Predicting a Crisis Norwegian GDP in the Growth-at-Risk Framework

Predicting a Crisis

Norwegian GDP in the Growth-at-Risk Framework

Maja Olderskog Albertsen

Thesis submitted for the degree of Master of Economics

30 credits

Department of Economics Faculty of Social Sciences

University of Oslo

May 2020

Predicting a Crisis

Norwegian GDP in the Growth-at-Risk Framework

Maja Olderskog Albertsen

c 2020 Maja Olderskog Albertsen

Predicting a Crisis. Norwegian GDP in the Growth-at-Risk Framework.

http:/www.duo.uio.no

Printed: Reprosentralen, University of Oslo

Acknowledgements

Writing this thesis has been much more fun than expected. I thoroughly enjoyed finally having full responsibility over the outcome, and the urge to always understand more that came with it. The practical experience of using so much of what I’ve learned has been invaluable. I have many people I wish to thank for their help in this process.

First and foremost, I would like to thank Mikkel Øien, for staying patient and ever supportive of me and my studies. Thank you for always believing in me, especially when I didn’t. I don’t know what I would have done without you, and your motivation as well as your proofreading has aided me through the entire degree, as well as this thesis.

A special thanks to my supervisor, Karsten R. Gerdrup in Norges Bank, who has been very patient with my endless questions, and provided excellent guidance. I would also like to thank Ragnar Nymoen, Mari Gjerdåker and Nina Larsson Midthjell, for taking the time to read my thesis and give feedback.

This thesis has been a part of the Norges Bank project "GDP-at-Risk", in the depart- ment for Financial Stability. Being included in this project has taught me a lot about research, coding and writing, and I want to thank Karsten R. Gerdup, Elif C. Arbatli-Saxegaard and Rønnaug M. Johansen for this. Although this thesis has been written in collaboration with Norges Bank, who provided both data and supervisor, any remaining errors are my own. All views of this thesis are my own, and do not necessarily reflect those of Norges Bank.

Abstract

In this thesis I apply the growth-at-risk framework to analyse how a component rep- resenting core financial indicators signals increasing tail risks in Norwegian medium-term GDP. I use principal component analysis to construct the component, which represents five core financial indicators for credit and debt service ratio in Norway, using data from 1975Q1-2019Q2. This component and the lagged dependent variable comprise my baseline model for GDP growth. I evaluate the performance by testing in-sample and out-of-sample predictions, and compare the results to actual observations of GDP, and test the probability integral transforms of the predictions. I find that the baseline model performs well in signalling changes in tail risks. I also make a comparison between the results for the quantile regression estimates of the baseline model, and the results of a model of the conditional mean (i.e., a regression model estimated by ordinary least squares). This allows for an informal evaluation of the added value of using quantile regression to examine changes in the lower tail predictions.

Keywords: Financial Stability, growth-at-risk, principal component analysis, quantile regression, GDP forecasting.

Contents

1 Introduction 1

2 Literature 3

2.1 Indicators of Financial Vulnerabilities . . . 3

2.2 Growth-at-Risk . . . 4

3 Empirical Methodology 6 3.1 Quantile Regression . . . 8

3.2 The Model . . . 11

3.3 Evaluation of Financial Indicators . . . 12

4 The Data Set 13 4.1 GDP . . . 13

4.2 Financial Indicators . . . 15

4.2.1 Conditional GDP distribution . . . 18

4.3 Principal Component Analysis . . . 20

5 Results 24 5.1 Model Choice . . . 24

5.2 Model Predictions . . . 28

5.3 Out-of-Sample Model Performance . . . 30

5.3.1 Probability Integral Transform . . . 32

6 Conclusion 34

References 36

Appendix A: Summary Statistics 39

Appendix B: Normality Tests 41

List of Figures

1 Annualised 3 year growth in GDP. 1975Q1 - 2016Q2 . . . 14

2 Standardised financial indicator time series . . . 17

3 Empirical conditional distributions for 3 year average GDP growth . . . 19

4 Eigenvalues of components . . . 21

5 Predicted time series of PC1 and PC2 . . . 23

6 Regression lines for PC1 of the model . . . 26

7 Empirical conditional distribution for 3 year average GDP growth . . . 27

8 In-sample principal component predictions for GDP at risk . . . 29

9 Timeseries for recursively constructed principal component 1 (PC1) . . . 31

10 Recursive out-of-sample quantile regression predictions for fifth percentile 3 year average GDP growth . . . 32 11 Probability integral transform of baseline model quantile regression predictions 34

List of Tables

1 Mean and median of a small sample . . . 7

2 Correlations between financial indicators . . . 18

3 Component 1 and 2 . . . 22

4 Coefficients and standard deviations of different models . . . 25

5 Summary statistics of the data . . . 39

6 Overview of data sources . . . 40

7 Shapiro-Wilk test of normality of GDP series . . . 41

8 Skewness and kurtosis test for normality . . . 41

1 Introduction

The financial crisis of 2007 provoked a shift in the attention of macro economists. Having previously focused on monetary policy almost exclusively, financial stability became increas- ingly acknowledged as a central part of macroprudential authorities’ responsibility. Identi- fying sources of risk to the financial system and the economy became vital for authorities wishing to ensure economic and financial stability. Recognizing rising vulnerabilities can help governments and policy makers correctly respond through regulation and surveillance, possibly preventing a crisis or cushioning the potential impact. Since the financial crisis, macroprudential policy has become an important policy area (Galati et al., 2013). This is reflected in the EU regulation for bank capital and liquidity, in particular the extensive use of capital buffers that can be increased or decreased to reflect current systemic risk ((ESRB), 2014).¹

Research on financial stability and crises following the financial crisis identified credit and asset prices as important financial indicators signalling rising financial vulnerabilities (Jordà et al., 2015). Credit-to-GDP ratios and debt service ratios were found to signal vulnerabilities steadily before crises, early and clearly enough to be dubbed early-warning indicators (Drehmann et al., 2014). These indicators are now monitored closely by macro- prudential policy makers attempting to avoid the next financial crisis. Research on risks to GDP has in recent years been expanded on in papers adopting quantile regression methods, conditioning GDP on financial indicators found to signal crises. These papers endeavour to analyse the movements in the prediction for the fifth percentile of the conditional distribution of GDP, the growth-at-risk. The fifth percentile of a distribution is the observation for which five percent of all observations in the sample have a lower or equal value. The 50^th percentile is the middle observation, or the median. Analysing the predicted value for the fifth percentile of GDP growth, Adrian et al. (2019) found stronger effects from financial indicators on the lower tail (the fifth percentile) of the predicted GDP distribution, as compared to the same

1See the ESRB handbook for operationalising macroprudential policy in the banking sector for a descrip- tion of the tools for regulation ((ESRB), 2014)

indicators’ effects on the median prediction. This implies that the independent variables can signal build-ups in financial vulnerabilities. Several papers have since then applied the same method to analyse the predicted distribution of GDP.² This method allows for a more direct and targeted estimation of the different parts of the distribution of possible outcomes than e.g. ordinary least squares (OLS) (Koenker et al., 1978).

