and higher statistical precision than the previous single-frequency approach that aggregates price data into a quarterly level. In particular, the MIDAS approach allows for heterogeneous effects of monthly prices on real consumption within each quarter. In empirical applications we find that the heterogeneous effects indeed exist in Japan and the U.S. Description The main paper is Motegi and Sadahiro (2015) ”Testing for Money Illusion Hypothesis in Aggregate Consumption Function: Mixed Data Sampling Approach”. This supplemental material contains deeper literature review, proofs of theorems, local power analysis, detailed Monte Carlo evidence, and more empirical results. For empirical applications, the main paper analyzes Japan only but the supplemental material analyzes Japan and the United States. Keywords: Aggregate consumption function, Hypothesis testing, Local asymptotic power, Money il- lusion, Mixed Data Sampling (MIDAS), Temporal aggregation. JEL classification: C12, C22, E21. ∗ Faculty of Political Science and Economics, Waseda University. E-mail: [email protected]† Faculty of Political Science and Economics, Waseda University. E-mail: [email protected]
34
Embed
Supplemental Material for ’Testing for Money Illusion ...motegi/Supp_Mat_EL_TMIHACF_v1.pdfSupplemental Material for ’Testing for Money Illusion Hypothesis in Aggregate Consumption
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Supplemental Material for ’Testing for Money Illusion Hypothesis in
Aggregate Consumption Function: Mixed Data Sampling Approach’
Kaiji Motegi∗ Akira Sadahiro†
First Draft: November 29, 2014
This Draft: November 27, 2015
Abstract
Testing for the money illusion hypothesis in aggregate consumption function generally involves
a regression model that projects real consumption onto nominal disposable income and a consumer
price index. Price data are usually available at a monthly level, but consumption and income data
are sampled at a quarterly level in some countries like Japan. This paper takes advantage of mixed
data sampling (MIDAS) regressions in order to exploit monthly price data. We show via local power
analysis and Monte Carlo simulations that our approach yields deeper economic insights and higher
statistical precision than the previous single-frequency approach that aggregates price data into a
quarterly level. In particular, the MIDAS approach allows for heterogeneous effects of monthly prices
on real consumption within each quarter. In empirical applications we find that the heterogeneous
effects indeed exist in Japan and the U.S.
Description
The main paper is Motegi and Sadahiro (2015) ”Testing for Money Illusion Hypothesis in Aggregate Consumption
Function: Mixed Data Sampling Approach”. This supplemental material contains deeper literature review, proofs
of theorems, local power analysis, detailed Monte Carlo evidence, and more empirical results. For empirical
applications, the main paper analyzes Japan only but the supplemental material analyzes Japan and the United
lusion, Mixed Data Sampling (MIDAS), Temporal aggregation.
JEL classification: C12, C22, E21.
∗Faculty of Political Science and Economics, Waseda University. E-mail:[email protected]†Faculty of Political Science and Economics, Waseda University. E-mail:[email protected]
1 Introduction
The history ofmoney illusiondates back to Fisher (1928), who defined it as ’failure to perceive that the
dollar, or any other unit of money, expands or shrinks in value’ (p. 4). Testing for money illusion serves
as an assessment of the fundamental assumption in economics that agents should be rational enough to
distinguish nominal and real values of money. See Howitt (2008) and Pochon (2015, Section 3.1) for the
historical development of the money illusion literature.
There are two research fields that test for money illusion empirically. One is behavioral economics
where money illusion at anindividual level has been tested extensively. See Shafir, Diamond, and Tver-
sky (1997) and Fehr and Tyran (2001) for seminal experiments that suggest the presence of money
illusion at the individual level. The other field is time series analysis where money illusion at anaggre-
gatelevel is tested via hypothesis testing. This paper focuses on the latter, in particular aggregate goods
markets.1 Since consumption is the largest component of gross domestic product (GDP) in virtually all
countries, it is of interest to analyze how consumption reacts to a change in nominal and real values of
money.
