Fuzzy Moderation and Moderated-Mediation Analysis of Real Estate Variables in South Korea

Dae Jong Kim; Yoo Young Koo; Jin Hee Yoon

doi:10.5391/IJFIS.2023.23.4.375

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 375-388

Published online December 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.4.375

Fuzzy Moderation and Moderated-Mediation Analysis of Real Estate Variables in South Korea

Dae Jong Kim¹, Yoo Young Koo², and Jin Hee Yoon³

¹Department of Business Administration, Sejong University, Seoul, Korea
²University College, Yonsei University, Incheon, Korea
³Department of Mathematics and Statistics, Sejong University, Seoul, Korea

Correspondence to :
Jin Hee Yoon (jin9135@sejong.ac.kr)

Received: July 6, 2023; Accepted: November 10, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

In a causal relationship involving independent and dependent variables, a mediator is a variable that influences both, sitting between them in the causal chain. Conversely, a moderator is a variable that affects the dependent variable but is not part of the direct causal relationship. This study analyzes variables affecting real estate using fuzzy moderation and fuzzy moderated-mediation analyses, which apply fuzzy theory to data observed with ambiguous information. Among the various variables that affect real estate, this study analyzes the relationship between variables that affect real estate prices, particularly house prices, by analyzing stock price, foreign exchange reserve, and so on, which are various variables that can affect real estate sales by loans. Given that these data are collected over a month or more, using observations at a single price point for analysis inevitably leads to a significant amount of information being condensed or summarized. Therefore, this study applies fuzzy moderation and fuzzy moderated-mediation analyses using fuzzy data. In addition to the existing fuzzy moderated-mediation analysis, a model for additional possible moderation was proposed. For the data analysis, six real estate-related variables from the Bank of Korea and KB Data Bank are analyzed. Two fuzzy moderation models and three fuzzy moderated-mediation models are proposed for the given dataset.

Keywords: Fuzzy data, Real estate data, Fuzzy moderation analysis, Fuzzy moderated-mediation analysis

1. Introduction

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

When analyzing the causal relationships between variables, regression analysis or deep learning techniques are commonly used. For this purpose, the relationship between the independent and dependent variables was analyzed; however, typically other variables indirectly affected the results. If the variable is directly involved in a causal relationship, it is called a mediator; if it is not placed in a causal relationship but affects the dependent variable, it is called a moderator. Mediation and moderation analyses, first proposed by Baron and Kenny [1], have been applied to various fields, including social science [2–5]. A mediator is involved in a causal relationship that directly affects the model. If moderation occurs in this relationship, that is, when there are moderators involved in the mediation model, it is called a moderated-mediation model.

Various studies have proposed mediation, moderation, and moderated-mediation models [2–5], that have been applied in various fields, such as psychology, education, and medicine, to analyze causal relationships. However, for variables used in these fields, observations are often collected in an ambiguous state. For example, in psychology, when expressing a person’s mood as a number, the levels of very good, good, normal, bad, and very bad are expressed by only five numbers, from one to five. However, the level of a person’s mood includes ambiguity, and the loss of information from the actual data is sufficient to compress and express the person’s mood with five numbers.

Likewise, there is considerable information loss in data analysis used not only in psychology but also in various other fields. In this study, we analyzed various variables that affect real estate prices. We use variables such as house price, stock price (KOSPI), total amount of loans, and consumer price index.

When variables are investigated in units of a specific period, it is common to use the average value for that period. For example, if monthly data are used, there will be changes in house and stock prices even within a month. This is typically averaged and used; however, ambiguous information is lost. Therefore, soft methods can be used, and the fuzzy theory introduced by Zadeh [6] is the most suitable among these methods. Mediation and moderating analyses using fuzzy theory were recently conducted by Yoon [7, 8]. In this study, we analyzed the variables that affect real estate prices, particularly Korean house prices, using fuzzy numbers. Real estate prices in South Korea are extremely sensitive to interest rates and various financial variables, and Korean financial variables, including interest rates, are significantly influenced by the movement of financial variables in the United States. Some studies have examined the relationships between variables related to real estate in South Korea [9–13].

By 2022, real estate prices in Korea will fall by 50%. The greatest reason for the decline in real estate was the increase in interest rates on bank loans. As the US inflation rose to 9.1%, the US Fed raised its benchmark interest rate from 0% to 4.5% to curb inflation [14]. The Korean government also increased the bank loan interest rate from 3% to 8%. Real estate in Korea fell by 50% owing to the increase in bank loan interest rates. The decline in real estate leads to bankruptcy, which lends 50%–90% of the house price. Therefore, the government has lifted real estate regulations to prevent real estate prices from falling. Many regulations were lifted to prevent a drop in real estate prices, such as the abolition of interim payment loan restrictions and mandatory residency.

Despite deregulation, real estate prices are expected to decline by 2023. In all countries, including Korea, the United States, Europe, and Asia, real estate prices fell sharply due to an increase in bank loan interest rates.

This study analyzed the major variables affecting Korean real estate prices using fuzzy-adjusted and moderated-mediation analyses.

This study contributes by suggesting that the government should develop strategies to prevent and stabilize real estate price declines. It does this by examining the causal relationships among various variables affecting real estate prices, utilizing moderation and moderated-mediation analyses. Real estate accounts for 80% of Korean national assets. The government should prevent a sharp decrease in housing prices and induce soft landings. Real estate prices do not rise, but a fall leads to the insolvency of financial institutions.

The US interest rate increase will continue until December 2023. The US is expected to raise its benchmark interest rate to 5.25% to lower the 7% inflation level by 2%.

The contribution of this study is to help the government devise measures to prevent and stabilize real estate price decline through a mediation analysis of the causal relationship of various variables that affect real estate prices.

In Korea, real estate prices are expected to drop by approximately 10% by 2023 because of the US interest rate hike. When the interest rate on loans exceeds 8%, real estate prices fall as interest rates increase more than housing prices do. Because of analyzing the real estate trend in Korea over the past 30 years, it has risen with 90% probability.

Since 80% of national assets are invested in apartments and real estate, a sharp drop in real estate indicates the collapse of the middle class. The government should stabilize the real estate market through sophisticated financial policies and deregulation. Korea’s household debt is expected to reach 2,000 trillion Korean won by 2023. Rising interest rates on loans, falling real estate prices, and rising national debt have all led to financial crises. The government should begin with deregulation to maintain the status quo of real estate prices. Six real estate-related variables, six variables that is house price index (HPI), consumer price index (CPI), Korea Composite Stock Price Index (KOSPI), interest rate (IR), loan amount (LA), foreign exchange reserve are analyzed. Five data analyses were conducted.

This paper is organized as follows: Section 2 introduces some preliminaries for the proposed method and proposes fuzzy moderation analyses for real estate-related variables. Section 3 provides a detailed explanation of the real estate-related variables used in the study. For data analysis, two simple mediation analyses and three moderated-mediation analyses with the given data are presented in Section 4, and Section 5 concludes the paper.