In this thesis, I discuss the use of quantile regression as a tool when predicting real 3-year growth in GDP for Norway conditional on financial indicators. The analysis will be performed on quarterly data on GDP and a set of financial indicators for Norway from 1975Q1 to 2019Q2. The group of independent variables contain 10 financial indicators found to signal build-ups in financial vulnerability (Arbatli-Saxegaard et al., 2020). These indica- tors are a group of credit variables for households, non-financial enterprises and the private economy as a whole, as well as debt service ratio, real estate variables and equity prices. All 10 indicators are highly correlated with each other, so to simplify the analysis without losing important signals, I conduct a principal component analysis (PCA), which returns two rele- vant components, explaining an aggregate of approximately 80 percent of the total variation in the 10 variables. Although not yet commonly used in the growth-at-risk research, PCA is useful for extracting trends from large groups of indicators, and is conceptually explored in the growth-at-risk analysis guide by Prasad et al. (2019). With the components retained from this analysis, I determine a baseline model, based on one component and a lag of GDP.

Additionally, I examine the added value of using quantile regression for this purpose.

To do this, I compare the results and predicting powers of the quantile regression to an OLS regression based on the same baseline model. I do not test the difference between these results formally, but rather evaluate the individual performances of the model in the two methods in an in-sample and out-of-sample prediction. To test the robustness of the predictions, I follow Adrian et al. (2019) and compare the out-of-sample predictions to the in-sample predictions, and analyse the probability integral transforms of the quantile regression results. First, I find that my principal component is successful in capturing financial vulnerabilities by forecasting lower GDP growth prior to the financial crisis. Second, I find that the quantile regression

2See (Adrian et al., 2019), (Adrian et al., 2018), (Aikman et al., 2019) and (Arbatli-Saxegaard et al., 2020)

quantifies more downside risks, defined as the difference between the median forecast and the fifth percentile, ahead of this episode. This is not a feature that my simple OLS allows for. The estimations in this thesis have been performed in the statistical program STATA 16, and can be provided on request.

The thesis will proceed in the following manner. Section 2 will discuss the relevant literature preceding this thesis, and the contribution of the thesis to the existing literature.

Section 3 discusses the empirical methodology, and gives a simplified explanation of quantile regression as a method. Section 4 gives a description of the data and the choices made when choosing explanatory variables, as well as the entire principal component analysis.

In section 5 I choose which components to retain in my baseline model, and evaluate the performance of said model. Finally, in section 6 I conclude and discuss relevant continuations and modifications to the analysis conducted.

2 Literature

2.1 Indicators of Financial Vulnerabilities

The literature on financial stability and risk is vast, with macro economists world-wide at- tempting to make financial systems ever more robust by analysing previous economic down- turns. The prediction of crises, and identification of building risks in the financial systems has become an important part of central banking, which is reflected by the research done by central banks and macroprudential institutions, such as the European Systemic Risk Board (ESRB) summarising the key instruments for macroprudential policy and a frame- work ((ESRB), 2014).

Research following the financial crisis found building financial imbalances to be leading predictors of financial crises, with especially credit having a tendency to be high in periods before banking crises, as found by Schularick et al. (2012) in their panel study of 14 countries.

Drehmann et al. (2014) found credit-to-GDP ratios and debt service ratio to be particularly

strong early-warning indicators for banking crises, with persistent signals of rising vulnera- bilities arriving early enough for macroprudential authorities to act. Analysing a panel of countries using data for 140 years, Jordà et al. (2015) found that the combination of credit and asset prices together significantly increase the chance of a crisis, more so than either type of indicator on its own. Periods of both bubble-like behaviour in asset prices and rising credit were found to lead into deeper and more prolonged recessions (Jordà et al., 2015).

Findings such as these have shaped the way macroprudential policy-makers operate, and guided the financial indicators inspected in financial stability risk analyses. In this thesis, financial indicators refer to variables such as real credit, credit-to-GDP ratios, debt service ratio and asset prices.

2.2 Growth-at-Risk

The growth-at-risk literature originates from the literature on financial vulnerabilities, but separates itself in the choice of econometric method, namely quantile regression. It concerns itself with analysing lower tail predictions for GDP, or some other measure for economic activity. The literature on quantile regression is, compared to most other statistical methods, quite recent. Koenker and Bassett first proposed a theory for regression quantiles in 1978 (Koenker et al., 1978), commencing the use of quantile regression as an alternative to the ordinary least squares approach. Since then, several authors have expanded on the subject and theory, many of the main works produced by or in collaboration with Robert Koenker (see e.g. (Koenker et al., 2006), (Ghysels, 2014), (Koenker, 2015), (Schmidt et al., 2016) and (Yu et al., 2003)). Although new, quantile regression stems from median regression, a much older theory. In 1760, Ruđer Josip Bošković, posed a problem with least squares regression to Thomas Simpson, who solved it using the weighted median, as opposed to the typically used mean (Stigler, 1984). This is the first known instance of using the median in the fitting of a regression line. For a gaussian normal distribution, the least squares estimator has been shown to be the most efficient unbiased estimator (Stock et al., 2014).

However, as the mean is very sensitive towards outliers it is a poor estimator in situations of non-gaussian distribution (Koenker et al., 1978). The need for a robust alternative was

recognized, and while the theory behind median regression began long before, it was not until the theory for linear programming and later the production of powerful computers that median regression was possible in practice, as calculations were tedious even for small samples and models (Leknes, 2008). Nearly 200 years after the first theories on the subject, it was finally possible to use the method in practice. Shortly thereafter, Koenker et al. (1978) introduced quantile regression, using the concepts applied for calculating the median for the calculation of any percentile in a predicted distribution. This allowed researchers to examine distributions without making assumptions about the variance, or even the full distribution of the disturbances, thus allowing for non-linear associations in regression.