Typically, testing for the money illusion hypothesis in aggregate consumption function involves a
regression model that projects real consumption onto nominal disposable income and a consumer price
index (CPI). If the loadings of nominal disposable income and CPI have opposite signs and the same
magnitude, then the consumption function is homogeneous of degree zero in income and price and
therefore the money illusion hypothesis is rejected.
Branson and Klevorick (1969), one of the earliest attempt to test for money illusion, uncover the
existence of money illusion in the U.S. aggregate consumption in 1955-1965. Succeeding discussions
of Cukierman (1972), Branson and Klevorick (1972), and Craig (1974) confirm the presence of money
illusion. A recent work by Pochon (2015, Ch.3) adds a further empirical evidence for money illusion in
the aggregate U.S. consumption.
We can find empirical studies on non-U.S. countries also. Koskela and Sullstrom (1979) use quarterly
and annual data of Finland, and conclude that money illusion exists at a quarterly level but not at an
annual level. For the Japanese economy, Economic Planning Agency (1995) and Hayashi (1999) find
non-illusion while Nagashima (2005) finds illusion. Their opposing evidence may be due to different
methodologies, data types, or sample period. Overall, a majority of applied papers support the money
illusion hypothesis but some papers cast a doubt on those results.
There are two issues when we interpret the empirical evidence of the previous papers. First, Lewbel
(1990) shows that money illusion in the aggregate level does not necessarily imply each economic in-
dividual’s irrationality. Lewbel (1990) derives a necessary and sufficient condition calledmean scaling,
under which aggregate consumption function suffers from money illusionif and only if each household
is irrational. This paper refrains from exploring this issue further in order to focus on another research
gap in the existing literature.
The second issue, which this paper resolves, is the sampling frequencies of relevant data. Price data
1Other markets are often analyzed as well. See Cohen, Polk, and Vuolteenaho (2005) for stock markets and Brunnermeierand Julliard (2008) for housing markets.
1
are usually available at a monthly level, but consumption and income data are sampled at a quarterly
level in some countries like Japan. Until recently, all time series models had been required to have a
single sampling frequency for all variables. Hence the applied papers above use quarterly or even annual
datasets with temporally aggregated price series.2 Temporal aggregation may produce misleading or
inaccurate results due to information loss (cfr. Silvestrini and Veredas (2008)).
Based on the growing literature of Mixed Data Sampling (MIDAS) regressions, we propose a new
testing strategy for money illusion in order to obtain deeper economic implications and sharper statistical
inference. We regress quarterly real consumption growth onto quarterly nominal disposable income
growth andmonthlyinflation (not aggregated quarterly inflation). Here the growth rate is taken in order to
make each variable stationary. A regression model on levels of variables would be more favorable if there
existed a cointegrated relationship among the levels of real consumption, nominal disposable income, and
prices. To focus on the implications of mixed frequency approaches, this paper lets cointegration be an
open question.3
MIDAS regressions (also called mixed frequency regressions) are put forward by Ghysels, Santa-
Clara, and Valkanov (2004), Ghysels, Santa-Clara, and Valkanov (2006), and Andreou, Ghysels, and
Kourtellos (2010).4 As demonstrated in Ghysels, Hill, and Motegi (2014) and Ghysels, Hill, and Motegi
(2015), the MIDAS approach improves the accuracy of hypothesis testing by exploiting all observable
data. They show that Granger causality tests with mixed frequency data achieve higher power in local
asymptotics and finite sample than single-frequency tests (also called low frequency tests) that aggregate
all series to the least frequency sampling.
We show via local power analysis and Monte Carlo simulations that the MIDAS approach allows
for heterogeneous effects of monthly inflation on real consumption growth within each quarter. This is
clearly a new contribution since the low frequency approach essentially assumes that monthly inflation
should have a homogeneous impact on real consumption growth. Our empirical study on Japan and the
U.S. indicates that the heterogeneous effects of inflation indeed exist. Money illusion does not exist at a
quarterly level, but monthly inflation has heterogeneous impacts on quarterly real consumption growth.
