metadata: authors: 0: author: Ahamada, Ibrahim
1: author: Diaz Sanchez, Jose Luis
count: United States
docna: 0: docna: A retrospective analysis of the house prices
            macro-relationship in the United States
docty: Policy Research Working Paper
subtopic: Economic Conditions and Volatility,Urban Housing and Land Settlements,Urban Governance and Management,Urban Housing,Housing Finance,Capital Markets and Capital Flows,Non Bank Financial Institutions,Capital Flows,Macroeconomic Management,Municipal Management and Reform,Financial Sector Policy
teratopic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
abstracts: cdata!: This study provides empirical evidence
            on the strengthening of the impact of house prices on the US
            macroeconomy. The stability of the house prices macro-link
            is tested in a small-dimensional vector autoregressive model
            over the last fifty years. The estimated break-points are
            used to split the sample into different segments and a
            multivariate time series analysis is performed within
            subsamples. The paper finds a robust structural break in the
            mid-1980s. In addition, time series analysis across segments
            provides evidence that the effect of house prices, not only
            on private consumption, but also on economic activity, has
            intensified since the mid-1980s.
geo_reg_key: 80475,80450,80465
repnme: repnme: A retrospective analysis of the house
            prices macro-relationship in the United States
display_title: A retrospective analysis of the
            house prices macro-relationship in the United States
listing_relative_url: /research/2013/07/18052787/retrospective-analysis-house-prices-macro-relationship-united-states
topicv3: Macroeconomic Vulnerability and Debt
            m1326057 1104,Housing Mortgage Finance Products m1326719 2183,Monetary Policy m1327609 444,Land and Housing m1327648 817,Financial Sector Development m1327681 1920
publishtoextweb_dt: 2013-07-01T00:00:00Z
historic_topic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
no_of_pages: 19
model_extractions: WPS6549 Policy Research Working Paper 6549 A Retrospective Analysis of the House Prices Macro-Relationship in the United States _Ibrahim Ahamada Jose Luis Diaz Sanchez_ The World Bank Development Economics Vice Presidency Operations and Strategy Unit July 2013
Policy Research Working Paper 6549

# **Abstract**

This study provides empirical evidence on the strengthening of the impact of house prices on the US macroeconomy. The stability of the house prices macrolink is tested in a small-dimensional vector autoregressive model over the last fifty years. The estimated break-points are used to split the sample into different segments and a multivariate time series analysis is performed within subsamples. The paper finds a robust structural break in the mid-1980s. In addition, time series analysis across segments provides evidence that the effect of house prices, not only on private consumption, but also on economic activity, has intensified since the mid-1980s.

This paper is a product of the Operations and Strategy Unit, Development Economics Vice Presidency. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at jdiazsanchez@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
A Retrospective Analysis of the House Prices Macro-Relationship in the United States<sup>_∗_</sup> Ibrahim Ahamada<sup>_a_</sup> and Jose Luis Diaz Sanchez<sup>_b,c,d_</sup>

> _a_ International Monetary Fund _b_ Paris School of Economics

> _c_ University of Paris 1 Pantheon-Sorbonne _d_ The World Bank

This study provides empirical evidence on the strengthening of the impact of house prices on the US macroeconomy. The stability of the house prices macro-link is tested in a small-dimensional vector autoregressive model over the last fifty years. The estimated break-points are used to split the sample into different segments and a multivariate time series analysis is performed within subsamples. The paper finds a robust structural break in the mid-1980s. In addition, time series analysis across segments provides evidence that the effect of house prices, not only on private consumption, but also on economic activity, has intensified since the mid-1980s.

JEL Code: C32, E20, E44, R30. Keywords: House prices, mortgage market deregulation, vector autoregressive.

Sector Board: Financial Sector (FSE).

# **1. Introduction**

A decade ago, Alan Greenspan suggested in his testimony to the Joint Economic Committee of the US Congress (2002) that the strong appreciation of house prices after the crash of the stock market in 2001 might have helped save the US economy from a more serious recession, a point of view shared by most of the economists

> _∗_ The views expressed in this article are those of the authors and do not necessarily represent those of the International Monetary Fund or the World Bank. The authors would like to thank John Williams (editor of the _International Journal of Central Banking_ ), two anonymous referees, Joerg Breitung, Jean-Bernard Chatelain, Jordi Gali, Jean-Olivier Hairault, Juergen von Hagen, Aristomene Varoudakis, seminar participants at the Eurostat 6th Colloquium on Modern Tools for Business Cycle Analysis, the Bonn Graduate School of Economics Macro-Workshop, the Central Bank of Peru research seminar, the PSE Macro-Workshop and the IMF Institute research seminar for their helpful comments and suggestions. Authors e-mail: iahamada@imf.org and jdiazsanchez@worldbank.org.
at that time. A few years later, a large crash in housing prices triggered -in all likelihood- a financial and economic crisis. Inevitably, both events have contributed to the consensus among economic practitioners and many researchers that fluctuations in housing prices are a key factor in the US business cycle.