In recent years the growth-at-risk topic has developed, merging the use of quantile regression to examine distributions, and the analysis of risk to GDP through financial indi- cators, specifically focusing on the lower tails of the distributions (growth-at-risk). Adrian et al. (2019) used quantile regression to predict distributions of GDP conditional on financial indicators, analysing their predicting abilities on upcoming crises in GDP, as well as the development of the distribution over time. They found that financial indicators had stronger effects on the lower tail of the distribution of the prediction for GDP, and measured this by specifically looking at the predictions for the fifth percentile (Adrian et al., 2019). The fifth percentile of the distribution³ had a different development from the median, and they found that downside risks⁴ increased in the periods preceding the financial crisis of 2007 (Adrian et al., 2019). Giglio et al.(2016) constructed a systemic risk index and analysed how systemic risk and developments in the financial market affects the distribution of shocks to GDP. They analysed how fluctuations in systemic risk affect probabilities of recession, and found that increases in their systemic risk index were associated with a widening of the left tail of the prediction for GDP (Giglio et al., 2016). Expanding on these papers, Adrian et al. (2018) considered a panel of 22 countries, and found that the conditional distribution of GDP relied upon financial conditions, and that growth-at-risk responded more strongly than the median and upper percentiles to change in financial conditions (Adrian et al., 2018).

3Commonly referred to as growth-at-risk

4The difference between the prediction for the median and the prediction for the fifth percentile. 50^th-5^th.

Prasad et al. (2019) review the methods and procedure of growth-at-risk, providing practical guidance on the process of this analysis. They argue that "a quantile approach is appropriate for evaluating the potentially asymmetric and non-linear association between systemic risks and the macroeconomy" (Prasad et al., p. 458). This non-linearity has been pointed out in previous financial stability and systemic risk literature, by among others Bernanke et al. (1998), Gertler et al. (2010) and He et al. (2012). In 2020, Norges Bank published a staff memo applying the framework of Growth-at-risk for a panel of 22 countries, focusing on the implications for Norwegian mainland GDP growth in the medium term (Arbatli-Saxegaard et al., 2020).

The contribution of this thesis to the existing literature is twofold. First, I perform a principal component analysis to construct a financial indicator for Norway, representing the variation in five early warning indicators.⁵ I then evaluate the financial indicators per- formance in predicting building financial vulnerabilities. To the best of my knowledge, this thesis is the first to do this for Norway. The second contribution is that of comparing the performance of my baseline model, containing lagged GDP, and the financial indicator, us- ing both quantile regression and a simple OLS regression. Quantile regression is a slower, more laborious analysis for both the economist and the computer involved, so to justify the extra work and time going in to an analysis of this type, the quantile regression predictions should do a better job than OLS regression. I check this by comparing in-sample predictions, out-of-sample predictions and coefficient estimates.

3 Empirical Methodology

The mean and median are two different ways of measuring the center of a data set. Quantile regression uses the median as its central measure, whereas OLS regression uses the mean.

This choice has implications for the analysis, since the two measures often result in different expectations about the center of the population. Take the following example:

5real private credit, private credit-to-GDP ratio, real household credit, household credit-to-GDP ratio and household debt service ratio

- Sample A : 4, 6, 6, 7, 7, 9 - Sample B : 4, 6, 6, 7, 7, 16

Table 1: Mean and median of a small sample

Mean Median Sample A 6.5 6.5 Sample B 7.67 6.5

Table 1 illustrates the sensitivity of the mean towards outliers. Sample A and sample B are identical, apart from one observation. Conveniently, the mean and median for sample A are the same. This is because the sample is symmetrically distributed. The median for sample B is also 6.5, but the mean is now 7,67. In other words, a person using the mean as a measure of the center of a distribution expects most of the observations from this population to have values around 7,67. The fact is however that our sample tells quite a different story.

All of the observations in sample B are lower than 7.67, except for one. This illustrates the point that the median is more robust towards outliers than the mean. The mean is very sensitive towards large outliers, and just a few outlying observations can significantly skew the mean. Linear regression models are usually mean based regressions. In this paper I will base my comparison of quantile regression vs linear regression with ordinary least squares (OLS) estimation, as it is the most commonly known and used method.

OLS has the same problem with outliers as the mean, because it uses the mean Y¯, or the conditional expectation E[Y|X =x], when minimising the expected square loss function E[(Y −Y¯)²|X =x](Yu et al., 2003).

n

P

i=1

r(yi−x^T_i β)is the sample estimate of the mean (Yu et al., 2003). A small number of outliers could shift the mean considerably, and consequently the estimated coefficients (beta). Another disadvantage of the OLS regression is that it assumes and requires normally distributed and independent error terms and homoskedastic variance. It is important to note that OLS can account for heteroskedastic variance under the condition that the change in the variance is symmetric (Stock et al., 2014). This can

for example mean that the uncertainty of the estimate increases in both directions, meaning that the direction of the risk is not specified. In cases where either of these assumptions fail, OLS estimation may no longer be the most efficient estimator (Stock et al., 2014). For example, if the increase in a financial indicator leads to an increased risk of lower values to GDP growth, this implies a change in the variance that is not symmetric, as we are not equally uncertain of a sudden boom in GDP as we are of a decline. When one suspects one or more of the requirements set for OLS estimation are violated, alternative analysis tools may be relevant and useful. When one cannot reasonably assume normal distribution of the error term, it is of interest to examine the separate quantiles of a distribution. Two different distributions may have the same expected value (mean) and the same standard deviation, but different skewness. OLS and other mean based regression analysis would not be able to pick up on this difference (Leknes, 2008). In such cases quantile regression can be a good alternative. Quantile regression makes no assumptions with respect to the error terms’ variance or distribution, but assumes that the expected value of the error term for each quantile equals 0 (Koenker et al., 1978). This thesis will assume that the reader is familiar with OLS regression, and not go into further detail on this method. The focus will be to explain the less familiar method of quantile regression.

3.1 Quantile Regression

So what does a quantile regression do? It is commonly applied in treatment effect analysis, where the researcher recognises that the sample affected by the treatment can be divided into different groups, or quantiles. Consider for example a class of children in the third grade of primary school. What are the effects on the students’ mathematical skills of hiring another teacher to help in the mathematics classes? Perhaps the children who were in no need of particular help in maths have very little positive effect from this new teacher, while the weaker students have a great benefit from it. Regressing this effect using mean based regression would return a coefficient somewhere between these two effects. However, this does not really show the full picture. Quantile regression allows for dividing the class into groups, and studying the effect of the new teacher. In this example, the regression would

perhaps return a strong coefficient for the lower percentiles, a medium coefficient for the median student, and a weak coefficient for the stronger students. One could argue this gives a more nuanced picture of the situation.⁶

Quantile regression is a statistical tool to estimate and make inference about condi- tional quantile functions, meaning the estimated effect on each quantile separately. This takes into account the fact that movement in the explanatory variable may not affect the entire distribution equally. The empirical median is by definition the 0.50^th quantile, as half of all observations are greater than it, and the other half smaller. Quantile regression pro- vides estimates for the full range of conditional quantile functions (Koenker et al., 1978), to which the same definitions apply. The 0.25^th quantile is the number where 25 percent of the observations of the sample or population are smaller than or equal to said number, and 75 percent are larger. Estimating effects on each quantile of interest allows us to in- spect the effects of the explanatory variables on the distribution of the dependent variable.

