Macroeconomy and yield curve : the Dynamic Nelson Siegel approach

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Macroeconomy and yield curve: The

Dynamic Nelson Siegel approach

Yupei Zhang

2014/8/22

Supervisors:

Mr. Pieter Marres Dr. Zhenzhen Fan

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Acknowledgements

Foremost, I would like to express my sincere gratitude to my supervisors Pieter Marres and Zhenzhen Fan for the continuous support of my thesis, for their patience, motivation, enthusiasm, and great inspiration. Their guidance helped me in all the time of research and writing of this thesis.

My sincere thanks also goes to my family and my dear friends for supporting me spiritually throughout my life.

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Abstract

This paper examines the importance of incorporating macroeconomic data when forecasting the term structure of U.S. interest rates. Yield data and macro data are described and analyzed before introducing econometrical models and Dynamic Nelson-Siegel models. The variations on Nelson-Siegel exponential components framework are used to model the whole yield curve, one period at a time, as a dynamically evolving three-dimensional parameter. Prediction ability of models alters over time period. The results on parsimonious models and Dynamic Nelson-Siegel model are encouraging. Although, limited promising outcomes over Augmented Dynamic Nelson-Siegel model (macro data involved), macroeconomic information is believed containing prediction power over the movement direction of interest rates.

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1. Introduction

Interest rate is well aware of being closely watched by the markets, and extracts information about current state of economy. Numerous literatures have demonstrated that yield curve is a reliable predictor of recession and future economic activity more generally. Indeed, studies have discovered the slope of the yield curve to subsequent changes in GDP, industrial production, consumption and investment. It will be in great use if one can predict the movement of the yield curve.

The interplay of interest rate and macroeconomic factors is in the core of economics study. Central bank can directly manipulate interest rates in pursue of controlling credit availability and inflation level. In crisis period, central bank can use bailout or quantitative easing approaches to resolve the market. The Federal Reserve – the central bank of America can affect the volume of market credit in three ways: first, banks are given bank reserve requirement. Second, the Federal Reserve can lend money to financial institutions with their own securities or loans as collateral. Financial institutions, thus, pay their interests to the Federal Reserve at a discount rate. By this discount rate approach, the Federal Reserve can either encourage or discourage borrowing and hence the supply of bank reserves. Third, the most frequently used and flexible monetary policy tool is open market operations conducted by Federal Open Market Committee. By selling U.S. Treasury bills to the securities dealers, Federal Reserve collects from financial institutions -- whom pay the checks to securities dealers by deducting their reserve accounts at the Federal Reserve. In this way, the Federal Reserve reduces the lending power of the financial institutions largely more than the amount of the payment because of the transaction fees over the trading procedures. Financial institutions can regain their lending ability when the Federal Reserve buys securities and pays for them by crediting its bank reserve accounts. These exercises affect the real economy by selling or buying the funds.

Government also has their power over interest rates. Government policies and their impact on deficit is an indirect indicator of interest rate term structure. Government will have to borrow money from market to fund its deficits. A rising fiscal deficit means government needs to buy more credit from the market. This puts an upward pressure on market interest rates. Government can also increase taxes or lower government spending in order to manage a fiscal contraction. Fiscal contraction will lower real output since less government spending means less disposable income for consumers. And because tax rates go up, demand and output will decrease.

Corporations have to fund their own business. They can fulfill their funding requirement through equity expansions in the financial market or take loans from financial institutions. Bullish trends in the financial markets imply companies to go in for the equity expansion direction. This reduces the demand for funds through borrowing. Therefore, interest rate is expected to go down. A negative correlation between interest rate and household consumption is within the range of imagination. Rational customers would save more and consume less out of current income when interest rise. Moreover, a rising interest rate can be the signal of falling stock, real estate and

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bond prices. If households’ wealth largely consists of those assets, then higher interest rates would lead to a decline in household wealth. With the perception of getting poorer, households would reduce their consumption rate. Consumption is a large proportion in GDP, thus, interest rates would influence real economy through households consumptions. Another approach that interest rates affect macroeconomy would be through mortgages. Investors who are risk-takers and want a slightly higher return will buy mortgages. However, instead of buying them directly, they normally buy securities backed by them, called mortgage-backed securities. When treasury yields rise, these home loans also have to provide higher returns to attract investors which ultimately leads to higher interest burden of house owners. And finally, this kind of securities became the triggers of financial crisis. But it will not discuss in details in this paper.