The question of the extent to which the deep transformation of mortgage markets in recent years has modified the house prices macro-relationship arose naturally. A consensus rapidly emerged in the literature that the effect of house prices on the macroeconomy has intensified since the mid-1980s, a time when important deregulations took place in the mortgage market along with the development of secondary markets and the increasing role of the Government Sponsored Enterprises (GSE) in mortgage financing.<sup>1</sup> Consequentially, a persistent fall in mortgage transaction costs occurred along with a strong relaxation of credit conditions. Homeowners were allowed to easily extract equity from their houses and loan-to-value ratios associated with mortgage loans rose, increasing the ability of landlords to borrow against their real estate.<sup>2</sup> In the literature, theoretical models showing how a relaxation of credit constraints on the mortgage sector amplifies the effect of housing prices on the macroeconomy have been developed by Miles (1992), Kiyotaki and Moore (1997), Iacoviello (2005), and Iacoviello and Neri (2010), among others. The primary mechanism of these models is that a reduction of credit constraints allows economic agents to increase the amount that they are able to borrow against their home equity, and use these additional resources to finance extra consumption and investment. Thus, because of the multiple signs of a relaxation in credit conditions in the US, we can expect that the impact of house prices in the macroeconomy has become stronger over time.

The important role of housing prices in the US business cycle in recent years does not provide sufficient evidence to conclude that the house prices effect in the macroeconomy has strengthened. Indeed, one should take into consideration the magnitude of the last boom and bust in house prices; according to the Shiller real house prices index, the boom and bust in house prices has been the largest at least since the end of the 19th century. Although time series and micro-econometric evidence largely show that housing prices have a positive impact on the US economy (see for example Green 1997, Case, Shiller, and Quigley 2005, Haurin and Rosenthal 2006, Bostic, Gabriel, and Painter 2008, and Miller, Peng, and Sklarz 2011), only a few papers provide empirical evidence on the strengthening of the effect of house prices on the macroeconomy over time. Muelbauer (2007), using a structural single equation estimation, finds that before the US credit market liberalization in the 1980s there was no significant housing wealth effect on consumption, however, since then, the effect has become positive. Altissimo et al

> 1See Gerardi, Rosen, and Willen (2010) for a brief review on the modifications of the U.S. mortgage market in the last forty years.

> 2Taking data from the Federal Home Loan Mortgage Corporation, we calculate a substantial fall in fees and points for a regular 30-year fixed-rate mortgage. We obtain an average of 2.3 for the 1980s, 1.5 for the 1990s and 0.7 for the 2000s. Greenspan and Kennedy (2008) find a sustained increase in both their gross and net equity-extraction measures (relative to disposable income) since the beginning of the 1990s.
(2005) estimate a marginal propensity to consume (MPC) out of wealth from aggregate US data. Their results suggest that the MPC appears not to have statistically changed since the beginning of the 1970s. They find, however, a big increase in the MPC at the end of the 1960s.

In this paper, we provide empirical evidence of the strengthening of impact of house prices on the US macro-variables. We proceed in two steps. First, we test the stability of the house prices macro-link in the US during the last 50 years. A recursive Vector Autoregressive (VAR) model including gross domestic product (GDP) or private consumption, interest rates and house prices, is chosen to account for the house prices macro-relationship. The methodology developed in Qu and Perron (2007) is applied to identify breaks in our multivariate system. In this method, the dates and the number of breaks are entirely determined by the algorithm, rather than being imposed ex-ante, like most of the methods used in the time series literature.<sup>3</sup> Thus, this methodology does not require to set _a priori_ the mid-80s -a period systematically found as a structural break in the literature on the “Great Moderation”- as a unique break-point in the house prices macro-link. Indeed, a rupture in this relationship could have occurred as early as the end of the 1960s, when the privatization of the Federal National Mortgage Association (“Fannie Mae”) in 1968 and the creation of the Federal Home Loan Mortgage Corporation (“Freddie Mac”) in 1970 established the foundation of what would become the renovated US mortgage market. A break might also be found as recently as the early 2000s, where a deepening of markets for securitized contracts and derivatives occurred and subprime mortgage loans developed at a very strong pace. Furthermore, the method allows us to consider the fact that the reduction in credit constraints has been a gradual process and that the modification in the house prices macro-link did not occur all at once. The implication of this hypothesis is that the afore proposed break-dates are not exclusive and that more than one break could exist in the link. In the second step, we use the break-points estimated to split the sample in different segments and, on each subsample, multivariate time series analysis, including impulse-responses and Granger causality tests are performed.