While OLS regressions minimise the sum of squared residuals in order to estimate models for conditional mean functions, quantile regression offers a method for estimating models for conditional median functions by minimising the sum of absolute residuals. This means that outlying observations are not penalised as strongly as when all residuals are squared (as in OLS regression), which puts a larger weight on considerable deviations.

- Sample C : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Just like OLS regression however, quantile regression attempts to fit regression lines to a scatter plot, but in a way that is more robust to outliers, for a more robust measure of the centre of a distribution, as well as a measure of each selected quantile. Each quantile regression line will always represent a range of quantiles, but as the sample size increases, these ranges will narrow down. As a simple example of this, the uniformly distributed sample C only has 10 observations. The 0.10 quantile is the number 1, but the 0.05 quantile

6For a more formal and mathematical explanation than this thesis will provide, I recommend reading Koenker et al. (1978), Koenker (2015) or Koenker (2005). For an in depth explanation for people who are not statisticians, I recommend the masters thesis by Leknes(2008), which thoroughly reviews the basic statistics terms needed to understand quantile regression, and takes the reader through the mathematics of quantile regression.

is also the number 1. The number one covers quantiles in the range from 0 to 0.1. As sample sizes increase the range become shorter. In theory there are an infinite number of quantiles, depending only on which quantiles are of interest. It is however common to settle for somewhere between 1 (the median) and 19 quantiles (every fifth from the 5th to the 95th percentile)⁷. Sample C is however so simple that no calculations are really required.

Imagine instead a more complicated sample, with over 200 observations for two vari- ables over time, t. If we wanted to find the conditional distribution (Y|x), where Y can be predicted by the independent variable x, plus a constant-term, we can find the quantiles of the distribution using quantile regression. An equivalent linear regression model of this problem can be shown by Y_t = ˆα + ˆβx_t where αˆ is the estimated constant, and βˆ is the estimated coefficient for x on y. However, as we are using quantile regression to estimate the conditional distribution instead, equation (1) shows the calculation of each quantile (τ) for Y conditional on x.

Q_τ(Y|x) = ˆα_τ+ ˆβ_τx. (1) Finding each of the quantiles,τ, in the conditional distribution ofY is achieved by minimising the sum of weighted absolute errors (SWAE) with respect to αˆτ and βˆτ as shown in equation (2). The resulting coefficients for each quantile are then inserted back into equation (1) which returns the observation marking quantile τ.

argmin

ˆ ατ,βˆτ

SW AE = (1 − τ) X

yt<ατ+xtβτ

|y_t−αˆ_τ −βˆ_τx_t |+τ X

yt≥ατ+xtβτ

|y_t−αˆ_τ −βˆ_τx_t| (2) When minimising the sum of weigthed absolute errors (SWAE), for each quantile τ, the coefficients minimising the sum of absolute residuals are chosen. The equation weights ob- servations beneath the regression line by (1−τ), and observations above the regression line byτ. As an example, consider the fifth percentile, which is the focus of this thesis,τ = 0.05.

The weight 0.95 will be given to any observations where y_i < αˆ_τ +x_tβˆ_τ, while 0.05 is the weight for any observations where y ≥ αˆ_τ +x_tβˆ_τ. Meaning that the function penalises all

7Percentiles are the percentage of the distribution covered by a quantile. The 0.05 quantile is at the observation where 5% of the distribution is lower or equal to this observation, while 95% of the distribution is higher. It is called the fifth percentile.

observations below the line harshly, which makes sense when your looking for a number un- der which only approximately 5 percent of all observations should be located. All of the observations are weighted and summed together, and the optimisation finds the coefficients minimising this sum. These coefficients give us the resulting Q_τ(Y|x) through equation (1).

As you go through the selected quantiles up towards the highest you wish to define, a lower weight will be given to the observations beneath the regression line, and a higher weight will be given to the observations above the line. The median marks the point where observations above and below the regression line are weighted equally by 0.5.

After running the regression and having determined the coefficients signalling the effects of x on Y, one can make predictions, just like in regular linear regression. The predicted value for each quantile τ of Y, can be found in the following way:

E[y_t,τ] = ˆα_τ + ˆβ_τ ·x_t (3)

3.2 The Model

In the case of this thesis, quantile regression is applied to explore how changes in financial indicators affect the future medium-term real GDP growth distribution. Medium term is here defined as a 3 year horizon. The dependent variable of my analysis is the future 3 year average annual growth in real Norwegian GDP, ∆y_t, in order to focus on persistent declines in growth. It is defined by equation(4):

∆y_t=

GDPt+12−GDP_t GDPt

3 ∗100 (4)

Thus examining the medium term growth in real GDP, which is the horizon of interest when trying to gauge the level of risk to financial stability (Prasad et al., 2019). Macroprudential authorities need time to implement rules and regulations set in place to mitigate build-ups in financial vulnerabilities. The dependent variable ∆y_t is a leading indicator, while all explanatory variables will be lagged indicators, meaning that X_t is based on data up until period t.⁸ This will allow for predictions about future real GDP growth, without requiring

8More on the transformations of the variables in section 4.

projections for financial indicators. From a policy and regulatory point of view, this is more useful than contemporaneous regressions (Giglio et al., 2016).

The models estimated in this thesis can be generalized by equation (5), showing the predicted real GDP growth distribution for each quantile τ. The vector of explanatory variables, X_t = [F I_t C_t], is composed of a component for financial indicators (F I_t) and a macroeconomic control variable (C_t). These will all also be medium-term variables.

∆y_t,τ = ˆα_τ + ˆβ_τX_t (5) In equation (5), ∆y_t,τ is the estimated quantile τ of the dependent variable, for each time period t. αˆτ is a constant. Both αˆτ and βˆτ can vary over quantiles, allowing the quantile models to give a rich picture of the distribution of ∆y_t, as the conditioning data shifts more than just the location of the distribution (Giglio et al., 2016). The coefficients vectors, αˆτ

and βˆ_τ, are estimated by the SWAE equation (2), simply by running the optimisation for more coefficients.

3.3 Evaluation of Financial Indicators

As a way to ensure the value added of quantile regression as a method, I will evaluate the financial indicators’ performance as explanatory variables using the same criterion as used by Arbatli-Saxegaard et al. (2020):

1. Financial indicators have a significant negative effect on the fifth percentile (growth- at-risk, or tail risks) of the growth distribution.