These four economic forces play important roles in influencing interest rates. There are other factors which have significant effects on interest rate as well, such as inflation, unemployment rate, industrial manufacturing output etc. But this paper will only focus on the cash flows between government, central bank, business sector (financial and non-financial) and household, and how do these cash flows influence interest rate movements.

Now it comes to the problem of building a good model to forecast interest rate trends. Abounding models are been examined in literature, but they are very different in form and fit (Diebold et al., 2006). A substantial gap exists between economists that they concentrate on different aspects of macroeconomics in determination of yields, thus, the models they built are very different from each other. The dynamic Nelson-Siegel model bridges this gap by formulating and estimating a yield curve model that integrates macroeconomic factors (Diebold et al., 2006). In this paper, Nelson-Siegel exponential components framework is employed to extract the entire yield curve, one period at a time, into a three-dimensional parameter that evolves dynamically. Those three time-varying parameters can be interpreted as factors. Different from factor analysis in which one estimates both the unknown factors and the factor loadings, the Nelson-Siegel framework specifies structure on the factor loadings (Diebold and Li, 2006). The advantages of doing this are speeds up highly precise estimation of the factors and allow interpretation of factors as level, slope and curvature.

The structure of this paper is as following. Section 2 presents what have literature done in forecasting interest rate with macro variables. Second 3 introduces comparative models, dynamic Nelson-Siegel model and Augmented Dynamic Nelson-Siegel model (DNSX). Next section analyzes sample data used in this paper. The following section shows the estimation procedures and accuracy. Section 6 reports the forecasting results and different models are compared. Section 7 demonstrate outcome of evaluation step. Finally, some recommendations are mentioned according to the development of this paper.

2. Literature review

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yield curve statistics, it has little saying on the nature of the underlying economic factors which drive the movements of yield curve. To examine the influence of macro indexes on interest rate, recent literatures have actively taken account of the interaction between interest rate and macroeconomy activities. The most considered macroeconomic variables in yield curve study are inflation, unemployment, consumption and manufacturing output as they closely related to GDP, and GDP is the most important indicator in macroeconomics. For instance, Diebold et al. (2005) originates an interpretation of the Nelson-Siegel representation by jointly employed VAR dynamics for the macroeconomy. They utilized maximum likelihood estimation approach to extracts three latent factors (level, slope and curvature) from U.S. treasury securities with 17 maturities and related the three latent factors to known macroeconomic factors (inflation, monetary policy instrument etc).

Andrew et al.(2003) and Ang et al. (2004) use inflation and real activity as macroeconomic factors. They employ VAR model to capture the dynamics of macro factors and additional latent factors. By using first principle component of bunch of series of macroeconomic data, Ang and Piazzesi (2003) avoid relying on particular macro variables. This sheds light on filtering macro data in this paper.

Nelson-Siegel factors indeed have relation with macroeconomic variables (Diebold et al., 2005). The level factor moves the similar pattern with inflation while the slope factor moves the similar pattern with real activity. There is no correlation found between the curvature factor and any of the main macroeconomic variables. Rudebusch and Wu (2014a) also found the comparable results. They discover the relationship between the level factor and market participants’ perception about the underlying or medium-term inflation target of the Federal Reserve. They also state that the cyclical response of the Federal Reserve has close relation with the slope factor. Central bank manipulates the short-term yield to accomplish their targets of stabilizing the real economy and keeping inflation rate to target. Interestingly, the level factor shocks feed back to the real economy through the interest rate beforehand.

