Fractional Integration and Volatility Transmission Between Real Estate and Stock Markets: Novel Evidence from a FIGARCH-BEKK Approach

Abstract

This paper examines the non-linear integration between the real estate and stock market for a series of developed markets namely UK, Germany, Australia, Hong-Kong, Japan, Singapore and the US. The period of analysis covers different market phases for these countries. We examine the volatility dynamics of the real estate and stock market in the UK and Germany within a novel FIGARCH-BEKK model. Our results reveal evidence of a common long-term fractional integration between real estate and stock market for these two countries. Moreover, when there is a lower common order of fractional integration, there might also be a significant bilateral or unilateral volatility spillover effect between real estate and stock market. Robustness tests confirm the consistency of the FIGARCH-BEKK model even during the global financial crisis. Additional tests capture the existence of volatility spillovers and fractional integration in the rest of countries (Australia, Hong-Kong, Japan, Singapore and the US) under examination. Our findings entail significant implications for investors and policy makers.

Appendix

A1. The FIGARCH-BEKK approach is modeled as:

$$\Phi (L){\left(1-L\right)}^d\ast {U}_t=C{C}^{\prime }+\left(1-A{A}^{\prime }(L)\right)\ast {V}_t<=>$$

where, ${V}_{t}={U}_{t}-{H}_{t}$

$${\displaystyle \begin{array}{c}\Phi (L){\left(1-L\right)}^d\ast {U}_t=C{C}^{\prime }+\left(1-A{A}^{\prime }(L)\right)\ast \left({U}_t-{H}_t\right)<=>\\ {}\Phi (L){\left(1-L\right)}^d\ast {U}_t=C{C}^{\prime }+{U}_t-A{A}^{\prime }(L){U}_t-{H}_t+A{A}^{\prime }(L){H}_t<=>\\ {}{H}_t=C{C}^{\prime }+A{A}^{\prime }(L){H}_t- AA^{\prime }(L){U}_t+{U}_t-\Phi (L){\left(1-L\right)}^d\ast {U}_t<=>\\ {}{H}_t=C{C}^{\prime }+A{A}^{\prime }(L){H}_t-A{A}^{\prime }(L){U}_t+{U}_t-{\mathrm{UFILTER}}_t<=>\\ {}{H}_t=C{C}^{\prime }+{AH}_t(L){A}^{\prime }-{AU}_t(L){A}^{\prime }+{BU}_t{B}^{\prime }-{\mathrm{UFILTER}}_t<=>\\ {}{H}_t=C{C}^{\prime }+{AH}_{t-1}{A}^{\prime }-{AU}_{t-1}{A}^{\prime }+{BU}_t{B}^{\prime }-{QU}_{t-2}{Q}^{\prime }-{\mathrm{UFILTER}}_t\end{array}}$$

where,

$${\mathrm{UFILTER}}_{\mathrm{t}}=\Phi (\mathrm{L}){(1-\mathrm{L})}^{\mathrm{d}}*{\mathrm{U}}_{\mathrm{t}}$$

Our FIGARCH-BEKK (1, d, 2) model takes the following form:

$${H}_{t}=C{C}^{^{\prime}}+{A}^{^{\prime}}{H}_{t-1}A+{{BU}_{t}}^{2}{B}^{^{\prime}}-{AU}_{t-1}^{2}{A}^{^{\prime}}-{QU}_{t-2}^{2}{Q}^{^{\prime}}-{\text{KUFILTER}}_{t}{K}^{^{\prime}}$$

(11)

where,

$${\mathrm{KUFILTER}}_{t}{K}^{^{\prime}}=\left(\begin{array}{cc}{\kappa }_{11}& 0\\ 0& {\kappa }_{22}\end{array}\right)*\left(\begin{array}{cc}{\left(1-L\right)}^{d1}& 0\\ 0& {\left(1-L\right)}^{d2}\end{array}\right)*{U}_{t}*{\left(\begin{array}{cc}{\kappa }_{11}& 0\\ 0& {\kappa }_{22}\end{array}\right)}^{^{\prime}}$$

From the above it follows naturally to test the hypothesis d₁ = d₂ within this framework.

