Volatility Modeling of Regional Norwegian Housing Prices

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Volatility Modeling of Regional Norwegian Housing Prices

A GARCH Analysis

Eirik Lading

Master Thesis

Master of Philosophy in Economics 30 ECTS

Department of Economics Faculty of Social Sciences UNIVERSITY OF OSLO

November 2017

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Volatility Modeling of Regional Norwegian Housing Prices Eirik Lading

http://www.duo.uio.no/

Trykk: Reprosentralen, Universitetet i Oslo

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Abstract

The thesis analyzes the volatility profiles of the housing price growth of Norwegian counties. GARCH models estimate level and volatility separately, allowing for analysis of variant volatility. They are used to analyze the heteroskedasticity of the housing price growth in Norwegian counties. A linear model is applied to correct for month specific effects and exogenous variables, rendering residuals that are analyzed with a GARCH model. The growth rates of the housing price indexes of Norwegian counties are found to be heteroskedastic.

After correcting for month specific effects, periods of volatility clusters are identified for most Norwegian counties. This result is robust to inclusion of other variables and different samples. Volatility clusters are found to be prominent in Oslo, Hordaland and Sør-Trøndelag.

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This thesis has been written as part of my master’s degree at the Department of Economics at the University of Oslo.

I would like to thank my supervisor Ragnar Nymoen for sharing his insight into econometric modeling and for helpful guidance on thesis writing.

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List of Tables

4.1 GARCH coefficients of observable housing price growth rates . . . 26

4.2 GARCH coefficients residuals of AR(12)-model . . . 28

4.3 Augmented Dickey Fuller test . . . 35

4.4 Jarque-Bera normality test . . . 38

4.5 ARCH LM test . . . 40

4.6 GARCH coefficients of shorter samples . . . 42

4.7 Coefficients for monthly dummies on housing price growth . . . 44

4.8 GARCH coefficients of residuals of linear model with dummies . . . 46

4.9 GARCH coefficients of residuals of multivariate linear model . . . 48

4.10 GARCH coefficients of residuals of multivariate linear model, with seasonal corrections . . . 50

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List of Figures

2.1 Housing price indexes for Norwegian counties . . . 6

2.2 Housing price indexes for Norwegian regions, deviation from mean . . . 7

2.3 Monthly annualized growth rate for housing prices in Norway . . . 9

4.1 Volatility for counties without volatility clusters . . . 30

4.2 Volatility for counties with volatility clusters . . . 31

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Chapter 1 Introduction

This thesis is an analysis of the volatility of housing prices in Norway. Specifically, it looks at the monthly fluctuations of housing price growth rates. In more general terms, one could refer to this as the uncertainty of the housing market. GARCH is a class of econometric models suitable for analysis of volatility. It is a particular extension of the ARCH model, which was introduced as a tool for analyzing variations in volatility.Traditionally, the housing price levels themselves are the main object of analysis in econometrics. This thesis analyses instead the associated uncertainty and fluctuations of the housing market.

Most of this thesis examines the characteristics of the growth of a housing price index for each of the 19 counties in Norway. As all of them are standardized to the initial value of 100 in January 2003, comparing the difference in levels between them is pointless. However, their growth rates are the same as the observable growth rates of the housing prices. Hence the analysis will be the same.

The housing market is important to financial economics and macroeconomics. Housing price growth is itself analogous to the return a financial object as the profits made by house price increases is similar in structure to the returns for other financial objects, such as stocks and bonds. Housing is also deeply intertwined with macroeconomic as its prices tail the

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movements in domestic growth and international financial markets. Volatility models are applied to this market to increase insight in the fluctuations of this market. A particular emphasis is put on the regional dimension of the Norwegian housing market.

Two approaches are employed. The first is to model each observable regional growth rate time series with the aid of a GARCH model. The observable growth rate time series are the monthly annualized growth rates, derived from the house price indexes. They are found to lack obvious volatility clusters without correcting for monthly effects. The other is referred to as the two stage analysis, the GARCH model is applied on the OLS residuals of a linear model. The latter approach is found to give important insights, highlighting the advantages of a multi-stage approach.

The two stage analysis is a means of regressing out seasonal effects and effects stemming from other variables. The first stage of the procedure is the application of a linear ordinary least squares (OLS) model. Here, different dummies or lags are regressed on the housing price growth rate in order to control for month specific effects. This first stage renders a time series of residuals which are in turn analyzed with a volatility framework. Distilling away the seasonal effects are found to be important for the analysis of the volatility of the Norwegian housing market.

A multivariate model is also employed in the two stage analysis. Here, the first stage regression includes two exogenous variables, along with an autoregressive term. The first exogenous variable of the first stage regression is the unemployment rate for the given region.

The other exogenous variable is the real interest rate, constructed from the KPI-JAE inflation rate and the central bank key rate.

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Existing Literature

This thesis is inspired by two papers, Crawford and Fratantoni (2003) and Miles (2008).

Both analyze the US housing markets using the GARCH volatility model. The flexibility of the GARCH framework allows for multiple uses. This thesis applies a similar model on the Norwegian housing market. The emphases in those theses are forecasting and identifying the presence of volatility clusters. This thesis expands on the latter by analyzing the specific volatility clusters that has occurred in Norway.

Crawford and Fratantoni (2003) is a starting point for many GARCH analyses of housing markets. The paper compares ARIMA, GARCH and regime-switching models with respect to how well they forecast aggregate housing prices. They analyze the price indexes for five US states with each model and compare their performance using AIC and RMSE. They conclude that regime-switching performs best among the three. This thesis uses the GARCH model to analyze the historical conditional volatility and make inference on a regional level, and not forecast.

Miles (2008) expands on Crawford and Fratantoni (2003) by including all 50 US states instead of only a few selected. Unlike Crawford and Fratantoni (2003), Miles seeks to find indication of ARCH effects with a test, rather than simply determine the model’s ability to forecast. Miles concludes that the difference in GARCH effects between states imply that the markets should be analyzed separately. My thesis is similar as counties are analyzed, instead of an aggregate approach. There is also a further analysis of the development of historical volatility.

As a primer on the models, the results for Miles (2008) were replicated. This was both a way of attaining the necessary programming skills to model AR and GARCH, as well as a means of ensuring that that my framework comparable. Another important take from this article was the notion of using annualized housing price growth rates. After the results were replicated, a similar model was applied to the data for the Norwegian housing price indexes

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from Eiendom Norge.

There are several articles analyzing the characteristics of the levels of housing prices in Norway, such as Anundsen (2010). That article examine the variables affecting the price level.

Specifically, it analyzes how debt and credit affect the amount of real estate mortgages, as well as the opposite effect, how housing prices affect the credit market. While this thesis has a similar approach by analyzing housing price growth rate with a multivariate linear model and, the housing price volatility is the focal point.

