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# Scottish housing market: tax revenue forecasting models – review

**21 Apr 2017**

Findings of an independent literature review of tax revenue forecasting models for the housing market.

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66 page PDF

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### 3. Model assessment

Table 2 lists the broad model classes that we cover.
^{
[4]
} We focus the review on model classes and methods within
model classes that are likely to be appropriate for the housing
market in Scotland in a public budgeting context.

We start by describing the simple assumptions that many practitioners use in place of elaborate models. These include using a rule of thumb, a growth accounting framework (decomposing the series into its main drivers and projecting them forward mechanically), or a consensus forecast (an average of external non-government forecasters).

We then introduce univariate time series models that predict future values of housing prices and transactions using only the past statistical properties of the series itself. These include models that use the tendency of a series to return to its path following shocks ( ARIMA modelling), models that estimate a series' tendency to behave differently over different periods (regime-switching models), and models that forecast a series' volatility ( GARCH).

Next, we review multivariate econometric models grounded in theoretical relationships between the housing market and other economic drivers (such as employment growth, household incomes, and population). These models attempt to predict both the path of the series and why it will take that path.

We also look at models that combine both time series and multivariate approaches in a multiple equation simultaneously determined framework: the vector autoregression approach.

We follow this with error-correction models, which were created to address technical issues with non-stationary time series data, and large-scale macroeconometric models, which use a combination of techniques to forecast the housing market as a key component of a system of equations for the economy as a whole.

We then look at theory-based models that use optimising agents (micro-foundations) to arrive at a forecast: dynamic stochastic general equilibrium models.

Finally, we assess microsimulation models that use survey data and tax return samples (or sometimes the entire universe of transactions) to build up a complete description of taxpayers in the economy.

**Table 2: Assessed model classes**

Description |
Class |
---|---|

Assumption or rule based |
Rules of thumb Growth accounting models External consensus forecasts |

Univariate time-series |
ARIMA GARCH Regime-switching models |

Multivariate approaches |
Multivariate regression models Vector autoregressive models Error-correction models Large-scale macroeconometric models |

Theory-based micro-founded |
Dynamic stochastic general equilibrium models |

Policy-focused models |
Microsimulation models |

**3.1 Forecasting by technical
assumption**

Practitioners often use approaches that are mechanical in nature, requiring little to no judgment or estimation of model parameters. We refer to these methods as technical assumptions. There were three general forms of technical assumptions that appeared frequently in budget backgrounders and practitioner interviews: 1) rules of thumb, 2) growth accounting models, and 3) an average of external forecasts.

**Rules of thumb** are simple approaches that are
grounded in theory, have shown value in practice, or are chosen
subjectively because forecasters have judged that it is not worth
applying significant modelling effort (for example, if the series
is negligible as a percentage of the overall budget or
GDP). They can be
applied to both prices and transactions. They often resemble
techniques in other model classes, with the difference that they
generally do not contain estimated parameters or are estimated
mechanically. Rules of thumb can take a range of
sophistication:

- holding the variable constant in the future based on its last observation
- projecting it forward using its simple historical average (the mean)
- using a constant trend growth assumption (for example, its historical average growth rate)
- using an exponential smoothing model
- a simple one-to-one growth relationship with other economic variables, such as GDP

Holding a variable constant based on its last observation would
be appropriate if, for example, examination of the series suggests
that it follows no predictable pattern, with long periods trending
up or down-that is, it is a random walk. If, however, it seems to
fluctuate randomly around a stable value, then it may make sense to
use its historical mean, or a moving average (the sample over which
the mean is estimated goes back a limited number of periods and
moves along as more observations are added).
^{
[5]
} These rules may be appropriate for housing transactions if
new housing development is restricted and population, real incomes,
and demographics are stable.

Similarly, projecting the series with a constant growth rate could be appropriate if historically prices have grown at roughly the same rate as general CPI price inflation, or for transactions if there are few restrictions on development and population is growing steadily.

Exponential smoothing models resemble a moving average, but
values further in the past are given decreasingly smaller weights
in its calculation. Exponential smoothing techniques have been
developed that can handle trending and seasonal data, and are
easily applied in a push-button manner by spreadsheet programs and
statistical software.
^{
[6]
}

Rules of thumb could also be based on rough economic relationships, such as assuming the nominal housing tax base grows with the growth of nominal GDP. Growing the tax base with the growth rate of nominal GDP is the same as assuming it grows by population, inflation and real incomes (productivity). Equivalently, this assumes the average consumer will spend the same proportional amount of their income on housing services over time

Rules of thumb can be decided by forecasters using basic descriptive statistics and judgment, or they can be institutionalised to impart a degree of independence by being imposed by an arm's-length body such as the Auditor General for Scotland.

Alternatively, forecasters could use a
**growth accounting model** that decomposes prices and
transactions into their main cost drivers. For example, Moro and
Nuño (2012) assume that average house prices in a period (
*P
_{t}*,) are directly related to construction costs.
They decompose costs into the cost of capital (the interest rate on
borrowing,

*R*) and the cost of labour (the wage rate,

_{t}*W*) in the construction sector relative to the rest of the economy, using a growth equation like the following:

_{t}^{ [7] }

where
*x
_{t}* is a residual growth factor capturing growth in
excess of capital and labour costs. All variables are expressed as
a ratio of prices in the construction sector to prices in the
general economy.

This framework can be used to project housing prices using
forecasts taken from a macroeconomic model for
*R
_{t}* and

*W*. Practitioners typically hold future values of

_{t}*x*constant at its historical average, unless there is a strong reason to suspect otherwise. Accounting methods incorporate economic drivers, but often assume away business cycle considerations (such as short-run deviations of supply and demand from equilibrium). For this reason, they are generally referred to as

_{t}*projections*rather than forecasts (Belsky, Drew, and McCue (2007) discuss this distinction). Accounting projections are nonetheless common in forecasting frameworks reported by practitioners, especially when the series is volatile and difficult to predict, or for projections beyond five years (for example, as part of long-term debt sustainability projections that assume markets are in equilibrium).

Finally, the Scottish forecasting framework could use an average
of
**non-government forecasts**. This would involve
regularly surveying private sector banks, real estate industry
firms and experts, think tanks, and universities for their outlook
for the housing market. The individual forecasts would then be
combined using a simple average for each year of the forecast. If
not used to generate the housing market forecast itself, the survey
could be used to generate exogenous economic assumptions for more
sophisticated models.

**Assessment**

**Application (forecasting):** good. Examples of rules
of thumb come mostly from practitioner literature and interviews.
Prior to the creation of the Office for Budget Responsibility (
OBR), the
UK budget was prepared in
part by using a set of basic assumptions audited by the National
Audit Office, including assumptions for trend
GDP growth,
unemployment, equity prices, oil prices, and tobacco consumption
trends.
^{
[8]
}

The OBR uses a rule of thumb for its forecasts of the devolved Scottish LBTT and Welsh SDLT taxes, assuming they remain at a constant share of their forecast for the UK as a whole ( OBR, 2016).

Practitioners reported that assuming a constant share of GDP is the "go-to" assumption for small taxes, or taxes for which data on fundamentals is limited.

The Scottish Government forecast housing prices in Draft Budget 2016-17 using a rule of thumb for the outer years of the outlook, interpolating between model results after the second year of the outlook to the historical average growth rate for prices in the fifth year. For transactions, the Scottish Government used a linear interpolation rule of thumb between the last historical value of the turnover rate to the historical average turnover rate imposed on the fifth year of the forecast horizon.

A growth accounting framework is used to project medium-run demand for homes based on fundamentals by the Joint Center for Housing Studies of Harvard University and presented most recently in Belsky et al. (2007). This approach models US housing demand with a simple accounting relationship based on three factors: 1) net household growth (itself projected using headship rates and immigration), 2) the net change in vacant units (calculated with demand for for-sale vacancies, for-rent vacancies, and second and occasional use homes-in turn projected by the age distribution of population, household wealth, and preferences), and 3) replacement of units lost on net from existing stock as a result of disaster, deterioration, demolition, and conversion to non-residential units. Although the framework is for new home construction, it could be extended to include turnover for existing homes for a projection of total transactions.

Belsky et al. also present an alternative and simpler approach to projecting total demand for new housing using the historical ratio of household growth to completions (the two most reliable housing data sources according to the authors).

If short-run dynamics are desired, McCue (2009) provides an extension of the Belsky et al. framework to compare the demand projections to actual supply to arrive at a short-run forecast of excess new supply and inventories. This excess (or deficit) supply measure could be used to introduce short-run cyclicality in prices using a multivariate regression model (discussed further in Subsection 3.3) while still being anchored at the end of the medium run by the accounting framework.

Growth accounting models may, however, be less suited to forecasting the UK housing market than the US market. There is convincing research that housing supply in the UK is inelastic (see Subsection 3.3), which suggests cyclical demand considerations could have a significant impact on prices.

