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Master Thesis
Determinants of the real long term interest rate in
the Netherlands
Development of a model for forecasts of future interest rates
Abstract
This study investigates the impact of different variables on the 10-year real interest rate in the Netherlands in 1971-2015. Stepwise least square regression selects only the statistically significant variables among the eight possible variables. I repeat this methodfor three different models, selecting a preferred specification for every model. Based on goodness of fit, I choose the best out of the three models. The chosen model is the levels model with government balance, consumption growth and the lagged real interest rate as explanatory variables. The levels model can forecast the real interest rates up to 2021. For this purpose, I use forecasts of the explanatory variables as inputs.
Field key words: real interest rate, Error Correction Model, variable selection JEL Classification: E44; E47; E52; G12; J11
June, 2017
Melissa Brockmann S2203405
MSc Finance
Supervisor: Lammertjan Dam
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1. Introduction
This research investigates the real interest rate in the Netherlands and eight possible factors influencing this rate. The research question is: which factors influence the real long term interest rate and what is an estimate of the future value of this rate. To answer the research question I test which variables have a significant impact on the real interest rate. Hereby, I focus on finding the minimal set of explanatory variables necessary to explain developments in the real interest rate. The interest rate I study is the 10-year real interest rate on Dutch government bonds. The final purpose of this research is to use the selected explanatory variables to predict future values of the real interest rate.
Nowadays, predicting future values of the real interest rate is both important and difficult. Extremely low nominal interest rates are common and these rates imply in some cases a negative real interest rate. In the Netherlands, a decline from around 12% to around 0.5% in the nominal long term rate has been experienced since the beginning of the 80’s (OECD). There is a huge strand of literature on several determinants of interest rates, however the impact of individual factors does not lead to a reliable future interest rate. Therefore, this research fills this gap as it selects economically and statistically significant variables and predicts a future interest rate. The real interest rate is important for almost all financial decisions made by (financial) firms but also by individuals. Interest rates influence the cost of borrowing for both individuals and corporations. The real interest rate on government bonds is also a conventional proxy for the risk free component in returns on capital. Both borrowing costs and returns on capital are important determinants for the returns on portfolios, so the interest rate is crucial for financial managers. Besides the practical relevance, a useful model for determining interest rates has academic relevance. Academic investigators use interest rates intensively in their research, therefore it is valuable to have a better understanding of the mechanisms and factors that drive these interest rates.
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levels form and include a lagged value of the interest rate. The second model is the difference model and this model includes all explanatory variables in their first difference form and additionally the lagged real interest rate. The error correction model assumes that there is a long term relationship between the dependent and independent variables but that in the short term deviations from this relationship are possible. This model combines elements of the levels and difference model. The stepwise least square regression method produces a single best estimation for all three models. These three models are compared on their goodness of fit, so the Schwarz Bayesian Information criterion chooses the best model. I use the chosen model for the final purpose of this study, predicting interest rates for the future. I obtain forecasts of the selected explanatory variables and use these in the model.
This research uses data in the period 1971-2015 and the focus is on the Netherlands. This paper uses the longest available data sample as interest rates move slowly. I focus on the Netherlands as this country has a specific pension system, most pensions are based on defined benefit. This increases risks for pension funds, therefore a good proxy of the future interest rate is highly important. The data is yearly, which means that the sample contains 45 observations. I investigate eight possible explanatory variables. The variables I choose are the most commonly used in empirical research and cover demographic, macroeconomic and fiscal forces driving the real interest rate. All variables impact the interest rate through an impact on saving or investment. Four variables impact the real interest through an impact on saving, namely the dependency ratio, the saving population ratio, consumption growth and income inequality. Three variables impact the real interest rate through investment, namely government debt, government balance and interest rate spread. The last variable, real gross domestic product (GDP) growth, impacts the real interest rate both through savings and investment. Data for two variables contains a shorter sample, the data on income inequality is available for 1977-2015 and the sample for interest rate spread is available for the period 1978-2012. Therefore, I use a shorter time period in estimations including these two variables.
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for years after 2021 are not possible. The forecast results indicate a further decline in the real interest rate between 2016-2021. Some commentary is in place regarding this result as the forecast suffers from much uncertainty and it should be treated with caution. This uncertainty results from uncertainty about forecasts of the independent variables and the assumptions made regarding these forecasts. In addition, the model does not perform very well when I test it for out-of-sample performance.
The main contribution of this research, is that it investigates different variables and models and selects the best estimation that can explain fluctuations in interest rates. This result is useful for financial managers and individuals, as their financial decisions depend on the (future) real interest rate. As described before, the study shows that government balance has a negative impact on interest rates, consumption growth a positive effect and that the lagged real interest rate has also influence on the real interest rate. Therefore, managers and individuals can expect changes in the interest rate following changes in the explanatory variables. They can use this knowledge to optimize their financial planning.
This paper proceeds as follows, Section 2 will focus on the relevant literature concerning possible determinants of the real interest rate. From the literature, I derive hypotheses that can answer the research question. Section 3 outlines the method used to investigate different variables and models, where Section 4 shows data and descriptives. Section 5 reveals the results and Section 6 includes the conclusion and discussion.
2. Literature review
In this section, I explain the mechanisms behind the possible determinants of the real interest rate. The saving-investment framework explains the influence of the different factors, therefore I introduce this framework first. Thereafter I describe for every variable the concerning literature and the expected relationship based on earlier research. The expected relationships are the same for all three models, whether the variables are in difference form or levels form. 2.1 Saving-Investment framework (SI-framework)
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addition, they observe the desired investment curve shifted inwards, implying less demand for investment at the same interest rate. The intersection of the two curves is lower, implying a decrease in the interest rate.
Figure 1: Saving-investment framework
This figure presents an illustration of the desired savings curve, the desired investment curve and the shifted desired savings and investment curve. The horizontal axis gives the percentage of GDP that is desired to save or invest at a certain level of the real interest rate. This figure and the theoretical background is based on Rachel and Smith (2015).
Rachel and Smith also describe different factors that contribute to the shift in savings and investment over the last 30 years. I include some of these factors in this thesis, however the analysis of Rachel and Smith focuses on world interest rates so not all factors apply to the real interest rate of the Netherlands.
2.2 Potential determinants of the real interest rate
The potential determinants of the interest rate have an impact via the savings curve or the investment curve. Based on this distinction I discuss several papers that investigate the impact of potential factors on the real interest rate. First, I describe the impact of several factors via the savings curve, after that I explain the impact of some factors through the investment curve.