Baba et al. (1989) developed the GARCH-BEKK (1, 1) model as follows:

$$\begin{array}{c}{\mathrm{h}}_{11,\mathrm{t}} = {\upomega }_{1} + {\mathrm{\alpha }}_{11}^{2}*{\upvarepsilon }_{1,\mathrm{t}-1}^{2} + 2*{\upbeta }_{11}*{\upbeta }_{21}*{\mathrm{h}}_{1,\mathrm{t}-1}*{\mathrm{h}}_{2,\mathrm{t}-1} + {\mathrm{\alpha }}_{21}^{2}*{\upvarepsilon }_{2,\mathrm{t}-1}^{2}\\ {\mathrm{h}}_{22,\mathrm{t}} = {\upomega }_{1} + {\mathrm{\alpha }}_{12}^{2}*{\upvarepsilon }_{1,\mathrm{t}-1}^{2} + 2*{\upbeta }_{22}*{\upbeta }_{12}*{\mathrm{h}}_{1,\mathrm{t}-1}*{\mathrm{h}}_{2,\mathrm{t}-1} + {\mathrm{\alpha }}_{22}^{2}*{\upvarepsilon }_{2,\mathrm{t}-1}^{2}\\ {\mathrm{h}}_{12,\mathrm{t}} ={\upomega }_{12} +{\mathrm{\alpha }}_{11}{*\mathrm{\alpha }}_{12}*{\upvarepsilon }_{1,\mathrm{t}-1}^{2} +{(\upbeta }_{21}*+{\upbeta }_{12}+{\upbeta }_{11}*{\upbeta }_{22})*{\mathrm{h}}_{1,\mathrm{t}-1}*{\mathrm{h}}_{2,\mathrm{t}-1}+{\mathrm{\alpha }}_{21}*{\mathrm{\alpha }}_{22}*{\upvarepsilon }_{2,\mathrm{t}-1}^{2}\end{array}$$

(12)

We extend the multivariate GARCH-BEKK (1, 1) model to the FIGARCH-BEKK (1, d, 1) motivated by two specifications:

• 1. The variance–covariance matrix is positive definite.

• 2. Stationarity is ensured by restrictions in the variance–covariance matrix.

The bivariate FIGARCH-BEKK (1, d, 1) model may be represented as follows:

$$\begin{array}{c}{\mathrm{h}}_{\mathrm{jj},\mathrm{t}}=({{\uplambda }_{\mathrm{jj}}+\upvarepsilon }_{\mathrm{j},\mathrm{t}}*{{\uplambda }_{\mathrm{ij}}+\upvarepsilon }_{\mathrm{i},\mathrm{t}}{)}^{2}*\left(\mathrm{L}\right)+[{\upomega }_{\mathrm{j}}/(1-{\upbeta }_{\mathrm{jj}}(1))+({\upomega }_{\mathrm{j}}/1-{\upbeta }_{\mathrm{ij}}(1))] <=>\\ {\mathrm{h}}_{\mathrm{jj},\mathrm{t}}=({{\uplambda }_{\mathrm{jj}}*+\upvarepsilon }_{\mathrm{j},\mathrm{t}}*{{\uplambda }_{\mathrm{ij}}*\upvarepsilon }_{\mathrm{i},\mathrm{t}}{)}^{2}*\left(\mathrm{L}\right)+[{2\upomega }_{\mathrm{j}}/(1+{\upbeta }_{\mathrm{ij}}{\upbeta }_{\mathrm{jj}}((1))\end{array}$$

(13)

$${\mathrm{h}}_{12,\mathrm{t}} ={\upomega }_{12} +{\mathrm{\alpha }}_{11}{*\mathrm{\alpha }}_{12}*{\upvarepsilon }_{1,\mathrm{t}-1}^{2} +{(\upbeta }_{21}*+{\upbeta }_{12}+{\upbeta }_{11}*{\upbeta }_{22})*{\mathrm{h}}_{1,\mathrm{t}-1}*{\mathrm{h}}_{2,\mathrm{t}-1}+{\mathrm{\alpha }}_{21}*{\mathrm{\alpha }}_{22}*{\upvarepsilon }_{2,\mathrm{t}-1}^{2}$$

(14)

It follows from the results in Bollerslev and Mikkelsen (1996) that positive definiteness in the bivariate diagonal FIGARCH-BEKK (1, d, 1) model is ensured if

$${\uplambda }_{\mathrm{jj}}=1-\{[{\Phi }_{\mathrm{jj}}(\mathrm{L})(1-\mathrm{L}{)}_{j}^{d}]/[1-{\upbeta }_{\mathrm{jj}}(\mathrm{L})]\}$$