2. Fuzzy Mediation Analysis

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

2.1 Fuzzy Numbers

In this section, the basic concept of fuzzy numbers is introduced. Fuzzy sets were first proposed by Zadeh [6]. A fuzzy number using a fuzzy set is defined as follows:

Definition 1 (Fuzzy numbers)

For the fuzzy membership function μ_A : R → [0, 1] of A, A fuzzy set Ã, Ã is a fuzzy number if Ã satisfies following properties:

1) (Normality) There exist x₀, which satisfies

(1)μA˜(x0)=1.
2) (Fuzzy convexity) For given x, y ∈ R, and t ∈ R,

(2)μA˜ (tx+(1-t)y)≥min (μA (x),μA (y)).
3) (Upper semi-continuity) For any x₀ ∈ R, if

(3)μA˜(x0)≥limx→x0+ μA˜(x),

then Ã is a fuzzy number.

The α-level set of a fuzzy set Ã is defined by

(4)Aα˜:={x∈R∣μA˜(x)≥α}.

In addition, the membership function of an LR fuzzy number Ã = (a, l_a, r_a)_LR is defined as

(5)μA˜(x)={LA˜(a-xla), for 0≤a-x≤la,RA˜(x-ala), for 0≤x-a≤ra,0,o.w,

where L_Ã and R_Ãare nonincreasing functions satisfying L_Ã (0) = R_Ã (0) = 1 and L_Ã (1) = R_Ã (1) = 0. Here, a is the mode of Ã and l_a, r_a are the left and right spreads, respectively. Especially, L_Ã and R_Ã are linear functions, and the fuzzy number Ã is called a triangular fuzzy number and is written as Ã = (l_a, a, r_a), where a − l_a and r_a − a are the left and right spreads, respectively. Figure 1 shows various types of fuzzy numbers.

Based on the extension principle in [6], the following two operations for fuzzy triangular numbers are defined:

(6)X˜⊕Y˜=(lx+ly,x+y,rx+ry),kX˜={(klx,kx,krx),k≥0,(krx,kx,klx),k<0,

where X̃ = (l_x, x, r_x), Ỹ = (l_y, y, r_y) ∈ F_T for k ∈ ℝ.

In Sections 2.2, and 2.3, some basic concepts for fuzzy moderation and moderated-mediation analysis introduced in [7, 8] are provided.

2.2 Fuzzy Moderation Analysis

In the causal relationship, if X̃ is a fuzzy predictor and Ỹ is a fuzzy response variable, then W̃ is a fuzzy moderator variable that influences the causal relationship of X̃ and Ỹ, then the coefficient of X̃ can be assumed to be affected by W̃, which means it can be expressed by a function of W̃. Let us consider X̃ ‘s effect can be expressed by fuzzy data as a function of W̃, as in

(7)Y˜^=b0⊕f (W˜)⊛X˜⊕b2W˜.

If f (W̃) is given by a linear function of W̃, then f (W̃) = b₁ ⊕ b₃W̃.

(8)Y˜^=b0⊕(b1⊕b3W˜)⊛X˜⊕b2W˜.

This can be rewritten as

(9)Y˜^=b0⊕b1X˜⊕b2W˜⊕b3X˜⊛W˜.

This model allows X̃’s effect on Ỹ to depend linearly on W̃. Here, δ _X̃_→_Ỹ is the “conditional effect” of X̃ on Ỹ defined by the function δ _X̃_→_Ỹ = b₁⊕b₃W̃. Also, δ _W̃_→_Ỹ is the conditional effect of W̃ on Ỹ defined by the function δ _W̃_→_Ỹ = b₂ ⊕ b₃X̃.

2.3 Fuzzy Moderated-Mediation Analysis

The simplest moderated-mediation analysis model combines simple mediation and moderation of the fuzzy conditional direct effect (FCDE) of X̃ on Ỹ.

Assuming linear moderation of the direct effect of X̃ by W̃, the model in Figure 2 is represented by

(10)M˜^=a0⊕aX˜,Y˜^=b0⊕c1.X˜⊕c2.W˜⊕c3.X˜⊛W˜⊕bM˜.

Or, equivalently,

(11)Y˜^=b0⊕δX˜→Y˜X˜⊕c2.W˜⊕bM˜,

where δ _X̃_→_Ỹ is the conditional direct effect of X̃ on Ỹ, defined as δX˜→Y˜=c1.⊕c3.W˜. where ci. is the coefficient that constitutes the direct conditional effect (Figure 3).

The mechanism linking X̃ to Ỹ can be considered conditional if the indirect effect of X̃ on Ỹ through M̃ is contingent on moderator W̃.

Above model in Figure 4 can be expressed by

(12)M˜^=a0⊕aX˜,Y˜^=b0⊕c˙X˜⊕b1M˜⊕b2W˜⊕b3M˜⊛W˜.

Or, equivalently,

Y˜^=b0⊕c˙X˜⊕δM→YM˜⊕b2,

where δ _M̃_→_Ỹ = b₁ ⊕ b₃W̃. The indirect effect of X̃ on Ỹ via M̃ is aδ _M̃_→_Ỹ but as δ _M̃_→_Ỹ is a conditional effect of M̃, then

(13)aδM˜→Y˜=a(b1⊕b3W˜)=ab1⊕ab3W˜

is the fuzzy conditional indirect effect (FCIDE) of X̃ on Ỹ via M̃.

The following fuzzy moderated-mediation model is proposed in this study. This model deals with the case in which moderation affects the causal relationship between an independent variable and a mediator.

Above model in Figure 5 can be expressed by

M˜^=a0⊕a1X˜⊕a2W˜⊕a3X˜⊛W˜ =(a0⊕a2W˜)⊕(a1⊕a3W˜)⊛X˜,

and

Y˜^=b0⊕c˙X˜⊕bM˜.

Here, the FCIDE is defined by

(14)b (a1⊕a3W˜)=a1b⊕a3bW˜.

And a₃b is called the index of moderated-mediation.

Generally, the distance between two fuzzy numbers is based on the distance between their α-cuts. A useful metric can be defined using support functions. The support function of any compact convex set A ∈ ℝ^d is defined as a function s_A : S^d⁻¹ → ℝ given by for all r ∈ S^d⁻¹.

sA(r)=supa∈A ⟨r,a⟩,

where S^d⁻¹ is the unit sphere of (d − 1) dimension in ℝ^d and ⟨·, ·⟩ represents the scalar product on ℝ^d. The support function s_A is uniquely determined for convex and compact A ∈ ℝ^d. A metric on a fuzzy number set is defined by the L₂-metric on the space of Lebesgue integrable as follows:

δ2(A,B)=[d·∫01∫s0∣sA(α,r)-sB(α,r)∣2μ(dr)dα]1/2.

Based on this equation, an L₂-metric for fuzzy numbers can be defined by

(15)d2(X˜,Y˜)=D22(supp X˜,supp Y˜)+[ml(X˜)-ml(Y˜)]2+[mr(X˜)-mr(Y˜)]2.

A fuzzy regression model introduced previously [15, 16] is suggested as follows:

(16)Y˜i=β0⊕β1X˜i1⊕β2X˜i2⊕⋯⊕βpX˜ip⊕E˜i,

where X̃_ij = (l_x_{_ij}, x_ij, r_x_{_ij}), Ỹ_i = (l_y_{_i}, y_i, r_y_{_i}) (i = 1, · · ·, n; j = 1, · · ·, p), and Ei˜ are presumed as fuzzy error terms that indicate fuzziness.

