ISSN (Print) 0473-453X

Discussion Paper No. 1125

ISSN (Online) 2435-0982

THE DYNAMICS OF THE HOUSE
PRICE-TO-INCOME RATIO:THEORY AND EVIDENCE

Charles Ka Yui Leung
Edward Chi Ho Tang

March 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Dynamics of the House Price-to-Income Ratio: Theory and EvidenceCharles Ka Yui LEUNG, City University of Hong Kong
Edward Chi Ho TANG, Hong Kong Shue Yan University
This version: March 2021
Abstract
The house price-to-income ratio (PIR) is widely used as an affordability indicator. This paper
complements the cross-sectionally focused literature by proposing a tractable model for the PIR
dynamics. Our model predicts that the PIR is very persistent and is correlated to the lagged aggregate
output. Cross-country analysis confirms this prediction and provides evidence for a long-term, positiveand significant relationship between PIR and aggregate production. Our results hint at the constructionof an early warning system for housing market mispricing. Our tractable formulation of a stochastic
money growth rule may carry independent research interest.
JEL Classification Number: E30, O40, R30
Keywords: housing affordability, output dynamics, endogenous house price, wage rigidity, monetary
policy rule.
Acknowledgment:This project is motivated by a conversation with John Quigley, whose generosity, insights andwisdom will always be missed. The authors are grateful to Wing Leong Teo for his participationin an earlier stage, and Nan-Kuang Chen and Fred Kwan, for many discussions. The authors also
want to thank (alphabetical order) Ken Chan, Tommy Chan, Kuang-Liang Chang, Mario Crucini,
Kristopher Gerardi, Richard Green, Cheuk-Yin Ho, Mingming Jiang, Ronald Jones, Yuichiro
Kawaguchi, Calvin Lin, Joe Ng, Shin-Kun Peng, Daniel Preve, Zhigang Tao, Byron Tsang, Ping
Wang, Jing Wu, Zan Yang, Matthew Yiu, Eden Yu, Ziqi Zheng, seminars participants of
AREUEA International Conference, Asian Econometric Society meeting, CEANA meeting,Conference on Real Estate and Financial Stability, Economic Linkage Conference, Midwest
Macro Conference, City University of Hong Kong, Tsinghua University, especially two
anonymous referees as well as the financial support from the City University of Hong Kong, theJoint Usage/Research Center at ISER, Osaka University and Grant-in-aid for Research Activity,
Japan Society for the Promotion of Science (15H05728, 20H05631), and the International Joint
Research Promotion Program of Osaka University. Part of the research is conducted when Leung
visited ISER, Osaka University, and Hoover Institution. Their hospitality is gratefully
acknowledged. The usual disclaimer applies.
Correspondence:
Leung: Department of Economics and Finance, City University of Hong Kong, Kowloon Tong, Hong Kong,
email: kycleung@cityu.edu.hk; Tang: Department of Economics and Finance, Hong Kong Shue Yan
University, Hong Kong, email: chtang@hksyu.edu.

1Introduction

The importance of House Price-to-Income-Ratio (PIR), which is also called theHouse Price-to-Earning Ratio, can hardly be overstated. For instance, Demographia (2016) reports that “the Median Multiple (a house price-to-income
ratio) is widely used for evaluating urban markets, and has been recommended
by the World Bank and the United Nations and is used by the Joint Center for Housing Studies, Harvard University. Similar house price-to-income ratios... are used to compare aﬀordability between markets by the Organization
for Economic Cooperation and Development, the International Monetary Fund,international credit rating services, media outlets (such as The Economist) andothers.” Yet, despite its importance, formal modeling of PIR is relatively rare.1
This paper constructs a simple dynamic, stochastic general equilibrium model(DSGE), where we can analytically link the PIR to the output growth. Sinceboth variables are observable, we also bring this testable implication of themodel to the data.
An essential application of the model developed here is housing aﬀordability.
PIR is one of the most commonly used measures of "aﬀordability."2 The reasonsare clear. The data requirement for PIR is minimal, and the ratio easy to
calculate and interpret. Thus, PIR is often computed and compared acrosscountries or regions, or cities within the same state. In addition to the crosssectional comparison, it is sometimes calculated for a fixed area or country
across diﬀerent periods. It indicates whether housing aﬀordability improves (ordeteriorates) over time. Since the housing aﬀordability literature mostly takes areduced-form approach and focuses on cross-sectional relationships, this paper
1 See Chen and Cheng (2017), Leung (2004), Leung and Ng (2019), among others, for areview of the literature.
2 In the literature, “aﬀordability” may carry diﬀerent meanings in diﬀerent contexts andcan be measured diﬀerently. Among others, see Hulchanski (1995), Quigley and Raphael(2004).

can complement the literature in the following ways. First, we provide a simple
DSGE model and derive the equilibrium level of PIR in the model. Since prices(e.g., house prices, wages) and quantities (e.g., physical capital stock, housing
stock) are endogenous in DSGE models, the equilibrium PIR is naturally tied
to the movement of "economic fundamentals." As a result, the dynamics of PIR
become predictable. This model also enables us to address concerns such as the"deterioration of housing aﬀordability" (DHA) in diﬀerent countries.3
Second, we study how rigid wages may aﬀect housing aﬀordability. Someauthors argue that the existence of wage rigidity, which some empirical works
confirmed, would worsen housing aﬀordability. It is because the wage cannot
respond fast enough with the flexible house price. Our model can allow for both
flexible and rigid wages.
Third, some authors claim that monetary policies would aﬀect the housing
market.4 We propose a formulation of the monetary policy in this paper. The
money growth rate is no longer a constant but instead a function of the previous
money growth rate and other macroeconomic variables.5 It generalizes someprevious work and may carry an independent research interest.
Fourth, we confront our theoretical model with data. More specifically, weshow a robust relationship between the PIR and real GDP in a dynamic panel
data setting. We also perform a panel cointegration test across countries. Thus,
it provides not only an empirical validation but also a suggestion for future
research directions. In the past, housing aﬀordability studies tend to focus on
the cross-sectional diﬀerence. The dynamics of housing aﬀordability, on theother hand, maybe under-explored. This study complements the literature byconsidering PIR dynamics. More specifically, we establish a coherent theoretical
3 The word “deterioration” suggests a comparison across diﬀerent periods.

Hence, a dynamic model may be more appropriate for the analysis of DHA.
4 The literature is too large to be reviewed here. Among others, see Jordà et al. (2015).
5 The previous literature, such as Friedman (1969), focuses on the case of a constant money
growth rate.

framework where the PIR changes over time. The model finds support fromcross-country data. It complements the previous work, such as Chen and Cheng(2017), which focuses on the United States’ case. It may also suggest that a
dynamic equilibrium perspective on housing policy could provide additional
insights.
This paper is related to several strands of the literature. For instance, academic researchers, media, and policy share the concerns of housing aﬀordability and its potential deterioration (DHA) (Asal, 2019; Australian Government,
2021; CBC News, 2015; McGee, 2009; Moody, 2015; National Housing Conference, 2012; National Housing Federation, 2012; RBC, 2015; Walks, 2014). Thereare concerns for housing aﬀordability in Australia, Canada, Sweden, U.K., andU.S.. Edvinsson et al. (2021) construct the real estate price index for Sweden
from 1818 to 2018. They show that the current house price cycles in Sweden
share some similarities with her history and suggest that government intervention may be useful. Second, there is extensive literature on wage rigidity(Barattieri et al., 2014; Bils et al., 2013; Dickens et al., 2007). Third, there
are studies on the empirical determinants of housing prices (Oikarinen, 2009;
Stadelmann, 2010; Agnello and Schuknecht, 2011). While this paper is built on
their insights, it has a very diﬀerent focus. Rather than searching for the empirical determinants of housing price, which may include aggregate output, labor
wage, monetary policy, etc., this paper attempts to relate the PIR and the production in a dynamic equilibrium model when all these variables are determined
endogenously. Furthermore, since our paper establishes a panel cointegration
relationship between PIR and macroeconomic variables, we can measure thelong-run relationship’s short-run deviations. If the short-run deviations are persistent or even growing over time, they should alert both academic researchersand policy-makers. In other words, it might be a preliminary step towards theconstruction of an early warning system for the housing market mispricing.6
The organization of this paper is simple. We first study a tractable dynamic,
stochastic general equilibrium (DSGE) model with flexible wage and examine
how the house price-to-wage ratio will evolve in the model economy. We thencompare the case with short-term sticky wages. We then confront the model
predictions concerning the PIR and output dynamics with data. The last section
concludes with all proofs reserved in the appendix.

