# Prosperity or Real Estate Bubble? Exuberance Probability Index of Real Housing Prices in Chile

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Regime Change Model with Predetermined Variables

#### 2.2. Estimation Strategy: Expectation–Maximization Algorithm

## 3. Data

## 4. Results

#### Real Property Value Index Simulation and Probability Index

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

`R`codes used in this paper are available upon request from the corresponding author.

## Conflicts of Interest

## Note

1 |

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**Figure 1.**Evolution of the real property value index in Chile’s Metropolitan Region and its annual growth rate (1994–2020).

**Figure 2.**Monthly real property value index and its annual growth rate (1994–2020), disaggregated by type of housing (house and apartment) and according to geographic area: (

**a**) house prices in Metropolitan Region northwest, (

**b**) in the northeast, (

**c**) in the south; (

**d**) apartment prices in Central Santiago, (

**e**) in Metropolitan Region northeast, (

**f**) in the northwest, (

**g**) the south; (

**h**) house prices in the Metropolitan Region; (

**i**) apartment prices in the Metropolitan Region.

**Figure 3.**Real property value index, credit rate in inflation-linked units for housing and economic activity index (1995–2020).

**Figure 4.**Evolution of the convergence rates of the EM algorithm. (

**a**) Annual real property value index growth (in %); (

**b**) Coefficient of the regression constant; (

**c**) Coefficient that accompanies annual economic activity index growth; (

**d**) Coefficient that accompanies the mortgage interest rate.

**Figure 5.**Evolution of the convergence rates of the EM algorithm. (

**a**) Standard deviation (volatility) of annual real property value index growth (in %); (

**b**) probability of observing the alternative state through which the real property value index is passing.

**Figure 6.**Mixed distribution of the annual growth rate of the real property value index (1994–2020). Distribution function of real property value index annual growth and its alternative states.

**Figure 7.**Conditional probabilities for each state of annual real property value index growth (January 1995 to December 2020). (

**a**) Conditional probabilities for the expansive regime, (

**b**) the medium growth regime, (

**c**) the contractive regime.

**Figure 8.**Real property value index evolution and its simulation based on the estimated parameters of the regime change model. (

**a**) Observed and simulated real property value index; (

**b**) observed real property value index and a 95% confidence interval of its simulation.

**Figure 10.**Evolution of the annual real property value index growth rate and its exuberance probability index.

Variable | Notation | Definition | Source |
---|---|---|---|

Real property value index in Santiago, Chile | ${p}_{t}$ | Measures the evolution of real housing price index methodology and is based on hedonic price estimation techniques. | Idrovo and Lennon (2011) |

Economic activity index | ${y}_{t}$ | Variable that reflects the evolution of Chile’s economic cycle and the life cycle income of consumers and investors in the real estate market. | Escandón et al. (2005); BCCh (2006) |

Mortgage interest rates denominated in inflation-linked units | ${r}_{t}$ | Variable that includes mortgage credit costs and financial conditions for accessing housing in Chile. | BCCh (2021) |

**Table 2.**Statistical summary of the variables to be considered in the regime change model and creation of the probability index (1994–2020).

Real Property | Credit Rate | Economic Activity | |
---|---|---|---|

Value Index (Annual | Mortgage | Index (Annual | |

Variation, in %) | (in %) | Variation, in %) | |

Mean | 2.92 | 5.65 | 4.09 |

Median | 2.01 | 4.53 | 4.20 |

Maximum | 12.72 | 13.54 | 23.41 |

Minimum | −3.34 | 1.99 | −15.52 |

Stand. Dev. | 3.99 | 2.57 | 4.39 |

Asymmetric coef. | 0.59 | 0.88 | −0.45 |

Kurtosis | 2.41 | 2.43 | 7.25 |

Jarque-Bera (p-value) | 0.00 | 0.00 | 0.00 |

**Table 3.**Information criteria for fitting the switching model with predetermined variables, according to alternative states of nature.

Information Criteria | Alternative States or Regimens | |||
---|---|---|---|---|

$\mathit{M}=2$ | $\mathit{M}=3$ | $\mathit{M}=4$ | $\mathit{M}=5$ | |

Akaike | 4.41 | 4.05 | 5.10 | 5.10 |

Hannan–Quinn | 4.46 | 4.14 | 5.19 | 5.22 |

Schwarz | 4.53 | 4.27 | 5.33 | 5.39 |

**Table 4.**Convergence rate for different annual growth regimes of the real property value index. The values in parentheses correspond to the standard error of the coefficients that accompany economic activity index and loan interest rates, respectively.

Description | Parameters | Expansive | Medium | Contractive |
---|---|---|---|---|

$({\mathit{S}}_{\mathit{t}}=1)$ | $({\mathit{S}}_{\mathit{t}}=2)$ | $({\mathit{S}}_{\mathit{t}}=3)$ | ||

Conditional growth (in %) | ${\overline{\widehat{p}}}_{t}={\overline{\mathbf{x}}}_{t-1}{\widehat{\mathbf{\beta}}}_{t}^{\left({k}_{0}\right)}\left(s\right)$ | 6.314 | 3.310 | −0.373 |

Conditional Stand. Dev. (in %) | ${\widehat{\sigma}}_{t}^{2\left({k}_{0}\right)}\left(s\right)\in {\widehat{\mathbf{\theta}}}^{\left({k}_{0}\right)}$ | 2.077 | 1.738 | 1.353 |