All cases can be encompassed by

lxij={xij-ξlij,if βj≥0,xij+ξrij,if βj<0,rxij={xij+ξrij,if βj≥0,xij-ξlij,if βj<0,

where ξ_l_{_ij} and ξ_r_{_ij} are the left and right spreads of X̃_ij, respectively. We can now obtain the estimators by minimizing the following function

(17)Q(βk0,βk1,⋯,βkp)=∑i=1nd2 (Y˜i,∑j=0pβkjX˜ij)=(lyi-∑j=0pβkjlxij)2+(yi-∑j=0pβkjxij)2+(ryi-∑j=0pβkjrxij)2,

for k = 1, 2, · · ·, h, where is the number of the regression model in this fuzzy analysis. This function can be acquired using the L₂-metric in (24) as follows:

To minimize (21), we obtain the normal equation applying ∂Q∂βkl=0 for each k = 1, 2, · · ·, h and for l = 0, 1, · · ·, p as follows:

(18)∂Q∂βkl=2∑i=1n(lyi-∑j=0pβkjlxij) lxil+2∑i=1n(yi-∑j=0pβkjxij) xil+2∑i=1n(ryi-∑j=0pβkjrxij) rxil,

and for each k = 1, 2, · · ·, h the normal equation which has β̂_kl as solutions can be obtained as follows:

(19)∑j=0pβ^kj∑i=1n(lxillxij+xilxij+rxilrxij)=∑i=1n(lxillyi+xilyi+rxilryi).

To determine the solution vector, a triangular fuzzy matrix (t.f.m.) is defined as follows:

X˜=[(1,1,1)(lx11,x11,r11)⋯(lx1p,x1p,r1p)⋮⋮⋱⋮(1,1,1)(lxn1,xn1,rn1)⋯(lxnp,xnp,rnp)].

This matrix is denoted as X̃ = [X̃_ij]_n_×₍_p₊₁₎ simply, where X̃_ij is a triangular fuzzy number for i = 1, · · ·, n; j = 1, · · ·, p. Also, a tringular fuzzy vector Ỹ = [Ỹ_i]^t is defined.

To minimize the objective function above, previously studied fuzzy operations [15, 16] were applied.

X˜⋄Y˜=lxly+xy+rxry,X˜⊗Y˜=(lxly, xy, rxry).

For given two n × n t.f.m’s, Γ̃ = [ X̃_ij ], Λ̃ = [Ỹ_ij ], and a crisp matrix Ã = [a_ij ], the operations are defined as follows:

Γ˜⋄Λ˜=[∑k=1nX˜ik⋄Y˜kj],Γ˜⊗Λ˜=[⊕k=1nX˜ik⋄Y˜kj],A˜Γ˜=[⊕k=1naikX˜kj], kΓ˜=[⊕k=1naikX˜ij],X˜A˜=[aijX], X˜⋄Γ˜=[X⋄Xij],X˜⊗Γ˜=[X⊗Xij].

Using the above operations and algebraic properties, we can obtain solutions of the normal equation fuzzy estimators for each k = 1, 2, …, h by

(20)βk^=(X˜t⋄X˜)-1X˜t⋄y˜,

where

X˜t⋄X˜=[∑i=1n(lxillxij+xilxij+rxilrxij)](p+1)×(p+1),

and

X˜t⋄y˜=[∑i=1n(lxillyi+xilyi+rxilryi)](p+1)×1,

for l = 0, 1, · · ·, p. Note that 22 exists if det( X̃^t ⋄ X̃ ) ≠ 0.

3. Real Estate Price and Related Variables

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

This section describes the data characteristics used in this study. To analyze the fuzzy moderation and fuzzy moderated-mediation models of variables affecting real estate prices, we used the following variables in South Korea. Nationwide housing prices are used as indicators of real estate prices.

• House price index (HPI): monthly national average housing price
• Consumer price index (CPI): monthly price index calculated by updated costbase period cost×100
• Korea Composite Stock Price Index (KOSPI): monthly Korean stock index
• Interest rate (IR): monthly certification of deposit (CD) interest rate
• Loan amount (LA): monthly total loans to individuals and businesses nationwide
• Foreign exchange reserve (FER): monthly foreign exchange reserve (US dollar)

Monthly data from February 2000 to December 2022 were used in this study. In South Korea, the ratio of homeowners to nonowners is 50%. Single-family households account for 30% of the total population and have continued to increase. The KOSPI is obtained by calculating the average prices of 2,200 listed companies in South Korea in terms of market capitalization. The KOSPI positively correlates with real estate prices, whereas real estate prices negatively correlate with interest rates.

Based on data analysis, house prices in Seoul moved in the same direction as those nationwide. FER data show South Korea’s stability in the international financial markets.

Various economic indices represent indices that shows the boom, recession, and recession periods.

The USD-to-KRW FER is a variable that shows the value of the Korean won against the US dollar.

Many variables affect housing prices. The descriptive statistical analysis in Table 1 is performed on the variables that affect house prices: the HPI, CPI, KOSPI, IR, LA, and FER.

Inflation and current accounts also have significant impacts on property prices. However, these two variables are excluded from this study because their impact on real estate prices is similar to that of the CPI.

Variables such as inflation and exchange rates tend to rise in the long term and rarely fall. South Korea has the world’s second-largest trade dependence; therefore, its current account balance is often in surplus. However, the biggest impact was a significant decrease in South Korean exports in 2020 and 2022, when global trade stagnated. South Korea’s current account balance also has a significant impact on real estate prices.

The HPI set the nationwide average house prices in January 2022 to 100 and used a figure expressing the relative prices for other months. This figure is often used to understand the relative movement of housing prices in Korea and is provided by KB Bank [17] in South Korea. The reference timepoint set to 100 may vary depending on the time period data is used.

Generally, real estate prices and the KOSPI exhibit similar trends. Eighty percent of the assets of the people of Korea’s assets are real estate. Korea’s stock investment ratio was 20%. There are many assets in real estate because of the belief that it will continue to rise as Korea has lived for an extended period on a limited piece of land where 50 million people live.

During the Asian financial crisis of 1997 and the US financial crisis of 2008, the international financial market greatly impacted the real estate market.

Normally, the CCI has a similar trend to real estate prices. Stock price is classified as a CLI and real estate price as a CCI.

Figure 6 is a time series of HPI for the past 25 years in Korea. The HPI is a value relative to the house price in 2022 of 100 made by the KB Bank [17,18]. Housing prices in Korea increase by 0.18% every month. The coefficient of determination was 0.94, which indicates a 94% probability of rising over 25 years.

Although many variables affect house prices, the results of analyzing long-term trends are upward-sloping. It declines significantly during the 1997 and 2008 financial crises. However, in the 25-year long-term, it increased with a 94% probability.

In the correlation analysis shown in Table 2, the variables that had a significant effect on national housing prices were FER (0.96), LA (0.98), CPI (0.96). In contrast, IR showed the opposite correlation at −0.82. Correlation analysis explains that bank interest has the greatest impact on house price decline. Correlation analysis is an analysis method that shows the influence of direction on the rise and fall. The major variables in the national housing prices showed a considerably high correlation. IRs, conversely, exhibit a strong inverse correlation.

However, when we analyze a moderation model, the independent variable that has the greatest influence may not necessarily be suitable as an independent variable for a moderation model. In some cases, independent variables with a small influence may be more suitable for a moderating analysis model because the degree to which they are influenced by the moderating variable is large. This was also shown in the data analysis.

4. Data Analysis

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The data used are monthly data from February 2000 to November 2022, where n = 274. The HPI was collected from KB Bank [17], and the remaining data were provided by the Bank of Korea [18]. In addition, all crisp time series data {x_t} can be fuzzified symmetrically with spread s_t, which is defined by

st=xt+1-xt2,

where t = 1, · · ·, n, which is much more reasonable to understand.