2. Financial indicators have a larger effect on the fifth percentile of the distribution than on the median (50^th percentile). Under this criterion financial indicators do not simply shift the entire growth distribution; they also have an effect on the size of the left tail and lead to an increase in downside risks.

This implies growth-at-risk coefficients with larger values ⁹ than the coefficients for the me-

9In absolute terms.

dian, and thus a steeper downward sloping regression line for growth-at-risk as compared with that for the median. For forecasting this implies a change in the tail of the forecast distribution, meaning the distance from the fifth percentile to the median. In-sample pre- dictions will be evaluated on its performance in signalling historic episodes of considerable downturns in real Norwegian GDP. A good model would see a decline in its fifth percentile, or in other words longer downside tails in the periods preceding a crisis. Main examples of such events in Norway being the recent financial crisis (2008-09) and the banking crisis (1988-93) (Arbatli-Saxegaard et al., 2020). Out-of-sample predictions will be judged similarly to the in-sample predictions. Quantile regression predictions and coefficients will be compared to the coefficients and predictions made by the corresponding OLS model. If there is indeed any gain from using quantile regression for the analysis, one would expect more precise predic- tion from the quantile regression than from the OLS. This is due to OLS regression’s equal coefficients across percentiles, and thus symmetrical distribution across all predictions. If these criterion are not met, it may mean that there is no added value from using quantile regression instead of OLS based time series regression.

4 The Data Set

This section will provide details about the dataset, motivation behind the choice of variables, and some properties of the data, before conducting a principal component analysis. The dataset is comprised of Norwegian data for the period 1975Q1 - 2019Q2. Details on sources of the data and summary statistics are provided in Appendix A.

4.1 GDP

The dependent variable in this thesis is Norwegian mainland GDP, which consists of Nor- wegian gross domestic product excluding shipping and petroleum. This is used because it is a measure of economic activity which is more relevant than total GDP, because a large share of Norway’s total GDP consists of its oil sector. The oil sector is very volatile, and to

Figure 1: Annualised 3 year growth in GDP. 1975Q1 - 2016Q2

8

6

4

2

0

-2 8

6

4

2

0

1975q1-2 1985q1 1995q1 2005q1 2015q1

a large extent outside macroprudential authorities influence. From 1978Q1 the series from Statistics Norway and Norges Bank is used. However, data for real total Norwegian GDP from 1970Q1 is available from the OECD. To merge these two series, the growth rate of the OECD data series is calculated, and applied backwards from 1978Q1 on the main series, thus assuming the same growth rates in mainland GDP as in total. Data for nominal GDP, used to construct credit to GDP ratios, did on the other hand not require such a merging. Data for this series was collected from Statistics Norway and Norges Bank.

Historically, Norway has seen a number of major crises resulting in considerable de- clines in GDP growth, as shown by Riiser (2005) in her analysis of Norwegian banking crises using data from 1819 to 2004. The data available to me for this analysis does, however, not include most of these, as one can see in figure 1. It does, however, include the two most recent severe crises of Norwegian history, namely the banking crisis of 1988-93 and the finan- cial crisis of 2008-09. ¹⁰ The banking crisis is the most severe crisis in the sample, with a prolonged negative growth in 3-year GDP growth for almost 2 years.

10Dates referencing the duration of these crises from a Norwegian perspective. For example, the financial crisis of 2008 hit several other countries earlier than Norway, and lasted longer.

4.2 Financial Indicators

The choice of financial indicators for this thesis is influenced by the existing growth-at-risk literature, with particular emphasis on the paper by Arbatli-Saxegaard et al. (2020). They use a broad set of indicators, spanning credit and asset price developments. Both credit and asset prices have been identified as key drivers of the financial cycle (Drehmann, Borio, et al., 2012). Specifically for Norway, Riiser (2005) finds that house prices, real equity prices and credit were useful in predicting banking crises over the last 150 years . Moreover, Gerdrup (2003) studies three major banking crises in Norway’s history (1899-1905, 1920-28 and 1988-92), and finds that development in the financial sector seem to be closely linked with booms and recessions in economic activity for the dataset spanning over 130 years.

Gerdrup (2003) found that the periods of economic upturn preceding each of the three crises have some features in common: each were characterized by considerable bank expansion, high asset price inflation and increased indebtedness. Lastly, Anundsen et al. (2016) find that the probability of a crisis increases considerably when bubble-like behavior in house prices coincides with high household leverage. Specifically analysing Growth-at-Risk, several papers have found that credit and property price booms pose considerable downside risks to 3-year growth in real GDP (Aikman et al., 2019) (Adrian et al., 2019) (Adrian et al., 2018).

The financial indicators included in my analysis are based on the findings of these papers.

The financial indicators have been transformed either by annualised average change in equation (6) or annualised average growth in equation (7), over 5 or 3 years, depending on their performance when predicting medium-term risks to real GDP growth.

∆X_t= X_t−Xt−h

h/4 (6)

∆Xt = X_t−X_t−h Xt−h

∗100 (7)

Where his the number of quarters set as horizon for the variable. Data for annualised 3 year average growth of any variable in 1975Q1 is based on data from 1972Q1 to 1975Q1 when available. Similarly, annualised 5 year average growth for any variable in 1975Q1 is based on data from 1970Q1 to 1975Q1 when available. Some of the financial indicators have shorter

data samples, and are regrettably not available for the entire period. The financial indicators are the following:

Real equity price growth is a data series collected from Thomson Reuters Datas- tream, showing real equity prices from the Oslo stock exchange. Real equity price growth performed best when using a 3 year horizon, and this transformation was applied as described by equation (7).

Real private creditandreal household creditwere constructed using their respec- tive nominal series, divided by average 4-quarter consumer price index. They both performed best using 5 year growth, defined by equation (7).

Non-financial enterprise (NFE) credit to GDP ratio and real non-financial enterprise credit both performed best on a 3 year horizon, which is shorter than all the other credit variables. Therefore, these financial indicators were applied to the analysis, with transformations 3 year change and 3 year growth respectively. See equations (6) and (7).

Private credit to GDP ratio and household credit to GDP ratio are ratios of credit to GDP. They were constructed taking each of their respective nominal series, dividing by nominal GDP and multiplying by 100. Both financial indicators performed best when using 5 year change, as defined in equation (6).

