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Testing for rational bubbles in the US housing market
Journal of Macroeconomics 38 (2013) 369–381
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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro
Testing for rational bubbles in the US housing market Bjørnar Karlsen Kivedal ⇑ Norwegian University of Science and Technology, Department of Economics, Dragvoll, NO-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 21 February 2013 Accepted 27 August 2013 Available online 19 September 2013 JEL classification: E31 G12 R21 C32 Keywords: Rational bubbles Price-rent ratio House prices Cointegration Vector autoregression
a b s t r a c t The presence of a bubble in the US housing market prior to the 2007 subprime mortgage financial crisis is investigated. This is done by looking into the relationship between house prices and rental prices, known as the price–rent ratio, which is an important measure of a potential deviation between house prices and its fundamental value. Additionally, the interest rate is taken into account since it is an important factor in determining demand for housing mortgages and thereby influencing house prices, and explosive behavior of house prices is considered. These relationships are investigated through a theoretical and econometrical framework. The empirical evidence suggests that there was a bubble in the housing market prior to the financial crisis, even when controlling for the decreasing interest rate and the fundamental information given by the rental price in the period. Explosiveness was the main source of the price increase, such that a bubble was present in the housing market after correcting for other fundamental factors. The econometric procedures used in the analysis may therefore be relevant for monitoring the housing market. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction As seen in the turmoil following the subprime mortgage financial crisis which started in the US in 2007, there seems to be an important relationship between house prices and the real economy. An investigation of a selection of 18 financial crises from the postwar period in various countries, carried out by Reinhart and Rogoff (2008), shows that there was a significant run-up in house prices prior to each of the financial crises. Additionally, on average for the financial crises investigated in Reinhart and Rogoff (2008, p. 170), the real stock of debt nearly doubled after a financial crisis. If a house price increase can be explained by fundamental factors, a financial crisis is not very likely to occur since house prices will not fall unless fundamental factors change. However, if psychological factors are the reason behind the price increase, a bubble may be present. If the bubble should burst, there is a possibility of a financial crisis because of a rapid decrease in house prices, which particularly will lead to large negative wealth effects for the investors, i.e. households owning a house. Monitoring whether house price increases are caused by fundamental factors or by a bubble is therefore important in order to assess the possibility of a financial crisis, which may cause several negative effects on the real economy. This implies that if the house price increase cannot be explained by the theoretical model containing these fundamental factors, we may have a bubble. The purpose of this analysis is to investigate whether we can find evidence of a bubble in US house prices by using observations prior to the burst of the bubble in 2007. This is done by using a simple theoretical model for house prices which includes the possibility of a bubble. The model is then incorporated into an econometric framework which allows testing for ⇑ Current address: Østfold University College, Faculty of Business, Languages and Social Sciences, P.O. Box 700, NO-1757 Halden, Norway. Tel.: +47 69215250. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2013.08.021
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the presence of such a bubble by taking explosive roots into account. It is also separated between a rational and an irrational bubble in order to determine whether the efficient market hypothesis holds for the housing market. This implies that if a bubble is found prior to the burst of the subprime mortgage bubble in 2007, the framework applied on the housing market here may be used in order to identify a house price bubble. The severe negative effects which a bubble may cause when bursting may then be dampened or avoided, since precautionary moves can be made. One of the important reasons behind the recent financial crisis, was the increased degree of subprime mortgages, leading to increased house prices. This was mortgages for households who were considered to be risky borrowers, e.g. because their future income was low or uncertain. These households were then able to obtain larger loans than what they should have had the possibility to, according to their financial situation. In addition to increasing the risk of the banks’ outstanding claims, this could work as an accelerator on house prices. Households with low income could then have larger loans and thereby afford more expensive houses or be able to increase their bids on houses. This would lead to a growth in aggregate house prices. A house price increase will increase households’ wealth, which may facilitate a further increase in house prices because of the increased wealth. This is known as the financial accelerator effect, which links changes in financial wealth to the real economy, see e.g. Bernanke et al., 1999. The collateral value for loans will also increase when house prices increase, which will amplify the effect of house price growth (Kiyotaki and Moore, 1997). Furthermore, the growth in mortgage equity withdrawal, where households could use house price appreciation in order to obtain larger loans, is also a factor that may have led to larger consumption because of increased wealth, working as a demand amplifier (Greenspan and Kennedy, 2008). Increased house prices may thereby cause large effects on the real economy. Hence, a potential bubble should be identified in order to avoid the accelerating effects which will work negatively on the real economy in the case of rapidly decreased house prices occurring from a bursting bubble. For instance, decreased house prices will cause negative wealth effects for the households which may initiate a recession if the effect is large enough to influence the entire economy. Stiglitz (1990) defines a bubble as a price being high only because investors believe that the selling price is high tomorrow, even though fundamental factors do not seem to justify the high price. This kind of psychology is then the main reason behind price increases during a bubble, and economic fundamental values play a smaller role as a driving force. An important fundamental value explaining house prices is the rental price. Investigating the ratio between house prices and the rental price (the price–rent (PR) ratio) might therefore be useful for detecting bubbles, as studied e.g. by Himmelberg et al. (2005). These prices should move together in the long run since the alternative to purchasing a house is to rent one. By comparing the series in Fig. 1, we see that the large increase in the house prices prior to the 2007 financial crisis is not followed by a similar increase in the rental price. Some other factors than the rental price must therefore exist in order to explain the large increase in house prices. These could be psychological factors causing a bubble. If a bubble exists, it may be either irrational or rational regarding the type of psychological factors driving the price. Irrational bubbles may result from investors being driven by irrationally optimistic expectations, fashion, or fads (Shiller, 2000), while rational bubbles occur when asset prices continue to rise because investors believe that they will be able to sell the overvalued asset at a higher price in the future (Flood and Hodrick, 1990). We should separate between these types of bubbles in order to conduct a proper analysis of the housing market. Additionally, we may view a house as an asset being purchased by an investor. The payoff from the asset will then be the rental price which is received by the owner (investor) when he lets out the house. Hence, the relationship between house prices and rental prices will be an analog to the relationship between asset prices and dividends in a financial market. Furthermore, the fundamental value of an asset will be the present value of all the future cash flows of the asset, such that a potential divergence of the actual price of the asset from its fundamental value is a speculative bubble (Brooks et al., 2001). Various tests for bubbles in asset markets found in the literature may therefore be of relevance when testing for bubbles in the housing market. I will use the econometric procedures presented in Engsted and Nielsen (2012), who tests for rational bubbles in the US stock market for the period 1974–2000. We will then be able to investigate how these methods are able to test for a housing bubble prior to the financial crisis. Phillips and Yu (2011) test for bubbles in the US housing market, where they find an explosive root in house prices corrected for rental prices using a recursive test mainly following Phillips et al. (2011). Hence, there are some similarities between the tests conducted in this paper and their test. However, a multivariate framework is used here, where the
Fig. 1. US real average house prices (deflated by CPI less shelter), seasonally adjusted (solid line, values on left vertical axis). US real average rental prices per quarter (deflated by CPI less shelter), seasonally adjusted (dotted line, values on right vertical axis). Data source: Davis et al., 2008.
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theoretical model of the housing market is tested. This allows us to investigate whether there is a bubble in house prices, as well as testing the model explaining house prices from fundamental factors and a bubble factor. Other tests for rational bubbles, such as the CUSUM test in Homm and Breitung (2012) and the Markov-switching ADF test of Hall et al. (1999) could also have been used in order to test for rational bubbles in US house prices. However, these are univariate tests who does not provide the possibility of testing the theoretical model as being done here. Testing and imposing cointegration between house prices and fundamental factors is possible in this multivariate framework (as opposed to in the univariate procedures), such that the theoretical model is tested statistically by imposing restrictions on the cointegrating vectors. Distinguishing between a rational and an irrational bubble is also possible, since we may test the efficient market hypothesis in the econometric framework as well. When housing mortgages are given based on the collateral value and the expected future collateral value of the house, it is important to know how this price is justified. If the price cannot be explained by fundamental factors, lenders should take this into account when submitting loans to households. High house prices because of psychological factors should therefore not be considered to yield a high future collateral value, since the price would be corrected back to its fundamental value eventually. Monitoring whether increased house prices may be explained by a bubble is therefore important for lenders, since the loans given should be restrained or limited. If signs of a bubble could be found prior to the subprime mortgage crisis, loans could have been limited, dampening the price increase and limiting the negative consequences of the bursting bubble in 2007. Furthermore, as argued e.g. in Leamer (2002), a high ratio between prices and earnings of an asset may be justified if other assets also are highly priced, for example if bond yields and mortgage rates are low. Hence, we should correct for changes in the interest rate when doing this analysis. An increase in house prices could be a result of decreasing interest rates, such that a speculative bubble is not the reason for the house price increase unexplained by the growth in the rental price. I will therefore take the interest rate into account, since it is an important factor for the cost of a mortgage, and it is clearly not approximately constant for the sample analyzed here. The results found here suggest that even when considering fundamental factors such as the rental price and the decreased interest rates, the house price increase before the subprime mortgage crisis is not explained. This shows that there had to be a bubble in the US housing market in order to explain the house price increase because of the explosive growth in house prices. This analysis therefore combines e.g. the analysis in Himmelberg et al. (2005) which focuses on the house price increase being explained by fundamental factors, Campbell et al. (2009) which considers low interest rates to be an explanation for the house price increase, and Shiller (2007) who suggests that the house price increase was due to bubbles. All of these factors are combined in the theoretical and the econometrical model used in the analysis here. 2. Modeling house prices The house price today should reflect the value of next period’s payouts (rents) and possible increase in house prices. This is modeled by
Pt ¼
1 Et ðPtþ1 þ Rtþ1 Þ; 1þi
ð1Þ
as e.g. in Campbell et al. (2009), where Pt is the real house price, Rt is the real rental income and i is the discount factor which is assumed constant (a constant discount factor is common in the empirical bubble literature (Engsted and Nielsen, 2012)). This is in line with the reasoning that the alternative to owning a house is to rent the house. Additionally, the owner of a house (assuming he uses the house as his primary residence himself) saves the rent he alternatively had to pay in order to live in the house if he did not own it (Himmelberg et al., 2005). If we write this out for the one-period gross return to house prices, as shown e.g. in Campbell et al. (2009), we have
ð1 þ iÞ ¼ Et
Ptþ1 þ Rtþ1 : Pt
ð2Þ
We may additionally formulate (1) as
Mt ¼ Pt þ Rt ð1 þ iÞPt1 ;
ð3Þ
where Et1Mt = 0 is a martingale difference, corresponding to the efficient market hypothesis as outlined e.g. in LeRoy (1989). The efficient market hypothesis implies that an expected value does not depend on its prehistory and that all public information is reflected in the price as soon as it is known. Additionally, the ‘‘spread’’ (as defined in Campbell and Shiller (1987, 1988)),
St Pt Rt =i;
ð4Þ
expresses the relationship between real house prices and the real rental price, and corrects for i (which is assumed constant over time here). We then have
Mt ¼ ð1 þ iÞD1 Pt i St ;
ð5Þ
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where D1 is the standard first-difference operator (D1Xt Xt Xt1). This will be used later to test the efficient market hypothesis. This framework is equivalent to the dividend valuation model for stock prices, see e.g. Mishkin (2007). The relationship between house prices and rent should therefore be similar to the relationship between stock prices and dividends. Methods for analyzing bubbles in the stock market using the dividend valuation model may therefore also be applied to house prices in order to test for bubbles in the housing market, simply by replacing asset prices with house prices and dividends with rental prices. The framework outlined by Engsted and Nielsen (2012), who perform tests for bubbles in the US stock market, will be used in order to investigate whether there was a bubble in the housing market prior to the financial crisis that started in 2007. 2.1. Rational bubbles in the housing market As shown e.g. in Gilles and LeRoy (1992), every continuous dynamic price system can be divided into two parts; a fundamental part and a bubble component. We should therefore distinguish between these two to see if there is evidence of a bubble in the data. If such bubbles are present, Pt will be determined as
Pt ¼
j 1 X 1 Et Rtþj þ bBt ; 1þi j¼1
ð6Þ
where Bt = (1 + i)1EtBt+1, i.e. Bt+1 = (1 + i)Bt + nt+1 where nt+1 is a rational forecast error. Additionally, we know that i > 0 such that Bt is explosive and an explosive component is added to Pt if b – 0. If homeowners are willing to pay inflated prices for houses today because they expect unrealistically high appreciation of house prices in the future, there exists a housing bubble (Case and Shiller, 2003). This is indicated by Bt > 0 in (6) which will increase house prices in period t through this psychological factor driving house prices to grow explosively. There may be an explosive root in house prices in addition to a unit root because of speculative bubbles, as shown in Engsted (2006). This can be investigated in the statistical framework of a a vector autoregressive (VAR) model and a cointegrated VAR model, as shown in Section 2.3, where house prices and rental prices are included. 2.2. A non-constant interest rate The discount factor used in the dividend valuation model above is assumed constant as is common in much of the empirical literature on bubbles in the asset market. However, as shown in Fig. 2, interest rates (which may be used as the relevant discount factor) varied a lot during the sample period. Assuming a constant interest rate is therefore an unrealistic assumption and may cause a loss of valuable information in our model. The interest rate is important for the fundamental value of house prices since low interest rates may boost demand for house mortgages and increase house prices. The increase in house prices in the period before the crisis may therefore be partially explained by the decrease in interest rates. Following e.g. Rosen (1979), a net imputed rental income for the homeowner is calculated as
RNt ¼ RGt Mt Dt T t MIt ;
ð7Þ
where RNt is the net imputed rent, calculated as the gross rental income RGt, less maintenance Mt, depreciation Dt, taxes Tt and mortgage interest MIt. This implies that an increase in the interest rate will increase the mortgage interest payments if
Fig. 2. Real 30-year conventional mortgage rate.