4.1 Simple Moderation Analysis

4.1.1 Simple moderation analysis I

Interest rates have a significant impact on housing prices. The loan amount can change as interest rates fluctuate. However, the CPI may affect the amount of loan money that can be invested in real estate. Therefore, we propose the following model Figure 7.

The moderation model in Figure 7 is as follows:

(21)Y˜^=b0⊕b1X˜⊕b2W˜⊕b3X˜⊛W˜.

Based on classical mediation analysis (CMA) and fuzzy mediation analysis (FMA), the model in Figure 7 is estimated as follows: Table 3 shows the parameter estimates of the proposed model. Table 4 summarizes the model.

For a moderating effect, the relationship between the dependent and independent variables must vary according to the level of the moderator variable. In other words, in this model, the relationship between IR and HPI should vary according to the CPI. By substituting a random CPI value into the regression equation, we can see how the relationship between IR and HPI changes according to the CPI. If the mean value of the CPI is w̄ and the standard deviation is s, then Table 5 presents the regression equations for the specific moderated values. These three values represent the cases in which CPIs are substituted with w̄ (Effect 1), w̄ − s (Effect 2), and w̄ + s (Effect 3). Figure 8 shows the moderating effects of the CMA and FMA.

The moderating effect of FMA was similar to that of CMA. Therefore, this graph has been omitted. From the table and graph above, we can see that the CPI has a moderating effect on the causal relationship between IR and HPI in both the CMA and FMA.

4.1.2 Simple moderation analysis II

Even when households opt to invest in real estate through loans, rising stock prices may lead some to invest a portion in stocks. In other words, fluctuations in stock prices can influence people’s psychology and their demand for real estate. Therefore, the following model was considered:

The moderation model in Figure 9 is as follows:

(22)Y˜^=b0⊕b1X˜⊕b2W˜⊕b3X˜⊛W˜.

Based on CMA and CFA, the model in Figure 9 estimated as follows. Table 6 shows the parameter estimates of the proposed model. Table 7 summarizes the model.

As shown in Table 6, because the difference between the results of the parameter estimates of the CMA and FMA is negligible, we obtain similar results by calculating the moderating effect calculated in Section 4.1.1. As shown in Table 5, if the KOSPI value is substituted, then Table 8 is obtained.

The following graph shows the moderating effects (Figure 10).

Figure 10 also shows that there is no significant difference between the CMA and FMA cases and the difference in the HPI decreases as LA increases.

4.2 Moderated-Mediation Analysis

4.2.1 Moderated-mediation analysis I

Often, people take out loans to invest in stocks, indicating a potential causal relationship between the total amount of loans and stock prices. In addition, the decision whether to invest in stocks can be affected by the CPI. As inflation increases, the amount of money available for investment decreases. Therefore, the following model can be considered:

The moderation model in Figure 11 is as follows:

(23)M˜^=a0⊕a1X˜⊕a2W˜⊕a3X˜⊛W˜,

(24)Y˜^=b0⊕c˙X˜⊕bM˜.

Based on CMA and CFA, the model in Figure 11 is estimated as follows: Table 9 shows the parameter estimates of the proposed model. Tables 10 and 11 summarize the models.

If the mean value of CPI is w̄ and the standard deviation is s, Table 12 shows the conditional indirect effect in (14) for the specific moderated values. These three values represent the cases in which CPIs are substituted with w̄ (Effect 1), w̄ − s (Effect 2), and w̄ + s (Effect 3).

4.2.2 Moderated-mediation analysis II

In the above model Figure 11, because the KOSPI can also be affected by foreign investment capital, we consider the following case, where FER acts as a moderator variable for the causal relationship between the total amount of loans and the KOSPI.

The moderation model in Figure 11 is as follows:

(25)M˜^=a0⊕a1X˜⊕a2W˜⊕a3X˜⊛W˜,

(26)Y˜^=b0⊕c˙X˜⊕bM˜.

Based on CMA and CFA, the model in Figure 12 is estimated as follows: Table 13 shows the parameter estimates of the proposed model. Table 14 summarizes the model.

The second equation, Eq. (26) of this model, is the same as that of the previous model in Figure 11; therefore, the summary of the model is the same as that shown in Table 11.

If the mean value of FER is w̄ and the standard deviation is s, Table 15 shows the conditional indirect effect in (14) for the specific moderated values. These three values represent the cases in which FERs are substituted with w̄ (Effect 1), w̄ − s (Effect 2), and w̄ + s (Effect 3).

4.2.3 Moderated-mediation analysis III

As in the model in Figure 11, the loan amount can act as a moderating variable for the causal relationship between the KOSPI and CPI. However, it is also possible to affect the causal relationship between the KOSPI and HPI. Therefore, the following model can be considered:

The moderation model in Figure 13 is as follows:

(27)M˜^=a0⊕aX˜,

(28)Y˜^=b0⊕c˙X˜⊕b1M˜⊕b2W˜⊕b3M˜⊛W˜.

Based on CMA and CFA, the model in Figure 13 is estimated as follows: Table 16 shows the parameter estimates of the proposed model. Tables 17 and 18 summarize the models.

If the mean value of the CPI is w̄ and the standard deviation is s, Table 19 shows the conditional indirect effect in (13) for the specific moderated values. These three values represent the cases in which CPIs are substituted with w̄ (Effect 1), w̄ − s (Effect 2), and w̄ + s (Effect 3).

5. Conclusion

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

The United States, Europe, and other countries, including South Korea, will face falling real estate prices by 2022 and 2023. An increase in bank interest rates has a significant impact not only on real estate, but also on stock markets. In this study, the variables affecting real estate prices in South Korea were examined using fuzzy moderation analysis. We applied a simple fuzzy moderation analysis and three fuzzy moderated-mediation analyses. In this study, we propose a new fuzzy moderated-mediation analysis when there is a moderated-mediating effect between the independent variable and the mediator. Six real estate-related variables, six variables that is HPI, CPI, KOSPI, IR, LA, and FER are analyzed. Five data analyses were conducted. Therefore, the KOSPI, CPI, and FER can be moderators when considering various independent variables with moderators. In this study, only some real estate-related financial variables were analyzed, but this analysis can be extended to any causal effect analysis when there are complicated causal relations.

Conflict of Interest

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
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Tables
References
Biographies

Jin Hee Yoon serves as (an) editor(s) of the International Journal of Fuzzy Logic and Intelligent Systems, but has no role in the decision to publish this article. Except for that, no potential conflict of interest relevant to this article was reported.

Figures

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4. Data Analysis
5. Conclusion
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Fig. 1.

Various types of fuzzy numbers.

Fig. 2.

Effects of X̃ and W̃ on Ỹ in simple moderation model: (a) conceptual diagram and (b) statistical diagram.

Fig. 3.

Moderated-mediation analysis for conditional direct effect: (a) conceptual diagram and (b) statistical diagram.

Fig. 4.

Moderated-mediation analysis for conditional indirect effect: (a) conceptual diagram and (b) statistical diagram.

Fig. 5.

Moderated-mediation analysis for conditional direct effect: (a) conceptual diagram and (b) statistical diagram.

Fig. 6.

House price index of South Korea over years.

Fig. 7.

Simple moderation model: (a) conceptual diagram and (b) statistical diagram.

