A look into German housing markets: A bubble call?

Abstract

The steep rise in German house prices in recent years raises the question of whether a speculative bubble has already emerged. Using a modified present-value model, we estimate the size of speculative house price bubbles in the German housing market. We do not find evidence for positive bubble accumulation in recent years, and interpret the current bullish run as reflecting the correction of house prices that have been undervalued for more than 10 years. With house prices close to their fair values as of 2018:Q1, our answer to the question is, ‘Not yet, but it is likely soon’.

Keywords

German housing market present-value model price–rent ratio fundamentals bubble

Introduction

Historically, the movement in the German housing market has been fairly decoupled from those in other advanced countries. Figure 1 provides an illustration: over the period 1995 to 2007, real house prices in the UK and US increased at an annual rate of 8.9% and 3.8%, respectively, whereas the German housing market was stable and even registered a slight decrease at the rate of 1.7%. During the subsequent housing market downturn from 2008 to 2011, the decrease in the real house price amounted to 3.4% and 5.6% in the UK and US, respectively, over a year on average. In contrast, the German housing market remained dormant again, with a slight increase in prices at the annual rate of 0.58% during the same period.

Figure 1.

Real house price indexes (1971:Q1-2018:Q1).¹

In assessing the cause of the unusual dormancy in the German housing market, a few institutional features peculiar to Germany have been pointed out. For example, Maclennan et al. (1998) emphasize that, unlike other advanced countries, Germany is characterized by fixed interest mortgage rates, relatively low loan-to-value ratios, high transaction costs and a smaller owner-occupied sector. According to Dahl and Góralczyk (2017), the home ownership share in Germany was only 52% in 2015, compared to 67% in the other Euro zone countries. The existence of a well-developed rental housing market is a reason for the prevalence of the prudent home mortgage system in Germany, as shown in Voigtländer (2014).

In recent years, since the end of the global financial crisis, however, the German housing market has come to be increasingly ‘synchronized’ with those in other advanced countries, as the global economy and housing markets have recovered from the recession: the annualized rate of house price increases in Germany since 2012 has been as high as 3.2%, readily comparable to 3.9% and 4.8% in the UK and US, respectively. Such a steep increase in house prices has sparked concerns about a potential price bubble, especially in major metropolitan areas, despite the considerable increase in the housing supply in the past few years.² The Bundesbank (Deutsche Bundesbank, 2018) therefore states that ‘in urban areas, the prices of housing continue to be well above the level that would appear justified in terms of the longer-term economic and demographic determinants, with house prices in the big cities overvalued by 35%’.

In contrast to the continued warnings about potential housing bubbles on the policy front, empirical analyses of the German housing market in recent years have been relatively scant and have yielded mixed results. Some earlier studies – for example, Chen and Funke (2013), Kholodilin et al. (2014), and an de Meulen and Micheli (2013) – attribute the house price inflation largely to fundamental factors, such as the increase in the number of households and monetary conditions favorable to housing market investors, and not to speculative exaggerations. Standing in sharp contrast, however, Kajuth et al. (2013) find that, although house prices reflect quite well the economic conditions in most of the regions they examine, there is a substantial overvaluation of up to 25% in metropolitan housing markets. With few studies examining the strong housing market development in recent years, therefore, this paper examines the possibility that current (as late as in 2018:Q1) house prices in Germany are ridden with speculative exaggeration. More specifically, by decomposing the movements in house prices into responses to market fundamentals and speculative bubbles, we identify the main driver of the German housing market.

This task in turn requires us to take a stance on the relationship between housing prices and their fundamental determinants. Our analytical workhorse is the present-value model of asset pricing in the spirit of Campbell and Shiller (1988a, 1988b), tying an asset’s worth to the expected value of the future payoff stream accruing to the asset. This type of model has been frequently used in the literature on housing market studies. For example, Meese and Wallace (1994) derive a present-value relation among house price, rents and the costs of capital for home owners (relative to renters) in the San Francisco area, and find appreciable upward deviations of house prices from what is predicted by the relation. In more recent studies, Case and Shiller (2003) and Gallin (2008) also find that US house prices are significantly above what is justified by the present-value model. Using a structural present-value relation explicitly derived from a dynamic equilibrium model, Ayuso and Restoy (2006) find evidence of overvaluation in the house prices of the US, the UK and Spain.

