# The housing bubble in real-time: the end of innocence

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DOI: 10.1007/s12197-010-9165-4

- Cite this article as:
- Peláez, R.F. J Econ Finan (2012) 36: 211. doi:10.1007/s12197-010-9165-4

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## Abstract

Market agents suffering through unanticipated boom-bust cycles would find extremely useful analytical techniques capable of serving as an early warning system. Unobserved components models and cointegration analysis are valuable in this respect. The stylized facts from unobserved components models alone do not suffice, but coupled with results from the Johansen cointegration test provided early evidence of the housing bubble and of its denouement. The paper uses real-time data vintages and shows that by 1998 the relationship between the smoothed growth rates of house prices and of per capita income was in uncharted territory. Moreover, the actual growth rates are cointegrated. This is important, as it establishes that any disequilibrium between the two becomes less tenable as its magnitude increases. By 2003, the disequilibrium was spectacular, yet it grew for another 4 years. In effect, we did not have to wait until 2008; the gruesome ending was predictable ex ante. Ironically, the greatest financial delusion of all occurred in an age that revered rationality, market efficiency, and the financial enlightenment of the TBTF actors. The empirical findings of this paper are a major problem for the rational expectations hypothesis and the remnants of the EMH.

### Keywords

CointegrationUnobserved ComponentsRationalityMarket Efficiency### JEL Classification

C220G010D8## 1 Introduction

The conventional view holds that bubbles are only recognizable ex post; otherwise, they would not occur (Kindleberger and Aliber 2005). However, Reinhart and Rogoff (2009), Borio and Lowe (2002), and Shiller (2000, 2003), document cases in the early stages of financial bubbles when deviations from empirical regularities signaled the looming disaster. It is extremely important to develop empirical methodologies capable of identifying asset price booms before they mature into full-blown bubbles.

This paper shows that an early warning system was available that could have probably averted the greatest destruction of wealth in history. The data and analytical techniques were available in real time as the bubble inflated. As early as 2001-02 the relationship between the growth rates of house prices and of income deviated sharply from the historical pattern of more than 25 years. Moreover, the quarterly growth rates of the house price index and of per capita disposable personal income are cointegrated. Therefore, as the disequilibrium grew the more certain and violent the correction had to be. Tragically, we did not have to wait until 2008 to know the denouement. The empirical findings cannot be reconciled with the rational expectations hypothesis or the efficient markets hypothesis (EMH).

By 2003, the disequilibrium was unprecedented, yet it would grow for another 4 years of delirious speculative fever. It is important to show that the crisis was not a random event like an unpredictable 100-year flood; instead, as the disequilibrium grew the more predictable was the disaster. The bubble swept into the graveyard of ideas the last remaining illusion about financial market macro-efficiency, vindicating Samuelson’s (1998) dictum.

The plan of the work is as follows. Section 1 describes the data and its sources. Section 2 describes a basic structural time series model. Sections 3 shows the smoothed growth rates that market participants could have used to formulate expectations about the growth of house prices in real time. Section 4 shows that the growth rates of the house price index and of disposable personal income per capita are cointegrated. The final section concludes.

## 2 Data

This paper is about inflation in house prices, thus clearly all variables are nominal. The Federal Housing Finance Agency (http://www.fhfa.gov) publishes quarterly and monthly house price indices for the USA, nine U.S. Census divisions, 50 states and the District of Columbia. We use the quarterly USA HPI index. The HPI is a weighted repeat-sales index for mortgages of single-family properties. Its methodology is similar to one developed by Bailey et al. (1963) and later refined by Case and Shiller (1989). However, it differs in scope and coverage from the S&P/Case-Shiller index.^{1} It is worth noting that the S&P/Case-Shiller index shows an even larger drop in prices from their peak than the HPI (OFHEO 2008). Calhoun (1996) provides a detailed technical description of the HPI. With each release, the entire HPI is revised from its inception in 1975 to the present.^{2} Since each subsequent vintage differs from previous ones, it is important to use the actual data vintages available in historical time as the bubble inflated.