Fig. 8.

Moderating effects on CMA and FMA for model in Figure 7.

Fig. 9.

Simple moderation model: (a) conceptual diagram and (b) statistical diagram.

Fig. 10.

Moderating effects on CMA and FMA for model in Figure 9.

Fig. 11.

Moderated-mediation model: (a) conceptual diagram and (b) statistical diagram.

Fig. 12.

Moderated-Mediation model. (a) Conceptual diagram. (b) Statistical diagram.

Fig. 13.

Moderated-mediation model: (a) conceptual diagram and (b) statistical diagram.

Tables

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1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. Descriptive statistics of HPI-related variables.

Descriptive statistics	HPI	KOSPI	IR	FER	LA	ER
Average	66.7541	1689.83	3.18128	2.90E+08	1066884	1135.14
Median	68.8476	1906	2.85	3.10E+08	1028823	1131.69
Standard deviation	14.9864	667.953	1.63402	1.10E+08	507666	102.534
Minimum value	38.9954	479.68	0.63	8.00E+07	259582	915.86
Maximm value	100.869	3296.68	7.17	4.70E+08	2158296	1461.98

Table. 2.

Table 2. Correlation Coefficients of HPI-related variables.

Correlation coefficient	HPI	KOSPI	IR	FER	LA	CPI
HPI	1
KOSPI	0.928071	1
IR	−0.82423	−0.77625	1
FER	0.96285	0.95528	−0.86182	1
LA	0.98249	0.91163	−0.85049	0.96156	1
CPI	0.96975	0.92913	−0.86337	0.98406	0.96700	1

Table. 3.

Table 3. Parameter estimates of simple moderation model in Figure 7.

	CMA	FMA
b₀	−0.0765	−0.0756
b₁	0.1292	0.1281
b₂	0.9771	0.9761
b₃	−0.1603	−0.1596

Table. 4.

Table 4. Model summary of simple moderation model in Figure 7.

	CMA	FMA
R²	0.9433	0.9432
MSE	0.0034	0.0100
F	1498.4	1494.7
df1	3	3
df2	270	270
p	0.0000	0.0000

Table. 5.

Table 5. Moderating effects of CPI for model in Figure 7.

Effect	CPI	CMA	FMA
1	0.238	0.156 + 0.091X	0.156 + 0.090X
2	0.511	0.423 + 0.047X	0.422 + 0.047X
3	0.784	0.690 + 0.004X	0.688 + 0.003X

Table. 6.

Table 6. Parameter estimates of simple moderation model in Figure 9.

	CMA	FMA
b₀	0.0337	0.0351
b₁	0.8434	0.8379
b₂	0.2283	0.2246
b₃	−0.1662	−0.1556

Table. 7.

Table 7. Model summary of simple moderation model in Figure 9.

	CMA	FMA
R²	0.9736	0.9734
MSE	0.0016	0.00472
F	3325	3296.24
df1	3	3
df2	270	270
p	0.0000	0.0000

Table. 8.

Table 8. Moderating effect of KOSPI for model in Figure 9.

Effect	KOSPI	CMA	FMA
1	0.193	0.078 + 0.811X	0.079 + 0.808X
2	0.431	0.132 + 0.772X	0.132 + 0.771X
3	0.668	0.186 + 0.732X	0.185 + 0.734X

Table. 9.

Table 9. Parameter estimates of moderated-mediation model in Figure 11.

	CMA	FMA
a₀	−0.0180	−0.0180
a₁	0.9368	0.9315
a₂	0.4179	0.4203
a₃	−0.5668	−0.5635
b₀	0.0533	0.0534
c	0.7391	0.7413
b	0.1930	0.1902

Table. 10.

Table 10. Model summary of (23) for moderated-mediation model in Figure 11.

	CMA	FMA
R²	0.8737	0.8711
MSE	0.0072	0.0217
F	622.6546	610.49
df1	3	3
df2	270	270
p	0.0000	0.0000

Table. 11.

Table 11. Model summary of (24) for moderated-mediation model in Figure 11.

	CMA	FMA
R²	0.9719	0.9717
MSE	0.0017	0.005
F	4682.1	4658.4
df1	2	2
df2	271	271
p	0.0000	0.0000

Table. 12.

Table 12. Conditional indirect effect of CPI for moderated-mediation model in Figure 11.

Effect	CPI	CMA	FMA
1	0.238	0.1548	0.1517
2	0.511	0.1250	0.1224
3	0.784	0.0950	0.0931

Table. 13.

Table 13. Parameter estimates of moderated-mediation model in Figure 12.

	CMA	FMA
a₁	0.0300	0.0300
a₁	−0.4163	−0.4193
a₂	0.9339	0.9353
a₃	0.2662	0.2680
b₀	0.0533	0.0534
c	0.7391	0.7413
b	0.1930	0.1902

Table. 14.

Table 14. Model summary for moderated-mediation model in Figure 12.

	CMA	FMA
R²	0.916	0.914
MSE	0.0048	0.0145
F	983.9	958.9364
df1	3	3
df2	270	270
p	0.0000	0.0000

Table. 15.

Table 15. Conditional indirect effect of FER for moderated-mediation model in Figure 12.

Effect	FER	CMA	FMA
1	0.247	−0.0677	−0.0672
2	0.533	−0.0530	−0.0526
3	0.819	−0.0383	−0.038

Table. 16.

Table 16. Parameter estimates of model in Figure 13.

	CMA	FMA
a₀	0.0875	0.0874
a	0.8072	0.8074
b₀	0.0270	0.0271
c	0.7166	0.7166
b₁	0.2170	0.2148
b₂	0.1439	0.1450

Table. 17.

Table 17. Model summary of (27) for moderated-mediation model in Figure 13.

	CMA	FMA
R²	0.830	0.827
MSE	0.0096	0.02915
F	1324.27	1305.656
df1	1	1
df2	272	272
p	0.0000	0.0000

Table. 18.

Table 18. Model summary of (28) for moderated-mediation model in Figure 13.

	CMA	FMA
R²	0.975	0.9746
MSE	0.0015	0.0045
F	2591	2581.24
df1	4	4
df2	269	269
p	0.0000	0.0000

Table. 19.

Table 19. Conditional indirect effect of CPI for moderated-mediation model in Figure 13.

Effect	CPI	CMA	FMA
1	0.238	0.0340	0.0331
2	0.511	0.025	0.0243
3	0.784	0.0160	0.0154