Adapted to the housing market, the present-value model predicts that house prices and rents should move in tandem, yielding a stable price–rent ratio. The movements in the actual price–rent ratio in Germany are, however, difficult to justify by the predictions of the standard present-value model. As shown in Figure 2, the price–rent ratio exhibits a long downswing, starting in the late 1970s, and a steep upswing since the late 2000s, rendering the price–rent ratio deviant from its historical average for extended periods.

Figure 2.

Price–rent ratio in Germany.

To the extent that the ‘fundamental’ forces in the housing market are reflected in the stable price–rent ratio, one way to address the long swing in the ratio is to append to the standard present-value relation an additional ‘non-fundamental’ component. In this regard, we follow Balke and Wohar (2009) and augment the standard present-value model with a special class of bubbles; that is, those that periodically gestate, bust and then reappear. Once the present-value model thus modified is estimated, we can decompose the movements in the price–rent ratio into what is attributable to the housing market fundamentals and the bubble, so that we can address the main questions posed above. Yielding the estimated probability of the explosive bubble regime for each sample period, our methodology can also be of help in judging whether or not the housing market is currently in the bubble regime. To the best of our knowledge, no previous studies have examined the possibility of a periodically collapsing bubble in the German housing market.

A few interesting findings emerge from our study. First, the bubble component captures a considerable portion of the dynamics in the German housing market, over the whole sample in general and in recent years in particular. Second, the German housing prices were increasingly ‘undervalued’ for more than two decades since the late 1990s. Third, the surge in house prices as well as their ratio to rent during the past seven to eight years mainly reflects the correction of the housing market towards fair valuation. Although our results do not preclude the possibility of the self-reinforcing evolution of a bubble down the road, and therefore support the need for pre-emptive actions by the Bundesbank, the German housing market as of 2018:Q1 appears more or less fairly-valued and has not entered the territory of a bubble – at least, not yet.

The model, data and key estimates

A present-value model with collapsing bubbles

We define $H_{t + 1} = (P_{t + 1} + R_{t + 1}) / P_{t}$ as the one-period gross real return on housing, where $P_{t + 1}$ is the real house price at the end of period t and $R_{t + 1}$ is the real rent payment during period t. Resorting to the Campbell–Shiller approximation, we have

p r_{t} = K + ρ p r_{t + 1} + Δ r_{t + 1} - h_{t + 1}

(1)

where $p r_{t} = \log (P_{t} / R_{t}),$ $Δ r_{t + 1} = \log (R_{t + 1} / R_{t}),$ $h_{t + 1} = \log (H_{t + 1}),$ $ρ = e^{\bar{p r}} / (1 + e^{\bar{p r}}),$ $\bar{p r}$ is the average of the log of the price–rent ratio over the sample, and $K$ is a linearization constant. In the absence of any explosive behavior in $p r_{t}$ , we obtain the standard present-value formula

p r_{t} = \frac{K}{1 - ρ} + E_{t} {\sum_{j = 0}^{\infty} ρ^{j} (Δ r_{t + j + 1} - h_{t + j + 1})}

(2)

which implies that the log of the price–rent ratio is a weighted discounted sum of the expected future rent growth $Δ r_{t + j + 1}$ and gross real return $h_{t + j + 1}$ , for $j ⩾ 0$ .

We introduce a pair of modifications to equation (2). First, following previous studies (e.g. Campbell and Ammer, 1993; Campbell et al., 2009), we decompose the log of gross real return, $h_{t}$ , into the risk-free real interest rate, $i_{t}$ , and the excess returns, π_t $π_{t}$ , which reflects the risk premium for housing investment. Second, the price–rent ratio is allowed to deviate from what is postulated by (2), so that

p r_{t} = \frac{K}{1 - ρ} + E_{t} {\sum_{j = 0}^{\infty} ρ^{j} (Δ r_{t + j + 1} - i_{t + j + 1} - π_{t + j + 1})} + b_{t} = p r_{t}^{f} + b_{t}

(3)

where $p r_{t}^{f}$ is the fundamental part of the price–rent ratio. An extraneous component of the ratio, the term $b_{t}$ captures the non-fundamental deviations of the actual ratio. More detailed specification of $b_{t}$ is discussed later.

