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3. THE THEORETICAL FRAMEWORK

3.1 Approaches to modelling the housing system

Until relatively recently, most published studies of the national and regional UK housing markets concentrated heavily on cycles in housing demand and house prices, driven partly by the lack of supply-side data available over long time periods. Indeed, there is a long tradition of housing market models constructed as inverted demand functions in the UK, justified mainly on the grounds of weak (short-run) supply elasticity. As a consequence, many published housing models see house price growth as a function of lagged house price growth, real house price levels and real household incomes. The appearance of lagged price growth in models is often justified on the basis that it captures a momentum effect while coefficients on real house price and household income levels capture the long-run relationship between prices and earnings, with periodic tendency for the market to correct over-shoots.

During the past two decades econometric models of the housing market have become increasingly sophisticated in their treatment of house price dynamics, including the role of expectations and of new-build supply. Many recent examples of published models involve the application of co-integration or error correction methods in an attempt to capture long-run relationships (see, for example, Meen 1996, 1999, 2001; Meen et al 2005; Giussani and Hadjimatheou 1991; Muellbauer and Murphy 1997). This recent move away from econometric approaches that have emphasised the short-run dynamics of house prices has been inspired, in part, by

growing policy recognition that the supply of housing in the UK is lower than required by rates of household formation and, in particular, rather unresponsive to price (in the long run). These factors created the momentum ultimately leading to the Barker Review of Housing Supply (Barker, 2004).

The Barker (2003, 2004) reports define the problems facing the British housing market as low and declining levels of housebuilding associated with weak supply responsiveness to house price levels and change. The knock-on impacts are high long run real house price rises with deteriorating affordability. Clearly, the most prominent economic impacts include escalating latent housing demand as well as restricted labour mobility, which itself is then associated with poor regional housing and labour market adjustment, inimical to Scottish Government economic growth objectives. The key recommendations of the Barker reports focus particularly on the planning system, acknowledging that land supply is the main constraint to increasing the supply of housing. Among other things, the review proposed regional affordability targets and called for a stronger evidence base to monitor the levels of construction output required to deliver improved affordability.

It is in this context that significant developments in our understanding of how to conceptualise and model the housing market have occurred. In part, these developments have been marked by a debate between "stock" and "flow" perspectives on the housing system. Modelling approaches that emphasise the importance of the housing stock, and subsequent price adjustment processes, are widely accepted in the economics community and this approach is implicit in a range of models, particularly more recently published works, including the work of Meen (1996, 1999, 2001; Meen et al 2005).

The essence of the "stock" perspective is that housing demand is seen as a derivation from the population, including its demographic and socio-economic characteristics. These give rise to processes of household formation and migration which, in turn, determine housing demand. Short-run effects alter the level of housing demand - for example, a rise in earnings or a fall in interest rates will increase demand in the short-run but the long-run level is determined by population, households, rates of formation, earnings, house price levels and the extent of the housing stock. Key features of modelling approaches emphasising the importance of housing stock include the greater attention paid to long-run relationships rather than short-run dynamics and the fact that supply (in a given year) cannot respond very quickly to short-run fluctuations in housing demand or prices.

The "flows" perspective emphasises the importance of observed supply and demand, i.e. households that actually move and actively participate in the housing market in a given time period, rather than all households irrespective of whether they are actively demanding housing. The attraction of this approach is that it appears to relate more clearly to the market as it actually operates in Britain - if households are not active in the housing market, it is difficult to see how their actions or thoughts can influence market outcomes. One of the counter arguments is that households' housing consumption and move decisions are partly based on the level of house prices and house price growth and so there is a two way interaction (the amount of housing market activity determines house price levels and growth while house price levels and growth are likely to then exert an influence on the number and type of households actively participating in the market by moving or trading).

In practice, published models in both the stock and flow traditions have made valuable contributions to our understanding of housing market processes in recent years. Work by Meen (1996, 1999, 2001; Meen et al 2005), in particular, has emphasised the interaction of population, demographics, labour markets, migration, house prices and construction activity in the long run while shorter timescale flows models (see Bramley and Leishman, 2005a,b) have emphasised housing market processes including household migration and construction at lower levels of geography. The latter have emphasised spatial interaction and the determination of housing outcomes at sub-regional levels.

