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:motegi@aoni.waseda.jp†Faculty of Political Science and Economics, Waseda University. E-mail:sadahiro@waseda.jp

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

y(τL) = a0 + axL(τL) +3∑

j=1

bjxH(τL, j) +X2(τL)′θ0,2 + ϵL(τL)

= X(τL)′θ0 + ϵL(τL),

(2.1)

whereX(τL) = [1,X1(τL)′,X2(τL)

′]′, X1(τL) = [xL(τL), xH(τL, 1), xH(τL, 2), xH(τL, 3)]′, θ0 =

[a0,θ′0,1,θ

′0,2]

′, andθ0,1 = [a, b1, b2, b3]′. X2(τL) is ak2 × 1 vector of extra regressors like net worth or

unemployment rate. Since our main focus lies on nominal disposable income growthxL and inflationxH ,

we refrain from discussing which variables should be included inX2(τL) here. That will be discussed

in more detail in Section 5.

We impose the following assumptions so that standard chi-squared asymptotics apply to our tests.

Assumption 2.1. (i) {y(τL),X(τL)} is jointly stationary and ergodic. (ii)ΣXX = E[X(τL)X(τL)′]

is nonsingular and finite. (iii)E[ϵL(τL)] = 0 andE[ϵL(τL)2|X(τL)] = E[ϵL(τL)

2] = σ2L < ∞. (iv)

{X(τL)ϵL(τL)} is a martingale difference sequence.

We do not discuss a possible cointegrated relationship amonglevelsof real consumption, nominal

disposable income, and prices. To focus on the implications of mixed frequency approaches, we let

3

cointegration be an open question and analyze growth rates of the variables which are supposed to be

stationary.

Assumption 2.1 excludes endogeneity (i.e. correlation betweenX(τL) andϵL(τL)) so that parame-

ters can be estimated via least squares. This assumption can be relaxed if we find valid instruments and

implement instrumental variable estimation. While it is easy to extend our asymptotic theory to the case

of instrumental variable estimation, finding valid instruments is likely a challenging empirical problem

as in many other economic applications. To keep focusing on the aspect of sampling frequencies, this

paper assumes that there is no endogeneity.

A primal definition of money non-illusion is thata, the marginal propensity to consume, equals

−(b1 + b2 + b3), the sum of the loadings of monthly inflation. Under this equality a proportional in-

crease in nominal income and price results in unchanged real consumption. An advantage of the mixed

frequency DGP (2.1) is that we can identifyb1, b2, andb3 separately. Using this feature, we can distin-

guish two forms of money non-illusion hypothesis. The first form is what we callstrong non-illusion

hypothesisHs0 :

Hs0 : b1 = b2 = b3 = −a/3 or Rsθ0 = 03×1, Rs︸︷︷︸

3×(k2+5)

=

0 1 3 0 0 01×k2

0 1 0 3 0 01×k2

0 1 0 0 3 01×k2

. (2.2)

Strong non-illusion assertsb1 = b2 = b3, namely the homogeneous impact of monthly inflationxH on

real consumptiony. This excludes seasonality or lagged information transmission within each quarter.

The second form of non-illusion hypothesis is what we callweak non-illusion hypothesisHw0 :

Hw0 : b1 + b2 + b3 = −a or Rwθ0 = 0, Rw︸︷︷︸

1×(k2+5)

= [0, 1, 1, 1, 1,01×k2 ]. (2.3)

Weak non-illusion doesnot imposeb1 = b2 = b3, allowing for possibly heterogeneous impacts ofxH

ony.

A fixed alternative hypothesis for strong non-illusion is written as

Hs1 : bj = −a

3+

csj3

for j = 1, 2, 3, or Rsθ0 =

cs1cs2

cs3

≡ cs ∈ R3. (2.4)

A fixed alternative hypothesis for weak non-illusion is written as

Hw1 : b1 + b2 + b3 = −a+ cw or Rwθ0 = cw ∈ R. (2.5)

In view of (2.2) - (2.5), it is straightforward to show that strong non-illusion is a special case of weak

non-illusion. First, it follows trivially that

Rw =1

3× ι′3Rs, where ι3 = [1, 1, 1]′. (2.6)

4

Hence we have thatcw = (1/3)∑3

j=1 csj . Take arbitrary(cs1, cs2, cs3) ∈ R3, then strong non-illusion

requires thatall of them should be equal to zero while weak non-illusion requires that themeanof them

should be equal to zero. In this sense, weak non-illusion can be expressed as amean-zero deviation from

strong non-illusion.

Example 2.1. Consider(cs1, cs2, cs3) = (0.6,−0.3,−0.3). It is clear that strong non-illusion does not

hold becausecs = 03×1, but weak non-illusion still holds since(1/3)∑3

j=1 csj = 0. Assume further

thata = 0.6, then we have that(b1, b2, b3) = (0,−0.3,−0.3) in view of (2.4).xH(τL, 1) has no impact

on y(τL), while xH(τL, 2) andxH(τL, 3) have an impact of−0.3 each. Intuitively, strong non-illusion

requiresboth the equality between the overall impact ofxH on y and the impact ofxL on y and the

homogeneous impact ofxH(τL, 1), xH(τL, 2), andxH(τL, 3) on y(τL). Weak non-illusion requires the

former but not the latter.

We summarize the results above in Lemma 2.1.

Lemma 2.1. The following are equivalent:

(i) Hw0 : Rwθ0 = 0 (i.e. weak non-illusion).

(ii) Hs1 : Rsθ0 = cs with 1

3

∑3j=1 csj = 0 (i.e. a mean-zero deviation from strong non-illusion).

We now formulate a MIDAS regression model which is correctly specified relative to (2.1).

y(τL) = α0 + αxL(τL) +

3∑j=1

βjxH(τL, j) +X2(τL)′θ2 + uL(τL)

= X(τL)′θ + uL(τL),

(2.7)

whereθ = [α0,θ′1,θ

′2]′ and θ1 = [α, β1, β2, β3]

′. Note that we treatxH(τL, 1), xH(τL, 1), and

xH(τL, 3) as if they were distinct regressors in order to avoid temporal aggregation ofxH . Below we

construct a Wald test for the strong non-illusion hypothesisHs0 . We then construct a Wald test for the

weak non-illusion hypothesisHw0 . For each case our goal is to prove the asymptotic chi-squared property

under the null and consistency under the fixed alternative.

Strong Non-Illusion Run ordinary least squares (OLS) on the mixed frequency model (2.7) to get

θ = Σ−1XX sXy, (2.8)

where

ΣXX =1

TL

TL∑τL=1

X(τL)X(τL)′ and sXy =

1

TL

TL∑τL=1

X(τL)y(τL). (2.9)

TL is sample size in terms of quarters. Letσ2L = (1/TL)

∑TLτL=1[y(τL) − X(τL)

′θ]2 and Σs =

σ2LRsΣ

−1XXR′

s. Using these quantities, formulatethe mixed frequency Wald statistic with respect to

5

strong non-illusion (MF-S statistic):

Ws = TLθ′R′

sΣ−1s Rsθ. (2.10)

By the standard chi-squared asymptotics, it is straightforward to prove thatWsd→ χ2

3 underHs0 . The

degrees of freedom are 3 sinceHs0 imposes three parametric restrictions (bj = −a/3 for j = 1, 2, 3). We

also have thatWsp→ ∞ underHs

1 (consistency).

Theorem 2.1. Impose Assumption 2.1. (a) UnderHs0 : Rsθ0 = 03×1, we have thatWs

d→ χ23. (b)

Under the fixed alternative hypothesisHs1 : Rsθ0 = cs for any cs = 03×1, we have thatWs

p→ ∞(consistency).

Proof . See Appendix B.

Weak Non-Illusion The weak non-illusion hypothesisHw0 : Rwθ0 = 0 can be handled in the same

way as the strong non-illusion hypothesisHs0 . Let Σw = σ2

LRwΣ−1XX R′

w. In view of (2.6), we have that

Σw = (1/9)ι′3Σsι3. That is,Σw is a mean of all elements ofΣs. Formulatethe mixed frequency Wald

statistic with respect to weak non-illusion (MF-W statistic):

Ww = TLθ′R′

wΣ−1w Rwθ. (2.11)

The asymptotic distribution ofWw underHw0 is χ2

1 since the number of parametric restrictions is just

one (b1 + b2 + b3 = −a). Consistency holds by the standard asymptotic argument.

Theorem 2.2. Impose Assumption 2.1. (a) UnderHw0 : Rwθ0 = 0, we have thatWw

d→ χ21. (b) Under

the fixed alternative hypothesisHw1 : Rwθ0 = cw for anycw = 0, we have thatWw

p→ ∞ (consistency).

Proof . See Appendix C.

As indicated in Theorems 2.1 and 2.2, straightforward chi-squared asymptotics can be applied to

the mixed frequency Wald tests. This is simply because the mixed frequency regression model (2.7) is

correctly specified relative to DGP (2.1).

2.2 Low Frequency Approach

This section considers testing for non-illusion hypothesis with the low frequency approach. Keep the

true DGP (2.1) and aggregatexH(τL) = (1/3)∑3

j=1 xH(τL, j). Formulate a low frequency regression

model:

y(τL) = α0 + αxL(τL) + βxH(τL) +X2(τL)′θ2 + uL(τL)

= X(τL)′θ + uL(τL),

(2.12)

6

whereX(τL) = [1,X1(τL)′,X2(τL)

′]′, X1(τL) = [xL(τL), xH(τL)]′, θ = [α0,θ

′1,θ

′2]′, andθ1 =

[α, β]′. We assume that the extra regressorsX2 are common in the mixed and low frequency regression

models. This is a simplifying assumption that allows us to focus onxH .