2.2.1 Factors influencing through savings curve
The first two factors that influence the real interest rate through the savings curve are demographic variables. The life cycle hypothesis (LCH) explains the impact of both variables, therefore I describe this hypothesis first, before turning to the individual factors. The LCH focuses on the saving pattern of people through their life and states that this pattern depends on the different needs at different moments in their life (Jappelli, Modigliani, 2003). According to
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Re al in tere st ra te (% ) Desired saving/investment (% of GDP) Savings Investment
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the life cycle hypothesis, people borrow when they are young, save in their working period and dis-save in their retirement period. Hassan, Salim, Bloch (2011) provide an overview of the existing literature regarding the influence of demographics on savings. They find that most papers support the life cycle hypothesis, although for the dis-saving of retired people there is less evidence. As the LCH links demographic variables to savings and the SI-framework links savings to interest rates, the LCH deems an important theory in describing developments in the interest rate.
Dependency ratio
The dependency ratio measures the dependency of the retirees on the working population (Carvalho, Ferrero, Nechio, 2016). The dependency ratio has increased in developed countries over the last years due to two forces. Population growth has slowed down and the average age of the population has increased. Carvalho, Ferrero, Nechio (2016) describe the results of these two forces and link this to the relationship between the increased dependency ratio and decreased interest rates. They state that an increase in life expectancy puts downward pressure on interest rates, as people save more to prepare for the longer retirement period. They also describe two (opposite) effects resulting from a lower population growth. Firstly, the impact on the interest rate can be positive. A lower population growth leads to more retirees, who are dis-savers according to the LCH. As a result of more retirees, savings will decrease. According to the SI-framework, the interest rate will rise. However, on the other hand, as there are less workers, the capital per worker increases. The increased capital/labor ratio will lower the marginal product of capital and so the real interest rates follow. The authors find evidence for the higher dependency ratio as a cause of one third of the decline in the real interest rate. So the evidence supports the two explanations that a drop in population growth and an increase in life expectancy (both captured by the increased dependency ratio) put downward pressure on real interest rates. Therefore, the hypothesis for the relationship between the dependency ratio and the real interest rate is negative.
Saving population ratio
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comparable age composition on the interest rate. They use the MY-ratio as a proxy for the age composition, this is the ratio of middle aged (40-49) to young people (20-29). The hypothesis is that a higher MY-ratio will lead to more savings, as middle-aged people save more compared to young people. Increased savings imply a lower interest rate, according to the SI-framework. Their international panel study confirms this hypothesis, they find a negative relationship of -0.044 between the MY-ratio and the interest rate. Based on the explanations given above and the earlier research, the expected relationship is negative.
Consumption growth
There is a distinction between current consumption and future consumption, where future consumption is financed by saving today (Rachel, Smith, 2015). Consumption growth measures whether future consumption is higher than current consumption. High (expected) consumption growth is associated with high real interest rates and this relationship works two-ways. First, a high expected consumption growth, for example from an expected increase in income, causes a decrease in savings. This is a result of lower precautionary savings as there is less uncertainty in consumption (Cochrane, 2001). As stated by the SI-framework, an inward shift of the desired savings curve leads to a higher interest rate. On the other hand, a high interest rate increases the motivation for savings, as the reward is bigger. Higher savings will increase expected consumption growth. So, the high interest rate also leads to higher expected consumption growth. The two relationships are in equilibrium, if the interest rate is high enough, this prevents people from raising current consumption even if their expected consumption growth is high, and they save instead. The above reasoning concerns expected consumption growth, which is not easily observable. Therefore, this research uses consumption growth as a measure of expected consumption growth. I justify this with the rational expectation hypothesis, agents have rational expectations and consumption growth is a good proxy for expected consumption growth (Poghosyan, 2014). I am not aware of any empirical research concerning the impact of consumption growth on the real interest rate, therefore, based on the explanations above the expected relationship between consumption growth and the real interest rate is positive.
Income inequality
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wealthy people are for a bigger part fulfilled than the consumption needs of the poorer people. So if the income of the wealthy people increases, they might save a bigger part of it. In addition, if the income of poorer people decreases, they might need to decrease their consumption, leaving total savings unchanged. On an aggregated base, savings can increase as a result of higher income inequality. As a result of higher savings, the interest rate decreases. There is not much empirical work on this relationship. Rachel and Smith (2015) provide one of the exceptions, their research confirms the reasoning above. They find a decrease of 45 basis points in the real interest rate over the past 30 years as a result of increased income inequality. Based on the explanations above and the earlier research, the expected relationship between income inequality and the real interest rate is negative.
2.2.2 Factors influencing through the investment curve
This research investigates two factors that proxy for the fiscal and debt policy of the government. Besides the impact via the investment curve, Ciocyte, Muns, Lever (2015) provide another explanation for the impact of these two factors. They state that government debt and deficits have a positive impact on interest rates. They follow the logic that more debt or a higher deficit decreases the creditworthiness of the country. A lower creditworthiness requires a higher return on capital, which increases government bond interest rates.
Government balance
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relationship between the government deficit and the short term real interest rate. They acknowledge that their findings do not correspond to the earlier research in this field. The above reasoning leads to a positive expected relationship between government deficit and the interest rate. As a deficit is a negative balance, the expected relationship between government balance and the real interest rate is negative.
Government debt
The government debt is the second measure of the fiscal policy of the government. Poghosyan (2014) gives two arguments for a positive relationship between the real interest rate and the government debt. The first explanation is the risk premium, as governments with more debt are more risky, more return is required as compensation. The second argument is that more government debt will crowd out private investment, which leads to a lower capital stock in equilibrium. The marginal product of capital is therefore higher, resulting in a higher real interest rate. This explanation does not exactly fits the SI-framework, but it supplements the reasoning that more government debt increases the demand for capital and hereby the interest rate. Upper and Worms (2003) investigate government debt as a factor that influences the real interest rate. They find a positive (0.0262), however not very large relationship between the two variables. Poghosyan (2014) confirms this positive relationship, using an error correction model to test different short run and long run factors influencing the real government bond yield. The magnitude of the government in the long term relationship is 0.021 and in the short run relationship it is 0.081. The expected relationship between the government debt and the real interest rate is positive, based on the explanations and empirical research above.
Interest rate spread
10 Real GDP growth
Gross domestic product (GDP) is a measure of the volume of the economy, but it is also a measure of the income of households. This household income can have an impact on savings, if the households are credit constrained, changes in current income can impact savings and consumption. A higher household income can shift the savings curve outwards (Upper, Worms, 2003). But on the other hand, (real) GDP growth can also impact interest rates through the investment curve, a growing economy can increase its level of investment. So there are two offsetting effects: a shift in the savings curve leads to a lower equilibrium interest rate, while a shift in the investment curve leads to a higher equilibrium interest rate. The paper of Upper and Worms finds a positive impact of 0.0115 from real GDP growth on the interest rate, so their argument is that the effect of real GDP growth works mostly through the investment channel. Based on the above reasoning and previous research, the expected relationship is positive.