2

A Benchmark Model

This section will present a simple DSGE model. We will first provide an informal description, followed by the introduction of a mathematical model. Loosely
speaking, the model developed here is a combination of Greenwood and Hercowitz (1991) and Benassy (1995), and hence a brief explanation will be suﬃcient.7 Time is discrete, and the horizon is infinite. The population is assumed
to be constant to simplify the exposition, and one can comfortably relax thisassumption. There are several goods in the economy: the non-durable consumption goods  , the business (or physical) capital  , residential capital(or housing)   The representative agent derives utility from the level of nondurable consumption, the amount of housing, and also the number of labor 8hours  and the amount of real cash balance 
The government prints
 .

nominal money  according to a specific money supply rule, which will be
6 Early warning system (EWS) has been extensively discussed in the context of the banking
crisis and financial crisis. See Bussière and Fratzscher (2006), Cumperayot and Kouwenberg(2013), Lang and Schmidt (2016), and the reference therein.
7 See also Leung (2007, 2014) for related studies.
8 The money-in-utility-function formulation can be easily justified as the transaction demand for money. There is extensive literature on this, and interested readers could consult
Croushore (1993), Feenstra (1986) for more details.

explained later. Firms combine labor and business capital to produce output
 . All agents in the model economy maximize their utility or profit.
Our formal description of the model begins with the household side. The
representative household in the model that maximizes the expected value of the
discounted sum of utility:max 0

∞
X
=0



µ

   +    




¶



(1)

where  is the amount of non-durable consumption,  is the amount of housing
stock,  is the amount of housing that are rented,  is the amount of laborhours (or eﬀorts) devoted in goods production,  is the real money balance,as  denotes the nominal money balance and  is the general price level. In
this paper, a simple utility functional form is assumed,
µ
¶



    +    = ln  +  1 ln ( +  ) +  2 ln (1 −  ) +  3 ln


(2)
   0,  = 1 2 3 are parameters governing the relative importance of housing,leisure, and money holding in the utility function. The maximization problemof the representative household is subject to several constraints. First, boththe business capital  and housing stock  are durable and can only adjust
gradually.9 The following equations capture this observation that the future
amount of stock (whether the business capital or housing) depends positively
on the amount of current stock level and investment,+1 = 1− 

+1 = ( +  )

1−

(3)




(4)

9 There is a vast literature on the gradual adjustment of capital stock and housing stock.
Among others, see Cooley (1995), Hanushek and Quigley (1979).

where      are the depreciation rates of business capital and housing stock,0 ≤      ≤ 1,  ,  are the investment in business capital and housing.
Second, this formulation also captures the idea that holding fixed the amountof existing stock, the marginal rate of return of investment on the future capitalis diminishing, which can also be interpreted as a form of adjustment cost.10
Notice also that in (4), the amount of housing stock purchased from the secondary market,   can also influence the amount of future housing stock.
Thus, this formulation also captures the idea that one can accumulate housing
stock through investment and direct purchase from the market. On top of therestrictions of (3) and (4), the household is also subject to the usual budget
constraint,  +   + −1≥  +  +  +   +   +



(5)

where  is the real rental rate of capital,  is the real wage rate,  and are the real price and real rental rate of housing, respectively. Following
Benassy (1995),  a stochastic multiplicative monetary shock.
This dynamic optimization problem above can be solved using the Dynamic
Programming method,



µ
µ
¶
µ
¶
¶
−1


   = max     +    +  +1  +1 



+1

subject to (3), (4), (5). The first order conditions are easy to derive and details
are provided in the appendix.
To be compatible with the literature, the rest of the model is straightforward.
1 0 In practice, the adjustment cost would include all the growth management policies, physical constraints such as cliﬀs on real estate development, and the regulations on permits. See
Leung and Teo (2011), Saiz (2010) for more discussion.

There is an aggregate production technology, which exhibits constant returns
to scale in capital and labor, =   1−



(6)

where  is the capital share, 0    1, and  is the productivity shock The
logarithm of the productivity shock follows an AR (1) process, = −1 +  (7)

where  = ln  ,  measures the persistence of the productivity shock, 0 ≤  ≤1. To further simplify the exposition, we assume that ( ) = 0,  ( ) = 2 , which is a constant, and (  ) = 0 whenever  6= . With competitive factor markets, the real rental rate and wage rate will be equalized to thecorresponding marginal product,


= 



(8)



= (1 − ) 



(9)

 =
 =

And it is easy to see that the economic profit is zero in this model economy,Π ≡  −   −   = 0(10)Following Benassy (2002), we assume that the monetary stock’s growth rate
is a random variable. The household will hold all the money printed by thegovernment. Mathematically, it means that =  −1 (11)We will provide details on the monetary growth rate  in a later section. At
this point, we can take it as given.11 Now, combining (6), (8), (9), (10) and(11), (5) can be simplified as: =  +  +  (12)

To solve the model, we need to impose some market-clearing conditions. Following Lucas (1978), there is no net trade among identical households, whether
in the sale market or rental market of housing. It follows that =  = 0(13)

For future reference, we use small letter to denote the natural log of capitalletter variables. For instance,  = ln  ,  = ln  ,  = ln  , etc. The
following proposition summarizes the equilibrium dynamics of the model (theproof can be found in the appendix).
Proposition 1 With flexible prices and wages, the dynamic system depends on
the joint dynamics of output and capital stock, which can be summarized by thefollowing vector dynamics equation,−→→→  = 0 + 1 − −1 + −
 

(14)

→
where 0 , 1 are matrices of constant, the transpose of the vector −  is (   ),
→
→
and −
  represents a vector of shock. In addition, we can show that −  is seriallycorrelated,→−→ − +  =  −

−1
X

→
 − 
 −

(15)

=0

1 1 Since all prices are flexible in this section, monetary policy will be neutral. Among others,
see Walsh (2010). In the following section, with the sticky wage, monetary policy will not be
neutral, and we will explicitly formulate the monetary policy.

¡→0
¢
¡→0 −
¢
where  −
 − = (0, 0),  −  →
  = 0 whenever  6= .12

Besides, we can also study the house price dynamics in this model.

Proposition 2 In this model, the house price positively correlates to the output
and negatively to the housing stock. Formally, it is characterized by the following
equation, =  +  −  

(16)

Notice that we have already derived the output level, the real wage, andthe model’s house price equation. We now focus on the real wage rate-to-house
price ratio, which is often used as an "aﬀordability index."13
Proposition 3 The real wage-to-house price ratio (in log and in real terms)
can be expressed as the follows,

ln

µ




¶

=  − =  +  

(17)

where  is a constant.
Notice that in log form, the commonly used house price-to-income ratio(PIR) is simply ln

³




´

=  −  = − ( −  ). Thus, our real wage-

to-house price ratio will inform us directly about the widely discussed PIR.
For mathematical convenience, we would proceed with the real wage-to-house
price ratio.14 Several observations are immediate from (17). First, even in a→→
1 2 Throughout this paper, we use − 0 to represent the transpose of the vector −.

1 3 The real wage rate-to-house price ratio measures how many hours (or any time units) oflabor a household needs to give up to exchange for a housing unit. Sometimes people would
use the reciprocal of it, i.e., the house price-to-wage rate ratio.
1 4 Some researchers argue that a more appropriate measure would be the wage income-tohouse price ratio, i.e.,    , rather than the wage rate-to-house price ratio   . As
we have shown in the appendix, the labor hours are constant at the flexible wage equilibrium.
Hence, the two ratios would only diﬀer by a fixed factor. We will revisit the diﬀerence between
the two ratios when the wage is rigid in the short run.

stationary environment, the real wage-house price is not constant but would
vary according to the housing stock. Second, if the stock of housing suddenly
decreases (say, due to unexpected natural disasters), the real wage-to-house
price ratio will also decrease. The intuition is simple. If some disasters destroy
some housing stock, agents need to be re-allocated to existing shelters. However,
housing stock cannot adjust soon enough to meet the demand, which increases
the house price. Other things being equal, the real wage-to-house price ratio
will drop.
In principle, one can estimate equations such as (14), (16), or (17) directly.
In practice, variables such as the business capital stock  and housing stock 
are not available in some countries. Even if the data on capital stock and housing
stock is available, they adjust slowly and are typically measured infrequently.
Therefore, we need to derive other testable implications of the model. The
following proposition takes a step in this direction.
Proposition 4 The real wage-to-house price ratio is related to lagged output
level of the economy,

 −  = 0 +  

−1
X
=0

(1 −   ) −1− + (1 −   )−1 0 

(18)

where 0 is a constant, 0 is the amount of initial housing stock in the model
economy.

The intuition behind this proposition is simple. Capital accumulation andhouse construction are both endogenous in the model. Therefore, the consumerworkers optimally allocate the resource in the two activities. Since house priceis related to the construction activities on the one hand, and the output andequilibrium wage depends on the amount of physical capital, on the other
hand, the PIR and GDP are naturally related.

Notice that as  −→ ∞,(1 −   )−1 0 −→ 0 as 0  (1 −   )  1 Thus, the importance of the initialstock of housing diminishes over time, and the real wage-to-house price ratio
will depend on the series of lagged output, {−1− }. In particular, an increasein the previous period’s real output will increase the real wage-to-house pricein some subsequent periods. The intuition is simple. Higher levels of previous
periods’ real output will lead to higher demand for housing and a higher levelof business capital, and the latter tends to lift the wage. Given the assumptions
made in this model economy, the wage eﬀect dominates the housing demand
eﬀect and the wage-to-house price ratio increases. We will further examine thistheoretical prediction’s robustness in the following section and then will confront
the theory with data.

