**Chapter 2: Are the responses of real estate activities to macroeconomic changes**

**2.2 Methodology**

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time. The case of Japan reminds us the changes in interest rate have different impacts on real estate prices during the expansionary and recessionary states.

The work of Seslen (2003) and Abelson et al. (2005) provide an explanation on Japan’s case. They claim that households are eager to get into the housing market during the upswing of housing cycle. Oppositely, loss aversion leads the households less likely to trade up during the downswing. Additionally, Kim and Bhattacharya (2009) point out the participants in real estate markets will not respond symmetrically over the real estate cycle. Consequently, the same interest rate policy doesn’t has the same impacts on expansionary and recessionary real estate market.

In addition to the Markov switching model, there are many candidate approaches useful
to analyze the asymmetric response of real estate market on macroeconomic changes.^{6}
In contrast to other nonlinear models, Markov provides us more information about real
estate cycle and its turning point. Accordingly, this paper focus on using Markov
switching model to evaluate the real estate cycle in China and further discussing the
empirical results and its implications. The rest of this Chapter is structured as follows.

Section 2.2 describes the methodology. Section 2.3 describes the dataset in details.

Section 2.4 performs the methodology and presents detailed description of the empirical results. The final section concludes the Chapter.

**2.2 Methodology **

In the original Markov switching model developed by Hamilton (1989), the change of regime depends only on the history of the process. We therefore present the

6 e.g., Smooth transition regression model developed by Lüükkonen et al. (1988), threshold regression model proposed by Hansen (1999), and smooth threshold approach proposed by Gonzàlez et al. (2005).

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corresponding model as follows:

### 𝐻

_{𝑡}

### =𝛼(𝑠

_{𝑡}

### ) + ∑

^{𝐾}

_{𝑘=1}

### 𝛾

_{𝑘}

### 𝐻

_{𝑡−𝑘}

### + 𝜀

_{𝑡}

### , 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### ) (1)

### 𝛼(𝑠

_{𝑡}

### ) = 𝛼

_{0}

### (1 − 𝑠

_{t}

### ) + 𝛼

_{1}

### 𝑠

_{𝑡}(2)

where 𝐻_{𝑡} denotes the real estate climate index, 𝑠_{𝑡} is the unobserved latent variable.

𝑠_{𝑡}= 1 is interpreted as an expansion while 𝑠_{𝑡}= 0 is interpreted as an contraction, the
corresponding intercepts (or the mean of 𝐻_{𝑡}) are 𝛼_{1} and 𝛼_{0}, and 𝛼_{1}＞𝛼_{0}. The
parameters of Markov switching model can be estimated through maximum likelihood.

The log likelihood function of 𝐻_{𝑡} is:

### 𝑙𝑛𝐿 = ∑

^{𝑇}

_{𝑡=1}

### 𝑙𝑛[∑

^{1}

_{𝑠}

_{𝑡}

_{=0}

### 𝑓( 𝐻

_{𝑡}

### |𝑠

_{𝑡}

### , 𝐼

_{𝑡−1}

### ) × 𝑃𝑟 (𝑠

_{𝑡}

### | 𝐼

_{𝑡−1}

### ) ]

(3)### where 𝑃𝑟 (𝑠

_{𝑡}

### | 𝐼

_{𝑡−1}

### ) is

prediction probability, and is also interpreted as weighting probability. Using the filtered probability### 𝑃𝑟 (𝑠

_{𝑡}

### | 𝐼

_{𝑡}

### ) as the initial value,

prediction probability### 𝑃𝑟 (𝑠

_{𝑡}

### | 𝐼

_{𝑡−1}

### )

can be computed recursively by applying Bayes’ Rule.Next, the transition probabilities can be described as follows:

### 𝑃𝑟(𝑠

_{𝑡}

### = 0|𝑠

_{𝑡−1}

### = 0) = 𝑝

^{00}

### , 𝑃𝑟(𝑠

_{𝑡}

### = 0|𝑠

_{𝑡−1}

### = 1) = 1 − 𝑝

^{11}

### 𝑃𝑟(𝑠

_{𝑡}

### = 1|𝑠

_{𝑡−1}

### = 0) = 1 − 𝑝

^{00}

### , 𝑃𝑟(𝑠

_{𝑡}

### = 1|𝑠

_{𝑡−1}

### = 1) = 𝑝

^{11}(4)

where 𝑝^{00} is the probability that real estate market remains staying in the contraction
state, while 𝑝^{11} is the probability that real estate market remains staying in the
expansionary state. Finally, we can adopt the approximation of Kim (1994) to compute
the smooth probabilities 𝕡(𝑠_{𝑡}= 𝑖| 𝐼_{𝑇}; 𝛼_{0}, 𝛼_{1}, 𝛾_{0}, … , 𝛾_{𝑘}, 𝑝^{00}, 𝑝^{11}), for 𝑖 = 0, 1. The
smooth probabilities enable us to recognize the periods of recession and expansion.