After running OLS on model (2.12), the low frequency approach implements a Wald test with respect

to β = −α, usingχ21 as the asymptotic null distribution. We elaborate the asymptotic property of this

test. Letθ be the OLS estimator forθ:

θ = Σ−1XX sXy, (2.13)

where

ΣXX =1

TL

TL∑τL=1

X(τL)X(τL)′ and sXy =

1

TL

TL∑τL=1

X(τL)y(τL). (2.14)

The parametric constraintβ = −α can be rewritten asRθ = 0, whereR = [0, 1, 1,01×k2 ]. Formulate

the low frequency Wald statistic (LF statistic):

W = TLθ′R′ σ−2Rθ, where σ2 = σ2

LR Σ−1XX R′. (2.15)

Under strong non-illusionHs0 : Rsθ0 = 03×1, we can show thatW

d→ χ21. Intuitively, the flow

aggregationxH(τL) = (1/3)∑3

j=1 xH(τL, j) does not cause any information loss whenxH(τL, 1),

xH(τL, 2), andxH(τL, 3) have homogeneous impacts ony(τL). See Appendix D for a complete proof.

Theorem 2.3. Impose Assumption 2.1. Under strong non-illusionHs0 : Rsθ0 = 03×1, we have that

Wd→ χ2

1.

Proof . See Appendix D.

Theorem 2.3 indicates that the LF test is correctly sized relative to strong non-illusion. The classical

approach usesχ21 for inference, andW indeed converges toχ2

1 underHs0 .

When strong non-illusion does not hold, the asymptotic property of the LF test is generally in-

tractable. A notable exception is whencs1 = cs2 = cs3 ≡ cs = 0. We call this case ahomogeneous

deviation from strong non-illusion. Even weak non-illusion does not hold under homogeneous devia-

tions, because weak non-illusion is equivalent tomean-zerodeviations from strong non-illusion (recall

Lemma 2.1). We can prove thatWp→ ∞ whencs1 = cs2 = cs3 ≡ cs = 0. An intuition is same as The-

orem 2.3; the flow aggregation ofxH does not cause any information loss whenxH(τL, 1), xH(τL, 2),

andxH(τL, 3) have homogeneous impacts ony(τL).

Theorem 2.4. Impose Assumption 2.1. Then we have thatWp→ ∞ underHs

1 : Rsθ0 = cs with

cs1 = cs2 = cs3 ≡ cs = 0.

Proof . See Appendix E.

Except for the case of homogeneous deviations, we do not have any general results concerning the

asymptotic properties ofW . This is essentially because model (2.12) is misspecified relative to DGP

(2.1). To fill this gap, Section 3 conducts local power analysis with some numerical examples.

7

3 Local Power Analysis

In Section 2 we have constructed MF-S, MF-W, and LF tests. This section computes the local asymptotic

power of each test. The MF-S and MF-W tests are discussed in Section 3.1, while the LF test is discussed

in Section 3.2. We present numerical examples in Section 3.3.

3.1 Mixed Frequency Tests

Consider a local alternative hypothesis

Hsla : Rsθ0 = (1/

√TL)νs, whereνs = [νs1, νs2, νs3]

′. (3.1)

νs is calledthe Pitman drift. AsTL → ∞, Hsla approaches strong non-illusionHs

0 .

MF-S Test In view of the standard chi-squared asymptotics, the MF-S statisticWs converges to a

noncentral chi-squared distribution underHsla. Characterizing the noncentrality parameter requires some

population moments. DefineΣXX = E[X(τL)X(τL)′] andΣs = σ2

LRsΣ−1XXR′

s.

Theorem 3.1. Impose Assumption 2.1. UnderHsla : Rsθ0 = (1/

√TL)νs, we have thatWs

d→χ23(ν

′sΣ

−1s νs), i.e. the noncentral chi-squared distribution with degrees of freedom 3 and noncentral-

ity parameterν ′sΣ

−1s νs.

Proof . See Appendix F.

Let α be a nominal size. The local asymptotic power of the MF-S test can be computed from the

definition of power:

P = 1− F1

[F−10 (1− α)

], (3.2)

whereF0 is the cumulative distribution function (c.d.f.) of the asymptotic distribution under the null,

whileF1 is the c.d.f. of the asymptotic distribution under the alternative. In the MF-S testF0 is the c.d.f.

of χ23, whileF1 is the c.d.f. ofχ2

3(ν′sΣ

−1s νs).

MF-W Test Under the local alternative hypothesisHsla, the MF-W statisticWw also follows a non-

central chi-squared distribution asymptotically. DefineΣw = (1/9)ι′3Σsι3. Σw is simply a mean of all

elements ofΣs.

Theorem 3.2. Impose Assumption 2.1. UnderHsla : Rsθ0 = (1/

√TL)νs, we have thatWw

d→χ21(

19ν

′sι3Σ

−1w ι′3νs).

Proof . See Appendix G.

Remark 3.1. When weak non-illusion holds, we have thatι′3νs = 0 and henceWwd→ χ2

1 (see Lemma

2.1 and Theorem 3.2).

The local asymptotic power of the MF-W test can be computed from (3.2).

8

3.2 Low Frequency Tests

We next characterize the asymptotic distribution of the LF test underHsla. DefineΣXX =E[X(τL)X(τL)

′].Also defineW andWX as follows.

[xL(τL)

xH(τL)

]︸ ︷︷ ︸=X1(τL)

=

[1 0 0 0

0 1/3 1/3 1/3

]︸ ︷︷ ︸

=W

xL(τL)

xH(τL, 1)

xH(τL, 2)

xH(τL, 3)

︸ ︷︷ ︸

=X1(τL)

and

1

X1(τL)

X2(τL)

︸ ︷︷ ︸

=X(τL)

=

1 01×4 01×k2

02×1 W 02×k2

0k2×1 0k2×4 Ik2

︸ ︷︷ ︸

=WX

1

X1(τL)

X2(τL)

︸ ︷︷ ︸

=X(τL)

. (3.3)

The2×4 matrixW transformsX1(τL) toX1(τL), while the(k2+3)×(k2+5) matrixWX transforms

X(τL) to X(τL). Finally, define

δs = RΣ−1XXWXΣXXνs and σ2 = σ2

LRΣ−1XXR′, (3.4)

whereνs = [0, 0, νs1/3, νs2/3, νs3/3, 01×k2 ]′. σ2 equals the probability limit ofσ2 defined in (2.15).

We are now ready to derive the asymptotic distribution of the LF test statisticW underHsla.

Theorem 3.3. Impose Assumption 2.1. UnderHsla : Rsθ0 = (1/

√TL)νs, we have thatW

d→χ21(δ

2s/σ

2).

Proof . See Appendix H.

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):

y(τL) = α0 + αxL(τL) + βxH(τL) + γ∆UR(τL + 4) + uL(τL). (5.1)

Year-to-Year growth rates are taken fory(τL), xL(τL), andxH(τL) in order to eliminate potential sea-

sonal effects. We expectα > 0 andβ < 0. Change from previous quarter is taken for seasonally-

adjusted unemployment. The four quarters of lead are taken for unemployment in order to approximate

households’ expectation of future labor market conditions. We expectγ < 0 since higher unemploy-

14

ment implies lower consumption in general. We implement the low frequency Wald test with respect to

β = −α.

As a supplemental analysis, we consider a more general model that has the year-to-year growth rate

of nominal net worth,GNW :

y(τL) = α0 + αxL(τL) + δGNW (τL − 1) + βxH(τL) + γ∆UR(τL + 4) + uL(τL). (5.2)

One quarter of lag is taken forGNW in view of timing consistency. Note thatGNW (τL−1) represents

the growth rate of net worth at theendof quarterτL − 1 from net worth at theendof quarterτL − 5.

This can be thought of as the most recent reference point for households when they decide how much to

consume in quarterτL. The null hypothesis of non-illusion is nowβ = −(α+δ). If this condition holds,

then consumption function is homogeneous of degree zero inxL, GNW , andxH . If it does not hold,

then there exists money illusion in income or net worth. The impact of real net worth on consumption is

well known as real balance effects (cfr. Tanner (1970) and Patinkin (2008)).

We next run mixed frequency regression models. The model without net worth is

y(τL) = α0 + αxL(τL) +3∑

j=1

βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL). (5.3)

The difference between (5.1) and (5.3) is that the aggregatedxH(τL) is included in the former while

monthlyxH(τL, 1), xH(τL, 2), andxH(τL, 3) are included as separate regressors in the latter. We test

for the strong non-illusion hypothesisβ1 = β2 = β3 = −α/3 and the weak non-illusion hypothesis

β1 + β2 + β3 = −α.

The mixed frequency model with net worth is

y(τL) = α0 + αxL(τL) + δGNW (τL − 1) +

3∑j=1

βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL) (5.4)

We test for the strong non-illusion hypothesisβ1 = β2 = β3 = −(α + δ)/3 and the weak non-illusion

hypothesisβ1 + β2 + β3 = −(α+ δ).

United States We now explain regression models for the U.S. The low frequency model without net

worth is exactly same as the case of Japan (cfr. (5.1)). The low frequency model with net worth is

y(τL) = α0 + αxL(τL) + δGNW (τL − 2) + βxH(τL) + γ∆UR(τL + 4) + uL(τL).