One of the most significant conceptual developments during the past decade or so has been the move away from the short-run treatment of house price determination as a closed system. Meen et al (2005) include separate modules for household formation, labour market, housing activity, tenure choice and migration. Bramley and Leishman (2005a,b) estimate separate equations for price determination, construction output and household migration in a panel econometric framework. Other recent innovations include measures of housing stock size relative to the number of households and household wealth (Meen, 1998). Muellbauer (2005) refers to the "financial accelerator" or the interaction between housing wealth (i.e. housing as a means of collateral and a measure of wealth), credit and consumer spending, a relationship that probably exacerbates the amplitude of economic and housing cycles.

3.2 The structure of the Scottish Model of Housing Supply and Affordability

The modelling approach largely follows that first set out by Meen et al (2005), with some adaptations for both the Scottish context and for empirical reasons. The model comprises separate modules (or econometric models) explaining rates of household formation, determination of labour market behaviour, housing market activity, households' tenure choice decisions and intra-Scotland household migration. The model is designed to allow significant interactions between the five modules. Together, the five modules comprise a simulation model which allows changes to both exogenous and endogenous variables to feed through to alter outcomes predicted by all of the elements of the system.

The issue of geography is a critical one and is examined in some detail in the next section of this report. However, at this stage it is worth noting that eight sub-national units of geography were adopted, and several of the modules are estimated on this sub-national basis (labour market outcomes, inter sub-regional household migration and housing market activity). The household formation and tenure choice modules are estimated at Scottish level and later applied to the sub-regions, for reasons discussed in more detail later in the report.

The remainder of this section examines the design and structure of the five econometric modules. Throughout the discussion, we begin by examining the requirements of the module as part of the Scotland level affordability model. We then consider the implications of applying the approach at the sub-Scotland level, if applicable.

3.2.1 Household formation ( HF)

The household formation or demographic module in the Meen et al (2005) work begins with official population projections and predicts population in each of a range of age groups by adopting the identity: population age (r) year (i) = population age (r-1) year (i-1) + net interregional migration + net international migration + births - deaths (see Meen et al, 2005, main report: 12). Since these definitions are set up as an identity, the predicted population in each age group should match official projections until net interregional migration is treated as an endogenous variable. At this point, assumptions about construction output feed through the migration ( MIG) and housing market ( HSG) modules, discussed below, and alter inter-regional migration. Thus, despite the fact that this element of the HF module is set up as an identity, it permits the affordability model as a whole to begin with the official population projections and then feed through the simulated impact of policy interventions over a long forward time period.

Meen et al (2005) discuss their use of modelling results set out by Andrew and Meen (2003) in what is effectively the second part of their demographic module. The Andrew and Meen (2003) results are based on estimation of a model of household formation based on British Household Panel Survey ( BHPS) data. The model effectively predicts the probability of an individual forming a new household based on their income, marital status, age, gender, number of children and a selection of economic and labour market variables. The latter may pose some difficulties in applying or re-estimating the model in a Scottish context since the Andrew and Meen (2003) model includes regional house prices as a proportion of those in the South East as well as the availability of new housing. The house price term allows for a form of regional ripple effect but it should be noted that studies of the regional ripple effect have largely found that the phenomenon stops short of Scotland. Meen et al (2005) also report that the availability of new housing variable is significant only in southern regions and, therefore, there is some doubt about the validity of including this term in the Scottish version of the model.

In the final Scottish affordability simulation model, the HF module describes linkages between births / deaths, population and household formation. International migration and that from elsewhere within the UK also feeds into population change. Meanwhile, the predicted number of households feed directly into the determination of owner occupied housing demand (i.e. the HF module feeds into the HSG module). Finally, the number of households feeds into the labour market ( LM) module, i.e. it impacts on the determination of labour supply. These relationships are shown in figure 1 below:

Figure 1 Household formation ( HF) module flow chart

Figure 1 deliberately abstracts only an element of the affordability model (for the purpose of demonstration). The starting position is that births / deaths feed into official projections of population. An econometric model of headship provides the parameters for converting these projections into predicted households. In the affordability model, there are feedback effects from new housing supply and migration (both of which can affect population directly and the propensity of different population groups to form new households).

The flow chart demonstrates that the formation / number of households then impacts on both labour supply and directly on owner occupied housing demand. These linkages lead off the flow chart and their impacts are shown in figure 2, which deals with the rest of the system.

Notwithstanding reservations about the housing cost and new housing supply variables, the Meen et al (2005) HF module also includes variables measuring the unemployment rate and the nominal mortgage interest rate. This means that the HF module contains interactions with the HSG module but it also means that the HF module is not simply static but its predictions change over time in the ultimate affordability simulation model (because the macroeconomic variables change over time).