^{3}The 30-year conventional mortgage rate is the average contract rate on commitments for fixed-rate first mortgages reported in the H15 Release of the Board of Governors of the Federal Reserve System. Figure 1a plots the logs of HPI (left axis), and DPI (right axis). An underlying economic relationship links the two since income determines debt-carrying capacity and is a loan qualifier.

The growth rates in Fig. 1b exhibit the noisy behavior typical of the quarterly growth rates of nominal variables. Directly computed growth rates reflect the behavior of components with different time-series properties, such as trend, cyclical, seasonal, and irregular. In order to observe more clearly the stylized facts of the underlying or core growth rates it is necessary to filter-out the higher frequency irregular and cyclical components.

## 3 Structural time series models (STMs)

Identifying the stylized facts of time series is an important part of research in economics, see e.g., Harvey (1989, 1997), Hodrick and Prescott (1980), Nelson and Plosser (1982), and Blanchard and Fisher (1989). Various techniques are available to separate the high-frequency components of a series (seasonal, irregular, and cyclical) from the more slowly evolving trend and its slope. The empirical analysis of business cycles provided the initial motivation for much of the early work. For example, it is important to know if a change in productivity is a temporary blip or a more permanent development. The Hodrick-Prescott filter (HP) was popular, but it may produce spurious cyclical behavior and distortion near the sample end-points (Cogley and Nason 1995, and Baxter and King 1999; Harvey and Jaeger 1993). Baxter and King proposed a band-pass filter to extract the business cycle components of macroeconomic activity. The band-pass filter approximates a two-sided moving average that retains components of a quarterly time series with periodic fluctuations between 6 and 32 quarters, while removing components at lower and higher frequencies.

^{4}A basic STM views a time series y

_{t}as the sum of four components, trend μ

_{t}, cycle ψ

_{t}, seasonal γ

_{t}, and a white noise irregular term, ε

_{t},

_{t}is the slope or one-period growth in μ

_{t}, while the disturbances η

_{t}and ζ

_{t}are normally and independently distributed with variances, \( \sigma_{\eta }^2 \) and \( \sigma_{\zeta }^2 \) respectively. The presence of η

_{t}allows the level to shift up or down, while ζ

_{t}allows the slope to change. The model in Eqs. (1)–(3) is essentially a time-varying parameter model.

Less general models arise depending on the variances of the disturbance terms. If \( \sigma_{\eta }^2 = 0 \) and \( \sigma_{\zeta }^2 > 0 \), we have a smooth trend model with a stochastic slope. Alternatively, if \( \sigma_{\eta }^2 > 0 \) and \( \sigma_{\zeta }^2 = 0 \), the trend is a random walk with drift. Finally, a deterministic trend emerges if both \( \sigma_{\eta }^2 \) and \( \sigma_{\zeta }^2 \) equal zero. Model selection is data dependent and is based on a measure of goodness of fit, such as prediction error variance, or R^{2} with respect to first differences \( \left( {{\hbox{R}}_d^2} \right) \).

A rich set of dynamic models results from including explanatory variables in an STM; such models combine time series and regression. For example, including the mortgage rate as an explanatory variable in a model of the HPI allows the level to reflect the mortgage rate, while the stochastic trend captures the effects of changes in tastes, government policy, expectations, or simply the *animal spirits* of market participants.

The HPI is not seasonally adjusted, hence we tested for possible seasonal effects within a local linear trend model, but the seasonal factors were not statistically significant. The X-12 ARIMA model of the U.S. Department of Commerce confirmed the absence of verifiable seasonality, thus the finally selected local linear trend model did not include seasonal factors. STAMP 8 of Koopman et al. (2007) was used in this part of the work. The model of log(HPI) included the contemporaneous logged value of the mortgage rate as an explanatory variable. In all cases, the coefficient of the mortgage rate was negative and statistically significant. A univariate model was chosen for the local linear trend model of log DPI.

