# RECORD bbc85c0376ef4c6974813a189124a40708684478
# TITLE: PRICE CHANGES OF REPEAT-SALES HOUSES IN KAOHSIUNG CITY: ANALYSES BASED ON HIERARCHICAL LINEAR GROWTH MODELS
# YEAR: 2023.0  VENUE: International Journal of Strategic Property Management  DOI: 10.3846/ijspm.2023.19935
# AUTHORS: Chun-Chang Lee; Yu-Chen Wang; Chih-Min Liang; Zheng Yu
# PAGES: 14  CANDIDATE TABLE PAGES (1-based): [9, 11]
# ABSTRACT: This study adopts the hierarchical linear growth modeling approach to analyze the differences in the changes of repeat-sales house prices in Kaohsiung City from 2012 to 2020. The Level 1 time-varying factors include house age and the time of repeat-sales; the Level 2 factors include house attributes such as house area, house type, and house location. Based on the results of the null model, the estimated variance is 0.42816, with a 1% level of significance. This shows that significant differences exist in the mean repeat-sales prices between houses. The interclass correlation coefficient is 91.65%, showing that the interclass variation and intraclass variation of the mean repeat-sales prices are 91.65% and 8.35%, respectively. The estimation results of the non-randomly varying slope model indicate that the sales time and sales time squared significantly affect repeat-sales prices. The annual growth rate and quadratic growth of sales prices do not differ by house type (luxury condominiums and apartment buildings) but are affected by house area and house location. The effect of house age on repeat-sales prices is moderated by house area, house type, and house location.


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Copyright © 2023 The Author(s). Published by Vilnius Gediminas Technical University
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unre-
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Introduction
On August 1, 2012, Taiwan began to implement a real 
estate actual price registration scheme. Before this, the 
previous sales prices of a new or pre-owned house had 
remained unavailable to housebuyers. However, the actual 
price registration scheme is unable to present the repeat-
sales prices of the same house at different points in time. 
Therefore, the actual price registration scheme does not 
reveal whether changes in the price of a single house over 
time are in line with market conditions.
The repeat-sales model was first proposed by Bailey 
et al. (1963) and was later reinterpreted and applied by 
Case and Shiller (1987). This model centers on analyzing 
the selling price of a single house at different points in 
time; it is generally used to formulate a real estate prop-
erty price index. The interference of external factors can 
be reduced by observing the same property at different pe-
riods, assuming that the structural, material, and external 
quality of the property remains unchanged. By combin-
ing the repeat-sales approach with weighted indices and 
the hedonic price model, Shiller (1991) examined house 
International Journal of Strategic Property Management
ISSN: 1648-715X / eISSN: 1648-9179
2023 Volume 27 Issue 5: 290–303
https://doi.org/10.3846/ijspm.2023.19935
*Corresponding author. E-mail: lcc@mail.nptu.edu.tw
PRICE CHANGES OF REPEAT-SALES HOUSES IN KAOHSIUNG CITY: 
ANALYSES BASED ON HIERARCHICAL LINEAR GROWTH MODELS
Chun-Chang LEE1,*, Yu-Chen WANG1, Chih-Min LIANG2, Zheng YU3
1 Department of Real Estate Management, National Pingtung University, Pingtung City, Taiwan
2 Department of Public Finance and Tax Administration, National Taipei University of Business, Taipei City, Taiwan
3 Department of Land Economics, National Chengchi University, Taipei City, Taiwan
Received 18 January 2023; accepted 7 September 2023
Abstract. This study adopts the hierarchical linear growth modeling approach to analyze the differences in the changes of 
repeat-sales house prices in Kaohsiung City from 2012 to 2020. The Level 1 time-varying factors include house age and 
the time of repeat-sales; the Level 2 factors include house attributes such as house area, house type, and house location. 
Based on the results of the null model, the estimated variance is 0.42816, with a 1% level of significance. This shows that 
significant differences exist in the mean repeat-sales prices between houses. The interclass correlation coefficient is 91.65%, 
showing that the interclass variation and intraclass variation of the mean repeat-sales prices are 91.65% and 8.35%, respec-
tively. The estimation results of the non-randomly varying slope model indicate that the sales time and sales time squared 
significantly affect repeat-sales prices. The annual growth rate and quadratic growth of sales prices do not differ by house 
type (luxury condominiums and apartment buildings) but are affected by house area and house location. The effect of 
house age on repeat-sales prices is moderated by house area, house type, and house location.
Keywords: hierarchical linear growth modeling, repeat-sales, house prices, house area, house type, house location.
price indices in Atlanta, Chicago, Dallas, and San Francis-
co from 1970 to 1986. Clapp and Giaccotto (1998) wrote 
that the repeat-sales model increases the precision validity 
of the standard error in an estimator indicator. Recently, 
Beltrán et al. (2019) adopted the repeat-sales model to an-
alyze the effects of flooding in England on property prices. 
The results show that property prices in completely inun-
dated areas fell by 21%, although this reduction was only 
significant within five years. On the other hand, the effects 
of flooding on lower-priced houses lasted for 6 to 7 years. 
Ryu and Song (2020) applied the repeat-sales model to 
exploring the price of office spaces in Seoul. According to 
their results, the price of office spaces per 3.3 m2 per year 
has risen continuously by 12% from 2000 to 2019.
However, many studies have analyzed the factors af-
fecting house prices using the hedonic price model (Ros-
en, 1974; Nelson & Schumaker, 2001; Umanailo et  al., 
2019). Hox (1995) uniquely pointed out that estimating 
different levels of characteristic variables at the same in-
dividual level will lead to fallacies in statistical inference. 
Giuliano et al. (2010) suggest that hierarchical linear mod-
eling (HLM) can control the variation in research data 