3.2.2 Housing market activity ( HSG)

In the Meen et al (2005) England model, the focus is on change in log real house prices, i.e. broadly real house price growth. Models are estimated separately for each region, although there is an interaction between regional prices and Greater London prices, particularly with respect to Southern regions. The construction of the house price model reflects macro-economic factors (credit market conditions, nominal and real mortgage rates, log real FTSE as a wealth effect, inflation and non-property income); and regional / demographic drivers (regional share of working population in the 20-39 age group and change in ratio of working population to housing stock). The weighting of population to households in the 20-39 age group is a popular characteristic of macro housing models and reflects the significant housing market activity demonstrated by this age group. The ratio of working population to housing stock is a key driver of house prices in the long-run and also constructs an important linkage between the housing market model and the household formation module. Finally, a number of time series devices are used to capture structural breaks and isolated shocks. For example, the models include a region specific time trend (defined as year-1990); there are positive dummies for 1988 and 1989 and a negative dummy for the year 2001. Time dummy variables may be used in this manner in time series models if there are reasons to suppose that outcomes in a particular time period were caused partly by unique circumstances that cannot be properly described by economic variables. Finally, the Meen et al (2005) housing market model also includes negative downside risk, proxied using a four period moving average ( MA4) term on lagged negative real rates of return on housing.

The majority of the key drivers are either national / macroeconomic by definition or are readily measurable regional drivers of the housing market. Where differences can occur is in terms of the responsiveness of the market to some of these variables. For example, it is possible to envisage (pre-estimation) that there may be a weaker linkage between mortgage interest rates and house prices in Scotland given the historically relatively lower gearing of households. However, these are differences in magnitude of effect rather than fundamental differences in model structure.

A further significant difference between the Scottish and English models may ultimately lie in the importance of spatial interaction terms. In the English models house prices in Greater London are very important in driving prices in southern regions. Meanwhile, prices in contiguous neighbouring regions are used as an explanatory variable in all other regional house price equations. So, there are two forms of regional spatial interaction in the English model - distance from London and the course of prices in neighbouring regions. Previous studies of the regional ripple effect find that the phenomenon is strongest in southern regions and the effect becomes weaker with distance from London. It is generally accepted that southern English regions are quite heavily driven by the London housing market while there is a weaker, and more lagged, ripple effect affecting the north of the country. This issue is not tackled directly in the Scottish model, but may require some thought at any future development stage of this model. Although it is not generally desirable that the model should seek to establish the Scottish market as "an island", i.e. with little interaction from the rest of the UK, the evidence for spatial interactions between Scotland and the rest of the country is weak.

The simulation model (representing the simultaneous working of the five constituent econometric modules) is designed to predict the housing affordability outcome resulting from joint determination of housing market, household formation, tenure choice, labour market and migration processes. The relationship between the five modules is summarised in figure 2.

Figure 2 Housing market ( HSG) module flow chart

Figure 2 shows the linkages between household formation (including household migration), the labour market and housing market activity, including new construction. New construction is shown as having influences on the level of owner occupied housing supply and on population change, i.e. constraining the supply impacts on population change and vice versa. Meanwhile, owner occupied housing demand is influenced by demographic factors (provided by the HF module) and by macro / regional economic factors discussed earlier. Average earnings estimates are provided by the LM module.

Figure 2 therefore shows the operation of the final affordability model. Affordability is the result of interactions between housing demand and supply - giving house prices - and earnings. Although it should be noted that there are many envisaged secondary interactions as mentioned above.

3.2.3 Sub-national household migration ( MIG)

At the outset of the modelling work, it is assumed that inward and outward migration at the level of Scotland is not an important driver of housing affordability (i.e. inward household migration to Scotland from the rest of the UK and vice versa are not postulated as a key driver of affordability). At the sub-national level, household migration is much more important because:

• it is responsive to differential rates of affordability within the country;
• it describes the movement of labour and can therefore be seen as an equilibrating force within the labour and housing markets;
• flows are likely to be relatively more important between Scotland's sub-regions than between Scotland and the rest of the UK.

The results of the Meen et al (2005) study suggest that inter-regional migration is driven by differential rates of unemployment and earnings (comparing each region to its contiguous neighbours). In addition, disparity between the housing stock and the population of working age is important, with a "surplus" obviously facilitating migration more easily and resulting in a high turnover of housing (transactions volume), which facilitates higher migration ceteris paribus. High weighting towards the industrial sector reduces migration (reflecting long-term decline in that sector). Finally, relative house prices are included in the model on the basis that high prices in contiguous regions encourage migration into this region (since people can live in one and commute into the other).