3

The case with short-term rigid wages

The previous section studies a model with perfectly flexible prices and wages,
and hence the monetary policy is neutral. Recent evidence, however, suggeststhat nominal wages are sticky. For instance, based on the micro-evidence fromthe 1970s to the early 2000s in twelve countries, Dickens et al. (2007) conclude
that the nominal wages are indeed sticky rather than flexible.15 The diﬀerential
flexibility between the house price and the wage may influence our conclusionin the present context. Therefore, we introduce sticky wages in this section. To
facilitate the comparison, the model economy we would consider is the same asin the previous section, except for the short-term nominal rigidity in wages. As
shown in Walsh (2010), monetary policy would aﬀect real economic activities
if wages are sticky. Here our focus is on how the equilibrium dynamics of PIR
could interact with the monetary policy. We adopt a tractable formulation
1 5 The twelve countries are Austria, Belgium, Denmark, Finland, France, Germany, Italy,
Norway, Portugal, Sweden, Switzerland, and the United Kingdom.

of Benassy (2002), assuming that wages are set one period in advance. Morespecifically, the contract wage is set equal to the expected value of the Walrasian
wage (−1 ∗ ), and hence the labor market will clear ex-ante. Mathematically,
the nominal wage becomes, = −1 ∗ =  + −1  

(19)

We will provide more discussion on the details of the expected value of money
supply −1  later.16 We will proceed as if −1  is known to the agent.
With short-term nominal wage rigidity, the real wage in the economy becomes, =  −  = −1 ∗ −  =  + −1  −  

(20)

After all the shocks realize, the firms will hire labor at the pre-committed wage.
Profit maximization will imply that the factor returns (wage and capital rentalrate) are still equalized to the corresponding marginal products, and hence (8),(9) are still valid. The following proposition dictates the equilibrium labor
supply under this slight change in the economic environment,
Proposition 5 If (19) holds, the equilibrium labor supply under short-term
wage rigidity depends on the "forecast error" in monetary supply, =  +  (21)

1 6 Notice that this formulation implies that the nominal wage rigidity is symmetric. Abbritti

and Fahr (2013) argue that the nominal wage is downward rigid but upward flexible. Babecký
et al. (2012) find that when employers do not cut "wages," they would cut other benefits
to reduce labor costs. Hence, from the employee perspective, the "net income" is reduced.
Hofmann et al. (2012) find that the US wage indexation degree varies across diﬀerent periods.
Elsby and Solon (2019) study the microdata across countries and find that a nominal wage
cut is typical annually. Thus, given the diverse opinions on downward wage rigidity, it might
not be a bad idea to study flexible and rigid wages and show how they might aﬀect the house
price-to-income ratio dynamics.

where  is the forecast error term in money supply at time , ≡  − −1  

(22)

Furthermore, we can show that the joint dynamics of output and capital stock
is summarized by the following vector dynamics equation,
→
−
→
−
→
 −1 +    = 0 + 1 −(23)→
where 0 , 1 are matrices of constant, the transpose of the vector −  is (   ),
→
−
→
−
and  represents a vector of shock. And  is serially correlated.
Notice that the form of (23) is identical to (14). The only diﬀerence is thatthe forecast error term in money supply at time ,  , is a part of the vector of→−shocks,   It is reasonable to expect that the forecast error term  depends on
how the monetary policy is conducted. Thus, to fully understand the dynamics
of the system, we must formally introduce the monetary policy.

4

Monetary policy

In the literature, a famous formulation of the monetary policy is to adopt a
version of the Taylor rule (e.g., Koenig et al., 2012; Walsh, 2010). While it has
many merits, a drawback is that an analytical solution is typically unavailable,
and the model would need to be solved numerically. To keep the model tractable,
we formulate the monetary policy as a money growth rule.17 On the other
hand, a stochastic money growth policy rule (SMG) seems to be under-explored.
1 7 In the literature, researchers discussed whether the Taylor rule and a constant money

growth rule suggested by Friedman (1969) are equivalent. For instance, see Carlstrom andFuerst (1995), Schabert (2003), Auray and Feve (2008), among others. See also Nelson (2008)
for a review.

Drawing lessons from the literature, our formulation of SMG is analogous to the
Taylor rule. In particular, we assume that the growth rate of the nominal money
supply  depends on its lag, the inflation rate  −1 , and the output level
 , relative to the corresponding steady-state values. Formally, it means that
µ




¶

=

µ

−1


¶ µ

 −1


¶ µ




¶

 

(24)

where    are the steady-state level of    −1   respectively, with  , ,  being the policy parameters. For instance, if the inflation rate  −1
deviates from the steady-state inflation rate , the money growth rate would
respond, and the parameter  governs how much the money supply should react
to the “excessive” inflation rate. Similarly, if the aggregate output level  falls
short of its steady-state level  , the money growth rate  might increase relative
to its steady-state level . The parameter  captures how sensitive is themonetary growth rate to the deviation of the aggregate output from its steadystate level Clearly, if  =  =  = 0 the monetary growth rate is an i.i.d.
process. For simplicity, we assume that the “innovation term” of the monetary
policy  has a zero mean and constant volatility, ( ) = 0,  ( ) =  2 .
The innovation terms are also uncorrelated over time, (  ) = 0 whenever 6= . Furthermore, we assume that { } and { } are independent. A meritof this formulation is that the model remains very tractable. Another meritof using SMG is that the gross monetary growth rate  =  −1 is, bydefinition, positive. Hence, we do not worry about the zero lower bound (ZLB)
of the nominal interest rate.18 Like other results, the following lemma is provedin the appendix.
Lemma 6 Given (24) holds, the forecast error of money supply in (22),  , is
1 8 Recently, Hirose and Inoue (2016) show that if we ignore the ZLB in an estimated DSGE
model with the usual interest rate-rule-type monetary policy, other parameters may be estimated with bias.

shown to be an “weighted sum” of the forecast error of productivity, ( − −1  ) and the monetary innovation term  ,≡  − −1 =   ( − −1  ) +   =    +    (25)

where   ,   are constant.
Equipped with these results, we can prove our main result, which related thereal wage-to-house price ratio to the output dynamics and other random terms
in this economy.
Proposition 7 If (19) and (24) hold, the real wage-to-house price ratio (WPR)
is related to lagged output level of the economy, as well as the forecast error of
productivity shock  , and that of money supply   −  = 0 +  

−1
X
=0

(1 −   ) −1− + (1 −   )−1 0 − b (26)

where 0 is a constant, 0 is the amount of initial housing stock in the model
economy, b is a stochastic residual term.

Clearly, (18) and (26) are very similar. In other words, the wage rigidity

does not significantly alter the PIR dynamics in this setup. To put it anotherway, the PIR does not provide “extra information” about the housing market;
it is merely a “summary statistics” of the aggregate output in previous periods.
In principle, as both PIR and aggregate output are observable in the data, wewill seek empirical confirmation of (26). In practice, however, the right-handside of (26) contains the whole series of past output { }−1
=0 , and it would pose achallenge in applied work. We, therefore, derive the following proposition from(26), which are much easier to implement in empirical works, as we will further
discuss it in the subsequent empirical section.
Proposition 8 The following equations characterize the real wage-to-house priceratio (WPR):(a) The variance of the WPR is a “weighted sum” of the variance of output
and some residual term, ( −  ) = e  ( ) +  (b ) (27)

where e  0 depends on  and is non-stochastic.

(b) The dynamics of the WPR is a “weighted average” of its lagged value

and output level:
0 +   + (1 −  ) ( −  ) + [g ] (+1 − +1 ) = g
 



+1

(28)

0 is a constant term, g is a stochastic term. It can be shown that
where g
+1

g
+1 is serially correlated, i.e. E[g
+1 e ] 6= 0

Notice that the lagged output coeﬃcient is    which is positive but small.
On the other hand, the coeﬃcient of lagged WPR is (1 −   )  which is lower
than but close to unity. The dynamics of WPR should be persistent.
Some researchers argue that a more appropriate measure would be the wage
income-to-house price ratio, i.e. (   )  rather than the wage rate-to-house
price ratio (  ). The following result shows that the dynamics of the two
ratios are very similar in log form.
Corollary 9 In log form, the wage income-to-house price ratio is very similarto the wage rate-to-house price ratio,

+1 + +1 − +1= 00 +  

−1
X
=0

(1 −   ) −1− + (1 −   )−1 0= e00 +    + (1 −   ) ( +  −  ) (29)Therefore, our empirical work’s choice variable will depend on data availability, present in the next section.

5

Empirical Evidence

The theoretical analysis has provided several testable implications, and thissection verifies them with international data. Since the model presumes a wellfunctioned capital market, it may be more appropriate to employ data from
more developed economies. Moreover, time-series data on house prices from developed countries are more accessible. Thus, our data on real GDP (in millions
of US dollars) and wage index are collected from OECD.stat, while the housing price indices are obtained from the Bank of International Settlements.19
Altogether, there are 15 countries in our study, including Australia, Canada,
Denmark, Finland, France, Germany, Italy, Japan, Netherlands, New Zealand,
Norway, Spain, Sweden, UK, and the US.20 Our data is in quarterly frequency
and covers the period from 1997Q1 — 2018Q4. Following Cooley (1995), theseasonal component is removed from the time series. Our choice of the sampling period balances the desire to maximize the number of countries included
and the discipline to include countries that can meet some econometric test
requirements. Figure 1 provides a visualization of the evolution of PIR across
1 9 The data can be downloaded from https://www.bis.org/
2 0 In an earlier version, we also include Ireland. However, we find that the results dramatically change once Ireland is removed. Hence, following the suggestion of a referee, we remove
Ireland from our sample.

