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英国论文指导-非对称变动对英国房地产价格的影响The asymmetric volatility of house prices in the UK
Abstract
Purpose – The purpose of this paper is to show an indication that the asymmetric volatility between house price movement may account for the defensiveness of the housing market.
Design/methodology/approach – First the UK nation-wide house price data from the last quarter
(Q4) of 1955 to the last quarter of 2005 are used and then the most suitable mean and variance
equations to estimate the conditional heteroscedasticity volatilities of the returns of house prices are
selected. Second, a variable that examines the leverage effect of volatility is incorporated into the
model. The GJR-GARCH model is used.
Findings – The results of the empirical test show that while the lagged innovations are negatively
correlated with housing return, that is when there is bad news, the current volatility of housing return
might decline.
Research limitations/implications – The results indicate that the volatilities between house
prices moving up and moving down are asymmetric.
Practical implications – The results show that there is a defensive effect in the UK housing market
during the data periods used.
Originality/value – Although several articles have documented that there is heteroscedasticity and
autocorrelation in the volatilities of real estate prices, few of those papers have noted one of the most
important advantages of the housing market, its defensiveness, from the viewpoint of volatile
behavior.
Keywords Real estate, Prices, Cyclical demand, Residential property, United Kingdom
Paper type Research paper
Introduction
Over the past few decades, the highly volatile behavior of house price series has been
recognized in a number of studies, and several empirical articles have tried to capture
the short-term adjustment process of house prices (Meen, 1990; Drake, 1993; Heiborn,
1994; Eitrheim, 1995; Abraham and Hendershott, 1996; Malpezzi, 2001, Kapur, 2006).
Most of these studies try to model price behavior based on factors related to the
housing demand and supply, and further use traditional regression for analysis. But,
most of these economic models do not come up with satisfying performance capturing
the volatile behavior of the housing markets.
Several methodologies and models are now being used in order to reduce the
heteroscedasticity problems of house price and estimate this volatile process. For
example, house prices were always converted to natural logarithms in empirical tests
in previous house price studies in order to reduce the heteroscedasticity variance
problems in ordinary least squares (OLS) regression. On the other hand, Hendry (1984)
and Giussani and Hadjimatheou (1990) have tried to model the volatility by using
The current issue and full text archive of this journal is available at
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Received December 2007
Revised October 2008
Accepted January 2009
Property Management
Vol. 27 No. 2, 2009
pp. 80-90
q Emerald Group Publishing Limited
0263-7472
DOI 10.1108/02637470910946390
non-linear specifications to capture extreme movements in house prices. Hendry (1984)
used a cubic approximation function, calculated as the cubic term of house price
changes. Giussani and Hadjimatheou (1990) used both square and cubic terms to
capture the rapid adjustment of house price.
Because high volatility is very common in financial data, the family of ARCH
(Engle, 1982) and GARCH (Bollerslev, 1986) models were developed and are widely
applied to model the variance of financial variables. These types of models allow the
conditional variance of a series to depend on the past realizations of the error process
and simultaneously model the time-dependent mean and variance. Because of their
excellence in capturing volatility, these types of models are applied to various areas,
including housing studies (Dolde and Tirtitoglu, 2002; Miller and Peng, 2006; Tsai et al.,
2008).
However, the volatility characteristic of housing markets might be distinct from
that of other financial markets, and it is surprising that few previous studies have
documented the main features of housing market volatility. However, similar studies
have been done in the securities market, and several articles have indicated that there
is a leverage effect in the volatile behavior of securities; that is, the impact on future
assets’ return when unexpected bad news is announced is larger than when unexpected
good news is made public. Black (1976) pointed out that an unexpected decline of stock
prices can result in an increase of the ratio of debt to equity, leading to an increase in
the financial risk of corporations and eventually a higher fluctuation of securities’
price. But it is not yet known whether there is also a leverage effect in the volatility of
housing market.
The main purpose of this paper is to study the volatility properties of the UK house
price series, in particular, to examine whether or not there is a leverage effect in the
volatility of house prices. To consider the asymmetric volatility, the GJR-GARCH
model is used. Glosten et al. (1993) and Zakoian (1994) suggest a GJR-GARCH model in
which the relation between lagged error term and current volatility may be dependent
on the sign of lagged error. The error is the difference between true value and
estimated value, hence, when lagged error is negative, it means that unexpected bad
news has come out, whereas when lagged error is positive, it indicates good news.