GARCH models for the Norwegian housing market was used in Skarbøvik (2013), similarly to Crawford and Fratantoni (2003). The paper concludes that GARCH models perform better than ARIMA models in forecasting housing prices in Norway. It also expands on the model by utilizing a hybrid model, combining and weighting models, attempting to improve forecasting ability. The data used in the paper is also from Eiendom Norge, which was then named Eiendomsmeglerforetakenes Forening. Their data set reports price per square meter, which was available from 2001, while my thesis uses the general index for the Norwegian counties. Another difference is the data transformation, where Skarbøvik uses first difference while the monthly annualized growth rate is used in this thesis.

This thesis is divided into five chapters in addition to the appendices and the reference section. Chapter 2 introduces the Norwegian housing market with a brief description of the Norwegian housing market and the housing price indexes. Chapter 3 explains the econometric terminology and models used throughout this thesis. The results for each model are presented in section 4, after a brief introduction of the data sets used. The estimated GARCH coefficients for all counties are listed for each model. Specification test results for the two main models are shown. Chapter 5 summarizes the findings.

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Chapter 2 The Norwegian Housing Market

2.1 History and Motivation

One could regard the modern housing market as beginning in the early 1990s after a wave of deregulations taking place the previous two decades. Starting in the 1970s, the credit markets were gradually liberalized. Throughout the following decades, financial regulations such as interest rate controls, quantitative controls and foreign exchange controls were phased out until the late 1980s and early 1990s. Since then, the market has been recognizable when compared to the current regulatory regime. This thesis analyzes the housing market, starting a decade after the implementation of the current financial regulation.

Norway has a high home ownership rate, which makes homes the most important financial asset for many families. In Norway, most real estates are financed with mortgage loans. Home ownership is associated with high returns and considered cheaper than renting. The interest payments are deductible, further increasing incentives to follow this model. This implies that housing price volatility is tied to the wealth of Norwegian households, as well as the portfolio of real estate investors.

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2.2 Regional Differences in Housing Price Growth

The time series used goes back to 2003, which acts as a starting point for the index, as every entry is 100 in January of 2003. Using 2003 as a starting point has the advantage of being in a state of stability in the business cycle, while partially preceding the oil price hike of the early 2000. The following years shows a substantial increase in housing prices from its lowest point in 2003, thus also housing returns. Throughout the period, the domestic housing price index had more than doubled, ending in an index of 231 by January 2016.

Figure 2.1: Housing price indexes for Norwegian counties

Between 2003 and 2015 the Norwegian housing markets experienced two major shocks that caused significant price decreases throughout the country. This emphasizes the importance of volatility analysis of the market. The first, and largest, of these was in late 2008 as part of the global financial crisis. After hitting a peak august 2007, the housing market had decreased 12% by the end of the following year. The prices had recovered by the fall of 2009.

The price drop could be the result of a credit crunch or an overall recession in the economy.

The second was in late 2013. This was substantially smaller of the two and the prices has already recovered a year later. Paradoxically, this price drop preceded the oil price drop of 2014.

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While each county largely followed the domestic development, there are important regional differences to be observed.

Figure 2.2: Housing price indexes for Norwegian regions, deviation from mean

The graph shows how each the price index for each region deviates from the mean. This underlines the differences between the regions, showing that the responses to shocks differ.

The time series are constructed as an average of the index for each county included in the region. The indexes are not weighted with respect to population. The counties are analyzed separately in later chapters.

The importance of regional dimension of the housing market is apparent as the different regions have had distinct price trajectories.

The highest price growth occurred in Rogaland and Hordaland, presumably as a result of the oil boom. The counties that had the lowest price growth were all situated in Eastern Norway, having consistently below-average indexes. Oslo and its adjacent municipalities are exceptions to this. Aust-Agder and Vest-Agder were the least affected by the 2008 turmoil, resulting in an above-average index in the following six years. However, this was followed by a steep downturn compared to the rest of the country, resulting indexes as low as Eastern Norway by the end of 2016. Northern Norway, Finnmark in particular, had the highest growth before the 2008 bust. In 2014 and 2015, there was an overall convergence of the

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indexes throughout the country.

The oil price shock of 2014 was followed by a significant price hike in most counties in the early 2015, with Rogaland being a notable exception. This could indicate that a currency depreciation led to an increased demand in domestic housing purchases. In Rogaland, this effect may have been offset by the negative income effect of decreased activity in the petroleum industry. Hence, this could be seen a result of the differences in industrial differences between regions.

Another explanation is that it could have been offset by an existing regional housing bubble, in line with the already high price level.

The largest discrepancy between the regions occurred before the two major busts. The prices then converged in the aftermath of these. This indicates that the shocks overall had the largest effects on the regions that had experienced the highest price growth, and the lowest effect on regions of low housing price growth.

Most of the analysis in this thesis is based on the price growth, and not the overall level.

The monthly annualized growth rate for Norway as a whole is plotted below.

This is a representative case as the plots of the monthly annualized growth rates for housing prices are fairly similar across counties. There seem to be some patterns repeating each year as well as some periods of particularly high or low growth levels. The decrease during the financial crisis stands out. The plots for each county are in the appendix.

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Figure 2.3: Monthly annualized growth rate for housing prices in Norway

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Chapter 3 Econometric Modeling

This thesis is about a particular volatility structure. To clarify the essence and implications of the results, some econometric concepts and models are explained and described in this chapter.

3.1 Volatility and Heteroskedasticity

The distinction between first and second order moments is important when modeling time series. First order moments involves looking at factors determining the level or mean of a process or variable. Second order moments are the variance and autocovariances of a process, the spread of observations. This thesis has both approaches, with a particular emphasis on the latter. Looking at the factors explaining housing price growth is an example of first order analysis, while modeling volatility is an example of the latter. This is not a rigid distinction, however, as looking for the factors explaining volatility is itself akin to a first order analysis.

Also, the models employed in this thesis provide both first order and second order analysis of the time series.

When analyzing stochastic process, such as housing price growth, one utilizes stochastic

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models that distinguish the deterministic and random components. Financial time series, such as housing price growth, consist of both elements. These processes are partially stochastic, founded upon an underlying long-term trend or an otherwise seemingly deterministic process. Econometrics offers many tools to distinguish these elements.

Constant volatility is referred to as homoskedasticity, while volatility that changes level over the period indicates the presence of heteroskedasticity. Due to their erratic nature, it is reasonable to expect that housing price growth is heteroskedastic.

The unpredictability of heteroskedastic processes makes them more complicated to model.

Homoscedasticity is often an easier case to assume in models, as it decreases the number of variables to consider. But homoskedasticity might be a too strong assumption in some models.