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C.-C. Lee et al. Price changes of repeat-sales houses in Kaohsiung city: analyses based on hierarchical...
transferred house area among the 3,167 pieces of data is 
125.079 m2 (SD = 60.879 m2); houses located in the sub-
urbs accounted for 8.6% (273 pieces) of the sample, while 
houses located in the CBD accounted for 91.4% (2,894 
pieces) of the sample; condominiums, luxury condomini-
ums, and apartment buildings accounted for 20.1% (637 
pieces), 10.3% (325 pieces), and 69.6% (2,205), respectively.
In the same data period, there were 136,913 pieces of 
single sales data, and the mean house area was 134.48 m2 
while the mean house age was 19.95 years. In terms of 
house type, condominiums, luxury condominiums, and 
apartment buildings accounted for 19.3%, 9.1%, and 
71.6%, respectively. In terms of location, houses in the 
suburbs accounted for 9.1% while houses in the CBD ac-
counted for 90.9%. The house attribute statistics did not 
vary greatly between the two samples.
4. Empirical results and analysis
4.1. Null model
A null model was developed for this study to check for sig-
nificant differences between the mean sales price of each 
house, as well as to examine the extent to which the vari-
ation in house prices is caused by variation between the 
houses. The estimation results obtained through HLM6.08 
software are presented in Table 4.
As shown in Table 4, the estimated coefficient of the 
grand mean of the first sales price of each house (b00) is 
6.132 and attains a 1% level of significance. In terms of 
random effects, the variance in Level 2 derived from the 
restricted maximum likelihood approach is 0.428, with 
a chi-square value of 80,509; it attains a 1% level of sig-
nificance. This suggests that the mean house prices differ 
significantly. To understand whether the empirical results 
can be estimated through the HLGM approach, we first 
computed the intraclass correlation coefficient (ICC). The 
ICC was determined to be 0.91648, indicating that the in-
terclass variance and intraclass variance of the repeat-sales 
prices account for 91.65% and 8.35% of the total variance 
respectively. In other words, the variance between the 
sales prices of a single house at different sales times ac-
counts for 8.35% of the total variance. Therefore, it is suit-
able to use the HLGM approach for subsequent analysis.
4.2. Non-randomly varying slope model
The empirical results of the non-randomly varying slope 
model are presented in Table 5. In terms of fixed effects, 
the estimated coefficient of the grand mean of the first 
sales price of each house (b00) is 6.0497, attaining a 1% 
level of significance. The estimated coefficient of house 
area (b01) is 0.0088, attaining a 1% level of significance, 
which suggests that every 1-m2 increase in house area in-
creases the sales price by 0.88% (converted using (eb –1) 
hereafter). The estimated coefficients of the two house 
types (b02 and b03) are 0.2395 and 0.4180, respectively, 
and both attain a 1% level of significance. This suggests 
that luxury condominiums and apartment buildings are 
27.06% and 51.89% more expensive, respectively, than 
condominiums. The estimated coefficient of house loca-
tion is 0.5478, attaining a 1% level of significance. This 
shows that houses in the CBD are 72.94% more expensive 
than those in the suburbs. Arribas et al. (2016) have dem-
onstrated that, regardless of the regression model used 
for estimation, house attributes significantly affect sales 
prices. Therefore, the results of our study align with those 
of Arribas et al. (2016).
Thhe estimated coefficient of the time of repeat-sales 
(b10) is 0.0382, attaining a 1% level of significance. This 
shows that the price of a house has grown annually by 
3.89% from 2012 onwards. The estimated coefficient of the 
time of repeat-sales squared (b20) is –0.0035, attaining a 
5% level of significance. This suggests that the curve of 
the quadratic growth of house prices in Kaoshiung City 
opens downward and that the growth rate of house prices 
gradually falls over time. In other words, an increase in 
the repeat-sales time increases the sales price of a house, 
and the margin diminishes with the sales time. In terms 
of moderating effects, the estimated interaction coef-
ficient between house area and the time of repeat-sales 
(b11) is –0.0005 and attains a 5% level of significance. The 
interaction coefficient between house area and the time of 
repeat-sales squared (b21) is 0.0001, attaining a 1% level 
of significance. These findings indicate that an increase 
in house area reduces the growth rate of sales prices and 
that the quadratic growth gradually reduces over time. The 
estimated interaction coefficients between the two house 
types (Type1 and Type2) and the time of repeat-sales (b12 
Table 4. Analysis results of the null model
Fixed effects
Coefficient
Standard error
t-ratio
p-value
Grand mean of the first sales price of 
each house b00 
6.1317
0.0118
516.458
0.001***
Random effects
Variance
Degree of freedom
Chi-square
p-value
Level 2 t00
Interclass variance
0.4282
3,166
80,509.915
0.001***
Level 1 s2
Intraclass variance
0.0394
Deviance
7,354.474
Number of estimated parameters
2
Note: *** denotes p < 0.01. Standard errors are presented as robust standard errors.