OECD countries over our sampling period. Several observations are in order.
While there are outliers over the years, they are not that many. The median
PIR fluctuates over the years in a relatively smooth manner. The range of PIR
between the 25th and 75th percentile tends to increase over time, which seems
consistent with OECD countries’ diverse economic performance during and after
the Great Recession.
(Figure 1, 2, Table 1 about here)
For testable implications, we begin with (27). Notice that the stochastic
residual term b is not directly observable. Thus, other things being equal, (27)predicts that the variance of wage-to-house price ratio and the variance of output

are positively correlated.21 Since the original time series may not be stationary,
we employ the Hodrick-Prescott (HP) filter to extract the cyclical components
for regression.22 Figure 2 visually suggests that a country with a volatile output
also tends to have a less volatile PIR. However, the coeﬃcient is not significant.
We then compute the correlation between the output variance and PIR variancefor the full sample and many sub-sample, each with one country removed. The
idea is to detect whether any outlier drives the correlation. Table 1 reports theresult. The correlation coeﬃcients are all negative, as predicted by the theory,
but none is statistically significant. The insignificance result may be due to oursmall sample size, or some ECB policies have distorted the real estate markets
after the economic crises (Acharya et al., 2019).
Now, we turn to the relationship between PIR and output dynamics. While
the two formulations, (26) and (28), are equivalent mathematically, it is better
to use the latter. Testing (26) directly would introduce many lags on the right2 1 We have conducted further analysis on the residual terms, based on our estimation of
the production function and monetary policy function across countries. Those results will beavailable upon request.
2 2 In an earlier version, we used the Christiano-Fitzgerald bandpass filter. The results are
similar. See Christiano and Fitzgerald (2003), Hodrick and Prescott (1997) for more details.

hand, which makes the estimation diﬃcult given the relatively short time series.
On the other hand, equation (28) expresses the wage-to-house price ratio as aweighted average of its lag and lagged output level. Its data requirement is lessstringent, and therefore, we prefer to estimate (28).
We consider the dynamic panel data approach (DPDA) as an appropriate
econometric methodology. It enables us to identify and measure eﬀects thatare not detectable in pure cross-section or pure time-series data. It also allows
us to control for individual heterogeneity. As equation (28) contains a lagged
dependent variable as an explanatory variable, strict exogeneity of the regressors
no longer holds (Hsiao, 2015). Hence, it will be proper to apply the dynamic
panel data model.23 Notice also that equation (28) is directly derived from oursimple DSGE model. Hence, our dynamic panel regression can be interpreted
as a "structural estimation" (of one aspect of a dynamical system). However, it
can also be interpreted as a "reduced-form estimation," where the PIR is related
to its past value and economic fundamental, proxied by the GDP.24
Table 2 confirms the theoretical predictions. It shows that the coeﬃcients oflagged output and lagged dependent variables are found to be positive and significant. Consistent with the theory, the coeﬃcient of the lagged PIR is smaller
than but close to unity. On the other hand, the lagged output coeﬃcient, which
is supposed to be   , is much lower. This result holds for the full sample andmost of the sub-samples, each with one being removed. Moreover, it is worthy
to note that DPDA requires instruments. Consistently, J-statistics probabilities are always between 0.34 and 0.46, suggesting that the instruments are all
2 3 Among others, see Arellano and Bond (1991), Baltagi (2013) for more discussion.
2 4 Also, to keep our model tractable, we abstract away from many institutional details,
varying across countries. They include whether public housing units and housing vouchers
are provided, whether (and how much) mortgage payments are tax-deductible, etc. Amongothers, see Green (2014), Malpezzi (1999b).

valid.25(Table 2 about here)

Finally, we would like to examine a long-run relationship between the wageto-house price ratio and the real GDP (both in log form). The justification
is clear. If we can identify a long-run relationship between the wage-to-house
price ratio and GDP, we can also detect "short-run deviations from the long-run
relationship." Such deviations might be used as one of the proxies to measure
whether the whole housing market "deviates" from its long-run situation.26
To examine whether a long-run relationship exists between the wage-to-house
price ratio and GDP (both in logarithm), we proceed in several steps. First, we
check the stationarity of the series. As suggested by Cheng and Kwan (2000),
Kwan (2007), performing a panel unit root test is more powerful than the unit
root test for individual time series. All the panel unit root tests suggest that the
two log series are indeed I(1).27 Second, we proceed to the panel cointegration
test. It adopts three types of panel cointegration tests, including Pedroni (1999,2004), Kao (1999), and Johansen Fisher Panel Cointegration Test (Maddala
2 5 There is an additional issue here. The theory predicts that the sum of coeﬃcients forthe lag ( − ) term and  term should sum to unity. Table 2 shows that in the full sample(i.e., when all countries are included), the coeﬃcients’ sum is equal to (0.94+0.11), which islarger than unity. However, when we turn to diﬀerent sub-samples, we might have a diﬀerent
conclusion. While the lag ( − ) coeﬃcients are all close to unity, the  term’s coeﬃcientsvary across diﬀerent sub-samples. For instance, when we exclude Canada, the coeﬃcient for
the lag ( − ) term is 0.93, but the coeﬃcient of the  term is insignificant, and hence thesum is 0.93, which is less than unity. A similar pattern is found when Finland, New Zealand,or the U.S. is excluded from the sample.
We then turn to Table 3a, when the size of government is controlled for. Again, thecoeﬃcients’ sum is equal to (0.94+0.16), which is larger than unity in the full sample. The
coeﬃcients of the lag ( − ) are close to unity in all sub-samples. However, when Canada, or
New Zealand, or the U.S. is excluded from the sample, the  term’s coeﬃcient is insignificant,and hence the sum of the coeﬃcients is less than unity. Thus, we conclude that whether thesum of the two coeﬃcients is larger than or less than unity might depend on the inclusion ofa few countries in the sample.
2 6 Clearly, this is the idea behind the “error correction model.” Among others, see Engle
and Granger (1987), Malpezzi (1999a).
2 7 See the appendix for details.

and Wu, 1999). The majority of the result suggests a long-run relationshipbetween ( −  ) and  . Third, it estimates a long-run relationship using thegroup-mean fully-modified OLS method (FMOLS), group-mean dynamic OLS(DOLS), and Static OLS method (SOLS). FMOLS employs a semi-parametriccorrection to eliminate the problems caused by the long-run correlation between
the cointegrating equation and stochastic regressors’ innovation. DOLS involves
augmenting the cointegrating regression with leads and lags for the change ofvariables. The resulting cointegrating equation error term is orthogonal to theentire history of the stochastic regressor innovations. When the leads and lags
are set to none, it becomes SOLS. Since SOLS produces biased, super-consistentestimates (Tsionas, 2019), it is used as a supplement.28 The coeﬃcients of
FMOLS and SOLS are significant and positive, verifying a long-run relationshipbetween PIR and the real GDP (Table 3a). To ensure the robustness of ourresults, we exclude a country one-at-a-time. The results are presented in Table
3a. Our results of the panel cointegration are robust.
(Figure 3, Table 3a about here)

We should interpret the panel cointegration test results with cautions. It
shows that the countries in our sample, as a group, display a cointegration
relationship. However, some sub-samples may not have such a cointegration
relationship (Pesaran, 2015, chapter 31). To understand the evolution of thewage-to-house price ratio in diﬀerent countries, we study the residual terms
from our Group-mean Fully-modified OLS. We take the first-diﬀerence of residuals and then divide each series by the corresponding standard deviation to
become comparable across countries. Figure 3 shows that these detrended andnormalized residual terms are stationary. For most countries, these terms do
2 8 FMOLS can be interpreted as an extension of Phillips and Hensen (1990). Among others,
see Pedroni (2000, 2001) for more discussion.

not significantly and persistently deviate from zero, suggesting that the PIR’s
growth rate is roughly constant in the long run.29 In other words, we do notfind evidence of persistent bubbles.

6

Robustness Checks

The previous section has empirically confirmed our theoretical results. As ourevidence comes from a panel dataset, the concern is the existence of "outlier(s),"which might impact the results. Constrained by data availability, we attempt
to address this concern by performing the following tests.30 First, we re-run the
dynamic panel regression, controlling for government size (captured by the government expenditure ratio to GDP ratio in log). Second, we exclude a country
one-at-a-time. The results are presented in Table 3b. It is safe to conclude thatthe results from the dynamic panel regression are robust.
(Table 3b about here)

7

Concluding Remarks

The house price-to-income ratio (PIR) is widely used in the media and policy
institutions to indicate the property market’s condition. Yet formal modelingis disproportionately rare. Existing studies also incline to concentrate on crosssectional, reduced-form regression. This paper attempts to bridge the gap.
First, it constructs a simple DSGE model and studies the endogenous dynamics
of the house price-to-wage ratio. We confirm the prediction that the PIR is very
persistent (close to the unit root) and is positively related to the previous period
GDP with the data of OECD countries. We show that the empirical result is
2 9 Australia, Canada, Sweden seem to be the exception.
3 0 As it is well known, many cross-country macroeconomic variables are in annual frequency,
while our dataset is quarterly.

robust. We further identify a long-run relationship between PIR and GDP. Our
robustness checks also indicate the importance of semi-parametric correction,
which means that some nonlinearity may exist in the data that our current
model has yet to capture. Perhaps more importantly, our panel cointegration
results on the long-run relationship between PIR and GDP imply that we cantrack the short-run deviations (SRD) in each period. If the SRD is persistentand even growing over time, it might suggest that more careful investigationis necessary for policy considerations. In other words, our cointegration results
might provide another indicator for housing market mispricing and could be
included in the "early warning system" for a possible housing-related crisis.31
Further research can extend the model in several ways. We can develop
an environment where some agents may be subject to collateral constraints.
Another possibility is to consider inventory accumulation in a sticky-price environment. Furthermore, we can build models in which agents live in diﬀerent
cities or have diﬀerent income paths. One would model information frictions inboth housing and labor markets as well. Finally, it can consider a richer set ofgovernment policies and compare their costs and benefits. The pursuit of these
possibilities would further enrich our understanding of housing aﬀordability.32

3 1 There is emerging literature on the "early warning system" for the housing-related crisis.
For instance, Yiu et al. (2013) propose a time series test for real-time bubble detection. Basedon a search-theoretic model, Leung and Tse (2017) suggest that the increase in the cost offunds for speculators could lead to a significant house price adjustment. Huang et al. (2018)
build another search-theoretic model and suggest that the price-rent ratio and turnover rateare essential indicators for a housing-related crisis.
3 2 Among others, see Chen and Cheng (2017), Teo (2009), Leung and Teo (2011), Lubik andTeo (2012).