This thesis employs models that examine the characteristics of the heteroskedasticity in the Norwegian housing market. While processes might have either high or low volatility in general, they could be a more interesting structures. For instance, the variance in income for a city could vary over time as the population level changes. Or the temperature fluctuations changes between seasons throughout the year.

One interesting volatility structure examined in this thesis is the volatility clustering.

This is characterized by having periods of particularly high volatility. One could reasonably assume to find volatility clustering in housing markets. In these markets, heteroskedastic price growth fluctuations could stem from periods of particularly unstable price dispersions, i.e. volatility clustering.

Volatility of returns has important implications for prices of financial objects, and interpreted widely this includes housing and real estate. Investors are usually risk averse and prefer a certain return over uncertainty, and thus would want to be compensated for taking risks. This trade-off and accompanying compensation is the basis for numerous common finance models, such as CAPM or Arrow-Debreu pricing. The prospect of higher returns

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on investments should increase the asset prices, while increased risk should decrease prices.

Using volatility models such as GARCH could give further insight about the characteristics of the risk profile for financial objects. Other use of these models is the possibility of more accurate forecasting. In general, extending the model with volatility terms could more accurately describe the data generating process.

3.2 Linear Regression

The linear regression model is a simple model describing the relationship between two stochastic variables. It effectively represents the conditional distribution of an endogenous response variable given a set of values for exogenous independent variables. With the use of ordinary least squares (OLS), one can estimate the linear model and calculate predicted values and residuals. In this thesis the model is used in the first stage of the two stage analysis. The residuals that are rendered in this process are further analyzed in stage two.

A standard equation representation of the linear model is

y_i =α+βX_i+u_i u_i ∼IID(0, σ²) (3.1) wherey is the regressand,X are explanatory variables,α is an intercept term whileβ is the coefficient describing the effect from X on y. ui is the error term which the OLS method minimize in order to attain the OLS estimators, and by extension predicted variables.

This simple linear framework can be extended by allowing for dynamic effects. One gains new insights by exploiting the naturally decided order of observations across time. Numerous variables in economics are partly determined by dynamic effects, such as business cycles, momentum, convergence and delayed effects.

An important role the model has in this thesis, is in the first stage, where is cleans out

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month specific effects. These are effects associated with a specific month through the year.

With housing prices, this is the tendency prices have to rise or fall in a similar fashion during the same month each year. In Norway, for instance, prices tend to rise more during the summer. These monthly effects are often too strong to discern additional effects. Hence, in order to emphasize volatility effects in this thesis, econometric tools are used to clean out the month specific effects.

Two methods of correcting for monthly effects are made use of. The first method uses a dynamic linear model where the 12th lag of the growth rate variable is included as a regressor. By correcting for the variation that occurred 12 periods prior, the effect of being in a particular month is in part controlled for. The second method uses monthly dummy variables. 11 variables are added, each of which have the value 1 during a particular month, and 0 otherwise. Thus the effect of being in a particular month is controlled for as part of the first stage.

An equation representation of this model could be

y_t=α+

11

X

i=1

γ_iM_it+u_t (3.2)

where M are the dummies having the value 1 for a given month and 0 otherwise. They are 11 in total in order to avoid perfect multicollinearity.

Using dummies instead of controlling for the 12th lag has the advantage of not necessitating 12 lags prior to an observation. In the 12 lag model, 11 periods must have passed in order to observe a variable. Hence the 12 lag model does not make predicted values for the first year of the time series. By using dummies, the model can predict a variable from the first observation.

Another advantage is that the model with dummies calculates a coefficient for each specific month. This yields an easily interpreted result that describes the effect of each specific month.

The coefficient of a dummy is the growth rate associated with the corresponding month. For

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instance, one can compare the growth rate of June for each county by observing the sixth monthly dummy. Also, as the same dummy variables are used for all counties, one can easily compare the coefficients for a month between counties in order to see if the month specific effects are similar across the country.

One may also employ exogenous variables as regressors, constructing a multivariate model.

Including variables reveal their effects the regressand. Also, accounting for other variables holds their effects constant in an estimation, which underlines the effect of other variables on the regressand.

In this thesis, the model is applied to gain insights in how unemployment and interest rate affects housing price growth and then obtain the residuals. Hence it as a first stage of the two-stage framework. These variables are expected to be correlated with housing price growth as they affect the ability to service housing debt and by extension affect the housing price markup. The regressions yields a function for the housing price growth for each region.

This is part of the first stage of the two-stage framework. The residuals are then extracted and analyzed with the ARCH/GARCH model in order to make inference about the volatility of housing price growth. This is referred to as the second stage.

3.3 Autoregressive (AR) and Autoregressive Moving Average (ARMA) models

Autoregressive models (AR) is a starting point for time series analysis. They can be used to examine the degree of correlation in movements a variable has with its own past values. The accompanying coefficient describes degree of autocorrelation. As many effects in financial economics have some degree of persistence, taking into account lagged variables often improve time series descriptions and forecasts.

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A general equation for the model could be y_t=c+

p

X

i=1

θ_iyt−i+ε_t ε_t∼IID(0, σ_ε²) (3.3) where θ is the AR coefficient and c is the constant.

The AR coefficients can reveal important information about the nature and stability of the housing growth rates. 0 indicates intertemporal independence, meaning that the growth rate for one month is not correlated with the growth rate of the preceding month. If the absolute value of the coefficient exceed 1, the system is explosive and thus unstable, rendering this an unlikely case for housing price growth. If the coefficient is negative, the price tend to alternate between growing and declining between periods. As the AR(1) coefficients for housing price growth are found to be between 0 and 0.25, their movements seem partially self-enhancing but remain stable.

Stationarity of time series variables are an important issue in time series modeling. David- son and MacKinnon (2004) defines it as when unconditional expectation, unconditional variance and the covariance are independent of time, referred to as covariance stationarity.

Stationarity requires that the roots of the characteristic polynomial associated with the autoregressive part of the process are different from 1 in magnitude. Stationary processes are considered causal when all roots are less than 1 in magnitude. If processes are found to be stationary, they are also stable in the meaning of dynamic analysis. They tend to predictably revert back to the mean, unlike the unstable random walk or explosive processes.

Autoregressive processes assume correlations with a variable’s immediate past observations and may be extended to the long term by adding lags. But one may also have a longer-term view using a moving averages approach. These processes take into account the average of past observations. These are more fitting when describing trends and past tendencies and how they affect present observations.

With Autoregressive Moving Average (ARMA) models, a process is modeled with both AR-terms and a number of lags for the error terms as regressors. This allows for the process

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to have a longer memory than AR processes with few lags. The model captures the effect of longer-term adjustment for the variable level as well as short term deviations.