This paper is organized as follows. In Section 2 we elaborate asymptotic theory on both mixed
frequency tests and conventional low frequency tests. In Section 3 we conduct local power analysis in
order to compare the relative performance of mixed frequency tests and low frequency tests. In Section
4 we run Monte Carlo simulations in order to examine finite sample properties of the tests. Section
5 implements empirical analysis on Japanese and U.S. economies. Finally, Section 6 provides some
concluding remarks. Tables and figures are displayed after Section 6. Proofs of theorems are presented
in Technical Appendices.
2The only exception is Nagashima (2005), who implements a rather ad-hoc interpolation of monthly income series basedon actual quarterly series.
3Cointegration with mixed frequency data is a relatively new research topic. See Ghysels and Miller (2015) for an earlycontribution.
4See Andreou, Ghysels, and Kourtellos (2011) and Armesto, Engemann, and Owyang (2010) for surveys.
2
2 Testing for Money Illusion
We describe the previous low frequency approach and our mixed frequency approach, using standard
notations in the MIDAS literature. Suppose that real consumption growthy and nominal disposable
income growthxL are sampled at a quarterly level while inflationxH is sampled at amonthly level
(subscripts ”L” and ”H” signify low and high frequencies, respectively). This is a realistic assumption
for some countries like Japan. See Section 5 for more details. A complete dataset available for each
quarterτL is {y(τL), xL(τL), xH(τL, 1), xH(τL, 2), xH(τL, 3)}, wherexH(τL, j) is inflation at thej-
th month of quarterτL (e.g. xH(τL, 1), xH(τL, 2), andxH(τL, 3) are respectively inflation in January,
February, and March whenτL signifies the first quarter of a year).
Since the previous literature was forced to work with a single-frequency dataset, they aggregate
the monthly inflation into a quarterly level according toxH(τL) = (1/3)∑3
j=1 xH(τL, j). A crucial
difference between the mixed frequency and low frequency approaches lies in how to incorporatexH in
regression models. The former uses{xH(τL, 1)}, {xH(τL, 2)}, and{xH(τL, 3)} separately as if they
were distinct quarterly variables. The latter usesxH(τL) only, which essentially means thatxH(τL, 1),
xH(τL, 2), andxH(τL, 3) have the same impact ony(τL). This implicit restriction masks potentially
heterogeneous impacts of monthly inflation on real consumption.
We discuss the mixed frequency approach in Section 2.1 and then the low frequency approach in
Section 2.2.
2.1 Mixed Frequency Approach
Assume that the true data generating process (DGP) is
Remark 3.2. Under strong non-illusion, we have thatνs = 03×1 and thusδs = 0 in view of (3.4).
Hence we verify from Theorem 3.3 thatWd→ χ2
1 underHs0 (cfr. Theorem 2.3).
The local asymptotic power of the LF test can be computed from (3.2).
3.3 Numerical Examples
In the previous sections we have derived the local power of MF-S, MF-W, and LF tests. In this section
we assume a specific DGP and compute local power numerically. This exercise is useful for two reasons.
First, we can verify and interpret the theoretical results obtained in Section 2. Second, we can compare
the local power of the three tests visually.
Suppose that the true DGP is
y(τL) = a0 + axL(τL) +
3∑j=1
bjxH(τL, j) + ϵL(τL), σ2L ≡ E[ϵ2L(τL)] = 1. (3.5)
This is a simplified version of (2.1) where there are no extra regressorsX2. We do not need to assume
specific values forθ0 = [a0, a, b1, b2, b3]′ since local power does not depend onθ0.
