**4. Robustness checks and further empirical evidence**

#### **4.1 Linear and nonlinear forecasting regressions**

This section provides additional empirical evidence and explores the nature of the relationship between London and other UK regional house prices. Therefore, it complements the results of Granger causality and serves as a useful robustness check. To this end, we initially focus on the following forecasting regression:

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

$$\boldsymbol{y}\_{t+h} = \boldsymbol{a} + \beta \mathbf{x}\_t + \sum\_{i=0}^{\mathcal{T}} \boldsymbol{\gamma}\_i \boldsymbol{y}\_{t-i} + \boldsymbol{e}\_{t+h},\tag{7}$$

where yt+h refers to the changes in the London house prices, *yt*þ*<sup>h</sup>* <sup>¼</sup> <sup>400</sup> *<sup>h</sup>*þ<sup>1</sup> *ln yt*þ*<sup>h</sup> Yt* � �, with forecast horizon, h > 0 and x represents the changes in the regional house prices. The null hypothesis of β =0 is tested here to observe the predictability of changes in the London house prices using the other regional house prices. The corresponding results for h = 1 are presented in **Table 3**.

We report that the other regional house prices are a significant short-term predictor of the changes in the London house prices in most of the cases, with the exception of Northern Ireland, Scotland and Wales. These forecasting results reaffirm and strengthen the evidence against the 'ripple effect' hypothesis in the literature.

We further extend the forecasting model to show more evidence of nolinear relationship between the regional house prices. For this purpose, we use smoothtransition threshold (STR) models [40–44]. Simple threshold model can trigger an abrupt change in the parameter values, however, STR models are capable to allow smooth transition between different regime states. Following Smooth Transition Threshold model is used:

$$\mathbf{y}\_{t+h} = \mathbf{a} + \beta \mathbf{x}\_t + \sum\_{i=0}^{p} \gamma\_i \mathbf{y}\_{t-i} + \left(\rho\_0 + \rho\_1 \mathbf{x}\_t + \sum\_{i=0}^{p} \theta\_i \mathbf{y}\_{t-i}\right) \mathbf{F}(\mathbf{y}\_{t-d}) + \mathbf{e}\_{t+h},\tag{8}$$

*F yt*�*<sup>d</sup>* � � is the transition function and *yt*�*<sup>d</sup>* is the transition variable, whereas remaining variables are as defined in Eq. (7). Based on the existing literature, we first consider the logistic form of transition function (LSTR) as shown in Eq. (9) [40, 42–44]:


*Notes: This table presents the results from the linear forecasting regressions described in Section 4.1 (Eq. (7)). \*\*\*, \*\* and \* imply significant causality at the 1%, 5% and 10% levels, respectively.*

#### **Table 3.** *Linear forecasting results.*

$$F(\boldsymbol{y}\_{t-d}) = \left(\mathbf{1} + \exp\left(-\lambda(\boldsymbol{y}\_{t-d} - \boldsymbol{c})\right)\right)^{-1}, \lambda > \mathbf{0},\tag{9}$$

Where *λ*, d and c are the smoothing, delay and transition parameters, respectively. This function is monotonically increasing in *yt*–*d*. Note that when *<sup>λ</sup>* ! þ∞, *F yt*�*<sup>d</sup>* becomes a Heaviside function: *F yt*�*<sup>d</sup>* <sup>¼</sup> 0 when *yt*�*<sup>d</sup>* <sup>≤</sup> *<sup>c</sup>* and *F yt*�*<sup>d</sup>* <sup>¼</sup> 1 when *yt*�*<sup>d</sup>* <sup>&</sup>gt;*c*.

Monotonic transition may not always be successful in empirical applications. Therefore, we consider exponential transition function (ESTR) [42–44]:

$$F(\mathbf{y}\_{t-d}) = \mathbf{1} - \exp\left(-\lambda \left(\mathbf{y}\_{t-d} - c\right)^2\right), \lambda > 0. \tag{10}$$

Here, the transition function is symmetric around c. This model emplies that expansion and contraction have similar dynamics while the these vary for the middle ground [44]. STR module can have some issues involving the smoothing parameter *λ*, therefore, we follow the literature and using variation of the transition varible *λ* is scaled in both of the models [44].

In this case, the transition function is symmetric around *c*. The ESTR model implies that contraction and expansion have similar dynamic structures while the dynamics of the middle ground differ [43, 44]. Hence, we have the obtain the following versions of transition functions, respectively:

$$F(\mathbf{y}\_{t-d}) = \left(\mathbf{1} + \exp\left(-\lambda(\mathbf{y}\_{t-d} - \mathbf{c})/\sigma(\mathbf{y}\_{t-d})\right)\right)^{-1}, \lambda > \mathbf{0},\tag{11}$$

$$F(\boldsymbol{y}\_{t-d}) = \mathbf{1} - \exp\left(-\lambda \left(\boldsymbol{y}\_{t-d} - c\right)^2 / \sigma^2(\boldsymbol{y}\_{t-d})\right), \lambda > \mathbf{0}.\tag{12}$$

**Table 4** presents the results of the LSTR and the ESTR models testing the changes in London house prices as a predictor for changes in the regional house prices. In the LSTR model results, the estimated transition parameter *c*, which


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


#### **Table 4.**

*Nonlinear forecasting results (regional house prices* ➔ *London house prices).*



**Table 5.**

*Nonlinear forecasting results (London house prices* ➔ *regional house prices).*

marks the half-way point between the two regimes, is significantly different from zero in most of the cases. Moreover, we observe that most of the estimated betas are positive and significant (at 1% and 5% levels, depending on the case), suggesting that higher regional house prices boost London house prices in the following quarter. Further, the estimates of *φ*<sup>1</sup> in the upper regime significance are found in nine out of 13 regions, revealing the importance of regional house prices as an explanatory variable for changes in the London house prices. Insignificant results are found for Northern Ireland, Scotland and Wales. **Table 5** shows the results based on the LSTR and ESTR models confirming as expected that changes in the London house prices are a significant predictor of house price changes in other regions in the UK.

Results for the estimated ESTR models are very similar to the LSTR results. This reaffirms the significance of the regional house prices as a short-term predictor of future changes in the London house prices in a nonlinear context and complements the previously reported results under the linear and nonlinear frameworks. Thus, ESTR and LSTR results reinforce the idea that the regional house prices have a feedback effect or connectedness to the London house prices. This shows evidence against the ripple effect where a unidirectional impact of changes in the London house prices on other regions is reported.

## **5. Conclusion and implications**

This chapter investigates the transmission mechanism driving the UK regional house prices using the linear causality model, the nonlinear Granger causality model, and the impulse response process. We employ quarterly housing prices data ranging from Q4–1973 to Q2–2018 from 13 regions from the UK. Results show bidirectional dependence between the London prices and other regions' prices except for Northern Ireland and Wales. This result is confirmed by the linear causality, the nonlinear causality and the impulse response tests. Further empirical examination applying linear and non-linear forecasting tests support the linear and non-linear causality results. Thus, we provide that London is not always important for the other UK regions over time, as well as that London itself may also receive shocks from other regions. Impulse response 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 spill over effect on London house prices. Identification of regional disparities can help policymakers to achieve a more balanced growth across the country. These results underline the importance of establishing appropriate regulations and stabilisation policies in the housing sector of the economy. Further, the interdependence between regional housing prices might provide significant insight regarding efficient diversification of investments across mortgage-backed securities.

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

**JEL Classification**: R2, R21, R31