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the house is financed by debt. This will lower the net rental income since mortgage costs needs to be payed by the investor. Depreciation, maintenance costs and taxes are held constant in the estimation of the US housing market in André (2010) such that including these variables will only involve a different scaling of the net rental income leading to changes in the coefficient on net rental income. These are therefore not considered here since it will not affect the results of the tests conducted below. Hence, we will use the model
Rnet ¼ Rt MRt ; t Rnet t
ð8Þ M Rt
where is real net rental income, is the real mortgage payments, and Rt is the real actual gross rental income as used above. This is therefore a real version of the model in (7) where maintenance costs, depreciation costs and taxes are left out. If the homeowner lives in the house, this is the net real value of what he saves by living in the house instead of renting a similar house, since the disadvantage of mortgage costs are taken into consideration. The real mortgage payments are calculated as the real mortgage rate multiplied by real house prices and the average ratio between outstanding debt and house prices (e.g. M Rt ¼ mt DTV t P t where mt is the real mortgage interest rate, DTVt is the ratio of debt to the market value of the housing stock and Pt is the real house price) such that it depends on the mortgage rate and the amount of outstanding debt. An increase in house prices will not affect total mortgage costs, since the value of debt relative to the house price (DVTt) is reduced such that only the amount of outstanding debt will influence what is paid interest on. For the period used in the estimation below (1986Q1–2005Q1), the average ratio of debt to the market value of houses is 0.446 and relatively stable, implying that 44.6% of the house wealth is financed by debt and therefore subject to mortgage interest costs. However, house prices have increased during this period. If DTVt is constant and house prices increase, this suggests that outstanding debt have increased. A decreased mortgage interest rate will therefore have a smaller effect on total mortgage costs. The calculated real net rental income is shown in Fig. 3. It is far more volatile than the gross rental income, but is seems to increase more in the period prior to the subprime mortgage crisis. This may suggest that the declining interest rate causes an increase in the net rental income, such that the decreased interest rate can explain the increase in house prices in the same period because of lower mortgage costs and a higher net rental income. Using the net rental income gives the following model for house prices:
Pt ¼
j 1 X 1 Et Rnet tþj þ bBt ; 1þi j¼1
ð9Þ
such that we still have a constant discount factor even though changes in the mortgage interest rate is taken into account through changes in Rnet t . The calculated net real rental income will then take into consideration the real cost of a mortgage. This allows using only two observable variables when estimating the model since the interest rate will be incorporated into the rental price. The alternative would be to use the interest rate as a third variable in the estimation by allowing a non-constant discount factor corresponding to the interest rate in (2) which would complicate the coexplosive VAR model. The interest rate is now considered to affect mortgage costs, but not the discount factor and the expected rate of return. In the following section, I will first investigate if there is a bubble in the housing market using the actual rental price, and secondly I will use the net rental price such that the declining interest rate in the period before the crisis is taken into consideration. This may enable us to see whether a declining interest rate was an important driver for the house price increase. If
Fig. 3. Real average quarterly rental prices (dotted line) as in Fig. 1 and net real quarterly rental income (solid line). Both are deflated by CPI less shelter and seasonally adjusted.
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it was, evidence in favor of a speculative bubble in the housing market could change when correcting for decreasing interest rates. 2.3. The vector autoregressive model The economic model outlined in (1) may be estimated by assuming that Xt = (Pt, Rt)0 is a vector autoregressive (VAR) model of order k where Xt = (Pt, Rt)0 (alternatively, Rnet may be used in place of Rt in order to allow for a varying interest rate as shown t in (8)). See e.g. Campbell and Shiller (1988).
Xt ¼
k X Aj X tj þ l þ et : j¼1
In equilibrium correction form, a reformulation of this yields
D1 X t ¼ PX t1 þ
k1 X
Cj D1 X tj þ l þ et ;
ð10Þ
j¼1
where Cj = (Aj+1 + Aj+2 + + Ak) and P = (I A1 Ak). This is labeled model M for forthcoming references, and it may be reformulated in order to obtain the coexplosive model outlined in Nielsen (2010). A reformulation of (10) yields (given k P 2)
D1 Dq X t ¼ P1 Dq X t1 þ Pq D1 X t1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
j¼1
Pk1 j Pk1 jl P 1 whereDq X t ¼ X t qX t1 ; P1 ¼ 1 j¼1 q Cj Þ and Uj ¼ l¼jþ1 q Cl . Additionally, it is assumed that Xt q ; Pq ¼ q ðI p þ P1 has one unit root and one explosive root through reduced rank restrictions. The explosive root, q > 1, is a freely varying parameter obtained from the estimated characteristic polynomial if the house price is found to be explosive. A long-run restriction for the coexplosive model is that the rank is set to r = 1, i.e. a long-run stationary relationship between the rental price and house price after correcting for explosiveness. This restriction is denoted H1, and we get
D1 Dq X t ¼ a1 b01 Dq X t1 þ aq b0q D1 X t1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
ð11Þ
j¼1
labeled model M1, where a constant is also included in the cointegrating space. Furthermore, b01 Dq X t ; b0q D1 X t and D1DqXt can be given a stationary distribution such that b1 is the cointegrating vector and bq is the coexplosive vector. The cointegration rank is determined through the likelihood test procedure by Johansen (1995). The rental price is assumed to be integrated of order one such that it is non-explosive. This can be tested through the hypothesis on the coexplosive vector that
HR : bq ¼ ð0; 1Þ0 : The equation for the coexplosive model is then reduced to
D1 Dq X t ¼ a1 b01 Dq X t1 þ aq D1 Rt1 þ
k2 X
Uj D1 Dq X tj þ l þ et :
ð12Þ
j¼1
For a given value of q, the likelihood is maximized by reduced rank regression of D1DqXt on DqXt1 correcting for lagged rental price growth D1Rt1 and lagged differences D1DqXtj. This restricted model is labeled M1R. By imposing two additional restrictions on the model in (12), we get the bubble model. The first restriction is that the ‘‘spread’’ St = Pt Rt/i, as shown in (4), is a cointegrating relation. The coefficient i, which represents the discount factor (assumed constant) or the expected one-period return, is then linked to the explosive root, q, through q = 1 + i. This gives the hypothesis 0
HS : b1 ¼ ð1; 1=iÞ : Imposing this yields the model
D1 Dq X t ¼ a1 Dq St1 þ aq D1 Rt1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
ð13Þ
j¼1
labeled M1RS. In addition, the martingale restriction following the efficient market hypothesis outlined in (3) should be imposed as a restriction. This implies that (5) is imposed, such that (12) may be re-written as
D1 Dq ðPt þ Rt Þ ¼ i0 a1 Dq St1 þ i0 aq D1 Rt1 þ i0
k2 X
Uj D1 Dq X tj þ i0 a1 f1 þ i0 et ;
j¼1
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where i0 = (1, 1) such that it reduces to Mt = i0 et denoted eM if the following hypothesis is accepted
HB : i0 a1 ¼ 1;
i0 aq ¼ ð1 þ iÞ2 =i; i0 Uj ¼ 0; f1 ¼ 0:
The model M1R restricted by HS and HB may be reparameterized. x shown below denotes the population regression coefficient of eR,t = (0, 1)et on eM,t = (1, 1)et. The model is then rewritten in terms of the marginal equation for Mt and the conditional equation for D1Rt. This yields the model M1RSB:
Mt ¼ eM;t ;
D1 Rt ¼ a1;R Dq St1 þ ðaq;R þ qÞD1 Rt1 þ
ð14Þ k2 X
Uj;R D1 Dq X tj þ xMt þ eRM;t ;
ð15Þ
j¼1
where eRM,t = eR,t xeM,t is uncorrelated with eM,t, as shown in Engsted and Nielsen (2012). For a known value of i (set to i = 1 q), (14) and (15) are unrelated such that the likelihood is maximized by maximizing over i using a profile argument, i.e. estimation conditional on a known value of i. This implies that i should be set to the value that maximizes the likelihood function such that the likelihood is estimated for several values of i, where the maximum likelihood value yields the chosen value of i. The likelihood profile for various choices of the discount factor must be investigated when estimating the parameter values such that we use the value of i that provides the highest maximum likelihood value. This applies for all of the estimated restricted models outlined above. 3. Estimation The two data series used for house prices and rents are real house prices and real rental prices for the US. The data are collected from Davis et al. (2008), where quarterly time series from 1960Q1 are still being updated. These series are estimated rents and house prices for the aggregate stock of houses in the US, and I choose to use the S&P Case–Shiller price index for the house prices after 2000, since it includes all home sales, in contrast to the FHFA index which only includes conforming home mortgages. The rental price is taken from the quarterly index for the rent of primary residence published by the Bureau of Labor Statistics (BLS), and it is used to interpolate average net rent from the decennial census of housing and extrapolated beyond 2000. Since the rental price is measured as a total cost per year for each quarter, I have divided it by four in order to obtain quarterly rental costs as a measure for Rt. The rental price measures gross rental income less expenses for utilities. I have seasonally adjusted the data using the U.S. Census Bureau’s X12 seasonal adjustment program in order to remove the cyclical seasonal movements from the series prior to the estimation. In order to obtain the real house price and the real rental price, the nominal price series (after seasonally adjusting them) are deflated by the national consumer price index excluding shelter. As mentioned previously, the net rental income is calculated by using the real mortgage interest rate multiplied by the real house price and the average ratio of debt to housing wealth in order to calculate real mortgage costs. This is