Our results can be summarized as follows: the application of Qu and Perron’s methodology in a VAR with linear restrictions concludes that there are two breakpoints in the house prices macro-relationship when GDP is introduced in the VAR specification: one at the end of the 1960s and the second in the mid-1980s, the latest rupture being the strongest among the two break-dates. We also find that there is only one structural break in the link when private consumption is included in the multivariate system, in the mid-1980s. Time series analysis across subsamples finds that the overall impact of house prices, not only on private consumption but also on economic activity has intensified since the mid-1980s. Our results give support to the hypothesis that the modification of the real estate market in the mid-1980s seems to have been a key factor in the strengthening of the effect of house prices on the macroeconomy.<sup>4</sup>

> 3Bai and Perron (1998) also develop a methodology to detect multiple breaks occurring at unknown dates, but the method only considers the case of a single linear equation.

> 4Generally, the “Great Moderation” literature finds reduced responses of GDP to different
The paper is organized as follows. In Section 2, we present Qu and Perron’s methodology which is used to detect breaks. Section 3 applies this methodology to a small-dimensional VAR model using US macro-data. In Section 4, we split the initial sample into different segments using the identified break-points. We then perform impulse-responses and Granger causality tests in each of the segments from the initial VAR model specification. Section 5 contains the concluding remarks and proposes avenues of future research.

# **2. Testing Structural Changes in a Multivariate Model**

The various methods of identifying structural changes in a single regression model have been well-documented in the literature. However, in the multivariate case, only few papers have dealt with structural breaks. Qu and Perron (2007) consider issues related to estimation, inference, and computation when multiple structural changes occur at unknown dates in a system of equations including a vector autoregressive model. The changes here can pertain to the regression coefficients and/or the covariance matrix of the errors. This section presents a brief description of Qu and Perron’s method before proceeding to an application in a VAR model using US macro-data in the next section.

# _2.1 The Model and Estimator_

In the presentation below, we consider a VAR model with three components, _yt_ = ( _y_ 1 _t, y_ 2 _t, y_ 3 _t_ )<sup>_′_</sup> and one lag, as in our application in Section 3. This can be extended to a general VAR model without any major difficulties. The dates and the number of structural changes in the parameters are unknown. We use _m_ to denote the total number of structural changes and _T_ to denote the sample size. The unknown break dates are denoted by the _m_ vector Γ = ( _T_ 1 _, ..., Tm_ ) and we use the convention that _T_ 0 = 1 and _Tm_ +1 = _T_ . Hence there are _m_ + 1 unknown sub-periods, _Tj−_ 1 + 1 _≤ t < Tj , j_ = 1 _, ..., m_ + 1. The model can be written as: where _yt_ = ( _y_ 1 _t, y_ 2 _t, y_ 3 _t_ )<sup>_′_</sup> . In each sub-period _Tj−_ 1 + 1 _≤ t < Tj_ : _πj_ 0 = ( _πj_<sup>(</sup><sup>_i_</sup> 0<sup>))</sup><sup>_i_=1</sup><sup>_,...,_3</sup> is the vector of constant parameters; _πj_ 1 = ( _πj_<sup>(</sup><sup>_kl_</sup> 1<sup>))</sup><sup>_k_=1</sup><sup>_,..,_3;</sup><sup>_l_=1</sup><sup>_,..,_3indicatesthe3X3</sup> matrices of the VAR parameters; and _εt_ is the vector of the residuals with mean _O_ and covariance matrix denoted by Σ _j_ . Here we are interested in the estimation of Λ = _m,_ � _T_<sup>�</sup> 1 _, ..., T_<sup>�</sup> _m,_ Σ<sup>�</sup> _j_ =1 _,...,m_ +1 . We suppose in this paragraph that _m_ is known, � � and will discuss later the estimation of _m_ . For the estimation of model (1) it is convenient to re-write the model as: shocks in the post mid-1980s period (see Stock and Watson 2002 for a survey), not increased responses as in this paper. However, the papers related to that literature do not provide evidence of a change on the effect of house prices shocks on the economy.
where _x_<sup>_′_</sup> _t_<sup>= (</sup><sup>_I_3</sup><sup>_⊗_(1</sup><sup>_, y_1</sup><sup>_t−_1</sup><sup>_, y_2</sup><sup>_t−_1</sup><sup>_, y_3</sup><sup>_t−_1))and</sup> _′ βj_ = _πj_<sup>(1)</sup> 0<sup>_, π_</sup> _j_<sup>(11)</sup> 1 _,_<sup>_π_</sup> _j_<sup>(12)</sup> 1<sup>_, π_</sup> _j_<sup>(13)</sup> 1<sup>_, ..., π_</sup> _j_<sup>(3)</sup> 0<sup>_, π_</sup> _j_<sup>(31)</sup> 1 _,_<sup>_π_</sup> _j_<sup>(32)</sup> 1<sup>_, π_</sup> _j_<sup>(33)</sup> 1 . The estimation method we � � consider is restricted quasi-maximum likelihood. Conditional on the given break dates Γ = ( _T_ 1 _, ..., Tm_ ), the Gaussian quasi-likelihood ratio is: where _f_ ( _yt|xt_ ; _βj,_ Σ _j_ )=(2 _π_ )<sup>_−n/_2</sup> _|_ Σ _j|_<sup>_−_1</sup><sup>_/_2</sup> exp � _−_<sup><u>1</u></sup> 2<sup>[</sup><sup>_yt −x_</sup> _t_<sup>_′βj_]</sup><sup>_′_Σ</sup><sup>_−_</sup> _j_<sup>1</sup> [ _yt − x_<sup>_′_</sup> _t_<sup>_βj_]</sup> � and _n_ = 3 is the number of equations in the VAR model. Γ<sup>0</sup> = ( _T_ 1<sup>0</sup><sup>_, ..., T_0</sup> _m_<sup>),</sup><sup>_β_</sup> _j_<sup>0</sup> and Σ<sup>0</sup> _j_<sup>indicatethetrueunknownparameters.QuandPerron’s(2007)approach</sup> introduces a restriction in _β_ = ( _β_ 1<sup>_′, ..., β_</sup> _m_<sup>_′_</sup> +1<sup>)</sup><sup>_′_and Σ = (Σ1</sup><sup>_, ...,_Σ</sup><sup>_m_+1).For example,</sup> we obtain a partial structural-change model when the restriction imposes that a particular subset of _βj_ is the same for all _j_ . We denote by _g_ ( _β, vec_ (Σ)) = 0 the form of the restrictions in _β_ and (or) Σ, where _g_ ( _._ ) is an _r_ -dimensional vector and _r_ the number of restrictions. Hence the Restricted Log-Likelihood Ratio is given by:

and the estimates are The maximization (5) is taken over all partitions Γ = ( _T_ 1 _, ..., Tm_ ) such that _|Tj − Tj−_ 1 _| ≥_ [ _εT_ ] and _Tm ≤_ [ _T_ (1 _− ε_ )], where _ε_ is an arbitrarily small positive number and [] denotes the integer part of the argument. The parameter _ε_ plays a trimming role and imposes a minimal length for each regime. One important result is that under more general assumptions the estimates of the break dates Γ = ( _T_ 1 _, ..., Tm_ ) and the coefficients ( _β,_ Σ) are asymptotically independent and valid restrictions on the latter do not affect the distribution of the former. Qu and Perron (2007) also discuss a method based on the principle of dynamic programming which searches for the optimal partition Γ = ( _T_ 1 _, ..., Tm_ ), i.e. the partition that yields the optimal value of the Log-Likelihood _rlr_ Γ. They argue that this method is efficient as it only requires least-squares calculations of order _O_ ( _T_ ) and matrix inversions of order _O_ ( _n_ ).

# _2.2 Selection of the Number of Breaks_

To determine the number of breaks _m_ we can appeal to the likelihood-ratio test of no change versus some specific number of changes, _k_ . As proposed by Qu and Perron (2007), this statistic can be constructed as follows: Here log _L_<sup>�</sup> _T T_ �1 _, ..., T_ � _k_ is the maximum of the log-likelihood obtained with the � � optimal partition _{T_<sup>�</sup> 1 _, ..., T_<sup>�</sup> _k}_ , log _L_<sup>�</sup> _T_ the maximum of the log-likelihood under the
null hypothesis of no structural change, _pb_ the total number of coefficients that are allowed to change (not including the coefficients of the variance-covariance matrix), and _nbd_ and _nbo_ indicate respectively the number of parameters that are allowed to change amongst the diagonal and off-diagonal coefficients of the variance-covariance matrix. As noted above, _ε_ is a parameter that allows us to impose a minimal length for each regime. The limiting distribution of sup _LRT_ is discussed in details by the authors, and depends on the aforementioned parameters. It is important to note that this testing procedure is particularly flexible. The test can be used with respect to many cases of structural change: (a) changes only in the coefficients of the conditional mean ( _nbd_ = 0 _, nbo_ = 0); (b) changes only in the coefficients of the covariance matrix of the residuals ( _pb_ = 0); and (c) changes in all of the coefficients ( _pb_ = 0 _, nbd_ = 0 _, nbo_ = 0).

The test for no change versus an unknown number of breaks can also be considered given some upper bound _M_ for _k_ . This class of tests are called double maximum tests, and the statistic is defined for some fixed weights _W_ = _{a_ 1 _, ..., aM }_ as: For uniform weights, i.e. _ai_ = 1 _, i_ = 1 _, ..., M.,_ the statistic is always denoted by _UD_ max _LRT_ ( _M_ ). The second test is noted by _WD_ max _LRT_ ( _M_ ) and applies weights to the individuals tests such that the marginal p-values are equal across values of _m_ . Bai and Perron (1998) provide more discussion of this class of tests and their critical values.