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PIR Figures and Tables

Table 1. Correlation between variance of PIR and variance of output (Cross-sectional)

Full sample

All countries
exceptAustralia
CanadaDenmark
FinlandFranceGermany
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US

Correlation between var(wt – pht) and var (‫ݕ‬௧ )-0.2917-0.2984-0.3197-0.4300-0.2584-0.3293-0.2175-0.3015-0.2736-0.2903-0.2569-0.2941-0.2938-0.2772-0.3018-0.2624

Note: All coefficients are insignificant. Cyclical components (from HP filter) are used.

Table 2. Dynamic Panel Data Regression Result
Dependent variable: wt+1 – ph,t+1

Instruments – Lags 2 to 4 of dependent variable included
Prob. (J-stat)
wt – phtyt
J-statistics
Full sample

All countries except

Australia
CanadaDenmark
FinlandFranceGermany
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US0.9440 ***0.9562 ***0.9363 ***0.9365 ***0.9460 ***0.9505 ***0.9548 ***0.9643 ***0.9527 ***0.9481 ***0.9556 ***0.9529 ***0.9502 ***0.9544 ***0.9523 ***0.9381 ***0.1127 **0.1357 ***0.06940.1013 **0.05590.1391 ***0.1235 *0.1737 ***0.1062 *0.1433 ***0.09080.1262 ***0.1029 ***0.1274 ***0.1064 *0.0663

13.89
13.11
11.83
12.80
12.48
13.32
13.38
13.47
12.47
12.90
12.83
12.99
12.72
13.41
12.67
12.62

Note: ***, ** and * denotes 1%, 5% and 10% statistical significance respectively.
Cyclical components (from HP filter) are used.

0.38
0.36
0.46
0.38
0.41
0.35
0.34
0.34
0.41
0.38
0.38
0.37
0.39
0.34
0.39
0.40

Table 3a. Panel Co-integration Test Between PIR and GDP (Full Sample and Robustness
Check)

Full sample

All countries except

Australia
CanadaDenmark
FinlandFranceGermany
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
USGroup-mean Fullymodified OLS
0.0024 ***0.0017 **0.0016 **0.0024 ***0.0025 ***0.0026 ***0.0029 ***0.0035 ***0.0031 ***0.0033 ***0.0015 **0.0019 **0.0036 ***0.00110.0022 ***0.0025 ***Group-mean
Dynamic OLS
0.00030.0004-0.00020.00020.00020.00050.00060.00120.00080.0011-0.0007-0.0004
0.0015-0.00100.00010.0003

Note: *** and ** denote 1% and 5% statistical significance respectively.

Static OLS
0.0040 ***0.0032 ***0.0030 ***0.0041 ***0.0041 ***0.0040 ***0.0046 ***0.0051 ***0.0048 ***0.0048 ***0.0030 ***0.0034 ***0.0051 ***0.0026 ***0.0037 ***0.0041 ***Table 3b. Dynamic Panel Data Regression Result (Robustness check)
Dependent variable: wt+1 – ph,t+1

Instruments – Lags 2 to 4 of dependent variable included
wt – phtyt
govt
J-statistics Prob. (J-stat)
Full sample

All countries
exceptAustralia
CanadaDenmark
FinlandFranceGermany
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US0.9446 ***0.9568 ***0.9312 ***0.9368 ***0.9382 ***0.9461 ***0.9469 ***0.9393 ***0.9485 ***0.9421 ***0.9499 ***0.9481 ***0.9459 ***0.9553 ***0.9521 ***0.9380 ***0.1569 **0.1891 ***0.08980.1369 **0.1380 *0.2135 ***0.1970 **0.2270 **0.1582 **0.1954 ***0.07790.1852 ***0.1497 **0.1856 ***0.1963 *0.11870.0451
0.05320.04920.0332
0.0966 *0.10110.10840.1629 **0.06350.1046 *0.14820.10680.0694 **0.0617
0.09350.0695

13.92
13.16
11.55
12.87
12.26
12.92
12.35
11.67
12.42
12.16
10.66
12.37
12.64
13.51
12.64
12.47

Note: ***, ** and * denotes 1%, 5% and 10% statistical significance respectively.
Cyclical components (from HP filter) are used.

0.31
0.28
0.40
0.30
0.34
0.30
0.34
0.39
0.33
0.35
0.47
0.34
0.32
0.26
0.32
0.33

Figure 1. Boxplot of PIR

Figure 2. Variance of Wage-to-House Price ratio and Variance of real GDP

Variance of ln(wage‐to‐house price ratio)

0.003

0.00250.002

0.00150.001

0.0005000.00005

0.00010.000150.00020.00025

Variance of ln(real GDP)

0.00030.000350.0004

Figure 3. Detrended residual (based on our Group-mean Fully-modified OLS)
The residuals terms are obtained after performing group-mean fully-modified OLS. Then,
we take the first-difference and divide each series by the corresponding standard
deviation so that they become comparable.

Australia
4
3
2
1
0
‐1
‐2
‐3

Canada
3
2
1
0
‐1
‐2
‐3
‐4

Denmark
4
3
2
1
0
‐1
‐2
‐3
‐4

Finland
5
4
3
2
1
0
‐1
‐2
‐3
‐4

France
3

2

1

0

‐1

‐2

‐3

Germany
3
2
1
0
‐1
‐2
‐3
‐4

Italy
3
2
1
0
‐1
‐2
‐3
‐4
‐5

Japan
3
2
1
0
‐1
‐2
‐3
‐4

Netherlands
4
3
2
1
0
‐1
‐2
‐3
‐4

New Zealand
3
2
1
0
‐1
‐2
‐3
‐4

Norway
4
3
2
1
0
‐1
‐2
‐3

Spain
3
2.5
2
1.5
1
0.5
0
‐0.5
‐1
‐1.5
‐2
‐2.5

Sweden
3

2

1

0

‐1

‐2

‐3

UK
4
3
2
1
0
‐1
‐2
‐3
‐4

US
4

3

2

1

0

‐1

‐2

This appendix has several parts.
Appendix A provides the proofs of the analytical results.
Appendix B provides additional empirical results.
Appendix C provides the results when Ireland is included in the sample.

A

Proof

A.1

Solution of the representative household problem andthe proof of (14)First, we will show that solution of the dynamic programming problem of therepresentative household can be summarized by the following proposition.
Proposition 10 When the wages and prices are all flexible, the equilibrium
dynamics of the model is characterized by the following equations:



 − 

==== +  
 +  
 +  
 +  (30)

where  ,  =    , are all constant. Eﬀort is constant over time, = 

(31)

The log real wage  and log nominal wage  depend on the money supply, =  +  −  (32)

 =  +  

(33)

where  is a constant.
Once we prove these statements, we can easily prove (14). We now provide
the proof of (30), (31), (32), (33), (14).
The solution method adopted here is similar to that in Leung (2007). To
prove all these results, we need to first obtain the first order conditions. Then
combine them with the market clearing conditions. And then we will manipulate
the algebra and obtain the equilibrium dynamics of the model. Recall that the
dynamic maximization problem of the representative agent household can beformulated as,
µ
µ
¶
µ
¶
¶
−1



    
= max     +    +  +1  +1 



+1
(34)
subject to (3), (4), (5).
The first order conditions are easy to derive,FOC 

1 = −1(35)

 2 (1 −  )−1 = 1 

(36)

µ
∙
¶¸
3+1 +2
3 =  1+1 +1 + (1 −   )
+1

(37)

FOC 
FOC +1

FOC 
1 = 3  

+1


(38)

FOC  1 = 2 (1 −   )FOC  1 =+1
 + 

(39)

1
 + 

(40)

FOC +1
2 = ∙2+1 +2
1
+ (1 −   )+1 + +1
+1 + +1

¸

(41)

FOC 
1 = 2  

+1


(42)

FOC wrt 
3
1
=
+ 



µ

1+1 +1
+1

¶



(43)

At the equilibrium, (5) holds with equality,  +   + −1=  +  +  +   +   +




(44)

Recall also that on the firm side, the production technology is constantreturns to scale, =   1−where  is the capital share, 0    1 The logarithm of the productivity shock
follows an AR (1) process, = −1 +  
where  = ln  ,  measures the persistence of the productivity shock, 0 ≤  ≤1, and  is a white noise (zero mean and constant variance). With competitive
factor markets, we have (8), (9),
 =
 =



= 





= (1 − ) In addition, we have market clearing conditions (for rental and secondary
sale markets), (13): =  = 0On the government side, the money printed will be held by the household
eventually, (11): =  −1 
Equipped with all these equations, we are now ready to solve the model.
Using (13), (41) can be written as
2 +1 =  [ 1 + (1 −   ) 2+1 +2 ](45)

Iterating the equation above forward, we have2 +1 =1 

1 −  (1 −   )(46)

where we impose a transversality condition
lim [ (1 −   )]  2++1 ++2 = 0

→∞

(47)

Substituting (35),(46) into (42), we have 1 =1 −  (1 −   )(48)

Substituting (35) into (38), we have3 +1 = 1  

(49)

Substituting (35), (8) (12) (48) and (49) into (37), we have3 +1
 1
  
 1
  



=
=
=
=
=

∙
¸
+1
 
+ (1 −   ) 3+1 +2 
+1
∙ µ
¶
¸
+1
+1 1
+1  1 ++
+ (1 −   )
+1
+1
+1  ∙ µ¸¶ 1
+1 1
+1  1 ++
+ (1 −   )+1
1 −  (1 −   )+1  µµ¶¶ 1
+1  1 ++ [  +  (1 −   )] 
1 −  (1 −   )+1
1
1 −  (1 −   ) +    1 
 

(50)
1 −  (1 −   )1 −   −  (1 −   )where we impose transversality condition
lim [  +  (1 −   )] →∞µ++1
++1

¶

= 0

(51)


Notice that as 0    1, [  +  (1 −   )]  1, and [  +  (1 −   )] → 0
as  → ∞
Using (48) and (50), (12) can be re-written as