In terms of the estimation direction, Ang et al. (2003) and Diebold et al. (2005) follow different approaches. Ang et al. (2003) only examine macro factors’ influence over yield curve, they simplify their arbitrage-free model by not considering the yields’ influence over macro factors. On the other hand, Diebold et al. (2005) take into account of a bidirectional dynamic interaction between macroeconomics and yield curves. They conclude that the influence of macro factors over yield curves is stronger than the opposite way. Ang et al. (2004) use a bidirectional direction as well. Their conclusion is that the amount of yield errors can be attributed to macro variables depends on the system allows for bidirectional linkages or not. If the estimation only limited to unidirectional path, the amount of yield errors can be explained by macro factors is very restricted. The bidirectional model, however, attributes over 50% of the variance of long term yields to macro factors.

Interest rate fitting is being widely modeled and tested; accurate results are generated with small amount of errors. However, predicting interest rate is still a difficult task to challenge. Since the nature of yields under all maturities is close to being non-stationary, thus, a simple random walk

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model can outperform any other models. This will elaborate further in the coming section. Ang and Piazzesi (2003) and Duffee (2002) point out that it is hard to outperform random walk model in terms of forecasting accuracy, especially vector autoregressive model. In recent decade, the forecasting accuracy has improved. Diebold and Li (2006) and Christensen et al. (2009) claim that dynamic Nelson-Siegel style factor models have more accurate results. Thus, this paper will discover further insight information of yield curves on Nelson-Siegel models.

3. Yield curve models

A range of models that are commonly used in research are applied in this section to forecast future yield curves and results of each model are being compared to test their performance. Earlier literature has shown that parsimonious models have better performance than sophisticated models, therefore, unrestricted linear econometrics models are considered as comparison models. The benchmark model is random walk model. Parametric structured model – Nelson-Siegel model is the main model of this thesis.

3.1 Random walk model

First model in this section is benchmark model – Random walk. For each maturity𝜏𝑖, 𝑖 = 1, … , 𝑁,

𝑦𝑡(𝜏𝑖)= 𝑦𝑡−1(𝜏𝑖)+ 𝜀𝑡(𝜏𝑖), 𝜀𝑡(𝜏𝑖)~𝒩(0, 𝜎(𝜏𝑖)2) (3.1) It can be deduced that one step ahead forecast in random walk model is equal to the last observation, i.e. 𝑦̂𝑇+1(𝜏𝑖)= 𝑦̂𝑇(𝜏𝑖). Random walk model is widely used in literature as the benchmark against which to access the forecast power of other models. The nature of yield is certainly non-stationary as the first-order autocorrelation coefficients are all very close to one. The non-stationary assumption is hard to interpret in economic field and the natural of non-stationary yields that they can move freely and do not revert back to a long-term mean, contradicts monetary policies set by the Federal Reserve.

3.2 AR model

In this paper, only first lag is considered. Therefore, first-order autoregressive model is presented in the following manner.

𝑦𝑡(𝜏𝑖)= 𝑐(𝜏𝑖)+ 𝜙(𝜏𝑖)𝑦𝑡−1(𝜏𝑖)+ 𝜑(𝜏𝑖)′𝑋𝑡+ 𝜀𝑡(𝜏𝑖), 𝜀𝑡(𝜏𝑖)~𝒩(0, 𝜎(𝜏𝑖)2) (3.2) where 𝑐(𝜏𝑖)_{, 𝜙}(𝜏𝑖)_{and 𝜎}(𝜏𝑖)_{are scalar parameters and 𝜑}(𝜏𝑖)′ _{is the parameter of}

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macroeconomic factors. In next section, AR model with and without macro factors are applied, with only yield data expressed as AR and with macro factors expressed as ARX. One step ahead forecast is constructed by regressing 𝑦𝑡(𝜏𝑖). Then obtain the parameters and use the parameters on 𝑦𝑡(𝜏𝑖) to acquire 𝑦̂𝑇+1(𝜏𝑖). Moving window forecast is employed in prediction practice. Two-thirds of the data are used to predict the rest one-third of the data. The remaining models are using the same regression method and moving window forecast. 𝑐̂1𝑇 and 𝜙̂1𝑇 denote estimated parameters, the following equation represents the 1 quarter ahead forecasting procedure.

𝑦̂𝑡+1𝑇 = 𝑐̂1𝑇+ 𝜙̂1𝑇𝑦1𝑇 (3.2.1) The augmented AR model (ARX) forecasting procedure can be expressed as the following.