We can also consider the sequential test based on the null hypothesis of, say, _l_ break dates versus _l_ + 1 breaks. The general form of this statistic is: Here _lrT_ ( _._ ) denotes the log of the likelihood, _{T_<sup>�</sup> 1 _, ..., T_<sup>�</sup> _l}_ the optimal partition if _l_ breaks are supposed, and Λ _j,ε_ = _{τ_ ; _T_<sup>�</sup> _j−_ 1+( _T_<sup>�</sup> _j −T_<sup>�</sup> _j−_ 1) _ε ≤ τ ≤ T_<sup>�</sup> _j −_ ( _T_<sup>�</sup> _j −T_<sup>�</sup> _j−_ 1) _ε}_ . In addition, the limiting distribution of the test depends on the number of coefficients that are allowed to change.

In practice, the preferred strategy to determine the number of breaks is first to look at the _UD_ max _LRT_ ( _M_ ) or _WD_ max _LRT_ ( _M_ ) tests to see if at least one structural break exists. We can then decide the number of breaks based on the _SEQT_ ( _l_ + 1 _|l_ ) statistic. We select _m_ breaks such that the _SEQT_ ( _l_ + 1 _|l_ ) tests are insignificant for any _l ≥ m_ . Bai and Perron (2003) conclude that this method leads to the best results and recommend it for empirical applications. However, in the case of a VAR with restrictions on the coefficients, Qu and Perron (2007) consider only the _SupLRT_ test to obtain the number and the location of the breaks.

# _2.3 Method of Lag Selection_

The presence of structural changes in multivariate models introduces non-linearity. As such, the usual model-selection criteria (Akaike and Bayesian information cri-
teria) are not applicable. One practical result in Qu and Perron (2007) is that the limited distribution of the estimates of the break-points is not affected by the imposition of valid restrictions on the parameters. Hence, a non-linear VAR model with a number of lags, say p, can be considered as a valid restriction of a non-linear VAR model with p+1 lags, if the break-dates in the two models are significantly unchanged. We use this result to decide on the number of lags in the application section.<sup>5</sup>

# **3. House Prices Macro-Link in the US: Evidence of Instability**

Under this section we apply the Qu and Perron’s algorithm to the US macrodata. We consider a VAR Model with three endogenous variables: real gross domestic product (GDP) or real private consumption (C), real house prices (HP) and the Fed Funds rate (I), as a proxy for the monetary policy stance. Closed specifications accounting for the house prices macro-relationship have been used by Iacoviello (2005), Goodhard and Hoffman (2007, 2008), Oikarine (2009), and Calza, Monacelli, and Stracca (2013) among others.

We work with two different VAR specifications, one including GDP and the other private consumption, to examine whether the reduction in credit constraints in the US could have modified only the impact of housing prices on consumption, but not its effect on overall economic activity.

Our data is in quarterly frequency from 1960q1 to 2009q3. GDP, private consumption, and house prices are measured in logarithms, with the log values detrended via the first-differencing filter.<sup>6</sup> Private consumption and house prices are deflated using the consumer price index. The Fed Funds rate is the average value in the first month of each quarter. The house price series comes from the Shiller real home price index. Figure 1 shows the data series used.

- _3.1 Application of the Qu and Perron’s Algorithm in a VAR with no Linear_ . _Constraints_

The Qu and Perron’s procedure to identify breaks is applied to the following model:

> 5An alternative method of lag selection in a non-linear VAR is proposed in Kurozumi and Tuvaandorj (2011). The authors discuss a modified AIC and BIC to be used in the case of a multivariate model with multiple structural changes.

> 6 We use the first-differencing filter instead of a two-sided moving average filter (for example, Hodrick-Prescott or Baxter-King) in order to test for Granger causality in Section 4. Indeed, it is well known in the econometric literature that the use of a two-sided moving average filter biais the Granger causality test towards a false finding of Granger causation (see for instance Cogley 2008).
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<!-- Start of picture text --> House Prices<br>0.035<br>0.015<br>-0.005<br>-0.025<br>0.045<br>-0.065<br>0.085<br>7&» Ff&»Y PP&“a rod» oesY eeRoNa &» es&Y FFsY &»<br><!-- End of picture text --> <!-- Start of picture text --> Gross Domestic Product<br>0.035<br>0.025<br>0.015<br>0.008<br>0.0050.015<br>0.025<br>FKLSSSoS o SSMP‘a SSEEEESKL KP KM SL<br><!-- End of picture text --> <!-- Start of picture text --> Interest Rates<br>0.04<br>0.035<br>0.03<br>0.025<br>b=<br>0.01<br>0.005<br>0<br>PPPsNa ge+ Asa Peaevd s&y ee&vd sy SPSsa Ssy Ss»<br><!-- End of picture text -->
and 1999q1. Table 2 shows the results of the _SEQT_ ( _l_ + 1 _|l_ ) test along with the estimated break dates with their 95% confidence intervals.