=
=
=

=


=

 
+
+1


1
 1 1 −  (1 −   ) +    1 
1  ++1
1 −  (1 −   )1 − 1   −  (1 −   ) 1 −  (1 −   )1   (1 −  (1 −   ) +    1 ) + [1 − 1   +  (1 −   )]    1 
[1 −  (1 −   )] [1 − 1   −  (1 −   )][1 −  (1 −   )] [1 − 1   +  (1 −   )]+[1 −  (1 −   )] [1 − 1   −  (1 −   )][1 −  (1 −   ) +    1 ] [1 −  (1 −   )][1 −  (1 −   )] [1 − 1   −  (1 −   )]  
(52)

where
 =

µ
1−

 
1 −  (1 −   )¶µ1 −  (1 −   )1 −  (1 −   ) +  1  

¶



(53)

Clearly, we need 0     1 so that 0      The following lemma
formulates the idea.
Lemma 11 =    , where 0     1(54)

1−(1− )Proof: Notice that as 0      1,  1  0, we have 0  1−(1− )+ 1   
1. It suﬃces to show that 0  1−(1− 1. Notice that 0  , ,    1) 
Thus, 0  1−(1−. Notice that as 0  ,   ,)  +  (1 −   )   +  (1 −   ) as   1=   1  1In other words,   +  (1 −   )  1, which means that 1−(1−
)
³
´³
´
³
´
1−(1−  )

 

1 − 1−(1− ) ,Since  = 1 − 1−(1− )1−(1− )+ 1   and 0 ³´
1−(1−  )
1−(1− )+ 1   1 It follows that 0    1Equipped with these results, we can prove the following lemma:

Lemma 12 =      =    , where 0        1(55)

Proof: We combine (50) and (52), and we have= 1  =   1
1 −  (1 −   ) +    1 

1 −  (1 −   )1 − 1   −  (1 −   )(56)

whereµ¶1 −  (1 −   ) +    1 1
1 −  (1 −   )1 −   −  (1 −   )µ¶1 −  (1 −   ) +    1 
1=  1 −  (1 −   )1 −   −  (1 −   )
¶
µ
¶µ
 
1 −  (1 −   )∗ 1−
1 −  (1 −   )1 −  (1 −   ) +  1  
 
=

1 −  (1 −   )≡  (57)
 1 Thus, we have 0     1.
and we have just shown that 0  1−(1−)Similarly, (48) can be written as:= 1

1 −  (1 −   )=    

(58)

where


 1 1 −  (1 −   )¶µ¶µ 1  
1 −  (1 −   )=1−
1 −  (1 −   )1 −  (1 −   )1 −  (1 −   ) +  1  µ¶¶µ
 1  
 
=
1−

(59)
1 −  (1 −   )1 −  (1 −   ) +  1  ≡ 
We have just proved that 0  1−(1− 1. In addition, 0      1,  1  0,)
 1 
we have 0  1−(1− 1. Thus, we have 0     1.
 )+ 1  

By the same token, we can prove the following results,
Lemma 13

µ




¶

=    , where 0    (60)Proof: Substituting (35), (11) into (43), we have
1


 

 



=
=
=
=

µ
¶
 3  +1+ +1 +1 µ¶+1 3 + +1 +1
3
1−3
 =   
1−

where
 ≡

3 
 
1−

(61)

(62)

with   is defined by (53), and we have implicitly assumed the transversality
condition,µ¶+lim   
= 0
(63)
→∞
+ +
Since 0    ,  3  (1 − )  0, it follows that 0    
We now take natural log of (54), (55) and (60), and use small letter variables
to denote log of capital variables, and we have (30).
Now we move to derive results regarding the labor market, including both
equilibrium labor supply and wages. Notice that the values of consumption,
investment, real money balance all depend on the output level   and by taking
log of (6), we have =  +  + (1 − )  
(64)
and thus to determine the dynamics of the output, we need to know the dynamics of both labor hours as well as business capital stock. In the case of flexible
wages, we can dictate the equilibrium labor hours by substituting (35), (9) into(36) and make use of (52)

 2 (1 −  )−1

=

−1

=

 2 (1 −  )  22
2 +    2
2


=
=
=

1

2
 
2 1
  
1 − 

1

2
= 
2 +    2(65)Taking log of (65) gives us (31), = 
where = lnµ22 +    2

¶



(66)

We then combine (61), (65) with (9) and obtain the following expression, = 2

1 1    (67)

Taking log on both sides and we verify (32). In addition, it is straightforward
to show that


µ
¶
1 1
= ln 2 = ln 2 −  − ln   (68)

where  is defined by (66) and   is defined by (62).
Recall that  is the real wage. Let  be the nominal wage. By definition,
the real wage is defined as the nominal wage divided by the price level, (67)
implies that = 2

1 1
 
 

(69)

So the equilibrium nominal wage is proportional to the money supply. Or,in log form,
 =  +  
which is (33).
Now we need to study the dynamics of the capital stock. Take log of (3) andmake use of (56), we have

+1

= (1 −   )  +   ¡¢= (1 −   )  +   ln   + 

= 0 + (1 −   )  +    (70)

where 0 is a constant. Notice that when we combine (31) with (64), we have =  +  +  

(71)

where  ≡ (1 − ) which is a constant. Putting (70) and (71) together, withappropriate adjustment in the time period, we have
⎛

⎞⎛
⎞ ⎛
⎞ ⎛

1
−


⎜
⎟⎜  ⎟ ⎜
⎟ ⎜ 0
⎝
⎠⎝
⎠=⎝
⎠+⎝
0 1


0

⎞ ⎛
⎞


⎟ ⎜ −1 ⎟ ⎜  ⎟
⎠+⎝
⎠⎝
⎠
(1 −   )
−1
0
0

⎞⎛

or,
⎛

⎞

⎛

⎞−1 ⎧⎛
⎞ ⎛
⎞ ⎛
⎞⎛
⎞⎫
⎪
⎪

⎨
⎬
− ⎟
0
⎜  ⎟ ⎜ 0
⎟ ⎜ −1 ⎟ ⎜  ⎟
⎝
⎠
⎠+⎝
⎠+⎝
⎠⎝
⎠
⎪
⎪
⎩ 0
⎭
1
  (1 −   )
−1
0
⎞ ⎧⎛
⎞ ⎛
⎞ ⎛
⎞⎛
⎞⎫
⎪
⎪
⎬
 ⎟ ⎨⎜  ⎟ ⎜ 00⎟ ⎜ −1 ⎟ ⎜  ⎟
⎠ ⎝
⎠+⎝
⎠+⎝
⎠⎝
⎠
⎪
⎪
⎩ 0
⎭
1

(1 −  )
0⎜  ⎟
⎜ 1
⎝
⎠ = ⎝
0

⎛
⎜ 1
= ⎝
0
⎛
⎞ ⎛

0
⎜  +  ⎟ ⎜  
= ⎝
⎠+⎝

0
which is (14). Thus,

By (7),





−1

⎞⎛

⎞

⎞

⎜  ⎟
→
−
 =⎝
⎠
0
 = −1 +  

Thus, we have
−1X→ − 
 −

=0

where

⎛

⎞

 (1 −   ) ⎟ ⎜ −1 ⎟ ⎜  ⎟
⎠+⎝
⎠⎝
⎠
(1 −   )
−1
0

⎛

→
−
→
 − +  =  −⎛⎞⎜  ⎟→−
  = ⎝
⎠
0

(72)

and by assumption,  ( ) = 0,  (   ) = 0 whenever  6= . Thus, we have
proved (15).

A.2

Proof of (16), (17) and (18)

The proof will proceed in a few steps. First, we will describe the relationshipbetween house price and house rent. Then we will relate the house price to the
output and housing stock.
Using (39) (40), (13) and (46), we have the relationship between housing
rent and housing price,





=

1
1
1 −  (1 −   )==1 
2 (1 −   ) +1
 (1 −   )(1 −   ) 1−(1−)=

1 −  (1 −   ) 
 (1 −   )Since house price and rent are proportional to each other, it suﬃces to study
the house price.
Combining (39) and (42), and using (13), we have =1 −     

and when we combine it with (55) and take log, we have (16), =  +  −  
The proof of (17) is easy. It can be obtained by simply combining (30), (32),(67) and (16).
To derive (18), we first recall (17) that

ln

µ




¶

=  −  =  +  And we can combine (4), (13) and (30),+1= (1 −   )  +   ¡¢= (1 −   )  +    + 

=    +    + (1 −   ) £¤=    +    + (1 −   )    +   −1 + (1 −   ) −1=    (1 + (1 −   )) +    + (1 −   )   −12

+ (1 −   ) −1
= 

X
(1 −   ) − + (1 −   ) 0 
= 0 +  

(73)

=0

where 0 is a constant, 0 is the amount of initial housing stock in the model
economy. We then substitute the last expression into (17) and we get −  = 0 −  

−1
X
=0

(1 −   ) −1− − (1 −   )−1 0 

where 0 is a constant, and we have completed the proof of (18).