References

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Baron, RM, and Kenny, DA (1986). The moderator–mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology. 51, 1173-1182. https://doi.org/10.1037/0022-3514.51.6.1173
James, LR, and Brett, JM (1984). Mediators, moderators, and tests for mediation. Journal of Applied Psychology. 69, 307-321. https://doi.org/10.1037/0021-9010.69.2.307
Frazier, PA, Tix, AP, and Barron, KE (2004). Testing moderator and mediator effects in counseling psychology research. Journal of Counseling Psychology. 51, 115-134. https://doi.org/10.1037/0022-0167.51.1.115
Hayes, AF (2022). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. New York, NY: Guilford Publications
Muller, D, Judd, CM, and Yzerbyt, VY (2005). When mediation is moderated and moderation is mediated. Journal of Personality and Social Psychology. 89, 852-863. https://doi.org/10.1037/0022-3514.89.6.852
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Yoon, JH (2020). Fuzzy mediation analysis. International Journal of Fuzzy Systems. 22, 338-349. https://doi.org/10.1007/s40815-019-00727-6
Yoon, JH (2020). Fuzzy moderation and moderated-mediation analysis. International Journal of Fuzzy Systems. 22, 1948-1960. https://doi.org/10.1007/s40815-020-00848-3
Han, KY, and Kim, HJ (2022). A perception of individual variables on housing price in housing policy. Journal of the Korean Housing Association. 33, 53-60. https://doi.org/10.6107/JKHA.2022.33.3.053
Song, IH (2012). A study on mutual effects between house prices and macroeconomic variables. Journal of Real Estate Policy Research. 13, 1-21.
Ko, PS, and Park, CS (2010). A study about the real estate’ policy impact on house prices: focusing on the time series analysis and regression. The Journal of the Korea Institute of Electronic Communication Sciences. 5, 205-213.
Choi, MK, and Lee, SH (2021). The effect of macroeconomics factor and real estate measures on the Gangnam 4 District apartment market. Journal of the Korean Society of Cadastre. 37, 105-114. http://doi.org/10.22988/ksc.2021.37.3.008
Kim, HJ, and Cho, IC (2019). A study of the effect of housing policy and macroeconomic factors on apartment. Journal of the Korea Real Estate Industry Society. 3, 1-27. http://kreis.kr/bbs/board.php?bo_table=journal_reading&wr_id=25&sfl=wr_subject&sop=and
CountryEconomy. (c2023) . US Federal Reserve Bank Key Rates. [Online]. Available: https://countryeconomy.com/key-rates/usa
Yoon, JJH, and Choi, SH (2013). Fuzzy least squares estimation with new fuzzy operations. Synergies of Soft Computing and Statistics for Intelligent Data Analysis. Heidelberg, Germany: Springer, pp. 193-202 https://doi.org/10.1007/978-3-642-33042-1_21
Yoon, JH, and Grzegorzewski, P (2020). On optimal and asymptotic properties of a fuzzy L2 estimator. Mathematics. 8, article no 1956. https://doi.org/10.3390/math8111956
KB Bank. (c2023) . KB real estate data hub. [Online]. Available: https://data.kbland.kr/
Bank of Korea. (c2023) . Economic statistics system. [Online]. Available: http://ecos.bok.or.kr

Biographies

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Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion
Conflict of Interest
Figures
Tables
References
Biographies

Dae Jong Kim received B.A. degree in Economics from Hankuk University of Foreign Studies, Economics, MBA Korea University, Ph.D. degree in Economics from Sogang University, South Korea. He is currently a faculty member in the Department of Business School at Sejong University, Seoul, South Korea. He is currently at the Korea Institute for Business and Economics and is listed on the Marquis Who’s Who. His research interests include fuzzy regression analysis, fuzzy time series, optimization, intelligent systems, and applied economics.

Yoo Young Koo received B.S., M.S., and Ph.D. degrees in Mathematics from Yonsei University, South Korea. She is currently a faculty member at the College of Yonsei University, Incheon, South Korea. Her research interests include statistical and data analyses, machine learning, and deep learning.

Jin Hee Yoon received B.S., M.S., and Ph.D. degrees in Mathematics from Yonsei University, South Korea. She is currently a faculty member of the Department of Mathematics and Statistics at Sejong University in Seoul, South Korea. Her research interests include fuzzy regression analysis, fuzzy time series, optimization, intelligent systems, and machine learning. She is a board member of (Korean Institute of Intelligent Systems) and has been working as an associate editor, guest editor, and editorial board member for several journals, including SCI and SCIE. She has regularly worked as an organizer and committee member at several international conferences.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 375-388

Published online December 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.4.375

Fuzzy Moderation and Moderated-Mediation Analysis of Real Estate Variables in South Korea

Dae Jong Kim¹, Yoo Young Koo², and Jin Hee Yoon³

Correspondence to:Jin Hee Yoon (jin9135@sejong.ac.kr)

Received: July 6, 2023; Accepted: November 10, 2023

Abstract

Keywords: Fuzzy data, Real estate data, Fuzzy moderation analysis, Fuzzy moderated-mediation analysis

1. Introduction

This study analyzed the major variables affecting Korean real estate prices using fuzzy-adjusted and moderated-mediation analyses.

The US interest rate increase will continue until December 2023. The US is expected to raise its benchmark interest rate to 5.25% to lower the 7% inflation level by 2%.

2. Fuzzy Mediation Analysis

2.1 Fuzzy Numbers

In this section, the basic concept of fuzzy numbers is introduced. Fuzzy sets were first proposed by Zadeh [6]. A fuzzy number using a fuzzy set is defined as follows:

Definition 1 (Fuzzy numbers)

For the fuzzy membership function μ_A : R → [0, 1] of A, A fuzzy set Ã, Ã is a fuzzy number if Ã satisfies following properties:

1) (Normality) There exist x₀, which satisfies

(1) $μ_{\tilde{A}} (x_{0}) = 1.$
2) (Fuzzy convexity) For given x, y ∈ R, and t ∈ R,

(2) $μ_{\tilde{A}} (t x + (1 - t) y) \geq min (μ_{A} (x), μ_{A} (y)) .$
3) (Upper semi-continuity) For any x₀ ∈ R, if

(3) $μ_{\tilde{A}} (x_{0}) \geq lim_{x \to x_{0}^{+}} μ_{\tilde{A}} (x),$

then Ã is a fuzzy number.

The α-level set of a fuzzy set Ã is defined by

(4)

\tilde{A_{α}} : = {x \in R ∣ μ_{\tilde{A}} (x) \geq α} .

In addition, the membership function of an LR fuzzy number Ã = (a, l_a, r_a)_LR is defined as

(5)

μ_{\tilde{A}} (x) = {\begin{array}{l} L_{\tilde{A}} (\frac{a - x}{l_{a}}), for 0 \leq a - x \leq l_{a}, \\ R_{\tilde{A}} (\frac{x - a}{l_{a}}), for 0 \leq x - a \leq r_{a}, \\ 0, o.w, \end{array}

Based on the extension principle in [6], the following two operations for fuzzy triangular numbers are defined:

(6)

\begin{array}{l} \tilde{X} \oplus \tilde{Y} = (l_{x} + l_{y}, x + y, r_{x} + r_{y}), \\ k \tilde{X} = {\begin{array}{l} (k l_{x}, k x, k r_{x}), & k \geq 0, \\ (k r_{x}, k x, k l_{x}), & k < 0, \end{array} \end{array}

where X̃ = (l_x, x, r_x), Ỹ = (l_y, y, r_y) ∈ F_T for k ∈ ℝ.

In Sections 2.2, and 2.3, some basic concepts for fuzzy moderation and moderated-mediation analysis introduced in [7, 8] are provided.

2.2 Fuzzy Moderation Analysis

(7)

\hat{\tilde{Y}} = b_{0} \oplus f (\tilde{W}) ⊛ \tilde{X} \oplus b_{2} \tilde{W} .

If f (W̃) is given by a linear function of W̃, then f (W̃) = b₁ ⊕ b₃W̃.

(8)

\hat{\tilde{Y}} = b_{0} \oplus (b_{1} \oplus b_{3} \tilde{W}) ⊛ \tilde{X} \oplus b_{2} \tilde{W} .

This can be rewritten as

(9)

\hat{\tilde{Y}} = b_{0} \oplus b_{1} \tilde{X} \oplus b_{2} \tilde{W} \oplus b_{3} \tilde{X} ⊛ \tilde{W} .