Drehmann and Juselius (2012) found that debt service costs have an important role as an early warning indicator of impending systemic banking crises. Household debt service ratio(DSR) is defined as interest expenses and estimated principal payments on an 18-year mortgage as a percentage of after-tax income, while taking into account tax deductibility for interest expenses and the variation in this over time(Arbatli-Saxegaard et al., 2020).

Household DSR performs best in predicting risk to real GDP growth when defined as 5 year change as defined in equation (6)

In Norway, around 77 percent of all households own their own home, while the rest of the households rent (Statistics Norway, 2017). House prices are thus an important part of the wealth of most households, and is important to the banks, as housing is used as

collateral. Rising house prices lead to a higher share of income being spent on housing.

Anundsen et al. (2016) found evidence of bubble-like behaviour in the credit and household markets, strikingly so when accompanying periods of high household leverage. An increase in this ratio could signal a forthcoming bursting of the bubble, leading to a weakening of GDP growth through a decrease in household wealth and a decrease of the value of bank’s collateral (Anundsen et al., 2016). House price to disposable income ratiois therefore included in the analysis. This financial indicator performs best when using 5 year change as defined in equation (6).

Loans to commercial real estate make up a considerable part of Norwegian bank loans in Norway, and was also included by (Arbatli-Saxegaard et al., 2020) and found to be significant in their GDP-at-risk analysis of Norway. Commercial real estate prices is therefore included in the analysis. For this financial indicator, 3 year growth proved to be the most efficient horizon and transformation. See equation (7).

Figure 2: Standardised financial indicator time series

-4 -2 0 2 4

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter

Real Equity Prices House Price-to-Income Commercial Real Estate Prices Real Private Credit Private Credit to GDP Real Household Credit Household Credit to GDP Household DSR

Real NFE Credit NFE Credit to GDP

To compare results, coefficients and time series across indicators, all of the financial indicators are standardised, meaning that the mean is set equal to 0, and standard deviations set to 1. Figure 2 plots the standardised time series of all financial indicators. The figure shows clearly that there is a large degree of co-movement among the indicators. It also shows that most indicators increase considerably in the periods preceding GDP crises, and fall from

the beginning of the crises. This gives a clear indication of the presumed predicting abilities on risk to GDP growth of these financial indicators. Furthermore, table 2 shows a high degree of correlation between most indicators, except for real equity prices, which both from figure 2 and table 2 seem to co-move mostly with commercial real estate prices.

Not all variation in real GDP growth can be explained by financial indicators, so to avoid omitted variable bias, I include laggedaverage annual 3-year growth in real GDP as a control variable in every quantile regression. The GDP control variable is defined by equation (8), and is the dependent growth variable, lagged 12 quarters.

∆GDP_t= GDP_t−GDPt−12

GDPt−12

∗100 (8)

Table 2: Correlations between financial indicators

Correlation between 1 2 3 4 5 6 7 8 9 10

standardised time series

1 Real Equity Price 1

2 House Price/Disp. Income 0,20 1

3 Commercial Real Estate 0,66 0,42 1

4 Real Private Credit 0,08 0,71 0,30 1

5 Private Credit/GDP -0,12 0,48 0,05 0,85 1

6 Real Household Credit 0,25 0,59 0,35 0,88 0,81 1

7 Household Credit/GDP 0,05 0,30 0,08 0,64 0,88 0,83 1

8 Household Debt Service Ratio -0,04 0,29 0,07 0,64 0,83 0,79 0,89 1 9 Real Non-Financial Enterprise Credit 0,41 0,75 0,69 0,77 0,51 0,73 0,38 0,50 1 10 Non-Financial Enterprise Credit/GDP 0,19 0,71 0,46 0,78 0,69 0,74 0,56 0,71 0,90 1

Sample: 1975Q1 - 2019Q2. Transformations as explained in chapter 4 or in table 5 in Appendix A.

4.2.1 Conditional GDP distribution

Conditioning the distribution of 3-year GDP growth on a characteristic of an explanatory variable is done by fitting a kernel distribution to all observations of GDP, given that the cri- teria set is complied with. For example, GDP conditional on total private credit being above

average is found by only taking observations for GDP in the periods when standardised total private credit was higher than 0. All observations for GDP in the periods when standardised total private credit was below its mean are not included in the distribution. The distribution is made using the Epanechnikov kernel density estimation.

Figure 3: Empirical conditional distributions for 3 year average GDP growth

0 .1 .2 .3 .4 .5 .6

Density

-4 -3 -2 -1 0 1 2 3 4

Standard deviation Unconditional distribution

Conditional on high real equity prices (3 year change) Conditional on high house prices/ disp. income (3 year change) Conditional on high real CRE prices (3 year change)

(a) Asset prices

0 .1 .2 .3 .4 .5 .6

Density

-4 -3 -2 -1 0 1 2 3 4

Standard deviation Unconditional distribution

Conditional on high real total credit (3 year change) Conditional on high total credit/GDP (3 year change)

(b) Total credit

0 .1 .2 .3 .4 .5 .6

Density

-4 -3 -2 -1 0 1 2 3 4

Standard deviation Unconditional distribution

Conditional on high real HH credit (3 year change) Conditional on high HH credit/GDP (3 year change) Conditional on high HH debt service ratio (3 year change)

(c) Household indicators

0 .1 .2 .3 .4 .5 .6

Density

-4 -3 -2 -1 0 1 2 3 4

Standard deviation Unconditional distribution

Conditional on high real NFE credit (3 year change) Conditional on high NFE credit/GDP (3 year change)

(d) Non-financial credit

Distributions are estimated using a kernel smoother. GDP growth and financial indicators are standardised. "High" defined as above 0 (the mean).

Figure 3 shows the empirical distribution of 3 year average GDP growth, compared to the same variable’s conditional distribution when conditioning on each of the financial indi- cators being higher than its historical mean. Both the financial indicators and the dependent

variable have here been standardised. The empirical unconditional distribution shows a slightly left skewed distribution. The empirical distribution for mainland GDP for Norway is in fact significantly different from normally distributed, as seen from Table 7 and Table 8 in Appendix B, where the null hypothesis of normal distribution is rejected. Consequently, the OLS assumption of normality is violated, and OLS estimation is not necessarily the "best"

estimation. As previously mentioned, quantile regression makes no assumption with respect to the distribution of the data, and does not require normality.

The unconditional empirical distributions display that higher real equity prices and commercial real estate prices are associated with noticeably higher tail risks (growth-at-risk) but also shifts the mode downwards. House prices to disposable income also leads to higher tail risks, but does not seem to shift the mode much. Meaning that higher values of this indicator does not shift the entire distribution to the left, but rather flattens the curve, giving higher values of GDP-at-risk, while not changing upside tail-risks much. The same can be seen for most credit indicators. This implies that criterion 1 and 2 from section 3 could be met by the financial indicators, an increase in financial indicators could affect the fifth percentile of GDP growth more than the median.