A.3

Proof of (21), (23)Recall that under the modified economic environment, we assume that wages
are set one period in advance. The contract wage is fixed one period in advancein such a manner that it will clear the market ex ante. This means that the contract wage is set equal to the expected value of the Walrasian wage (−1 ∗ ),i.e. (19), = −1 ∗ =  + −1  and the real wage in this case is described by (20), =  −  = −1 ∗ −  =  + −1  −  
where the formula of  is given by (68).
Notice that the factor return is still equalized to the corresponding marginal
product as the firms maximize profit and the factor market is competitive. Inparticular, (9) still holds. In log form, it can be expressed as
 =  −  + ln 2 

(74)

At the same time, money market clearing ensures that (60) holds. In log
form, it is expressed as −  = ln   +  
(75)
We then combine (20), (74), (75), with (68), and obtain =  +  
which is (21), when we define  according to (22), ≡  − −1  
To derive (23), we first recall that the aggregate production function can be
written in log form, which is (64), =  +  + (1 − )  
Combine it with (21), we have= (1 − )  +  +  + (1 − ) =  +  +  + (1 − )  (76)

Thus, to understand the dynamics of   we also need to understand the
dynamics of   We begin our investigation with the law of motion of the businesscapital   Taking log of (3) and combine it with (30), we have

+1

¡
¢
= (1 −   )  +   ln   + 

=   ln   + (1 −   )  +    
which is (70), with   is deinfed in (57).
Now we can combine (76) with (70), and we have

⎛

⎞⎛
⎞ ⎛
⎞ ⎛

1
−


⎜
⎟⎜  ⎟ ⎜
⎟ ⎜ 0
⎝
⎠⎝
⎠=⎝
⎠+⎝
0 1


0

⎞ ⎛
⎞
+
(1
−
)



 ⎟
⎟ ⎜ −1 ⎟ ⎜ 
⎠+⎝
⎠⎝
⎠
(1 −   )
−1
0
0

⎞⎛

⎛

(77)
⎞

⎜  + (1 − )  ⎟which is (23). Notice also that the “residual term” is ⎝⎠0which is serially correlated because  is serially correlated, by (7).

A.4

Proof of (25)Recall from (24) that the money growth rate is now a function of other economic
variables,
µ ¶ µ
¶ µ
¶ µ ¶−1 
 −1    
 
=




which, combined with (11), can be re-written in log form,( − −1 )−ln  =  (−1 − −2 − ln )+ ( − −1 − ln )+ ( − )+ (78)
When we take expectation on both sides based on the period (−1) information,
we will get−1 = −1 +  (−1 − −2 ) + (1 −  ) ln  +  (−1  − −1 − ln )+ (−1  − ) 
(79)

Therefore, by (78) and (79), we have≡  − −1 =  ( − −1  ) +  ( − −1  ) +  (80)

In words, it means that the “forecast error” on the money growth rate is a"weighted sum" of the forecast error of the general price level ( − −1  )  theforecast error of the aggregate output level ( − −1  )  and the innovation
term  . Thus, to gain a better understanding of the  term, we need tocompute the two forecast errors, ( − −1  ) and ( − −1  ) Based on (76), =  +  +  + (1 − )  Thus,( − −1  ) = ( − −1  ) + (1 − )  (81)

Now we need to compute ( − −1  )  Recall from (30) that −  =  +  
which means that( − −1  ) − ( − −1  ) = ( − −1  ) 
By definition, it means that( − −1  ) =  − ( − −1  ) (82)

Now substitute (81), (82) into (80), we have =  ( − ( − −1  )) +  ( − −1  ) +  or(1 −  ) 

which means that

or

£

¡
¢
 −  ( − −1  ) + ¡¢=  −  [( − −1  ) + (1 − )  ] +  
=

¡
¡
¢
¤
¢
(1 −  ) −  −  (1 − )  =  −  ( − −1  ) +  



¡
¢
 − 
¡¢¤ ( − −1  )= £(1 −  ) −  −  (1 − )1¡¢¤  +£(1 −  ) −  −  (1 − )In words, it means that the “forecast error” of money supply  , is a weighted
sum of the forecast error of productivity, ( − −1  ), and the innovation
term in monetary growth  . In fact, we can give even more details about theforecast error of productivity, ( − −1  ). Recall from (7) that = −1 +  

It is straightforward to show that( − −1  ) =  
Thus, we complete the proof of (25), with





≡
=



A.5

≡

¡
¢
 − 
¡£¢¤(1 −  ) −  −  (1 − )1¸∙(1− )−(1−)( − )
1
¡
£
¢
¤
(1 −  ) −  −  (1 − )Proof of (26)Before we prove the desired result, we need to have some technical results.
Corollary 14 Given (24) holds, the forecast error of money supply,  , shares
the same properties of  in the following sense, ( ) = 0,   ( ) is a constant,  (  ) = 0 whenever  6= (83)

The proof is simple.
•  ( ) =  (   +    ) =    ( ) +    ( ) = 0
•   ( ) =   (   +    ) = (  )2   ( ) +(  )2   ( ) since{ } and { } are independent. And since both   ( ),   ( ) are constant, so is   ( ).
•  (  ) =  [(   +    ) (   +    )] = (  )2  (  ) +2(  )  (  ) since { } and { } are independent. By assumption,  (  ) = (  ) = 0, whenever  6=  Thus,  (  ) = 0.
Equipped with all these results, we know that the forecast error of money
supply,  , behaves like an i.i.d. process. In other words, we do not need toworry about the serial correlations. This piece of knowledge will simplify thecomputations significantly.
Now recall (77) that
⎛

⎞⎛
⎞ ⎛
⎞ ⎛

1
−


⎜
⎟⎜  ⎟ ⎜
⎟ ⎜ 0
⎝
⎠⎝
⎠=⎝
⎠+⎝
0 1


0

⎞ ⎛
⎞

+
(1
−
)


 ⎟
⎟ ⎜ −1 ⎟ ⎜ 
⎠+⎝
⎠⎝
⎠
(1 −   )
−1
0
0

⎞⎛

And by (25), we have =    +    
Combine the two expressions, we can describe the dynamics of the output. Our
focus, however, is the real wage-to-house price ratio. Thus, we should try to
relate the real wage and the house price to the output and innovation terms.
We begin with the real wage. Recall that if (19) holds, then the real wage
process is described by (20), =  −  = −1 ∗ −  =  + −1  −  
Since (75) holds, we also have =  − ln   −  
Combine the two expressions, we have =  −  =  + −1  −  + ln   +  
By definition and (25), we have − −1  =  =    +    
Thus, the real wage (in log) can be expressed aswhere =  −  = ( + ln   ) +  − b b ≡    +    =  

(84)

(85)

Now we need to derive the equilibrium house price  in this model economy.
It is not diﬃcult to see that the house price equation changes very little withrigid wage. One can start from the individual dynamic optimization problem,
i.e. to maximize (34) subject to (3), (4), (5), as before. The only change isthat the wage is pre-set by the wage contract. From the perspective of therepresentative household, however, the real wage and other prices are taken as
given anyway. Thus, the short run wage rigidity may not have much impact on
the housing market.
In the first glance, this result may be surprising. Notice that the nominal
rigid wage will impact the ex post labor supply, as shown in (21). However,since consumption, housing and leisure all separable to each other in the utility
function, this change would not have a direct eﬀect on the marginal utility ofconsumpton or that of housing. In fact, following the proof of proposition 1 and
2, we find that the equation (16) still holds,
 =  +  −  
Now since (35) to (43) still apply, following the proof of proposition 1, it isPstraightforward to show that (73) still applies, and hence  = 0 +  −1
=0 (1 −   ) −1− +−1(1 −   )0 
 = 0 +  −  

−1
X
=0

(1 −   ) −1− − (1 −   )−1 0 

(86)

¡
¢
where 0 ≡  − 0 , which is a constant.
Combining (84) and (86) delivers (26).

A.6

Proof of (28)The proof is simple. Recall (26) that −  = 0 +  

−1
X
=0

(1 −   ) −1− + (1 −   )−1 0 − b 

Update one more period and we get+1 − +1= 0 +  


X
=0



= 0 +    +  


X
=1

(1 −   ) − + (1 −   ) 0 − d+1= 0 +    + (1 −   )  
= 

0



(1 −   ) − + (1 −   ) 0 − d
+1
−1
X
=0

(1 −   ) −1− + (1 −   ) 0 − d+1hi
+    + (1 −   ) ( −  ) − 0 − (1 −   )−1 0 + b+ (1 −   ) 0 − d+1
=   0 +    + (1 −   ) ( −  ) + [(1 −   ) b − d+1 ] which is (28), if we define
0
g
g
+1

≡   0 ≡ (1 −   ) b − d
+1 

Notice that by (85), we have b =   and by (83), we have  (  ) = 0
whenever  6=h Therefore, [g+1 e ] =  [[(1 −   ) b − d+1 ] [(1 −   ) d
−1 − b ]]i2= (1 −   )  (b )  0.

A.7

Proof of (27)The proof is simple.33 Recall (26) that
−1X

 −  = 0 +  

=0(1 −   ) −1− + (1 −   )−1 0 − b 

For un-conditional moment, we know that  ( ) =  (− ),  = 1 2 
Thus, taking variance on both sides, we haveÃ ( −  ) =  0 +  Ã=   2

−1X

(1 −   ) −1− + (1 −   )(1 −   ) −1− − b

=0+ (b )

=0





=0
Ã −1
X

= (  ) −1X

Notice that if we can show 
we will have

(1 −   ) −1−!

!
− 2  

−1

0 − b

Ã −1
X
=0

!



(1 −   ) −1−  b

³P
−1

´

(1
−

)



b
= 0, then by (87),

−1−

=0

3 3 The authors are deeply indebted to Fred Kwan, whose suggestions lead to this result.

!

(87)

 ( −  )
Ã −1
Ã −1
!
!
X
X
2


= (  ) (1 −   ) −1− − 2  
(1 −   ) −1−  b
=0+ (b )2= (  )  ( )

=0

Ã −1
X
=0

(1 −   )2!

= e  ( ) +  (b ) +  (b )which is (27), wheree≡ (  )= (  )

2

Ã −1
X
=0

2

(1 −   )2 1 − (1 −   )

!