subtracted from the actual gross rental price. Furthermore, the real mortgage interest rate is calculated by subtracting the four-quarter average seasonally adjusted consumer price index inflation for all urban consumers for all items published by the BLS, from the nominal 30-year conventional mortgage rate obtained from the Board of Governors of the Federal Reserve System. The 30-year conventional mortgage rate is used since it is the type of mortgage loan the majority of homeowners obtain when purchasing a house (Nakajima, 2011). In order to calculate the ratio of debt to housing wealth, the additive inverse of the fraction between owner’s equity in real estate and real estate at market value for households and nonprofit organizations is computed. Both series are from the Board of Governors of the Federal Reserve System. I choose to use the sub-sample 1986Q1–2005Q1. The beginning of the sample coincides with the tax reform act of 1986 which stimulated investing in owner-occupied housing after the real estate boom pertaining to the first half of the 1980s, and starting in 1986Q1 avoids the presence of multiple house price bubbles in the sample since we exclude the period of increasing house prices before the savings and loans crisis. The sample ends in 2005Q1, since the growth in house prices declines after this period as shown in Fig. 1. This is needed since the sample has to end before the potential bubble bursts, and also illustrates that this method is useful for identifying bubbles before they burst. Ending the sample anywhere between 2001Q3 and 2005Q4 also seems to give the result of explosive behavior in house prices such that the results should not be very sensitive to the choice of the sample end period, as shown in Appendix A. By estimating the model for the sample 1960Q1–1985Q4 or for the full sample, it is not possible to find cointegration, and a lot of intervention dummy variables needs to be added in order to get a well specified model. All of the variation in house prices for the full sample is therefore hard to explain by only using data for house prices and the rental price. The sample used here (1986Q1–2005Q1) provides a well specified model without adding any intervention dummy variables, such that the housing market is properly explained by the observations on house prices and the renal price for this period. First, the unrestricted VAR will be estimated. The model needs to be well specified in order for the various tests carried out below to be valid. As shown by the misspecification tests in Table 1, the model does not contain residual autocorrelation or non-normality using a 3.9% level of significance. The test for autocorrelation is proven to be valid when the model contains explosiveness (Nielsen, 2006), and the tests for normality and autoregressional conditional heteroskedasticity (ARCH) are believed to be valid (Engsted and Nielsen, 2012). The residuals may contain ARCH, but the rank test should be robust to
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B.K. Kivedal / Journal of Macroeconomics 38 (2013) 369–381 Table 1 Multivariate misspecification tests. Residual autocorrelation AR 1-5 Test for normality Test for ARCH LM(1)
F(20, 118) = 1.379 (0.147) v2(4) = 10.086 (0.039) F(24, 186) = 3.544 (0.000)
Table 2 Restricted models and their maintained hypotheses. Model
Hypothesis
Description
M1 M1R M1RS M1RSB
H1, H1, H1, H1,
Impose rank r = 1 Rental price is non-explosive St = Pt Rt/i is a cointegrating relation Efficient market hypothesis
r=1 HR HR , HS HR , HS , HB
moderate ARCH effects (Rahbek et al., 1998) and statistical inference is quite robust to residual heteroskedasticity (Juselius, 2006, p. 47). 1 Additionally, the proper lag length for the VAR is set to two, since this is the smallest lag length which still provides no residual autocorrelation. The largest root needs to be larger than one, q > 1, such that the co-explosive model may be formulated. The rank should be r = 1, and the largest root should still be larger than unity after imposing this. Next, the various hypotheses related to the house price model and the bubble model are tested in the coexplosive VAR framework. These hypotheses were outlined in Section 2.3, and they are summarized in Table 2. First, HR is tested given r = 1 and q > 1. Then HS is tested with i = q 1, and finally HB is tested. First, we may estimate the ECM from (10) labeled model M. This may, given rank r = 1, be reparameterized to the coexplosive model M1 from (11), where the explosive root q > 1 is a freely varying parameter obtained from the estimated characteristic polynomial. Next, we test the hypothesis that Rt is non-explosive by imposing bq = (0, 1)0 . Imposing this gives the model M1R as shown in (12). For a given value of q, the likelihood is maximized by reduced rank regression of D1DqXt on DqXt1 correcting for D1Rt1, lags of D1DqXtj and a constant. The likelihood is then maximized by a profile argument over q. The hypothesis pertaining to the ‘‘spread’’, HS, may be tested by imposing b1 ¼ 1; 1 , where i = q 1 such that we get the i model M1RS shown in (13). The likelihood is again maximized over q. Finally, the efficient market hypothesis may be tested by imposing HB. 3.1. Results First, by looking at Fig. 1, we clearly see signs of explosiveness in the house price for the period before the subprime mortgage crisis while the rental price shows no such signs. Section 3.1.1 shows the results from the tests when the interest rate is not taken into account, i.e. assumed constant, when estimating the model, while Section 3.1.2 shows the results when the declining interest rate during the sample period is taken into account by using the net rental price. 3.1.1. Assuming a constant interest rate The initial model is a VAR with two lags (which is enough lags needed to avoid autocorrelated residuals). The characteristic roots are 1.104, 1.037, 0.613 and 0.136, which indicates two explosive roots in the system (although one of them cannot be rejected as a unit root). The cointegration rank test is shown in Table 3, and indicates a rank of r = 1, although we cannot reject the hypothesis of r = 0 if we have a significance level smaller than 2%. If we impose a unit root, i.e. set the rank to r = 1, the largest root is still high at 1.113, while the second to largest is imposed to be 1.000 since cointegration is imposed. This points to the anticipated result after investigating the data graphically, namely that there is one explosive root and one unit root. The results in Table 4 report whether the various hypotheses pertaining to the restrictions on the VAR model may be rejected or not.2 We see that the hypothesis of a non-explosive rental price cannot be rejected such that the explosive component of the estimated VAR model belongs to house prices through the p-value pertaining to HR. The hypothesis HS gives p-values of 0.10 and 0.15 when tested against M1R and M1, respectively, such that the hypothesis of a stationary spread between house prices and the rental price cannot be rejected either. 1 It is not proven that the same holds when explosiveness is present in the model. The tests related to the theoretical model are nevertheless carried out below, and it is assumed that the model is sufficiently well-specified based on the results from the misspecification tests even though all of them are not proven to be valid. Additionally, the quantile-quantile (QQ) plots in Appendix B does not provide evidence against the model being well-specified. 2 The test statistic is assumed to be asymptotically v2 distributed under the null hypothesis, as proven to be asymptotically valid for most of the cases here as shown in Engsted and Nielsen (2012). Hence, the p-values here are calculated from the v2 distribution. This also applies for the p-values in Table 5
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B.K. Kivedal / Journal of Macroeconomics 38 (2013) 369–381 Table 3 Rank test. Critical values for the test statistic and the p-values from Table 15.2 in Johansen (1995). Hypothesis
Likelihood
Test statistic
p-Value
r62 r61 r=0
783.811 786.491 795.213
LR(r 6 1jM) = 5.36 LR(r = 0jM) = 22.81
0.26 0.02
Table 4 Tests of the rational bubble restrictions. Model
Hypothesis
Log-likelihood
M1 M1R M1RS
H1 , r = 1 H1 , HR H1, HR, HS
786.491 787.016 788.366
M1RSB
H1, HR, HS, HB
837.900
Test statistic LR(M1RjM1) LR(M1RSjM1R) LR(M1RSjM1) LR(M1RSBjM1RS) LR(M1RSBjM1R) LR(M1RSBjM1)
1.05 2.70 3.75 99.07 101.77 102.82
d.f.
p-Value
1 1 2 3 4 5
0.31 0.10 0.15 0.00 0.00 0.00
When testing the hypothesis HS, the imposed b vector is b1 ¼ 1; 1 . Furthermore, since i = q 1 where q = 1.081 (the i root which yields the maximum likelihood), we have that the restricted estimate of i under H1R is i = 0.081. The unrestricted b vector is b1 = (1, 94.25) such that the unrestricted estimate of i is î = 0.011. Since HS is not rejected even with this large deviation between the restricted and the actual value, there is little information in the data on this parameter which measures the expected return. Additionally, the imposed value of i = 0.081 is quite high, since this implies an expected return of 8.1% per quarter. The hypothesis pertaining to the bubble, HB, is rejected. This indicates that the standard house price model with a constant discount rate and a rational bubble is not supported and excess returns do not behave as a martingale difference. This may indicate that the rental price is not fully able to explain the movements in house prices even when correcting for explosiveness. Additionally, this may be evidence of an ‘‘irrational bubble’’, since there clearly is an explosive component in house prices but the efficient market hypothesis does not hold when we correct for this. This suggests that agents do not have rational expectations but instead are driven by an irrational optimistic view of the future which influences their information set. Assuming that all individuals do not have different comparative information in acquiring information, is perhaps a strong assumption for the housing market. It is more likely that this is the case in the stock market than the housing market, since the stock market to a larger extent consists of professional investors and the housing market have a large share of private households buying and selling homes. Individual preferences and different abilities to obtain information about the market may be an important factor in the housing market. This will then act as evidence against the efficient market hypothesis, indicating an irrational bubble.