The modelling framework accounts for international migration (defined as migration to / from England and Scotland's share of migration originating from beyond the UK). The primary mechanism for accounting for international migration is through the population projections supplied by the General Register Office for Scotland ( GROS). These feed into the household formation ( HF) module, causing further changes in housing market activity and labour market outcomes.

3.2.4 Labour market ( LM)

The purpose of the labour market module is to explicitly capture the interactions that occur between housing and labour markets, influencing the determination of house prices, earnings, employment, and unemployment. Bover et al (1989) and Cameron and Muellbauer (2001) examine these interactions at the national and regional level and identify four sets of effects directly associated with house prices. First, there is a positive relationship between house prices and the cost of living within a region. This is a supply side influence which has a direct impact on the commuting patterns of workers already living in a region, as well as on potential migrants. Secondly, there is also an indirect effect of house prices on land prices which subsequently impacts on the costs of firms locating within or to a region. Together these two effects give rise to spatial interaction between and within regions. The third and fourth effects are wealth and expectation effects. Changes in house prices may impact on regional spending as credit-constrained households and small businesses use their housing equity to release capital to help finance spending. The expectation of future changes in earnings may also be capitalised into house and land prices. These interactions will arguably not only be found at the regional level but the sub-regional and local level. A further link between labour market and housing market activity relates to how tenure structures impact on the mobility of labour and their propensity to migrate, although Cameron and Muellbauer (2001) argue that this effect is driven by national forces and tends to be less obvious at disaggregated levels of analysis.

The labour market module in the Meen et al (2005) model aims to capture the four regional labour and housing market mechanisms and spatial interactions between neighbouring regions. There are three reduced-form equations within the system to account for full time average earnings, employment and unemployment. All three of these variables are expressed as deviations from the GB average which, in turn, is assumed to be exogenously determined. The Meen et al (2005) study also described a more complex version of a labour market module. However, there are only very limited conceptual or empirical advantages in this innovation and we do not propose to follow this more complex formulation in this project.

In general, the availability of data at the sub-Scotland spatial level constrains our use of the Meen et al (2005) specification. Therefore, our approach differs in respect of the LM module and we follow the methods specified in Brown et al (2003). Their approach is more supply side focused and involves splitting the individuals within the labour market into four "labour market states" and estimating the percentage of the working age population in each of these. These labour market states are: full time employment, part time employment, unemployed and economically inactive.

Earnings in each labour state are estimated as average income with some accounting and imputation to capture other, non-work related, earnings. An indirect measure of income can be estimated by a method based on the qualifications held by the population in each area, labour market status for individuals with different levels of qualification and average wage estimates. This approach was proposed by Brown et al (2003) who argue it is a technique commonly used in US models of household formation and tenure choice to overcome problems of simultaneity. In the modelling approach followed here, changes in employment are also included to capture the demand side effects in the model.

3.2.5 Tenure choice ( TC)

The choice between housing tenure options is a critical one made by households. Tenure and housing outcomes are closely intertwined because of spatial concentrations of specific tenures, the tenure-specific nature of new development and because of the imperfect substitutability between different housing options (e.g. owning versus buy-to-let renting). Affordable housing has specific tenure flavours, be it low cost home ownership or social renting or indeed mid-market renting. The English regional affordability model recognised the importance of tenure choice and tenure-specific outcomes but, because of their focus on owner-occupation and data issues, they treated rents as exogenous and rental housing as an aggregate combining market and social rented housing - albeit in a rather cursory way. Meen et al also recognised the importance of ideally including a fully working tenure choice module within the overarching system being developed.

Allocating households by tenure in the model identifies demand for different tenures and allows potentially for a more sophisticated and realistic approach to supply, construction and affordable housing. Tenure choice modelling at a three way level (owning versus market rent versus social rent) is hamstrung by data issues, particularly at lower spatial scales because of the small and dynamic nature of the rental market. A pragmatic approach is to collapse rented housing together and model as a discrete choice problem the determinants of ownership versus renting in order to derive the parameter estimates for the simulation model, effectively generating a further dimension of possible outcomes for households in each iteration of the model. If the data supports this it could then be further developed in the future as a hierarchical structure where once the decision is made to rent, a second decision branch is required for the household to choose between market and social renting. This module is estimated at the Scotland level.

A priori, the likely drivers of tenure choice may be expected to include age or life-cycle stage, income, wealth/deposit constraints, other factors that are correlated with mobility decisions, household characteristics (e.g. recent in-migrants require easy access housing and are likely to rent initially), previous/current tenure and, importantly, the relative cost of owning versus renting (usually a comparison of the user cost of capital versus a market/social rent). The precise configuration of the model depends partly on data availability and quality, but also on the empirical choices made in other modules as, in principle, several variables that might explain household formation or migration may also influence tenure choice.

