**2.2 Bivariate and multivariate linear causality**

In order to examine the linear relationship between various UK regional house prices with the London house prices, we consider the widely accepted vector autoregression (VAR) specification and the corresponding Granger causality test [20]. This approach enables us to assess whether there is a causal relationship between the variables in terms of time precedence and in which direction the causality flows. This will help us to test whether there is a ripple effect or connectedness between the prices UK regional house prices. The specification of the applied bivariate VAR model can be expressed as follows:

<sup>3</sup> ADF test [20] and KPSS test [21] show that the changes in the house prices (first difference series) are stationary. Results for these tests are available from the authors on request.

$$\mathbf{x}\_{t} = \rho\_{1} + \sum\_{i=1}^{n} a\_{i} \mathbf{x}\_{t-i} + \sum\_{i=1}^{n} \beta\_{i} \mathbf{y}\_{t-i} + \mathbf{e}\_{1t} \tag{1}$$

$$\mathcal{y}\_t = \varphi\_2 + \sum\_{i=1}^n \gamma\_i \mathbf{x}\_{t-i} + \sum\_{i=1}^n \delta\_i \mathbf{y}\_{t-i} + \varepsilon\_{2t} \tag{2}$$

where, in our case, *xt* represents house prices in London in first differences, *yt* is the log-difference of the respective UK regional house prices. *φ*<sup>1</sup> and *φ*<sup>2</sup> are the constants, whereas *αi*, *βi*, *γ<sup>i</sup>* and *δi*, *i =* 1,...,*n*, are the parameters for linear relationships between the underlying variables. Ripple effect hypothesis can be tested if only London house prices affect other regions, but not vice versa. On the other hand, connectedness can be shown if bidirectional causality exists between London house prices and other regions. In the next sections, we present the nonlinear approach adopted in our study and describe the relevant tests employed.

#### **2.3 Bivariate nonlinear causality**

Arrival of new information and dynamics of economic fluctuations cause changes in the security prices. Campbell et al. [21] describe these processes as nonlinear. Furthermore, many other researchers have highlighted the existence of nonlinear features in macroeconomic variables and models [22–27]. Hiemstra and Jones [15] reported nonlinear causality in financial variables using a correlation integral based approach. Subsequent research papers have provided more evidence on nonlinear modelling of various financial variables [28–34]. Market frictions such transaction cots and information asymmetries could be associated with the nonlinear dynamics and can cause non-convergence towards the long-term equilibrium. Anderson [35] reports that the transaction costs in the asset pricing literature could be one of the factor for disequilibrium error. He further demonstrates that nonlinear models which consider the transaction costs often outperform the parametric models. Some of the studies have identified heterogenous investors' beliefs as one of the sources for nonlinearities in macro-financial time series [36]. This heterogeneity exisit mainly

**Figure 2.** *Regional house prices.*

*UK House Prices – Connectedness or Ripple Effect? DOI: http://dx.doi.org/10.5772/intechopen.98868*

due to differences in investor horizonds, risk profiles [37] and herding behaviour [38]. Due to the above, we study the Granger causality in using nonlinear framework.

Correlation integral based nonlinear Granger causality was introduced by Baek and Brock [39] and was further developed by Hiemstra and Jones [15]. This research studies nonlinear causality between the UK regional house prices, using the Hiemstra and Jones [15] test statistic.