3.1.2. Correcting for a varying interest rate When using the net rent as shown in (8) instead of the actual rental price when testing the restrictions on the model, we get the results as shown in Table 5 for a VAR model with two lags. Also in this case, the model is econometrically well-specified and the largest root is larger than unity using k = 2 lags. The characteristic roots are 1.078, 0.69, 0.40 and 0.40, so there still seems to be an explosive root in the system which is only slightly smaller than in the model without the interest rate. The same pattern emerges in this case, namely that there is a rank of r = 1 and there is an explosive component that belong to house prices. The largest root is 1.056 when imposing rank r = 1, and the second to largest root is 1.000 such that there seems to be an explosive root and a unit root. HR and HS are not rejected at a 1.2% significance level, such that the explosive component belongs to house prices and there is cointegration between house prices and rental prices (which now also
Table 5 Tests of the rational bubble restrictions when the interest rate is included in the estimation. Model
Hypothesis
Log-likelihood
M1 M1R M1RS
H1 , r = 1 H1 , HR H1, HR, HS
966.726 768.147 971.168
M1RSB
H1, HR, HS, HB
1021.785
Test statistic LR(M1RjM1) LR(M1RSjM1R) LR(M1RSjM1) LR(M1RSBjM1RS) LR(M1RSBjM1R) LR(M1RSBjM1)
2.84 6.04 8.88 101.23 107.28 110.12
d.f.
p-Value
1 1 2 3 4 5
0.092 0.014 0.012 0.000 0.000 0.000
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includes effects from the interest rate). The estimated unrestricted b vector is b1 = (1, 55.50), such that the unrestricted estimate of i is î = 0.018. Hence, it is still quite small compared to the restricted value under H1R which is i = 0.081, such that the estimation still provides little information on the expected return. Furthermore, the bubble model is rejected, which indicates that the efficient market hypothesis does not hold even when we correct for the non-constant interest rate. The inclusion of the interest rate in order to use the information in its dynamics over the sample is therefore not helpful in supporting the efficient market hypothesis. Correcting for explosiveness, cointegration and the interest rate is not sufficient to obtain a result that supports the efficient market hypothesis. Hence, the difference between the stock market and the housing market still seems to be evident. However, the perhaps most interesting finding when correcting for the interest rate is that we still find evidence of a bubble. This indicates that the explosive growth in house prices prior to the subprime mortgage crisis was not a result of lower interest rate and a possibly higher credit demand boosting demand for housing. At least this was not enough to explain the explosive behavior. Psychological factors therefore seems to be the driver behind house prices when we correct for rental prices and the interest rate, such that a bubble is found in the data. This is clearly indicated by the explosive root in house prices. The lack of evidence supporting the efficient market hypothesis may also be a result of the difference between rational bubbles and irrational bubbles. Since we reject the efficient market hypothesis, this may indicate that the bubble we found was an irrational bubble. Furthermore, one of the most important differences between stock markets and housing markets is the participants in the markets. Since the housing market to a larger extent consists of unprofessional investors, it is more likely that incorrect judgments are made in the housing market since households may not obtain all available information about the market. Professional investors that to a large extent populates the stock market should be expected to be more rational. In the housing market, an irrational bubble may be caused by increased demand for houses because households have visions of improving their life when buying an expensive house, or by a trend that drives the market towards owning rather than renting houses. This trend may be because of expected increases in house prices in the future such that the households may sell their home with a profit, which will indicate a rational bubble. However, such behavior is more relevant from an investors point of view, since an investor seeks to buy a house today and sell it later with a profit. If a household should behave as a profit maximizing investor, it needs to alternate between owning and renting which is not that likely to be the case for the majority of households since a household mainly buys a house in order to live in it. When households sell their house, it is more likely to buy a new house than to rent a house. Increases in the value of their first home is then used for buying a new house. If this type of behavior is more probable than profit maximizing behavior, a potential bubble should be regarded as an irrational bubble. The difference between the results here and the results in Engsted and Nielsen (2012) therefore seems to support the difference between stock and housing markets, since the efficient market hypothesis holds when the bubble model is tested on the stock market but not on the housing market. However, as argued in Dale et al. (2005), the distinction between rational and irrational bubbles is not very clear since the misjudgment of the investors in the market needs to be investigated in order to make the distinction. Further analysis of the behavior of the agents in the housing market therefore needs to be done if one should find proper evidence for whether the bubble is rational or irrational. Another important difference between the cost of owning versus renting, is that renting is paid by after-tax income while e.g. the cost of a housing mortgage is tax deductible. However, this advantage of owning is constant for the sample period used here such that the PR-ratio should not been affected by this. Additionally, even though including the declining interest rate during the sample period do not affect the evidence of a bubble, it may cause changes in household behavior. A relatively low interest rate over a long period of time may increase the price range of houses they can afford. This can change the way households distributes their income such that a shift towards investing in more expensive houses becomes the norm, and may be viewed as behavior in line with an irrational bubble. As shown in Table 6 in Appendix A, there seems to be an explosive root if we end the sample in any period between 2001Q3 and 2005Q4.3 This may suggest that we could have been able to spot a bubble in house prices earlier than what we did when using the sample which ended in 2005Q1. The model has not been re-estimated for the various sample sizes, primarily since the model is not well specified for all periods . This would require e.g. increasing the lag length or adding intervention dummies for certain periods in order to obtain an econometrically well specified model, possibly because of the shortening of the sample size which causes a loss of information.
4. Conclusion The empirical evidence presented in this paper suggests that there is an explosive root in house prices, while the rental price does not contain explosive elements. This also holds in the case where the net rental price is used, indicating that the declining interest rate in the period before the subprime financial crisis is not a strong enough effect to explain the large 3 Although it is not known whether the root is significantly above unity since we do not know their estimated standard errors. The model is not re-estimated for different end points of the sample since the loss of information when shortening the data set would require increasing the econometric model with more lags and/or with dummy variables pertaining to institutional events.
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increase in the house price that exceeds the increase in the rental price. Hence, a speculative bubble might have been an important driving force behind the increase in house prices when we look at data prior to the crisis. The hypotheses of the theoretical model is also tested through this framework, where the equilibrium relationships between the housing market and the rental market are not rejected when tested by the econometrical model after correcting for the explosive root which takes the bubble into account. Additionally, the bubble seems to be irrational rather than rational, which indicates that there is a difference between how the housing market behaves compared to the stock market. Since the econometric methods used here were able to find evidence for a bubble in the early 2000s, it may be an important instrument for monitoring the housing market. The relevance of the method is particularly important since it does not take into consideration the timing regarding the starting point of the bubble or try to pin point the time period when the bubble bursts. If house prices deviate from their fundamental value and there is an explosive root in a VAR model estimated on the basis of these two variables (and possibly also including the interest rate), this framework should be able to find evidence for a bubble through the various tested hypotheses based on the theoretical model. Furthermore, this framework was applied to the US housing market for the sample 1986Q1–2005Q1. This implies that the estimation and test of hypotheses could be conducted the first quarter of 2005, which was before the subprime financial crisis. A bubble could then be identified such that precautionary moves, e.g. using macro prudential or monetary policy tools influencing the demand side of the housing market, could be implemented in order to dampen the growth of the bubble. The explosive behavior of house prices would then be reduced, such that a smoother adjustment towards fundamental values could be made instead of a burst of the bubble which would make the same correction more rapidly. When lenders considered the the size of mortgages to households based on their collateral value, the part of the increase in house prices unexplained by fundamental factors should not have been taken into account for determining the collateral value. This would have lowered the purchasing power of households in the housing market, such that the growth in house prices could have been decreased. The possibility of mortgage equity withdrawal would then also have been reduced, dampening its accelerating affect which influenced the real economy. This indicates that the burst of the bubble could have been avoided, or the growth in house prices prior to the burst could have been dampened. The negative effects on the real economy following the burst could also have been dampened, because house prices would have returned to a level consistent with their fundamental value by using various policy actions such as reducing the size on loans to households or the degree of subprime mortgages. This highlights the importance of separating between fundamental and psychological factors behind house price increases, in order for the housing market to function properly and thereby to avoid a financial crisis. It also highlights the relevance of using policy actions in order to correct for imbalances in the housing market. Acknowledgments The author thanks Gunnar Bårdsen, Bent Nielsen, John Leahy and four anonymous referees for helpful comments and suggestions. I would also like to thank participants at the second workshop in dynamic macroeconometrics at the University of Oslo, and discussant Hans-Martin Straume and other participants at the Norwegian PhD workshop in economics 2012 for helpful comments. This research was conducted while I was affiliated with the Department of Economics at the Norwegian University of Science and Technology. I am grateful for the PhD scholarship from the Norwegian University of Science and Technology funding my research. Appendix A. The largest unrestricted root for different end periods of the sample See Table 6. Table 6 The largest unrestricted root for the estimated model M in (10) using k = 2 lags for different sample end points for the model without including the interest rate (column ‘‘A’’) and the model using net rental price (column ‘‘B’’). End period
Largest unr. root
End period
A
B
2000Q1 2000Q2 2000Q3 2000Q4
0.9847 0.9617 0.9382 0.9207
1.013 1.007 1.020 1.020
2001Q1 2001Q2 2001Q3 2001Q4
0.9208 0.9955 1.037 1.043
2002Q1 2002Q2 2002Q3 2002Q4
1.072 1.071 1.067 1.047
Largest unrestr. root A
B
2003Q1 2003Q2 2003Q3 2003Q4
0.9583 1.054 1.013 1.080
1.049 1.070 1.066 1.079
0.9982 1.048 1.065 1.060
2004Q1 2004Q2 2004Q3 2004Q4
1.057 1.067 1.058 1.067
1.071 1.080 1.075 1.066
1.078 1.082 1.086 1.076
2005Q1 2005Q2 2005Q3 2005Q4
1.104 1.060 1.059 1.045
1.078 1.072 1.057 1.058
2006Q1
1.006
1.050
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Fig. 4. QQ plots (residuals against normal distribution) for the unrestricted ECM when the interest rate is assumed constant.
Fig. 5. QQ plots (residuals against normal distribution) for the unrestricted ECM when the interest rate is included in the net rental value.