The equation

yt=c+

p

X

i=1

θiyt−i+

q

X

j=1

φjεt−j+εt (3.4)

is a representation of the ARMA model, generalized to including multiple lags. The number of AR lags is pand the number of MA lags is q, with their respective coefficients, θ and φ.

The ARMA framework is mainly applied in this thesis as part of the GARCH model. While it is commonly used as a tool determining the factors explaining the level of a variable, this is not the focus of this thesis, which primarily deals with second moment analysis. Instead, the ARMA model is applied as a tool in modeling the volatility and not as a standalone model describing housing price growth level.

3.4 Generalized Autoregressive Conditional Heteroscedas- ticity

The Autoregressive Conditional Heteroscedasticity (ARCH) model is a framework primarily describing the second order moment, volatility, as opposed to the linear or ARMA models.

While it may be used in calculating a conditional mean for a time series, its strength is modeling conditional volatility. It is thus a departure from the first order moment tradition that the previous models are part of.

The model was introduces in Engle (1982), as an attempt to move away from constant variance. The paper introduced the ARCH model, and by extension an entire class of models with conditional variance. Specifically, the variance was conditional on past observations of the modeled variable.

The paper was a departure from the leading approach to correct for heteroskedasticity.

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Heteroskedasticity is in many cases the result of a missing variable. Hence one could often remove heteroskedasticity by including the variable causing heteroskedastic results in the regression. A drawback from this approach is the lack of joint analysis of mean and variance.

Also, by predicting outlier clustering, one could minimize their effect on first order estimates, possibly improving predictions.

In addition to the model, the paper included a method of testing for the presence of ARCH effects in a time series. This ARCH LM test is applied in order to examine indications of volatility clustering.

ARCH models yields an estimator of coefficients between variables and the volatility of a time series. The regressors of the volatility portion of the standard ARCH(1) model is simply the lagged value for volatility, effectively making the volatility an AR(1)-process.

y_t =σ_tε_t ε_t ∼IID(0,1) (3.5)

σ_t² =ω+αy_t−1² (3.6)

where y_t is a financial returns series or a series of residuals, ω is time invariant volatility and α is the AR term. The latter captures the volatility clustering effects, also referred to as the ARCH-effect.

Generalized Autoregressive Conditional Heteroscedasticity (GARCH) expands on the volatility equation by including a term for the lagged εt. This equational form is similar to an ARMA model, effectively rendering the volatility function as an ARMA process. An important difference between the models, however, is that the endogenous variable of the GARCH volatility equation is unobservable, unlike the regressand in an ARMA model.

This thesis applies this class of volatility models to the Norwegian housing market. This model estimates effects of variables on housing price volatility in Norway. These are found

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by estimating a GARCH model on the housing price growth rates of all Norwegian counties.

GARCH is widely used in modeling volatility. Models for volatility are important as many processes in finance are stochastic. Heteroskedastic models are often employed where homoskedastic models are insufficient. One such instance could be the housing markets, where volatility might be a heteroskedastic process, in part conditional on other variables or its own past.

The endogenous variable in ARCH/GARCH models is either a financial returns process or regression errors. The growth rate of an asset price is a financial return, while the residuals of the linear model is an example of the regression errors. Hence the models are appropriate both when applied to the annualized housing price growth rate and the residuals of the linear regression. The model could also be further generalized to including exogenous variables directly, a specification not utilized in this thesis.

Employing the ARCH and GARCH models on the housing market is an attempt to describe the volatility as a process determined by its own past values. The assumption is that if volatility is high one month, it is more likely to be high the next month or in the future.

The ARCH term describes the immediate increase in volatility from a sudden spike, while the GARCH term describes the longer-term adjustment. ARCH models with few lags might not capture non-constant volatility if the correlation with immediate past is low. GARCH, however, captures periods of generally increased volatility even if it is not observable by looking at the short term past. Hence it represents a framework well suited for the modeling of the heteroskedastic process.

The model consists of two main equations. The first of this is referred to as the mean equation. This may include exogenous variables or variables based on past values of the regressand, such as AR or MA terms. This thesis sticks to the simple constant trend model, y_t=µ+σ_tε_t ε_t∼IID(0,1) (3.7) where y_t is a financial returns series or a series of residuals, µ is the growth rate trend, and

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σ_tis the conditional volatility. µis included when modeling observable housing price growth rates, and omitted when the model is applied on regression residuals in this thesis.

The second main equation is the conditional volatility equation, which is what distinguishes ARCH/GARCH from earlier models.

σ_t² =ω+αy²_t−1+βσ²_t−1 ω > 0 α, β ≥0 (3.8) Here, ω is time invariant volatility of the growth rate, α is the ARCH term, and β is the GARCH term. Both their absolute and relative magnitudes determine the nature of the estimated volatility.

GARCH is renowned for its parsimonious nature. It relies only on a few equations and parameters, which facilitates modeling volatility. The univariate model relies solely on past observations on one variable, making it particularly simple to estimate. The GARCH specification used throughout this thesis, is the simple GARCH(1,1). In this specification, the ARCH and GARCH variables of the volatility equations are each lagged once. This is the most commonly used specification for the volatility portion of the GARCH models. The mean portion of the model includes a constant trend when estimating a financial returns series. The trend term is omitted when estimating conditional volatility of regression residuals in the second stage of the two stage framework.

As GARCH models are flexible and general, they are occasionally found to fit better than simpler AR, ARMA and ARCH models. Extending the models by allowing endogenous volatility as an ARMA process facilitates modeling instances where foreseeable volatility explains outliers in the data. By including more explanatory variables, GARCH models could render forecasts that are more accurate than simpler models.

This facilitates using the model for analyzing volatility clustering and long term volatility effects in finance.

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In a little more detail, the main concepts and model elements applied in this thesis are the following:

µ is in the conditional mean portion of the model and describes the underlying trend. The larger this term is, the more prices growth increases throughout the period. The standard GARCH model does not have this term, but it is a common generalization of the model, especially when the endogenous variable is a financial return.

σ_t² is the conditional variance, which is included as an equation in ARCH and GARCH models. It contains the entire history of ε_t, not just a single lag, which is an improvement compared to ARCH and thus could render more precise regressions than ARCH.

ω Is the portion of the volatility that is invariant across time. It is restricted to being positive, ensuring that conditional variance is positive.

α is the ARCH term, which captures the short term volatility clustering effects

β is the coefficient of volatility on past volatility. This is the element that distinguish GARCH from ARCH and captures longer-term volatility effects. An absence of this, i.e. if it is not significantly different from 0, indicates lack of specific GARCH effects and the model become equivalent to ARCH. An interpretation is the persistence of shocks on future volatility. It smooths effect from the one period outliers, but emphasize periods with many outliers.