3.3 Choice of geography

There have been numerous studies of the regional housing market in the UK, largely following the rehearsed macro housing market modelling framework (see Giussani and Hadjimatheou, 1991; Muellbauer and Murphy, 1997). The approach adopted by Meen (1999) is interesting, as he tries to examine the main features of regional house price dynamics, including the "ripple effect" in a more rigorous way than attempted in many previous studies. He concludes that the ripple effect is caused by adjustments within regions rather than between them, i.e. it is not migration or spatial arbitrage which explains the ripple effect but simply amplified differences in economic performance.

In general, most previous regional housing market studies tend to treat regional markets as small versions of the UK housing market and thus lose the distinctive regional dimension. Despite this, it is clear from previous work focused on regional house prices, migration and affordability that housing market activity, and hence affordability, are determined at higher than local spatial levels. Leishman and Bramley (2005), for example, find very significant interaction between land supply, construction output and migration. Their simulation model suggests that the impact of increased land supply in a given local housing market is likely to be in terms of greater sub-regional household migration rather than any significant price effects (i.e. if land supply were to increase in a particular local market then a net increase to household inward migration is much more likely than any significant drop in house prices).

This logic is consistent with our view that the appropriate level of geography for this modelling work is not as small scale as the local authority or local housing market area, but perhaps a geographical unit that captures the essence of the housing market as a functional system in which house price levels and growth, household migration and labour market conditions are jointly determined. Given the importance and urgency of the decision over choice of geography, particular focus was placed on the task of selecting an appropriate unit. In February 2008 a half day workshop involving the research team and the Scottish Government discussed the political, economic and empirical issues in some depth, following up earlier circulation of two discussion papers (one each by the research team and Communities Analytical Services Division). A broad conclusion was reached at the half-day workshop and a work plan established to allow examination of the practicalities of adopting the proposed sub-national units of geography.

A number of existing definitions of sub-national geography were considered for possible adoption. These included:

• Pieda housing market areas:DTZCommunities Scotland / The term housing market area ( HMA) refers to a functional area in which most households (other than those moving due to a change in lifestyle such as a change in work, retirement to the country etc) are willing to search for housing. HMAs must be "anchored" on some centre. In this instance, HMAs were built up around Scotland's largest employment centres using Sasines data for 1996 to 1999 ¹ to analyse the origins and destinations of house purchasers in the private market. Communities Scotland's set of 13 HMAs include the four city regions, eight other cities/towns and their hinterlands, and one grouping of three towns in North Ayrshire (see appendix A for further details).
• The strategic development planning authorities (the 4 city regions):
  As part of the modernisation of Scotland's planning system introduced by the 2006 Planning Act, four strategic development planning authorities ( SDPAs) covering the regions around Glasgow, Edinburgh, Aberdeen, and Dundee have been established. Nineteen of Scotland's 32 local authorities are covered by these four city regions.
• Scottish Ministers have determined the precise SDPA boundaries in Planning Circular 3/2008 (http://www.scotland.gov.uk/publications/2008/11/25145654/15). These are similar to the Communities Scotland HMA boundaries, and generally follow local authority boundaries, but excluding national parks. Fife Council's area is split between the Dundee, Angus, Perth and North SDPA, and the Edinburgh & South East Scotland SDPA
• Scottish Government's six regional enterprise areas: In September 2007 the Scottish Government announced organisational changes to the Scottish Enterprise Network ( SE), Highlands and Islands Enterprise ( HIE) and VisitScotland ( VS). As part of these changes, SE's network of local enterprise companies ( LECs) will be replaced by 5 regional offices plus a sixth regional office that will serve the Highlands & Islands. In addition, VisitScotland will align its operations with these six areas. The local authorities that fall into each regional office are detailed in appendix A. It is not clear which regional office will be responsible for Arran.
• 3 areas:NUTSNUTS (Nomenclature of Units for Territorial Statistics) is a three tier based geography that was created by the European Office for Statistics (Eurostat). Scotland is one of the 12 NUTS1 areas in the UK. Scotland contains four NUTS2 areas that are further sub-divided into 23 NUTS 3 areas. Each NUTS 2 and 3 area is comprised as a combination of council areas and in some instances, (mainly in the case of the Highlands and Islands Enterprise area) LEC areas. The NUTS spatial hierarchy for Scotland summarised in appendix A includes amendments that came into effect in January 2008.
• Local authority areas: At present there are 32 local authority areas in Scotland.
• sTTWATravel to work areas ( ): The 2001 Census based TTWAs were published at the end of October 2007 and split Scotland into 48 distinct areas within which 67+% of the workforce in employment live and work. The Scottish TTWAs were constructed from datazones. As appendix A indicates, several TTWAs extend across two or more local authority areas. Two TTWAs also cut across the Scotland and England border. The Berwick TTWA includes the south east coast of Scotland (from Dunbar southwards) and the Berwick on Tweed district of Northumberland. The mainly English based Carlisle TTWA extends into the Langholm area.