## 4 Expectations and the smoothed growth rates of HPI and DPI

Local linear trend models were estimated with successive data vintages from 2003Q1 through 2006Q1 at annual intervals, in order to extract the smoothed growth rates of house prices and income that real time agents would have seen as the bubble inflated. A vintage shows the actual data available in real time. Only the graphical output is shown below due to the number of models and that estimation in STAMP produces a large amount of output. However, all results are available from the author.

Christopher Dodd (2007), Chairman of the U.S. Senate Committee on Banking, Housing, and Urban Affairs indicated that the regulatory agencies first noticed credit standards deteriorating late in 2003. Historically, the borrower’s ability to repay framed the lending decision, but this age-old practice receded as predatory lenders pushed interest-only loans, Alt-A loans, and option-ARM loans on wage earners, elderly families on fixed income, lower-income families, and others eager to “invest” in real estate.

## 5 Testing for cointegration

A finding of cointegration between the actual growth rates Dlog(HPI), and Dlog(DPI) in Fig. 1b explodes the notion that market participants behaved rationally.^{5} Testing for a unit root is the first step in testing for cointegration. It may appear unlikely for the growth rates to be integrated of order one, I(1), as this requires that the level variables be integrated of order two, I(2). Nevertheless, Juselius (2008) notes that nominal variables in levels often exhibit I(2) behavior. A non-stationary series (X_{t}) is integrated of order one, I(1), if its first difference (X_{t}–X_{t-1}), is stationary I(0). The order of integration is the number of times that a non-stationary series is differenced to obtain a stationary I(0) series. Stationarity also occurs in fractionally integrated I(d) series in which the order of integration, d, is less than 0.5. Besides differencing, other transformations may induce stationarity. In the context of this paper, if the growth rates Dlog(HPI) and Dlog(DPI) are non-stationary due to unit roots, there may exist a cointegration vector such that Dlog(HPI)–βDlog(DPI) is stationary. Cointegration vectors are of great interest because short-run disequilibria between cointegrated variables are temporary, and the long-run relationship is one of equilibrium.

The presence of a unit root in a time series is central to the issue of the persistence of shocks, i.e., the effect of a current shock on the forecast of the series. A large amount of literature develops tests of the null hypothesis of a unit root. Initially, the Dickey and Fuller (1979), augmented Dickey and Fuller (1981), and Phillips and Perron (1988) tests were popular, but Cochrane (1991) and others warned about drawing strong inferences from those tests due to their low power. Maddala and Kim (2000) more bluntly state that the Dickey-Fuller and Phillips-Perron tests should not be used any more. A related critique is that the augmented Dickey-Fuller (ADF) test often fails to reject the unit root null when the time series is fractionally integrated (see e.g., Diebold and Rudebusch 1991; Hassler and Wolters 1994; and Lee and Schmidt 1996).

In recent years significantly more powerful tests have been developed, see e.g., Elliott et al. (1996), Perron and Ng (1996), and Ng and Perron (2001). Elliott, Rothenberg, and Stock (ERS) show via Monte Carlo experiments that local Generalized Least Squares (GLS) detrending of the data together with a data-dependent lag-length selection procedure, yields substantial power improvements over the widely used ADF test. Their DFGLS test in effect supersedes the ADF test. Perron and Ng (1996) developed modified versions (MZα, MZt, MSB, and MP_{t}) of the Phillips and Perron (1988) Zα and Zt tests, Bhargava’s (1986) R1 test, and the feasible Point Optimal test of Elliott et al. (1996). Ng and Perron (2001), extended those four tests to allow for GLS detrending of the data, and introduced a class of data-dependent Modified Information Criteria for selecting the lag length of the autoregressive process. These refinements yield tests with desirable size and power properties (Ng and Perron 2001).