GARCH allows for estimation of conditional volatility, in addition to coefficients. These estimated volatility functions are plotted along the GARCH coefficients for the estimation done in section 4.3.1. Fluctuating conditional volatility functions are characterized by large α’s and their corresponding plots show short term volatility spikes. The counties with large β’s does not show this heteroskedastic characteristic. Rather, those counties display almost linear volatility functions. This could be a result of an omitted variable that makes the volatility appear more stable. Or, the function could be eligible for analyses using persistent

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volatility models, such as Integrated GARCH (IGARCH) models, which was introduced by Engle and Bollerslev (1986). Further analysis of this aspect of the Norwegian housing prices are beyond the scope of this thesis.

3.5 GARCH modeling of observable data and for resid- uals

The GARCH model can be used on a financial returns series or a series of residuals. This thesis applies the model to both type of time series. Using the observable financial returns series, the housing price growth rates does not give any clear indication of heteroskedasticity. But when applied to a series of residuals based on a model of financial returns, more interesting heteroskedastic results are found. As the residuals are retrieved from a financial returns, it could be considered a hybrid of two of the classes of time series that the GARCH model is suitable for.

In the first applications, only the time series data of the observable housing price growth are applied. Each of the 19 county house price time series are analyzed by itself and the result are derived from models utilizing autoregressive terms in different structures/specifications.

Later, more exogenous variables are added, making the analysis more comprehensive.

In all, five distinct GARCH models are estimated. The first one is based on the observable housing price growth rates and the remaining four are based on OLS residuals. Two of the residual models are based on month correction while the other two includes exogenous variables. All these models are univariate in the sense that the GARCH model is estimated on a single time series.

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Chapter 4 Data and Empirical Results

The main results of this thesis follows, after a brief description of the data sets. Results from five different models are presented. The first results are derived from a GARCH estimation of the observable housing price growth rates, which are found to lack ARCH effects. The second model includes an important correction for month specific effects. This makes it possible to identify volatility clusters that have occurred in the Norwegian counties. The volatility spikes in Norwegian counties in 2004-2015 are then described and analyzed. The plot of estimated conditional volatilities from these two models are in the appendices. The importance of the correction for month specific effects are underlined in the specification tests that are conducted on the two models. The third model has an alternate correction for seasonal effects and yields similar results. The last two models expand on the former by including additional variables to the first stage.

4.1 Data

The data is limited to the time frame starting in January 2003 ending December 2014 due to availability of data. Housing prices starts in 2003 as noted and the monthly reported

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unemployment rate are available throughout 2014.

The data set for housing prices are the Eiendom Norge monthly registered housing price index. It is a collective effort by Eiendom Norge, Eiendomsverdi and Finn.no, covering about 70 percent of sold real estate in Norway. The indexes are presented at differing regional levels, but available for each county. The data set was downloaded from Eiendom Norge’s website February 2016. The indexes are scaled as a monthly annualized growth rate for this thesis.

The growth rates are constructed in accordance with the formula p_t=

P_t Pt−1

12

−1

!

∗100 (4.1)

where P_t is the housing price index level for a country at time t. As noted, the monthly annualized growth rate can be interpreted as a times series of financial returns.

The data for the unemployment are the Statistics Norway monthly unemployment for 15-74 year olds, at county level. Specifically, it is the registered unemployment rate at the Employment Office, Statistics Norway table 10540. After 2014, they ceased reporting the monthly unemployment, opting for yearly reporting. As the unemployment rate is available at county levels, the regressions are also conducted at this level, despite availability of city and district level data for housing prices.

The inflation data used is the SSB KPI-JAE total index, which is registered monthly. The total index is annualized in the same fashion as the house price index in order to render the monthly annualized inflation rate. This scaling makes it possible to see how the inflation rate affects house price growth rates, as opposed to the overall price level.

The interest data is the Norges Bank policy rate, also referred to as the sight deposit rate.

The rate is reported as monthly averages, hence the months with an interest rate change have an interest rate level between the preceding and following month. This rate is chosen due to availability of data in the time frame 2003 - 2014. This overlaps with the availability of the main housing price data used.

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The real interest rate is constructed by subtracting the inflation variable from the central bank key interest rate. As the interest rate is the more volatile element of the two series, it is the main driver of fluctuations for this variable.

4.1.1 Software

The R package used for fitting a GARCH model to the data isrugarch by Alexios Ghalanos.

It is one of the most used GARCH packages available for R. The package is available in two separate versions, a univariate and a multivariate. Only the univariate version is used throughout this thesis, as the first stage regressions are conducted outside this framework.

The R package allows a number of different specifications beyond the default model of a trend model with a GARCH(1,1) structure, which is useful for analysis on residuals derived from other models. The model is modified slightly from the default by omitting theµ term as it is obsolete when analyzing residuals as their means are 0 by construction. For the same reasons, it is futile to calculate with AR or MA terms in the mean model. It analyses a vector, finding the best fit for the GARCH specification. An advantage to this particular software is the easily rendered variance-covariance matrices and conditional SD time series compared to other tried GARCH packages for R.

Additional econometrics oriented packages includes tseries, which was used to conduct the Augmented Dickey-Fuller test and Jarque-Bera test. fGarch is an alternate GARCH package used for comparison and controlling. dynlm was used to estimate linear models with lag terms easily. FinTS has an ARCH LM test function which does not necessitate constructing a VAR model.

Some plotting and graphics packages for R were used. ggplot2 and egg rendered some of the graphs, while stargazer facilitated making some of the tables.

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4.2 GARCH Estimation on Observable Housing Prices

Without correcting for the seasonality by including the 12th lag as a variable, estimation of the GARCH model gives little indication of heteroskedasticity. With a few exceptions, most Norwegian county housing markets exhibits a structure where the GARCH coefficients may be effectively 1. Also, the ARCH effects are negligible. The accompanying small slopes of the estimated conditional volatility indicate that volatility is almost constant. This means that the time series are close to homoskedasticity.

The counties Hedmark, Oppland, Buskerud and Vestfold seem to have a somewhat lower β coefficient, indicating a weaker GARCH effect. Also, unlike the rest of the counties, the conditional standard deviation functions are increasing. This implies a slightly higher volatility in the latter half of the time series. However, these effects are not significant and are not present in later models. Homoskedasticity remains the most convincing description.

Also notable is that the GARCH effect dominates the ARCH effect in all of the markets.

By itself, this also indicates that housing prices growth processes are in general less prone to volatility clustering than long term volatility stability. The relatively weak ARCH effects reported in throughout the country could indicate a lack of volatility clustering, meaning that volatility is not self-enhancing in the short term.