3. Methodology
I compare three different models that all test the relationship between the dependent and independent variables. These models are: 1. a model in levels form (including one lag of the dependent variable), 2. a model in first differenced form, also with one lag of the dependent variable and 3. an error correction model. I compare different models, as it is upfront not clear which model is most suitable. Some authors use the error correction model and others use the levels model. The error correction model is a combination of the levels and difference model, therefore I also include the difference model for comprehensiveness. In every model, I select statistically significant variables for the best specification. After this variable selection, I choose the best model. This section starts with the method used for variable selection. Thereafter, I discuss all three models, model comparison and I conclude with the method for out-of-sample illustration and forecasting.
3.1 Variable selection
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the highest p-value at every step until all variables are statistically significant at 5%. After every step, the software checks the removed variables against the forward criterion, hence whether re-adding them gives a p-value lower than 5%. So this method considers many specifications and comes up with one that fits best. More details on this method are given in Appendix 1. The selected significant variables can differ between the three models.
3.2 Levels model
The first model is a model in levels form. For this model I regress the dependent variable (Rt) on the independent variables in their level form and the lagged real interest rate. This model is most useful when all variables are stationary.
𝑅𝑡 = 𝛼 + ∑𝑛𝑘=1 𝑘𝛽 𝑓𝑘,𝑡+ 𝜃𝑅𝑡−1+ 𝑣𝑡 k=1,…n, t=1,….T (1) K refers to the specific independent variable, where I use n variables total in this research. vt is the error term and is assumed to be independent and identically distributed 𝑁~ (0, 𝜎2). I test these assumptions in Appendix 7.
3.3 Difference model
The second model is a model in difference form, also including the lagged real interest rate. This model is useful if the variables and residuals are non-stationary. If data is I(1), taking the first difference solves the non-stationarity, the data becomes I(0).
∆𝑅𝑡 = 𝛼 + ∑𝑛𝑘=1𝛽𝑘∆𝑓𝑘,𝑡+ 𝜃𝑅𝑡−1+ 𝑣𝑡 k=1,…n, t=1,….T (2) ∆ denotes a first difference of a variable. K refers to the specific independent variable, where I use n variables total in this research. vt is the error term and is assumed to be independent and identically distributed 𝑁~ (0, 𝜎2). I test these assumptions in Appendix 7.
3.4 Error correction model
The third model I estimate combines some elements from both the levels model and the difference model. The error correction model captures the influence of both short term factors and long term factors. I use this model because some authors believe that the real interest rate has a long term equilibrium relationship with the explanatory variables, but that deviations from this relationship are possible in the short run (Orr, Edey, Kennedy, 1995). The short term relationship is more comparable to the difference model, whereas the long term relationship is more similar to the levels model, both without the lagged dependent variable.
I estimate the following equation:
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Appendix 5 outlines the specific procedure of building the ECM, I discuss only the highlights here. The first part (after the intercept α1) of the equation measures the short term influence from changes in the independent variables on the dependent variable. The term ∆fk,t includes the independent variables in their first differenced form. The corresponding β measures the impact of the specific independent variable on the real interest rate. K refers to the specific variable, where there are n variables in total in this research.
The second part of the equation measures the deviation of the real rate (Rt) from the equilibrium value. Rt is expected to change between t-1 and t, in response to a disequilibrium in period t-1. The parameter 𝜆 measures the speed of adjustment. If this term is statistically insignificant, the model actually converges to the model in difference form. The underlying long run equilibrium relationship between Rt and its explanatory variables is:
𝑅𝑡= 𝛼 + ∑𝑛 𝛾𝑘
𝑘=1 𝑓𝑘,𝑡+ 𝑤𝑡 k=1,…n, t=1,….T (3.2) In this formula, fk,t includes the same independent variables, however this equation uses not their difference but their level. The parameter 𝛾𝑘 measures the long term relationship between the specific independent variable and Rt. The third and last part of equation 3.1 is the error term (vt) and is assumed to be independent and identically distributed 𝑁~ (0, 𝜎2). I test this assumption in Appendix 7.
Cointegrating relationship
From the nature of the error correction model, I deduce the hypothesis for the cointegrating term. The expected sign for the cointegrating factor (speed of adjustment) is negative. If the value at Rt-1 is higher than it should be according to the equilibrium relationship (so the deviation from equilibrium is positive), the change in Rt should be negative to compensate for this excess.
3.5 Model comparison
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does well in-sample but underperforms out-of-sample. Appendix 6 outlines the calculation of SBIC.
3.6 Out-of-sample illustration
As the forecast of a real interest rate is the final purpose of this paper, it is useful to illustrate the out-of-sample performance of the chosen model. I use a recursive forecasting model, a form of a multi-step-ahead forecast method. I use the first part of the sample to forecast the next observation and then I increase the sample with one year and repeat the forecast. In this way, every year I update the model and use it to forecast one year ahead, with one additional observation. For every new sample created, I conduct a complete stepwise least squares regression, which means that not only the coefficients of the variables can vary with the time sample, but also the variables selected by the software. I calculate the root mean square error to evaluate the out-of-sample forecasts. The following formula is used:
𝑅𝑀𝑆𝐸 = √∑𝑛𝑖=1(𝑦̂𝑖−𝑦𝑖)2
𝑛 i=1,…n (4)
Here, 𝑦̂𝑖 is the predicted real interest rate by the model and 𝑦𝑖 is the actual observed real interest rate, i denotes the specific forecast and n is the total number of forecasts.
3.7 Forecast
I use the chosen model to forecast future values of the real interest rate. For this purpose, I need forecasts of the selected explanatory variables. I plug this values into the model and hereby calculate future real interest rates.
4. Data
This section covers two main topics, variable definition and summary statistics. I first focus on the definition and calculation of the variables. Thereafter, I report summary statistics of all dependent and independent variables. This section concludes with a correlation matrix. 4.1 Variable definition
14 Table 1: Overview of variables
This table presents the symbols, calculation and possible remarks for the dependent variable and the eight independent variables. The variables are calculated yearly where t stands for the specific year, representing the period 1971-2015. Income inequality data is only available for the period 1977-2015 and data for the interest rate spread only for 1977-2012.
4.1.1 Real interest rate
The yearly rate on 10-year Dutch government bonds proxies for the nominal interest rate. I retrieve this rate from the Organisation for Economic Co-operation and Development (OECD). This interest rate is an average of daily market information, the OECD calculates this rate using prices at which the government bonds trade. I calculate the real interest rate (Rt) by subtracting the inflation (πt) from the nominal interest rate (it). OECD also provides data on inflation.
𝑅𝑡= 𝑖𝑡− 𝜋𝑡 t=1,….T (5)
Note that the original formula (Fisher, 1930) contains expected inflation instead of actual inflation. As it is very difficult to obtain expected inflation for the Netherlands, this research uses actual inflation. According to the rational expectation hypothesis this alternative should not impose any problems as agents have rational expectations (Poghosyan, 2014).