Selection of a suitable unit of geography for the sub-national level of analysis was not straightforward. As Government guidance on defining strategic housing markets stresses ², there is no single or right way to define the sub-national geography. Given

the strong theoretical economic foundations of the work, there should be an expectation that the overriding criterion will be economic coherence. However, there are other relevant arguments and criteria. For example, the chosen unit of geography should be meaningful (and useful) in political and administrative terms and it should be possible to access and replicate data at the chosen unit of geography. In the context of this study, economic coherence has two dimensions: spatial and temporal. The latter is relevant because the estimated simulation model resulting from this work will be expected to provide meaningful simulations over a 30 year forward period. It is therefore important that any unit of geography chosen is sufficiently robust over time that it is unlikely to quickly date beyond the immediate life of the project. Considering political, economic and empirical factors jointly, it is possible to identify 7 criteria that can be used to guide the selection of the most appropriate sub-national geography:

• Economic coherence: The sub-national geography should (as far as practicable) reflect functional areas rather than simply rely on administrative area boundaries. By this we mean that the areas identified should have some intrinsic rationale in terms of clear links between where people live and work and the housing choices of households.
• over timeHMAPossible expansion of a : In the last 20 years the numbers of travel to work areas in the UK has reduced considerably ^3.. Assuming the underlying trend towards more and longer distance commuting continues, the numbers of TTWAs and HMAs are likely to continue to fall. In particular, the boundaries of the city-region HMAs that include local authorities where GROS project significant population growth are likely to spread outwards ⁴.
• Policy relevance: The sub-national geography should be broadly consistent with administrative geography, particularly for the city region areas where strategic decisions on housing, planning, and economic development need to be co-ordinated to tackle problems associated with decentralisation of the urban core, affordability pressures, and various supply constraints.
• Political acceptability: Consideration must be given to the local political acceptability of the proposed sub-national geography provided they do not exclude functional areas that form part of the HMA area.
• Comprehensiveness and clarity of coverage: The sub-national geography should cover the whole of Scotland. The complexity of the model means that it is not practical to define overlapping boundaries and that all parts of the country should be included in a single spatial unit. Thus each spatial unit (e.g. TTWA or LA area) used as a building block should be allocated to one 'housing market area'.
• Data availability: The variables selected for the econometric modelling work depend partly on the availability, relevance, and reliability of secondary data sources. Ideally we would wish to construct the agreed sub-national geography from smallest practical standardised spatial scale (such as datazones or TTWA). However, in some instances data are only collected at or released at local authority area.
• Minimum scale: In terms of identifying the sub-national geography it is important to ensure that the selected spatial units contain sufficient population to ensure there is reliable data to permit econometric modelling. We therefore believe that each agreed spatial unit should contain at least 40,000 adults of working age in employment, which we estimate on average is equivalent to a population of around 80,000 or more.

Each of the sub-national geography options identified is subject to strengths and limitations in terms of these 7 selection criteria. These can be summarised as shown in table 1.

Table 1 Strengths and limitations of existing sub-national geographies

Given that none of the existing sub-national spatial frameworks met all 7 criteria, the choice of geography required careful consideration of the trade-offs between the criteria. A leading option for defining an appropriate sub-national geography for urban and commuting areas of Scotland was to base the sub-national geography on combinations of TTWAs. This approach begins with the TTWAs that most closely match Communities Scotland's HMAs for the four city regions. This is consistent with previous studies (see for example Halden 2002) that suggest there is a reasonably close relationship between the boundaries of HMAs and TTWAs. The approach involves allocating an entire TTWA to a HMA in instances where the TTWA includes a medium or large sized town that falls within one of the four city region HMAs. For instance, the links between Linlithgow and the Edinburgh HMA are such that including Linlithgow necessitates including the whole of the Falkirk TTWA.