We assume that regressorsX1(τL) = [xL(τL), xH(τL, 1), xH(τL, 2), xH(τL, 3)]′ follow Ghysels’
9
(2015) structural mixed frequency vector autoregression (MF-VAR) of order 1:1 0 0 0
0 1 0 0
0 −ϕH 1 0
0 ϕH/2 −ϕH 1
︸ ︷︷ ︸
=N
xL(τL)
xH(τL, 1)
xH(τL, 2)
xH(τL, 3)
︸ ︷︷ ︸
=X1(τL)
=
ϕL d3 d2 d1
e1 ϕH/3 −ϕH/2 ϕH
e2 0 ϕH/3 −ϕH/2
e3 0 0 ϕH/3
︸ ︷︷ ︸
=M
xL(τL − 1)
xH(τL − 1, 1)
xH(τL − 1, 2)
xH(τL − 1, 3)
︸ ︷︷ ︸
=X1(τL−1)
+
ξL(τL)
ξH(τL, 1)
ξH(τL, 2)
ξH(τL, 3)
︸ ︷︷ ︸
=ξ(τL)
(3.6)
or NX1(τL) = MX1(τL − 1) + ξ(τL).5 We assume thatE[ξ(τL)ξ(τL)′] = I4. The low frequency
AR(1) coefficient ofxL is ϕL ∈ {0.5, 0.8}. The high frequency AR(3) coefficients ofxH areϕH ,
−ϕH/2, andϕH/3, whereϕH ∈ {0.5, 0.8}. The case of(ϕH , ϕL) = (0.8, 0.8) is closest to the reality
since nominal disposable income growth and inflation are well known to be persistent. We also consider
(ϕH , ϕL) = (0.5, 0.5), (0.8, 0.5), (0.5, 0.8) in order to see how local power depends on the persistence
of xH andxL.
Granger causality fromxH to xL is governed byd = [d1, d2, d3]′. We consider what Ghysels, Hill,
and Motegi (2014) call thedecaying causality, i.e. dj = (−1)j−1 × 0.2/j for j = 1, 2, 3. As time lag
gets larger, the impact ofxH onxL decays geometrically with the alternating signs. Ghysels, Hill, and
Motegi (2014) consider other causal patterns. The present paper focuses on the decaying causality only,
because our main interest does not lie on Granger causality. In extra simulations not reported here, local
power was nearly same across different choices ofd.
Granger causality fromxL to xH is governed bye = [e1, e2, e3]′. We again consider the decaying
causalityej = (−1)j−1 × 0.2/j for j = 1, 2, 3. As in causality fromxH to xL, local power is nearly
same across differente’s.
The reduced form of (3.6) is written asX1(τL) = A1X1(τL−1)+η(τL), whereA1 = N−1M and
η(τL) = N−1ξ(τL). The eigenvalues ofA1 all lie inside the unit circle for any choice of(ϕH , ϕL,d, e)
discussed above. The stability condition is therefore always satisfied.
To compute local power, we elaborate the covariance matrix ofX1(τL). LetΥ0 =E[X1(τL)X1(τL)′].
Using the discrete Lyapunov equation, we have that
vec[Υ0] = [I16 −A1 ⊗A1]−1 vec
[E[η(τL)η(τL)
′]]= [I16 −A1 ⊗A1]
−1 vec[N−1E
[ξ(τL)ξ(τL)
′]N−1′]
= [I16 −A1 ⊗A1]−1 vec
[N−1N−1′
].
We next characterize the mixed frequency population momentΣXX = E[X(τL)X(τL)′]. Since
X(τL) = [1,X1(τL)′]′, we have that
ΣXX =
[1 E [X1(τL)
′]
E [X1(τL)] E [X1(τL)X1(τL)′]
]=
[1 01×4
04×1 Υ0
]. (3.7)
UsingΣXX , we can characterizeΣs = σ2LRsΣ
−1XXR′
s in terms of the underlying parametersN and
5Equation (3.6) is a common type of structural form considered in the MIDAS literature. See e.g. Ghysels, Hill, and Motegi(2014), Gotz and Hecq (2014), Ghysels (2015), and Ghysels, Hill, and Motegi (2015).
10
M . σ2L = 1 as stated in (3.5), andRs is defined in (2.2).
For a given value of Pitman value ofνs we calculate the noncentrality parameter of MF-S test,
ν ′sΣ
−1s νs (cfr. Theorem 3.1). Finally, we use (3.2) to get the local power of MF-S test. Similar proce-
dures hold for the MF-W test (cfr. Theorem 3.2).