Test statistics and 5% probability values, in parentheses, for tests of the unit root null hypothesis

Variable | Dlog(HPI) | Dlog(DPI) | ||
---|---|---|---|---|

Exogenous variables: | Constant | Const. & trend | Constant | Const. & trend |

ADF | −1.72 (−2.88) | −2.25 (−4.03) | −1.26 (−2.88) | −2.81 (−3.44) |

DFGLS (CV5%) | −1.05 (−1.94) | −2.45 (−2.99) | 0.71(−1.94) | −1.89 (−3.00) |

MZα (CV 5%) | −5.06 (−8.1) | −11.16 (−17.3) | 0.48 (−8.1) | −0.56 (−17.3) |

MZt (CV 5%) | −1.35 (−1.98) | −2.33 (−2.91) | 0.82 (−1.98) | −0.42 (−2.91) |

MSB (CV 5%) | 0.27 (0.23) | 0.21 (0.168) | 1.72 (0.23) | 0.75 (0.168) |

MPT (CV 5%) | 5.44 (3.17) | 8.35 (5.48) | 170.5 (3.17) | 107.8 (5.48) |

Each I(1) growth rate may meander without converging to a long-run level. However, if they are cointegrated a linear combination of the two is stationary, i.e., the growth rates move towards a state of long-run equilibrium with each other. The economic significance is that any disequilibrium between cointegrated variables becomes more unlikely as its magnitude grows. In the bubble’s context, cointegration implies predictability. Short-run disequilibria induced by a sharp rise in the growth of the HPI must eventually ebb as the growth of the HPI returns to its long-term cointegrating equilibrium with the growth rate of DPI.

The EMH has retreated from the 1970s Panglossian view of “the price is right,” to admitting that investors may behave irrationally, at times under-reacting or over-reacting and unwittingly creating bubbles. However, if these bouts of irrationality are random and thus unpredictable like a 100-year flood, the EMH allegedly survives. Fama (1998) admits that researchers have documented irrationalities and biases, but claims that critics have not shown how to exploit the irrationality of others to earn an abnormal return, i.e., irrationality has to be systematic enough to make prices predictable to the point of economic significance. According to this reasoning, a finding of cointegration is a problem for then the bubble’s inevitable and ruinous denouement would have been predictable.

Johansen (1988) developed a likelihood ratio test for cointegration; Johansen and Juselius (1990), and Johansen (1991) refined and extended the test. In a VAR system consisting of *n* I(1) variables, the cointegration rank *r* (the number of cointegration equations), is bounded by the interval 0≤*r*≤(n–1).^{6} The first step is to test the null of no-cointegration (*r* = 0); the null is rejected if the likelihood ratio (LR) exceeds some critical value, say, 1%. One then tests whether there is one cointegration equation, i.e., *r* = 1, and continues testing ever higher *r* until the null is not rejected.

The Johansen test allows for the inclusion of intercepts, linear or quadratic trends, seasonal factors, intervention variables, and exogenous variables; typically, an information criterion selects the lag length in the VAR. The finally selected VAR system included a restricted constant since the growth rates do not have linear trends, three lags of Dlog(DPI) and DlogHPI), and five intervention dummy variables to account for transitory shocks.^{7}

Other minor shocks occurred earlier. The 1980 recession stretched from January through July. In 1980:Q4 employment, average weekly hours, and real GDP rebounded strongly, inducing a blip in the growth rate of DPI that is dummied-out by D804_{t} = 1 for *t* = 1980:Q4, and zero otherwise. In 1992, real GDP growth averaged over 4.3%, but in 1993:Q1 growth decelerated sharply to 0.7%. The sudden downshift induced a sharp drop in the growth rate of DPI on 1993:Q1 which was dummied out via, D93_{t} = 1 for (*t* = 1993:Q1), D93_{t} = −1 for (*t* = 1993:Q2), and D93_{t} = 0 otherwise. Another blip occurred on 2005:Q1, when the growth rate of DPI dropped from 2.2% in 2004:Q4 to −0.006% on 2005:Q1; hence D05_{t} = 1 for 2005:Q1, and D05_{t} = 0 otherwise.

*r*= 0), (P-value = 0.000), whereas the null that

*r*= 1 is not be rejected in favor of

*r*= 2, (P-value = 0.855).

^{8}In short, the growth rates tend to gravitate to a long-run equilibrium, underscoring the incongruity of the bubble that nearly destroyed the U.S. economy.