**Table 1. WDmax Test for up to 4 Breaks. VAR with no Linear Restrictions**

||VAR with X=GDP|VAR with X=C|
|---|---|---|
|WDmax statistic|54.13|50.75|

Critical value at 5%: 30.62 Next we apply the procedure setting X=C. The _WD_ max _LRT_ ( _M_ ) statistic allows us to reject the null hypothesis at the 5% level (Table 1). The value of the _WD_ max _LRT_ ( _M_ ) statistic is 50.75, whereas the critical value at the 5% level is 30.62. We conclude then that there is at least one structural break in the considered VAR model. The _SEQT_ ( _l_ + 1 _|l_ ) test allows us to reject the null hypothesis of one break against the alternative of two breaks as _Seq_ (2 _|_ 1) = 39 _._ 32 with the 5% critical value being 32.60. Moreover, _SEQT_ ( _l_ + 1 _|l_ ) also allows us to reject the null hypothesis of two breaks against the alternative of three breaks as _Seq_ (3 _|_ 2) = 37 _._ 88 with the 5% critical value being 33.96. Nevertheless, the null hypothesis of three breaks against four breaks is not rejected as _Seq_ (4 _|_ 3) = 0 _._ 00 where the 5% critical value is 34.89. Thus, the performed tests allow us to conclude that there are three structural breaks in the considered VAR specification which includes a private consumption variable: 1970q2, 1985q1 and 1999q1. Table 3 shows the 95% confidence intervals for each break.

**Table 2. The** _SEQT_ ( _l_ + 1 _|l_ ) **Test for a VAR with no Linear Restrictions with X=GDP**

||Statistic|Critical value<br>at 5% level|Signifcant break|
|---|---|---|---|
|Seq( 2_|_1|)<br>40.84|32.60|-|
||||1969q3 (1968q1-1971q1);|
|Seq( 3_|_2|)<br>37.88|33.96|1982q4 (1981q2-1984q2);<br>and 1999q1 (1997q1-2001q1)|
|Seq( 4_|_3|)<br>0.00|34.89|-|

Afterwards, we apply the tests using a VAR with a lag fixed at _p_ = 2. The number and location of the break-dates remain significantly unchanged.<sup>8</sup> In both cases, the VAR model with one lag can then be considered as a valid and parsimonious restriction of the VAR model with two lags.

_3.2 Application of the Qu and Perron’s Algorithm in a VAR with Linear_ . _Constraints_

> 8The test results are not reported because of space considerations but are available by request.
**Table 3. The** _SEQT_ ( _l_ + 1 _|l_ ) **Test for a VAR with no Linear Restrictions with X=C**

||Statistic|Critical value<br>at 5% level|Signifcant break|
|---|---|---|---|
|Seq( 2_|_1|)<br>39.32|32.60|-|
||||1970q2 (1967q4-1972q4);|
|Seq( 3_|_2|)<br>38.82|33.96|1985q1 (1983q4-1986q2);<br>and 1999q1 (1998q2-1999q4)|
|Seq( 4_|_3|)<br>0.000|34.89|-|

The estimated break-dates in the first part of this section are submitted to a robustness exercise in this sub-section. We would like to dismiss the possibility that the breaks found are not primarily driven by changes in the effect of house prices on the macroeconomy. For instance, the new monetary policy paradigm of the Federal Reserve -nominal policy interest rates reacting more than proportionaly to inflation rate changes- could be the main factor causing the rupture detected in the mid-80s. To avoid this issue, we constrain our VAR to only allow a change in the set of parameters related directly to the impact of house prices in the macrovariables. Thus, the Qu and Perron’s algorithm is applied to the following VAR model: where _m_ denotes the unknown breaks and _εt_ = _{εt_ 1 _, εt_ 2 _, εt_ 3 _}_ is the error term with mean _0_ and constant variance-covariance matrix. Then, in the above VAR with linear restrictions, only the parameters _β_ 3 _j_ , _β_ 7 _j_ and _β_ 11 _j_ are allowed to change. As proposed by Qu and Perron (2007), the _SupLRT_ test is performed to obtain the number and the location of breaks in a VAR with linear constraints.