One can conclude that, without any modification or corrections, housing price growth in most Norwegian counties are almost homoskedastic. And in all counties, the heteroskedasticity is a strictly increasing or decreasing function. The dominant seasonal effect supersedes other effects. These effects could lead to other volatility structures, such as volatility clustering.

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µ ω α β

Østfold 6.80 0.29 0.00 0.999

Akershus 7.46 0.24 0.00 0.999

Oslo 7.73 0.00 0.00 0.999

Hedmark 7.55 12.62 0.00 0.963

Oppland 7.28 10.38 0.00 0.971

Buskerud 7.69 13.55 0.00 0.957 Vestfold 6.63 14.44 0.00 0.955

Telemark 7.49 0.28 0.00 0.999

Aust-Agder 7.89 0.35 0.00 0.999 Vest-Agder 7.86 0.23 0.00 0.999 Rogaland 10.15 0.22 0.00 0.999 Hordaland 9.00 0.00 0.00 0.999 Sogn og Fjordane 9.20 0.00 0.00 0.999 Møre og Romsdal 8.35 0.26 0.00 0.999 Sør-Trondelad 8.61 0.07 0.00 0.999 Nord-Trøndelag 9.00 0.29 0.00 0.999

Nordland 8.81 0.22 0.00 0.999

Troms 9.14 0.19 0.00 0.999

Finnmark 9.43 0.00 0.00 0.999

Table 4.1: GARCH coefficients of observable housing price growth rates

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4.3 GARCH with Seasonal Correction

4.3.1 Results with 12th lag correction

The adjustment is made by including a lag term in the first stage. This is effectively an alteration of the GARCH mean model. The first stage equation becomes

y_t =c+θ₁₂yt−12+ν_t ν_t=σ_tε_t ε_t ∼IID(0,1) (4.2)

The accompanying GARCH(1,1) volatility function is the same as in equation (3.8).

The most prominent ARCH effects are found in Hordaland, Oslo, Sør-Trøndelag and Rogaland.

Interestingly, these are also the counties with the largest cities. This could indicate that the intertwined housing markets within the larger cities experience periods of parallel volatility.

As cities experience the same volatility waves, this spills over to nearby municipalities, resulting in a county-wide volatility wave. The other counties, however, might have a structure of smaller, disparate housing markets. The result is weaker ARCH effects as the volatility clustering effects are not as prominent.

This is analogous to diversification. The separate elements making up the index have a tendency to have increased volatility during the same time periods. If an investor have 19 portfolios, one for each county, the portfolios with large, dominant cities are more prone to volatility clustering. They are in other words less diversified against this specific type of volatility.

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ω α β

Østfold 0.061 0.000 0.999

Akershus 8.759 0.094 0.839

Oslo 12.617 0.169 0.762

Hedmark 0.071 0.000 0.999

Oppland 20.623 0.104 0.768 Buskerud 13.147 0.048 0.854 Vestfold 8.545 0.070 0.865 Telemark 0.000 0.000 0.999 Aust-Agder 0.000 0.000 0.998 Vest-Agder 8.045 0.079 0.879 Rogaland 10.981 0.113 0.821 Hordaland 8.039 0.170 0.780 Sogn og Fjordane 0.000 0.000 0.997 Møre og Romsdal 9.402 0.078 0.862 Sør-Trøndelag 13.249 0.161 0.739 Nord-Trøndelag 0.000 0.000 0.998 Nordland 5.801 0.057 0.908

Troms 15.995 0.073 0.833

Finnmark 5.665 0.077 0.901

Table 4.2: GARCH coefficients residuals of AR(12)-model

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4.3.2 Volatility spikes with seasonal correction

Having estimated volatility functions with significant ARCH effects allows for analysis of the development of housing price volatility in the period 2004-2015.

Temporary volatility spikes can be observed from the trajectories of the conditional volatility of the price growth of most of Norwegian counties. These are sudden increases in conditional volatility as calculated from the GARCH specification of the time series variance.

They usually last from one to four months followed by a longer decrease until reaching pre- spike levels about one year after the onset. By and large, the counties follow the same trends, as seen from the overall similar movements throughout the country, with minor differences distinguishing the counties. There are interesting patterns in which counties respond the most to nationwide volatility spikes.

A few counties are estimated to have strictly decreasing conditional standard deviation functions, similar to the model without corrections. The other counties have fluctuating trajectories for volatility, characterized by periods of increased and decreased volatility levels.

The five counties without fluctuating conditional standard deviation functions are those that had largeβ’s and did not appear to have volatility clustering effects. They are still arguably heteroskedastic, as their slopes are negative, which signifies decreasing volatility. This implies that volatility was overall higher during the first half of the time period than the last. This appears reasonable as the financial crisis, along with most of the volatility spikes, took place during the first half of the time period.

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Figure 4.1: Volatility for counties without volatility clusters

The regional inference to be made from these counties, is that Eastern Norway display signs of lower and more stable volatility. Østfold, Hedmark and Telemark have the lowest measured level of volatility and the lowest absolute value changes during the time period.

The remaining 13 counties are found to have fluctuating estimated conditional volatility, which are plotted below. By observing these trajectories, one can identify periods of increased price uncertainty in the housing market.

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Figure 4.2: Volatility for counties with volatility clusters

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The first country-wide volatility spike occurred during fall of 2005. Easter Norway was the least affected by it, along with Hordaland.

The largest volatility spike occurred during the international financial crisis, beginning in late 2007. This was also apparent when analyzing the index levels and growth earlier. The most affected counties in this period was Hordaland, Oslo and Sør-Trøndelag. These counties overlaps with the previously found counties with the highest volatility cluster effects. As this is the most profound event of the time period, it is likely a main driver of developments in volatility.

The third volatility spike occurred a year later, between the fall of 2008 until summer 2009.

As with the previous spike, Oslo, Hordaland and Sør-Trøndelag has the largest increases in conditional volatility. Also notable is the large spike of Vest-Agder, a county that also had a larger volatility increase than Eastern Norway during the 2007 spike. This further strengthen the hypothesis of large cities driving volatility clusters in counties. The implication is that the volatility of Kristiansand could spill over to the other counties of Vest-Agder. The weaker effect in Rogaland during the financial crisis could stem from the reliance upon petroleum, and thus less exposure to the financial crisis.

The last nationwide spike occurred in early 2011. It is notable for being more sudden than the preceding, and for the fact that it happened to all included counties simultaneously during January 2011. As always, the counties with the five largest cities had the largest spikes, as well as the fastest recover to pre-spike levels.

There is a period of increased volatility towards the end of the time period, but not as sudden as the previous spikes. This coincides with the countrywide fall in prices observed earlier. Interestingly, this turmoil preceded the oil price fall of 2014, indicating a less intuitive explanation. It could stem from the general downturn of the business cycle following the 2012-2013 peak.