4.1.2 Dependency ratio
The dependency ratio measures the dependency of the retirees on the working population. As it is very difficult to measure the actual size of the working population and the number of
Variable Symbol Definition/calculation Remarks
Dependent variable:
Real interest rate Rt Rt = it – πt it = nominal interest
rate in year t, πt = inflation in year t Independent variables: fk,t:
Dependency ratio drt People 65+ over people
15-64 in year t
Saving population ratio spt People 25-64 over total population in year t Government balance (%
of GDP)
gbt EMU-balancet (% of GDPt) Negative balance implies deficit Government debt (% of
GDP)
gdt EMU-debtt (% of GDPt)
Consumption growth (%) cgt Consumption growtht (%) Real GDP growth (%) rgdpgt
Income inequality gct Gini coefficient measures
the deviation of the Lorenze curve from perfect equality Interest rate spread (%) st Spreadt between return on
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retirees, I use age as a proxy. People over 65 proxy for the retired population, people between 15 and 64 are representative for the working population. The following formula defines the dependency ratio:
𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑐𝑦 𝑟𝑎𝑡𝑖𝑜 (𝑑𝑟𝑡) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 65+𝑡
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 15−64𝑡 (6)
4.1.3 Saving population ratio
The saving population ratio measures the share of the population that is part of the saving population. The share of the population in the age range 25-64 is a proxy of the saving population ratio. I calculate the saving population ratio with the following formula:
𝑆𝑎𝑣𝑖𝑛𝑔 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 (𝑠𝑝𝑡) =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 25−64𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑡
𝑡 (7)
4.1.4 Government balance
I measure government balance by the EMU-balance as a percentage of GDP. The EMU-balance is the measure of the European Monetary Union for the difference between governments earnings and expenditures. EMU-balance includes also financials from local governments and the social sector. A negative balance represents a deficit and a positive balance a surplus. The balance is measured at transaction base, meaning that the that act leads to a receipt or payment is the measurement moment. The measurement moment is the criterion on which to decide in which year the payment or income should be reported.
4.1.5 Government debt
A factor related to the government balance is government debt as a percentage of GDP. In years of a deficit, the government might raise its borrowing and this will increase government debt. I measure government debt as EMU-debt as a percentage of GDP.
4.1.6 Consumption growth
Consumption growth is a measure to compare current and future consumption. If consumption growth is positive, future (expected) consumption is higher than current consumption. Consumption growth is measured as a percentage. As stated in the literature review, I use consumption growth instead of expected consumption growth, as expected consumption growth is not observable.
4.1.7 Real gross domestic product growth
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can be seen as a measure of the size of the economy. If real GDP increases, production increases so the economy is growing. As this research focuses on the real interest rate, real GDP gives a better view than nominal GDP. I use growth of GDP, as real GDP is only increasing over time and therefore not a very good descriptor of fluctuations in interest rates. The following formula calculates the percentage growth in real GDP:
𝑅𝑒𝑎𝑙 𝐺𝐷𝑃 𝑔𝑟𝑜𝑤𝑡ℎ (%) (𝑟𝑔𝑑𝑝𝑔𝑡) = 𝑅𝑒𝑎𝑙 𝐺𝐷𝑃𝑅𝑒𝑎𝑙 𝐺𝐷𝑃𝑡−𝑅𝑒𝑎𝑙 𝐺𝐷𝑃𝑡−1
𝑡−1 𝑥 100% (8)
4.1.8 Income inequality
This paper investigates another macroeconomic factor, namely income inequality. The Gini coefficient is the most commonly used measure of income inequality. The Gini coefficient uses the Lorenze curve, a visualization of the distribution of income. Figure 3 presents a (fictive) Lorenze curve and the line of equality. The horizontal axis presents the cumulative proportion of the population ranging from the poorest to richest people. The vertical axis presents the percentage of income earned by the specific percentage of the population indicated on the horizontal axis. The diagonal dashed line represents the line of equality, which means that at this line income is equally spread across people. The Lorenze curve (solid line) represents the actual distribution of income.
Figure 2: Fictive Lorenze curve and line of equality
This figure presents the fictive Lorenze curve and the line of equality. These two lines are elements to calculate the Gini coefficient. The horizontal axis presents the part of the population that earns a certain percentage of the income, the vertical axis gives this certain % of income. The Gini coefficient measures how much the Lorenze curve deviates from equality by the following formula: 0 10 20 30 40 50 60 70 80 90 100 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 In come (% ) Population (%)
Line of equality Lorenze curve
A
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𝐺𝑖𝑛𝑖 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑡= 𝐴𝑟𝑒𝑎 𝐴𝑡
𝐴𝑟𝑒𝑎 (𝐴+𝐵)𝑡 (9)
Here Area A is the area between the equality line and the Lorenze curve and Area B is the area below the Lorenze curve. The Gini-coefficient can take a value between zero and one, zero implying no income inequality and one implying perfect inequality (Gini, 1912). From now on, I refer to income inequality with the Gini coefficient.
4.1.9 Interest rate spread
The last variable I investigate is the spread between the return on capital and the risk free rate. For this variable, different definitions are possible (Rachel and Smith, 2015). A bank credit spread measures the difference between bank lending rates and deposit rates, this is also called interest rate spread. A fixed income spread measures the difference between yields on corporate and government bonds and equity market spreads measure the difference between earning yields and government bond yields. I select the interest rate spread (bank lending spread) in this research to measure the spread between return on capital and the risk free rate.
4.2 Summary statistics
Table 2 presents summary statistics for all the dependent and independent variables.
Table 2: Summary statistics, for the interest rate and determining factors for the Netherlands, 1971-2015
This table presents summary statistics for yearly observations of the interest rate and its potential determinants. *The series for the Gini coefficient contain only values for the period 1977-2015 and the series for interest rate spread only for the period 1978-2012. Sources: World Bank, OECD, CPB, CBS, Eurostat, SWIDD, Gini research.org. For detailed variable definition, see § 4.1.
Mean Median St. Dev Minimum Maximum Stationary
Nominal interest rate (%) 6.2 6.4 2.6 0.7 11.6 No
Inflation (%) 3.4 2.5 2.6 -0.7 10.2 No
Real interest rate (%) 2.8 2.7 2.3 -1.4 7.1 No
Dependency ratio 0.195 0.188 0.029 0.164 0.279 No
Saving population 0.523 0.535 0.035 0.449 0.560 No
Government balance (% of GDP) -2.5 -2.2 2.1 -8.6 1.9 Yes
Government debt (% of GDP) 56.9 57.8 12.9 36.6 75.9 No
Consumption growth (%) 2.1 2.1 1.6 -1.2 5.2 No
Real GDP growth (%) 2.3 2.3 1.9 -3.8 5.1 Yes
Gini coefficient* 0.268 0.270 0.015 0.236 0.290 No
Interest rate spread (%)* 3.7 3.4 3.4 -1.1 11.0 No
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time period it can be statistically non-stationary. The same holds for the real interest rate, we do not expect this rate to wander very far from its mean. The reason I report the results on the stationarity test, is that the non-stationarity in this short time period can cause problems in statistical procedures. Appendix 3 reports the p-values of the stationarity tests.