Appendix B. Quantile-Quantile (QQ) plots See Figs. 4 and 5. References André, C., 2010. A bird’s eye view of OECD housing markets. OECD Economics Department Working Papers, 746. Bernanke, B., Gertler, M., Gilchrist, S., 1999. The financial accelerator in a quantitative business cycle framework. Handbook of Macroeconomics 1, 1341– 1393. Brooks, C., Katsaris, A., McGough, T., Tsolacos, S., 2001. Testing for bubbles in indirect property price cycles. Journal of Property Research 18 (4), 341–356. Campbell, J.Y., Shiller, R.J., 1987. Cointegration and tests of present value models. The Journal of Political Economy 95 (5), 1062–1088. Campbell, J., Shiller, R., 1988. Interpreting cointegrated models. Journal of Economic Dynamics and Control 12 (2-3), 505–522. Campbell, S., Davis, M., Gallin, J., Martin, R., 2009. What moves housing markets: a variance decomposition of the rent–price ratio. Journal of Urban Economics 66 (2), 90–102. Case, K.E., Shiller, R.J., 2003. Is there a bubble in the housing market? Brookings Papers on Economic Activity 2003 (2), 299–362. Dale, R., Johnson, J., Tang, L., 2005. Financial markets can go mad: evidence of irrational behaviour during the South Sea Bubble. The Economic History Review 58 (2), 233–271. Davis, M., Lehnert, A., Martin, R., 2008. The rent-price ratio for the aggregate stock of owner-occupied housing. Review of Income and Wealth 54 (2), 279– 284,
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Gilles, C., LeRoy, S.F., 1992. Bubbles and charges. International Economic Review 33 (2), 323–339. Greenspan, A., Kennedy, J., 2008. Sources and uses of equity extracted from homes. Oxford Review of Economic Policy 24 (1), 120–144. Hall, S., Psaradakis, Z., Sola, M., 1999. Detecting periodically collapsing bubbles: a markov-switching unit root test. Journal of Applied Econometrics 14 (2), 143–154. Himmelberg, C., Mayer, C., Sinai, T., 2005. Assessing high house prices: bubbles, fundamentals and misperceptions. The Journal of Economic Perspectives 19 (4), 67–92. Homm, U., Breitung, J., 2012. Testing for speculative bubbles in stock markets: a comparison of alternative methods. Journal of Financial Econometrics 10 (1), 198–231. Johansen, S., 1995. Likelihood Based Inference in Cointegrated Vector Autoregressive Models. Advanced Texts in Econometrics. Oxford University Press. Juselius, K., 2006. The Cointegrated VAR Model. Methodology and Applications. Oxford University Press. Kiyotaki, N., Moore, J., 1997. Credit cycles. Journal of Political Economy 105 (2), 211–248. Leamer, E., 2002. Bubble trouble. Your Home Has a P/E Ratio Too. UCLA Anderson Forecast Quarterly. LeRoy, S., 1989. Efficient capital markets and martingales. Journal of Economic Literature 27 (4), 1583–1621. Mishkin, F., 2007. The Economics of Money, Banking, and Financial Markets. The Addison-Wesley Series in Economics. Pearson Addison Wesley. Nakajima, M., 2011. Understanding house-price dynamics. Business Review, Federal Reserve Bank of Philadelphia 2, 20–28. Nielsen, B., 2006. Order determination in general vector autoregressions. In: H.-C.Ho, C.-K.Ing, T. L.Lai (Eds.), Time Series and Related Topics: In Memory of Ching-Zong Wei, vol. 52, pp. 93–113. IMS Lecture Notes and Monograph Series, Beachwood, Ohio, USA: Institute of Mathematical Statistics . Nielsen, B., 2010. Analysis of coexplosive processes. Econometric Theory 26 (03), 882–915. Phillips, P.C.B., Yu, J., 2011. Dating the timeline of financial bubbles during the subprime crisis. Quantitative Economics 2 (3), 455–491. Phillips, P., Wu, Y., Yu, J., 2011. Explosive behavior in the 1990s Nasdaq: when did exuberance escalate asset values? International Economic Review 52 (1), 201–226. Rahbek, A., Hansen, E., and Dennis, J.G., 2002. ARCH innovations and their impact on cointegration rank testing. Technical report, Preprint no. 12, 1998, Department of Theoretical Statistics. Working paper no. 22, Centre for Analytical Finance. Reinhart, C., Rogoff, K., 2008. Is the 2007 US sub-prime financial crisis so different? an international historical comparison. The American Economic Review 98 (2), 339–344. Rosen, H.S., 1979. Housing decisions and the us income tax: an econometric analysis. Journal of Public Economics 11 (1), 1–23. Shiller, R.J., 2000. Irrational Exuberance. Princeton University Press. Shiller, R., 2007. Understanding recent trends in house prices and home ownership. NBER Working Paper Series, 13553. Stiglitz, J., 1990. Symposium on bubbles. The Journal of Economic Perspectives 4 (2), 13–18.
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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro
Testing for rational bubbles in the US housing market Bjørnar Karlsen Kivedal ⇑ Norwegian University of Science and Technology, Department of Economics, Dragvoll, NO-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 21 February 2013 Accepted 27 August 2013 Available online 19 September 2013 JEL classification: E31 G12 R21 C32 Keywords: Rational bubbles Price-rent ratio House prices Cointegration Vector autoregression
a b s t r a c t The presence of a bubble in the US housing market prior to the 2007 subprime mortgage financial crisis is investigated. This is done by looking into the relationship between house prices and rental prices, known as the price–rent ratio, which is an important measure of a potential deviation between house prices and its fundamental value. Additionally, the interest rate is taken into account since it is an important factor in determining demand for housing mortgages and thereby influencing house prices, and explosive behavior of house prices is considered. These relationships are investigated through a theoretical and econometrical framework. The empirical evidence suggests that there was a bubble in the housing market prior to the financial crisis, even when controlling for the decreasing interest rate and the fundamental information given by the rental price in the period. Explosiveness was the main source of the price increase, such that a bubble was present in the housing market after correcting for other fundamental factors. The econometric procedures used in the analysis may therefore be relevant for monitoring the housing market. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction As seen in the turmoil following the subprime mortgage financial crisis which started in the US in 2007, there seems to be an important relationship between house prices and the real economy. An investigation of a selection of 18 financial crises from the postwar period in various countries, carried out by Reinhart and Rogoff (2008), shows that there was a significant run-up in house prices prior to each of the financial crises. Additionally, on average for the financial crises investigated in Reinhart and Rogoff (2008, p. 170), the real stock of debt nearly doubled after a financial crisis. If a house price increase can be explained by fundamental factors, a financial crisis is not very likely to occur since house prices will not fall unless fundamental factors change. However, if psychological factors are the reason behind the price increase, a bubble may be present. If the bubble should burst, there is a possibility of a financial crisis because of a rapid decrease in house prices, which particularly will lead to large negative wealth effects for the investors, i.e. households owning a house. Monitoring whether house price increases are caused by fundamental factors or by a bubble is therefore important in order to assess the possibility of a financial crisis, which may cause several negative effects on the real economy. This implies that if the house price increase cannot be explained by the theoretical model containing these fundamental factors, we may have a bubble. The purpose of this analysis is to investigate whether we can find evidence of a bubble in US house prices by using observations prior to the burst of the bubble in 2007. This is done by using a simple theoretical model for house prices which includes the possibility of a bubble. The model is then incorporated into an econometric framework which allows testing for ⇑ Current address: Østfold University College, Faculty of Business, Languages and Social Sciences, P.O. Box 700, NO-1757 Halden, Norway. Tel.: +47 69215250. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2013.08.021
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the presence of such a bubble by taking explosive roots into account. It is also separated between a rational and an irrational bubble in order to determine whether the efficient market hypothesis holds for the housing market. This implies that if a bubble is found prior to the burst of the subprime mortgage bubble in 2007, the framework applied on the housing market here may be used in order to identify a house price bubble. The severe negative effects which a bubble may cause when bursting may then be dampened or avoided, since precautionary moves can be made. One of the important reasons behind the recent financial crisis, was the increased degree of subprime mortgages, leading to increased house prices. This was mortgages for households who were considered to be risky borrowers, e.g. because their future income was low or uncertain. These households were then able to obtain larger loans than what they should have had the possibility to, according to their financial situation. In addition to increasing the risk of the banks’ outstanding claims, this could work as an accelerator on house prices. Households with low income could then have larger loans and thereby afford more expensive houses or be able to increase their bids on houses. This would lead to a growth in aggregate house prices. A house price increase will increase households’ wealth, which may facilitate a further increase in house prices because of the increased wealth. This is known as the financial accelerator effect, which links changes in financial wealth to the real economy, see e.g. Bernanke et al., 1999. The collateral value for loans will also increase when house prices increase, which will amplify the effect of house price growth (Kiyotaki and Moore, 1997). Furthermore, the growth in mortgage equity withdrawal, where households could use house price appreciation in order to obtain larger loans, is also a factor that may have led to larger consumption because of increased wealth, working as a demand amplifier (Greenspan and Kennedy, 2008). Increased house prices may thereby cause large effects on the real economy. Hence, a potential bubble should be identified in order to avoid the accelerating effects which will work negatively on the real economy in the case of rapidly decreased house prices occurring from a bursting bubble. For instance, decreased house prices will cause negative wealth effects for the households which may initiate a recession if the effect is large enough to influence the entire economy. Stiglitz (1990) defines a bubble as a price being high only because investors believe that the selling price is high tomorrow, even though fundamental factors do not seem to justify the high price. This kind of psychology is then the main reason behind price increases during a bubble, and economic fundamental values play a smaller role as a driving force. An important fundamental value explaining house prices is the rental price. Investigating the ratio between house prices and the rental price (the price–rent (PR) ratio) might therefore be useful for detecting bubbles, as studied e.g. by Himmelberg et al. (2005). These prices should move together in the long run since the alternative to purchasing a house is to rent one. By comparing the series in Fig. 1, we see that the large increase in the house prices prior to the 2007 financial crisis is not followed by a similar increase in the rental price. Some other factors than the rental price must therefore exist in order to explain the large increase in house prices. These could be psychological factors causing a bubble. If a bubble exists, it may be either irrational or rational regarding the type of psychological factors driving the price. Irrational bubbles may result from investors being driven by irrationally optimistic expectations, fashion, or fads (Shiller, 2000), while rational bubbles occur when asset prices continue to rise because investors believe that they will be able to sell the overvalued asset at a higher price in the future (Flood and Hodrick, 1990). We should separate between these types of bubbles in order to conduct a proper analysis of the housing market. Additionally, we may view a house as an asset being purchased by an investor. The payoff from the asset will then be the rental price which is received by the owner (investor) when he lets out the house. Hence, the relationship between house prices and rental prices will be an analog to the relationship between asset prices and dividends in a financial market. Furthermore, the fundamental value of an asset will be the present value of all the future cash flows of the asset, such that a potential divergence of the actual price of the asset from its fundamental value is a speculative bubble (Brooks et al., 2001). Various tests for bubbles in asset markets found in the literature may therefore be of relevance when testing for bubbles in the housing market. I will use the econometric procedures presented in Engsted and Nielsen (2012), who tests for rational bubbles in the US stock market for the period 1974–2000. We will then be able to investigate how these methods are able to test for a housing bubble prior to the financial crisis. Phillips and Yu (2011) test for bubbles in the US housing market, where they find an explosive root in house prices corrected for rental prices using a recursive test mainly following Phillips et al. (2011). Hence, there are some similarities between the tests conducted in this paper and their test. However, a multivariate framework is used here, where the
Fig. 1. US real average house prices (deflated by CPI less shelter), seasonally adjusted (solid line, values on left vertical axis). US real average rental prices per quarter (deflated by CPI less shelter), seasonally adjusted (dotted line, values on right vertical axis). Data source: Davis et al., 2008.