Examples of government forecasters using the private sector
average include the
OBR
and
HM Treasury (
HMT), who, until
December 2013, used an average of private sector forecasts for
their 2-year ahead outlook for house price inflation (Auterson,
2014). Eighteen of the 38 external forecasters in
HMT's October
2016 survey provided forecasts for housing price inflation for the
UK as a whole for at
least five quarters.
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[9]
}

Canada's federal Department of Finance uses the average of
private sector forecasts for all key macro variables including real
GDP growth,
GDP inflation,
real interest rates, and exchange rates. However, they use an
internal macroeconometric model constrained to the consensus growth
rates to decompose
GDP into its
components and income shares for fiscal forecasting, including
housing prices and transactions. They have maintained the internal
capacity for complete macro forecasting, but impose the private
sector average as a basis for fiscal planning to "introduce an
element of independence into the Government's fiscal forecast."
^{
[10]
}

**Application (policy):** fair. Neither of the three
broad technical assumptions lend themselves to policy costings.
However, there may be some scope to adjust assumptions to produce
sensitivity tables and assess alternative assumptions-for example,
if the growth rate of
GDP or the
consensus forecast of house price inflation were used, the rates
could be adjusted by one percentage point and the consequences to
the fiscal outlook could be reported. Alternative capacity for
policy costings would need to be developed.

**Accuracy (short run):** fair. Forecasting by a
technical assumption does not necessary mean sacrificing
forecasting performance. On the strength of academic research on
the unpredictability of many economic time series, practitioners
are increasingly foregoing sophisticated forecasting techniques in
favour of simple assumptions. For example, recent research such as
Alquist and Vigfusson (2013) found that the common approach to
forecasting the price of oil using the oil market futures curve
cannot beat a simple no-change assumption. Their work has led many
practitioners to abandon sophisticated oil price models.
^{
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}

However, for the housing market, there is sufficient research to reject the idea that sophisticated models cannot improve upon technical assumptions. Researchers as early as Case and Shiller (1988) convincingly demonstrated that housing markets are not efficient (that is, they do not follow a random walk and observant investors can earn returns above a safe rate). This suggests housing time series are predictable. For example, if they experience one year of above-average growth, the next year tends also to be above average. Therefore, simple rules are unlikely to do well in cross-model comparisons based on accuracy.

Moro and Nuño (2012) tested their growth accounting framework empirically over the 1980s to late 2007 in four countries: the UK, US, Spain, and Germany. They found that it provides a useful description of housing price movements only in the US.

There is considerable evidence that averaging forecasts, as in the consensus approach, can provide superior forecasting performance (for example, Meulen et al. (2011) and Granziera, Luu, and St-Amant (2013)). However, in the case of many economic and fiscal variables, governments may have more timely and accurate information than private sector forecasters (for example, real-time VAT receipts). Private sector economic forecasters may also have biases unique to their circumstances. For example, there may be a bias in the public forecasts of private sector investment banks as a result of financial incentives (few would be enthusiastic to invest if the economic outlook is grim) and for mechanical reasons related to lags in recognising downturns (Batchelor, 2007).

**Accuracy (medium run):** fair. The medium run should
benefit from forecast averaging or simple anchors based on
high-level economic trends or the variable's history. However,
there is some evidence that suggests otherwise. Tsolacos (2006)
evaluates a consensus forecast of real estate rents (a returns
index) and finds that while the consensus forecast for rents is
best at a one-year forward horizon, simple time series approaches
and regression models with interest rates outperform the consensus
forecast two years out.

**Communication (story telling):** fair. Technical
assumptions vary in their ability to tell a story-for example,
using the historical mean or average growth rate would not reflect
economic fundamentals, but growing prices or transactions with
GDP may capture
general economic trends. Growth accounting models allow broad
trends to be discussed, but may not be able to explain short-run
changes related to the business cycle.

The private sector average can tell a story fitting both overall economic trends and trends in the housing market, depending on how detailed the survey is; however, budget-to-budget revisions may be impossible to explain.

**Communication (transparency):** good. Presenting and
explaining technical assumptions is relatively straightforward. All
three approaches can be made independent and transparent. If the
consensus forecast is viewed as a form of externally-imposed rule,
it is transparent. However, the underlying methodology and
assumptions that non-government forecasters use to produce the
forecast would typically not be available.

**Data compatibility:** fair. Generally, technical
assumptions have few data requirements, and the ones that do (such
as for growth accounting models) are at a high level that will be
available to Scottish forecasters. However, there are some
limitations that reduce this score to
*fair*.

First, using a private sector average may prove challenging for
a forecast of the Scottish market. Relatively few institutions
produce Scottish forecasts, and even fewer offer detailed forecasts
down to the residential housing sector level, especially for a
five-year forecast horizon. However, there are surveys of
professional housing forecasters that may satisfy the requirements.
^{
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}

The OBR suggests that although the consensus forecast was effective for communicating with stakeholders, it was abandoned largely because the data was not timely and definitions were problematic. This is summarised by Auterson (2014):

[the consensus forecast] had the advantage of being simple and transparent, but the disadvantage that there can be a significant lag between new information becoming available and external forecasts being updated, collated and published. This problem is particularly apparent when house price inflation is changing rapidly. Another drawback was that external forecasts reference a number of different house price indexes, meaning only a subset are directly relevant to the ONS house price series we forecast. (p. 1)

Finally, rules of thumb that rely on basic statistics such as historical means do not work well with trending data, seasonality, or level shifts (Makridakis, Wheelwright, and Hyndman, 2008). This will largely rule out these approaches for Scottish data in levels, but may be suitable for transformed data. This will require further evaluation.

**Resources:** good
**.** Technical assumptions are cost effective and
require few analytical resources and little to no specialist skills
to apply. However, internal modelling capacity may still be
required for policy analysis. Technical assumptions are easily
estimated or imposed in spreadsheets and statistics software
packages.

Our assessment of forecasting by technical assumption is summarised in Appendix Table A1.

**3.2 Univariate time series approaches**

Univariate time series models predict housing prices and
transactions based solely on their own statistical properties,
particularly the relationships of the variable with its values in
the past (that is, the correlations and partial correlations
estimated by regressing the variable on time-lagged values of
itself). For example, if residential prices rose more quickly this
year than their trend, it may be likely that they will also rise
more quickly next year. These properties can be used to predict
future values of prices and transactions without considering the
wider economy.
^{
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}

We look at three univariate forecasting approaches. First, the tendency for price or transaction shocks to dampen or persist can be modelled using a generalised approach called ARIMA modelling. Second, the behaviour of a series may change over different periods, such as when it is trending up or down (for example, during a housing boom or bust). This type of behaviour can be modelled using regime-switching methods. Third, univariate models can predict another property of the series: its volatility. This can be modelled using a GARCH process and may be useful in forecasting the risk to the budget outlook.

**
ARIMA
**

The general form of a univariate time series model is the
autoregressive integrated moving average (
ARIMA)
model, attributed to Box and Jenkins (1976), who first described
the technique in detail and created a systematic approach for its
estimation.
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}

ARIMA modelling attempts to capture two basic types of time series behaviour and their combination:

1. Autoregression

2. Moving average

The autoregressive (
AR) component presumes
that housing prices or transactions are a function of lagged values
of themselves. That is, future values can be forecast with current
and past values by estimating the correlation of the series' value
in time
*t* with its values in time
*t-1*,
*t-2, t-3,*
etc. The autoregressive model has
the following general form (as given by Enders (2014)):

where
*c* is a constant (often not required),
*e
_{t}* is a random shock with constant variance, and

*p*is the last lagged value that affects the current realisation.

The moving average ( MA) component presumes that housing prices or transactions are a function of random surprises in previous years - that is, the difference, or errors, between the model and actual observations as time advances.

The general form of an MA process is:

The MA in an ARIMA process is a different concept than the moving average smoothing techniques discussed in Subsection 3.1. Here it is a behaviour of the error term of the model-a similar averaging process, but applied to the forecast errors of the series, instead of its past values.

AR and MA models can be combined in a general univariate time series model-the ARMA model. A simple ARMA model that depends only on its value and error in the previous round takes the form:

An
ARMA model can be used only if the
series is
*stationary,* a condition unlikely to be met by housing
market variables (an overview of Scottish housing market data along
with a definition of stationarity is given in Box 2). For example,
prices are likely to increase each year to some extent along with
other prices in the economy (general price inflation). In this
case, the mean (average) of house prices would grow over time. An
ARMA process on the level of house
prices would generally under-predict prices. But the series can be
transformed to be stationary. If the trend is steady and
predictable (in the inflation example, prices may follow the Bank
of England's inflation target), the variable can be made stationary
by detrending the data (regressing the series on a time trend). If,
however, the data has a stochastic trend (that is, it is
unpredictable), the variable must be transformed by differencing.
^{
[15]
} The latter case means the variable is integrated, which is
the abbreviated letter
*I* in
ARIMA.
^{
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}

**Regime-switching models**

Regime-switching models allow parameters to take different values in each of a fixed number of historical intervals. A change in the model's parameters may arise from a range of causes, such as changes to monetary policy or changes to the exchange rate regime (Stock, 2002).

Regime-switching models fall into two categories:
*threshold* models and
*Markov switching* models. In threshold models, regime
changes arise from the behaviour of an observed variable relative
to some threshold value. These models were first introduced by Tong
(1983). They are formulated in general terms in Stock (2002)
as:

Where
*α*
*(L)* and
*β*
*(L)* create coefficients and lags of the variable against
which they are multiplied, and
*d
_{t}* is a non-linear function of past data that
switches between the parameter regimes

*α*

*(L)*and

*β*

*(L).*Different functional forms of

*d*

_{t}determine how the model transitions between regimes.