Contrary to the case of Japan, we take two quarters of lags for net worthGNW . This is an admittedly ad-

hoc attempt to improve a model fit. It is rather an empirical question at which reference point households

use when they decide how much to consume.

The mixed frequency model without net worth is exactly same as (5.3). The model with net worth

15

usesGNW (τL − 2) again:

y(τL) = α0 + αxL(τL) + δGNW (τL − 2) +

3∑j=1

βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL).

For all cases, Wald tests can be implemented as in the case of Japan.

5.2 Data and Preliminary Analysis

Sample period is the first quarter of 1981 through the first quarter of 2013 (written as 1981Q1-2013Q1)

for Japan and 1981Q1-2014Q2 for the U.S. For each country, we need historical data of real consumption,

nominal disposable income, inflation, unemployment, and nominal net worth. We first explain the data

source of Japan. For real consumption we use a seasonally-adjusted real series of ”Private Consumption”

of the System of National Accounts (93SNA), published by the Economic and Social Research Institute,

Cabinet Office. For nominal disposable income we retrieve a seasonally-unadjusted nominal series of

”Disposable Income” of 93SNA and then fit X-12-ARIMA to remove seasonality. For inflation we use

the seasonally-adjusted consumer price index (all items) announced by the Statistics Bureau, the Ministry

of Internal Affairs and Communications. We also retrieve seasonally adjusted unemployment data from

the same data source.

Historical data of Japan’s net worth can be found at the Flow of Funds Accounts, Bank of Japan.

Those data however have a point of discontinuity at 1997Q4. Moreover, they used to be announced

at the first and fourth quarters only until 1995Q1. Hence we interpolated the raw data to the quarterly

frequency for 1981Q1-1995Q1. Interpolation is made so that there is a constant growth at the second

and third quarters. We then connected the interpolated old series and the new series at 1997Q3.

We now explain the data source of the U.S. Seasonally adjusted series of real consumption and

nominal disposable income can be found at the Bureau of Economic Analysis, Department of Commerce.

Seasonally adjusted series of consumer price index (all items) and unemployment can be found at the

Bureau of Labor Statistics, Department of Labor. Finally, net worth data are available at the Financial

Accounts of the United States, the Board of Governors of the Federal Reserve System.

See Figure 2 for time series plots of the Japanese and U.S. data. Panel A.1 displays quarterly real

consumption, quarterly nominal disposable income, and monthly inflation of Japan. In Panel A.2 the

nominal disposable income is replaced with quarterly nominal net worth. Panels B.1 and B.2 have the

same structure with the U.S. data. Shaded areas represent official recession periods. It is clear from the

shaded areas that the overall performance of Japanese economy is worse than the U.S. economies in our

sample period. As many as 136 out of 387 months lie in recession periods for Japan (35.1%), while

54 out of 402 months lie in recession periods for the U.S. (13.4%). For both countries we see strong

positive correlation between disposable income and consumption. Correlation between net worth and

consumption seems present as well, although the net worth is much more volatile than the consumption.

In particular, the U.S. net worth declined by 16.7% from previous year in 2008Q4, while the real con-

sumption declined by only 2.0%. Correlation between inflation and consumption seems low, especially

in the second half of the sample period.

16

See Table 3 for sample statistics. They confirm the visual patterns observed in Figure 2. The mean of

real consumption, nominal disposable income, and net worth are respectively 2.0%, 1.9%, and 5.4% for

Japan, while they are 3.0%, 5.5%, and 6.2% for the U.S. The U.S. has higher growth rates than Japan in

all three variables. Average inflation is below 1% for Japan while it is above 3% for the U.S., reflecting

the prolonged deflation during Japan’s Lost Decade.

The correlation coefficient between disposable income and consumption is moderately high at 0.575

for Japan and 0.463 for the U.S. (Panel B, Table 3). Similarly, the correlation coefficient between net

worth and consumption is 0.615 for Japan and 0.609 for the U.S. The correlation coefficient between

inflation and consumption is close to zero for the U.S. and even positive for Japan. These results suggest

that a key driver of real consumption is nominal disposable income and net worth, while inflation plays

a minor role.

5.3 Empirical Results

Now we report empirical results. Unless otherwise specified, a nominal size is fixed at the 5% level.

Table 4 presents empirical results on Japan. We first discuss low frequency models. Without net worth,

the null hypothesis of non-illusion is not rejected with p-value 0.135. With net worth, p-value decreases

to 0.058 but the null hypothesis is still not rejected at the 5% level. We thus do not observe a strong

evidence for money illusion at the quarterly level.

We next discuss mixed frequency models. Without net worth, the coefficients of monthly inflation

series are 0.067, -1.267, and 0.876 with standard errors 0.371, 0.522, and 0.420 respectively. This

suggests that there is a deviation from strong non-illusion with positive and negative signs. In fact,

the strong non-illusion hypothesis is rejected but the weak non-illusion hypothesis is not at the 5%

level (p-value is 0.035 for strong non-illusion and 0.106 for weak non-illusion). Similar results appear

with net worth (p-value is 0.042 for strong non-illusion and 0.053 for weak non-illusion). We thus

conclude that non-illusion holds at a quarterly level but monthly inflation has heterogeneous impacts on

real consumption.

Table 5 presents empirical results on the U.S. We first discuss low frequency models. Without net

worth, the null hypothesis of non-illusion is not rejected with p-value 0.466. A similar result follows

when net worth is included (p-value is 0.592). Hence we do not observe any evidence for money illusion

at the quarterly level.

We now turn on to mixed frequency models. Without net worth, the coefficients of monthly inflation

series are -0.596, -0.223, and 0.200 with standard errors 0.299, 0.543, and 0.335 respectively. The strong

non-illusion hypothesis is rejected but the weak non-illusion hypothesis is not (p-value is 0.046 for strong

non-illusion and 0.253 for weak non-illusion). Similar results appear with net worth (p-value is 0.020 for

strong non-illusion and 0.950 for weak non-illusion). As in Japan, non-illusion holds at a quarterly level

but monthly inflation has heterogeneous impacts on real consumption.

In summary, there likely exists mean-zero deviation from strong non-illusion in both Japan and the

U.S. As we show in the local power analysis and Monte Carlo simulations, the MF-S test is powerful

while the LF test loses power under mean-zero deviations. Our empirical results are exactly consistent

17

with those observations. Rejection of the MF-S test and non-rejection of the MF-W test indicate a

practical importance of distinguishing strong and weak non-illusion.

6 Conclusions

This paper proposes a mixed data sampling (MIDAS) methodology that improves statistical tests for

money illusion in aggregate consumption function. Typically, money illusion in aggregate consumption

is tested by projecting real consumption onto nominal disposable income and a consumer price index.

While consumption and income data are often sampled at a quarterly level, price data are available at

a monthly level. This paper takes advantage of MIDAS regressions in order to exploit monthly price

data. We regress quarterly real consumption growth onto quarterly nominal disposable income growth

and monthly inflation. A regression model on levels of variables (i.e. cointegration) remains as a future

task.

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 ag-

gregates monthly inflation into a quarterly level. In particular, the MIDAS approach allows us to dis-

tinguish strong non-illusion and weak non-illusion. Under strong non-illusion, the overall impact of

monthly inflation equals the impact of nominal disposable income growthand an individual impact of

monthly inflation is all equal. Under weak non-illusion, the former condition holds but the latter does

not. Distinguishing strong and weak non-illusion is of practical interest because heterogeneous effects

of monthly inflation on real consumption growth may well exist due to seasonality or lagged information

transmission.

In empirical applications on Japan and the U.S., we find that strong non-illusion is rejected but weak

non-illusion is not. Hence we conclude that the heterogeneous effects of monthly inflation in fact exist.

This is clearly a new finding because the classical single-frequency approach cannot distinguish strong

and weak non-illusion.

References

ANDREOU, E., E. GHYSELS, AND A. KOURTELLOS(2010): “Regression Models with Mixed Sampling

Frequencies,”Journal of Econometrics, 158, 246–261.

(2011): “Forecasting with Mixed-Frequency Data,” inOxford Handbook of Economic Forecast-

ing, ed. by M. Clements,andD. Hendry, pp. 225–245.

ARMESTO, M., K. ENGEMANN, AND M. OWYANG (2010): “Forecasting with Mixed Frequencies,”

Federal Reserve Bank of St. Louis Review, 92, 521–536.

BRANSON, W. H., AND A. K. K LEVORICK (1969): “Money Illusion and the Aggregate Consumption

Function,”American Economic Review, 59, 832–849.

18

(1972): “Money Illusion and the Aggregate Consumption: Reply,”American Economic Review,

62, 207–210.

BRUNNERMEIER, M. K., AND C. JULLIARD (2008): “Money Illusion and Housing Frenzies,”Review

of Financial Studies, 21, 135–180.

COHEN, R. B., C. POLK , AND T. VUOLTEENAHO (2005): “Money Illusion in the Stock Market: The

Modigliani-Cohn Hypothesis,” NBER Working Paper No. 11018.

CRAIG, G. D. (1974): “Money Illusion and the Aggregate Consumption: Note,”American Economic

Review, 64, 195–199.

CUKIERMAN , A. (1972): “Money Illusion and the Aggregate Consumption: Comment,”American Eco-

nomic Review, 62, 198–206.