𝑦̂𝑡+1𝑇 = 𝑐̂1𝑇+ 𝜙̂1𝑇𝑦1𝑇+ 𝜑̂1𝑇𝑋1𝑇 (3.2.2)

3.3 VAR model

Vector autoregressive model allows for other maturities involved as additional information on top of any maturity’s own history. VAR model with one lag can be expressed as following.

𝑌𝑡= 𝑐 + Φ𝑌𝑡−1+ Ψ𝑋𝑡−1+ 𝜀𝑡, 𝜀𝑡~𝒩(0, 𝐼) (3.3) where 𝑐 and Φ scalar parameters and Ψ is the parameter for macro variables. The regression and forecasting approach is similar to AR model. The VAR model with and without macro factors is defined as VARX and VAR respectively.

Similar to AR model, the prediction procedure of VAR model is as following.

𝑌̂𝑡+1𝑇 = 𝑐̂1+ Φ̂1𝑌𝑡+ Ψ̂1𝑋𝑡 (3.3.1)

3.4 Dynamic Nelson-Siegel model

Diebold and Li (2006) state that on the base of static Nelson and Siegel (1987) model, a dynamic factor model produces highly accurate interest rate predictions. The representation of Dynamic Nelson-Siegel model is as following,

𝑦𝑡(𝜏) = 𝛽1,𝑡+ 𝛽2,𝑡[1−𝑒 −𝜏𝜆 𝜏 𝜆 ] + 𝛽3,𝑡[1−𝑒 −𝜏𝜆 𝜏 𝜆 − 𝑒−𝜏𝜆] (3.4)

where 𝜏 is maturities; the parameters 𝛽1,𝑡, 𝛽2,𝑡 and 𝛽3,𝑡 are determined by observations and 𝜆, the decay parameter, is fixed at 0.0609 which is the optimize result given by Diebold and Li

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(2006). This prespecified parameter allows calculating the values of beta’s loadings. The model is a widely used and uncomplicated method for term structure estimation. The beta loadings [1,1−𝑒𝜏−𝜏𝜆

𝜆

, 1−𝑒𝜏−𝜏𝜆 𝜆

− 𝑒−𝜏𝜆] enable the model to capture a range of monotonic, humped, curved

and S-type shapes frequently observed in yield data. The three loadings can be interpreted as short, medium and long-term components. As can be seen from the graph below, beta1’s loading is constant at value 1 while beta 2’s loading is exponentially decreased. Loading 3 is climbing at first then reaches its maximum point, and finally decrease to the same level as loading 2. The maximum value of loading 3 is determined by lambda value, and is the value where the first inflection point of yield curve occurs. The last term of loading 3 (𝑒−𝜏𝜆_{) is decreased to 0 when}

maturity 𝜏 grows larger and larger. Thus loading 2 and loading 3 are gradually moving to the same level.

Nelson Siegel parameters are estimated using Ordinary Least Square Regression. The long-term component is the loading on 𝛽1,𝑡 since it is constant at 1 thus the same for every maturity. 𝛽1,𝑡 is hence representing the long term interest rate level. From the result graph in estimation section, we can see that 𝛽1,𝑡 are very close to 30-year or 20-year interest rates to each yield curves. The loadings of second parameter is an exponential decay function, thus a positive 𝛽2,𝑡 value means downward sloping yield curve, vice versa. A positive 𝛽3,𝑡 represents a hump in the curve and a negative 𝛽3,𝑡 represents a trough in the curve in Nelson Siegel model. The larger the absolute value of 𝛽3,𝑡 is, the more pronounced the hump or trough.

Figure. 1 Factor loadings

0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1

Factor Loadings for Nelson Siegel Model with time factor of 0.0609

maturity

Beta1 Beta2 Beta3

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Forecasting steps of Nelson-Siegel models are similar to AR model with one exception that the target variable is not 𝑦 but 𝛽. With macro variable 𝑋, each of three factors has 𝑇 × 1 vector of macro factors 𝑋ℎ for each predict horizon ℎ.