In Markov-switching models, the regime change arises from the
outcome of an exogenous, unobserved, discrete random variable
assumed to follow a Markov process (that is, the history of the
variable does not offer any more information about its future than
its current value).
^{
[17]
} The general form of a Markov-switching model is similar to
the threshold model but the function
*d
_{t}* represents the unobserved Markov process
variable.

**Forecasting volatility**

The variance of an economic series is a measure of risk. If forecasters are interested in forecasting the risk to the LBTT outlook, or quantifying the variance of housing prices or transactions at different points in time, the series' own history can be used to forecast the variance. This is known as generalised autoregressive conditional heteroskedastic ( GARCH) modelling, where heteroskedastic means that the variable's volatility is not constant over time.

GARCH models were developed first by Engle (1982) and later refined by Bollerslev (1986). GARCH may be thought of as an extension of the ARIMA method that forecasts using the typical time series behaviour of both the values of a series, and its variance.

The following equation is a simplified GARCH model, based on Enders (2014):

where
*u* is the error term of a simple
*
AR(p)* process and
is the conditional variance of
*u* which depends on the information available at time
*t-1*.

**Box 2: Overview of Scottish residential prices and
transactions**

**Figure B1: Scottish residential housing prices and
transactions**

Figure B1 plots Scottish housing prices and transactions volumes from the Registers of Scotland, along with the seasonally adjusted series using the X-13 ARIMA- SEATS procedure of the US Census Bureau.

Housing prices and transactions contain a trend and seasonal pattern. Both series were affected by (and in turn affected) the downturn following the global financial crisis. Between late 2007 and early 2009 housing transactions collapsed to around a third of their pre-crisis levels. The fall in transactions coincides with a structural break in trend prices, ending the strong growth preceding the crisis.

A main concern when specifying a forecasting model is whether a series is stationary-that is, it has a constant mean and variance over time. Both smoothed series show a clear trend over time and transactions seasonality seems to widen after 2013. This suggests the data is non-stationary (and indeed housing market data in the UK and abroad is routinely found to be non-stationary (see Barari et al. (2014), among others). To use many of the forecasting techniques evaluated in our review, the data would need to be transformed, either by deseasonalising if they are found to be stationary in level terms (for example, through using dummy variables to capture seasonality), detrending if they are found to be stationary around a constant time trend, or differencing if they are found to be stationary around a stochastic trend (this can include seasonal differencing, that is, using the annual growth rate for each quarter). Alternatively, special techniques can be used to maintain the model in levels (see Subsection 3.5 - error-correction models).

Modelling is further complicated by policy changes such as the stamp duty holiday between September 2008 and December 2009, the introduction of graduated "slice" tax rates (similar to how personal income taxes are structured) to replace the previous "slab" rates on 4 December 2014, and the transition from the reserved stamp duty land tax ( SDLT) to the devolved land and buildings transaction tax ( LBTT) on 1 April 2015.

**Assessment**

**Application (forecasting):** good.
ARIMA,
regime-switching, and
GARCH
models have been applied to forecasting housing prices and
transactions in a wide range of academic literature.

Potentially useful applications of ARIMA models to prices include Barari, Sarkar, and Kundu (2014), Stevenson and Young (2007), and Crawford and Fratantoni (2003), among others. For applications to volumes measures for transactions and housing supply, see Fullerton, Laaksonen, and West (2001).

Among practitioners, ARIMA modelling was the approach used by the Scottish Government in Draft Budget 2016-17 for forecasting average house prices in the first two year of the outlook, and for the whole five-year forecast horizon in Draft Budget 2017-18. Outside of Scotland, it is not widespread, but is often used as a benchmark comparison. ARIMAs are, however, used widely for economic forecasting and for revenues that are a small percentage of the overall tax take, or that do not have a tax base that lends itself to modelling directly.

Although we heard of no regime-switching models applied among
practitioners, they are popular in academic research. For example,
Park and Hong (2012
*)* observed that monthly indexes of the
US housing market are
released at the end of the subsequent month. This, in turn, creates
a month-long standstill in making judgements about the housing
market. They exploit this interval to show how Markov-switching
models can be used to promptly forecast the prospects of the
US housing market within
the standstill period.

Enders (2014) suggests that GARCH models are particularly suited to asset markets such as the housing market, as prices should be negatively related to their volatility (if market participants are risk-averse). Further, referring again to Box 2, it seems that the variance of house transactions in Scotland changes over time. The housing forecast may therefore benefit from GARCH modelling

Univariate models can produce
*ex ante* forecasts without needing to be conditioned on
auxiliary forecasts of exogenous variables (that is, they can
forecast using only historical information available at the time of
the forecast).

**Application (policy):** poor. All three methods of
univariate forecasting are ill-suited for scenario analysis and
fiscal impact costing, as they are not specified with explanatory
variables to assess the impact of different economic assumptions
and risk scenarios on the housing market.
^{
[18]
}
ARIMA
and
GARCH
models may have some limited use in risk assessments:
ARIMA
models can assess how exogenous shocks in one period are
transmitted to future house prices and transactions in the future,
and
GARCH
models may be able to improve upon estimates of annual revenue at
risk.

Although of limited use for policy analysis themselves, univariate models may be a useful component of a broader policy analysis approach, such as for projecting components of the price distribution for the application of tax rates.

**Accuracy**
**(short run):** good.
ARIMA
models showed mixed but broadly positive performance in
out-of-sample forecasts, and in many cases, outperformed the more
sophisticated models they were compared against. However,
researchers are generally quick to assert that performance is
dictated by the specific regions and time periods under study.

For example, the ability of ARIMA models to forecast Irish housing prices was evaluated by Stevenson and Young (2007) for the period 1998 to 2001. They found that ARIMA models provided more accurate forecasts compared to two other models-a multivariate regression and VAR model-on five forecasting accuracy measures: mean error, mean absolute error, mean squared percentage error, and error variance.

Lis (2013) estimated ARIMA models and other classes for rolling forecasts of the Canadian real estate markets in Vancouver and Toronto. Lis found that no single forecasting model performed best in all situations, but rather concludes that a forecasting approach should be chosen using detailed diagnostics for each series and time under study.

Brown et al. (1997) compared the ability of a regime-switching to predict house prices in the UK in the 1980s and 1990s to an error-correction model, an AR and a VAR model. They found that the regime-switching model performed the best in out-of-sample forecasts.

Meulen et al. (2011) constructed a unique price index using online real estate listings to control for different housing characteristics. They estimated a simple autoregressive model, as well as a VAR that incorporated information about macroeconomic trends. They found that the macroeconomic variables only slightly reduced the forecast errors compared to the naïve autoregression.

Maitland-Smith and Brooks (1999) found that Markov switching models are adept at capturing the non-stationary characteristics of value indexes of commercial real estate in the US and UK. The researchers found that it provided a better description of the data than models that allowed for only one regime (for example, a simple AR model).

Barari et al. (2014) estimated an ARIMA model and two regime-switching models on a 10-city composite S&P/Case-Shiller aggregate price index they created for seasonally-adjusted monthly data from January 1995 to December 2010. They found that the ARIMA model performs as well as the regime-switching models in out-of-sample forecasts.

**Accuracy**
**(medium run):** fair. Forecast tests for univariate
models in the academic literature are rarely performed more than
two years into the future. The length of the useful forecast
horizon is determined by the speed of decay, which is rarely
significant beyond 8 quarters; however, when specified to decay to
a simple trend, they may prove sufficient for the medium run.

Larson (2011) provides useful benchmark comparisons between univariate and multivariate models for three years into the future, finding that for Californian housing prices univariate time series models were outperformed by multivariate over the 1970s to late 2000s.

**Communication (story telling):** poor. Univariate
time series models do not generally offer a direct causal
interpretation of coefficients and can be difficult to communicate.
That is, they predict what will happen, not why (Hyndman and
Athanasopoulos, 2012). However, a univariate equation need not be
entirely atheoretical, as a complex system of multivariate
explanatory equations can often be transformed into a univariate
reduced-form equation (Enders, 2014). In that manner, a univariate
time series estimated by Ordinary Least Squares (
OLS) can capture
the theoretical relationships of a wide assortment of underlying
economic relationships. Nonetheless, the signs and magnitudes lose
much of their ability to be interpreted, and the structural
properties are impossible to recover from the final estimated
equation.

**Communication (transparency):** fair. Equations and
estimated coefficients would need to be published frequently, as
specifications and estimates are likely to change with each
addition of new or revised data. Fiscal sensitivity tables could
not be estimated and published to provide a check on model
revisions given economic developments. However, their relative
simplicity lends them some merit, as scrutinizers with a general
economics background would largely be able to understand and test
the assumptions.

**Data compatibility:** good. The key advantage of
this method is that it does not place a large burden on data
collection-only historical data for the variable being modelled is
required. Newton (1988) recommends a minimum of 50 observations for
ARIMA
modelling, which is in-line with the given history of reliable and
detailed Scottish data. Univariate time series models are therefore
well-suited to the available data for the Scottish housing
market.