ECONOMIC PLANNING AGENCY (1995): Heisei 7 Nendo Nenji Keizai Hokoku (Economic Survey of

Japan (1995-1996)). Government of Japan.

FEHR, E., AND J.-R. TYRAN (2001): “Does Money Illusion Matter?,”American Economic Review, 91,

1239–1262.

FISHER, I. (1928):The Money Illusion. New York: Adelphi Co.

GHYSELS, E. (2015): “Macroeconomics and the Reality of Mixed Frequency Data,”Journal of Econo-

metrics, (forthcoming).

GHYSELS, E., J. B. HILL , AND K. M OTEGI (2014): “Simple Granger Causality Tests for Mixed Fre-

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(2015): “Testing for Granger Causality with Mixed Frequency Data,”Journal of Econometrics,

(forthcoming).

GHYSELS, E., AND J. I. MILLER (2015): “Testing for Cointegration with Temporally Aggregated and

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314–324, in Japanese.

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of Money, Credit and Banking, 2, 473–485.

20

Tabl

esan

dF

igur

es

Tabl

e1:

Loca

lAsy

mpt

otic

Pow

er

Cas

e1.

Str

ong

Non

-Illu

sion

:(νs1,ν

s2,ν

s3)=

(0,0,0)

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(νs1,ν

s2,ν

s3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.0

,0.0

,0.0

)0.

050

0.05

00.

050

0.05

00.

050

0.05

00.

050

0.05

00.

050

0.05

00.

050

0.05

0

Cas

e2.

Wea

kN

on-I

llusi

on:∑ 3 j=

1ν s

j=

0an

dνs=

03×1

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(νs1,ν

s2,ν

s3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(1.0

,1.0

,-2.

0)0.

096

0.05

00.

051

0.09

70.

050

0.05

20.

099

0.05

00.

052

0.09

90.

050

0.05

3(2

.0,2

.0,-

4.0)

0.26

70.

050

0.05

40.

268

0.05

00.

056

0.27

90.

050

0.05

70.

281

0.05

00.

060

(2.0

,-4.

0,2.

0)0.

156

0.05

00.

052

0.15

50.

050

0.05

20.

130

0.05

00.

051

0.13

10.

050

0.05

2(-

4.0,

2.0,

2.0)

0.26

70.

050

0.05

10.

266

0.05

00.

052

0.27

50.

050

0.05

20.

274

0.05

00.

053

(4.0

,-2.

0,-2

.0)

0.26

70.

050

0.05

10.

266

0.05

00.

052

0.27

50.

050

0.05

20.

274

0.05

00.

053

Cas

e3.

Illus

ion:∑ 3 j=

1ν s

j=

0

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(νs1,ν

s2,ν

s3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(2.0

,2.0

,-2.

0)0.

164

0.07

50.

092

0.17

20.

082

0.10

60.

185

0.09

10.

118

0.20

60.

109

0.15

0(2

.0,2

.0,2

.0)

0.19

20.

288

0.29

20.

235

0.34

90.

356

0.29

00.

423

0.43

10.

411

0.56

20.

576

(4.0

,4.0

,4.0

)0.

652

0.79

90.

806

0.76

60.

882

0.88

90.

864

0.94

20.

946

0.96

40.

988

0.99

0

Thi

sta

ble

pres

ents

loca

lasy

mpt

otic

pow

erof

the

mix

edfr

eque

ncy

test

with

resp

ectt

ost

rong

non-

illus

ion

(MF

-S),

the

mix

edfr

eque

ncy

test

with

resp

ectt

ow

eak

non-

illus

ion

(MF

-W),

and

the

low

freq

uenc

yte

st(L

F).

Pitm

anpa

ram

eter

s(ν

s1,ν

s2,ν

s3)

are

allz

eros

inC

ase

1(s

tron

gno

n-ill

usio

n),s

umup

toze

roin

Cas

e2

(wea

kno

n-ill

usio

n),a

ndsu

mup

toa

nonz

ero

valu

ein

Cas

e3

(illu

sion

).ϕH

andϕL

sign

ifyth

epe

rsis

tenc

eofx

Han

dxL

,res

pect

ivel

y.

21

Tabl

e2:

Rej

ectio

nF

requ

enci

es(A

.TL=

50)

Cas

e1.

Str

ong

Non

-Illu

sion

:(cs1,c

s2,c

s3)=

(0,0,0)

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.0

,0.0

,0.0

)(0

.3,-

0.1,

-0.1

,-0.

1)0.

086

0.06

80.

068

0.08

80.

068

0.06

80.

085

0.06

80.

067

0.08

60.

070

0.06

9(0

.6,0

.0,0

.0,0

.0)

(0.6

,-0.

2,-0

.2,-

0.2)

0.08

50.

066

0.06

60.

086

0.07

00.

067

0.08

70.

070

0.07

00.

084

0.07

20.

073

(0.9

,0.0

,0.0

,0.0

)(0

.9,-

0.3,

-0.3

,-0.

3)0.

083

0.06

50.

067

0.08

60.

070

0.06

90.

087

0.06

80.

070

0.08

50.

070

0.06

6C

ase

2.W

eak

Non

-Illu

sion

:∑ 3 j=1c s

j=

0an

dcs=

03×1

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

6)(0

.3,0

.0,0

.0,-

0.3)

0.10

90.

071

0.07

40.

117

0.07

20.

073

0.11

00.

066

0.06

80.

115

0.07

10.

076

(0.3

,0.3

,-0.

6,0.

3)(0

.3,0

.0,-

0.3,

0.0)

0.10

20.

075

0.07

40.

095

0.07

00.

071

0.09

50.

070

0.06

80.

095

0.06

60.

069

(0.3

,-0.

6,0.

3,0.

3)(0

.3,-

0.3,

0.0,

0.0)

0.11

00.

074

0.07

40.

107

0.07

10.

072

0.11

00.

068

0.07

00.

111

0.06

90.

067

(0.6

,0.6

,0.6

,-1.

2)(0

.6,0

.0,0

.0,-

0.6)

0.18

70.

073

0.07

50.

188

0.06

70.

073

0.19

50.

071

0.07

70.

196

0.07

20.

081

(0.6

,0.6

,-1.

2,0.

6)(0

.6,0

.0,-

0.6,

0.0)

0.14

30.

069

0.07

40.

137

0.06

90.

073

0.13

00.

065

0.06

60.

130

0.07

10.

076

(0.6

,-1.

2,0.

6,0.

6)(0

.6,-

0.6,

0.0,

0.0)

0.18

80.

072

0.07

60.

189

0.06

90.

077

0.18

90.

069

0.07

10.

196

0.07

10.

077

(0.9

,0.9

,0.9

,-1.

8)(0

.9,0

.0,0

.0,-

0.9)

0.33

00.

066

0.08

10.

328

0.06

40.

082

0.33

90.

066

0.07

90.

338

0.07

10.

086

(0.9

,0.9

,-1.

8,0.

9)(0

.9,0

.0,-

0.9,

0.0)

0.21

30.

069

0.07

30.

210

0.07

20.

079

0.18

60.

067

0.07

30.

182

0.06

70.

074

(0.9

,-1.

8,0.

9,0.

9)(0

.9,-

0.9,

0.0,

0.0)

0.33

10.

074

0.08

30.

325

0.07

00.

075

0.33

90.

071

0.07

80.

337

0.07

30.

084

Cas

e3.

Illus

ion:∑ 3 j

=1c s

j=

0

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

3)(0

.3,0

.0,0

.0,-

0.2)

0.09

80.

076

0.07

80.

100

0.07

40.

077

0.09

50.

071

0.07

50.

103

0.07

80.

082

(0.3

,0.3

,-0.

3,0.

3)(0

.3,0

.0,-

0.2,

0.0)

0.09

40.

068

0.07

00.

094

0.07

30.

070

0.09

10.

074

0.07

30.

088

0.07

50.

073

(0.3

,-0.

3,0.

3,0.

3)(0

.3,-

0.2,

0.0,

0.0)

0.09

50.

072

0.07

40.

100

0.07

40.

074

0.09

70.

073

0.07

20.

098

0.07

70.

074

(0.3

,-0.

3,-0

.3,-

0.3)

(0.3

,-0.

2,-0

.2,-

0.2)

0.09

90.

096

0.09

90.

107

0.10

00.

102

0.11

30.

109

0.11

10.

122

0.12

30.

129

(0.6

,0.6

,0.6

,-0.

6)(0

.6,0

.0,0

.0,-

0.4)

0.14

30.

076

0.08

70.

149

0.08

20.

096

0.15

70.

091

0.10

60.

160

0.09

50.

113

(0.6

,0.6

,-0.

6,0.

6)(0

.6,0

.0,-

0.4,

0.0)

0.11

10.

079

0.07

90.

120

0.09

00.

086

0.11

10.

090

0.08

40.

116

0.09

60.

092

(0.6

,-0.

6,0.

6,0.

6)(0

.6,-

0.4,

0.0,

0.0)

0.13

70.

082

0.08

20.

138

0.08

60.

086

0.13

70.

087

0.08

30.

139

0.08

90.

087

(0.6

,-0.

6,-0

.6,-

0.6)

(0.6

,-0.

4,-0

.4,-

0.4)

0.14

30.

168

0.17

60.

167

0.19

30.

205

0.18

60.

227

0.23

60.

221

0.26

70.

282

(0.9

,0.9

,0.9

,-0.

9)(0

.9,0

.0,0

.0,-

0.6)

0.21

30.

092

0.11

70.

234

0.10

00.