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4.3.3 Specification tests

Specification tests were conducted in order to verify the soundness of the main estimation results and findings. The tests are conducted on both observable housing price growth rates and the redisuals of the linear model with the AR(12) term. Comparing the results could verify the importance of the correction and underline the changes to the volatility structures of the times series.

Dickey-Fuller is a unit root test, testing if the data fits the notion of a unit root process against the alternate hypothesis of stationarity. In an AR(1)-process, this is when the coefficient of the first lag of a variable is 1. Due to the possibility of serial correlation in the error term for time series, the underlying assumptions might not be valid. For this reason an alternative statistic is proposed, referred to as the Augmented Dickey Fuller (ADF) test.

The critical values are obtained from the Dickey-Fuller distribution.

Specifically, the test calculates if the changes in the variable of interest is related 1-to-1 to the previous observation.

∆y_t = (γ−1)yt−1+u_t

H₀ :γ = 1, H₁ :γ <1

When the test is applied on the observable growth rates, the regression includes a constant variable.

Schwert (2002) suggests using the following as a rule of thumb when choosing lag length:

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p_max=

"

12 T

100 ¹₄#

(4.3)

This implies 12 lags for the data used in this thesis.

The tests do not reject the hypothesis of a unit root. This could be due to the known problem of low power with Dickey Fuller test.

When testing on the residuals of the first stage regression, the null hypothesis of stationarity is not rejected for some counties.

After testing on the conditional standard deviations of the counties, the volatility processes for most counties appear to be unit root.Also, the counties with the lowest ARCH coefficients have the lowest DF statistics.

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observable first stage residuals conditional SD DF statistic p-value DF statistic p-value DF statistic p-value

Østfold -1.71 0.70 -3.27 0.08 -3.02 0.15

Akershus -1.98 0.58 -3.32 0.07 -1.84 0.64

Oslo -2.25 0.47 -3.17 0.10 -1.70 0.7

Hedmark -1.93 0.61 -3.39 0.06 -4.16 0.01

Oppland -1.81 0.65 -3.34 0.07 -1.76 0.68

Buskerud -2.19 0.50 -3.39 0.06 -1.96 0.59

Vestfold -2.38 0.42 -3.59 0.04 -1.68 0.71

Telemark -2.20 0.49 -3.48 0.05 -3.28 0.08

Aust-Agder -2.15 0.51 -4.08 0.01 -2.90 0.20

Vest-Agder -1.89 0.62 -3.22 0.09 -1.89 0.62

Rogaland -1.91 0.61 -3.14 0.10 -2.22 0.48

Hordaland -1.95 0.60 -3.11 0.11 -1.84 0.64

Sogn og Fjordane -2.89 0.21 -3.91 0.02 -3.15 0.10

Møre og Romsdal -2.21 0.49 -3.32 0.07 -1.77 0.67

Sør-Trøndelag -2.25 0.47 -3.02 0.15 -1.91 0.61

Nord-Trøndelag -2.10 0.54 -3.33 0.07 -4.35 0.01

Nordland -1.63 0.73 -2.68 0.29 -2.35 0.43

Troms -1.69 0.70 -3.19 0.09 -2.36 0.43

Finnmark -1.93 0.60 -3.04 0.14 -2.53 0.36

Table 4.3: Augmented Dickey Fuller test

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Jarque-Bera is a normality test, calculating if the data has the skewness and kurtosis according to the normal distribution. Skewness is the degree of asymmetry of the data, if there is excess above or below average observations. The standard normal distribution has no skewness, hence the average and median are both 0. Kurtosis is the degree of fat tails, the probability of particularly high or low observations. It is measure by kurtosis in excess to the normal distribution. The null hypothesis is the lack of skewness and excess kurtosis, i.e. the data fitting the description of a normal distribution.

The test is it applied to see if the month correction makes the data seem less normally distributed. With volatility clusters, one should expect to find indication of systematic outliers, which goes against the assumptions of a normal distribution.

It is not an explicit heteroskedasticity test. But as normality is not compatible with this characteristic, one should expect to find an absence of normality for processes with volatility clusters.

The test statistic is derived from:

X_norm² =nκˆ²₃

6 +nκˆ²₄ 24

where κ3 is skewness and κ4 is excess kurtosis. The higher this statistic is, the weaker is the indication of normal distribution.

The observable housing price growth rates fits the notion of normality well. This would imply that housing price growth could be a normally distributed stochastic process. However, testing on the linear model residuals, the null hypothesis can not be rejected, with the exception of Vest-Agder and Sogn og Fjordane. This lack of normality is reasonable as housing price growth are not expected to be normally distributed. Comparing the test results of observable housing price growth and the linear model residuals indicates that the month effects correction have made the volatility more variant. This underlines the importance of

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regressing away these effects.

But the results does not show the same regional differences as the estimated GARCH model. The GARCH model applied on the residuals suggests that, in addition to Sogn og Fjordane, five other counties did not have clear volatility clusters. Also, Vest-Agder does not show indication of ARCH effects here, despite previously having fluctuating estimated volatility functions.

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Observable housing Linear model price growth residuals X² p-value X² p-value

Østfold 16.04 0.00 0.43 0.81

Akershus 21.68 0.00 0.02 0.99

Oslo 13.64 0.00 0.01 0.99

Hedmark 9.73 0.01 0.28 0.87

Oppland 11.63 0.00 2.02 0.36

Buskerud 16.44 0.00 0.1 0.95

Vestfold 14.50 0.00 0.61 0.74

Telemark 12.52 0.00 0.96 0.62

Aust-Agder 13.21 0.00 1.06 0.59

Vest-Agder 16.22 0.00 14.79 0.00

Rogaland 21.29 0.00 0.12 0.94

Hordaland 14.76 0.00 0.11 0.94

Sogn og Fjordane 25.79 0.00 49.7 0.00

Møre og Romsdal 19.39 0.00 0.7 0.70

Sør-Trøndelag 14.18 0.00 0.78 0.68

Nord-Trøndelag 15.04 0.00 1.81 0.41

Nordland 13.05 0.00 1.52 0.47

Troms 10.80 0.00 0.3 0.86

Finnmark 17.63 0.00 4.96 0.08

Table 4.4: Jarque-Bera normality test

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ARCH LM test calculates the presence of ARCH effects in the time series. It was introduced along with the initial description of the ARCH model in Engle (1982). This is a more explicit test of indication of ARCH effect than the Jarque-Bera test.

The null hypothesis is zero serial autocorrelation of the variance, indicating white noise.

By rejecting this hypothesis, there is an indication of heteroskedasticity in the ARCH-sense.

As with the Jarque-Bera test, the results conform to previous findings with respect to highlighting the role month specific effects. The null hypothesis of an absence of ARCH effects cannot be rejected for most the observable price growth rates.