**Resources:** good.
ARIMA
models are an accessible forecasting model for small teams with
limited technical background. Most software packages and
forecasting guides have detailed procedures for the Box-Jenkins
methodology that can guide the model selection procedure.

Our assessment of univariate time series models is summarised in Appendix Table A2.

**3.3 Multivariate regression models**

Rather than rely only on the past statistical behaviour of
housing prices and transactions, forecasters can look for other
factors that influence the housing market, such as interest rates
and population. Models that include other explanatory variables are
known as multivariate regression models.
^{
[19]
}

These models often use simple regression techniques such as
OLS to predict
future values of prices and transactions. They are similar to
cross-sectional econometric analysis, except that explanatory
variables are a function of time, and the estimated parameters can
vary over time. For example, a simple multivariate forecast of
housing prices may take the form:
^{
[20]
}

*where*:
*Δ*G
*DP
_{t}* = the change in gross domestic product,
representing general sentiment about the strength of property
markets and the wider economy.

*INT
_{t}* = some measure of interest rates, capturing the
cost of borrowing, the discount rate on future housing benefits,
and the risk-free rates on capital gains for competing investments,
among others.

The variables to include and the model specification are guided by economic theory, particularly the interaction of demand for housing by households and the supply of housing by land and building developers.

Importantly, forecasting with multivariate models requires forecasts of the future values of explanatory variables that will need to be provided by exogenous forecast models ( Subsection 3.4 considers a technique where the future values of explanatory variables can be endogenously forecast).

**Economic theories of the housing market**

Multivariate regression equations in academic literature and
practitioner research was most often based on
**asset-pricing theory**. This approach was presented
in an influential 1984 paper by Poterba. Asset-pricing models base
the level of housing prices on an equilibrium concept with the
'income' that houses generate.
^{
[21]
} This income includes the value of housing services to the
owner-occupier.

In Poterba's words:

A rational home buyer should equate the price of a house with the present discounted value of its future service stream. The value of future services, however, will depend upon the evolution of the housing stock, since the marginal value of a unit of housing services declines as the housing stock expands. The immediate change in house prices depends upon the entire expected future path of construction activity. The assumption that the buyers and sellers of houses possess perfect foresight ties the economy to this stable transition path and makes it possible to calculate the short-run change in house prices which results from a user cost shock. (1984, p. 730)

From another perspective, these models assume an arbitrage
opportunity between buying and renting: if the cost of renting a
house is lower than the expected cost of buying and occupying an
equivalent house (the
*user cost* of housing), then owners will sell and rent
instead, increasing the supply of homes for sale and reducing
vacant rentals until rents and user costs converge. A simplified
specification of the user cost of housing could look like the
following, based on Auterson (2014) and others:

*where P
_{t}* = the real price

*i
_{t}* = the net mortgage rate

*τ*
*
_{t}
* = the property tax rate

*δ*
*
_{t}
* = depreciation

*m
_{t}* = maintenance and repair expense

*g
_{t}* = expected capital gain (inflation plus the real
expected price increase)

The market clearing rental rate is taken as a function of housing supply, or other variables such as real incomes and demographics.

By setting the rental rate equal to the user cost equation and
manipulating the resulting equation, growth in housing prices can
be specified as an inverted demand function:
^{
[22]
}

where
*g** is now the real expected capital gain. Auterson
provides a description of how this model may be implemented in
practice in the
UK, including detailed
data sources.

Implementing multivariate models in a forecast can involve
simple applications of linear regression analysis with single
equation specifications, for example an equation with price on the
left-hand side (dependent variable). Models could also describe the
housing market more generally with several equations, having price
and transactions simultaneously determined, each driving one
another within a feedback loop. The latter could include both
structural and reduced-form systems of equations.
^{
[23]
}

Multivariate models could also be constructed using a bottom-up or top-down approach. In the bottom-up approach, relationships are estimated by looking at the factors that influence a house's value (such as the number of bedrooms, detached versus higher-density units) or real estate markets in different cities, and then aggregating to the national level based on population weights and construction trends. Models could also be applied separately for land and structures or for new construction versus existing turnover. Typically, however, structural relationships are examined at the top-down level. In a top-down approach, the relationship is examined at the national (aggregate) level using a model that relates aggregate average house price growth to macroeconomic variables such as growth in real incomes and employment.

**Assessment**

**Application (forecasting):** fair. There are many
examples of multivariate regression models applied to forecasting
housing prices and transactions in the
UK and abroad. For
example, Dicks (1990) extended the multiple equation demand and
supply models of earlier researchers in the
US to the
UK market to forecast
house prices for new and second-hand dwellings, as well as housing
completions and the uncompleted stock of dwellings.

There is also a strong base of multivariate model research for
the
US market, owing to the
richness and ease of access to data. In an influential paper, Case
and Shiller (1990) pooled data across four
US cities using
OLS regression to
examine how prices evolved based on explanatory variables such as
the change in per capita real income, employment expansion, the
change in the adult population, and changes in housing construction
costs.
^{
[24]
} They estimate several specifications of forecasting models
that prove to have significant forecast accuracy.

This approach is also common among practitioners. For example,
the
OBR's model
as presented in Auterson (2014) describes the multivariate model
they use to forecast the housing base for Stamp Duty Land Tax and
for constraining the housing sector of the macroeconomic model.
They model rental prices using real incomes, housing starts, and
demographics, modifying the equation above to include an estimate
of credit conditions,
*mrat*, as follows:
^{
[25]
}

The OBR's model is based on several papers by Meen (2013, 2009, and 1990, among others) who has undertaken a wide range of research on the UK housing market.

Multivariate models of the housing economy can be used as auxiliary models outside of the main macroeconomic forecasting model, and their outputs can be imposed on the macroeconomy model, or, where accounting concepts are different or aggregated accounting identities need to hold, are used to apply judgement to the central model's equations (for example, see Kapetanios, Labhard, and Price, 2008).

Because multivariate models are conditioned on exogenous
explanatory variables, they cannot produce
*ex ante* forecasts. This reduces their score compared to
models that can produce
*ex ante* forecasts.

**Application (policy):** good. Multivariate
econometric models are particularly relevant for policy assessments
and fiscal impact costings. They can include a wide range of
variables representing government policy and other explanatory
variables that can be changed to estimate the cost of policy or to
evaluate alternative assumptions. Makriditis et al. (2008) suggests
governments have "few alternatives other than econometric models if
they wish to know the results of changes in tax policies on the
economy (p. 301)."

**Accuracy (short run):** fair. Single equation models
can be tailored to fit the historical data perfectly, if enough
explanatory variables are added. This is not necessarily an
indication of useful forecasting performance, however, and in fact
can lead to the opposite-overfitting and poor out-of-sample
forecasts.

Dicks (1990) discussed the overfitting issue while estimating a number of simple demand and supply equations for the UK market based on earlier research by Hendry (1984). Dicks found that extending the demand and supply models to include the mortgage market, demographic factors, and construction costs can improve short-run forecasting results for prices and volumes and achieved reasonable results for the 1970s and 1980s, albeit with a tendency to under-predict the rate of house price increases.

Forecasting with a regression model requires conditioning the model on future values of explanatory variables. Other models will need to provide these variables, such as household income from a macroeconomic model. The forecast accuracy will be largely determined by these exogenous forecasts.

**Accuracy (medium run):** fair. Although forecasts
using explanatory variables introduce an additional source of
uncertainty, anchoring the medium run forecast to fundamentals may
nonetheless provide an improvement over naïve forecasts
(Lawson (2011) confirms this for the case of Californian housing
prices and provides a discussion). However, as most relationships
must be specified in terms of their growth rates to achieve
stationary data, the medium-run performance is likely to perform
poorer than other models that permit long-run levels relationships
(see
Subsection 3.5).

**Communication (story telling):** good. Multivariate
regression models must be conditioned on future values of exogenous
explanatory variables. This makes them well-suited for story
telling and integration within a wider budgetary framework to
provide a consistent budget narrative.

**Communication (transparency):** good. They can
provide a clear explanation for forecast errors. Equations can be
published and their specification (particularly model coefficients)
is unlikely to change frequently. Further, model coefficients have
intuitive interpretations that can be easily evaluated and repeated
by budget scrutinisers with a general background in economics.

**Data compatibility:** fair. Multivariate regression
models work well with the number of observations of quarterly data
available for the Scottish housing market. The data required for
the asset-price approach appears to be available, including housing
starts and completions, although there may be limitations on the
length of rental series. There is also sufficient data for
affordability models, including interest and total payment to
income ratios available from the Council of Mortgage lenders. That
said, there are likely to be some restrictions on the set of
explanatory variables for Scotland, rather than the
UK as a whole, and the
data requirements are more involved than univariate models,
resulting in a
*fair* score.