2
2 

(88)

1 − (1 −   )Notice that e depends on  in a deterministic manner. Clearly, since 0 ≤    1,
0  (1 −   )2  1. e  0
Thus, we focus on the co-variance term, 
By definition,



Ã −1
X
=0

=

−1
X
=0

=

−1
X
=0

³P
−1



=0 (1 −   )(1 −   ) −1−  b

!

³
´

 (1 −   ) −1−  b(1 −   )  (−1−  b ) ´−1−  b (89)

Notice that by definition, the “shocks” in period  b  should be independent ofprevious period output. In other words, we have (b |−1− ) = 0,  = 0 1 (90)

From (90), by the law of iterated expectation, we have the following (b ) =  ( (b |−1− )) = 0,  = 0 1 (91)

Given that, we can now compute the co-variance terms,

=
=
=
=
=
=

 (−1−  b ) (−1− b ) −  (−1− )  (b ) (−1− b ) by (91)
 ( (−1− b |−1− )) (−1−  (b |−1− )) (−1− ∗ 0) by (90)0Substitute (92) into (89), we have 
hence by (87), we have (27).

A.8

³P
−1

=0 (1 −   )

(92)
−1−  b´= 0 and

Proof of Variance Bound of ( −  )The proof is simple.34 Recall the formula (27), and (88),where ( −  ) = e  ( ) +  (b ) 2

2 1 − (1 −   )e = (  )2 
1 − (1 −   )and by (85), b ≡    +    =   which means that22 (b ) = (  )  ( ) + (  )  ( ) as the technological shock  and monetary policy innovation  are assumed
to be independent.
Clearly, since 0 ≤    1, 0  (1 −   )2  1. e  0 Moreover, e has a
3 4 The authors are deeply indebted to Richard Green, whose suggestions lead to this result.

limit,e ≡lim e = (  ) lim
2

→∞

→∞

1

2

= (  )1 − (1 −   )Ã

1 − (1 −   )2
1 − (1 −   )2

!

2



2 − 

= ≥ 0. In fact, it is easy to that e
 is also
Again, since 0 ≤    1  2, ee (  = 0) = 0, e (  = 1) = 1.
bounded. If we see  as a function of   , e−2e  = 2 (2 −   )  0, as 0 ≤    1  2 Thus, 0  e  1.

A.9

Proof of (29)

Notice that by definition,

=
=
=
=

µ

¶
 
ln

 +  − ( −  ) + ( −  ) +  +  , by (21)
Ã
!
−1
X

−1
0 +  (1 −   ) −1− + (1 −   )0 − b
=0where b+ +  , by (26)=  , by (85)−1X
= 00 +  (1 −   ) −1− + (1 −   )−1 0 
=0

which is very similar to (26), except for a diﬀerent constant term. Furthermore,+1 + +1 − +1= 00 +  


X
=0

(1 −   ) − + (1 −   ) 0= 00 +    +  


X
=1

00





(1 −   ) − + (1 −   ) 0

=  +    + (1 −   )  

−1
X
=0





(1 −   ) −1− + (1 −   ) 0

hi
−1
= 00 +    + (1 −   ) ( +  −  ) − 00 − (1 −   )0

+ (1 −   ) 0
=   00 +    + (1 −   ) ( +  −  ) which means that the wage income-to-house price ratio can also be written as
a moving average form, as the case of wage rate-to-house price ratio.

Appendix B
Table B1: Panel Unit Root Tests

IPS W-statistic
ADF-Fisher
Chi-Square
PP-Fisher ChiSquare

wt - pht
1.03
33.49

Δ(wt - pht)
-10.47 ***184.75 ***

yt
-1.48
34.13

Δ(yt)
-21.40 ***399.68 ***13.84

248.04 ***37.86

451.27 ***Notes: H0: Each country follows an individual unit root process. H1: At least one country’s
process is trend stationary. Exogenous variables: individual effects, individual linear trends.
The lag length is selected by Hannan-Quinn Criterion. *** denotes 1% statisticalsignificance.

Table B2: Panel Integration Test Results
Kao and Johansen tests confirm that PIR and output are cointegrated.
Pedroni tests have several versions. Some confirm that PIR and output are cointegrated.

(i)

Kao Residual Cointegration Test

ADF

Statistic-3.3909 ***(ii)

Johansen Fisher Panel Cointegration Test

Hypothesized Number of Cointegrating Equations

Trace test

Max. Eigen Test

None
At most 1

69.22 ***
39.23

67.87 ***
39.23

(iii)

Pedroni Residual Cointegration Test

(Individual intercept only)
Panel v-Statistic
Panel rho-Statistic
Panel PP-Statistic
Panel ADF-Statistic
Group rho-Statistic
Group PP-Statistic
Group ADF-Statistic

Statistic-0.64231.44771.3332
1.72251.28281.41881.0762(No intercept or trend)
Panel v-Statistic
Panel rho-Statistic
Panel PP-Statistic
Panel ADF-Statistic
Group rho-Statistic
Group PP-Statistic
Group ADF-Statistic

Statistic-1.52170.1961-1.3246 *1.7248 **2.5939-1.4091 *-2.1855 **Note: *** and ** denote 1% and 5% statistical significance respectively.

(Not for publications)
Appendix C: Results when Ireland is included
Table C1. Correlation between variance of PIR and variance of output (Cross-sectional)

Full sample

All countries
exceptAustralia
CanadaDenmark
FinlandFranceGermany
Ireland
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US

Correlation between var(wt – pht) and var (‫ݕ‬௧ )0.23080.31030.2226
0.29100.26040.22670.2977-0.2057
0.22870.23570.22000.21170.27070.21890.24470.22290.2715

Note: All coefficients are insignificant. Cyclical components are used.

Table C2. Dynamic Panel Data Regression Result
Dependent variable: wt+1 – ph,t+1

Instruments – Lags 2 to 4 of dependent variable included
Prob. (J-stat)
wt – phtyt
J-statistics
Full sample

All countries except

Australia
CanadaDenmark
FinlandFranceGermany
Ireland
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US0.9687 ***0.9800 ***0.9716 ***0.9622 ***0.9659 ***0.9704 ***0.9823 ***0.9623 ***0.9798 ***0.9781 ***0.9686 ***0.9696 ***0.9722 ***0.9736 ***0.9674 ***0.9712 ***0.9706 ***0.1895 ***0.2063 ***0.1853 ***0.1752 ***0.1743 ***0.1981 ***0.2479 ***0.1812 ***0.2313 ***0.2196 ***0.1985 ***0.1875 ***0.1969 ***0.2017 ***0.1886 ***0.1893 ***0.2032 ***Note: *** denotes 1% statistical significance. Cyclical components are used.

14.67
16.32
12.33
14.71
14.26
13.89
13.55
13.48
14.04
14.22
13.56
13.54
13.53
14.44
14.05
13.72
15.57

0.40
0.29
0.58
0.40
0.36
0.53
0.41
0.41
0.52
0.36
0.41
0.41
0.41
0.34
0.37
0.39
0.34

Table C3a. Dynamic Panel Data Regression Result (Robustness check)
Dependent variable: wt+1 – ph,t+1

Instruments – Lags 2 to 4 of dependent variable included
wt – phtyt
govt
J-statistics Prob. (J-stat)
Full sample

All countries
exceptAustralia
CanadaDenmark
FinlandFranceGermany
Ireland
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
US0.9663 ***0.9790 ***0.9734 ***0.9619 ***0.9664 ***0.9754 ***0.9768 ***0.9617 ***0.9859 ***0.9926 ***0.9690 ***0.9656 ***0.9732 ***0.9790 ***0.9674 ***0.9830 ***0.9749 ***0.2920 ***0.2553 ***0.3226 ***0.2668 ***0.2612 ***0.2508 ***0.3291 ***0.2075 ***0.3167 ***0.2926 ***0.2815 ***0.2674 ***0.3152 ***0.2965 ***0.2924 ***0.2868 ***0.2999 ***0.1100 ***0.04070.1385 ***0.08070.0683 **-0.00560.10130.02120.0189-0.0072
0.0671
0.0990 ***0.1197 ***0.0377
0.0937 ***0.02880.0697

14.22
13.86
11.48
13.66
12.84
12.76
13.98
13.46
12.82
8.52
13.33
14.59
13.90
12.92
14.01
12.82
15.44

Note: *** and ** denote 1% and 5% statistical significance respectively. Cyclicalcomponents are used.

0.43
0.38
0.57
0.32
0.46
0.39
0.38
0.33
0.38
0.81
0.35
0.33
0.38
0.37
0.37
0.38
0.28

Table C3b. Panel Co-integration Test Between PIR and GDP (Robustness Check)

Full sample

All countries except

Australia
CanadaDenmark
FinlandFranceGermany
Ireland
Italy
Japan
Netherlands
New Zealand
NorwaySpain
SwedenUK
USGroup-mean Fullymodified OLS
0.0016 ***0.0009 **0.0008 **0.0016 ***0.0017 ***0.0017 ***0.0021 ***0.0028 ***0.0026 ***0.0022 ***0.0024 ***0.00070.0010 ***0.0027 ***0.00030.0014 ***0.0017 ***Group-mean
Dynamic OLS-0.0008-0.0007-0.0014-0.0010-0.0009-0.0007-0.0006
0.00030.0000-0.0004-0.0001-0.0018-0.00150.0002-0.0021-0.0011-0.0009

Note: *** and ** denote 1% and 5% statistical significance respectively.

Static OLS
0.0028 ***0.0020 ***0.0018 **0.0028 ***0.0029 ***0.0028 ***0.0033 ***0.0040 ***0.0038 ***0.0035 ***0.0036 ***0.0019 **0.0022 ***0.0038 ***0.0015 **0.0025 ***00029 ***