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International Journal of Strategic Property Management, 2023, 27(5): 290–303
299
and b13) are 0.0049 and 0.0381, respectively, and do not 
attain a level of significance; the estimated interaction co-
efficients between the two house types and the time of 
repeat-sales squared (b22 and b23) are 0.0002 and –0.0025, 
respectively, and also do not attain a level of significance. 
These findings indicate that the growth rate and quadratic 
growth (changes in the growth rate over time) of house 
prices are not moderated by house type. In other words, 
the annual growth in house prices do not differ by house 
types. The estimated interaction coefficient between house 
location and the time of repeat-sales (b14) is –0.1335, at-
taining a 1% level of significance; the interaction coef-
ficient between house area and the time of repeat-sales 
squared (b24) is 0.0182, attaining a 5% level of signifi-
cance. These findings indicate that the growth rate and 
quadratic growth (changes in the growth rate over time) 
of house prices are moderated by house location. Houses 
in the CBD have a lower growth rate than those in the 
suburbs. Melser (2017) has demonstrated that houses in 
suburbs have a higher price appreciation than houses in 
the city center. Locational differences will diminish the 
differences in the growth rates of house prices over time.
The estimated coefficient of house age (b30) is –0.0211 
and attains a 1% level of significance; the estimated coeffi-
cient of house age squared (b40) is 0.0005 and attains a 1% 
level of significance. This shows that an increase in house 
age reduces the sales price, and that the decrease becomes 
more gradual with decreasing house age. Soltani et  al. 
(2021) show that house age has the most substantial effect 
on sales prices, compared to other house attributes. Shi-
mizu et al. (2010) concur that sales prices change over time 
and that such changes also vary continuously over time. 
Lyndsey et al. (2020) report that an increase in house age 
has diminishing effects on the sales prices of older houses 
and those suitable for redevelopment. In terms of moder-
ating effects, the estimated interaction coefficient between 
house area and house age (b31) is 0.0001, attaining a 1% 
level of significance; the interaction coefficient between 
house area and house age squared (b41) is 0.000002, at-
taining a 10% level of significance. These findings indicate 
that an increase in house area and house age will reduce 
the decrease in sales prices, and that the depreciation in 
sales prices accelerates with increasing house age. The es-
timated interaction coefficients between Type1 houses and 
house age (b32) and house age squared (b42) are 0.0342 
and –0.0008, attain levels of significance of 10% and 5%, 
respectively. This suggests that, compared to condomini-
ums, the decrease in the prices of luxury condominiums 
is smaller when the house age is higher, while the rate of 
depreciation falls with an increasing house age. The esti-
mated interaction coefficients between Type2 houses and 
house age (b33) and house age squared (b43) are 0.0278 
and –0.0012, attaining levels of significance of 5% and 1%, 
respectively. This suggests that, compared to condomini-
ums, the decrease in the prices of apartment buildings is 
smaller when the house age is higher, while the rate of 
depreciation falls with increasing house age. The estimated 
interaction coefficient between house location and house 
age (b34) is 0.0279 and attains a 1% level of significance; 
the interaction coefficient between house location and 
house age squared (b44) is –0.0008 and attains a 1% level 
of significance. These findings indicate that, compared to 
houses in suburbs, an increase in house age will reduce the 
decrease in the prices of houses in the CBD, while the de-
preciation diminishes with increasing house age. Xu et al. 
(2018) have found that the added value of older houses in 
Beijing between 2005 and 2012 stems from the land itself 
rather than buildings on the land, while the price growth 
rate and quadratic growth are affected by house location. 
Liou et al. (2016) report that, between 2004 and 2012, the 
annual quadratic growth of house prices in Taipei City 
was faster than that of houses in New Taipei City. These 
findings demonstrate that the quadratic growth of house 
prices is affected by house location.
In terms of random effects, p0i is a Level 2 variable 
used to interpret p0i based on the three Level 2 charac-
teristics (i.e., house area, house type, and house location). 
r0i is the error term with an estimated variance (t00) of 
0.0727 that has a significance of 1%. Therefore, p0i remains 
random even after controlling for the effects of the three 
Level 2 variables on house prices. In other words, other 
house attribute variables affect house prices.
We apply the HLGM approach to analyze repeat-sales 
data. Our empirical results suggest that the annual growth 
rate and quadratic growth of sales prices do not differ by 
house type. The effects of house age and the time of re-
peat-sales on sales prices are affected by house location 
and house area. Thus, the annual growth rate and qua-
dratic growth of repeat-sales prices are affected by house 
location and house area. Controlling for the effects of the 
Level 2 variables (house area, house type, and house loca-
tion) and a Level 1 variable (house age) on sales prices will 
reduce the random component of the initial price (p0i). 
Lastly, given that area is a continuous variable and its 
moderating effects cannot be plotted easily, and house 
type (Type1, Type2) did not significantly influence the re-
lationship between the time variables and house price, we 
did not plot the effects of these variables. We only plotted 
the effects of house location on sales time and price; house 
type and location affected the relationship between house 
age and price (see Figures 3 to 5).
Figure 3. The moderating effects of house location on the 
relationship between house price and sales time