**Resources:** fair. Econometric models are not
push-button and require more resources than purely statistical
models such as in
Subsection 3.2. They require fewer
resources than other techniques in the review such as
DSGE
models, but nonetheless require specialised knowledge about both
housing markets and econometric theory. They require frequent
maintenance, re-estimation, and re-specification. Makridakis et al.
(2008) provide a useful discussion of the resources devoted to
econometric models versus simpler univariate approaches:

Whether intended for policy or forecasting purposes, econometric models are considerably more difficult to develop and estimate than using alternative statistical methods. The difficulties are of two types:

1. Technical aspects, involved in specifying the equations and estimating their parameters, and

2. Cost considerations, related to the amount of data needed and the computing and human resources required. (p. 302)

On the question of whether the extra burden of multivariate models over univariate approaches is justified, Makridakis et al. provide an opinion based on their own experiences, that suggests the appropriateness of a multivariate econometric model will ultimately depend on its intended use within the Scottish Government's budget production framework:

The answer is
*yes*, if the user is a government,
*maybe* if it is a large organization interested in policy
considerations, and
*probably not* if it is a medium or small organization, or
if the econometric model is intended for forecasting purposes only.
(p. 302)

Our assessment of multivariate models is summarised in Appendix Table A3.

**3.4 Vector autoregressive models**

The basic vector autoregressive model ( VAR) is a collection of time series models for different variables, estimated at the same time as a system. VAR models offer a simple and flexible alternative to the multivariate regression models of Subsection 3.3. The VAR approach need not rely on economic theory to specify relationships between variables (though theory often drives the choice of variables to include). VARs are instead based on the idea that economic variables tend to move together over time and tend to be autocorrelated.

Sargent and Sims (1977) promoted
VARs as an
alternative to large-scale macro-econometric models. They
criticised macro models for the strong assumptions they imposed on
the dynamic relation between macroeconomic variables and for not
accounting for the forward-looking behaviour of economic agents.
^{
[26]
} They proposed an alternative that allows the data itself to
determine how macroeconomic aggregates interact.

In
VAR equations, the
time path of the variable of interest can be affected by its past
values and current and past values of other variables, while also
letting the other variables be affected by current and past
realizations of the variable of interest and each other-that is,
they allow feedback between variables.
^{
[27]
} For a simple case of two variables, it has the following
form, taken from Enders (2014):

While the VAR model does not need to make any assumptions (impose restrictions) about which variables affects the other, an economic theory-based model can be imposed on a VAR, along with other behavioural and reduced-form (no contemporaneous effects) specifications.

If some series are thought to be determined exogenously or the researcher is working with ragged edge data (releases of some series are available before others) their values can be imposed exogenously. An example may be the outlook for the policy rate path of the Bank of England. However, Brooks and Tsolacos (2010) note that comparisons between unconditional and conditional VARs find little improvement in forecast accuracy from using conditioned exogenous variables. VARs can also include exogenous variables such as time trends, seasonal dummies, and other explanatory variables. Non-stationary data (as is likely to be the case for Scottish prices and transactions) may need to be transformed (using logged differences or growth rates) before entering the VAR.

**Assessment**

**Application (forecasting)**: good.
VARs are
well-designed for forecasting and are commonly used across a wide
range of forecasting applications in the public and private sector.
Because they contain only lags of variables, future values can be
forecast without forecasting other determinants in separate models
and imposing them exogenously or assuming their time path-that is,
they are not conditioned on any future realisations of explanatory
variables.

Brooks and Tsolacos (1999) provide a useful example of a VAR applied to the UK housing market. They estimated the impact of macroeconomic and financial variables on the UK property market, as measured by a real estate return index. They chose monthly data to be comparable to US studies over the period December 1985 to January 1999. The variables were selected based on other non- UK studies that have determined the variables' relevancy under various theoretical and empirical specifications. The variables include: the rate of unemployment as a proxy for the real economy (as its available at the monthly frequency), nominal interest rates, and the spread between long- and short-run rates, unanticipated inflation, and the dividend yield. They find that macroeconomic factors offer little explanatory power for the UK real estate market, although the interest rate term structure and unexpected inflation have a small contemporary effect.

Even if not used as the main forecast, VARs frequently serve as a yardstick against which to measure the forecasting performance of other more resource-intensive models, such as large-scale macroeconometric models.

**Application (policy):** poor. It is generally
difficult or impossible to recover interpretable directional causal
relationships from a
VAR in practice.
VARs are therefore
not useful for scenario analysis or fiscal impact costings. Certain
structural forms of
VAR models have
some use for performing risk assessments, for example, the impulse
response of a real income shock on housing prices.

**Accuracy (short run):** good.
VARs are often
found to perform better than univariate time series and more
complex theory-based models. They are especially suited for the
short-run horizon. This is conditional on having sufficient
Scottish historical data that doesn't limit the
VAR's
specification.

**Accuracy (medium run):** fair. Because
VARs
specifications are not grounded in long-run causal relationships
based on theory, forecasts for the medium-term may suffer. There
were generally few applications of
VARs beyond 8
quarters.

**Communication (story telling):** poor. By not
imposing a strict theoretical structure,
VARs allow the
data to drive the forecast. Although this makes for a better
forecast, it makes interpretation difficult. The complex lag
structure (and contemporaneous impacts of variables if so
specified) makes it difficult or impossible to isolate the
influences of variables on each other to tell a story.

A VAR may have trouble being made consistent with other budget forecasts and the economic narrative, depending on the specification. Under certain conditions it can be constrained to other forecasts or conditioned on exogenous variables from the economic model or other fiscal variables.

**Communication (transparency):** fair.
VAR specifications
are likely to change frequently and would need to be published
frequently. External budget scrutiny and testing of equations would
be limited to specialists. However, the limited judgment involved
with running a
VAR model adds to
its transparency.

**Data compatibility:** fair. Because of the lag
structure in
VARs, adding
additional variables increases the number of parameters that the
number of parameters to estimate dramatically. More parameters
require more observations. To conserve degrees of freedom, standard
VARs are generally
quite small, with around six to eight variables (Bernanke, Boivin,
and Eliasz, 2005). Given the relative limitations of Scottish data
compared to
UK and
US data, this is likely to
be even fewer. Although the number can be expanded with Bayesian
techniques (see
Section 4), in practice it may be necessary
to discard potentially useful variables simply to estimate the
model. A problem may emerge for Scotland if there are sufficient
observations to estimate the
VAR, but not
enough to include enough lags to whiten residuals. This issue can
be surmounted by common factors analysis laid out by Bernanke et
al. (2005), Stock and Watson (2002) and discussed further in
Section 4.

**Resources:** good.
VARs can be
implemented quickly and largely programmatically in statistics
software packages, using automatic criteria for selecting the
model's lag length. They are unlikely to require significant
resources or specialists.

Our assessment of VARs is summarised in Appendix Table A4.

**3.5 Error-correction models**

The forecasting techniques discussed so far can be used only if
house prices, volumes, and explanatory variables are stationary or
transformed to be stationary-that is, their means and variances are
constant over time.
^{
[28]
} House prices and transactions in Scotland and elsewhere tend
to grow over time. Further, their quarterly fluctuations (variance)
tend to be different during different periods.

Differencing the series to apply
ARIMA
and
VAR approaches
allows model coefficients to be estimated with
OLS regression,
but may sacrifice explanatory power between variables in their
levels form. Further, there would no longer be a long-run solution
to the model. In economics applications, this long-run solution
means that the system is in equilibrium and there are no longer
short-run fluctuations.
^{
[29]
} This would be the case, for example, in situations where the
output gap is closed and economic variables have returned to their
long-run steady state (such as in the outer years of a five-year
budget forecast). For the housing market, this long-run solution is
generally considered as the horizon over which supply is
elastic.

Error-correction models were developed to overcome the limitations of differencing to preserve the long-run levels information and present both the short-run growth information and the long-run equilibrium relation in a single statistical model. This makes them particularly powerful for forecasting the real estate market, given that it has been shown that this long-run information contains useful information (as discussed above in Subsection 3.3).

Error-correction mechanisms are based on the concept of cointegration: two or more non-stationary variables may wander around, but will never be too far apart-that is, the gap between them is stable over time, and the gap is itself stationary.

This empirical connection is the result of a theoretical equilibrium market force or shared trend. Researchers applying error-correction models to the housing sector point to several such potential cointegrating relations:

- household incomes and house prices
- rental rates, the discount rate (interest rates) and house prices
- house prices, GDP and total employment

A basic error-correction model is represented by the following equation, based on the presentation in Enders (2014):

The two difference terms (differencing is represented by
*Δ*) are stationary. The term
is an algebraic manipulation of the long-run
levels model, and it represents that amount by which the two
variables were out of equilibrium the period before (that is, the
error). For example, this could be the amount by which imputed
rents from owner-occupied homes are out of sync with
quality-adjusted actual rents. Because
is stationary, the model can be estimated with
OLS and
statistical inference is valid. The coefficient
*β*
*
_{2}
* is the speed at which the disequilibrium is corrected (for
example, a coefficient of 0.5 would mean roughly half the gap
between imputed rents and quality-adjusted actual rents is closed
one period later.

Error-correction models rely on the same theoretical underpinnings as the multivariate regression models in Subsection 3.3. Indeed, many of the specifications and models discussed in 3.3 are more appropriately implemented as error-correction models.