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theoretical model of the housing market is tested. This allows us to investigate whether there is a bubble in house prices, as well as testing the model explaining house prices from fundamental factors and a bubble factor. Other tests for rational bubbles, such as the CUSUM test in Homm and Breitung (2012) and the Markov-switching ADF test of Hall et al. (1999) could also have been used in order to test for rational bubbles in US house prices. However, these are univariate tests who does not provide the possibility of testing the theoretical model as being done here. Testing and imposing cointegration between house prices and fundamental factors is possible in this multivariate framework (as opposed to in the univariate procedures), such that the theoretical model is tested statistically by imposing restrictions on the cointegrating vectors. Distinguishing between a rational and an irrational bubble is also possible, since we may test the efficient market hypothesis in the econometric framework as well. When housing mortgages are given based on the collateral value and the expected future collateral value of the house, it is important to know how this price is justified. If the price cannot be explained by fundamental factors, lenders should take this into account when submitting loans to households. High house prices because of psychological factors should therefore not be considered to yield a high future collateral value, since the price would be corrected back to its fundamental value eventually. Monitoring whether increased house prices may be explained by a bubble is therefore important for lenders, since the loans given should be restrained or limited. If signs of a bubble could be found prior to the subprime mortgage crisis, loans could have been limited, dampening the price increase and limiting the negative consequences of the bursting bubble in 2007. Furthermore, as argued e.g. in Leamer (2002), a high ratio between prices and earnings of an asset may be justified if other assets also are highly priced, for example if bond yields and mortgage rates are low. Hence, we should correct for changes in the interest rate when doing this analysis. An increase in house prices could be a result of decreasing interest rates, such that a speculative bubble is not the reason for the house price increase unexplained by the growth in the rental price. I will therefore take the interest rate into account, since it is an important factor for the cost of a mortgage, and it is clearly not approximately constant for the sample analyzed here. The results found here suggest that even when considering fundamental factors such as the rental price and the decreased interest rates, the house price increase before the subprime mortgage crisis is not explained. This shows that there had to be a bubble in the US housing market in order to explain the house price increase because of the explosive growth in house prices. This analysis therefore combines e.g. the analysis in Himmelberg et al. (2005) which focuses on the house price increase being explained by fundamental factors, Campbell et al. (2009) which considers low interest rates to be an explanation for the house price increase, and Shiller (2007) who suggests that the house price increase was due to bubbles. All of these factors are combined in the theoretical and the econometrical model used in the analysis here. 2. Modeling house prices The house price today should reflect the value of next period’s payouts (rents) and possible increase in house prices. This is modeled by
Pt ¼
1 Et ðPtþ1 þ Rtþ1 Þ; 1þi
ð1Þ
as e.g. in Campbell et al. (2009), where Pt is the real house price, Rt is the real rental income and i is the discount factor which is assumed constant (a constant discount factor is common in the empirical bubble literature (Engsted and Nielsen, 2012)). This is in line with the reasoning that the alternative to owning a house is to rent the house. Additionally, the owner of a house (assuming he uses the house as his primary residence himself) saves the rent he alternatively had to pay in order to live in the house if he did not own it (Himmelberg et al., 2005). If we write this out for the one-period gross return to house prices, as shown e.g. in Campbell et al. (2009), we have
ð1 þ iÞ ¼ Et
Ptþ1 þ Rtþ1 : Pt
ð2Þ
We may additionally formulate (1) as
Mt ¼ Pt þ Rt ð1 þ iÞPt1 ;
ð3Þ
where Et1Mt = 0 is a martingale difference, corresponding to the efficient market hypothesis as outlined e.g. in LeRoy (1989). The efficient market hypothesis implies that an expected value does not depend on its prehistory and that all public information is reflected in the price as soon as it is known. Additionally, the ‘‘spread’’ (as defined in Campbell and Shiller (1987, 1988)),
St Pt Rt =i;
ð4Þ
expresses the relationship between real house prices and the real rental price, and corrects for i (which is assumed constant over time here). We then have
Mt ¼ ð1 þ iÞD1 Pt i St ;
ð5Þ
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where D1 is the standard first-difference operator (D1Xt Xt Xt1). This will be used later to test the efficient market hypothesis. This framework is equivalent to the dividend valuation model for stock prices, see e.g. Mishkin (2007). The relationship between house prices and rent should therefore be similar to the relationship between stock prices and dividends. Methods for analyzing bubbles in the stock market using the dividend valuation model may therefore also be applied to house prices in order to test for bubbles in the housing market, simply by replacing asset prices with house prices and dividends with rental prices. The framework outlined by Engsted and Nielsen (2012), who perform tests for bubbles in the US stock market, will be used in order to investigate whether there was a bubble in the housing market prior to the financial crisis that started in 2007. 2.1. Rational bubbles in the housing market As shown e.g. in Gilles and LeRoy (1992), every continuous dynamic price system can be divided into two parts; a fundamental part and a bubble component. We should therefore distinguish between these two to see if there is evidence of a bubble in the data. If such bubbles are present, Pt will be determined as
Pt ¼
j 1 X 1 Et Rtþj þ bBt ; 1þi j¼1
ð6Þ
where Bt = (1 + i)1EtBt+1, i.e. Bt+1 = (1 + i)Bt + nt+1 where nt+1 is a rational forecast error. Additionally, we know that i > 0 such that Bt is explosive and an explosive component is added to Pt if b – 0. If homeowners are willing to pay inflated prices for houses today because they expect unrealistically high appreciation of house prices in the future, there exists a housing bubble (Case and Shiller, 2003). This is indicated by Bt > 0 in (6) which will increase house prices in period t through this psychological factor driving house prices to grow explosively. There may be an explosive root in house prices in addition to a unit root because of speculative bubbles, as shown in Engsted (2006). This can be investigated in the statistical framework of a a vector autoregressive (VAR) model and a cointegrated VAR model, as shown in Section 2.3, where house prices and rental prices are included. 2.2. A non-constant interest rate The discount factor used in the dividend valuation model above is assumed constant as is common in much of the empirical literature on bubbles in the asset market. However, as shown in Fig. 2, interest rates (which may be used as the relevant discount factor) varied a lot during the sample period. Assuming a constant interest rate is therefore an unrealistic assumption and may cause a loss of valuable information in our model. The interest rate is important for the fundamental value of house prices since low interest rates may boost demand for house mortgages and increase house prices. The increase in house prices in the period before the crisis may therefore be partially explained by the decrease in interest rates. Following e.g. Rosen (1979), a net imputed rental income for the homeowner is calculated as
RNt ¼ RGt Mt Dt T t MIt ;
ð7Þ
where RNt is the net imputed rent, calculated as the gross rental income RGt, less maintenance Mt, depreciation Dt, taxes Tt and mortgage interest MIt. This implies that an increase in the interest rate will increase the mortgage interest payments if
Fig. 2. Real 30-year conventional mortgage rate.
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the house is financed by debt. This will lower the net rental income since mortgage costs needs to be payed by the investor. Depreciation, maintenance costs and taxes are held constant in the estimation of the US housing market in André (2010) such that including these variables will only involve a different scaling of the net rental income leading to changes in the coefficient on net rental income. These are therefore not considered here since it will not affect the results of the tests conducted below. Hence, we will use the model
Rnet ¼ Rt MRt ; t Rnet t
ð8Þ M Rt
where is real net rental income, is the real mortgage payments, and Rt is the real actual gross rental income as used above. This is therefore a real version of the model in (7) where maintenance costs, depreciation costs and taxes are left out. If the homeowner lives in the house, this is the net real value of what he saves by living in the house instead of renting a similar house, since the disadvantage of mortgage costs are taken into consideration. The real mortgage payments are calculated as the real mortgage rate multiplied by real house prices and the average ratio between outstanding debt and house prices (e.g. M Rt ¼ mt DTV t P t where mt is the real mortgage interest rate, DTVt is the ratio of debt to the market value of the housing stock and Pt is the real house price) such that it depends on the mortgage rate and the amount of outstanding debt. An increase in house prices will not affect total mortgage costs, since the value of debt relative to the house price (DVTt) is reduced such that only the amount of outstanding debt will influence what is paid interest on. For the period used in the estimation below (1986Q1–2005Q1), the average ratio of debt to the market value of houses is 0.446 and relatively stable, implying that 44.6% of the house wealth is financed by debt and therefore subject to mortgage interest costs. However, house prices have increased during this period. If DTVt is constant and house prices increase, this suggests that outstanding debt have increased. A decreased mortgage interest rate will therefore have a smaller effect on total mortgage costs. The calculated real net rental income is shown in Fig. 3. It is far more volatile than the gross rental income, but is seems to increase more in the period prior to the subprime mortgage crisis. This may suggest that the declining interest rate causes an increase in the net rental income, such that the decreased interest rate can explain the increase in house prices in the same period because of lower mortgage costs and a higher net rental income. Using the net rental income gives the following model for house prices:
Pt ¼
j 1 X 1 Et Rnet tþj þ bBt ; 1þi j¼1
ð9Þ
such that we still have a constant discount factor even though changes in the mortgage interest rate is taken into account through changes in Rnet t . The calculated net real rental income will then take into consideration the real cost of a mortgage. This allows using only two observable variables when estimating the model since the interest rate will be incorporated into the rental price. The alternative would be to use the interest rate as a third variable in the estimation by allowing a non-constant discount factor corresponding to the interest rate in (2) which would complicate the coexplosive VAR model. The interest rate is now considered to affect mortgage costs, but not the discount factor and the expected rate of return. In the following section, I will first investigate if there is a bubble in the housing market using the actual rental price, and secondly I will use the net rental price such that the declining interest rate in the period before the crisis is taken into consideration. This may enable us to see whether a declining interest rate was an important driver for the house price increase. If
Fig. 3. Real average quarterly rental prices (dotted line) as in Fig. 1 and net real quarterly rental income (solid line). Both are deflated by CPI less shelter and seasonally adjusted.
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it was, evidence in favor of a speculative bubble in the housing market could change when correcting for decreasing interest rates. 2.3. The vector autoregressive model The economic model outlined in (1) may be estimated by assuming that Xt = (Pt, Rt)0 is a vector autoregressive (VAR) model of order k where Xt = (Pt, Rt)0 (alternatively, Rnet may be used in place of Rt in order to allow for a varying interest rate as shown t in (8)). See e.g. Campbell and Shiller (1988).