Consider two stationary time series f g *Xt* and f g *Yt* , for t = 1,2, … … An m-length lead vector of *Xt* is denoted by *X<sup>m</sup> <sup>t</sup>* wheras *XLx <sup>t</sup>*�*Lx* and *<sup>Y</sup>Ly <sup>t</sup>*�*Ly* are lag vectors of *Xt* and *Yt* as shown below:

$$\begin{array}{ll}X\_t^m \equiv (X\_t, X\_{t+1}, \dots X\_{t+m-1}), & m = 1, 2, \dots, t = 1, 2, \dots, \\ X\_{t-Lx}^{Lx} \equiv (X\_{t-Lx}, X\_{t-Lx+1}, \dots, X\_{t-1}), \\ Lx = 1, 2, \dots, t = Lx + 1, Lx + 2, \dots \\ Y\_{t-Ly}^{Ly} \equiv (Y\_{t-Ly}, Y\_{t-Ly+1}, \dots, Y\_{t-1}), \\ Ly = 1, 2, \dots, t = Ly + 1, Ly + 2, \dots \end{array} \tag{3}$$

Using the Hiemstra and Jones [15] framework, Y does not strictly Granger cause X if:

$$\begin{split} ⪻\left( \left|| \mathbf{X}\_{t}^{m} - \mathbf{X}\_{s}^{m} \right|| < \varepsilon \, \middle| \, \left|| \mathbf{X}\_{t-Lx}^{Lx} - \mathbf{X}\_{s-Lx}^{Lx} \right|| < \varepsilon, \, \left|| \mathbf{Y}\_{t-Ly}^{Ly} - \mathbf{Y}\_{s-Ly}^{Ly} \right|| < \varepsilon \right) \\ &= Pr\left( \left|| \mathbf{X}\_{t}^{m} - \mathbf{X}\_{s}^{m} \right|| < \varepsilon \, \middle| \, \left|| \mathbf{X}\_{t-Lx}^{Lx} - \mathbf{X}\_{s-Lx}^{Lx} \right|| < \varepsilon \right) \end{split} \tag{4}$$

Porbablity and maximum norm in Eq. (4) are denoted by Pr(.) and ||∙||, respectively. The conditional probability that the deviation between two arbitrary lead vectors of f g *Xt* of m-length is less than e, while deviation between the corresponding lag vectors of *XLx <sup>t</sup>*�*Lx* and *<sup>Y</sup>Ly <sup>t</sup>*�*Ly* is also less then e, is shown on the left hand side of the Eq. (4). The right hand side represent the conditional probability that two arbitrary m-length lead vectors of f g *Xt* are with a distance of e of each other, assuming that the corresponding lag vectors i.e. *XLx <sup>t</sup>*�*Lx* and *<sup>X</sup>Lx <sup>s</sup>*�*Lx* are also within a distance e of each other. For all regions, *Xt* represents the changes in the London housing prices and *Yt* represents the changes in the housing prices in other regions. Therefore, if Eq. (4) is true, this implies that the changes in the London housing prices do not affect the respective changes in regional housing prices. Nonlinear causality test proposed by Hiemstra and Jones [15] is based on the conditional probabilities using corresponding ratios of joint probabilities:

$$\frac{C\mathbf{1}(m+L\boldsymbol{\omega},L\boldsymbol{\upchi},\boldsymbol{e})}{C\mathbf{2}(L\boldsymbol{\upchi},L\boldsymbol{\upchi},\boldsymbol{e})} = \frac{C\mathbf{3}(m+L\boldsymbol{\upchi},\boldsymbol{e})}{C\mathbf{4}(L\boldsymbol{\upchi},\boldsymbol{e})}\tag{5}$$

where joint probabilities are denoted as *C1, C2, C3 and C4.*<sup>4</sup> Assuming f g *Xt and Y*f g*<sup>t</sup>* are strictly stationary and weakly dependent, if f g *Yt* does not strictly Granger cause f g *Xt* then,

$$\sqrt{n}\left(\frac{\text{Cl}(m+L\mathbb{X},L\mathbb{Y},e,n)}{\text{Cl}(L\mathbb{X},L\mathbb{Y},e,n)}-\frac{\text{C3}(m+L\mathbb{X},e,n)}{\text{C4}(L\mathbb{X},e,n)}\right)\to N\left(\mathbb{0},\sigma^{2}(m,L\mathbb{X},L\mathbb{Y},e)\right)\tag{6}$$

<sup>4</sup> See Hiemstra and Jones [15] for further details on correlation integrals and joint probabilities.