**Assessment**

**Application (forecasting):** good
**.** There is a vast literature that applies
error-correction models to the housing sector. An influential
error-correction model framework for housing supply and prices was
developed by Riddel (2000, 2004). It allows both disequilibrium in
housing supply and house prices to affect one another. Stevenson
and Young (2014) applied the model to the Irish housing market,
which may serve as a useful guide for modelling the Scottish
market. In this model, the long-run supply equilibrium is estimated
empirically by the equation

and the error-correction specification is

*where:*
*HC
_{t}* = housing completions

*HP
_{t}* = prices

*BC
_{t}* = real building costs

*r
_{t} =* the real after tax interest rate

*=* error terms

*ε*
*
_{t-1}
* = the lagged disequilibrium (error) from the long-run price
equation below

The long-run price equilibrium is an inverted demand function, estimated by the equation

and the error-correction specification of the price equation is

*where:*
*POP
_{t}* = population aged 25 to 44

*RDI
_{t}* = real disposable income per capita

*HS
_{t}* = is the per capita housing stock

*ω*
*
_{t}
* = error term

Addison-Smyth, McQuinn, and O'Reilly (2008) modelled Irish housing supply using error-correction models and found that developers do respond to disequilibrium. However, the findings also suggest the gap is slow to correct itself, with only roughly 10 per cent of the disequilibrium being corrected annually.

Error-correction models are perhaps the most common way to model the housing market in finance ministries and central banks. For example, all three major public macro forecasters in the UK ( HMT, OBR, and the Bank of England) rely on error-correction models.

**Application (policy):** fair. Error-correction
models rely strongly on economic theory and are relevant for policy
analysis. They can include a wide range of variables representing
government policy and other explanatory variables that can be
changed to evaluate alternative assumptions. That said, their focus
is on forecasting the dynamic impact of these variables, and they
rely on cointegrating relationships between variables that may not
exist between the policy levers of interest for fiscal impact
costing, and so are less relevant to policy than multivariate
regressions or microsimulation models with a fiscal impact costing
focus.

**Accuracy (short run):** fair. Error-correction
models generally perform well in both the short run and the medium
run. However, there is some evidence that in the
UK they may underperform
in the first eight quarters of the outlook. The
OBR
has found that the error-correction model it uses to model the
housing market (
HMT and the Bank
of England use similar approaches) is not well-suited for capturing
short-run dynamics, although it provides good forecasts in the
medium run (Auterson, 2014).

**Accuracy (medium run):** good
**.** Because of their basis in theory and being
ground in long run levels equilibrium relationships, error
correction models are likely to provide better forecasting
performance over years three to five than other methods. Lawson
(2011) found convincing evidence that error-correction models
outperform a number of other univariate and multivariate models
over a three-year horizon for Californian housing prices when
estimated over the period 1975 to 2006 and forecast over 2007 to
2009. Larson also found that error-correction models could predict
a housing price decline well in advance (the ability to forecast
the timing of the decline, however, was poor).

**Communication (story telling):** good. Like
multivariate regression models, error-correction models are
conditioned on future values of exogenous explanatory variables.
This makes them well-suited to story telling and integration within
a wider budgetary framework to provide a consistent budget
narrative.

**Communication (transparency):** fair. Equations can
be published and their specification (particularly model
coefficients) is unlikely to change frequently. Model equations are
more opaque to budget scrutinizers with only a general economics
background than more simple regression equations, but the equations
are nonetheless more economically intuitive than
VARs.

**Data compatibility:** fair. While it is possible to
estimate an error-correction model using Scottish data, there may
be some limits that could affect forecasting performance.
Practitioners suggest that housing cycles modelled with
error-correction models can last eight to ten years, and that these
dynamics will form the basis for estimating the error-correction
model's parameters. Suitable Scottish historical data may only
capture one cycle, and that cycle included the financial
crisis.

**Resources:** fair. Error-correction models are
likely to require greater expertise to develop, run, and maintain,
than many other options. However, the specification and forecasting
can be done easily in statistical software packages by specialists,
and is not likely to require significant time or effort following
the initial development period.

Our assessment of error-correction models is summarised in Appendix Table A5.

**3.6 Large-scale macroeconometric
models**

Forecasts of the housing market are produced within the macroeconomic models of budget forecasting frameworks to estimate the residential investment component of GDP, an important driver of business cycles.

Macroeconometric models simulate the economy as the interaction of aggregate supply and aggregate demand on the same basis as the National Accounts statistical framework. They use a mix of the techniques above to specify equations that describe the working of the entire economy. Although not employing new tools, the systems approach and the way it is implemented in practice-particularly national accounts data and identities and the goal of forecasting GDP-deserves special consideration as a model class on its own.

Macroeconometric models are loosely grounded in Keynes's General
Theory, which Hicks (1939) and later researchers formulated into
the well-known
IS-
LM
framework. The estimation of a system of econometric equations
estimated one equation at a time (ad hoc basis) related together
using national accounting identities was first undertaken by Klein
and Goldberger (1955) for the Cowles Commission (an initiative
running from 1939 to 1955 to apply mathematical and statistical
analysis to the economy).
^{
[30]
}

Models typically take a view of the output gap (the economy's actual output relative to its potential output, forecast separately) and combine it with other macroeconomic relationships such as the IS curve (aggregate demand and interest rates), Phillips curve (unemployment and inflation), a Taylor rule (monetary policy), and interest parity conditions (exchange rates). They strike a middle ground between theory-based and pure time-series models, taking aspects of both to capture both theoretical relationships and rich dynamics of variables over time for forecasting.

The housing sector (private sector investment in dwellings in
the
UK, often called
residential investment or residential gross fixed capital formation
elsewhere) includes purchases of new housing and major improvements
to existing dwellings by households (Carnot et al, 2014).
^{
[31]
} It is an important determinant of (and is determined by)
household wealth and consumption with the model framework. It is
estimated using aggregate behavioural equations within the
household sector to derive the wealth stocks of consumers that,
along with disposable income, guide the household sector's
consumption equations.

A typical equation for the real stock of housing resembles the following combination of a short run difference and long-run levels equation (error-correction model), from Carnot et al. (2014):

*where: y = real disposable income*

*k = residential investment*

*r = the real interest rate (usually a long rate, but Carnot
suggests a short rate in the
UK, where mortgages are
predominantly at variable rates)*

*z = other explanatory variables such as the relative price of
housing*

The justification for this specification is often grounded on the neo-classical standard model of life-cycle utility maximisation, where consumers choose between a mix of consumption goods and housing investment goods. This is, however, only a loose theoretical justification-it is not necessarily implemented by specifying a household utility function, which is the realm of DSGE models discussed in Subsection 3.8. The degree of foresight and optimisation of the household can differ. They can have perfect foresight (Robidoux and Wong, 1998) or include some rule of thumb consumers, to introduce an element of constrained rational expectations (Gervais and Gosselin, 2013).

Housing prices are usually modelled as a rental price for
housing services (for owner-occupied homes this is the best
estimate of what the owner would charge if she were renting it to
herself). The price then feeds into the rate of return on housing
investment, which drives investment in residential housing and
affects future supply.
^{
[32]
}

Traditional macroeconomic models continue to be the workhorse of macro modelling in government departments and central banks. Although DSGE models were implemented in many central banks and finance ministries, they are general used for scenario analysis in parallel with macroeconometric models and to challenge the forecast from an alternative perspective. Further, traditional large-scale macroeconometric models are experiencing a resurgence in popularity and credibility (for example, see the academic and online discussions generated by Romer (2016)).

**Assessment**

**Application (forecasting):** fair. Finance
ministries and central banks are moving toward making their
macroeconometric model documentation public. There are many
published examples that forecasters could use as a foundation on
which to build the model using Quarterly National Accounts
Scotland.

For example, the housing sector component of the
macroeconometric model shared by
HMT and
OBR
is described in
OBR
(2013). It forecasts private sector investment in dwellings (
*RES
_{t}* in £m and chained volume measures) using
the following relationship with real house prices, the real
interest rate, and the number of property transactions:

*where:*
*APH
_{t}* = an average house price index

*PCE
_{t}* = the consumer expenditure deflator

*RS
_{t}* =
UK three month inter-bank
rate

*PD
_{t}* = property transactions

Property transactions (particulars delivered) are assumed to be negatively related to the difference between actual and expected house prices, where expected house prices are determined by the user cost of capital, consumer prices, and real disposable income, given by:

*where:*
*RHHDI
_{t}* = real household disposable income

*RHP
_{t}* = real house prices,

*APH*/

_{t}*PCE*

_{t}
*UCH
_{t}* = user cost of housing (a function of mortgage
rates and the change in prices in the previous period as a proxy
for the expected capital gain)

*A2029
_{t}* = population of cohort aged 20-29

*D
_{t}* = a collection of dummy variables to control for
abnormal events

The remaining equations of the model estimate the other components of aggregate demand: consumption (durables and current), investment, government spending, exports, and imports (see OBR (2013) for the complete specification).

There is a robust literature demonstrating the importance of the housing sector's role in the macroeconomy and importance of considering it within this wider framework

For example, the correlation between the growth in housing prices and the growth in consumption and savings has been demonstrated by Meen (2012), who estimated the correlation coefficient as 0.74 on average over 1982 to 2008. The relationship has broken down somewhat since the turn of the century, however. The correlation in the period 1982 to 1999 was 0.81 and for 2000 to 2008 was 0.65.

In the US, Case et al. (2005) provide a useful summary and theoretical framework, finding house wealth to be more important than other forms of financial wealth for driving consumption patterns.