𝛽𝑡= 𝛿0+ 𝛿1𝛽𝑡−ℎ+ 𝛿3𝑋ℎ,𝑡−ℎ+ 𝜀𝑡 (3.4.1) Next equation describes the forecast procedure with 𝛿̂0, 𝛿̂1 and 𝛿̂3 represent estimated parameters.

𝛽̂𝑡+ℎ= 𝛿̂0+ 𝛿̂1𝛽 + 𝛿̂3𝑋𝑡,ℎ (3.4.2) Then, yield curve can be estimated.

𝑌̂𝑡+ℎ= [1,1−𝑒 −𝜏𝜆 𝜏 𝜆 , 1−𝑒𝜏−𝜏𝜆 𝜆 − 𝑒−𝜏𝜆] ∗ 𝛽̂_𝑡+ℎ_(3.4.3)

4. Data

In this section, empirical data is described and analyzed. This paper only considers U.S. economic data, the reasons are 1) U.S. data is easily accessed; 2) macroeconomic data is comprehensive and reliable.

4.1 Yield curve data

Quarterly secondary market rates on U.S. Treasury bill with maturities 3-month, 6-month and 1-year are used from the first quarter 1977 through first quarter 2014, taken from the federal reserve bank (FRED) of St. Louis. Yields with maturity horizons from 3-year, 5-year, 7-year, 10-year, 20-year and 30-year are extracted from the FRED Treasury constant maturity rate category, all rates are quarterly selected from first quarter 1977 to first quarter 2014. However, FRED is lacking of 20-year yields from first quarter 1987 to third quarter 1993 and 30-year yields from first quarter 2002 to the last quarter of 2005. The missing parts are replaced by Wright data (Gurkaynak, Sack, and Wringt 2007) to construct thorough yield curves.

The various yields will play an essential role in the sequel. Therefore, it is necessary to have a look at the details of yield curves. Figure.2 is a three dimensional plot of the data. As can be seen from the plot, from 1977 to 2014 yields are climbing at first. Then they start a long slump from late 80s until recently. From Table. 1 it is obvious that the yield curves are upward sloping which can be explained by market expectation theory. And long-term interest rates are less volatile then short term rates. It is interesting to notice that most curves have a little hump at the end of the curve – 20-year yields have higher average value than 30-year yields. A possible explanation of this

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situation is investors have had high demand for the 30-year long bond, which raises the price of bond value.

Figure. 2 Yield curves, 1977 Q1 – 2014 Q1. The sample consists of monthly yield data at maturities of 3, 6, 12, 24, 36, 60, 84, 120, 240 and 360 months.

3 month 6 month 1 year 5 year 10 year 20 year 30 year

Mean 5.00 5.12 5.56 6.35 6.78 7.19 7.14 Std. Dev 3.55 3.52 3.79 3.41 3.10 2.84 2.74 Minimum 0.01 0.04 0.10 0.67 1.62 2.31 2.70 Maximum 15.49 15.03 16.52 15.93 15.32 15.07 14.67 ρ̂(1) 0.95 0.96 0.96 0.97 0.97 0.98 0.97 ρ̂(12) 0.58 0.60 0.64 0.76 0.79 0.79 0.79 ρ̂(30) 0.66 0.67 0.69 0.74 0.75 0.75 0.75

Table. 1 Descriptive statistics of yield curve

4.2 Macroeconomic data

Macroeconomic factors are from U.S. Federal Reserve Board statistical releases. Five major economical categories are selected – Non-financial Business (NFB), State Governments (SG), Federal Government (FG), Households and Nonprofit Organizations (HNO) and Financial Business (FB). Under each category around 50 series are listed. For instance, income before taxes, gross

0 200 400 1975 1980 1985 1990 1995 2000 2005 2010 2015 0 2 4 6 8 10 12 14 16 Time Maturity Y ie ld (P e rc e n t)

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saving less net capital transfers paid, total capital expenditures etc. are under Non-financial Business category. All the macroeconomic data has the same time frame as yield curves – quarterly data date back to first quarter 1977.