$${\upbeta }_{\mathrm{jj}}-{\mathrm{d}}_{\mathrm{j}}\le \left(1/3\right)\left(2-{\mathrm{d}}_{\mathrm{j}}\right),{\mathrm{d}}_{\mathrm{j}}\left[{\Phi }_{\mathrm{jj}}-\left(1/2\right)\left(1-{\mathrm{d}}_{\mathrm{j}}\right)\right]\le {\upbeta }_{\mathrm{jj}}{(\Phi }_{\mathrm{jj}}-{\upbeta }_{\mathrm{jj}}+{\mathrm{d}}_{\mathrm{j}})$$

and

$${\uplambda }_{\mathrm{jj},\mathrm{t}}=1-\{[{\Phi }_{\mathrm{ij}}(\mathrm{L})(1-\mathrm{L}{)}_{\mathrm{j}}^{\mathrm{d}}]/[1-{\upbeta }_{\mathrm{ij}}(\mathrm{L})]\}$$

$${\upbeta }_{\mathrm{jj}}-{\mathrm{d}}_{\mathrm{j}}\le \left(1/3\right)\left(2-{\mathrm{d}}_{\mathrm{j}}\right),{\mathrm{d}}_{\mathrm{j}}\left[{\Phi }_{\mathrm{ij}}-\left(1/2\right)\left(1-{\mathrm{d}}_{\mathrm{j}}\right)\right]\le {\upbeta }_{\mathrm{ij}}{(\Phi }_{\mathrm{ij}}-{\upbeta }_{\mathrm{ij}}+{\mathrm{d}}_{\mathrm{j}}$$

(15)

The above processes are stationary for all 0 ≤ d_j ≤ 1, and j = 1, 2.

A2. We also model real estate and stock market returns with long memory in variance in a bivariate setting. The extension of the simple VECH-HYGARCH variance equation is as follows:

$$\begin{array}{c}{h}_{11,t}={c}_{11}*\left(1-{\beta }_{11}\right)+{\beta }_{11}*{h}_{11,t-1}+{\alpha }_{11}*{\varepsilon }_{11,t}^{2}-\left({\beta }_{11}-\left(1-{\alpha }_{11}\right)\right)*{\delta }_{11})*{\varepsilon }_{11,t-1}^{2}-\\ {\alpha }_{11}*{\text{ufilter}}_{11,t}+{\alpha }_{11}*{\delta }_{11}*{\text{ufilter}}_{11,t-1}\end{array}$$

(16a)

$$\begin{array}{c}{h}_{12,t}={c}_{12}*\left(1-{\beta }_{12}\right)+{\beta }_{12}*{h}_{12,t-1}+{\alpha }_{12}*{\varepsilon }_{12,t}^{2}-\left({\beta }_{12}-\left(1-{\alpha }_{12}\right)\right)*{\delta }_{12})*{\varepsilon }_{12,t-1}^{2}-\\ {\alpha }_{12}*{\text{ufilter}}_{12,t}+{\alpha }_{12}*{\delta }_{12}*{\text{ufilter}}_{12,t-1}\end{array}$$

(16b)

$$\begin{array}{c}{h}_{22,t}={c}_{22}*\left(1-{\beta }_{22}\right)+{\beta }_{22}*{h}_{22,t-1}+{\alpha }_{22}*{\varepsilon }_{22,t}^{2}-\left({\beta }_{22}-\left(1-{\alpha }_{22}\right)\right)*{\delta }_{22})*{\varepsilon }_{22,t-1}^{2}-\\ {\alpha }_{22}*{\text{ufilter}}_{22,t}+{\alpha }_{22}*{\delta }_{22}*{\text{ufilter}}_{22,t-1}\end{array}$$

(16c)

where,

$$\begin{array}{c}{\mathrm{\alpha }}_{11}*{\mathrm{ufilter}}_{11,\mathrm{t}} ={\mathrm{\alpha }}_{11}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{11,\mathrm{t}}^{2}\\ {\mathrm{\alpha }}_{22}*{\mathrm{ufilter}}_{22,\mathrm{t}} ={\mathrm{\alpha }}_{22}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{22,\mathrm{t}}^{2}\\ {\mathrm{\alpha }}_{12}*{\mathrm{ufilter}}_{12,\mathrm{t}} ={\mathrm{\alpha }}_{12}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{12,\mathrm{t}}^{2}\end{array}$$

and

$$\begin{array}{c}{\mathrm{\alpha }}_{11}*{\updelta }_{11}*{\mathrm{ufilter}}_{11,\mathrm{t}-1}={\mathrm{\alpha }}_{11}{\updelta }_{11}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{11,{\mathrm{t}}^{-1}}^{2}\\ {\mathrm{\alpha }}_{22}*{\updelta }_{22}*{\mathrm{ufilter}}_{22,\mathrm{t}-1}={\mathrm{\alpha }}_{22}{\updelta }_{22}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{22,{\mathrm{t}}^{-1}}^{2}\\ {\mathrm{\alpha }}_{12}*{\updelta }_{12}*{\mathrm{ufilter}}_{12,\mathrm{t}-1}={\mathrm{\alpha }}_{12}{\updelta }_{12}(1-\mathrm{L}{)}^{\mathrm{d}} {\upvarepsilon }_{12,{\mathrm{t}}^{-1}}^{2}\end{array}$$