4.2.1 Real interest rate
The real interest rate ranges between -1.4% and 7.1%, with a mean of 2.9%. Compared to the real interest rate, the nominal rate ranges between 0.7% and 11.6% and has an average of 6.2%. The standard deviation is pretty high for both rates, 2.3% for real rates, 2.6% for nominal rates. For inflation the standard deviation is 2.6%, quite high compared to the mean of 3.4% and median of 2.5%. Figure 2 illustrates the development of interest rates. The nominal interest rate has decreased primarily in this sample. The real interest rate exhibits a less pronounced trend. In the last 25 years we see mostly a decreasing trend, but before 1988 the rate was rising, coming from its lowest point in 1975. Inflation can be seen as an important and volatile component in determining interest rates. Inflation can often explain high nominal rates. It is also interesting that inflation was almost always positive, except for 1987. This deflation is mostly a result of the extreme decline in oil prices, an important determinant of the price level at that time. This same factor can be used in explaining the high inflation rates in the seventies.
Figure 3: Nominal, real interest rates and inflation
This graph presents the nominal and real interest rate and inflation for the years 1971-2015. Source: OECD
4.2.2 Dependency ratio
World Bank provides data for the dependency ratio. The dependency ratio mean is 0.195, see table 2. Figure A1 in Appendix 4 visualizes the trend in the data. Here, the upward sloping trend in the ratio is obvious, this indicates the increase of the average age of the population and the
-4% -2% 0% 2% 4% 6% 8% 10% 12% 14% 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015
Interest rate
19
slowdown in population growth. The dependency ratio increased from 0.164 to 0.279 in the period 1971-2015.
4.2.3 Saving population ratio
I retrieve information about the age distribution from the CBS (Statistics Netherlands). Section 4.1.3 describes how I calculate the saving population ratio from this data. Table 2 presents the summary statistics of this variable. The saving population ratio has a mean of 0.523 and values in the range of 0.449 to 0.560. Standard deviation is not that high, only 0.035. Figure A2 in Appendix 4 presents an illustration of the trends in the saving population ratio.
4.2.4 Government balance (% of GDP)
The CPB, the Bureau for Economic Policy Analysis in the Netherlands provides data on the EMU-balance as a percentage of GDP (the proxy for the government balance). Summary statistics for this variable can be found in table 2 and Appendix 4 presents a graph (figure A3). The mean (-2.5%) and median (-2.2%) are both negative, implying a government deficit. The observations support this finding, in 39 of the 45 years there was a deficit. The standard deviation is very high for this data, 2.1%. This high standard deviation is mostly due to the high deficit in 1995, when the Dutch government paid of the premiums of housing corporations. Another high deficit occurred in 2009, probably due to financial crisis. The government had to support the financial sector with a lot of capital injections, which causes the deficit. It is possible to remove the outliers from the sample, however there are more (negative) peaks in this dataset. So this variable is very volatile and the outliers support that. Therefore, I do not remove them.
4.2.5 Government debt (% of GDP)
CPB also measures government debt, labeled as the EMU-debt as a percentage of GDP. Table 2 presents the summary statistics for this variable. As can be seen, the mean is 56.9% and the standard deviation is 12.9%. Appendix 4 presents a graphical illustration of the data on government debt.
4.2.6 Consumption growth (%)
20 4.2.7 Real gross domestic product growth
I retrieve data about the real GDP in billion € from the OECD, and section 4.1.7 outlines the calculation of real GDP growth. Table 2 reports the summary statistics for real GDP growth. Real GDP growth is quite volatile, the mean is 2.3% and the standard deviation is 1.9%. Figure A6 in Appendix 4 illustrates the data of real GDP growth.
4.2.8 Gini coefficient
Historic data on the Gini coefficient is hard to obtain, there is no unique database that contains values from 1971 to 2015 for the Netherlands. Therefore, I compose a sample from different sources. A shorter sample is the consequence, as I find data for the period 1978-2012. Therefore, the estimations including Gini coefficient contain a shorter time sample. Appendix 2 lists the various sources and outlines their dissimilarities. Summary statistics for this variable can be found in table 2. The Gini coefficient ranges from 0.236 to 0.290, with a mean of 0.268 and a standard deviation of 0.015. The graph in Appendix 4 illustrates the data for the Gini coefficient.
4.2.9 Interest rate spread
In this research I use the bank credit spread and retrieve the data from World Bank, labeled interest rate spread. As this sample only contains data from 1978-2012, the estimations with this variable contain a smaller time frame than the other configurations. The mean of the interest rate spread is 3.7% but its standard deviation is 3.4%, quite high compared to the mean. The interest rate spread ranges in this sample from -1.1% to 11.0%.
4.2.10 Correlation
21 Table 3: Correlation between all variables
This table presents the correlations between the dependent and independent variables. For detailed variable definition, see § 4.1. Sources: World Bank, OECD, CPB, CBS, Eurostat, World databank: Microdata, Standardized World Income Inequality Database, Gini research.org.
Real interest rate Dependency ratio Saving population ratio Government balance (% of GDP) Government debt (% of GDP) Real GDP growth Consumption growth (%) Gini coefficient Interest rate spread (%)
Real interest rate 1
22
5. Results
This section presents the results of the stepwise least square regression for the three different models. Stepwise least regression causes some problems if the sample sizes of the different variables are unequal. As the samples of Gini coefficient and interest rate spread are smaller, dropping these variables during the stepwise regression will increase the sample size, which influences the significance of the selected variables. Two options are available; a larger time period without the Gini coefficient and interest rate spread or a shorter time period including the Gini coefficient and interest rate spread. As a large sample is important in the analysis of interest rate developments, I decide to use a large sample, excluding interest rate spread and Gini coefficient in this regression. As a robustness check, I repeat the stepwise least square regression for the shorter time period including the Gini coefficient and the interest rate spread.1 After the robustness check, I proceed with the model comparison to choose the best model. Then, I present an out-of-sample illustration, followed by the last part, the forecasting of future interest rates.
5.1 Levels model
I estimate the levels model with stepwise least square regression. The model includes all the variables in their levels form and the lagged value of the real interest rate. The table below reports the coefficients of the variables selected by the stepwise least squares. The table shows the standard errors (S.E.) between brackets in the row below the coefficient. The first column shows the regression results with all the variables, the second column shows the regression results with only the selected variables.