129

0.23

70.

110

0.14

50.

251

0.12

90.

168

(0.9

,0.9

,-0.

9,0.

9)(0

.9,0

.0,-

0.6,

0.0)

0.14

80.

099

0.09

40.

151

0.10

40.

097

0.14

70.

111

0.10

20.

155

0.11

90.

112

(0.9

,-0.

9,0.

9,0.

9)(0

.9,-

0.6,

0.0,

0.0)

0.19

50.

091

0.08

90.

203

0.10

50.

103

0.22

10.

113

0.10

70.

218

0.12

10.

113

(0.9

,-0.

9,-0

.9,-

0.9)

(0.9

,-0.

6,-0

.6,-

0.6)

0.23

80.

299

0.31

20.

274

0.34

40.

360

0.32

70.

413

0.43

30.

403

0.49

50.

522

Thi

sta

ble

pres

ents

reje

ctio

nfr

eque

ncie

sof

the

mix

edfr

eque

ncy

test

with

resp

ectt

ost

rong

non-

illus

ion

(MF

-S),

the

mix

edfr

eque

ncy

test

with

resp

ectt

ow

eak

non-

illus

ion

(MF

-W),

and

the

low

freq

uenc

yte

st(L

F).

Sam

ple

size

is50

,10

0,or

130

quar

ters

.D

evia

tions

(cs1,c

s2,c

s3)

are

allz

eros

inC

ase

1(s

tron

gno

n-ill

usio

n),

sum

upto

zero

inC

ase

2(w

eak

non-

illus

ion)

,and

sum

upto

ano

nzer

ova

lue

inC

ase

3(il

lusi

on).

ϕH

andϕL

sign

ifyth

epe

rsis

tenc

eofx

Han

dxL

,res

pect

ivel

y.

22

Tabl

e2:

Rej

ectio

nF

requ

enci

es(B

.TL=

100)

Cas

e1.

Str

ong

Non

-Illu

sion

:(cs1,c

s2,c

s3)=

(0,0,0)

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.0

,0.0

,0.0

)(0

.3,-

0.1,

-0.1

,-0.

1)0.

066

0.06

00.

060

0.06

90.

059

0.05

80.

067

0.05

90.

059

0.06

90.

058

0.05

8(0

.6,0

.0,0

.0,0

.0)

(0.6

,-0.

2,-0

.2,-

0.2)

0.06

80.

057

0.05

90.

066

0.06

40.

061

0.06

30.

058

0.05

70.

064

0.05

70.

059

(0.9

,0.0

,0.0

,0.0

)(0

.9,-

0.3,

-0.3

,-0.

3)0.

069

0.05

50.

055

0.06

40.

056

0.05

50.

063

0.05

80.

059

0.05

90.

059

0.06

1C

ase

2.W

eak

Non

-Illu

sion

:∑ 3 j=1c s

j=

0an

dcs=

03×1

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

6)(0

.3,0

.0,0

.0,-

0.3)

0.11

00.

056

0.05

90.

110

0.05

50.

059

0.11

80.

063

0.06

60.

117

0.06

10.

066

(0.3

,0.3

,-0.

6,0.

3)(0

.3,0

.0,-

0.3,

0.0)

0.08

80.

054

0.05

60.

088

0.06

20.

062

0.08

00.

058

0.05

80.

081

0.05

60.

057

(0.3

,-0.

6,0.

3,0.

3)(0

.3,-

0.3,

0.0,

0.0)

0.11

00.

059

0.05

90.

115

0.05

70.

056

0.11

80.

058

0.05

90.

114

0.05

60.

061

(0.6

,0.6

,0.6

,-1.

2)(0

.6,0

.0,0

.0,-

0.6)

0.27

10.

057

0.06

80.

268

0.05

80.

064

0.28

30.

061

0.07

00.

282

0.06

30.

075

(0.6

,0.6

,-1.

2,0.

6)(0

.6,0

.0,-

0.6,

0.0)

0.16

60.

061

0.06

30.

168

0.06

00.

064

0.14

50.

060

0.06

30.

147

0.05

90.

063

(0.6

,-1.

2,0.

6,0.

6)(0

.6,-

0.6,

0.0,

0.0)

0.27

10.

058

0.06

30.

273

0.06

20.

062

0.28

00.

061

0.06

60.

274

0.05

90.

064

(0.9

,0.9

,0.9

,-1.

8)(0

.9,0

.0,0

.0,-

0.9)

0.53

70.

060

0.07

60.

530

0.06

20.

083

0.55

50.

064

0.08

40.

557

0.05

90.

084

(0.9

,0.9

,-1.

8,0.

9)(0

.9,0

.0,-

0.9,

0.0)

0.31

40.

061

0.07

00.

316

0.06

10.

072

0.25

40.

063

0.07

10.

254

0.05

90.

068

(0.9

,-1.

8,0.

9,0.

9)(0

.9,-

0.9,

0.0,

0.0)

0.53

20.

059

0.06

80.

530

0.05

90.

069

0.54

50.

059

0.06

90.

545

0.05

70.

070

Cas

e3.

Illus

ion:∑ 3 j

=1c s

j=

0

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

3)(0

.3,0

.0,0

.0,-

0.2)

0.08

80.

061

0.06

50.

092

0.06

60.

072

0.09

40.

067

0.07

10.

097

0.07

00.

079

(0.3

,0.3

,-0.

3,0.

3)(0

.3,0

.0,-

0.2,

0.0)

0.08

00.

062

0.06

10.

080

0.06

50.

063

0.07

60.

066

0.06

40.

079

0.06

90.

069

(0.3

,-0.

3,0.

3,0.

3)(0

.3,-

0.2,

0.0,

0.0)

0.09

00.

064

0.06

30.

091

0.06

60.

066

0.09

10.

066

0.06

40.

096

0.07

60.

075

(0.3

,-0.

3,-0

.3,-

0.3)

(0.3

,-0.

2,-0

.2,-

0.2)

0.09

60.

108

0.11

00.

104

0.12

50.

126

0.11

10.

140

0.14

00.

137

0.17

40.

184

(0.6

,0.6

,0.6

,-0.

6)(0

.6,0

.0,0

.0,-

0.4)

0.17

30.

080

0.09

60.

180

0.08

10.

107

0.19

10.

094

0.12

20.

211

0.10

80.

146

(0.6

,0.6

,-0.

6,0.

6)(0

.6,0

.0,-

0.4,

0.0)

0.11

60.

082

0.07

40.

121

0.08

50.

076

0.11

80.

094

0.08

80.

121

0.10

90.

099

(0.6

,-0.

6,0.

6,0.

6)(0

.6,-

0.4,

0.0,

0.0)

0.16

90.

089

0.08

70.

161

0.08

70.

082

0.17

40.

098

0.09

00.

182

0.11

20.

102

(0.6

,-0.

6,-0

.6,-

0.6)

(0.6

,-0.

4,-0

.4,-

0.4)

0.20

90.

274

0.28

30.

231

0.31

90.

327

0.27

80.

378

0.39

10.

384

0.48

80.

507

(0.9

,0.9

,0.9

,-0.

9)(0

.9,0

.0,0

.0,-

0.6)

0.33

10.

114

0.15

60.

349

0.12

50.

181

0.37

50.

141

0.20

40.

411

0.16

70.

254

(0.9

,0.9

,-0.

9,0.

9)(0

.9,0

.0,-

0.6,

0.0)

0.18

70.

110

0.09

60.

192

0.12

50.

110

0.18

90.

142

0.12

90.

202

0.17

60.

151

(0.9

,-0.

9,0.

9,0.

9)(0

.9,-

0.6,

0.0,

0.0)

0.29

40.

107

0.10

60.

307

0.12

30.

113

0.31

90.

146

0.12

80.

333

0.17

70.

153

(0.9

,-0.

9,-0

.9,-

0.9)

(0.9

,-0.

6,-0

.6,-

0.6)

0.38

80.

517

0.53

10.

451

0.58

50.

600

0.55

90.

693

0.70

90.

688

0.80

30.

821

23

Tabl

e2:

Rej

ectio

nF

requ

enci

es(C

.TL=

130)

Cas

e1.

Str

ong

Non

-Illu

sion

:(cs1,c

s2,c

s3)=

(0,0,0)

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.0

,0.0

,0.0

)(0

.3,-

0.1,

-0.1

,-0.

1)0.

064

0.05

50.

057

0.06

60.

059

0.05

80.

062

0.05

60.

054

0.06

40.

060

0.06

0(0

.6,0

.0,0

.0,0

.0)

(0.6

,-0.

2,-0

.2,-

0.2)

0.06

50.

056

0.05

70.

061

0.05

60.

057

0.06

40.

054

0.05

40.

058

0.05

80.

057

(0.9

,0.0

,0.0

,0.0

)(0

.9,-

0.3,

-0.3

,-0.

3)0.

064

0.06

00.

059

0.06

10.

059

0.06

00.

063

0.05

60.

055

0.06

10.

058

0.05

6C

ase

2.W

eak

Non

-Illu

sion

:∑ 3 j=1c s

j=

0an

dcs=

03×1

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

6)(0

.3,0

.0,0

.0,-

0.3)

0.12

50.

059

0.05

90.

122

0.05

60.

058

0.12

20.

060

0.06

50.

127

0.05

60.

061

(0.3

,0.3

,-0.

6,0.

3)(0

.3,0

.0,-

0.3,

0.0)

0.09

40.

054

0.05

60.

091

0.06

00.