A3. The return equation of each country’s (Australia, Hong Kong, Japan, Singapore and the US) real estate and stock price indexes under study is influenced by the constant and previous day’s returns and has the following form for the constrained GARCH-BEKK model:

$${R}_{i,t =}{b}_{0}+{b}_{1}*{R}_{i,t-1}+{u}_{t }|{\Omega }_{t-1}\sim N(0,{H}_{t }) for i=\mathrm{1,2}$$

(17)

$${u}_{i,t}={R}_{i,t}-{b}_{i } for i=\mathrm{1,2}$$

(18)

where, ${R}_{i,t}$ is the real estate or stock price return for the five countries under study.

${b}_{i}$ is the martingale constant drift for real estate or stock price index returns.

Next, we apply the two-variable constrained and full GARCH-BEKK model (Engle & Kroner, 1995) for the variance as:

$${H}_{t }=C{C}^{^{\prime}}+{A}^{^{\prime}}{H}_{t-1 }A+{B}^{^{\prime}}{\varepsilon }_{t-1}*{\varepsilon }_{t-1}^{1}*{{\varepsilon }_{t-1}^{^{\prime}}}^{^{\prime}}B$$

(19)

where,

${H}_{t-1}$ is the volatility vector. $A and {A}^{^{\prime}}$ are the usual and the transposed constrained term respectively. ${\varepsilon }_{t}$ is the error term. $C and {C}^{^{\prime}}$ are the constant constrained vector terms, the first is the usual one and the second is the transposed term. $B and {B}^{^{\prime}}$ are the error coefficient constrained vectors, the first is the usual one and the second is the transposed term.

In the constrained GARCH-BEKK model some of the variance’s cross correlations were omitted as they have been set equal to zero. However, the full version of the GARCH-BEKK model has all the variance parameters without setting zero some of the cross-product correlations. This is the difference between these two models as applied on the article.

The parameters of the two-variable systems are estimated by computing the conditional log-likelihood function for each time period as:

$${L}_{t }\left(\Theta \right)=-\mathrm{log}2\uppi -\frac{1}{2}\mathrm{log}\left|{H}_{t+1 }\right|=\frac{1}{2}E{({\varepsilon }_{t})}^{^{\prime}}(\Theta ){H}_{t}^{-1}(\Theta )E({\varepsilon }_{t})(\Theta )$$

$${\mathrm{and} L}_{t }\left(\Theta \right)=\sum_{t=1}^{T}{L}_{t }\left(\Theta \right)$$

(20)

where, $\Theta$ is the vector of all volatility and error estimations parameters. The numerical maximization of the log-likelihood function follows the BHHH or the BFGS algorithm which accounts for the maximum likelihood estimates.

A4. Following Cotter and Stevenson (2006) we believe that the FIGARCH(1,d,1) model best captures the volatility in the real estate and also in stock price markets. The conditional variance of the FIGARCH (1,d,1) assumes the following form:

$${R}_{t =}{b}_{0}+{b}_{1}*{R}_{i,t-1}+{u}_{t}$$

(21)

$${u}_{t}={R}_{t}-{b}_{0}$$

(22)

$${\sigma }_{t}^{2}=c+\beta {\sigma }_{t-1}^{2}+[1-\beta L-(1-eL)(1-L{)}^{d}]{\varepsilon }_{t}^{2}$$

(23)

The FIGARCH (p,d,q) model contains two different models for two different values of d. Taking the value 0 to d we get the covariance-stationary GARCH(p,q) model while the IGARCH model results from d = 1. Values of d vary between 1 and 0 allowing us to account for the long-term dependence in the conditional variance. If 0 < d < 0.5, the series are long-term reverting with respect to covariance, and if 0.5 < d < 1, the series are then stationary, however the shocks die away in the short-run rather than in the long-run.

The long memory volatility model, the Fractional Integrated GARCH (FIGARCH), developed by Baillie et al. (1996) who claim that the FIGARCH (p,d,q) model can capture the long memory of financial volatility for daily equity returns through the fractional differencing parameter (d).