Therefore, the asymmetric volatility model, GJR-GARCH, can measure the concept of
leverage effect empirically. Following the work of Glosten et al. (1993) and Zakoian
(1994), we can determine the important characteristics of the volatile behavior in the
housing market.
This paper is structured as follows. The next section describes the methodologies,
while section three shows a review of our data and tests using time-series properties.
Estimation results are reported and discussed in section four, and the results of
robustness test are showed in section five. The last section provides a summary of the
main findings and draws some conclusions.
Methodology
To capture volatility of house prices, we employed the ARCH-type model to model the
volatility of house price changing over time, and used the GJR-GARCH model to see
whether or not there is a leverage effect in the variance process of house price. These
two kinds of models are discussed below:
Volatility of
house prices
81
Modeling volatility of house prices over time: ARCH and GARCH models
Many economic time-series do not have constant mean and volatility. Engle (1982)
showed that it is possible to simultaneously model the time-dependent mean and
variance through the widely known Autoregressive Conditional Heteroskedastic
(ARCH) model. This allows the conditional variance of a series to depend on the past
realizations of the error process. Bollerslev (1986) extended Engle’s original work by
developing the Generalized Autoregressive Conditional Heteroscedasticity (GARCH)
model that allows for both autoregressive and moving average components in the
heteroskedastic variance. We briefly illustrate the features of these two models in the
following.
ARCH model. Let yt denote the return of house price at time t. The error process is
obtained from a first-order autoregression for yt following the ARCH (q) model, and it
can be specified as:
yt ¼ a0 þ a1yt21 þ 1t
1t
V~j t21Nð0; htÞ
ht ¼ v0 þX
q
i¼1
ai12
t2i
where q is the number of ARCH terms, and ht is the heteroskedastic conditional
variance, which is correlated with the lagged error terms.
GARCH model. If the error process obtained from a first-order autoregression for yt
follows the GARCH(p,q) model then it can be specified as:
1t
V~j t21Nð0; htÞ
ht ¼ v0 þX
p
i¼1
biht2iX
q
i¼1
ai12
t2i
where ht is the heteroskedastic conditional variance, correlated with the lagged error
terms and conditional variance.
Asymmetric volatility in the housing market: GJR-GARCH
With the ARCH-type models we have shown above, the heteroskedastic conditional
variance is symmetrically correlated with the lagged error terms. Since it does not
matter whether the lagged error is negative or positive, the relation between the
heteroskedastic conditional variance and the lagged error terms is a constant
coefficient, namely, ai . Hence, those models may not be appropriate for a series that has
asymmetric volatility or one that arises from markets in which there is a leverage
effect. To deal appropriately with the series having asymmetric characteristics, the
Glosten, Jagannthan, and Runkle (GJR)-GARCH model is used. The features of the
GJR-GARCH model are briefly described as follows.
Let yt denote the return of house price at time t. The error process is obtained from a
first-order autoregression for yt following the GJR-GARCH (p, q) model, and it can be
specified as:
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yt ¼ a0 þ a1yt21 þ 1t
1t
V~j t21Nð0; htÞ
ht ¼ v0 þX#p#分页标题#e#
p
i¼1
biht2i þX
q
i¼1
ai12
t2i þg12
t21Dt21
where Dt21 is a dummy variable, when 1t21 , 0, Dt21 ¼ 1; otherwise, Dt21 ¼ 0. The
coefficient g represents the asymmetric feature of conditional variance. If the estimated
results show that g is significantly not equal to zero, it indicates that the volatile
behavior of house price is not symmetric. The relation between conditional variance
and lagged error is dependent on whether or not the lagged error is negative. If g is
positive, a leverage effect exists, because when bad news comes out (i.e. the lagged
error is negative), the variance will increase. In contrast, if g is negative, an
anti-leverage effect exists because following bad news (i.e. the lagged error is negative)
the variance will decline.