060

0.08

70.

061

0.06

10.

082

0.05

50.

055

(0.3

,-0.

6,0.

3,0.

3)(0

.3,-

0.3,

0.0,

0.0)

0.12

40.

057

0.05

70.

120

0.05

60.

057

0.12

70.

056

0.05

50.

126

0.05

60.

059

(0.6

,0.6

,0.6

,-1.

2)(0

.6,0

.0,0

.0,-

0.6)

0.33

00.

056

0.06

50.

334

0.05

90.

069

0.35

50.

058

0.07

00.

348

0.05

80.

070

(0.6

,0.6

,-1.

2,0.

6)(0

.6,0

.0,-

0.6,

0.0)

0.19

60.

057

0.05

90.

198

0.05

90.

059

0.16

70.

059

0.06

00.

159

0.05

50.

057

(0.6

,-1.

2,0.

6,0.

6)(0

.6,-

0.6,

0.0,

0.0)

0.33

20.

061

0.06

50.

336

0.05

40.

059

0.34

00.

059

0.06

30.

334

0.05

60.

063

(0.9

,0.9

,0.9

,-1.

8)(0

.9,0

.0,0

.0,-

0.9)

0.64

40.

057

0.07

70.

641

0.05

40.

082

0.66

80.

054

0.07

60.

674

0.05

40.

086

(0.9

,0.9

,-1.

8,0.

9)(0

.9,0

.0,-

0.9,

0.0)

0.39

10.

061

0.07

00.

386

0.05

80.

068

0.30

50.

053

0.06

10.

313

0.05

80.

067

(0.9

,-1.

8,0.

9,0.

9)(0

.9,-

0.9,

0.0,

0.0)

0.65

60.

064

0.07

20.

637

0.05

60.

063

0.66

00.

057

0.07

00.

662

0.05

80.

069

Cas

e3.

Illus

ion:∑ 3 j

=1c s

j=

0

(ϕH,ϕ

L)=

(0.5,0.5)

(ϕH,ϕ

L)=

(0.5,0.8)

(ϕH,ϕ

L)=

(0.8,0.5)

(ϕH,ϕ

L)=

(0.8,0.8)

(a,c

s1,c

s2,c

s3)

(a,b

1,b

2,b

3)

MF

-SM

F-W

LFM

F-S

MF

-WLF

MF

-SM

F-W

LFM

F-S

MF

-WLF

(0.3

,0.3

,0.3

,-0.

3)(0

.3,0

.0,0

.0,-

0.2)

0.09

40.

067

0.07

20.

100

0.06

80.

074

0.09

90.

067

0.07

50.

104

0.07

20.

086

(0.3

,0.3

,-0.

3,0.

3)(0

.3,0

.0,-

0.2,

0.0)

0.07

40.

067

0.06

60.

079

0.06

40.

063

0.07

60.

069

0.06

80.

075

0.07

00.

068

(0.3

,-0.

3,0.

3,0.

3)(0

.3,-

0.2,

0.0,

0.0)

0.09

20.

063

0.06

30.

094

0.06

60.

064

0.09

60.

067

0.06

20.

101

0.06

90.

065

(0.3

,-0.

3,-0

.3,-

0.3)

(0.3

,-0.

2,-0

.2,-

0.2)

0.10

20.

124

0.12

40.

111

0.14

40.

148

0.12

40.

166

0.17

10.

157

0.20

20.

209

(0.6

,0.6

,0.6

,-0.

6)(0

.6,0

.0,0

.0,-

0.4)

0.21

00.

089

0.11

10.

215

0.08

90.

115

0.23

00.

103

0.13

70.

252

0.12

30.

173

(0.6

,0.6

,-0.

6,0.

6)(0

.6,0

.0,-

0.4,

0.0)

0.12

50.

091

0.07

90.

133

0.09

00.

083

0.12

80.

104

0.09

70.

135

0.12

00.

108

(0.6

,-0.

6,0.

6,0.

6)(0

.6,-

0.4,

0.0,

0.0)

0.18

90.

088

0.08

30.

188

0.09

30.

086

0.20

10.

107

0.09

40.

212

0.12

50.

113

(0.6

,-0.

6,-0

.6,-

0.6)

(0.6

,-0.

4,-0

.4,-

0.4)

0.23

50.

332

0.34

00.

277

0.38

90.

404

0.34

70.

469

0.48

00.

463

0.59

70.

615

(0.9

,0.9

,0.9

,-0.

9)(0

.9,0

.0,0

.0,-

0.6)

0.40

20.

125

0.17

40.

426

0.14

40.

213

0.45

80.

166

0.24

40.

509

0.19

90.

312

(0.9

,0.9

,-0.

9,0.

9)(0

.9,0

.0,-

0.6,

0.0)

0.22

30.

121

0.10

60.

237

0.14

20.

122

0.21

90.

165

0.14

40.

247

0.21

20.

177

(0.9

,-0.

9,0.

9,0.

9)(0

.9,-

0.6,

0.0,

0.0)

0.37

00.

125

0.11

60.

376

0.14

10.

125

0.39

80.

165

0.14

30.

405

0.19

60.

169

(0.9

,-0.

9,-0

.9,-

0.9)

(0.9

,-0.

6,-0

.6,-

0.6)

0.46

80.

612

0.62

30.

566

0.69

80.

714

0.67

40.

797

0.81

20.

819

0.89

80.

913

24

Table 3: Sample Statistics

A. Individual Statistics

A.1. Japan (First Quarter, 1981 - First Quarter, 2013)Mean Median Min. Max. Std. Dev. Skew. Kurt.

Consumption (y) 2.039 1.876 -3.778 7.657 1.942 0.085 3.516Income (xL) 1.887 1.100 -3.257 9.436 3.072 0.534 2.271

Net Worth (GNW ) 5.398 4.889 -8.673 14.72 4.876 -0.241 2.872CPI1 (xH(·, 1)) 0.807 0.386 -2.557 7.025 1.526 0.891 4.219CPI2 (xH(·, 2)) 0.790 0.450 -2.256 6.094 1.489 0.784 3.426CPI3 (xH(·, 3)) 0.773 0.450 -2.256 5.937 1.464 0.786 3.393

CPIQ (xH ) 0.790 0.331 -2.258 6.352 1.476 0.827 3.675

A.2. United States (First Quarter, 1981 - Second Quarter, 2014)Mean Median Min. Max. Std. Dev. Skew. Kurt.

Consumption (y) 2.977 3.147 -2.731 6.476 1.705 -0.734 4.013Income (xL) 5.502 5.377 -1.717 11.84 2.322 -0.042 4.333

Net Worth (GNW ) 6.242 6.902 -16.70 15.38 5.469 -1.764 7.333CPI1 (xH(·, 1)) 3.155 2.870 -1.978 11.15 1.851 1.569 8.134CPI2 (xH(·, 2)) 3.130 2.983 -1.495 10.79 1.784 1.562 7.982CPI3 (xH(·, 3)) 3.105 2.923 -1.388 10.40 1.751 1.372 7.548

CPIQ (xH ) 3.130 2.913 -1.620 10.67 1.769 1.583 8.179

B. Correlation Coefficient Matrix

B.1. Japan (First Quarter, 1981 - First Quarter, 2013)Consumption Income Net Worth CPI1 CPI2 CPI3 CPIQ

Consumption (y) 1.000 - - - - - -Income (xL) 0.575 1.000 - - - - -

Net Worth (GNW ) 0.615 0.608 1.000 - - - -CPI1 (xH(·, 1)) 0.289 0.764 0.399 1.000 - - -CPI2 (xH(·, 2)) 0.289 0.787 0.393 0.970 1.000 - -CPI3 (xH(·, 3)) 0.311 0.771 0.400 0.954 0.976 1.000 -

CPIQ (xH ) 0.299 0.783 0.402 0.986 0.993 0.987 1.000

B.2. United States (First Quarter, 1981 - Second Quarter, 2014)Consumption Income Net Worth CPI1 CPI2 CPI3 CPIQ

Consumption (y) 1.000 - - - - - -Income (xL) 0.463 1.000 - - - - -

Net Worth (GNW ) 0.609 0.329 1.000 - - - -CPI1 (xH(·, 1)) -0.103 0.670 0.115 1.000 - - -CPI2 (xH(·, 2)) -0.074 0.679 0.153 0.974 1.000 - -CPI3 (xH(·, 3)) -0.021 0.691 0.200 0.922 0.975 1.000 -

CPIQ (xH ) -0.068 0.690 0.158 0.980 0.997 0.979 1.000

Panel A presents sample mean, median, minimum, maximum, standard deviation, skewness, and kurtosis of year-to-year growth

rates of real consumptiony, nominal disposable incomexL, nominal net worthGNW , and consumer price index. CPIj (or

xH(·, j)) represents the CPI at thej-th month of each quarter (e.g. CPI1 picks January, April, July, and October). CPIQ (or

xH ) represents a quarterly average of CPI. Sample size is 129 quarters for Japan and 134 quarters for the U.S. Panel B presents

a correlation coefficient matrix for each country.