2.3 Fuzzy Moderated-Mediation Analysis

The simplest moderated-mediation analysis model combines simple mediation and moderation of the fuzzy conditional direct effect (FCDE) of X̃ on Ỹ.

Assuming linear moderation of the direct effect of X̃ by W̃, the model in Figure 2 is represented by

(10)

\begin{array}{l} \hat{\tilde{M}} = a_{0} \oplus a \tilde{X}, \\ \hat{\tilde{Y}} = b_{0} \oplus \dot{c_{1}} \tilde{X} \oplus \dot{c_{2}} \tilde{W} \oplus \dot{c_{3}} \tilde{X} ⊛ \tilde{W} \oplus b \tilde{M} . \end{array}

Or, equivalently,

(11)

\hat{\tilde{Y}} = b_{0} \oplus δ_{\tilde{X} \to \tilde{Y}} \tilde{X} \oplus \dot{c_{2}} \tilde{W} \oplus b \tilde{M},

where δ _X̃_→_Ỹ is the conditional direct effect of X̃ on Ỹ, defined as $δ_{\tilde{X} \to \tilde{Y}} = \dot{c_{1}} \oplus \dot{c_{3}} \tilde{W}$ . where $\dot{c_{i}}$ is the coefficient that constitutes the direct conditional effect (Figure 3).

The mechanism linking X̃ to Ỹ can be considered conditional if the indirect effect of X̃ on Ỹ through M̃ is contingent on moderator W̃.

Above model in Figure 4 can be expressed by

(12)

\begin{array}{l} \hat{\tilde{M}} = a_{0} \oplus a \tilde{X}, \\ \hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus b_{1} \tilde{M} \oplus b_{2} \tilde{W} \oplus b_{3} \tilde{M} ⊛ \tilde{W} . \end{array}

Or, equivalently,

\hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus δ_{M \to Y} \tilde{M} \oplus b_{2},

where δ _M̃_→_Ỹ = b₁ ⊕ b₃W̃. The indirect effect of X̃ on Ỹ via M̃ is aδ _M̃_→_Ỹ but as δ _M̃_→_Ỹ is a conditional effect of M̃, then

(13)

a δ_{\tilde{M} \to \tilde{Y}} = a (b_{1} \oplus b_{3} \tilde{W}) = a b_{1} \oplus a b_{3} \tilde{W}

is the fuzzy conditional indirect effect (FCIDE) of X̃ on Ỹ via M̃.

Above model in Figure 5 can be expressed by

\begin{array}{l} \hat{\tilde{M}} & = a_{0} \oplus a_{1} \tilde{X} \oplus a_{2} \tilde{W} \oplus a_{3} \tilde{X} ⊛ \tilde{W} \\ = (a_{0} \oplus a_{2} \tilde{W}) \oplus (a_{1} \oplus a_{3} \tilde{W}) ⊛ \tilde{X}, \end{array}

and

\hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus b \tilde{M} .

Here, the FCIDE is defined by

(14)

b (a_{1} \oplus a_{3} \tilde{W}) = a_{1} b \oplus a_{3} b \tilde{W} .

And a₃b is called the index of moderated-mediation.

s_{A} (r) = \underset{a \in A}{s u p} ⟨ r, a ⟩,

\begin{array}{l} δ_{2} (A, B) \\ = {[d \cdot \int_{0}^{1} \int_{s^{0}} {∣ s_{A} (α, r) - s_{B} (α, r) ∣}^{2} μ (d r) d α]}^{1 / 2} . \end{array}

Based on this equation, an L₂-metric for fuzzy numbers can be defined by

(15)

d^{2} (\tilde{X}, \tilde{Y}) = D_{2}^{2} (s u p p \tilde{X}, s u p p \tilde{Y}) + {[m_{l} (\tilde{X}) - m_{l} (\tilde{Y})]}^{2} + {[m_{r} (\tilde{X}) - m_{r} (\tilde{Y})]}^{2} .

A fuzzy regression model introduced previously [15, 16] is suggested as follows:

(16)

{\tilde{Y}}_{i} = β_{0} \oplus β_{1} {\tilde{X}}_{i 1} \oplus β_{2} {\tilde{X}}_{i 2} \oplus \dots \oplus β_{p} {\tilde{X}}_{i p} \oplus {\tilde{E}}_{i},

where X̃_ij = (l_x_{_ij}, x_ij, r_x_{_ij}), Ỹ_i = (l_y_{_i}, y_i, r_y_{_i}) (i = 1, · · ·, n; j = 1, · · ·, p), and $\tilde{E_{i}}$ are presumed as fuzzy error terms that indicate fuzziness.

All cases can be encompassed by

\begin{array}{l} l_{x_{i j}} = {\begin{array}{l} x_{i j} - ξ_{l i j}, & if β_{j} \geq 0, \\ x_{i j} + ξ_{r i j}, & if β_{j} < 0, \end{array} \\ r_{x_{i j}} = {\begin{array}{l} x_{i j} + ξ_{r i j}, & if β_{j} \geq 0, \\ x_{i j} - ξ_{l i j}, & if β_{j} < 0, \end{array} \end{array}

where ξ_l_{_ij} and ξ_r_{_ij} are the left and right spreads of X̃_ij, respectively. We can now obtain the estimators by minimizing the following function

(17)

\begin{array}{l} Q (β_{k 0}, β_{k 1}, \dots, β_{k p}) \\ = \sum_{i = 1}^{n} d^{2} ({\tilde{Y}}_{i}, \sum_{j = 0}^{p} β_{k j} {\tilde{X}}_{i j}) \\ = {(l_{y_{i}} - \sum_{j = 0}^{p} β_{k j} l_{x_{i j}})}^{2} + {(y_{i} - \sum_{j = 0}^{p} β_{k j} x_{i j})}^{2} \\ + {(r_{y_{i}} - \sum_{j = 0}^{p} β_{k j} r_{x_{i j}})}^{2}, \end{array}

for k = 1, 2, · · ·, h, where is the number of the regression model in this fuzzy analysis. This function can be acquired using the L₂-metric in (24) as follows:

To minimize (21), we obtain the normal equation applying $\frac{\partial Q}{\partial β_{k l}} = 0$ for each k = 1, 2, · · ·, h and for l = 0, 1, · · ·, p as follows:

(18)

\begin{array}{l} \frac{\partial Q}{\partial β_{k l}} = 2 \sum_{i = 1}^{n} (l_{y_{i}} - \sum_{j = 0}^{p} β_{k j} l_{x_{i j}}) l_{x_{i l}} \\ + 2 \sum_{i = 1}^{n} (y_{i} - \sum_{j = 0}^{p} β_{k j} x_{i j}) x_{i l} \\ + 2 \sum_{i = 1}^{n} (r_{y_{i}} - \sum_{j = 0}^{p} β_{k j} r_{x_{i j}}) r_{x_{i l}}, \end{array}

and for each k = 1, 2, · · ·, h the normal equation which has β̂_kl as solutions can be obtained as follows:

(19)

\begin{array}{l} \sum_{j = 0}^{p} {\hat{β}}_{k j} \sum_{i = 1}^{n} (l_{x_{i l}} l_{x_{i j}} + x_{i l} x_{i j} + r_{x_{i l}} r_{x_{i j}}) \\ = \sum_{i = 1}^{n} (l_{x_{i l}} l_{y_{i}} + x_{i l} y_{i} + r_{x_{i l}} r_{y_{i}}) . \end{array}

To determine the solution vector, a triangular fuzzy matrix (t.f.m.) is defined as follows:

\tilde{X} = [\begin{matrix} (1, 1, 1) & (l_{x_{11}}, x_{11}, r_{11}) & \dots & (l_{x_{1 p}}, x_{1 p}, r_{1 p}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (1, 1, 1) & (l_{x_{n 1}}, x_{n 1}, r_{n 1}) & \dots & (l_{x_{n p}}, x_{n p}, r_{n p}) \end{matrix}] .