Data description
We use the UK nation-wide house price data starting from the last quarter of 1955 to
the last quarter of 2005. All house and new house price data are compared. The data set
used in our analysis consists of quarterly observations on all house prices (Allph) and
new house prices (Newph).
Table I presents a summary of the descriptive statistics for two house price
variables. It also reports the outcome of tests for stationarity. An Augmented
Dickey-Fuller (Said and Dickey, 1984) test and a Phillips and Perron (1988) test both
confirm that two house price variables are I(1). Evidently, the unit-root hypothesis
cannot be rejected at the 5 per cent significance level for two series in levels. In
AllPh NewPh
Variable
Number of observations 213 213
Mean 476.77 499.97
Std dev. 205.43 199.77
Skewness 1.24 0.72
Kurtosis 4.79 3.35
Variables in level
ADF test 0.79 0.31
0.99 0.98
PP test 0.83 0.39
(0.99) (0.98)
Variables in differenced
ADF test 24.85 27.68
(0.00) (0.00)
PP test 26.99 28.03
(0.00) (0.00)
Notes: One-sided p-values are reported in parentheses
Table I.
Descriptive statistics
Volatility of
house prices
83
addition, tests applied to differenced data favor the stationary alternative for two
series. To avoid the problem of spurious regression throughout the paper, we use the
first-difference data to estimate the empirical models.
To observe the volatility of house prices, Figure 1 plots the quarterly time series of
two house prices for the sample period. We can observe that the two price series have
increased in non-monotonic ways during the sample period. The two series seem to
have changes in volatility, showing up and down movements. Hence, this paper
emphasizes the volatile properties of house prices.
Empirical results
Modeling volatility of house prices over time: ARCH and GARCH models
Before estimating the house price volatility, we need to determine the mean equation.
We use the lagged data of house price as the independent variables, and choose the
model which can minimize the value of Akaike Information Criterion (AIC) and
Schwartz Bayesian Criterion (SBC) to decide the number of lag terms. Due to the
consideration of the degrees of freedom, only lags of length 1 to 4 are tested. The
empirical results of different AR models are shown in Table II.
Table II shows that the AR(1) model for new house prices is the most appropriate
because these two model selection criteria, derived from the first-order autoregression
model, perform better than the other AR models. For all house prices, the SBC criteria
suggest similar results, although AIC suggests that AR(4) might be better. Because the
SBC is asymptotically consistent, whereas the AIC is biased toward selecting an
over-parameterized model, we choose the AR(1) model for all housing markets.
Figure 1.
House prices in the all
housing and new housing
markets
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Before we use the ARCH and GARCH models, it is necessary to test whether or not
ARCH effects exist in the data. We use the formal Lagrange multiplier test for ARCH
disturbances, as proposed by Engle (1982), and results of the LM test are shown in
Table III.
Table III shows that the disturbances obtained from the first-order autoregression of
two series are autocorrelated, which means the variance (risk) in the two house markets
are time dependent. Then, we use theARCHandGARCHmodels to estimate the variance
of the two series at a particular point in time. We estimate three different ARCH and
GARCH models to determine the most appropriate model for volatilities of the two
housing markets. These are the AR(1)-ARCH(1), AR(1)-ARCH(2), andAR(1)-GARCH(1,1)
models. The results of these three-model estimations are shown in Table IV.
Three ARCH-family models for two series are estimated and the results are presented
in Table IV. The estimated coefficients of ARCH and GARCH effects are highly
significant in each series. The results show that the GARCH model might perform better
than the other models, since the lower values of AIC and SBC and the innovations are
closer to being white noise, because the Q statistics and Q-squared statistics are lowest.
Hence, the AR(1)-GJR-GARCH(1,1) models are used to estimate the leverage effect.
Furthermore, the three ARCH-family models might not be appropriate for the two house
markets, because the sum of estimated coefficients describing the relation between the
conditional variance and the lagged error and lagged variance are larger than one. The
inappropriateness of these models might because the models do not incorporate the
asymmetry of volatility. In observing the feature of the volatility, Figures 2and3showthe
returns of house price and the conditional volatilities as estimated by GARCH (1,1). The
returns of house price and the estimated volatilities seem to be positively correlated in the
two markets; that is, there is a higher return with higher volatility and a lower return with
lower volatility. Hence, asymmetric volatility might exist in the markets. The
GJR-GARCH model is used to deal with the inference more carefully.