Details on the definition and the estimator of the variance *<sup>σ</sup>*2ð Þ *<sup>m</sup>*, *Lx*, *Ly*,*<sup>e</sup>* are provided in an appendix of Hiemstra and Jones [15].

### **3. Results**

**Table 1** shows the results for linear Granger causality based on Eqs. (1) and (2). The results show that London predominantly affects the regional house prices except for Northern Ireland, Outer Metropolitan and Outer East, and Wales. Similarly, regional house prices affect London house prices in seven out of 13 regions with some of these showing a feedback effect from or connectedness to the changes in the London house prices. No evidence of price feedback is found in any direction for Northern Ireland and Wales. The Northern Ireland and Wales results may be due to the increased independence of these regions from the UK government and the far distance location from London. Scotland prices are affected by London but not vice versa. Surprisingly Outer Metropolitan and Outer East affect the London prices but not vice versa. These results confirming connectedness between the house prices may be due to geographically adjacent or economically linked regions.

**Table 2** shows results for the nonlinear Granger causality. This test is applied to the standardised residuals obtained from the VAR models after filtering any linear dependence among the underlying variables. The null hypothesis of no nonlinear Granger causality has been rejected in most of the cases except for Northern Ireland and Wales. This shows significant evidence of nonlinear interdependence among the housing prices of London and other regions in the UK. We report bidirectional dependence between London and the other regions except for Northern Ireland and Wales. These results evince the nonlinear feedback effect or connectedness. No evidence of any causality in any direction is found between London and Wales/ Northern Ireland.


*Notes: Table 1 shows linear Granger causality results based on Eqs. (1) and (2). \*\*\*, \*\* and \* imply significant causality at the 1%, 5% and 10% levels, respectively.*

#### **Table 1.**

*Linear causality results.*


#### *UK House Prices – Connectedness or Ripple Effect? DOI: http://dx.doi.org/10.5772/intechopen.98868*

*Notes: Table 1 shows test-statistic proposed by Hiemstra and Jones [27] using Eq. (2). \*\*\*, \*\* and \* imply significant causality at the 1%, 5% and 10% levels, respectively.*

#### **Table 2.**

*Nonlinear causality results.*

Further evidence is presented by means of the impulse response function. **Figure 3** shows the impulse response function to one-standard-deviation innovations to the housing prices originating in London and other regions, respectively. These graphs can be interpreted into two categories – i.e. i) house prices in other regions responding to the shocks to London house prices and ii) London house prices responding to the shocks occurring in other regions in the UK. Firstly, a one per cent shock to the London house prices shows an immediate impact on house prices in most of the regions within a range from 1.5–3% – e.g., East Anglia, East Midlands, West Midlands, Outer Metropolitan, Outer Southeast, South West and Yorkshire. Geographically speaking, with the exception of Yorkshire, these regions are close to London. In other regions, although the shocks are statistically significant, they are smaller in magnitude. Interestingly, in the second category, innovations that originate in regions like East Anglia, Outer Metropolitan, Outer Southeast, South West, West Midlands and Yorkshire affect the London house prices with shocks in the range of 1% to 2.5%. This shows that London remains the central focus in the overall UK housing market and any shocks occurring here transmit to most of the regions. However, local shocks in other regions also show a spillover effect on London house prices. Antonakakis et al. [5] also report that East Anglia, Outer South East and South West are the major transmitters of regional shocks.

The evidence of connectedness presented here implies that although London is important from the housing market perspective, other regions also transmit the shocks back to the London market. This may be due to the information spillover (investor expectations) between different regions [14] although this research does not explicitly test the information hypothesis. By taking into consideration the impact of the bidirectional spillover effect of price, appropriate regulations and policies for the UK housing sector should be formulated. The results further imply the importance of house prices in other regions when investing in houses in London, and vice versa.