1 Some observations of the Gini coefficient are lost in this process, as data on interest rate spread ranges from
23 Table 4: Results Stepwise Least Squares regression, levels model
This table presents regression results for the stepwise least square regression of the levels model. The estimated regression is: 𝑅𝑡 = 𝛼 + ∑𝑛𝑘=1 𝑘𝛽 𝑓𝑘,𝑡+ 𝜃𝑅𝑡−1+ 𝑣𝑡. The dependent variable is the real interest rate. For detailed variable definition, see § 4.1. This regression excludes the Gini coefficient and the interest rate spread, but includes the lagged real interest rate as independent variable. The sample contains yearly observations for the years 1971-2015, this results in 44 observations due to the use of one lag. Sources: World Bank, OECD, CPB, CBS. *Significant at 10% **Significant at 5% ***Significant at 1%
From the table we can see that the stepwise least square regression method selects the government balance, it is significant at 1%. The sign of the government balance is negative, implying that an increase in the balance of the government leads to a decrease in the real interest rate. This is in line with the literature concerning the relationship between government balance and the real interest rate. Most literature states that a government deficit has a positive influence on the real interest rate. As stated earlier, a government deficit is a negative balance, therefore, these results confirm the earlier research. The size of the coefficient is -0.213, so an increase of 1% in the government balance leads to a decrease of 0.213% of the real interest rate. This is larger than the effect that Cebula, Angjellari-Dajci, Foley (2014) find, namely 0.088. This is not surprising, their analysis includes more and different variables, which changes the magnitude of the impact of government balance. The stepwise least square regression method selects consumption growth as the second variable. This variable is significant at 5%, and it has a positive coefficient. This is in line with expectations, the relationship between
Dependent variable: real interest rate
All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept -0.003 -0.005 (0.029) (0.004) Dependency ratio -0.055 (0.125) Saving population ratio 0.032
(0.073) Government balance -0.253** -0.213*** (0.109) (0.086) Government debt -0.016 (0.022) Consumption growth 0.091 0.233** (0.179) (0.104) Real GDP growth 0.120 (0.125)
Lagged real interest rate 0.808*** 0.824**
(0.412) (0.071)
R2 0.869 0.839
24
consumption growth and the real interest rate is positive according to the (theoretical) literature. The magnitude of the coefficient is 0.233, this is not very large but also not negligible. It means that an increase of 1% in the growth of consumption leads to an increase of 0.233% in the real interest rate. The stepwise least square regression also selects the lag of the real interest rate, which is significant at 5% and has a positive coefficient. The coefficient is also quite high, namely 0.824. This means for example that if the interest rate in the previous year is 1% higher, the current interest rate is 0.824% higher as a result of this. For this regression, I test the Ordinary Least Square (OLS) conditions. Appendix 7 displays the results, I conclude that this specification does not violate the OLS conditions.
5.2 Difference model
The difference model uses all variables in their first difference form and additionally the previous value of the real interest rate (not in differenced form). Table 5 shows the results for the longer time period, excluding Gini coefficient and interest rate spread
Table 5: Results Stepwise Least Squares regression, difference model
This table presents regression results for the stepwise least square regression of the difference model. The estimated regression is: ∆𝑅𝑡= 𝛼 + ∑𝑛𝑘=1𝛽𝑘∆𝑓𝑘,𝑡+ 𝜃𝑅𝑡−1+ 𝑣𝑡. The dependent variable is the difference in the real interest rate. For detailed variable definition, see § 4.1. This regression excludes the Gini coefficient and the interest rate spread, but includes the lagged real interest rate as independent variable. The sample contains yearly observations for the years 1972-2015, this results in 44 observations due to the use of differences and one lag. Sources: World Bank, OECD, CPB, CBS. *Significant at 10% **Significant at 5% ***Significant at 1%
Dependent variable:
difference in real interest rate
All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept 0.006 0.008** (0.004) (0.003)
Difference dependency ratio -0.769 -1.104*
(0.872) (0.556)
Difference saving population ratio 0.435 (0.842) Difference government balance -0.139*
(0.080)
Difference government debt 0.024
(0.044)
Difference consumption growth 0.225 0.274*
(0.160) (0.141)
Difference real GDP growth 0.103
(0.084)
Lagged real interest rate -0.170** -0.159**
(0.076) (0.071)
R2 0.501 0.192
25
Stepwise least squares regression selects the difference in the dependency ratio, which is significant at 10% (almost at 5% as the p-value is 0.054)2. The difference in dependency ratio has a negative coefficient, this is in line with expectations. The coefficient is quite large, it is even greater than one. The result indicates that an increase of 1% in the difference of the dependency ratio leads to a decrease of 1.104% in the difference of the real interest rate. Besides this demographic variable, the method also selects the difference in consumption growth and the lagged real interest rate. The difference in consumption growth is significant at 10% (almost at 5% as the p-value is 0.058) and its sign is again positive. The lagged real interest rate is significant at 5% but it has a negative coefficient in this regression, while it was positive in the levels model. When taking into account the nature of differencing this is logic. As ∆𝑅𝑡 = 𝑅𝑡− 𝑅𝑡−1, it follows that the coefficient for the lagged value of the real interest rate in the levels model (here called 𝛽1) should be the coefficient of the difference model (𝛽2) plus one.3 The size of the coefficient is much smaller, namely -0.159. So the lag has more impact on the real interest rate itself than on the difference in the real interest rate. For this estimation, I also conduct diagnostic tests to secure the OLS conditions, see appendix 7. The test results point out that the regression does not violate the conditions of OLS.
5.3 Error correction model
For the error correction model first I estimate the long term relationship between the real interest rate and its explanatory variables (step 1). I use the residuals of the long term relationship in the total error correction model (step 2). Table 6 displays the results of the stepwise least squares regression of the large time period sample, excluding Gini coefficient and interest rate spread. The upper part of the table gives the results on the long term relationship, whereas the lower part of the table presents the complete ECM, using the residuals of step 1.
2 This is a consequence of the decreased sample size as I use differences and a lag. The sample size influences
the significance and therefore the selected variables are just insignificant.