AR(1) AR(2) AR(3) AR(4)
All house price
AIC 7.92 7.93 7.93 7.88
SBC 7.95 7.98 7.99 7.96
New house price
AIC 8.03 8.04 8.05 8.04
SBC 8.06 8.08 8.11 8.12
Table II.
Estimates of AR models
Lags of length 1 2 3 4
All house price
TR2 4.81 7.03 7.11 22.70
p-value 0.03 0.03 0.07 0.00
New house price
TR2 3.04 7.80 12.34 13.32
p-value 0.08 0.02 0.01 0.01
Note: H0, there are no ARCH effects
Table III.
Results of ARCH
effect test
Volatility of
house prices
85
Figure 2.
Returns and estimated
conditional volatilities for
the all house market
Model ARCH(1) ARCH(2) GARCH(1,1)
All house New house All house New house All house New house
Mean equation
a0 1.44 1.48* 1.76* 1.22* 0.67 0.64
(0.93) (0.37) (0.89) (0.43) (0.47) (0.48)
a1 0.66* 0.53* 0.70* 0.59* 0.61* 0.46*
(0.05) (0.02) (0.05) (0.04) (0.07) (0.08)
Variance equation
v0 114.40* 56.38* 75.20 14.22* 0.26 0.58
(8.67) (10.38) (7.52) (5.70) (0.48) (0.53)
a1 0.33* 1.61* 0.33* 1.60* 0.15* 0.17*
(0.12) (0.31) (0.12) (0.28) (0.04) (0.03)
a2 – – 0.34* 0.44* – –
– – (0.11) (0.14) – –
b1 – – – – 0.89* 0.87*
– – – – (0.03) (0.02)
Adj R-squared 0.42 0.31 0.41 0.31 0.41 0.29
AIC 7.89 7.98 7.84 7.84 7.43 7.62
SBC 7.95 8.04 7.92 7.92 7.51 7.70
Q(20) 211.79* 46.39* 219.82* 40.49* 131.08* 40.22*
Q2(20) 58.79* 19.43 47.89* 13.12 26.55 6.50
Notes: Model: Let yt denote the series of the differenced house price
yt ¼ a0 þ a1yt21 þ 1t
1t jVt21
~ jNð0; htÞ
ht ¼ v0 þX
p
i¼1
biht2iX
q
i¼1
ai12
t2i
* indicates significance at the 5 per cent level; numbers in parentheses are standard errors. Q(20) and
Q2(20) are the Ljung-Box statistic based on the standardized residuals and the squared standardized
residuals respectively up to the 20th order
Table IV.
Empirical results of
ARCH-family
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Asymmetric volatility in the housing market: GJR-GARCH
To see whether or not the conditional volatilities of house prices switch are dependent
on the news (i.e. bad or good), the GJR-GARCH (1,1) model is used, The results of
estimations are shown in Table V.
As we can see in Table V, the coefficient (g) that represents the asymmetric feature
of conditional variance is very significant, indicating there are asymmetric volatilities
in both housing markets. Furthermore, both g are negative, indicating that there are
anti-leverage effects in two markets. This is because when there is bad news (i.e. the
lagged error is negative) the variance will decrease, which is in contrast with the
findings of those empirical articles using data from stock markets. This research
shows that the asymmetric volatility between house prices moving up and down might
account for the defensiveness of the housing market.
In addition, the results inTableVindicate that theGJR-GARCHmodel performs better
than GARCH, which does not incorporate asymmetric volatility, since the values of AIC
and SBC are lower and the innovations are closer to white noise. Finally, the#p#分页标题#e#
inappropriateness of the ARCH and GARCH models might be because they do not
considerate asymmetric volatility, sinceg are negative. Therefore, when we consider the
anti-leverage effect, the sum of estimated coefficients describing the relation between the
conditional variance and the lagged error and lagged variance will decrease.
Robustness test
The results in Table V are obtained by assuming that the volatility of house price
would only change with innovations. In this section, we proceed to test whether the
asymmetric volatility can be influenced by the economic conditions by using the real
Gross Domestic Product (GDP) and the interest rate of three-month Treasury Bills in
the UK as proxy variables for the robustness test. These two variables are put into the
variance equation of the GJR-GARCH (1,1) model, with the following results.