The local power of LF test can be computed analogously. DefineΣXX = E[X(τL)X(τL)′], then
ΣXX = WXΣXXW ′X by (3.3). SinceΣXX is characterized in (3.7), we can characterizeΣXX in
terms ofN andM . Hence we can calculateδs andσ2 according to (3.4). It is now straightforward to
calculate the noncentrality parameterδ2s/σ2 and the local power of LF test (cfr. Theorem 3.3).
We takeνs1, νs2, νs3 ∈ {−4,−3.8, . . . , 3.8, 4} so that there are413 = 68, 921 combinations of
νs1, νs2, andνs3. Those combinations can be categorized into three cases. Case 1 is strong non-illusion:
(νs1, νs2, νs3) = (0, 0, 0). Case 2 is mean-zero deviations from strong non-illusion:(1/3)∑3
j=1 νsj = 0.
In Case 2 we assume that at least one of(νs1, νs2, νs3) is nonzero in order to avoid an overlap between
Cases 1 and 2. Put differently, Case 2 is when weak non-illusion holds but strong non-illusion does not.
1,260 combinations out of 68,921 are categorized in Case 2 (e.g.(νs1, νs2, νs3) = (1.0, 1.0,−2.0)).
Finally, Case 3 is when weak non-illusion does not hold (i.e.(1/3)∑3
j=1 νsj = 0). 67,660 cases out of
68,921 are categorized in Case 3.
In Case 1, local power must be equal to nominal sizeα = 0.05 for all three tests. This conjecture is
based on our theoretical results of chi-squared asymptotics under strong non-illusion (cfr. Theorems 2.1,
2.2, and 2.3).
In Case 2, the local power of MF-S test must be larger than 0.05 because of consistency (cfr. Theorem
2.1). The local power of MF-W test must be equal to 0.05 (cfr. Theorem 2.2). It is of interest to observe
how the LF test behaves because we do not have any analytical results for the LF test under Case 2.
In Case 3, MF-S and MF-W tests must have power larger than 0.05 because of consistency (cfr.
Theorems 2.1 and 2.2). It is again of interest to observe the local power of LF test. At least we know that
the LF test must have power larger than 0.05 whenνs1 = νs2 = νs3 ≡ νs = 0 (cfr. Theorem 2.4). We
have 40 combinations satisfying this condition (i.e.νs = −4,−3.8, . . . ,−0.2, 0.2, . . . , 3.8, 4). Other
than these homogeneous deviations, power properties of LF test are analytically unknown.
Results In Table 1 we pick some representative Pitman drifts out of all 68,921 combinations. In Case
1 (strong non-illusion), all tests have an exactly correct size of 0.05 as expected.
In Case 2 (weak non-illusion), the MF-S test has moderate power ranging between 0.096 and 0.267.
The MF-W test has an exactly correct size of 0.05 as expected. The local power of the LF test is close
to but slightly higher than 0.05 (ranging between 0.051 and 0.060). This result can be interpreted in two
ways. From a viewpoint of strong non-illusion, the LF test has clearly lower power than the MF-S test.
From a viewpoint of weak non-illusion, the LF test has asymptotically 100% size distortions against a
fixed alternative (although the rate of divergence is quite slow). These two scenarios hinder practical
interpretations of the LF test. The mixed frequency approach provides clearer interpretations because we
can distinguish strong and weak non-illusion by implementing MF-S and MF-W tests separately.
In Case 3 (illusion), all tests have moderate or high power. Under homogeneous deviations (e.g.
11
νs1 = νs2 = νs3 = 2), the LF test has higher power than the MF tests (the difference is about 10%
points). When Pitman parameters have positive and negative signs (e.g.(νs1, νs2, νs3) = (2, 2,−2)), the
MF-S test is more powerful than the LF test (the difference is about 7% points). An intuitively reason is
that temporal aggregation ofxH offsets positive and negative individual impacts. Case 2 can be thought
of as an extreme case of positive and negative Pitman drifts. It is therefore not surprising that the LF test
has very low power against strong non-illusion in Case 2. Finally, the power of MF-W test resembles the
LF test, but the latter is slightly more powerful.