The data is standardized before Principle Component Analysis (PCA) is employed. PCA helps to analyze and filter the data to ensure them well constructed. Only first two principle components are chosen over each of five categories. The first components account for over 70% of variation in all the macro panels while first two components together account for over 80% of variation. Hence, the use of first two PCAs over five panels should provide ample information on macroeconomic data set. Incomes play important role in Non-financial Business and Households and Non-profit Organizations while financial asset is an influential factor in Financial Business. Expenditure has its power on State governments and federal government’s cash flow.

The following graph reports the macro data used in models. Immense turbulence occurs at later period of time frame, which is possibly related to U.S. financial crisis happened at the end of 2008. Interesting to notice that during the financial crisis, the second PCA component of Financial Business sector’s cash flow plummets sharply to reach its bottom, while the second PCA components of Non-Financial Business and Federal Government sectors’ cash flow roar to their zeniths.

Figure. 3 First two PCA components of each category (standardized)

5. Estimation

First, betas are calculated using OLS regression on yield data with Nelson-Siegel equation. Some comparative graphs are presented to illustrate that Nelson-Siegel factors indeed capture the shape of yield curve.

OLS regression is employed on all models. Dataset is divided into two parts. First part starts from -12 -10 -8 -6 -4 -2 0 2 4 6 8 19 77Q1 19 78Q3 19 80Q1 19 81Q3 19 83 Q1 19 84Q3 19 86Q1 19 87Q3 19 89Q1 19 90Q3 19 92Q1 19 93Q3 19 95Q1 19 96Q3 19 98Q1 19 99Q3 20 01Q1 20 02Q3 20 04 Q1 20 05Q3 20 07Q1 20 08Q3 20 10Q1 20 11Q3 20 13Q1 NFB PCA 1 NFB PCA 2 SG PCA 1 SG PCA 2 FG PCA 1 FG PCA 2 HNO PCA 1 HNO PCA 2 FB PCA 1 FB PCA 2

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first quarter 1977 to fourth quarter 2001, includes 100 observations, is used as estimation sample. Second part from second quarter 2002 to first quarter 2014, includes 48 observations, is used as comparison sample. A recursive estimation procedure – moving window is applied. The moving step is one. The initial sample is from first season 1977 to fourth season 2001 (100 observations). Then this sample is used to estimate the parameters of next date point – first season 2002. Next section, the parameters obtained from last procedure is utilized to predict date point – second season 2002’s value.

Since Random walk model one period ahead prediction is simply the last observation, no estimation is needed in this model. In this section, the estimation results of AR, ARX, DNS and DNSX models are presented separately.

5.1 Fitting yield curves

OLS regression is employed to obtain a time series of estimates of {𝛽̂1𝑡, 𝛽̂2𝑡, 𝛽̂3𝑡} from the yield data. An assessment of model fitness can be seen from Figure. 8 where the average fitted yield curve and the average actual yield curve are plotted. The two lines are fairly close to each other. Figure. 4 is a comparative plot of three factors. Level factor shows high persistence in Figure. 4. It is climbing as yields rising at the beginning, then it decrease gradually with downtrend of the yields. Slope and curvature factors are more volatile than level factor. They can be both positive and negative. Figure. 5 to 7 are isolated factors with different empirical approximations.

Figure. 5 displays the estimated level factor and 30-year yield. The two lines support the interpretation of level factor as long-term interest rate level. Figure. 6 shows the estimated slope factor and the difference between 30-year and 3-month yields. The empirical proxy successfully resembles the slope factor. Finally, in Figure. 7 displays the estimated curvature and the difference between twice the 2-year yield minus the sum of 20-year and 3-month yield. Although variation occurs at some time period, it cannot be rejected that these two curves’ trend resembles each other.

Figure. 4 Estimates of level, slope and curvature -10 -5 0 5 10 15 20 19 77/ 3/ 1 19 78/ 8/ 1 19 80/ 1/ 1 19 81/ 6/ 1 19 82/1 1 /1 19 84/ 4/ 1 19 85/ 9/ 1 19 87/ 2/ 1 19 88/ 7/ 1 19 89/1 2 /1 19 91/ 5/ 1 19 92/1 0 /1 19 94/ 3/ 1 19 95/ 8/ 1 19 97/1/ 1 19 98/ 6/ 1 19 99/1 1 /1 20 01/ 4/ 1 20 02/ 9/ 1 20 04/ 2/ 1 20 05/ 7/ 1 20 06/1 2 /1 20 08/ 5/ 1 20 09/1 0 /1 20 11/ 3/ 1 20 12/ 8/ 1 20 14/ 1/ 1 Per ce n t Level factor Slope factor Curvature factor