25

Table 4: Empirical Results on Japan (First Quarter, 1981 - First Quarter, 2013)

A.1. Low Frequency Model (without Net Worth)y(τL) = α0 + αxL(τL) + βxH(τL) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β γ R2 = 0.418

Coef. 1.392 0.498 – -0.321 -2.554 R2= 0.404

S.E. 0.156 0.074 – 0.169 1.061 F = 29.70t-Stat. 8.917 6.714 – -1.899 -2.408 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ – 0.060∗ 0.018∗∗ DW = 1.261Wald Test for Non-Illusion

H0 : β = −α ⇒ W = 2.235, Pr(W ) = 0.135

A.2. Low Frequency Model (with Net Worth)y(τL) = α0 + αxL(τL) + δGNW (τL − 1) + βxH(τL) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β γ R2 = 0.530

Coef. 0.774 0.315 0.170 -0.284 -1.951 R2= 0.515

S.E. 0.181 0.075 0.031 0.153 0.963 F = 34.73t-Stat. 4.269 4.212 5.423 -1.860 -2.026 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.065∗ 0.045∗∗ DW = 1.352Wald Test for Non-Illusion

H0 : β = −(α+ δ) ⇒ W = 3.582, Pr(W ) = 0.058∗

B.1. Mixed Frequency Model (without Net Worth)y(τL) = α0 + αxL(τL) +

∑3j=1 βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β1 β2 β3 γ R2 = 0.447

Coef. 1.380 0.514 – 0.067 -1.267 0.876 -2.605 R2= 0.424

S.E. 0.154 0.074 – 0.371 0.522 0.420 1.059 F = 19.71t-Stat. 8.974 6.987 – 0.179 -2.426 2.085 -2.459 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ – 0.858 0.017∗∗ 0.039∗∗ 0.015∗∗ DW = 1.250Wald Test for Strong and Weak Non-Illusion

Hs0 : β1 = β2 = β3 = −α/3 ⇒ Ws = 8.636, Pr(Ws) = 0.035∗∗

Hw0 : β1 + β2 + β3 = −α ⇒ Ww = 2.615, Pr(Ww) = 0.106

B.2. Mixed Frequency Model (with Net Worth)y(τL) = α0 + αxL(τL) + δGNW (τL − 1) +

∑3j=1 βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β1 β2 β3 γ R2 = 0.547

Coef. 0.789 0.338 0.163 0.160 -1.060 0.604 -1.952 R2= 0.525

S.E. 0.180 0.075 0.031 0.338 0.476 0.385 0.970 F = 24.38t-Stat. 4.376 4.515 5.184 0.475 -2.226 1.568 -2.012 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.636 0.028∗∗ 0.120 0.046∗∗ DW = 1.305Wald Test for Strong and Weak Non-Illusion

Hs0 : β1 = β2 = β3 = −(α+ δ)/3 ⇒ Ws = 8.181, Pr(Ws) = 0.042∗∗

Hw0 : β1 + β2 + β3 = −(α+ δ) ⇒ Ww = 3.742, Pr(Ww) = 0.053∗

y is real consumption growth;xL is nominal disposable income growth;xH is inflation;GNW is net worth growth;∆UR is

change in unemployment rate. Three asterisks (∗∗∗) are put when the null hypothesis is rejected at 1% level; two asterisks (∗∗)

when the null is rejected at 5% but not at 1%; one asterisk (∗) when the null is rejected at 10% but not at 5%.

26

Table 5: Empirical Results on the United States (First Quarter, 1981 - Second Quarter, 2014)

A.1. Low Frequency Model (without Net Worth)y(τL) = α0 + αxL(τL) + βxH(τL) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β γ R2 = 0.517

Coef. 1.150 0.705 – -0.659 -0.748 R2= 0.505

S.E. 0.281 0.062 – 0.086 0.358 F = 45.63t-Stat. 4.088 11.30 – -7.656 -2.089 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ – 0.000∗∗∗ 0.039∗∗ DW = 0.696Wald Test for Non-Illusion

H0 : β = −α ⇒ W = 0.531, Pr(W ) = 0.466

A.2. Low Frequency Model (with Net Worth)y(τL) = α0 + αxL(τL) + δGNW (τL − 2) + βxH(τL) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β γ R2 = 0.721

Coef. 1.310 0.524 0.166 -0.716 -0.520 R2= 0.712

S.E. 0.215 0.051 0.017 0.066 0.274 F = 82.00t-Stat. 6.085 10.23 9.637 -10.87 -1.898 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.060∗ DW = 0.872Wald Test for Non-Illusion

H0 : β = −(α+ δ) ⇒ W = 0.288, Pr(W ) = 0.592

B.1. Mixed Frequency Model (without Net Worth)y(τL) = α0 + αxL(τL) +

∑3j=1 βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β1 β2 β3 γ R2 = 0.544

Coef. 1.119 0.692 – -0.596 -0.223 0.200 -0.823 R2= 0.526

S.E. 0.276 0.063 – 0.299 0.543 0.335 0.354 F = 30.04t-Stat. 4.055 11.05 – -1.990 -0.410 0.596 -2.326 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ – 0.049∗∗ 0.682 0.552 0.022∗∗ DW = 0.785Wald Test for Strong and Weak Non-Illusion

Hs0 : β1 = β2 = β3 = −α/3 ⇒ Ws = 8.013, Pr(Ws) = 0.046∗∗

Hw0 : β1 + β2 + β3 = −α ⇒ Ww = 1.306, Pr(Ww) = 0.253

B.2. Mixed Frequency Model (with Net Worth)y(τL) = α0 + αxL(τL) + δGNW (τL − 2) +

∑3j=1 βjxH(τL, j) + γ∆UR(τL + 4) + uL(τL)

EstimationParam. α0 α δ β1 β2 β3 γ R2 = 0.741

Coef. 1.280 0.515 0.163 -0.571 -0.221 0.110 -0.587 R2= 0.728

S.E. 0.210 0.051 0.017 0.227 0.411 0.254 0.269 F = 59.52t-Stat. 6.111 10.17 9.744 -2.521 -0.538 0.435 -2.183 Pr(F ) = 0.000∗∗∗

p-Val. 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗∗ 0.013∗∗ 0.592 0.664 0.031∗∗ DW = 0.908Wald Test for Strong and Weak Non-Illusion

Hs0 : β1 = β2 = β3 = −(α+ δ)/3 ⇒ Ws = 9.878, Pr(Ws) = 0.020∗∗

Hw0 : β1 + β2 + β3 = −(α+ δ) ⇒ Ww = 0.004, Pr(Ww) = 0.950

y is real consumption growth;xL is nominal disposable income growth;xH is inflation;GNW is net worth growth;∆UR is

change in unemployment rate. Three asterisks (∗∗∗) are put when the null hypothesis is rejected at 1% level; two asterisks (∗∗)

when the null is rejected at 5% but not at 1%; one asterisk (∗) when the null is rejected at 10% but not at 5%.

27

Figure 1: Histogram of Local Asymptotic Power under Weak Non-Illusion (Case 2)

A. (ϕH , ϕL) = (0.5, 0.5)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

A.1. MF-S

0.05 0.052 0.054 0.0560

0.05

0.1

0.15

0.2

0.25

0.3

A.2. LF

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

A.3. (MF-S) - (LF)

B. (ϕH , ϕL) = (0.5, 0.8)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

B.1. MF-S

0.05 0.052 0.054 0.056 0.0580

0.1

0.2

0.3

0.4

0.5

B.2. LF

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

B.3. (MF-S) - (LF)

C. (ϕH , ϕL) = (0.8, 0.5)

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

C.1. MF-S

0.05 0.052 0.054 0.056 0.0580

0.1

0.2

0.3

0.4

C.2. LF

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

C.3. (MF-S) - (LF)

D. (ϕH , ϕL) = (0.8, 0.8)

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

D.1. MF-S

0.05 0.055 0.06 0.0650

0.1

0.2

0.3

0.4

D.2. LF

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

D.3. (MF-S) - (LF)

This figure plots histograms of the local asymptotic power of the mixed frequency test with respect to strong non-illusion (MF-

S) and the low frequency test (LF). Their difference in power is also plotted in the third column. For each panel local power is

put on the horizontal axis and relative frequencies are put on the vertical axis.ϕH andϕL signify the persistence ofxH and

xL, respectively. Pitman parameters(νs1, νs2, νs3) sum up to zero (weak non-illusion).

28

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29

Technical Appendices

A Preliminaries

In Technical Appendices we prove each theorem in the main body. We first review the data generating process

(DGP) and the null hypotheses of interest. The DGP is given by

y(τL) = a0 + axL(τL) +3∑

j=1

bjxH(τL, j) +X2(τL)′θ0,2 + ϵL(τL)

= X(τL)′θ0 + ϵL(τL),

(A.1)

whereX(τL) = [1,X1(τL)′,X2(τL)

′]′,X1(τL) = [xL(τL), xH(τL, 1), xH(τL, 2), xH(τL, 3)]′, θ0 = [a0,θ

′0,1,θ

′0,2]

′,

andθ0,1 = [a, b1, b2, b3]′. The strong non-illusion hypothesisHs

0 and weak non-illusion hypothesisHw0 are written

as

Hs0 :

0 1 3 0 0 01×k2

0 1 0 3 0 01×k2

0 1 0 0 3 01×k2

︸ ︷︷ ︸

=Rs

a0

a

b1

b2

b3

θ2

︸ ︷︷ ︸=θ0

= 03×1 and Hw0 : [0, 1, 1, 1, 1,01×k2 ]︸ ︷︷ ︸

=Rw

a0

a

b1

b2

b3

θ2

︸ ︷︷ ︸=θ0

= 0. (A.2)

The fixed alternative hypotheses are written asHs1 : Rsθ0 = cs ≡ [cs1, cs2, cs3]

′ andHw1 : Rwθ0 = cw.