Xt ¼
k X Aj X tj þ l þ et : j¼1
In equilibrium correction form, a reformulation of this yields
D1 X t ¼ PX t1 þ
k1 X
Cj D1 X tj þ l þ et ;
ð10Þ
j¼1
where Cj = (Aj+1 + Aj+2 + + Ak) and P = (I A1 Ak). This is labeled model M for forthcoming references, and it may be reformulated in order to obtain the coexplosive model outlined in Nielsen (2010). A reformulation of (10) yields (given k P 2)
D1 Dq X t ¼ P1 Dq X t1 þ Pq D1 X t1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
j¼1
Pk1 j Pk1 jl P 1 whereDq X t ¼ X t qX t1 ; P1 ¼ 1 j¼1 q Cj Þ and Uj ¼ l¼jþ1 q Cl . Additionally, it is assumed that Xt q ; Pq ¼ q ðI p þ P1 has one unit root and one explosive root through reduced rank restrictions. The explosive root, q > 1, is a freely varying parameter obtained from the estimated characteristic polynomial if the house price is found to be explosive. A long-run restriction for the coexplosive model is that the rank is set to r = 1, i.e. a long-run stationary relationship between the rental price and house price after correcting for explosiveness. This restriction is denoted H1, and we get
D1 Dq X t ¼ a1 b01 Dq X t1 þ aq b0q D1 X t1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
ð11Þ
j¼1
labeled model M1, where a constant is also included in the cointegrating space. Furthermore, b01 Dq X t ; b0q D1 X t and D1DqXt can be given a stationary distribution such that b1 is the cointegrating vector and bq is the coexplosive vector. The cointegration rank is determined through the likelihood test procedure by Johansen (1995). The rental price is assumed to be integrated of order one such that it is non-explosive. This can be tested through the hypothesis on the coexplosive vector that
HR : bq ¼ ð0; 1Þ0 : The equation for the coexplosive model is then reduced to
D1 Dq X t ¼ a1 b01 Dq X t1 þ aq D1 Rt1 þ
k2 X
Uj D1 Dq X tj þ l þ et :
ð12Þ
j¼1
For a given value of q, the likelihood is maximized by reduced rank regression of D1DqXt on DqXt1 correcting for lagged rental price growth D1Rt1 and lagged differences D1DqXtj. This restricted model is labeled M1R. By imposing two additional restrictions on the model in (12), we get the bubble model. The first restriction is that the ‘‘spread’’ St = Pt Rt/i, as shown in (4), is a cointegrating relation. The coefficient i, which represents the discount factor (assumed constant) or the expected one-period return, is then linked to the explosive root, q, through q = 1 + i. This gives the hypothesis 0
HS : b1 ¼ ð1; 1=iÞ : Imposing this yields the model
D1 Dq X t ¼ a1 Dq St1 þ aq D1 Rt1 þ
k2 X
Uj D1 Dq X tj þ l þ et ;
ð13Þ
j¼1
labeled M1RS. In addition, the martingale restriction following the efficient market hypothesis outlined in (3) should be imposed as a restriction. This implies that (5) is imposed, such that (12) may be re-written as
D1 Dq ðPt þ Rt Þ ¼ i0 a1 Dq St1 þ i0 aq D1 Rt1 þ i0
k2 X
Uj D1 Dq X tj þ i0 a1 f1 þ i0 et ;
j¼1
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where i0 = (1, 1) such that it reduces to Mt = i0 et denoted eM if the following hypothesis is accepted
HB : i0 a1 ¼ 1;
i0 aq ¼ ð1 þ iÞ2 =i; i0 Uj ¼ 0; f1 ¼ 0:
The model M1R restricted by HS and HB may be reparameterized. x shown below denotes the population regression coefficient of eR,t = (0, 1)et on eM,t = (1, 1)et. The model is then rewritten in terms of the marginal equation for Mt and the conditional equation for D1Rt. This yields the model M1RSB:
Mt ¼ eM;t ;
D1 Rt ¼ a1;R Dq St1 þ ðaq;R þ qÞD1 Rt1 þ
ð14Þ k2 X
Uj;R D1 Dq X tj þ xMt þ eRM;t ;
ð15Þ
j¼1
where eRM,t = eR,t xeM,t is uncorrelated with eM,t, as shown in Engsted and Nielsen (2012). For a known value of i (set to i = 1 q), (14) and (15) are unrelated such that the likelihood is maximized by maximizing over i using a profile argument, i.e. estimation conditional on a known value of i. This implies that i should be set to the value that maximizes the likelihood function such that the likelihood is estimated for several values of i, where the maximum likelihood value yields the chosen value of i. The likelihood profile for various choices of the discount factor must be investigated when estimating the parameter values such that we use the value of i that provides the highest maximum likelihood value. This applies for all of the estimated restricted models outlined above. 3. Estimation The two data series used for house prices and rents are real house prices and real rental prices for the US. The data are collected from Davis et al. (2008), where quarterly time series from 1960Q1 are still being updated. These series are estimated rents and house prices for the aggregate stock of houses in the US, and I choose to use the S&P Case–Shiller price index for the house prices after 2000, since it includes all home sales, in contrast to the FHFA index which only includes conforming home mortgages. The rental price is taken from the quarterly index for the rent of primary residence published by the Bureau of Labor Statistics (BLS), and it is used to interpolate average net rent from the decennial census of housing and extrapolated beyond 2000. Since the rental price is measured as a total cost per year for each quarter, I have divided it by four in order to obtain quarterly rental costs as a measure for Rt. The rental price measures gross rental income less expenses for utilities. I have seasonally adjusted the data using the U.S. Census Bureau’s X12 seasonal adjustment program in order to remove the cyclical seasonal movements from the series prior to the estimation. In order to obtain the real house price and the real rental price, the nominal price series (after seasonally adjusting them) are deflated by the national consumer price index excluding shelter. As mentioned previously, the net rental income is calculated by using the real mortgage interest rate multiplied by the real house price and the average ratio of debt to housing wealth in order to calculate real mortgage costs. This is subtracted from the actual gross rental price. Furthermore, the real mortgage interest rate is calculated by subtracting the four-quarter average seasonally adjusted consumer price index inflation for all urban consumers for all items published by the BLS, from the nominal 30-year conventional mortgage rate obtained from the Board of Governors of the Federal Reserve System. The 30-year conventional mortgage rate is used since it is the type of mortgage loan the majority of homeowners obtain when purchasing a house (Nakajima, 2011). In order to calculate the ratio of debt to housing wealth, the additive inverse of the fraction between owner’s equity in real estate and real estate at market value for households and nonprofit organizations is computed. Both series are from the Board of Governors of the Federal Reserve System. I choose to use the sub-sample 1986Q1–2005Q1. The beginning of the sample coincides with the tax reform act of 1986 which stimulated investing in owner-occupied housing after the real estate boom pertaining to the first half of the 1980s, and starting in 1986Q1 avoids the presence of multiple house price bubbles in the sample since we exclude the period of increasing house prices before the savings and loans crisis. The sample ends in 2005Q1, since the growth in house prices declines after this period as shown in Fig. 1. This is needed since the sample has to end before the potential bubble bursts, and also illustrates that this method is useful for identifying bubbles before they burst. Ending the sample anywhere between 2001Q3 and 2005Q4 also seems to give the result of explosive behavior in house prices such that the results should not be very sensitive to the choice of the sample end period, as shown in Appendix A. By estimating the model for the sample 1960Q1–1985Q4 or for the full sample, it is not possible to find cointegration, and a lot of intervention dummy variables needs to be added in order to get a well specified model. All of the variation in house prices for the full sample is therefore hard to explain by only using data for house prices and the rental price. The sample used here (1986Q1–2005Q1) provides a well specified model without adding any intervention dummy variables, such that the housing market is properly explained by the observations on house prices and the renal price for this period. First, the unrestricted VAR will be estimated. The model needs to be well specified in order for the various tests carried out below to be valid. As shown by the misspecification tests in Table 1, the model does not contain residual autocorrelation or non-normality using a 3.9% level of significance. The test for autocorrelation is proven to be valid when the model contains explosiveness (Nielsen, 2006), and the tests for normality and autoregressional conditional heteroskedasticity (ARCH) are believed to be valid (Engsted and Nielsen, 2012). The residuals may contain ARCH, but the rank test should be robust to
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B.K. Kivedal / Journal of Macroeconomics 38 (2013) 369–381 Table 1 Multivariate misspecification tests. Residual autocorrelation AR 1-5 Test for normality Test for ARCH LM(1)
F(20, 118) = 1.379 (0.147) v2(4) = 10.086 (0.039) F(24, 186) = 3.544 (0.000)
Table 2 Restricted models and their maintained hypotheses. Model
Hypothesis
Description
M1 M1R M1RS M1RSB
H1, H1, H1, H1,
Impose rank r = 1 Rental price is non-explosive St = Pt Rt/i is a cointegrating relation Efficient market hypothesis
r=1 HR HR , HS HR , HS , HB
moderate ARCH effects (Rahbek et al., 1998) and statistical inference is quite robust to residual heteroskedasticity (Juselius, 2006, p. 47). 1 Additionally, the proper lag length for the VAR is set to two, since this is the smallest lag length which still provides no residual autocorrelation. The largest root needs to be larger than one, q > 1, such that the co-explosive model may be formulated. The rank should be r = 1, and the largest root should still be larger than unity after imposing this. Next, the various hypotheses related to the house price model and the bubble model are tested in the coexplosive VAR framework. These hypotheses were outlined in Section 2.3, and they are summarized in Table 2. First, HR is tested given r = 1 and q > 1. Then HS is tested with i = q 1, and finally HB is tested. First, we may estimate the ECM from (10) labeled model M. This may, given rank r = 1, be reparameterized to the coexplosive model M1 from (11), where the explosive root q > 1 is a freely varying parameter obtained from the estimated characteristic polynomial. Next, we test the hypothesis that Rt is non-explosive by imposing bq = (0, 1)0 . Imposing this gives the model M1R as shown in (12). For a given value of q, the likelihood is maximized by reduced rank regression of D1DqXt on DqXt1 correcting for D1Rt1, lags of D1DqXtj and a constant. The likelihood is then maximized by a profile argument over q. The hypothesis pertaining to the ‘‘spread’’, HS, may be tested by imposing b1 ¼ 1; 1 , where i = q 1 such that we get the i model M1RS shown in (13). The likelihood is again maximized over q. Finally, the efficient market hypothesis may be tested by imposing HB. 3.1. Results First, by looking at Fig. 1, we clearly see signs of explosiveness in the house price for the period before the subprime mortgage crisis while the rental price shows no such signs. Section 3.1.1 shows the results from the tests when the interest rate is not taken into account, i.e. assumed constant, when estimating the model, while Section 3.1.2 shows the results when the declining interest rate during the sample period is taken into account by using the net rental price. 3.1.1. Assuming a constant interest rate The initial model is a VAR with two lags (which is enough lags needed to avoid autocorrelated residuals). The characteristic roots are 1.104, 1.037, 0.613 and 0.136, which indicates two explosive roots in the system (although one of them cannot be rejected as a unit root). The cointegration rank test is shown in Table 3, and indicates a rank of r = 1, although we cannot reject the hypothesis of r = 0 if we have a significance level smaller than 2%. If we impose a unit root, i.e. set the rank to r = 1, the largest root is still high at 1.113, while the second to largest is imposed to be 1.000 since cointegration is imposed. This points to the anticipated result after investigating the data graphically, namely that there is one explosive root and one unit root. The results in Table 4 report whether the various hypotheses pertaining to the restrictions on the VAR model may be rejected or not.2 We see that the hypothesis of a non-explosive rental price cannot be rejected such that the explosive component of the estimated VAR model belongs to house prices through the p-value pertaining to HR. The hypothesis HS gives p-values of 0.10 and 0.15 when tested against M1R and M1, respectively, such that the hypothesis of a stationary spread between house prices and the rental price cannot be rejected either. 1 It is not proven that the same holds when explosiveness is present in the model. The tests related to the theoretical model are nevertheless carried out below, and it is assumed that the model is sufficiently well-specified based on the results from the misspecification tests even though all of them are not proven to be valid. Additionally, the quantile-quantile (QQ) plots in Appendix B does not provide evidence against the model being well-specified. 2 The test statistic is assumed to be asymptotically v2 distributed under the null hypothesis, as proven to be asymptotically valid for most of the cases here as shown in Engsted and Nielsen (2012). Hence, the p-values here are calculated from the v2 distribution. This also applies for the p-values in Table 5
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B.K. Kivedal / Journal of Macroeconomics 38 (2013) 369–381 Table 3 Rank test. Critical values for the test statistic and the p-values from Table 15.2 in Johansen (1995). Hypothesis
Likelihood
Test statistic
p-Value
r62 r61 r=0
783.811 786.491 795.213
LR(r 6 1jM) = 5.36 LR(r = 0jM) = 22.81
0.26 0.02
Table 4 Tests of the rational bubble restrictions. Model
Hypothesis
Log-likelihood
M1 M1R M1RS
H1 , r = 1 H1 , HR H1, HR, HS
786.491 787.016 788.366
M1RSB
H1, HR, HS, HB
837.900
Test statistic LR(M1RjM1) LR(M1RSjM1R) LR(M1RSjM1) LR(M1RSBjM1RS) LR(M1RSBjM1R) LR(M1RSBjM1)
1.05 2.70 3.75 99.07 101.77 102.82
d.f.
p-Value
1 1 2 3 4 5
0.31 0.10 0.15 0.00 0.00 0.00
When testing the hypothesis HS, the imposed b vector is b1 ¼ 1; 1 . Furthermore, since i = q 1 where q = 1.081 (the i root which yields the maximum likelihood), we have that the restricted estimate of i under H1R is i = 0.081. The unrestricted b vector is b1 = (1, 94.25) such that the unrestricted estimate of i is î = 0.011. Since HS is not rejected even with this large deviation between the restricted and the actual value, there is little information in the data on this parameter which measures the expected return. Additionally, the imposed value of i = 0.081 is quite high, since this implies an expected return of 8.1% per quarter. The hypothesis pertaining to the bubble, HB, is rejected. This indicates that the standard house price model with a constant discount rate and a rational bubble is not supported and excess returns do not behave as a martingale difference. This may indicate that the rental price is not fully able to explain the movements in house prices even when correcting for explosiveness. Additionally, this may be evidence of an ‘‘irrational bubble’’, since there clearly is an explosive component in house prices but the efficient market hypothesis does not hold when we correct for this. This suggests that agents do not have rational expectations but instead are driven by an irrational optimistic view of the future which influences their information set. Assuming that all individuals do not have different comparative information in acquiring information, is perhaps a strong assumption for the housing market. It is more likely that this is the case in the stock market than the housing market, since the stock market to a larger extent consists of professional investors and the housing market have a large share of private households buying and selling homes. Individual preferences and different abilities to obtain information about the market may be an important factor in the housing market. This will then act as evidence against the efficient market hypothesis, indicating an irrational bubble.