The counties with the most prominent ARCH effects have in general lower p-values, but the results are far from overlapping perfectly. For instance, Hedmark, Telemark and Aust- Agder seem to reject the null of no ARCH effect after correcting for month specific effects.

Oslo, Hordaland and Rogaland seem to not reject the test, despite having clear volatility spikes previously.

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Observable housing price growth rates Linear model residuals

χ² P-value χ² P-value χ² P-value χ² P-value 10 lags 10 lags 20 lags 20 lags 10 lags 10 lags 20 lags 20 lags

Østfold 31.3 0.00 84.5 0.000 17.3 0.07 23.6 0.26

Akershus 22.3 0.01 72.3 0.000 29.4 0.00 32.5 0.04

Oslo 32.6 0.00 61.5 0.000 20.0 0.03 27.5 0.12

Hedmark 30.3 0.00 75.4 0.000 6.0 0.81 15.6 0.74

Oppland 26.5 0.00 71.2 0.000 20.0 0.03 23.1 0.28

Buskerud 27.7 0.00 78.4 0.000 13.2 0.21 21.2 0.39

Vestfold 26.8 0.00 81.4 0.000 23.3 0.01 27.8 0.11

Telemark 21.0 0.02 69.6 0.000 8.6 0.57 15.2 0.77

Aust-Agder 16.7 0.08 54.8 0.000 12.3 0.27 26.2 0.16

Vest-Agder 16.9 0.08 65.5 0.000 10.4 0.41 22.1 0.33

Rogaland 17.4 0.07 65.3 0.000 23.4 0.01 30.2 0.07

Hordaland 34.1 0.00 65.9 0.000 26.5 0.00 33.7 0.03

Sogn og Fjordane 14.6 0.15 47.5 0.001 29.8 0.00 32.6 0.04

Møre og Romsdal 22.3 0.01 70.4 0.000 15.7 0.11 24.2 0.23

Sør-Trøndelag 33.1 0.00 71.4 0.000 28.3 0.00 39.1 0.01

Nord-Trøndelag 18.2 0.05 60.7 0.000 20.1 0.03 21.4 0.37

Nordland 22.1 0.01 66.4 0.000 9.8 0.46 20.7 0.42

Troms 30.6 0.00 66.2 0.000 19.0 0.04 26.5 0.15

Finnmark 25.4 0.00 50.5 0.000 16.1 0.10 39.5 0.01

Table 4.5: ARCH LM test

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Restricting the sample period to the first or last 100 observations for each county does not alter the main findings.

Three different sample periods are used. As the linear model needs a 12th lag, the first observation is in February 2004. The main specification uses the entire time span, resulting in 131 observations for each county. The two other samples use the 112 first and last time periods, resulting in 100 observations for each county. By comparing these, one may find the extent to which the coefficients are invariant across time.

The results of this specification is in line with previous findings. Østfold, Hedmark, Tele- mark, Aust-Agder, Sogn og Fjordane and Nord-Trøndelag still have the weakest indication of ARCH effects as the β’s are close to 1.

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All observations First 100 observations Last 100 observations

α β α+β α β α + β α β α+β

Østfold 0.000 0.999 0.999 0.000 0.969 0.969 0.000 0.998 0.998 Akershus 0.094 0.839 0.934 0.098 0.817 0.915 0.073 0.881 0.954 Oslo 0.169 0.762 0.931 0.151 0.820 0.970 0.143 0.747 0.891 Hedmark 0.000 0.999 0.999 0.000 0.964 0.964 0.000 0.998 0.998 Oppland 0.104 0.768 0.872 0.130 0.736 0.866 0.109 0.699 0.809 Buskerud 0.048 0.854 0.902 0.049 0.834 0.883 0.000 0.997 0.997 Vestfold 0.070 0.865 0.935 0.071 0.835 0.906 0.083 0.842 0.925 Telemark 0.000 0.999 0.999 0.000 0.967 0.967 0.047 0.884 0.931 Aust-Agder 0.000 0.998 0.998 0.000 0.999 0.999 0.000 0.997 0.997 Vest-Agder 0.079 0.879 0.957 0.067 0.834 0.901 0.053 0.922 0.975 Rogaland 0.113 0.821 0.934 0.060 0.856 0.915 0.068 0.894 0.962 Hordaland 0.170 0.780 0.950 0.163 0.779 0.942 0.117 0.847 0.964 Sogn og Fjordane 0.000 0.997 0.997 0.000 0.997 0.997 0.000 0.999 0.999 Møre og Romsdal 0.078 0.862 0.940 0.054 0.846 0.900 0.094 0.851 0.945 Sør-Trøndelag 0.161 0.739 0.901 0.152 0.759 0.910 0.217 0.674 0.891 Nord-Trøndelag 0.000 0.998 0.998 0.000 0.999 0.999 0.000 0.997 0.997 Nordland 0.057 0.908 0.965 0.000 0.999 0.999 0.030 0.955 0.985 Troms 0.073 0.833 0.906 0.090 0.779 0.869 0.094 0.797 0.891 Finnmark 0.077 0.901 0.978 0.148 0.639 0.787 0.014 0.980 0.994

Table 4.6: GARCH coefficients of shorter samples

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4.3.4 Specification with monthly dummies

In this specification, monthly dummies are applied instead of the 12th lag of th AR process.

This means that eleven variables are added to the linear model, with each one of them having the value 1 for a month and 0 otherwise. These values captures the effect of being in each month, highlighting other effects affecting volatility. Then the residuals are analyzed with the GARCH model.

Dummies are an alternative method to correcting for month specific effects. As expected, the results does not significantly differ from those from the model with the 12th lag correction.

The similarity of the results strengthen the hypothesis that the month specific effects are dampen indication of volatility clusters.

An important difference between the methods of correcting for month specific effects is that dummies does not need lags, which increases the sampling period. When using dummies, the first 11 observations are also valid as observations of the house price growth. This is an advantage as the number of observations for each county increases. Another important difference is that the 12th lag model only look at the previous instance of a month, while the dummy captures the month specific effect for each month from all years. Despite the differences, the models broadly attains the same results.

As all the dummies are valued at 0 the first time period, the intercept is interpreted as the price growth associated with February. This intercept is negative for all counties, indicating that prices usually fall this month. The coefficients are interpreted as the difference between the given month and the February reference point. On average, growth rates tend to be positive June throughout October and otherwise negative, with the exception of January.

There could be many reasons for this, such as the higher propensity to buy new homes during the summer and after Christmas due to more spare time. Further analysis of results from this model is beyond the scope of this thesis.

The intercept is the month specific effect of February, M₂ is the dummy for March and

Volatility Modeling of Regional Norwegian Housing Prices