4.3 Principal Component Analysis

The first contribution of this thesis lies in its construction of a set of components, using principal component analysis. This is unique in the growth-at-risk literature for Norway. As I have limited this thesis to only examine Norwegian data, I am restricted in the number of explanatory variables I can include in a multivariate regression. However, as a way to circumvent this restriction, I will use principal component analysis (PCA) as a data reduction tool to acquire one or two principal components to represent the core variation of the financial indicators.

Principal component analysis (PCA) attempts to capture the essential patterns of a matrix of data. Say we have a matrix with n rows of observations, for m rows of vari-

ables. Correspondingly, we would have n*m observations. PCA then generates a number of components¹¹ which give alternative representations to the data. Together, all the compo- nents encapsulate all the variation in the matrix, where some components are less important than others, and PCA ranks the components from most important (PC1) to least important (PCm). However, PCA is not meaningful for all data sets, and requires a large degree of correlation between the variables in the analysis (Wold et al., 1987). In order to more clearly see the variance of the variables in comparison to each other, they have all been standardised (Wold et al., 1987). This means that they all have a mean of 0, and a standard deviation of 1. Figure 2 shows the normalised time series for all the explanatory variables, which shows a pronounced co-movement of the variables, hinting of a certain degree of correlation between them. Table 2 confirms that the variables are highly correlated, and thus that the variables seem suitable for principal component analysis. In this case, PCA returns 10 components,

Figure 4: Eigenvalues of components

0 2 4 6

0 2 4 6 8 10

the number of explanatory variables in the analysis. However, a core assumption when using PCA is that the components with the largest eigenvalues contain the most useful information, and that the remaining components mostly contain noise (Wold et al., 1987). The objective of using PCA in this instance is to reduce the number of explanatory variables included in a multivariate quantile regression, without losing too much information. A common rule is

11PCA generates n or m components, whichever number is smallest.

Table 3: Component 1 and 2

Variable Component 1 Component 2 Unexplained

Real Equity Price 0.5112 0.3826

House Price/Disp. Income 0.3171 0.3918

Commercial Real Estate 0.5720 0.1829

Real Private Credit 0.3571 0.1569

Private Credit/GDP 0.4493 0.06843

Real Household Credit 0.3705 0.1105

Household Credit/GDP 0.4202 0.1952

Household Debt Service Ratio 0.4254 0.1692

Real Non-Financial Enterprise Credit 0.4200 0.08918

Non-Financial Enterprise Credit/GDP 0.17

Blanks spaces in eigenvectors for values < 0.3. Sample: 1985Q1 - 2019Q2. Transformations as explained in chapter 4 or in table 5 in Appendix 6

to retain all components with eigenvalues larger than unity. This implies that any retained component should account for at least as much variation as any of the original variables going in to the analysis. Another is to look for an "elbow" in the eigenvalue plot, which shows the eigenvalues of the correlation matrix in descending order. The eigenvalues are interpretable as the shares of total variation explained by each component, if you multiply each of the eigenvalues by 0.1. Meaning that component 1 (which has an eigenvalue of about 6) explains approximately 0.6 of the total variation in the matrix, where 1 would be explaining all the variation. Figure 4 shows that component 1 (PC1) and 2 (PC2) have eigenvalues above 1.

The elbow, where all following components add little more could either be at component 2 or 3. However, PC2 carries most of the variance in the asset price variables, see table 3. If only the first component was to be retained, important variation in asset prices would be lost. I have therefore decided to retain the two first components of the PCA, and further examine their usefulness in the regression.

Table 3 lists the variance explained by the two components, as well as the variance

left unexplained.¹² PC1 summarises the most important variation in the household variables and total private credit, while PC2 summarises the asset prices and non-financial enterprise credit. With the two components constructed, one can compute time series based on the weights assigned by the principal component analysis. The two time series are illustrated by figure 5. It is clear from the figure that both components were above their means ¹³ preceding both the crises of 1988 and 2008, however, PC1 decreased more following the crisis of 2008. This means that both credit and asset prices can provide signals of future GDP crises. However, PC2 also rises in the years leading up to the dot-com crisis in 2001, which makes sense, as it contains equity prices. Household variables and total credit seem to be picking up the longer cycles of GDP crises, while asset prices and non-financial enterprise credit pick up on shorter cycles.

Figure 5: Predicted time series of PC1 and PC2

-6 -2 0 2 4

-6 -2 0 2 4

1985q1 1995q1 2005q1 2015q1

Average annual 3-year GDP growth Principal Component 1

Principal Component 2

The dependent variable, average annual 3-year GDP growth has been lagged 12 quarters to simplify comparison, since PC1 and PC2 are backward-looking variables.

Finally, I evaluate the principal component analysis using the Kaiser Meyer Olkin

12These components have been rotated using orthogonal varimax rotation, which gave very similar results as oblique rotation. Since orthogonal rotation retains the independence between the components (Corner, 2009), it was chosen in this instance. The rotation specialises the two components, in this case the change is only marginal. For more information of rotation see Corner (2009)

13Both means equal 0

measure of sampling adequacy (KMO-MSA). As defined by Dziuban et al. (1974), "The index yields an assessment of whether the variables belong together psycho-metrically and thus whether the correlation matrix is appropriate for factor analysis" (p.359). A low KMO- MSA indicates that PCA will not be able to efficiently summarize the variation of the selected variables in a meaningful way, due to a low degree of correlation between the variables. A high KMO-MSA means that the variables can be represented by only a few components¹⁴, which together summarize nearly all variation in the variables. The PCA of this group of variables returns a KMO-MSA of 0.64, which is an adequate result. The use of PCA is justified for the group of financial indicators.

5 Results

In this section I evaluate the performance of PC1 and PC2 in potential models, to answer whether they can successfully predict increasing tail risks to GDP growth. In order to select a final model, I will first assess the different models by evaluating them against the criteria explained in section 3, and then examine the predictions made by the chosen model. These predictions will be compared to the predictions for the same model using OLS as a method instead, in order to assess whether there is any added value from using quantile regression as opposed to the more typical OLS regression. Finally, I will examine how the chosen model performs in an out-of-sample exercise, simulating a real time prediction made over many years.

5.1 Model Choice

In order to avoid an omitted variable bias, lagged GDP will be included in any relevant model.

PC1, being the component that summarizes approximately 60 percent of the variation of the financial indicators will also be included in any relevant model. The main question is therefore whether or not a good model needs to utilise PC2 as well. All of the variables going into

14Relative to the number of variables going in to the PCA.