To minimize the objective function above, previously studied fuzzy operations [15, 16] were applied.

\begin{array}{l} \tilde{X} ⋄ \tilde{Y} = l_{x} l_{y} + x y + r_{x} r_{y}, \\ \tilde{X} \otimes \tilde{Y} = (l_{x} l_{y}, x y, r_{x} r_{y}) . \end{array}

For given two n × n t.f.m’s, Γ̃ = [ X̃_ij ], Λ̃ = [Ỹ_ij ], and a crisp matrix Ã = [a_ij ], the operations are defined as follows:

\begin{array}{l} \tilde{Γ} ⋄ \tilde{Λ} = [\sum_{k = 1}^{n} {\tilde{X}}_{i k} ⋄ {\tilde{Y}}_{k j}], \tilde{Γ} \otimes \tilde{Λ} = [\oplus_{k = 1}^{n} {\tilde{X}}_{i k} ⋄ {\tilde{Y}}_{k j}], \\ \tilde{A} \tilde{Γ} = [\oplus_{k = 1}^{n} a_{i k} {\tilde{X}}_{k j}], k \tilde{Γ} = [\oplus_{k = 1}^{n} a_{i k} {\tilde{X}}_{i j}], \\ \tilde{X} \tilde{A} = [a_{i j} X], \tilde{X} ⋄ \tilde{Γ} = [X ⋄ X_{i j}], \\ \tilde{X} \otimes \tilde{Γ} = [X \otimes X_{i j}] . \end{array}

Using the above operations and algebraic properties, we can obtain solutions of the normal equation fuzzy estimators for each k = 1, 2, …, h by

(20)

\hat{β_{k}} = {({\tilde{X}}^{t} ⋄ \tilde{X})}^{- 1} {\tilde{X}}^{t} ⋄ \tilde{y},

where

{\tilde{X}}^{t} ⋄ \tilde{X} = {[\sum_{i = 1}^{n} (l_{x_{i l}} l_{x_{i j}} + x_{i l} x_{i j} + r_{x_{i l}} r_{x_{i j}})]}_{(p + 1) \times (p + 1)},

and

{\tilde{X}}^{t} ⋄ \tilde{y} = {[\sum_{i = 1}^{n} (l_{x_{i l}} l_{y_{i}} + x_{i l} y_{i} + r_{x_{i l}} r_{y_{i}})]}_{(p + 1) \times 1},

for l = 0, 1, · · ·, p. Note that 22 exists if det( X̃^t ⋄ X̃ ) ≠ 0.

3. Real Estate Price and Related Variables

• House price index (HPI): monthly national average housing price
• Consumer price index (CPI): monthly price index calculated by $\frac{updated cost}{base period cost} \times 100$
• Korea Composite Stock Price Index (KOSPI): monthly Korean stock index
• Interest rate (IR): monthly certification of deposit (CD) interest rate
• Loan amount (LA): monthly total loans to individuals and businesses nationwide
• Foreign exchange reserve (FER): monthly foreign exchange reserve (US dollar)

Based on data analysis, house prices in Seoul moved in the same direction as those nationwide. FER data show South Korea’s stability in the international financial markets.

Various economic indices represent indices that shows the boom, recession, and recession periods.

The USD-to-KRW FER is a variable that shows the value of the Korean won against the US dollar.

Many variables affect housing prices. The descriptive statistical analysis in Table 1 is performed on the variables that affect house prices: the HPI, CPI, KOSPI, IR, LA, and FER.

During the Asian financial crisis of 1997 and the US financial crisis of 2008, the international financial market greatly impacted the real estate market.

Normally, the CCI has a similar trend to real estate prices. Stock price is classified as a CLI and real estate price as a CCI.

4. Data Analysis

s_{t} = \frac{x_{t + 1} - x_{t}}{2},

where t = 1, · · ·, n, which is much more reasonable to understand.

4.1 Simple Moderation Analysis

4.1.1 Simple moderation analysis I

The moderation model in Figure 7 is as follows:

(21)

\hat{\tilde{Y}} = b_{0} \oplus b_{1} \tilde{X} \oplus b_{2} \tilde{W} \oplus b_{3} \tilde{X} ⊛ \tilde{W} .

4.1.2 Simple moderation analysis II

The moderation model in Figure 9 is as follows:

(22)

\hat{\tilde{Y}} = b_{0} \oplus b_{1} \tilde{X} \oplus b_{2} \tilde{W} \oplus b_{3} \tilde{X} ⊛ \tilde{W} .

Based on CMA and CFA, the model in Figure 9 estimated as follows. Table 6 shows the parameter estimates of the proposed model. Table 7 summarizes the model.

The following graph shows the moderating effects (Figure 10).

Figure 10 also shows that there is no significant difference between the CMA and FMA cases and the difference in the HPI decreases as LA increases.

4.2 Moderated-Mediation Analysis

4.2.1 Moderated-mediation analysis I

The moderation model in Figure 11 is as follows:

(23)

\hat{\tilde{M}} = a_{0} \oplus a_{1} \tilde{X} \oplus a_{2} \tilde{W} \oplus a_{3} \tilde{X} ⊛ \tilde{W},

(24)

\hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus b \tilde{M} .

Based on CMA and CFA, the model in Figure 11 is estimated as follows: Table 9 shows the parameter estimates of the proposed model. Tables 10 and 11 summarize the models.

4.2.2 Moderated-mediation analysis II

The moderation model in Figure 11 is as follows:

(25)

\hat{\tilde{M}} = a_{0} \oplus a_{1} \tilde{X} \oplus a_{2} \tilde{W} \oplus a_{3} \tilde{X} ⊛ \tilde{W},

(26)

\hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus b \tilde{M} .

Based on CMA and CFA, the model in Figure 12 is estimated as follows: Table 13 shows the parameter estimates of the proposed model. Table 14 summarizes the model.

The second equation, Eq. (26) of this model, is the same as that of the previous model in Figure 11; therefore, the summary of the model is the same as that shown in Table 11.

4.2.3 Moderated-mediation analysis III

The moderation model in Figure 13 is as follows:

(27)

\hat{\tilde{M}} = a_{0} \oplus a \tilde{X},

(28)

\hat{\tilde{Y}} = b_{0} \oplus \dot{c} \tilde{X} \oplus b_{1} \tilde{M} \oplus b_{2} \tilde{W} \oplus b_{3} \tilde{M} ⊛ \tilde{W} .

Based on CMA and CFA, the model in Figure 13 is estimated as follows: Table 16 shows the parameter estimates of the proposed model. Tables 17 and 18 summarize the models.

5. Conclusion

Abstract
1. Introduction
2. Fuzzy Mediation Analysis
3. Real Estate Price and Related Variables
4. Data Analysis
5. Conclusion

Fig 1.

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