As we can see in Table VI, although economic variables are incorporated in this
model, asymmetric volatility still exists in the two markets, and since the coefficient g
Figure 3.
Returns and estimated
conditional volatilities for
the new house market
Volatility of
house prices
87
is also significantly negative, we can say there is still a anti-leverage effect in the UK
housing market after controlling for the influence from macroeconomics.
Conclusion
This paper studies volatility properties in the UK house price series, in particular, to
examine whether or not there is a leverage effect in the volatility of house price. To
estimate the conditional volatility, the ARCH-type models are used, and to capture the
leverage effect (asymmetric volatility effect), a GJR-GARCH model is used.
The data are composed of UK nation-wide house prices from 1955 to 2005;
comparing all house and new house price data. Empirical tests show that the
coefficient representing the asymmetric feature of conditional variance is very
significant, indicating that there is asymmetric volatility in the two housing markets.
Furthermore, the results show that there are anti-leverage effects in both markets
because when bad news is announced, the lagged error is negative and the variance
will decrease. This is in contrast with the findings of empirical articles using data from
stock markets. This research shows that the asymmetric volatility between house price
movement might account for the defensiveness of the housing market.
GJR-GARCH(1,1)
Model All house New house
Mean equation
a0 0.83* 0.87*
(0.35) (0.16)
a1 0.52* 0.50*
(0.05) (0.06)
Variance equation
v0 0.41* 0.56*
(0.13) (0.11)
a1 0.08* 0.05*
(0.01) (0.00)
b1 1.01* 1.02*
(0.01) (0.01)
g 20.19* 20.16*
0.03 0.02
Adj R-squared 0.39 0.30
AIC 7.35 7.50
SBC 7.44 7.69
Q(20) 130.50* 31.15*
Q2(20) 21.05 6.50
Notes: Model, let yt denote the series of the differenced house price
yt ¼ a0 þ a1yt21 þ 1t
1t jVt21
~ jNð0; htÞ
ht ¼ v0 þb1ht21 þa112t
21 þgDt12t
21
* indicates significance at the 5 per cent level, where Dt21 is a dummy variable, when 1t21 , 0, then
Dt21=1, otherwise, Dt21=0; numbers in parentheses are standard errors; Q(20) and Q2(20) are the
Ljung-Box statistics based on the standardized residuals and the squared standardized residuals,
respectively, up to the 20th order
Table V.
Empirical results of
GJR-GARCH(1,1) model
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It is important to point out a limitation of this study, that house price series are difficult
to construct because housing is not a homogeneous asset. Houses differ according to a
variety of qualitative characteristics relating to their physical attributes, and so a good
adjustment for quality change in the series is desirable. However, our study requires a
relatively long-term data series for ARCH type analysis but the data we use are
compiled by weighted averages. The limitation in quality adjustment could result in
over-stating the possible fluctuation of the house price series. However, a good
adjustment for quality change was not an option available in this study and this will
leave further research.
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GJR-GARCH(1,1)
Model All house New house
Mean equation
a0 1.16 * 1.49 *
(0.14) (0.58)
a1 0.53 * 0.50 *
(0.05) (0.07)
Variance equation
v0 21.48 22.58
(1.71) (2.54)
a1 0.09 * 0.06
(0.01) (0.03)
a2 2156.43 29.69
(105.16) (289.16)
a3 0.69 0.71
(0.35) * (0.43)
b1 1.03 * 1.00 *
(0.02) (0.02)
g 20.30 * 20.17 *
(0.05) 0.04
Adj R-squared 0.38 0.29
AIC 7.44 7.69
SBC 7.57 7.82
Q(20) 161.62 * 33.83 *
Q2(20) 36.32 * 7.77
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yt ¼ a0 þ a1yt21 þ 1t
1t jVt21
~ jNð0; htÞ
ht ¼ v0 þb1ht21 þa112t
21 þa2DGDPt þa3it þgDt12t
21
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Corresponding author
Ming-Chi Chen can be contacted at: [email protected]
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