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C.-C. Lee et al. Price changes of repeat-sales houses in Kaohsiung city: analyses based on hierarchical...
Figure 4. The moderating effects of house type on the 
relationship between house price and age
Figure 5. The confounding effects of house location on the 
relationship between house price and age
Table 5. Analysis results of the non-randomly varying slope model
Fixed effects
Coefficient
Standard error
t-ratio
p-value
Grand mean of the first sales price of 
each house	
b00
6.0497
0.0186
325.730
0.001***
	
Area	
b01
	
Type1	
b02
	
Type2	
b03
	
Location	 b04
0.0088
0.2395
0.4180
0.5478
0.0003
0.0907
0.0698
0.0571
31.099
2.641
5.987
9.591
0.001***
0.009***
0.001***
0.001***
Time – T		
b10
	
Area	
b11
	
Type1	
b12
	
Type2	
b13
	
Location	 b14
0.0382
–0.0005
0.0049
0.0381
–0.1335
0.0090
0.0002
0.0336
0.0282
0.0436
4.260
–2.222
0.146
1.350
–3.060
0.001***
0.026**
0.884
0.177
0.003***
(Time – T)2	
b20
	
Area	
b21
	
Type1	
b22
	
Type2	
b23
	
Location	 b24
–0.0035
0.0001
0.0020
–0.0025
0.0182
0.0014
0.00004
0.0050
0.0042
0.0072
–2.542
3.358
0.411
–0.578
2.499
0.011**
0.001***
0.681
0.563
0.013**
Age	
	
b30
	
Area	
b31
	
Type1	
b32
	
Type2	
b33
	
Location	 b34
–0.0211
0.0001
0.0342
0.0278
0.0279
0.0030
0.00003
0.0179
0.0113
0.0052
–7.090
3.271
1.910
2.459
5.286
0.001***
0.001***
0.056*
0.014**
0.001***
Age2	
	
b40
	
Area	
b41
	
Type1	
b42
	
Type2	
b43
	
Location	 b44
0.0005
0.000002
–0.0008
–0.0012
–0.0008
0.00007
0.000001
0.0003
0.0002
0.0001
6.925
1.858
–2.450
–5.912
–5.247
0.001***
0.063*
0.015**
0.001***
0.001***
Variance
Degree of freedom
Chi-square
p-value
Level 2 t00
Interclass variance
0.0727
3,162
17,543.4250
0.001***
Level 1 s2
Intraclass variance
0.0369
Deviance
2,305.7188
Number of estimated 
parameters
2
Note: *** denotes p < 0.01; ** denotes p < 0.05; * denotes p < 0.1. Standard errors are presented as robust standard errors.


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International Journal of Strategic Property Management, 2023, 27(5): 290–303
301
Conclusions and recommendations
This study focuses on changes in the repeat-sales prices 
of houses over time in Kaohsiung City. The empirical re-
sults suggest that the HLGM approach is a feasible means 
of exploring house price changes. Notwithstanding the 
wide availability of other options, examining changes in 
the sales prices of houses over time through ANOVA can 
only reveal relative changes in the sales prices between 
houses; it is unable to directly estimate changes in the sales 
prices of houses. Moreover, ANOVA does not account for 
measurement errors–nor does it include variables at the 
aggregate level (Level 2) for estimation (Lee et al., 2013). 
Arribas et al. (2016) demonstrate that the variance of each 
coefficient in HLGM is always smaller than those in OLS. 
This is because the lower estimation efficiency of OLS will 
result in a higher variance than that from HLGM. Fur-
thermore, HLGM can also calculate the interclass vari-
ance, as well as the ratio of the intraclass variance to the 