As in van Binsbergen and Koijen (2010), we treat the expectations of the three housing market fundamentals, ( $Δ r, i, π$ ), as unobserved components following simple AR(2) processes:³

g_{t} - γ_{0} = γ_{1} (g_{t - 1} - γ_{0}) + γ_{2} (g_{t - 2} - γ_{0}) + ϵ_{t}^{g},

(4a)

μ_{t} - δ_{0} = δ_{1} (μ_{t - 1} - δ_{0}) + δ_{2} (μ_{t - 2} - δ_{0}) + ϵ_{t}^{μ}

(4b)

λ_{t} - θ_{0} = θ_{1} (λ_{t - 1} - θ_{0}) + θ_{2} (λ_{t - 2} - θ_{0}) + ϵ_{t}^{λ}

(4c)

where $g_{t} = E_{t} [Δ r_{t + 1}]$ , $μ_{t} = E_{t} [i_{t + 1}]$ , and $λ_{t} = E_{t} [π_{t + 1}]$ are the one-period-ahead expectations of rent growth, risk-free returns and excess returns. The innovations, $ε_{t} = (ε_{t}^{g}, ε_{t}^{μ}, ε_{t}^{λ})$ , in expectations reflect news on the housing market fundamental variables, and are assumed to follow an i.i.d. Gaussian process.

In principle, the non-fundamental deviation $b_{t}$ can reflect irrational behavior, such as fads (e.g. Summers, 1986; Shiller and Perron, 1985), as well as rational speculation. At the risk of over-emphasizing the role of speculation in driving housing prices, we interpret $b_{t}$ as representing a rational speculative bubble. More specifically, we follow Balke and Wohar (2009) and specify $b_{t}$ as a periodically collapsing bubble switching between two regimes – that is, a non-exploding regime and an exploding regime – so that the bubble can form and collapse sporadically. The actual regimes that prevail in each period are in turn determined by a hidden Markov-Switching variable, $S_{t}$ , with the time-invariant transition probabilities

Prob [S_{t} = 1 | S_{t - 1} = 1] = p, Prob [S_{t} = 0 | S_{t - 1} = 0] = q

(5)

In the non-exploding regime with $S_{t}$ =0, $b_{t}$ is a stationary AR process:

b_{t} = \bar{b} + ψ b_{t - 1} + ε_{t}^{b}, 0 < ψ < 1

(6a)

so that the bubble decays slowly in this regime.⁴ If the regime in the current period switches to the exploding one from the non-exploding one of the previous period (i.e. if $S_{t} = 1$ follows $S_{t - 1} = 0$ ), the law of motion for $b_{t}$ is

b_{t} = - \frac{q}{(1 - q)} \bar{b} + \frac{1}{1 - q} [\frac{1}{ρ} - q ψ] b_{t - 1} + ε_{t}^{b} .

(6b)

Finally, if the bubble continues to stay in the exploding regime (i.e. if $S_{t} = 1$ is preceded by $S_{t - 1} = 1$ ),

b_{t} = - \frac{(1 - p)}{p} \bar{b} + \frac{1}{p} [\frac{1}{ρ} - (1 - p) ψ] b_{t - 1} + ε_{t}^{b} .

(6c)

We close the model by relating the actual data and model variables via measurement equations. Equations (4) and (6) together link the actual price–rent ratio to the expectation and bubble terms $(g_{t}, μ_{t}, λ_{t}, b_{t}$ ). The actual data series of the rent growth,∆r_t $Δ r_{t}$ , real interest rate, $i_{t}$ , and excess returns, $π_{t}$ π_t, are the sum of respective one-step-ahead expectations and idiosyncratic disturbances:

Δ r_{t} = g_{t - 1} + u_{t}^{r}, i_{t} = μ_{t - 1} + u_{t}^{i}, π_{t} = λ_{t - 1} + u_{t}^{π}

(7)

where the disturbances, $u_{t} = (u_{t}^{r}, u_{t}^{i}, u_{t}^{π})$ , are a Gaussian i.i.d. distribution with a covariance matrix, $Σ_{u} = diag (σ_{g}^{2}, σ_{i}^{2}, σ_{λ}^{2})$ . Finally, we assume that the innovations $(ε_{t}^{g}, ε_{t}^{μ}, ε_{t}^{λ}, ε_{t}^{b})$ and the disturbances $u_{t}$ are independent of each other.

The present-value model described above can be cast into a state-space form subject to Markov-Switching.⁵ As such, the model is estimated by the maximum likelihood method of Kim and Nelson (1999).