3 ∆𝑅
𝑡= 𝑅𝑡− 𝑅𝑡−1= 𝛼 + 𝛽2𝑅𝑡−1
𝑅𝑡= 𝛼 + (𝛽2+ 1)𝑅𝑡−1
26 Table 6: Results Stepwise Least Squares regression, Error Correction Model
This table presents regression results for the stepwise least square regression of the error correction model. The estimated regression is: ∆𝑅𝑡 = 𝛼1+ ∑𝑘=1𝑛 𝛽𝑘∆𝑓𝑘,𝑡+ 𝜆(𝑅𝑡−1− 𝛼2−
∑𝑛𝑘=1 𝛾𝑘 𝑓𝑘,𝑡−1) + 𝑣𝑡. The dependent variable is the difference in the real interest rate. For
detailed variable definition, see § 4.1. The independent variables exclude the Gini coefficient and the interest rate spread. The upper part of the table represents the long term relationship, this regression uses the independent variables in their levels form. The lower part of the table represents the short term relationship, this regression uses the explanatory variables in their first difference form and includes the error correction term. The sample contains yearly observations for the years 1972-2015 this results in 44 observations due to the use of differences and one lag. Sources: World Bank, OECD, CPB, CBS
*Significant at 10% **Significant at 5% ***Significant at 1% Long term relationship (step 1)
Dependent variable: real interest rate
All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept -0.020 -0.041 (0.038) (0.033) Dependency ratio -0.636*** -0.584*** (0.104) (0.087)
Saving population ratio 0.255*** 0.262***
(0.084) (0.085) Government balance -0.242 -0.286** (0.148) (0.113) Government debt 0.064*** 0.069*** (0.024) (0.022) Consumption growth -0.373* (0.219) Real GDP growth 0.228 (0.169) Total ECM (step 2)
Dependent variable: difference in real interest rate
Intercept 0.002 -0.000
(0.004) (0.002)
Difference dependency ratio -0.660
(0.898) Difference saving population ratio 0.037
(0.850) Difference government balance -0.172**
(0.082)
Difference government debt 0.027
(0.048) Difference consumption growth 0.180
(0.168)
Difference real GDP growth 0.097
(0.087)
Error correction term -0.216
(0.138)
R2 0.233 0.000
27
The long term relationship is quite clear, the software selects the dependency ratio, the saving population ratio, government balance and government debt. The sign of the dependency ratio is negative, as expected. The sign of the saving population ratio is positive, which is opposite of the expected relationship. The sign on government balance is negative and the government debt has a positive impact, which is both conform expectations. This result indicates that in the long term there is an equilibrium relationship between the real interest rate and the four selected variables. For this relationship, an increase of 0.01 in the dependency ratio leads to decrease of 0.584% in the real interest rate. An increase of 0.01 in the saving population ratio leads to an increase of 0.262% in the interest rate. An increase of 1% in the government balance leads to a decrease of 0.286% in the interest rate. An increase of 1% in government debt leads to an increase of 0.069% in the real interest rate.
I test the residuals of the long term relationship for stationarity with the Augmented Dicky Fuller test (ADF).4 The p-value is 0.003 so I reject the null-hypothesis, concluding that the residuals do not contain a unit root, they are stationary. However, the augmented Dicky Fuller test gives a p-value based on critical values for data of a variable, not for residuals. Engle and Yoo (1987) tabulate new critical values for testing the stationarity of residuals. When using the Engle/Yoo critical values, the null-hypothesis of a unit root cannot be rejected. The critical value from Engle/Yoo for a research with 3 variables is -4.11, while the t-statistic from the test is -3.114. According to these critical values, the residuals do contain a unit root, so they are non-stationary.
As the result on the non-stationarity test of the residuals is ambiguous, I continue estimating the Error Correction Model. The lower part of the table displays the stepwise least squares regression for the total ECM. Interesting to see is that the stepwise least square regression removes all variables from the estimation, even the error correction term. This result seems to point out that this specification does a poor job describing the relationships and processes in interest rate development. As nothing useful results from this model, nothing can be said about the signs and there is also no need to conduct any diagnostic tests around OLS conditions. The only useful lesson to learn from this specification is that the long term relationship is enough to describe the dataset. This long term relationship is comparable to the levels model, which provides support for this model.
28 5.4 Robustness
For robustness, I repeat the stepwise least squares regression for the sample including the Gini coefficient and interest rate spread, this reduces the time period to 1978-2012. The goal is to compare the regressions and discover any dissimilarities in the selected variables. Therefore, I discuss these results only briefly. Table 7 displays the results of the levels model for this smaller sample.
Table 7: Results Robustness Stepwise Least Squares regression, levels model
This table presents regression results for the robustness check for the stepwise least square regression of the levels model. The estimated regression is: 𝑅𝑡 = 𝛼 + ∑𝑛𝑘=1 𝑘𝛽 𝑓𝑘,𝑡 + 𝜃𝑅𝑡−1+ 𝑣𝑡. The dependent variable is the real interest rate. For detailed variable definition, see § 4.1. This regression includes the Gini coefficient and the interest rate spread and the lagged real interest rate as independent variable. The sample contains yearly observations for the years 1978-2012, this results in 35 observations due to the use of one lag. Sources: World Bank, OECD, CPB, CBS, Giniresearch.org, World databank: Microdata, Eurostat, Standardized World Income Inequality Database.
*Significant at 10% **Significant at 5% ***Significant at 1%
The selected variables in the shorter time period differ slightly from the longer sample regression. The variable selection method chooses the dependency ratio in the new regression, where it was not in the longer time period sample. This extra variable changes the magnitude of the coefficients of the other variables slightly, but the regression result of this shorter time
Explanatory variables All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept 0.091 0.076*** (0.058) (0.020) Dependency ratio -0.349* -0.351*** (0.177) (0.086)
Saving population ratio -0.041 (0.167) Government balance -0.221** -0.233*** (0.102) (0.071) Government debt 0.006 (0.027) Consumption growth 0.225 0.244** (0.188) (0.108) Real GDP growth 0.033 (0.125) Gini coefficient 0.015 (0.246)
Interest rate spread -0.006
(0.089)
Lagged real interest rate 0.435** 0.473***
(0.193) (0.103)
R2 0.869 0.868
29
period is not much different from the regular analysis, it is quite robust. Table 8 shows the regression results for the difference model including the Gini coefficient and interest rate spread.
Table 8: Results Robustness Stepwise Least Squares regression, difference model
This table presents regression results for the robustness check for the stepwise least square regression of the difference model. The estimated regression is: ∆𝑅𝑡 = 𝛼 + ∑𝑛𝑘=1𝛽𝑘∆𝑓𝑘,𝑡 + 𝜃𝑅𝑡−1+ 𝑣𝑡. The dependent variable is the difference in the real interest rate. For detailed variable definition, see § 4.1. This regression excludes the Gini coefficient and the interest rate spread and includes the lagged real interest rate as independent variable. The sample contains yearly observations for the years 1978-2012, this results in 35 observations due to the use of differences and one lag. Sources: World Bank, OECD, CPB, CBS, Giniresearch.org, World databank: Microdata, Eurostat, Standardized World Income Inequality Database.
*Significant at 10% **Significant at 5% ***Significant at 1%
The difference model is not very robust as the specification with the shorter time period includes different variables. Instead of using the difference in the dependency ratio and consumption growth, the stepwise least square regression method selects the difference in saving population ratio. However, it should be taken into account that the saving population ratio is also a demographic variable, just like the dependency ratio and that their correlation is also pretty high, 0.607. The specification also includes the lagged real interest rate.
Explanatory variables All variables
Coefficient (S.E.)
Selected variables Coefficient (S.E.)
Intercept 0.015*** 0.015***
(0.005) (0.004)
Difference dependency ratio -0.331
(1.086)
Difference saving population ratio 3.279* 3.914***
(1.619) (0.867)
Difference government balance -0.137* (0.073)
Difference government debt 0.023
(0.041) Difference consumption growth 0.228
(0.177)
Difference real GDP growth 0.005
(0.092)
Difference Gini coefficient 0.157
(0.177) Difference interest rate spread 0.173*
(0.101)
Lagged real interest rate -0.571*** -0.615***
(0.204) (0.139)
R2 0.501 0.407
30
The table below gives the results of the robustness test for the error correction model, with the long term relationship in the upper part and the total ECM in the lower part.