total variance (i.e., the ICC). OLS, on the other hand, can 
only derive the variance between the horizontal intraclass 
residuals. Not only is HLGM able to estimate changes in 
the growth curve of each housing transaction at different 
points in time, it can also examine the differences between 
individual-level data, which facilitates investigations into 
how house attributes affect changes in sales prices (Hsieh, 
2015). After analyzing the advantages and drawbacks of 
each approach, showing that the HLGM approach is the 
appropriate method to examinine repeated observations in 
housing transactions over time. It can analyze the curves 
of the growth rate and quadratic growth of sales prices 
while elucidating how house attributes affect changes in 
the growth rate and quadratic growth of sales prices.
According to the empirical results of the null model, 
the interclass error attained significance, suggesting that 
major differences exist in the mean house prices in this 
study, with an ICC of 91.65%. This means that the dif-
ferences in housing prices caused by different house at-
tributes accounted for 91.65% of the variance in the re-
peat-sales data. The differences in housing prices of the 
same house caused by different sales periods accounted 
for 8.38% of the variance. This result cannot be obtained 
through a single-level repeat-sales model. The non-ran-
domly varying slope model results indicate that the annual 
growth rate and quadratic growth of sales prices failed to 
attain significance in terms of the interaction coefficient 
between sales time and house type. This suggests that the 
effects of sales time on sales prices do not differ by house 
type. House area and house location moderate the effects 
of sales time on sales prices (i.e., both variables affect the 
growth rate and quadratic growth). The interaction coef-
ficients between house area, house type, and house loca-
tion are all significant, which implies that all three vari-
ables moderate the effects of house age on sales prices. 
This finding shows that in repeat-sales data, the growth 
rate and the changes in the growth rate of the price of the 
same house are influenced by house area and house loca-
tion. This result cannot be obtained through a single-level 
repeat-sales model. The empirical evidence demonstrates 
that house attributes are an important factor and modera-
tor of the dynamic changes in repeat-sale house prices.
The current repeat-sales approach suffers from con-
ventional drawbacks, such as small sample sizes and sam-
ple selection bias (Guo et al., 2014). We recommend that 
future studies perform HLGM analyses using panel data-
based analytical methods. From July 1, 2021, onwards, 
house numbers must be disclosed on the actual price 
registration system. This measure would assist future re-
searchers in performing accurate validations and would 
reduce the need to compare and analyze repeat-sales data. 
Additionally, house price indices can be developed using 
the HLGM approach.
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