Although the correlation between housing prices and consumption and saving behaviour is well established, conclusive evidence of causality has remained elusive. Elming and Erlmler (2016) reviewed the relevant literature and found four main links between house prices and consumption: 1) the housing wealth effect, 2) housing equity serving as collateral and precautionary wealth, 3) the common factor of income expectation, and 4) the common factor of overall credit conditions and financial liberalisation. The authors provide convincing evidence in favour of a direct causal influence of housing prices on consumption. They do so by exploiting the natural variation between household price drops in different regions during the global financial crisis, using households with two public service income earners to control for income expectations (the public service salaries are set through strict collective bargaining arrangements).

Macro models could form part of the overall budget and help inform the housing market forecast; however, their applicability to the LBTT forecast itself may be limited. Macro models typically use a different concept of average house prices than would be useful for the tax base, and would need to be modified to serve that purpose. Many institutions use an auxiliary model, which is estimated outside the forecasting framework and they impose the model results exogenously (Auterson, 2014).

The aggregate time series in the national accounts, such as residential investments, may have limited ability to predict house transactions. Mankiw and Weil (1989) discuss how the noise (large standard errors) of national accounts data obscures relationships that show up when estimated using other time series grounded in administration or census data.

**Application (policy):** fair. Macroeconometric
models can be used for policy analysis. For example,
HMT used the
macro forecast and
OBR's
auxiliary housing market model to assess the impact of a vote in
favour of leaving the
EU on house prices (
HMT, 2016). They
can be used to model economic and fiscal sensitivities to shocks
such as an oil price decline, and prepare fiscal multiplier
estimates (Office of the Parliamentary Budget Officer, 2016).
However, they may be of limited use for fiscal impact costing,
because of the differences between housing investment in the
national accounts and the tax base.

**Accuracy (short run):** fair. Early macroeconomic
models that were driven by theory alone, ignoring the dynamics and
inter-temporal properties of the data, generally produced poor
forecasts.
^{
[33]
}

Forecast performance has since been improved with the introduction of better techniques to capture dynamics, and modern macroeconometric models should fare relatively well for their purpose. Granziera and St-Amant (2013) compares the forecast of their rational ECM framework of the housing markets to AR and regular ECM models and finds it performs better both four quarters and eight quarters ahead in rolling forecast experiments over 2002 to 2011.

As discussed, macroeconometric models are a blunt tool for forecasting tax bases. Conforming to national accounting identities restricts the specification of the tax base (housing transactions can relate to non-current production). Correspondence between tax bases and national accounts aggregates can be poor. They tend to perform more poorly than a model dedicated to the specific tax base and tax program parameters.

**Accuracy (medium run):** fair. Because of their
theoretical underpinnings, use of long-run equilibrium conditions
(closing of the output gap), and a greater ability to maintain
variables in a levels specification (making use of error correction
models), these models are likely to improve upon naïve
forecasts for the medium run. However, long-run macroeconometric
forecasts suffer from the same concerns regarding tax base
congruence as in the short run.

**Communication (story telling):** good. Large-scale
macroeconometric models offer relatively straightforward, intuitive
relationships that can be communicated easily. Parameter signs and
magnitudes should make intuitive sense and align with economic
theory. Because of their reduced-form specification, interactions
between equations and simultaneously determined outcomes are
limited relative to an unrestricted systems approach. Impulse
response charts can be produced and published publicly to
demonstrate the characteristics of the model.

**Communication (transparency):** fair. Model
documentation, equations, and datasets can be published online for
public scrutiny by interested parties such as academics, think
tanks, and private consultancies (for example, the
ITEM
group at Ernst and Young uses
HM Treasury's model for
its own consultancy purposes, including an annual report that
provides a check on the government's forecasts).
^{
[34]
} Parameter estimates and equation specifications should
remain relatively stable between forecast. That said, the
transition from history to model results involves a lot of
smoothing and adjustments to fit with economic monitoring. The
modeller's judgment plays a large role in the starting point, first
two quarters, end point, and dynamics in between. For this reason
the model only receives a
*fair* score.

**Data compatibility:** fair. Estimating a large-scale
macroeconometric model should be possible using the Scottish
Quarterly National Accounts. However, drawbacks related to
congruence with the tax base could be amplified as a result of
Scottish data limitations, as there are only high-level sector
accounts for residential gross fixed capital formation on an
experimental basis.

**Resources:** poor. One or two experienced and
skilled analysts are sufficient to maintain a large-scale
macroeconometric model once developed; however, the development is
a significant undertaking. Practitioners reported that it is useful
to have several less-experienced analysts in charge of economic
monitoring and to work independently on model components such as
import/export modules. Model development would require programmable
statistics software beyond Excel.

Our assessment of large-scale macroeconometric models is summarised in Appendix Table A6.

**3.7 Dynamic stochastic general equilibrium
models**

Dynamic stochastic general equilibrium ( DSGE) models are a systems framework for modelling the aggregate economy. They are relied upon heavily in modern macroeconomics. There is a broad spectrum of models falling under the DSGE label and an exact definition is difficult to pin down. De Vroey (2016) suggests that early DSGE models were defined by the following elements, based on his interpretation of Manuelli and Sargent (1987):

- a general equilibrium perspective (markets are not considered in isolation, but rather as a whole)
- dynamic analysis (the decisions and behaviour of economic agents depend are modelled over time, rather than a single period)
- rational expectations (the forecasts of agents within the model are assumed to be the same as the forecasts of the model (see Sargent, 1987))
- microfoundations (economic agents are modelled with optimizing behaviour, for example utility-maximizing and forward-looking behaviour in household decisions on saving, consumption, and labour supply subject to a budget constraint)
- markets clear (prices adjust to eliminate excess demand or supply)
- exogenous stochastic shocks (shocks come from outside the system rather than emerging within)

These features, along with other elements such as production technologies, budget constraints, and decision rules are formulated mathematically to represent the economy in a manner that a computer can simulate and solve. Equations are estimated all at once in a binding, unified way rather than the piecemeal equation-by-equation method of traditional macroeconometric models (Carnot et al, 2011).

Although early models assumed macroeconomic fluctuations were the result of random disturbances in technology and preferences, they were eventually modified to be based on frictions (price and wage rigidities) and to include monetary policy, with these new classes of models being called New Keynesian models (for example, see Christiano, Eichenbaum, and Evans (2005)).

DSGE models emerged out of the real business cycle literature, notably Kydland and Prescott (1982). Slanicay (2014) provides a history, describing them as a response to the forecasting failure and problematic theoretical underpinning of large-scale macroeconometric models-namely, the simultaneous high inflation and high unemployment of the 1970s (a breakdown in the Keynesian Phillips curve) and lack of microfoundations.

A technical specification of the equations of DSGE modelling would go beyond what is possible in this review, but we provide a qualitative description of a small-scale new-Keynesian DSGE model, based loosely on An and Schorfheide (2007) as presented by Herbst and Schorfheide (2015).

The basic
DSGE
economy is modelled using five agents. There are two types of
firms: a single representative
**final goods producing firm** that is perfectly
competitive, taking input prices and output prices as given. The
firm's inputs are supplied by
**intermediate goods producing firms** that are
monopolistically competitive, choosing labour inputs and output
prices to maximise the present value of future profits.
**Households** choose their labour supply,
consumption, and saving to maximise utility, subject to a budget
constraint that accounts for investment returns and tax payments. A
**central bank** uses an interest rate feedback rule
to respond to monetary policy shocks and targets a steady-state
inflation rate consistent with a level of output at its potential.
A
**fiscal authority** is assumed to consumes a fraction
of aggregate output subject to a budget constraint and levies a
lump-sum tax to finance any shortfalls in government revenues.

Examples of DSGE models that could be drawn upon for Scottish forecasting include the Bank of England's COMPASS, the Bank of Canada's ToTEM, and the New York Federal Reserve's FRBNY models. The IMF also uses two well-known DSGE models: MultiMod and GEM.

DSGE models have faced criticism following the financial crisis and their use and misuse is being keenly debated. Their opponents include Romer (2016), who criticises DSGE models broadly, attributing their popularity to "imaginary" constructs that succeeded through mutual loyalty among well-known economists and a departure from scientific principles.

A particular target of criticism is the model's rational expectations assumption, although methods to introduce frictions have been implemented to slow adjustments to reflect observed behaviour, and more recently models such as Slobodyan and Wouters (2012) have introduced bounded rationality.

**Assessment**

**Application (forecasting)**: poor. The focus of
DSGE
models is not forecasting, but rather simulating and tracking how
shocks are propagated through the economy. Slanicay (2014)
describes their range of application, from the models that central
banks use to discuss the transmission of monetary policy shocks
through the economy, to the more stylised academic models tailored
to test and demonstrate implications of particular economic
assumptions.

Applications of DSGE models to the housing market are limited. Basic DSGE models typically only operate in flow space (changes period-to-period) and residential investment stocks generally do not play a role. That said, there have been some efforts in recent years to incorporate a housing sector.

For example, Caldara, Harrison, and Lipinska (2012) developed a method to use the correlations between housing price shocks estimated in auxiliary VAR models with variables included in the DSGE model to assess the implications of shocks to US housing market data.