Let us consider the case of a VAR with X=GDP. The null hypothesis of no break is tested against the previously estimated three breaks. Results (Table 4) show that the _SupLRT_ test does not allow us to reject the null hypothesis of no structural break against the alternative of three breaks, as the value of the _SupLRT_ statistic is 29 _._ 92 while the critical value at the 5% level is 33 _._ 71. Thus, we cannot associate each of the three previously detected breaks as structural changes in the effect of house prices on the macroeconomy. We proceed to examine if two of the estimated breaks are robust to the imposed restrictions. The _SupLRT_ test allows us to reject the null hypothesis of no structural break against the alternative of two breaks, as the value of the _SupLRT_ statistic is 29 _._ 54 while the critical value at the 5% level is 28 _._ 17. The test gives us 1971q2 and 1983q1 as structural breaks. The confidence intervals associated to these two breaks, 1968q2-1973q2 and 1980q2-1985q4, cover the first two break-points obtained in the no restricted
VAR (1969q3 and 1982q4). We can then conclude that only the break-dates 1969q3 and 1982q4 can be associated with a change in the impact of house prices on the macroeconomy, when GDP is included in the VAR specification. In a final step, we use the _SupLRT_ test to detect the location of the strongest rupture in the system during the considered period. To do this, we apply the _SupLRT_ test with the null hypothesis of no structural break against the alternative of one break. The _SupLRT_ test allows us to reject the null hypothesis of no structural break against the alternative of one break, as the value of the _SupLR_ statistic is 25 _._ 08 while the critical value at the 5% level is 21 _._ 37. The test gives us 1984q4 as a break-point, and the confidence interval associated to this break, 1981q4-1987q4, covers the second break-date obtained in the no restricted VAR (1982q4). Thus, we conclude that the strongest rupture in the house prices macro-link, for the considered VAR specification, occurred in the mid-1980s.

**Table 4. The** _SupLRT_ **Test for a VAR with Linear Restrictions with X=GDP**

||Statistic|Critical value<br>at 5% level|Signifcant break|
|---|---|---|---|
|_SupLRT_ test ( 1_|_0|):<br>25.08|21.37|1984q4(1981q4-1987q4)|
|_SupLRT_ test ( 2_|_0|):<br>29.54|28.17|1971q2 (1968q2-1973q2);|
||||1983q1 (1980q2-1985q4)|
|_SupLRT_ test ( 3_|_0|):<br>29.92|33.71|-|

We now test the robustness of the three break-dates estimated in a VAR including private consumption as an endogenous variable (i.e, X=C). Results (Table 5) show that the _SupLRT_ test does not allow us to reject the null hypothesis of no structural break against the alternative of three breaks, as the value of the _SupLRT_ statistic is 24 _._ 40 while the critical value at the 5% level is 33 _._ 70. The _SupLRT_ test also does not allow us to reject the null hypothesis of no structural break against the alternative of two breaks, as the value of the _SupLRT_ statistic is 24 _._ 24 while the critical value at the 5% level is 28 _._ 17. Finally, the _SupLRT_ test allows us to reject the null hypothesis of no structural break against the alternative of one break. The test gives us 1984q3 as the only break in the system. The confidence interval associated with this break -1981q1-1988q1- covers the second break-point obtained in the non restricted VAR (1985q1). We can then conclude that only the break-point 1985q1 can be associated to a change in the effect of house prices on the macroeconomy, when private consumption is included in the VAR system.

Results in this section suggest that the housing prices impact on the selected US macro-aggregates experienced an important rupture in the mid-1980s. Even though the economic agents experienced a progressive relaxation on credit constraints specially during the last 30 years, the modification of the housing market in the mid-1980s seems to have been a determining factor of the change of the house prices macro-relationship.<sup>9</sup>

> 9Our results also suggest that the break estimated in the end of the 1960s is not due to a
**Table 5. The** _SupLRT_ **Test for a VAR with Linear Restrictions with X=C**

||Statistic|Critical value<br>at 5% level|Signifcant break|
|---|---|---|---|
|_SupLRT_ test|( 1_|_0 ):<br>23.05|21.37|1984q3 (1981q1-1988q1)|
|_SupLRT_ test|( 2_|_0 ):<br>24.24|28.17|-|
|_SupLRT_ test|( 3_|_0 ):<br>24.40|33.70|-|

# **4. Multivariate Time Series Analysis within Subsamples**

In Section 3 we have shown that the impact of house prices on the macroeconomy changed in the mid-1980s. In this section we look for time series evidence to find in which direction the house prices macro-link has moved. More specifically, we seek evidence that the effect of house prices on economic activity and on private consumption has been reinforced after the break. In this aim, we split the sample into two segments using the break-date estimated in the mid-1980s. Impulseresponses and Granger-causality tests are performed for both subsamples. The VAR specification used to identify the breaks in the previous section is also considered in each sub-period.<sup>10</sup>

# _4.1 Generalized Impulse-Response Functions_

In order to analyze the dynamics of a VAR system, most of the empirical papers in the literature use the “orthogonalised”impulse-responses where the underlying shocks are orthogonalised using the Cholesky decomposition methodology. The main problem with this method is that in practice, when the order of the variables in the VAR is modified, the responses are often significantly different. To avoid the issue of potentially different results, we apply the generalized impulse responses (GIRs) proposed by Pesaran and Shin (1998), which is invariant to the ordering of the variables in the VAR. For each of the VAR specifications, either including GDP or consumption, we perform GIRs and cumulative GIRs for each of the two sub-periods (Figure 2). Shocks represent an exogenous increase in one percent in the house prices variable.<sup>11</sup> change in the house prices effect on consumption, but it would be mainly driven by changes in the impact of house prices on investment. The increase in the competition on the secondary market due to the creation of “Freddie Mac” (1970) could have played a role in the modification of the link between housing investment and house prices, however more work would be needed to clarify this issue.