PC1 and PC2 were proved to be significant in explaining 3-year GDP growth on their own by Arbatli-Saxegaard et al. (2020). However, having incorporated the key variation in all of these financial indicators into two components, a decision must be made between keeping them both in the final model or just using one. Together, they cover both asset prices and credit, and as seen from figure 5, they represent two different degrees of volatility. This is unsurprising, as credit tends to move slowly, while asset prices, and especially equity prices are highly volatile. In order to decide on a preferred final model composition, I examine the coefficients and significance for each variable in a set of different models.

Table 4 shows the estimated coefficients of average annual 3-year GDP growth for the fifth percentile, and the 50^th percentile, for two model specifications. Accompanying each quantile regression model is the beta coefficient for the conditional mean, estimated by an OLS regression using the exact same variables. Every coefficient estimate is accompanied by its respective standard error and significance level. Each coefficient has been tested using a Wald test, providing individual p-values indicating whether or not the coefficients are indeed significantly different from 0. The answer to these tests are indicated by stars in the table.

Table 4: Coefficients and standard deviations of different models

Model variables 1 1 OLS 1 2 2 OLS 2

5th 50th Beta 5th 50th Beta

PC1 -0.43*** -0.39*** -0.36*** -0.45*** -0.36*** -0.37***

(0.03) (0.23) (0.03) (0.04) (0.17) (0.04)

PC2 0.24 -0.18 -0.55***

(0.04) (0.17) (0.21) Lagged GDP -0.31*** -0.14*** -0.18*** -0.51*** 0.01 0.27**

(3-year growth) (0.03) (0.23) (0.08) (0.04) (0.17) (0.14) The table shows coefficients showing the estimated average annual impact of a change of an increase on one in each variable for both the 5th and 50th percentile on average annualised 3-year GDP growth. Standard errors are in parenthesis. Stars denote the significance level at which the coefficient is different from 0, according to the Wald test. * - 10% level, ** - 5% level and *** - 1%

level.

Figure 6: Regression lines for PC1 of the model

-2 0 2 4 6

-2 0 2 4 6

Average 3year GDP growth

-6 -4 -2 0 2 4

Financial imbalances

Fifth percentile 50th percentile 95th percentile OLS Regression line

Criterion 1 from section 3 states that the coefficient for the fifth percentile should be significantly negative. Criterion 2 states that the difference between the 50^th percentile and the fifth percentile should be positive, implying a non-symmetrical effect on the quantiles as opposed to an OLS regression where coefficients are equal across quantiles. These criteria are summed up by equations (9) and (10). If both criteria are complied with, this indicated steeper downwards sloping regression lines for the fifth percentile as opposed to the median.

β_q50 <0 (9)

β_q50−β_q5 >0 (10)

Model 1 only consist of PC1 and the lagged GDP control variable. Both variables have significantly negative fifth and 50^th percentile coefficients. The fifth percentile coefficient for PC1 is somewhat more negative than the coefficient for the 50^th percentile. For lagged GDP growth however, the effect on the fifth percentile is estimated to be almost double that of the 50^th percentile. Both variables conform with criteria 1 and 2. The coefficient for PC1 from the OLS regression is close to the coefficients for the median. The coefficient for lagged GDP is close the to median coefficient, and is also significant.

The second model adds PC2. Adding PC2 only marginally affects the coefficients

Figure 7: Empirical conditional distribution for 3 year average GDP growth

0 .2 .4 .6

0 .2 .4 .6

Density

-4 -2 0 2 4

Standard deviation Unconditional distribution Conditional on high PC1

for PC1, but has considerable effect on the coefficients for lagged GPD. The fifth percentile coefficient is increased by 0.2, while the median coefficient is approximately 0. The difference between the median and the fifth coefficient is even larger than before. The OLS regression has, in contrast, a significant coefficient for PC2, as well as a strengthened coefficient for lagged GDP growth. The coefficient for PC1 is also in this instance unchanged. As the coefficient for PC2 is not significant in the quantile regression model, however, I will not include it in my model. Based on these results, I have chosen model 1 as my preferred model, including only PC1 and lagged GDP, both of which have significant and negative effects on both downside risk and GDP-at-risk.

The regression lines for the fifth, median and, for comparison, the 95^th percentile, are plotted in figure 6, together with the regression line for the mean by the OLS regression. As could be deduced from the coefficients in table 4, the regression line for the fifth percentile is steeper than that of the median, which is slightly steeper than the regression line for the mean. This is a sign of fatter lower tails. The 95^th percentile is even flatter than all the other regression lines, indicating a less responsive upper tail of the distribution. Plotting the conditional distribution for GDP on PC1 shows that PC1 indeed lowers both the fifth and the 50^th percentile noticeably, leading to increased values for growth-at-risk. See figure 7.

5.2 Model Predictions

Figure 8 shows the predictions for the fifth percentile and the median of 3-year annualised average GDP growth made by my baseline model. Figure 8a shows the in-sample prediction by the quantile regression of the model, while figure 8b shows the in-sample prediction made by the OLS regression of the model. For comparison, realised GDP growth has also been plotted.

The quantile regression prediction is estimated by equation (11), whereαis a constant,

∆GDP_t is the lagged annual average 3-year growth in real GDP in period t, as described in section 4.3. P C1t is the values for the component in period t made by the principal component analysis, and τ is the relevant quantile, i.e. the fifth and the 50^th percentile. The coefficients α,ˆ γˆ1,τ and βˆ1,τ are found by minimising the SWAE as explained in section 3.

∆y_t,τ = ˆα+ ˆγ_1,τ∆GDP_t+ ˆβ_1,τP C1_t (11) The OLS regression prediction is estimated by equation (12). The OLS prediction for the fifth percentile is calculated by taking the predicted mean value minus the standard error of the forecast times the critical value. This is exactly symmetrical to the calculation of the 95^th percentile, as the critical value marks the 90% confidence interval of the prediction.

∆y_t =α+γ∆GDP_t+β₁P C1_t (12)

In-sample prediction means that the model has been provided with all available data for the time interval for both dependent and explanatory variables to calculate its coefficients.

Those coefficients are then applied on the same explanatory variables, to give a prediction for the dependent variable. The predictions for the median and the fifth percentile are based on data up to the date of the prediction, and predicts the average annual GDP growth for the following three years. The series for average 3 year GDP growth, however, uses data for the next three years from the date of the prediction.¹⁵ For example, the prediction for 2008Q1 in figure 8a predicts a tail risk of -1.48 % growth in 3-year growth in GDP ahead. This prediction

15See section 3 and section 4 for the definition of the dependent and independent variables.