In Case 3, the local power of each test is generally increasing in persistence parameters(ϕH , ϕL). In
particular, the LF test is more sensitive toϕH than MF tests. A larger value ofϕH implies thatxH(τL, 1),
xH(τL, 2), andxH(τL, 3) take more similar values for eachτL, making the information loss by temporal
aggregation smaller. It is thus reasonable that the LF test has higher power whenϕH is larger.
Summarizing Table 1, the benefit of mixed frequency approach appears most when there exists weak
non-illusion (Case 2). The MF-S test has moderate power and the MF-W test has correct size, so we
can likely reach a truth that weak non-illusion holds but strong non-illusion does not. The LF test, on
one hand, is inferior to the MF-S test since it has lower power against strong non-illusion. The LF test,
on the other hand, is inferior to the MF-W test since it suffers from size distortions approaching 100%
(although the rate of divergence seems quite slow).
To further elaborate on Case 2, we draw histograms of the local power of the MF-S and LF tests in
Figure 1. Their difference is also plotted in another histogram. In most cases, the power of MF-S test
is around 0.1 or 0.2. In some case it exceeds 0.3 or even 0.4. The power of LF test is at most 0.06, and
the difference between the MF-S power and LF power is always positive (sometimes more than 0.3). In
general, the LF test suffers from low power when Pitman drifts have positive and negative signs. The
advantage of MF-S test is that its power is not substantially affected by the sign of Pitman drifts.
4 Monte Carlo Simulations
In this section we conduct Monte Carlo experiments in order to compare the MF-S, MF-W, and LF tests
in terms of finite sample performance.
4.1 Simulation Design
Our simulation design is basically analogous to the local power analysis in Section 3.3. First, assume
that the true DGP for regressorsX1(τL) = [xL(τL), xH(τL, 1), xH(τL, 2), xH(τL, 3)]′ is the structural
MF-VAR(1) appearing in (3.6):NX1(τL) = MX1(τL − 1) + ξ(τL). We assume thatξ(τL)i.i.d.∼
N(04×1, I4). Parameters on persistence and Granger causality are same as in Section 3.3:ϕH , ϕL ∈{0.5, 0.8} andd = e = [0.2,−0.1, 0.667]′.
12
Second, assume that the true DGP fory(τL) is
y(τL) = axL(τL) +
3∑j=1
bjxH(τL, j) + ϵL(τL)
= axL(τL) +
3∑j=1
(−a
3+
csj3
)xH(τL, j) + ϵL(τL), ϵL(τL)
i.i.d.∼ N(0, 10).
The second equality just rewritesbj as a deviation from−a/3, so there is not a loss of generality (cfr.
(2.4)). We are assuming thatσ2L = 10 so that rejection frequencies do not reach 1. (Ifσ2
L = 1 as in
Section 3.3, then rejection frequencies would reach 1 in many cases and therefore we could not compare
the MF and LF tests meaningfully.) Unlike local power analysis, we should actually generate samples
from DGPs. We thus need to set specific values for not onlycs = (cs1, cs2, cs3) but alsoa. We trya ∈{0.3, 0.6, 0.9}, which means that we consider(a, cs1, cs2, cs3) = (0.3, 0, 0, 0), (0.6, 0, 0, 0), (0.9, 0, 0, 0)
in Case 1 (strong non-illusion).
For Case 2 (weak non-illusion), we try nine representative combinations of(a, cs1, cs2, cs3) that
satisfy(1/3)∑3
j=1 csj = 0. One of them is(a, cs1, cs2, cs3) = (0.3, 0.3, 0.3,−0.6), which corresponds
to (a, b1, b2, b3) = (0.3, 0, 0,−0.3). In this examplexH(τL, 1) andxH(τL, 2) have no impacts ony(τL)
but the negative impact ofxH(τL, 3) exactly offsets the positive impact ofxL(τL).
For Case 3 (illusion), we consider twelve representative combinations of(a, cs1, cs2, cs3) that satisfy
(1/3)∑3
j=1 csj = 0. One of them is(a, cs1, cs2, cs3) = (0.3, 0.3, 0.3,−0.3), which corresponds to
(a, b1, b2, b3) = (0.3, 0, 0,−0.2). In this example, the negative impact ofxH(τL, 3) is not large enough
to offset the positive impact ofxL(τL). As a result the same amount of increase in{xL(τL), xH(τL, 1),
xH(τL, 2), xH(τL, 3)} raises real consumptiony(τL).