3.1.2. Correcting for a varying interest rate When using the net rent as shown in (8) instead of the actual rental price when testing the restrictions on the model, we get the results as shown in Table 5 for a VAR model with two lags. Also in this case, the model is econometrically well-specified and the largest root is larger than unity using k = 2 lags. The characteristic roots are 1.078, 0.69, 0.40 and 0.40, so there still seems to be an explosive root in the system which is only slightly smaller than in the model without the interest rate. The same pattern emerges in this case, namely that there is a rank of r = 1 and there is an explosive component that belong to house prices. The largest root is 1.056 when imposing rank r = 1, and the second to largest root is 1.000 such that there seems to be an explosive root and a unit root. HR and HS are not rejected at a 1.2% significance level, such that the explosive component belongs to house prices and there is cointegration between house prices and rental prices (which now also
Table 5 Tests of the rational bubble restrictions when the interest rate is included in the estimation. Model
Hypothesis
Log-likelihood
M1 M1R M1RS
H1 , r = 1 H1 , HR H1, HR, HS
966.726 768.147 971.168
M1RSB
H1, HR, HS, HB
1021.785
Test statistic LR(M1RjM1) LR(M1RSjM1R) LR(M1RSjM1) LR(M1RSBjM1RS) LR(M1RSBjM1R) LR(M1RSBjM1)
2.84 6.04 8.88 101.23 107.28 110.12
d.f.
p-Value
1 1 2 3 4 5
0.092 0.014 0.012 0.000 0.000 0.000
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includes effects from the interest rate). The estimated unrestricted b vector is b1 = (1, 55.50), such that the unrestricted estimate of i is î = 0.018. Hence, it is still quite small compared to the restricted value under H1R which is i = 0.081, such that the estimation still provides little information on the expected return. Furthermore, the bubble model is rejected, which indicates that the efficient market hypothesis does not hold even when we correct for the non-constant interest rate. The inclusion of the interest rate in order to use the information in its dynamics over the sample is therefore not helpful in supporting the efficient market hypothesis. Correcting for explosiveness, cointegration and the interest rate is not sufficient to obtain a result that supports the efficient market hypothesis. Hence, the difference between the stock market and the housing market still seems to be evident. However, the perhaps most interesting finding when correcting for the interest rate is that we still find evidence of a bubble. This indicates that the explosive growth in house prices prior to the subprime mortgage crisis was not a result of lower interest rate and a possibly higher credit demand boosting demand for housing. At least this was not enough to explain the explosive behavior. Psychological factors therefore seems to be the driver behind house prices when we correct for rental prices and the interest rate, such that a bubble is found in the data. This is clearly indicated by the explosive root in house prices. The lack of evidence supporting the efficient market hypothesis may also be a result of the difference between rational bubbles and irrational bubbles. Since we reject the efficient market hypothesis, this may indicate that the bubble we found was an irrational bubble. Furthermore, one of the most important differences between stock markets and housing markets is the participants in the markets. Since the housing market to a larger extent consists of unprofessional investors, it is more likely that incorrect judgments are made in the housing market since households may not obtain all available information about the market. Professional investors that to a large extent populates the stock market should be expected to be more rational. In the housing market, an irrational bubble may be caused by increased demand for houses because households have visions of improving their life when buying an expensive house, or by a trend that drives the market towards owning rather than renting houses. This trend may be because of expected increases in house prices in the future such that the households may sell their home with a profit, which will indicate a rational bubble. However, such behavior is more relevant from an investors point of view, since an investor seeks to buy a house today and sell it later with a profit. If a household should behave as a profit maximizing investor, it needs to alternate between owning and renting which is not that likely to be the case for the majority of households since a household mainly buys a house in order to live in it. When households sell their house, it is more likely to buy a new house than to rent a house. Increases in the value of their first home is then used for buying a new house. If this type of behavior is more probable than profit maximizing behavior, a potential bubble should be regarded as an irrational bubble. The difference between the results here and the results in Engsted and Nielsen (2012) therefore seems to support the difference between stock and housing markets, since the efficient market hypothesis holds when the bubble model is tested on the stock market but not on the housing market. However, as argued in Dale et al. (2005), the distinction between rational and irrational bubbles is not very clear since the misjudgment of the investors in the market needs to be investigated in order to make the distinction. Further analysis of the behavior of the agents in the housing market therefore needs to be done if one should find proper evidence for whether the bubble is rational or irrational. Another important difference between the cost of owning versus renting, is that renting is paid by after-tax income while e.g. the cost of a housing mortgage is tax deductible. However, this advantage of owning is constant for the sample period used here such that the PR-ratio should not been affected by this. Additionally, even though including the declining interest rate during the sample period do not affect the evidence of a bubble, it may cause changes in household behavior. A relatively low interest rate over a long period of time may increase the price range of houses they can afford. This can change the way households distributes their income such that a shift towards investing in more expensive houses becomes the norm, and may be viewed as behavior in line with an irrational bubble. As shown in Table 6 in Appendix A, there seems to be an explosive root if we end the sample in any period between 2001Q3 and 2005Q4.3 This may suggest that we could have been able to spot a bubble in house prices earlier than what we did when using the sample which ended in 2005Q1. The model has not been re-estimated for the various sample sizes, primarily since the model is not well specified for all periods . This would require e.g. increasing the lag length or adding intervention dummies for certain periods in order to obtain an econometrically well specified model, possibly because of the shortening of the sample size which causes a loss of information.
4. Conclusion The empirical evidence presented in this paper suggests that there is an explosive root in house prices, while the rental price does not contain explosive elements. This also holds in the case where the net rental price is used, indicating that the declining interest rate in the period before the subprime financial crisis is not a strong enough effect to explain the large 3 Although it is not known whether the root is significantly above unity since we do not know their estimated standard errors. The model is not re-estimated for different end points of the sample since the loss of information when shortening the data set would require increasing the econometric model with more lags and/or with dummy variables pertaining to institutional events.
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increase in the house price that exceeds the increase in the rental price. Hence, a speculative bubble might have been an important driving force behind the increase in house prices when we look at data prior to the crisis. The hypotheses of the theoretical model is also tested through this framework, where the equilibrium relationships between the housing market and the rental market are not rejected when tested by the econometrical model after correcting for the explosive root which takes the bubble into account. Additionally, the bubble seems to be irrational rather than rational, which indicates that there is a difference between how the housing market behaves compared to the stock market. Since the econometric methods used here were able to find evidence for a bubble in the early 2000s, it may be an important instrument for monitoring the housing market. The relevance of the method is particularly important since it does not take into consideration the timing regarding the starting point of the bubble or try to pin point the time period when the bubble bursts. If house prices deviate from their fundamental value and there is an explosive root in a VAR model estimated on the basis of these two variables (and possibly also including the interest rate), this framework should be able to find evidence for a bubble through the various tested hypotheses based on the theoretical model. Furthermore, this framework was applied to the US housing market for the sample 1986Q1–2005Q1. This implies that the estimation and test of hypotheses could be conducted the first quarter of 2005, which was before the subprime financial crisis. A bubble could then be identified such that precautionary moves, e.g. using macro prudential or monetary policy tools influencing the demand side of the housing market, could be implemented in order to dampen the growth of the bubble. The explosive behavior of house prices would then be reduced, such that a smoother adjustment towards fundamental values could be made instead of a burst of the bubble which would make the same correction more rapidly. When lenders considered the the size of mortgages to households based on their collateral value, the part of the increase in house prices unexplained by fundamental factors should not have been taken into account for determining the collateral value. This would have lowered the purchasing power of households in the housing market, such that the growth in house prices could have been decreased. The possibility of mortgage equity withdrawal would then also have been reduced, dampening its accelerating affect which influenced the real economy. This indicates that the burst of the bubble could have been avoided, or the growth in house prices prior to the burst could have been dampened. The negative effects on the real economy following the burst could also have been dampened, because house prices would have returned to a level consistent with their fundamental value by using various policy actions such as reducing the size on loans to households or the degree of subprime mortgages. This highlights the importance of separating between fundamental and psychological factors behind house price increases, in order for the housing market to function properly and thereby to avoid a financial crisis. It also highlights the relevance of using policy actions in order to correct for imbalances in the housing market. Acknowledgments The author thanks Gunnar Bårdsen, Bent Nielsen, John Leahy and four anonymous referees for helpful comments and suggestions. I would also like to thank participants at the second workshop in dynamic macroeconometrics at the University of Oslo, and discussant Hans-Martin Straume and other participants at the Norwegian PhD workshop in economics 2012 for helpful comments. This research was conducted while I was affiliated with the Department of Economics at the Norwegian University of Science and Technology. I am grateful for the PhD scholarship from the Norwegian University of Science and Technology funding my research. Appendix A. The largest unrestricted root for different end periods of the sample See Table 6. Table 6 The largest unrestricted root for the estimated model M in (10) using k = 2 lags for different sample end points for the model without including the interest rate (column ‘‘A’’) and the model using net rental price (column ‘‘B’’). End period
Largest unr. root
End period
A
B
2000Q1 2000Q2 2000Q3 2000Q4
0.9847 0.9617 0.9382 0.9207
1.013 1.007 1.020 1.020
2001Q1 2001Q2 2001Q3 2001Q4
0.9208 0.9955 1.037 1.043
2002Q1 2002Q2 2002Q3 2002Q4
1.072 1.071 1.067 1.047
Largest unrestr. root A
B
2003Q1 2003Q2 2003Q3 2003Q4
0.9583 1.054 1.013 1.080
1.049 1.070 1.066 1.079
0.9982 1.048 1.065 1.060
2004Q1 2004Q2 2004Q3 2004Q4
1.057 1.067 1.058 1.067
1.071 1.080 1.075 1.066
1.078 1.082 1.086 1.076
2005Q1 2005Q2 2005Q3 2005Q4
1.104 1.060 1.059 1.045
1.078 1.072 1.057 1.058
2006Q1
1.006
1.050
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Fig. 4. QQ plots (residuals against normal distribution) for the unrestricted ECM when the interest rate is assumed constant.
Fig. 5. QQ plots (residuals against normal distribution) for the unrestricted ECM when the interest rate is included in the net rental value.
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