Table 9: Results Robustness Stepwise Least Squares regression, Error Correction Model
This table presents regression results for the robustness check for the stepwise least square regression of the difference model. The estimated regression is: ∆𝑅𝑡 = 𝛼1+ ∑𝑛𝑘=1𝛽𝑘∆𝑓𝑘,𝑡 +
𝜆(𝑅𝑡−1− 𝛼2− ∑𝑛𝑘=1 𝛾𝑘 𝑓𝑘,𝑡−1) + 𝑣𝑡. The dependent variable is the difference in the real
interest rate. For detailed variable definition, see § 4.1. This regression includes the Gini coefficient and the interest rate spread and includes the lagged real interest rate as independent variable. The sample contains yearly observations for the years 1978-2012, this results in 35 observations due to the use of differences and one lag. Sources: World Bank, OECD, CPB, CBS, Giniresearch.org, World databank: Microdata, Eurostat, Standardized World Income Inequality Database.
*Significant at 10% **Significant at 5% ***Significant at 1% Long term relationship (step 1)
Dependent variable: Real interest rate
All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept 0.134** 0.138*** (0.060) (0.016) Dependency ratio -0.579**** -0.703*** (0.155) (0.069)
Saving population ratio -0.096**
(0.178) Government balance -0.185 (0.108) Government debt 0.046 0.059*** (0.021) (0.012) Consumption growth 0.098 (0.193) Real GDP growth 0.111 (0.129) Gini coefficient 0.110 (0.260)
Interest rate spread 0.012
(0.095) Total ECM (step 2) Dependent
variable: Difference in real interest rate All variables Coefficient (S.E.) Selected variables Coefficient (S.E.) Intercept 0.001 -0.001 (0.004) (0.001)
Difference dependency ratio -0.667
(0.990) Difference saving population ratio -0.604
(0.780) Difference government balance -0.209***
(0.068)
Difference government debt -0.023 0.079*
(0.041) (0.040)
31
(0.167)
Difference real GDP growth 0.014
(0.088)
Difference Gini coefficient -0.024
(0.168)
Difference interest rate spread -0.034 0.209**
(0.107) (0.094)
Error correction term 0.463*** -0.867***
(0.144) (0.187)
R2 0.537 0.434
SBIC -6.239 -6.659
The result on the robustness check for the error correction model indicates that the model is not very robust. In this smaller time period the stepwise regression method selects the difference in government debt, the difference in interest rate spread and the error correction term. This is quite different from the longer time period regression which includes no variables at all. However, the long term relationship given in the upper part of the table is quite robust, the regression method selects the exact same variables.
5.5 Model comparison
To compare the models, table 10 reports the R-squared and SBIC. The focus is on the large sample regressions, but for comparison the table also reports the results on the robustness regressions.
Table 10: Model selection
This table presents the R-squared (R2) and Schwarz Bayesian Information criterion (SBIC) for the three different models, for the sample including Gini coefficient and interest rate spread (labeled robustness) and for the sample without these two variables.
Model R2 SBIC Levels 0.839 -6.263 Difference 0.192 -6.254 ECM 0.000 -6.299 Levels (robustness) 0.868 -6.703 Difference (robustness) 0.407 -6.624 ECM (robustness) 0.434 -6.659
32
and long sample of the levels model, whereas for the difference and ECM, the R2 increases considerably in the shorter sample.
5.6 Out-of-sample illustration
As the levels model is the best model to forecast the real interest rate, it is useful to test its ability to forecast out-of-sample. I use the first part of the sample (around two-third, so 29 observations, from 1972-2000)5 to produce a forecast one-step ahead, so for 2001. Then I use information about the coefficients in the period 1972-2001 to produce a forecast for 2002 and so on. Section 3.6 gives more information on the methodology regarding the out-of-sample illustration. Table A9 in Appendix 8 reports the selected variables and their coefficients for every forecast step. The table below displays the forecasted real interest rate, the actual real interest rate and the difference between the forecast and actual value (residual).
Table 11: Out-of-sample performance levels model
This table presents the forecasted real interest rate, the actual real interest rate and the residuals at every forecast step. The total sample is 1972-2015, the forecast sample is initially 1972-2000, updating with new information each year. The recursive forecast method produces forecasts for the years 2001-2015, the method uses one extra observation at every forecast step.
Forecast year
Forecasted real interest rate
Actual real
interest rate Residual
2001 3,22% 0,80% 2,42% 2002 1,36% 1,60% -0,24% 2003 2,19% 2,03% 0,16% 2004 2,23% 2,83% -0,60% 2005 2,54% 1,69% 0,85% 2006 1,52% 2,68% -1,16% 2007 2,34% 2,67% -0,33% 2008 0,25% 1,74% -1,49% 2009 2,85% 2,50% 0,35% 2010 3,28% 1,71% 1,56% 2011 2,41% 0,65% 1,76% 2012 0,97% -0,52% 1,50% 2013 3,41% -0,55% 3,95% 2014 3,04% 0,48% 2,56% 2015 0,56% 0,09% 0,47%
This table shows that the residuals are quite large, there are outliers of almost 4%. The root mean square error is 1.65%. This seems quite substantial, however it is not possible to compare the root mean square error as I test only one model. Given the large deviations, the forecasting
33
should be done with caution. Besides this, it is interesting to see that the stepwise least squares method is quite consequent in selecting variables, for almost all forecast periods the method selects the lagged real interest rate and government balance. However, the levels model based on the complete sample also includes consumption growth, the stepwise least square regression only selects the consumption growth in the last forecast period. This inconsistency raises questions whether the final levels model is suitable for forecasting. It is possible that the relationship between dependent and independent variables changed over time.
5.7 Forecasting
As the levels model is the best estimation, I use this model for forecasting. Therefore, I need forecasts on the government balance and consumption growth. The CPB provides forecasts for several variables, but not for every year. For consumption growth little information is available. The only prediction the CPB makes is 1.0% annual growth for household consumption and 1.3% annual growth for government consumption between 2018 and 2021. As this thesis uses total consumption and no information is available on the distribution of consumption between household and government, the data of the CPB is not usable. Consumption growth is therefore locked at its last value, which was 1.03% in 2015. However, comparing this value to the predictions of CPB, it can at least be said that it is not an extreme value or outlier. The second variable for which I need forecasts is government balance. The CPB provides forecasts for 2016-2021 and 2040 and 2060. Data on this variable is quite optimistic, the CPB believes that the Dutch government will not have deficits in the future.
Predictions of the lagged real interest are not available as inputs, I use the lagged forecast of the real interest rate in this model. A problem arises here, as we can only forecast for the years 2016-2021 and 2040 and 2060 due to data limitations on government balance forecasts. It is not possible to forecast the real interest rate for 2039 and 2059 and as these values are inputs for the years 2040 and 2060, I cannot forecast for these years. Therefore, I only forecast for the years 2016-2021.