Data and key estimates

For actual estimation of the model, we use the quarterly housing market data of Germany spanning 1971:Q1 to 2018:Q1. The raw series of real house prices, real rents and price–rent ratio, all seasonally adjusted, are available from the OECD database. As the nominal interest rates, we use the 10-year Treasury Bond Yield Rates obtained from the Federal Reserve Economic Data of the St Louis Fed (FRED). The risk-free rate of return is then proxied by the real interest rates; that is, the nominal interest rates minus the inflation rates, where the latter are calculated from the German core CPI series obtained from the FRED. The rates of excess returns are then calculated as the log of the one-period return minus the real interest rate.

Before estimating the model, we check the low-frequency properties of the price and rent series constructed above. In Table 1, the ADF and Phillips–Perron tests (in the top panel) fail to reject the null of a unit root in the price–rent ratio at the 5% significance level, and the Johansen test (in the bottom panel) does not detect any cointegrating relation between house price and rents. We interpret these results as a validation of incorporating an extraneous term with potential non-stationarity into the standard price–rent formula.

Table 1.

Results of pre-tests.

Stationarity tests (H₀: unit root in the price–rent ratio)
Test	Test statistic	p-value
ADF	−1.136	0.702
Phillips–Perron	−1.022	0.745
Johansen test (H₀: no cointegration between house price and rent)
Test	Test statistic	p-value
Trace	5.780	0.723
Maximum eigenvalue	5.121	0.729

Some key estimation results are presented in Table 2. From the top panel, we can deduce the following: the long-run autoregressive coefficients $δ_{1} + δ_{2} = 0.8971$ and $θ_{1} + θ_{2} = 0.9292$ for the expectations of future real interest rates and excess returns, respectively, show that investors change their expectations very slowly. In contrast, the long-run AR coefficient of $γ_{1} + γ_{2} = 0.5394$ exhibits a modest degree of persistence in the expectation of future rent growth.

Table 2.

Key maximum likelihood estimates.

Fundamental part
$γ_{1}$	0.3404	(0.0484)
$γ_{2}$	0.1990	(0.0376)
$δ_{1}$	1.1969	(0.0489)
$δ_{2}$	−0.2998	(0.0411)
$θ_{1}$	0.6405	(0.0409)
$θ_{2}$	0.2887	(0.0388)
Bubble part
$q$	0.9403	(0.0078)
$p$	0.9751	(0.0042)
$\bar{b}$	−0.0039	(0.0008)
$ψ$	0.9059	(0.0011)

Note: All parameters in Table 2 are sharply estimated at the 5% critical level.

The properties of the bubble components are reported in the bottom panel. In terms of the expected duration the exploding regime turns out to be a lot more persistent, lasting 1/(1 − 0.9751) = 40.16 quarters, whereas the non-exploding regime continues for 1/(1 − 0.9403) = 16.75 quarters on average. The AR coefficient and transition probabilities for the bubble component depict how the bubble evolves, switching between the two regimes. More specifically, the bubble has a strong tendency of self-reinforcing once it has entered the explosive regime. Suppose that a mild overvaluation of house prices is developed in a non-explosive period t, and that the bubble regime switches to the explosive one in the next period t+1. The corresponding AR coefficient in period t+1 is $\frac{1}{1 - q} [\frac{1}{ρ} - q ψ]$ = 3.0842, which explains a sudden expansion of the bubble frequently observed in the earlier phase following its birth. If the exploding regime continues in the subsequent period, t+2, the AR coefficient is then $\frac{1}{p} [\frac{1}{ρ} - (1 - p) ψ]$ =1.0394, so that the bubble goes on even without any further news driving it. Now suppose that the regime switches back to the non-exploding regime and remains there. As the AR coefficient, $ψ$ = 0.9059, implies, the bubble that has developed so far goes a long way until it dies away, unless there is a sudden development of pessimistic innovation in $ε_{t}^{b}$ .

Bubble v. fundamentals: What has driven the German housing market?