Other research comes from Iacoviello and Neri (2010). The authors considered both the impact of macroeconomic shocks on the housing market and how shocks to the housing market affect the macroeconomy. The housing market is incorporated via a production function that produces houses using capital, labour, and land.

A recent innovation in DSGE models is the stock-flow consistent models of Burgess, Burrows, Godin, Kinsella, and Millard (2016). They show promising improvements on traditional DSGE models, incorporating the balance sheets of economic sectors including the housing sector. Stock-flow consistent DSGE models may be more suitable for housing market forecasting and policy in the future.

**Application (policy)**: fair.
DSGE
models may have use for modelling the transmission through the
economy of housing market scenarios.
DSGE
models are not appropriate for static fiscal impact costing
(costings that are estimated using a single market and do not
consider the feedback effects of the rest of the economy). However,
they can be used for dynamic scoring (modelling and costing the
feedback of government policy changes through the wider
economy).

**Accuracy (short run)**: fair. Although early
DSGE
models had poor forecasting performance, recent developments such
as Smets and Wouters (2007) have demonstrated refinements that can
offer better forecast properties. The
*fair* score has been given for the general forecasting
performance of
DSGE
models, but forecasters should consider that application to the
housing market is largely untested.

**Accuracy (medium run)**: fair.
DSGE
models may offer better performance in the medium run than other
models less grounded in economic theory. Iacoviello and Neri (2010)
find their housing-market augmented
DSGE model
is able to capture long-run trends and the medium-run business
cycle well. When choosing a
DSGE model
for the medium-run horizon, there is evidence that smaller is
better. For example, Del Negro and Schorfheide (2012) assess the
forecast performance of the large-scale Smets and Wouters (2003)
DSGE model
against a small-scale version. They find that although the
short-run forecast performance of the large-scale model is slightly
better than the smaller model, the medium-term forecast performance
of the compact model is superior.

**Communication (story telling)**: fair.
DSGE micro
foundations lend themselves to intuitive narratives. However,
complexity of interactions runs significant risks of becoming a
'black box' with difficult or no interpretation.

**Communication (transparency)**: poor. Caldara et al.
(2012) suggest that as layers of complexity and interaction are
added, the results and interactions become more opaque and harder
to explain to policymakers. Considerable judgment is applied
throughout estimation. External scrutiny would require specialist
training.

**Data compatibility**: good
**.** A
DSGE model
could be estimated using the Quarterly National Accounts of
Scotland. The Smets and Wouters (2007) model is constructed using
only seven data series (real
GDP, consumption,
investment, wages, hours worked, inflation, and interest rates). To
model the housing sector in the manner of Iacoviello and Neri
(2010) would require the addition of measures for capital and
land.

**Resources**: poor.
DSGE
models are generally an advanced forecasting technique requiring
specialised training and most likely the services of a PhD
economist, especially during development. However, the required
resources may not put
DSGE
models out of reach of the Scottish forecasting framework.

Our assessment of DSGE models is summarised in Appendix Table A7.

**3.8 Microsimulation models and micro
databases**

Microsimulation models use survey data to construct a
representative distribution of typical households and individuals
in the economy.
^{
[35]
} Weights are then used to scale the sample to the population
level. If linked to tax returns, microsimulation models can be used
to assess the fiscal and distributional consequences of changes to
the tax and transfer system.

Microsimulation models are not designed for forecasting-they are static accounting models that mechanically apply legislated or proposed tax and transfer parameters to the relevant characteristics of individuals and households. These may include characteristics such as an individual's net income, the number of children under a certain age that qualify for child benefits, or real estate purchases made throughout the year.

Certain properties of microsimulation models nonetheless allow
them to be used as a tool in a wider framework to arrive at
forecasts. Specifically, models can apply the current tax system to
the population characteristics in past years. In this manner,
forecasters can calculate the revenue elasticity (sensitivity of
growth) of a tax to the tax base. For a flat tax, this revenue
elasticity would be zero. However, for a graduated tax such as
personal income taxes or
LBTT,
revenues will increase faster than one-to-one with the base.
^{
[36]
} This is known as
*fiscal drag*. Because
LBTT
thresholds are not automatically indexed to inflation, fiscal drag
could be significant.

The elasticity of revenues to the tax base can then be applied to a forecast of the tax base to arrive at a forecast of revenues. Although the model may be built with the ability to grow the base itself to future years, the benefit of this approach is that the forecast doesn't have to correspond exactly to the tax base of revenues. For example, the microsimulation-based elasticity can be imposed on a macroeconometric model by using its historical correlation with the National Accounts elasticity.

Although not technically a microsimulation model, some practitioners use a database of the universe of tax returns that can be queried to retrieve variables of interest. These can include simple relationships that can be programmatically changed and aggregated to cost alternative policies, or can be assessed before and after a policy has been implemented to assess its impact (for example, see Matthews (2016)).

Microsimulation models and micro databases are also among the few ways to examine the distribution of housing prices over time. If clear trends are observed in the summary statistics of the distribution (for example, its mean, variance, and symmetry), these may be forecast using techniques described above such as univariate time series models.

**Assessment**

**Application (forecasting)**: poor. Microsimulation
models and micro databases cannot, on their own, provide forecasts
for the housing market. However, they can be used in conjunction
with other forecast models to incorporate fiscal drag effects into
the forecast. For example, their base years can be grown to future
years by imposing growth rates from auxiliary forecast models, and
tax rates and thresholds can then be applied to this uplifted base
to project tax receipts.

Microsimulation models can be used to estimate variables to include in other models, such as multivariate econometric models. For example, microsimulation models can be used to calculate the average effective tax rates of homebuyers over history and the outlook to use as an explanatory variable when estimating the net benefits of home ownership.

**Application (policy)**: good. Microsimulation models
and micro databases are particularly well-suited for policy
analysis. They are the main way that government budget authorities
translate forecasts of the tax base into forecasts of revenues.

There are many examples of microsimulation models in the UK applied to policy analysis. HMRC has a microsimulation model specifically for SDLT that covers the universe of transactions. Other microsimulation models in the UK include the Department of Work and Pensions' Pensim2 model, the IFS's TAXBEN model, and the London School of Economics' SAGE model. However, these models don't have a detailed specification for SDLT or Scottish LBTT.

The OBR uses HMRC's residential stamp duty plus model ( SDM+) to prepare its devolved forecasts for Scottish residential LBTT, Stamp Duty Land Tax, and Welsh residential SDLT. OBR (2016) describes the SDM+ model as follows:

[ SDM+] allows us to apply the tax schedules for LBTT and SDLT to a full sample of transactions from a given year and then grow them in line with our price and transactions forecasts for the residential property markets (p. 22).

Microsimulation models are general developed for social policy
analysis, personal income taxes, and consumption taxes. There are
few microsimulation models applied to housing.
^{
[37]
}

Many studies assessing the behavioural response of
UK policy measures simply
use micro data files with the universe of tax returns to evaluate
historical policy changes by looking at the period before and after
the intervention. For example, Best and Klevin (2013) used the
universe of
SDLT administration
data (that is, every property transaction) from 2004 to 2012 to
assess a number of tax changes, including the success of the stamp
duty holiday in 2008 to 2009 as fiscal stimulus.
^{
[38]
}

Microsimulation models are also useful for producing revenue sensitivity estimates, for example in the production of cyclically adjusted budget balanced and fan charts.

One drawback is that, because they are by nature mechanical, they require ad hoc adjustments to the simulation output to incorporate behavioural responses.

**Accuracy.** As the model itself cannot forecast,
this criterion is not applicable. However, if economic variables
forecast with auxiliary models are accurate, then the
microsimulation model should give accurate conditional estimates of
revenues.

**Communication (story telling)**: good. Because
microsimulation models are a rote imposition of the tax code on
real households, communication of microsimulation results are
intuitive.

**Communication (transparency)**: good. The underlying
equations are mechanical identities, and aside from weights to
scale results to the population level, little estimation and no
judgment is applied. Although model code can be a challenge to
access and examine, results can typically be tested against
back-of-the-envelope calculations.

**Data compatibility**: poor. There may be limitations
to Scottish forecasters' access to tax-payer level data.
Microsimulation model development may therefore only be possible if
appropriate data protocols can be put in place.

**Resources:** poor
**.** Microsimulation models require considerable
resources to develop and maintain. The initial design requires a
great deal of time and expertise on both tax policy and software
development. However, the model may not need to be built from
scratch. Li and O'Donoghue (2013) surveyed microsimulation models
and point to four generic software programs that can be adapted to
build new microsimulation models: ModGen (Wolfson and Rowe, 1998),
UMDBS
(Sauerbier, 2002),
GENESIS (Edwards,
2004) and
LIAM (O'Donoghue,
Lennon, and Hynes, 2009).

Parameters need to be updated to reflect new tax and transfer legislation once or twice annually within the budget cycle. This can be an involved, resource-intensive process if policies are implemented that aren't simple rate or threshold changes.

Our assessment of the qualities of microsimulation models in relation to the Scottish budget forecast process is summarised in Appendix Table A8.

### Contact

Email: Jamie Hamilton

Phone: 0300 244 4000 – Central Enquiry Unit

The Scottish Government

St Andrew's House

Regent Road

Edinburgh

EH1 3DG