> 10In the previous section we have found that a VAR with one lag is a valid and parsimonious restriction of a VAR with two lags. This result allows us to estimate for this section a VAR with either one or two lags. We choose a VAR with one lag for the sake of preserving an adequate number of degrees of freedom for the estimation.

> 11Because the housing prices series has been log-tranformed and first-differenciated, an increase of one percent of this variable represents an increase in one percent in the growth rate of
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percent after 10 quarters for this sub-period, which is approximately 100 percent stronger than the accumulated effect estimated for the first segment.

# _4.1.2 Second Case: GIRs for a VAR Including Private Consumption_

_First segment (1960q1-1985q1)_ . The effect of a one percent increase in the house prices variable increases private consumption by approximately 0.1 percent in the first quarter. The shock appears to be absorbed rapidly, implying that a change in housing prices has little influence on private consumption after two quarters. The cumulative impulse-responses for a 10 quarter horizon suggests that the accumulated effect of an increase in housing prices of one percent on consumption after 10 quarters is around 0.5 percent for this sub-period.

_Second segment (1985q2-2009q3)_ . For this segment, the rise in private consumption following a house prices increase of one percent is very close in the short run to the response estimated in the first sub-period. Nevertheless, the disturbance appears to impact consumption long after the initial impact. Moreover, the cumulative impulse-responses show that the accumulated effect of a rise of one percent in house prices on private consumption is around 0.8 percent after 10 quarters for this sub-period, which is more than 50 percent stronger than the accumulated effect found for the first segment.

In sum, generalized impulse-responses show that the effect of house prices on the economic activity and on aggregate private consumption has substantially increased since the mid-1980s.

# _4.2 Granger-Causality Test of Block Exogeneity_

Under this short sub-section multivariate Granger-causality of block exogeneity Wald tests are performed for each of the VAR specifications, either including GDP or private consumption, and for each sub-period (see Table 6).

Granger causality tests show that before the mid-80s, there is no unidirectional Granger causality running either from house prices to GDP or consumption, or in the opposite direction. However, a clear unidirectional Granger causality running from house prices to the economic activity and to private consumption is detected since the mid-1980s.

# **5. Conclusion**

This paper provides empirical evidence on the strengthening of the impact of house prices on the macroeconomy in the US. First, Qu and Perron’s methodology is applied to identify breaks in a VAR model. The choice of this method is guided by its key advantages compared to the alternative methods used in the literature: the method allows us to detect a break at unknown dates and it allows for multiple breaks. Second, the estimated break-points are used to split the sample in different segments, and the generalized impulse-responses proposed by Pesaran and Shin (1998) as well as Granger causality tests are performed within the subsamples.
**Table 6. Granger Causality Tests**

|**Results **|**for a VAR w**|**ith X= GDP**||
|---|---|---|---|
|||Chi-square|P-value|
|First segment|_HP_ ⇏_GDP_|0.81|0.37|
|(1960q1-1982q4)|_GDP_ ⇏_HP_|0.01|0.93|
|Second segment|_HP ⇒GDP_|10.86|0.00|
|(1983q1-2009q3)|_GDP_ ⇏_HP_|3.32|0.07|
|**Resul**|**ts for a VAR **|**with X=C**||
|||Chi-square|P-value|
|First segment|_HP_ ⇏_C_|1.16|0.28|
|(1960q1-1985q1)|_C_ ⇏_HP_|1.00|0.31|
|Second segment|_HP ⇒C_|20.74|0.00|
|(1985q2-2009q3)|_C_ ⇏_HP_|0.66|0.42|

_Note_ . The null hypothesis is the non Granger causality. The _⇒_ and ⇏ symbols denote the Granger causality and the non Granger causality, respectively, at 5% significance level.

Using macro-data from the last fifty years, this study finds that there is a robust structural break in the mid-1980s in the house prices effect on the macroeconomy, which is obtained when either GDP or private consumption is included in the multivariate specification. The paper also finds time series evidence that the overall impact of house prices on consumption and on economic activity has reinforced since the mid-1980s. Results in this study support the widespread hypothesis in the theoretical literature that the deep transformation in the mortgage market during the mid-1980s is the main cause of the intensification of the impact of housing prices, not only on household expenditures decisions, but also on the economic activity as a whole.

Future research could be focused on the consequences of the Dodd-Frank Act on credit conditions and to what extent it will affect the house prices macrolink. Indeed, a tightening in credit constraints could be expected following the Title XIV of the Dodd-Frank Act -“Mortgage Reform and Anti-Predatory Lending Act”- since its main objective is the imposition of restrictions on mortgage originators to only provide lending to borrowers who are likely to repay their loans.