B Proof of Theorem 2.1

The true DGP is given by (A.1). We fit a mixed frequency regression model:

y(τL) = X(τL)′θ + uL(τL), (B.1)

whereθ = [α0,θ′1,θ

′2]

′ and θ1 = [α, β1, β2, β3]′. We get the ordinary least squares (OLS) estimatorθ =

Σ−1XX sXy, whereΣXX = (1/TL)

∑TL

τL=1X(τL)X(τL)′ andsXy = (1/TL)

∑TL

τL=1X(τL)y(τL).

Substitute the true DGP (A.1) into the expression ofθ to get

θ = Σ−1XX × 1

TL

TL∑τL=1

X(τL) {X(τL)′θ0 + ϵL(τL)} = θ0 + Σ−1

XX × 1

TL

TL∑τL=1

X(τL)ϵL(τL).

Under the strong non-illusion hypothesisHs0 in (A.2), we have that

√TLRsθ = RsΣ

−1XX × 1√

TL

TL∑τL=1

X(τL)ϵL(τL). (B.2)

Under Assumption 2.1, we have thatΣXXp→ ΣXX and (1/

√TL)

∑TL

τL=1 X(τL)ϵL(τL)d→ N(0, σ2

LΣXX),

whereΣXX = E[X(τL)X(τL)′] andσ2

L = E[ϵ(τL)2]. By the continuous mapping theorem and Slutsky’s theo-

30

rem, it follows that √TLRsθ

d→ N(03×1, Σs), where Σs = σ2LRsΣ

−1XXR′

s. (B.3)

Let σ2L = (1/TL)

∑TL

τL=1[y(τL)−X(τL)′θ]2 andΣs = σ2

LRsΣ−1XXR′

s, thenΣsp→ Σs. Define the Cholesky

factorΣs = LsL′s, then it follows from (B.3) that

√TLL

−1s Rsθ

d→ N(03×1, I3). Hence, the mixed frequency

Wald statistic with respect to strong non-illusion (MF-S statistic)Ws = TLθ′R′

sΣ−1s Rsθ = (

√TLL

−1s Rsθ)

′

(√TLL

−1s Rsθ) converges in distribution toχ2

3.

Consider the fixed alternative hypothesisHs1 : Rsθ0 = cs with cs = 03×1. Then an extra nonzero term

√TLcs → ±∞ will be added to the right-hand side of (B.2). This makesWs

p→ ∞ and thus consistency holds.

C Proof of Theorem 2.2

DefineΣw = σ2LRwΣ

−1XXR′

w andLw = Σ1/2w . Formulate the mixed frequency Wald statistic with respect to weak

non-illusion (MF-W statistic):Ww = TLθ′R′

wΣ−1w Rwθ. Following the same steps as in Appendix B, we obtain

Wwd→ χ2

1 underHw0 whileWw

p→∞ underHw1 : Rwθ0 = cw for anycw = 0 (i.e. consistency).

D Proof of Theorem 2.3

Keep the DGP (A.1). Formulate a low frequency model:

y(τL) = α0 + αxL(τL) + βxH(τL) +X2(τL)′θ2 + uL(τL)

= X(τL)′θ + uL(τL),

(D.1)

whereX(τL) = [1,X1(τL)′,X2(τL)

′]′, X1(τL) = [xL(τL), xH(τL)]′, θ = [α0,θ

′1,θ

′2]

′, andθ1 = [α, β]′.

Let θ be the OLS estimator forθ. Then we have that

θ = Σ−1

XX sXy, (D.2)

where

ΣXX =1

TL

TL∑τL=1

X(τL)X(τL)′ and sXy =

1

TL

TL∑τL=1

X(τL)y(τL). (D.3)

To analyze the asymptotic property ofθ, defineΣXX andΣXX as follows.

[xL(τL)

xH(τL)

]︸ ︷︷ ︸=X1(τL)

=

[1 0 0 0

0 1/3 1/3 1/3

]︸ ︷︷ ︸

=W

xL(τL)

xH(τL, 1)

xH(τL, 2)

xH(τL, 3)

︸ ︷︷ ︸

=X1(τL)

. (D.4)

1

X1(τL)

X2(τL)

︸ ︷︷ ︸

=X(τL)

=

1 01×4 01×k2

02×1 W 02×k2

0k2×1 0k2×4 Ik2

︸ ︷︷ ︸

=WX

1

X1(τL)

X2(τL)

︸ ︷︷ ︸

=X(τL)

. (D.5)

31

Equation (D.5) implies that

ΣXX = WXΣXXW ′X . (D.6)

SinceΣXXp→ ΣXX , we have thatΣXX

p→ ΣXX , whereΣXX = E [X(τL)X(τL)′] = WXΣXXW ′

X .

Substitute the DGP (A.1) into (D.2):

√TLRθ =

√TLR Σ

−1

XX × 1

TL

TL∑τL=1

X(τL) {X(τL)′θ0 + ϵL(τL)}

=√TLR Σ

−1

XX

{1

TL

TL∑τL=1

X(τL)X(τL)′

}θ0 +R Σ

−1

XX

{1√TL

TL∑τL=1

X(τL)ϵL(τL)

}

=√TLR Σ

−1

XXWXΣXXθ0︸ ︷︷ ︸≡δTL

+R Σ−1

XXWX

{1√TL

TL∑τL=1

X(τL)ϵL(τL)

}.

(D.7)

The central limit theorem can be applied to the second term of the rightmost side of (D.7):

R Σ−1

XXWX

1√TL

TL∑τL=1

X(τL)ϵL(τL)

d→ N(0, σ2),

whereσ2 = σ2LRΣ−1

XXR′. Hence, the convergence of√TLRθ depends entirely on the convergence of the first

term of the rightmost side of (D.7),δTL.

We can show thatδTL = 0 under strong non-illusionHs0 : b1 = b2 = b3 = −a/3. Note that underHs

0

θ0 ≡

a0

a

b1

b2

b3

θ0,2

=

a0

a

−a/3

−a/3

−a/3

θ0,2

=

1 0 0 01×k2

0 1 0 01×k2

0 0 1/3 01×k2

0 0 1/3 01×k2

0 0 1/3 01×k2

0k2×1 0k2×1 0k2×1 Ik2

a0

a

−a

θ0,2

︸ ︷︷ ︸

≡a

= W ′Xa. (D.8)

Substitute (D.8) into the definition ofδTLto get

δTL =√TLR Σ

−1

XXWXΣXXW ′Xa =

√TLR Σ

−1

XXΣXXa =√

TLRa = 0, (D.9)

where the second equality follows from (D.6). The last equality holds sinceR = [0, 1, 1,01×k2] anda =

[a0, a,−a,θ′0,2]

′ by construction. Thus, the low frequency Wald statistic (LF statistic) converges toχ21 under

strong non-illusionHs0 :

W = TLθ′R′ σ−2Rθ =

(√TLσ

−1Rθ)′ (√

TLσ−1Rθ

)p→ χ2

1. (D.10)

E Proof of Theorem 2.4

ConsiderHs1 : Rsθ0 = cs with cs1 = cs2 = cs3 ≡ cs = 0. In this case we have from (D.8) thatθ0 =

W ′Xa+W ′

Xc, wherec = [0, 0, cs,01×k2 ]′. Hence,δTL defined in (D.7) can be simplified:

δTL=

√TLR Σ

−1

XXWXΣXX (W ′Xa+W ′

Xc) =√TLR Σ

−1

XXΣXXc =√TLRc =

√TLcs,

32

where the second equality follows from (D.6) and (D.9). Sincecs = 0, δTLdiverges to either∞ or−∞. Thus, the

LF statistic diverges underHs1 with cs1 = cs2 = cs3 ≡ cs = 0.

F Proof of Theorem 3.1

Consider the DGP (A.1) and the mixed frequency model (B.1). Under the local alternative hypothesisHsla :

Rsθ0 = (1/√TL)νs, (B.2) and (B.3) are replaced with

√TLRsθ = νs +RsΣ

−1XX × 1√

TL

TL∑τL=1

X(τL)ϵL(τL)d→ N(νs, Σs). (F.1)

Following the same steps as in Appendix B, we have thatWsd→ χ2

3(ν′sΣ

−1s νs).

G Proof of Theorem 3.2

We have by construction thatRw = (1/3)×ι′3Rs, whereι3 = [1, 1, 1]′. Hence,√TLRwθ = 1

3ι′3(√TLRsθ). Us-

ing (F.1), we have that√TLRwθ

d→ N((1/3)ι′3νs, Σw) under the local alternative hypothesisHsla, whereΣw =

(1/9)ι′3Σsι3. Following the same steps as in Appendix B, we conclude thatWwd→ χ2

1((1/9)ν′sι3Σ

−1w ι′3νs).

H Proof of Theorem 3.3

UnderHsla : Rsθ0 = (1/

√TL)νs, we have from (D.8) thatθ0 = W ′

Xa + (1/√TL) × νs, whereνs = [0, 0,

νs1/3, νs2/3, νs3/3, 01×k2 ]′. Hence, the key quantityδTL in (D.7) becomes

δTL=

√TLR Σ

−1

XXWXΣXX

(W ′

Xa+1√TL

νs

)= R Σ

−1

XXWXΣXXνs

p→ RΣ−1XXWXΣXXνs ≡ δs.

Hence,√TLRθ

d→ N(δs, σ2) and consequentlyW

d→ χ21(δ

2s/σ

2).

33