Results from baseline model

We now address the main question as to the sources of variations in the German price–rent ratio. Panel (a) of Figure 3 plots the estimated fundamental price–rent ratio with the solid line, along with the actual ratio, depicted by the shaded area. A few findings emerge. First, whereas the actual price–rent ratio exhibits a large swing, the fundamental ratio turns out to be considerably stable. Historically, the late 1970s and the early 1980s are deemed to have been periods in which there was overvaluation in the German housing market, as shown by Philiponnet and Turrini (2017). Our results lend further support to this view, showing that the surge and fall in the price–rent ratio in these periods is caused by the bubble component of the ratio. Second, the continued fall in the ratio from the mid-1990s and the equally long upturn that follows reflect undervaluation and the subsequent correction in the price–rent ratio towards its fundamental level. In particular, the rise in the ratio from 2011 is accompanied by increases in the fundamental ratio (except for its consecutive decline in 2017:Q4 and 2018:Q1), which supports the view that the current bullish run in the German housing market does not necessarily involve bubble formation and expansion.

Figure 3.

Decomposing the price–rent ratio (baseline model).

In panel (b) of Figure 3, the proportion of speculative bubble in the house prices is plotted against the estimated probability of the non-exploding bubble regime (shaded), from which more detailed features of the bubble emerge. House prices were overvalued up to 20% by speculative bubbles at the peak of the bullish run in 1981, but the bubble was subdued to zero by the mid-1980s. From the mid-1990s, however, the price–rent ratio continued to fall for more than two decades, owing to the ‘negative’ accumulation of the bubble.⁶ The estimated feature of the bubble component provides more detailed description of this phenomenon. A negative bubble was initially formed prior to 1995. Since the bubble remained in the exploding regime where the AR coefficient (1.0394) is larger than one, the rapid accumulation of the negative bubble incorporates not only the self-reinforcing nature of the speculative bubble, but also the effects of ‘news shocks’ which caused a stream of negative innovations in it. As a result, house prices were undervalued by around 37% over the period of the subprime financial crisis.

Given the concern about the overheated housing market during the current bullish run, the movement of the bubble component in the 2010s provides useful information. Although there has been a continued rise in the percentage of bubble, it is not justified to give a bubble call merely on that account. There are two reasons for this. First, the rise in the bubble is towards zero, not higher. As mentioned above, such a movement should be deemed not so much a bubble expansion but rather a correction towards fair valuation. Second, the increase in the bubble percentage (again, towards zero) has slowed considerably since 2012 compared with the preceding two to three years. All in all, the percentage of bubble as at the end of our sample period is nearly zero.

An extension: Incorporating macro fundamentals

We note one pitfall of the baseline model in evaluating the relative importance of the fundamental and bubble components of the price–rent ratio: lacking other variables that can affect the housing market, the baseline model may have falsely relegated the effects of the left-out fundamentals to the bubble term.⁷ To address this problem, we allow the one-period-ahead expectations $(g_{t}, μ_{t}, λ_{t})$ of investors to depend on the macro fundamental variables $X_{t}$ :

g_{t} - γ_{0} = γ_{1} (g_{t - 1} - γ_{0}) + γ_{2} (g_{t - 2} - γ_{0}) + β_{g}' X_{t} + ϵ_{t}^{g},

(8a)

μ_{t} - δ_{0} = δ_{1} (μ_{t - 1} - δ_{0}) + δ_{2} (μ_{t - 2} - δ_{0}) + β_{μ}' X_{t} + ϵ_{t}^{μ},

(8b)

λ_{t} - θ_{0} = θ_{1} (λ_{t - 1} - θ_{0}) + θ_{2} (λ_{t - 2} - θ_{0}) + β_{λ}' X_{t} + ϵ_{t}^{λ},

(8c)

where the augmented variables $X_{t}$ consist of the year-on-year growth rates of real GDP and dwellings and residential building permits, expected to capture the movement of demand and supply in the housing market, respectively.⁸

As shown in Figure 4, the results for the extended version turn out to be qualitatively invariant, except that the estimated fundamental ratio is slightly lower than in the baseline model.⁹ The continued increase in the actual price–rent ratio since the mid-2010s mainly reflects the correction towards its fundamental level after a decade of widening undervaluation. As a result, the estimated percentage of bubble is again negligible, accounting for a mere 1.43% of house price as of 2018:Q1.

Figure 4.

Decomposing the price–rent ratio (extended model).