Arguably, the most appropriate option for rural TTWAs that have little interaction with each other or with Scotland's major urban centres is to build on the Scottish Household Survey 8-fold rural typology and the rural typologies' framework prepared by Land Use Consultants (2005) on behalf of the Scottish Government. This logically leads to two groupings of TTWAs as follows:

• Remote and very remote rural housing market areas: This grouping would include TTWAs that for the most part include small settlements of less than 3,000 population and are a considerable drive time from a medium sized town with a population of around 10,000. This generally describes the TTWAs found in the Highlands and Islands Enterprise area of Scotland with the exception of the TTWAs that form the Inverness HMA and the Moray HMA.
• Accessible and intermediate rural housing markets: This grouping of TTWAs would generally consist of small settlements and small towns (3,000 to 10,000) within a reasonable drive time of a medium sized town. This generally describes the TTWAs found in the Scottish Border LA and Dumfries and Galloway LA with the exception of the Peebles and Galashiels TTWA.

One of the difficulties with these proposed groupings is that not all variables needed for the economic modelling work are available at units of geography below local authority. In addition, the discussion at the geography workshop emphasised the importance of selecting a unit of geography that yields boundaries meaningful to local authorities while, at the same time, retaining the concept of larger scale "sub regional", rather than local authority boundaries. The solution proposed, shown in table 2, involved a combination of economic with political, empirical and pragmatic factors yielding the following geographical units:

Table 2 Adopted sub-national units

In order to evaluate the stability and coherence of the spatial units from a modelling perspective, a number of additional checks were made focusing on data quality and key variable behaviour. Logically, separate consideration of a spatial area ("sub-region") can be justified according to two main criteria:

a. The data series are well behaved despite the disaggregation. This criterion would require the data to appear to closely measure the economic phenomenon that they are taken to represent, without evidence of spurious volatility likely to be related to small sample size rather than true underlying economic behaviour.

b. There is evidence of a degree of independence between the different spatial clusters. Logically, if two spatial areas show persistently similar outcomes over a lengthy time period (20-30 years) then it is difficult to justify their separate consideration.

The analysis in this section focuses on house prices, partly because these are an important equilibrating mechanism in the system being modelled, and partly because these data are likely to be more susceptible to the issue of small samples than many, or any, of the other data sources employed in the modelling. This is because the owner occupied house price series are generated from the Survey of Mortgage Lenders ( SML) returns which are, for most of the study period, based on a 5% sample of all lenders' mortgage transactions. These data were not designed to allow analysis at sub-national, or at worst sub-regional, levels of spatial aggregation. It is therefore possible that house price series for some of the clusters of local authority areas will exhibit greater volatility than is credible which, in turn, is suggestive of the presence of other forms of bias (sample selection bias) not effectively mitigated by the use of sample weights in the SML data.

An indication of the behaviour of the eight sub-regional house price series can derived from figure 3. These series represent mean house price series, mix adjusted using the SML analysis weights. The series are further transformed by deflating according to the non-housing retail price index and are then converted to natural logarithms. Despite this, the series are still strongly upward trending, suggesting that, even after removing inflation, there is still evidence of pronounced long-run growth in prices. The actions described should mean that the series do not have non-linear trends, i.e. they should be integrated of order 1 or I(1). In other words, a further single round of differencing should induce stationarity (broadly defined as the lack of a unit root or autocorrelation process).

Figure 3 Real mix adjusted mean price series (natural logs)

The visual impression of the price series is interesting, and it is reassuring to see that the price series behave roughly as expected, are not extremely volatile, show evidence of adhering to an underlying Scottish trend and show evidence of short-run cycles. However, it does not prove either that (i) the price series are sufficiently free from noise to conceivably be capable of being modelled, or (ii) it is valid to consider each of the 8 areas separately. To determine the answer to the first of these questions, a series of time series models was estimated to yield adjusted R square and a basic set of coefficients. The results are shown in table 3.

Table 3 Simplified regression results

Notes: the dependent variable in each case is ln(P _t)- ln(P _t-1) with column A representing ln(P _t-1)- ln(P _t-2), B representing ln(Y _t)- ln(Y _t-1) and column C representing ln(Y _t-1)- ln(Y _t-2).
P represents mix adjusted mean house price at sub-national level and Y represent mean household income at sub-national level.
* significant at 10%; ** significant at 5%; *** significant at 1%

The results suggest that it should be possible to achieve a fairly high model fit in the case of most sub-regions. The areas with the lowest adjusted R square are the areas for which there is greatest doubt regarding coherence as a stand-alone unit of geography - Borders/Dumfries, Highlands and Islands and Stirling. However, the adjusted R square statistics (column D) are all above 0.40 which, for a model in log differences, should be regarded as acceptable empirical performance. In addition to this, there is evidence of different dynamics between the sub-regions. For example, the elasticity on household income (column B) ranges from less than 0.7 in Dundee to 1.15 in Aberdeen. It is noticeable that the coefficients on lagged house price growth (column A) vary quite considerably between the sub-regions (and are positively signed in some, negative in others). One important reason for including this variable in the model specification is to reduce the possibility of first order serial correlation. The presence of the lagged house price variable (column A) improves the reliability of the other variables in the regression (i.e. the income variable in column B).