We generate 10,000 Monte Carlo samples ofX1 andy for each combination of(a, cs1, cs2, cs3).
For each sample we implement the MF-S, MF-W, and LF tests based on the chi-squared distributions
(cfr. Theorems 2.1, 2.2, and 2.3). The mixed frequency regression model isy(τL) = α0 + αxL(τL) +∑3j=1 βjxH(τL, j) + uL(τL), while the low frequency regression model isy(τL) = α0 + αxL(τL) +
βxH(τL) + uL(τL). Finally, we compute rejection frequencies in order to investigate empirical size and
power. Sample sizeTL is 50 quarters (small), 100 quarters (medium), or 130 quarters (large).6 Nominal
size is0.05.
4.2 Simulation Results
Table 2 presents rejection frequencies. We first focus on Case 1 (strong non-illusion) in order to check
empirical size of each test. Even in the small sampleTL = 50 (Panel A), we do not see severe size
distortions for any tests. Empirical size lies between[0.065, 0.088], fairly close to the nominal size 0.05.
WhenTL = 100 (Panel B) orTL = 130 (Panel B), empirical size gets even closer to 0.05. Since our
parametric restrictions are simple enough, the asymptotic chi-squared tests perform well in finite sample.
We next discuss empirical power. We observe similar results with the local power analysis in general,
6Sample size in empirical applications is approximately 130 quarters (cfr. Section 5).
13
but the advantage of mixed frequency approach is more emphasized. In Case 2 (weak non-illusion), the
MF-S test has moderately high power. It sometimes exceeds 0.3 forTL = 50; 0.5 forTL = 100; 0.65 for
TL = 130. Rejection frequencies of the LF test, in contrast, lie between[0.05, 0.10] regardless of sample
size. Hence the MF-S test achieves much higher power than the LF test in terms of strong non-illusion.
In Case 3 (illusion), the MF-S test is more powerful than the LF test whencs contains both positive
and negative signs. When(a, cs1, cs2, cs3) = (0.9,−0.9, 0.9, 0.9) with (TL, ϕH , ϕL) = (50, 0.5, 0.5),
for example, the rejection frequencies of MF-S and LF tests are 0.195 and 0.089, respectively. The
LF test is more powerful than the MF-S test whencs contains a single sign. When(a, cs1, cs2, cs3) =
(0.9,−0.9,−0.9,−0.9), for example, the rejection frequencies of MF-S and LF tests are now 0.238 and
0.312, respectively. The MF-W test has similar power with the LF test, but the former tends to be slightly
more powerful.
5 Empirical Applications
In this section we test for money illusion in aggregate consumption functions of Japan. In Japan, con-
sumption and income data can be collected only at a quarterly level. One might argue that monthly data
could be collected through Family Income and Expenditure Survey of the Statistics Bureau, the Ministry
of Internal Affairs and Communications. It is however well known that Family Income and Expendi-
ture Survey has a much smaller coverage than the System of National Accounts (SNA). Hence we use
SNA-based consumption and income data which are available only at a quarterly level.
As a supplemental analysis, this paper investigates the U.S. case as well. Strictly speaking, the
MIDAS approach is not required for the U.S. because monthly data of consumption and income can
be collected through Personal Income and Outlays of the Bureau of Economic Analysis, and these data
have large enough coverage. To evaluate the implications of MIDAS regressions, this paper compares a
MIDAS regression model and a low frequency regression model with all quarterly series.
Section 5.1 describes regression models. Section 5.2 presents data and preliminary statistics. Section
5.3 provides empirical results and discussions.
5.1 Models
Japan We first explain low frequency models of Japan. We regress real consumption growthy(τL)
onto nominal disposable income growthxL(τL), aggregated quarterly inflationxH(τL), and change in
unemployment rate with four quarters of leads∆UR(τL + 4):