Although there seems to be no sign of speculative exaggeration at the current juncture, a few words of warning are in order concerning the possibility of overvaluation around the corner. The estimated probabilities for the baseline model suggest that the bubble switches from the non-exploding regime to the exploding one in 2017:Q4, and that regime becomes more likely to prevail in the following quarter. If the current regime continues along with a few occurrences of positive innovation in the bubble component, the housing market may step into the territory of an explosive and self-reinforcing bubble. With the consecutive decline in the fundamental ratio in recent quarters, the warning of a bubble and the call for pre-emptive measures by the Bundesbank is deemed to be timely. Macro fundamentals taken into account, the extended model shows a similar yet stronger signal, stamping 2017:Q3 as the date of regime switching.

Conclusion

This paper employs the Campbell–Shiller present-value model to examine the sources of variation in the German housing market. In contrast to the prediction of the standard present-value formula, the German price–rent ratio since the 1970s has exhibited a large downswing followed by an equally long upturn. We therefore modify the Campbell–Shiller model and allow the price–rent ratio to be driven by a periodically collapsing rational bubble, in addition to the expectations of future housing market fundamentals.

When we decompose the price–rent ratio into the fundamental and bubble parts, a few important findings emerge. First, with the fairly stable fundamental ratio, the historical surge and fall in the price–rent ratio are caused mainly by the bubble component of the ratio. Second, the rise in the ratio in the 2010s is deemed as a correction towards the fair value after a decade of continued decline in the ratio below its fundamental level. Finally, although there seems to be no sign of speculative bubble build-up as of 2018:Q1, the German housing market is likely to have entered a regime where the bubble once formed exhibits explosive behavior. These results are corroborated by the extended model, incorporating macro fundamental variables intended to capture the supply and demand factors in the housing market.

Footnotes

Appendix

We first apply the law of iterated expectation to represent the fundamental part of the price–rent ratio in terms of expectations:

(A1)

p r_{t}^{f} = \frac{Κ}{1 - ρ} + \frac{γ_{0} - δ_{0} - θ_{0}}{1 - ρ} + B_{1} [\begin{matrix} g_{t} \\ g_{t - 1} \end{matrix}] - B_{2} [\begin{matrix} μ_{t} \\ μ_{t - 1} \end{matrix}] - B_{3} [\begin{matrix} λ_{t} \\ λ_{t - 1} \end{matrix}],

where

$B_{1} = ρ [10] {(I - ρ [\begin{matrix} γ_{1} & γ_{2} \\ 1 & 0 \end{matrix}])}^{- 1}, B_{2} = ρ [10] {(I - ρ [\begin{matrix} δ_{1} & δ_{2} \\ 1 & 0 \end{matrix}])}^{- 1},$ and $B_{3} = ρ [10] {(I - ρ [\begin{matrix} δ_{1} & δ_{2} \\ 1 & 0 \end{matrix}])}^{- 1}$ .

The transition equations of the model are cast into the following form:

(A2)

\begin{matrix} [\begin{matrix} \begin{matrix} g_{t} \\ g_{t - 1} \end{matrix} \\ \begin{matrix} μ_{t} \\ μ_{t - 1} \end{matrix} \\ \begin{matrix} λ_{t} \\ λ_{t - 1} \end{matrix} \\ \begin{matrix} u_{t}^{r} \\ \begin{matrix} u_{t}^{i} \\ \begin{matrix} \begin{matrix} u_{t}^{π} \\ ε_{t}^{g} \end{matrix} \\ ε_{t}^{μ} \\ \begin{matrix} ε_{t}^{λ} \\ b_{t} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] = [\begin{matrix} 0_{2 \times 1} \\ 0_{2 \times 1} \\ 0_{2 \times 1} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \bar{b} (S_{t}) \end{matrix}] + [\begin{matrix} [\begin{matrix} γ_{1} & γ_{2} \\ 1 & 0 \end{matrix}] & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 7} \\ 0_{2 \times 2} & [\begin{matrix} δ_{1} & δ_{2} \\ 1 & 0 \end{matrix}] & 0_{2 \times 2} & 0_{2 \times 7} \\ 0_{2 \times 2} & 0_{2 \times 2} & [\begin{matrix} δ_{1} & δ_{2} \\ 1 & 0 \end{matrix}] & 0_{2 \times 7} \\ 0_{1 \times 13} \\ 0_{1 \times 13} \\ 0_{1 \times 13} \\ 0_{1 \times 13} \\ \begin{matrix}  \end{matrix} & \begin{matrix} 0_{1 \times 13} \\ 0_{1 \times 12} \end{matrix} & \begin{matrix} F_{b} (S_{t}) \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix}] [\begin{matrix} g_{t - 1} \\ g_{t - 2} \\ \begin{matrix} μ_{t - 1} \\ μ_{t - 2} \end{matrix} \\ \begin{matrix} λ_{t - 1} \\ λ_{t - 2} \end{matrix} \\ \begin{matrix} u_{t - 1}^{r} \\ u_{t - 1}^{i} \end{matrix} \\ \begin{matrix} u_{t - 1}^{π} \\ ε_{t - 1}^{g} \end{matrix} \\ ε_{t - 1}^{μ} \\ \begin{matrix} ε_{t - 1}^{λ} \\ b_{t - 1} \end{matrix} \end{matrix}] + [\begin{matrix} C_{G} \\ C_{M} \\ C_{Λ} \\ \begin{matrix} I_{6} & 0_{6 \times 1} \end{matrix} \\ \begin{matrix} 0_{1 \times 6} & 1 \end{matrix} \end{matrix}] [\begin{matrix} u_{t}^{r} \\ \begin{matrix} \begin{matrix} u_{t}^{i} \\ u_{t}^{π} \end{matrix} \\ ε_{t}^{g} \\ ε_{t}^{μ} \end{matrix} \\ \begin{matrix} ε_{t}^{λ} \\ ε_{t}^{b} \end{matrix} \end{matrix}] \\ (X_{t} = A (S_{t - 1}) + F (S_{t - 1}) X_{t - 1} + C η_{t}) \end{matrix}