Long-term probability (in %) | $\mathrm{E}\left[\mathbb{P}\left({S}_{t}\right|{\widehat{\mathbf{\theta}}}^{\left({k}_{0}\right)})\right]$ | 0.358 | 0.338 | 0.304 |

Intercept | ${\widehat{\mu}}_{t}^{\left({k}_{0}\right)}\left(s\right)\in {\widehat{\mathbf{\beta}}}_{t}^{\left({k}_{0}\right)}\left(s\right)\subset {\widehat{\mathbf{\theta}}}^{\left({k}_{0}\right)}$ | 14.067 ** | 6.273 ** | 0.673 ** |

(0.464) | (0.050) | (0.080) | ||

Economic activity index coefficient | ${\gamma}_{{s}_{t}}^{{k}_{0}}\in {\widehat{\mathbf{\beta}}}_{t}^{\left({k}_{0}\right)}\left(s\right)\subset {\widehat{\mathbf{\theta}}}^{\left({k}_{0}\right)}$ | 0.0124 | 0.236 ** | 0.075 ** |

(0.302) | (0.052) | (0.032) | ||

Interest rate coefficient | ${\delta}_{t}^{\left({k}_{0}\right)}\left(s\right)\in {\widehat{\mathbf{\beta}}}_{t}^{\left({k}_{0}\right)}\left(s\right)\subset {\widehat{\mathbf{\theta}}}^{\left({k}_{0}\right)}$ | −1.381 ** | −0.695 ** | −0.239 ** |

(0.388) | (0.067) | (0.042) | ||

Iterations for convergence | ${k}_{0}$ = 92 | |||

Iterations considered | ${k}_{max}$ = 500 |

**Table 5.**Transition probabilities and average duration of the alternate stages of real property price growth. The values in brackets correspond to the average duration of the real property value index cycles, considering their historical evolution.

Expansive | Medium | Contractive | |
---|---|---|---|

$({\mathit{S}}_{\mathit{t}-\mathbf{1}}=\mathbf{1})$ | $({\mathit{S}}_{\mathit{t}-\mathbf{1}}=\mathbf{2})$ | $({\mathit{S}}_{\mathit{t}-\mathbf{1}}=\mathbf{3})$ | |

Expansive $({S}_{t}=1)$ | ${p}_{11}=0.836$ | ${p}_{12}=0.135$ | ${p}_{13}=0.043$ |

[6.1 months] | [1.2 months] | [1.0 months] | |

Medium $({S}_{t}=2)$ | ${p}_{21}=0.127$ | ${p}_{22}=0.768$ | ${p}_{23}=0.108$ |

[1.1 months] | [4.3 months] | [1.1 months] | |

Contractive $({S}_{t}=3)$ | ${p}_{31}=0.037$ | ${p}_{32}=0.097$ | ${p}_{33}=0.849$ |

[1.0 months] | [1.1 months] | [6.6 months] |

**Table 6.**Unit root statistics tests. (a) Equation includes only constant, (b) equation includes constant and trend. DFA: Augmented Dickey–Fuller test. PP: Phillips–Perron test. ${P}_{t}$ is the real property value index at time t; ${p}_{t}$ is the annual real property value index growth rate at time t; and ${y}_{t-1}$ is the annual economic activity index growth rate in $t-1$. ${r}_{t-1}$ is the mortgage interest rate in $t-1$ and ${\widehat{\epsilon}}_{t}$ is the estimated series of the switching model residuals.

Indicators | Levels (a) | Levels (b) | ||
---|---|---|---|---|

DFA | PP | DFA | PP | |

Real property value index $\left({P}_{t}\right)$ | 3.30 | 3.62 | −0.25 | 0.14 |

${\Delta}^{12}$ Real property value index $\left({p}_{t}\right)$ | −1.86 | −3.07 | −2.54 | −3.69 |

Economic activity index $\left({y}_{t-1}\right)$ | −4.34 | −4.16 | −5.05 | −4.89 |

Interest rate $\left({r}_{t-1}\right)$ | −1.21 | −0.89 | −2.35 | −2.01 |

Residuals $\left({\widehat{\epsilon}}_{t}\right)$ | −5.29 | −5.93 | −5.34 | −5.92 |

Critical Value | ||||

1% | −3.45 | −3.45 | −3.99 | −3.99 |

5% | −2.87 | −2.87 | −3.42 | −3.42 |

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**MDPI and ACS Style**

Idrovo-Aguirre, B.J.; Lozano, F.J.; Contreras-Reyes, J.E.
Prosperity or Real Estate Bubble? Exuberance Probability Index of Real Housing Prices in Chile. *Int. J. Financial Stud.* **2021**, *9*, 51.
https://doi.org/10.3390/ijfs9030051

**AMA Style**

Idrovo-Aguirre BJ, Lozano FJ, Contreras-Reyes JE.
Prosperity or Real Estate Bubble? Exuberance Probability Index of Real Housing Prices in Chile. *International Journal of Financial Studies*. 2021; 9(3):51.
https://doi.org/10.3390/ijfs9030051

**Chicago/Turabian Style**

Idrovo-Aguirre, Byron J., Francisco J. Lozano, and Javier E. Contreras-Reyes.
2021. "Prosperity or Real Estate Bubble? Exuberance Probability Index of Real Housing Prices in Chile" *International Journal of Financial Studies* 9, no. 3: 51.
https://doi.org/10.3390/ijfs9030051