The original Markov switching model represented by Eq. (1) to Eq. (4) somehow

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ignores the fact that some macroeconomic factors could be useful in forecasting the real
estate cycles.^{7} As a result, we next consider the Markov switching model with the
explicative variables. The first model is presented as follows:

**Model 1：** 𝐻

_{𝑡}

### =𝛼(𝑠

_{𝑡}

### ) + 𝛽

_{1}

### 𝐸𝐼𝐿𝐼

_{𝑡−1}

### + 𝛽

_{2}

### 𝑅𝐼𝑁𝑇

_{𝑡−1}

### + 𝛽

_{3}

### 𝐻

_{𝑡−1}

### + 𝜀

_{𝑡}

### , 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### )

(5)where 𝐸𝐼𝐿𝐼 is leading economic climate index, and 𝑅𝐼𝑁𝑇 is real interest rate. Model 1 implies the changes in

### 𝐸𝐼𝐿𝐼

_{𝑡−1}(

### 𝑅𝐼𝑁𝑇

_{𝑡−1}) has the symmetric effect on

### 𝐻

_{𝑡}

### , no matter the real estate market is in an expansion

or a contraction. The marginal effect of### 𝐸𝐼𝐿𝐼

_{𝑡−1}(

### 𝑅𝐼𝑁𝑇

_{𝑡−1}) on

### 𝐻

_{𝑡}d

### enoted by

𝛽_{1}(𝛽

_{2}). Model 1 is not reliable when there exists nonlinear relationship between real estate market and macroeconomic factors. We then extend model 1 to model 2.

**Model 2：**

𝐻_{𝑡}=

### 𝛼(𝑠

_{𝑡}

### ) + 𝛽

_{1}

### (𝑠

_{𝑡}

### )𝐸𝐼𝐿𝐼

_{𝑡−1}

### + 𝛽

_{2}

### (𝑠

_{𝑡}

### )𝑅𝐼𝑁𝑇

_{𝑡−1}

### + 𝛽

_{3}

### (𝑠

_{𝑡}

### )𝐻

_{𝑡−1}

### + 𝜀

_{𝑡}

### , 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### )

(6) In Model2, the marginal effect of 𝐸𝐼𝐿𝐼_{𝑡−1}on 𝐻

_{𝑡}is 𝛽

_{1}(𝑠

_{𝑡}), which means the degree of the impacts of 𝐸𝐼𝐿𝐼

_{𝑡−1}on 𝐻

_{𝑡}depends on whether the real estate market is in an expansion (𝑠

_{𝑡}= 1) or in a contraction (𝑠

_{𝑡}= 0).

It is worth noting that the conduction of climate economic index is through a combination of the single indicator chosen according to its relationship with the economic climate index. If we use the economic index as the main explicative variable, it is easy to ignore the effects of some index components on real estate. In particular, some indicators like M2 or stock turnover value may play a key role in forecasting the

7 Review the previous literature, the main macroeconomic factors that may influence the real estate market are (1) demographic factors (2) interest rate and monetary policy (3) income and investment (3) inflation. See Summers(1981), Kau and Keenan(1981), Brown(1984), Fortura and Kushner(1986), Darrat and Glasock(1993), Glasser and Gyourko(2003), and Neukirchen and Lange(2005) for more discussion.

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real estate cycle. Besides, an increase in the economic climate index doesn’t mean all index components are going an upswing, sometimes it is manipulated by the weighting scheme. In the next model, we use the components of leading index as the main explanatory variables rather than using the single composite economic index, described as follows:

**Model 3：** 𝐻

_{𝑡}

### =𝛼(𝑠

_{𝑡}

### ) + 𝜑

_{1}

### 𝑆𝑇𝑂𝐶𝐾

_{𝑡}

_{−1}

### + 𝜑

_{2}

### 𝑃𝑆𝑅

_{𝑡}

_{−1}

### + 𝜑

_{3}

### 𝐶𝐸𝐼

_{𝑡}

_{−1}

### + 𝜑

_{4}

### 𝑇𝐹𝑇

_{𝑡}

_{−1}

### +𝜑

_{5}

### 𝐹𝐴𝐼𝑁

_{𝑡}

_{−1}

### + 𝜑

_{6}

### 𝑀2𝑌

_{𝑡}

_{−1}

### + 𝜑

_{7}

### 𝑅𝐼𝑁𝑇

_{𝑡−1}

### + 𝜑

_{8}

### 𝐻

_{𝑡−1}

### + 𝜀

_{𝑡}

### ,

### 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### )

(7)**Model 4：** 𝐻

_{𝑡}

### =𝛼(𝑠

_{𝑡}

### ) + 𝜑

_{1}

### (𝑠

_{𝑡}

### )𝑆𝑇𝑂𝐶𝐾

_{𝑡}

_{−1}

### + 𝜑

_{2}

### (𝑠

_{𝑡}

### )𝑃𝑆𝑅

_{𝑡}

_{−1}

### + 𝜑

_{3}

### (𝑠

_{𝑡}

### )𝐶𝐸𝐼

_{𝑡}

_{−1}

### +𝜑

_{4}

### (𝑠

_{𝑡}

### )𝑇𝐹𝑇

_{𝑡}

_{−1}

### + 𝜑

_{5}

### (𝑠

_{𝑡}

### )𝐹𝐴𝐼𝑁

_{𝑡}

_{−1}

### + 𝜑

_{6}

### (𝑠

_{𝑡}

### )𝑀2𝑌

_{𝑡}

_{−1}

**+𝜑**_{7}

### (𝑠

_{𝑡}

**)𝑅𝐼𝑁𝑇+𝜑**

_{8}

### (𝑠

_{𝑡}

### )𝐻

_{𝑡−1}

### + 𝜀

_{𝑡}

### 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### )

(8)where 𝑆𝑇𝑂𝐶𝐾 is growth rate of Shanghai stock exchange turnover in value (A share),
PSR is product sales rate, 𝐶𝐸𝐼 is consumer expectation index, TFT is growth rate of
freight traffic, 𝐹𝐴𝐼𝑁 is growth rate of the number of Fixed asset investment project
newly started, and 𝑀2𝑌 is growth rate of M2.^{8}

In order to compare the forecast accuracy of economic leading indicator and economic coincident indicator, we then build Model 5, Model 6, Model 7, and Model 8, presented as follows:

**Model 5：** 𝐻

_{𝑡}

### =𝛼(𝑠

_{𝑡}

### ) + 𝜃

_{1}

### 𝐸𝐼𝐶𝐼

_{𝑡−1}

### + 𝜃

_{2}

### 𝑅𝐼𝑁𝑇

_{𝑡−1}

### + 𝜃

_{3}

### 𝐻

_{𝑡−1}

### + 𝜀

_{𝑡}

### , 𝜀

_{𝑡}

### ~𝑁(0, 𝜎

^{2}

### )

(9)

8 The components of economic leading index includes Shanghai stock exchange turnover in value (A share), product sales rate, consumer expectation index, freight traffic, volume of freight handled in major coastal ports, number of fixed asset investment project newly started, contracted value of foreign direct investment, interest-rate spread between domestic interest rate and foreign interest rate, and M2.

However, data acquisition has its limitations, and some variables are only available in quarterly frequency.

Therefore, we don’t use all components of the leading index in the model.

## ‧

industry, 𝐹𝐴𝐼 is growth rate of fixed asset investment, 𝑅𝑆𝐶𝐺 is growth rate of retail sales of consumer goods, 𝐸𝑋𝐼𝑀 is growth rate of value of export and import, 𝑇𝐴𝑋 is growth rate of government tax revenue, and 𝐷𝐼 is growth rate of urban disposable income per capita### .

^{9}

Besides focusing on the implications of estimated coefficients, we also concern on the
forecast accuracy among the model. As a result, we then use the Diebold Mariano test
(hereafter referred to as DM test) to make pairwise forecast accuracy comparisons
between the models.^{10}

9 Similarity, due to the limitation of data acquisition, we don’t use all components of the coincident index in the model.

10 Ma and Lin (2009) suggest we can use turning point error, share of correct identification and DM test simultaneously to compare the forecasting performances from different models. The assessment of turning point error and share of correct identification require formal announcements of business cycle turning points. However, there is no China’s formal announcements about the turning points in its real estate cycle. Consequently, we use DM test to select the most appropriate model. The DM statistic is:

𝐷𝑀 𝑠𝑡𝑎𝑡 = ^{𝑑̅}

√^{𝑉𝑎𝑟(𝑑)}_{𝑇}

~N(0,1)

Where 𝑑̅ is the sample mean of the loss differential series defined as ∑^{𝑇}_{𝑡=1}[𝑔(𝑒_{𝑖𝑡}) − 𝑔(𝑒_{𝑗𝑡})]/𝑇. 𝑔(𝑒_{𝑖𝑡})
and 𝑔(𝑒_{𝑗𝑡}*) are loss functions of model i and model j. The significant positive DM stat means the forecast *
*error of model i is statistically larger than the forecast error of model j. *

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