where

$C_{G} = [\begin{matrix} 0001000 \\ 0_{1 \times 7} \end{matrix}]$ , $C_{M} = [\begin{matrix} 0000100 \\ 0_{1 \times 7} \end{matrix}]$ , $C_{Λ} = [\begin{matrix} 0000010 \\ 0_{1 \times 7} \end{matrix}]$ , $\bar{b} (S_{t}) = \bar{b} (1 - S_{t}) - \frac{q}{1 - q} \bar{b} S_{t} (1 - S_{t - 1}) - \frac{1 - p}{p} \bar{b} S_{t} S_{t - 1}$ , and $F_{b} (S_{t}) = ψ (1 - S_{t}) + \frac{1}{1 - q} (\frac{1}{ρ} - q ψ) S_{t} (1 - S_{t - 1}) + \frac{1}{p} (\frac{1}{ρ} - (1 - p) ψ) S_{t} S_{t - 1}$

Putting $p r_{t} = p r_{t}^{f} + b_{t}$ , we write the observation equations as

(A3)

\begin{matrix} [\begin{matrix} Δ r_{t} \\ \begin{matrix} i_{t} \\ π_{t} \end{matrix} \\ p r_{t} \end{matrix}] = [\begin{matrix} γ_{0} \\ \begin{matrix} δ_{0} \\ θ_{0} \end{matrix} \\ \frac{Κ}{1 - ρ} + \frac{γ_{0} - δ_{0} - θ_{0}}{1 - ρ} \end{matrix}] + [\begin{matrix} 01 & 00 & 00 & 10 & 00 & \begin{matrix} 0 & 00 \end{matrix} \\ \begin{matrix} 00 \\ 00 \end{matrix} & \begin{matrix} 01 \\ 00 \end{matrix} & \begin{matrix} 00 \\ 01 \end{matrix} & \begin{matrix} 01 \\ 00 \end{matrix} & \begin{matrix} 00 \\ 10 \end{matrix} & \begin{matrix} \begin{matrix} 0 & 00 \end{matrix} \\ \begin{matrix} 0 & 00 \end{matrix} \end{matrix} \\ B_{1} & B_{2} & B_{3} & 00 & 00 & \begin{matrix} 0 & 01 \end{matrix} \end{matrix}] [\begin{matrix} g_{t} \\ g_{t - 1} \\ μ_{t} \\ μ_{t - 1} \\ λ_{t} \\ λ_{t - 1} \\ u_{t}^{r} \\ u_{t}^{i} \\ u_{t}^{π} \\ ε_{t}^{g} \\ ε_{t}^{μ} \\ ε_{t}^{λ} \\ b_{t} \end{matrix}] \\ (Y_{t} = Z + H X_{t}) \end{matrix}

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was supported by the Hankuk University of Foreign Studies Research Fund. We gratefully acknowledge this support. We are responsible for errors, if any.

Notes

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