The income coefficients in table 3 suggest independence between the sub-regions (which would itself be a good justification for separate estimation). This is investigated further by re-estimating the equations for Borders/Dumfries, Highlands and Islands and Stirling but with the inclusion of variables from neighbouring sub-regions (Edinburgh is used for the Borders/Dumfries model; Aberdeen is used for the Highlands and Islands model and Dundee and Edinburgh are each used separately in the Stirling model). The results for the Borders/Dumfries model are shown in table 4.

Table 4 Results of Borders/Dumfries re-estimation

*** significant at 1%; ** significant at 5%; * significant at 10%

Dependent variable is ln(P _t)- ln(P _t-1)

DLRMAP refers to log difference mix adjusted house price; DLRMHHY refers to log difference real mean household income

The results show that adding the Edinburgh house price index (contemporaneous and lagged) to the Borders/Dumfries model does not increase the explanatory power of the model. These variables are not statistically significant.

Table 5 Results of Highlands and Islands re-estimation

Dependent variable is ln(P _t)- ln(P _t-1)

Table 5 shows the results of the Highlands and Islands model re-estimation. Adding the Aberdeen contemporaneous and lagged price index increases the explanatory power of the model slightly but neither of the variables is statistically significant (lagged house price growth in Aberdeen is "nearly" significant which probably accounts for the slightly higher adjusted R square in this model).

The results for the Stirling re-estimations are less straightforward. These are shown in tables 6a and 6b.

Table 6a Results of Stirling re-estimation (with Edinburgh)

Dependent variable is ln(P _t)- ln(P _t-1)

Table 6b Results of Stirling re-estimation (with Dundee)

Dependent variable is ln(P _t)- ln(P _t-1)

The results for Stirling clearly show that there is a relationship between Stirling and both the Dundee and Edinburgh sub-regions. The adjusted R square is increased in both models and Dundee / Edinburgh house prices are significant in the Stirling model (contemporaneous in the case of Edinburgh; lagged one year in the case of Dundee). Thus, this gives rise to a very interesting situation in which Stirling does not appear to be independent of other sub-regions, but there is not an obvious candidate "parent" sub-region with which to merge it (because both Edinburgh and Dundee appear to be important).

To summarise, the preliminary analysis is not designed to uncover final versions of the house price model, but to yield some indications about the likely validity and probable robustness of this module. The housing market module was chosen as a test case because it relies on house price data series constructed from the Survey of Mortgage Lenders and, while very useful, this dataset is probably the most susceptible to error arising from disaggregation. The analysis examines the volatility of the sub-regional house price indices, coefficient similarity between sub-regions and evidence of independence (between sub-regions) in house price dynamics.

The results suggest that the house price model is likely to be associated with at least satisfactory model fit statistics for five of the eight sub-regions (Aberdeen, Ayrshire, Dundee, Edinburgh and Glasgow). For the remaining three (Borders/Dumfries, Highlands & Islands and Stirling), there is no evidence of unexplainable volatility in the mix adjusted house price series, but the adjusted R squares are lower. However, they are still satisfactory, particularly for a model estimated in log differences.

Consideration of the three least satisfactory sub-regions in more detail yields an interesting outcome. Borders/Dumfries and Highlands & Islands are shown to be statistically independent of their most important immediate neighbouring sub-regions, although this is not conclusive since further experimentation with lead-lag variables could reveal a more complex relationship. However, the apparently differential performance in the short-run (contemporaneously and subject to one lag), coupled with the satisfactory R squares, argues against combining either of these sub-regions with a larger adjacent area. Meanwhile, the Stirling sub-region appears to be influenced by two different neighbouring sub-regions (Edinburgh and Dundee). However, the bias that would result from including this sub-region in either one of these neighbours coupled with the satisfactory R square effectively argues against this. In conclusion, the simple preliminary analysis is supportive of the idea that the more detailed empirical analysis on the basis of the eight selected sub-national units (table 2) will be viable.

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