*r*= 1 (one cointegrating vector) is not be rejected.

This table shows the Johansen cointegration trace-test statistic and its probability value for various sample periods

Effective sample | Ho: rank<=0 Trace test [Prob.] | Ho: rank<=1 Trace test [Prob.] |
---|---|---|

1976:1–1998:4 | 25.246 [0.008]** | 3.6306 [0.481] |

1976:1–1999:4 | 25.963 [0.006]** | 3.8653 [0.445] |

1976:1–2000:4 | 27.306 [0.004]** | 4.4147 [0.366] |

1976:1–2001:4 | 22.360 [0.023]* | 4.5330 [0.35] |

1976:1–2002:4 | 23.243 [0.017]* | 4.8871 [0.307] |

1976:1–2003:4 | 25.880 [0.006]** | 5.2246 [0.269] |

1976:1–2004:4 | 25.226 [0.008]** | 4.9677 [0.297] |

1976:1–2005:4 | 23.199 [0.017]* | 5.2279 [0.269] |

1976:1–2006:4 | 25.942 [0.006]** | 6.3059 [0.174] |

1976:1–2007:4 | 25.486 [0.007]** | 5.2184 [0.27] |

1976:1–2008:4 | 23.898 [0.013]* | 2.6197 [0.659] |

1976:1–2009:4 | 26.040 [0.006]** | 2.2020 [0.737] |

## 6 Conclusions

The EMH claims that it does not matter if some investors are irrational some of the time, and if others are irrational *all* of the time, because the smart money takes advantage of the irrationality of others, thus keeping markets efficient. However, no amount of theorizing can bury the fact that the superbly informed agents of the TBTF created the greatest financial disaster since the 1930s.

Using the empirical evidence available in real-time this paper has demonstrated an early warning system of the housing price boom. Both the data and analytical techniques used were available in the early stages of the bubble to document the imbalances that eventually had to be resolved. The same techniques may serve in other cases.

The Johansen cointegration test shows that the relationship between the actual growth rates is stationary, i.e., that the disequilibrium term has time-invariant properties. As early as 2001-02, the gruesome ending was predictable ex ante; as the short-run disequilibrium grew, the more certain a drastic drop in house prices became.

In the wake of the Great Depression, economists created a revolution in macroeconomics. The greatest bubble in history put to rest the idea that the price is right in the macro sense, validating Samuelson’s (1998) dictum that financial markets are *macro*-*inefficient*. Its most durable legacy is likely to be that generations of economists will develop models in which irrationality plays a larger role in the determination of asset values.

The Case-Shiller index, unlike the HPI, is not national in coverage; it omits 13 states and has incomplete coverage for 29 other states (Leventis, 2007).

The HPI began publication on January 1975, by the Office of Federal Housing Enterprise Oversight (OFHEO)—an agency that operated within the Department of Housing and Urban Development. On July 2008, the Housing and Economic Recovery Act combined OFHEO and the Federal Housing Finance Board into the new Federal Housing Finance Agency.

STMs have been used to model the behavior of exchange rates (Harvey et al. 1992); forecast consumer expenditures (Harvey and Todd, 1983); model inflation and the output gap (Kuttner 1994; Domenech and Gomez 2006; and Harvey 2008). Other examples include modeling inflation persistence (Stock and Watson 2007, and Dossche and Everaert 2005); productivity growth (Peláez 2004, and Crespo 2008); business cycles (Clark 1987), earnings per share (Peláez 2007); permanent income (Huang et al. 2008); the U.S. regional housing market (Fadiga and Wang 2009); modeling the core unemployment rate, (Harvey and Chung 2000); and testing for deterministic trends (Nyblom and Harvey 2005).

Dlog(HPI) is the first-difference of the natural logarithm of HPI. Clearly, this section does not use the smoothed growth rates.

The Schwarz Criterion and the Hannan-Quinn Criterion reached their minima for a lag length of three quarters. The finding of cointegration is robust to an unrestricted constant and to the exclusion of any intervention variables as well.