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Figure. 5 Estimates of level factor and 30-year yield

Figure. 6 Estimates of slope factor and the difference between the 30-year and 3-month yields (Slope factor multiplies by -1 to make a comparable curve)

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 3/ 1 /7 7 9/ 1 /7 8 3/ 1 /8 0 9/ 1 /8 1 3/ 1 /8 3 9/ 1 /8 4 3/ 1 /8 6 9/1 /87 3/1 /89 9/1 /90 3/1 /92 9/ 1 /9 3 3/ 1 /9 5 9/ 1 /9 6 3/ 1 /9 8 9/ 1 /9 9 3/ 1 /0 1 9/ 1 /0 2 3/ 1 /0 4 9/ 1 /0 5 3/ 1 /0 7 9/ 1 /0 8 3/ 1 /1 0 9/ 1 /1 1 3/ 1 /1 3 p ER CE N T 30 year Level factor -4 -3 -2 -1 0 1 2 3 4 5 6 3/1 /77 8/1 /78 1/1 /80 6/1 /81 11 /1/82 _4/1/84 _9/1/85 _2/1/87 _7/1/88 12 /1/89 _5/1/91 10 /1/92 _3/1/9 4 8/1 /95 1/1 /97 6/1 /98 11 /1/99 _4/1/01 _9/1/02 _2/1/04 _7/1/05 12 /1 /0 6 5/1 /08 10 /1/09 _3/1/11 _8/1/12 _1/1/14 Per ce n t

Positive Slope Factor 30-year - 3-month

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Figure. 7 Estimates of 0.3 times curvature and the difference between twice the 2-year yield minus the sum of the 3-month and 20-year yields

Figure. 8 Actual and fitted average yield curve

5.2 AR and VAR results

The estimation results presented below use full sample data from first quarter 1977 to first quarter 2014 to give an impression of the quality of estimation. Statistical descriptions of AR and ARX are reported. VAR model has the similar parameters with AR model. The constant parameter can be seen as the sample mean of interest data under different maturities. The second parameter(s) is the autocorrelation coefficient of the yields. As can be seen from the table, the results are justified. The second parameters are very close to 1, which is the same as statistical description of yield data listed earlier this paper. The p-values with 90% confidence all give credits to significant parameters. -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 3/1 /77 9/1 /78 3/1 /80 9/1 /81 3/ 1 /8 3 9/1 /84 3/1 /86 9/1 /87 3/1 /89 9/1 /90 3/1 /92 9/1 /93 3/ 1 /9 5 9/1 /96 3/1 /98 9/1 /99 3/1 /01 9/1 /02 3/1 /04 9/1 /05 3/1 /07 9/1 /08 3/1 /10 9/1 /11 3/1 /13 Per ce n t 0.3Curvature

2*(2-year ) - 20-year - 3-month

0 50 100 150 200 250 300 350 400 4.5 5 5.5 6 6.5 7 Maturity(Months) Y ie ld (P er ce nt ) Actual Fitted NS

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From Table. 3 and Table. 4 we can see that augmented AR model has higher long-term mean under all maturities than AR model. The autocorrelation of yield data with 1 displacement has weaken. All the parameters are significant according to p-value under 90% confidence.

AR 𝑐 𝜏 1 − 𝜙𝜏 𝜙𝜏 p-value 3 month 4.39 0.95 0.0000 6 month 4.35 0.96 0.0000 1 year 4.60 0.96 0.0000 2 year 4.59 0.97 0.0000 3 year 4.61 0.97 0.0000 5 year 4.94 0.98 0.0000 7 year 5.13 0.98 0.0000 10 year 5.25 0.98 0.0000 20 year 5.70 0.98 0.0000 30 year 5.79 0.98 0.0000

Table. 2 AR(1) estimated parameters and p-value. The second column is the approximation of each maturity’s long-term mean. The third column is the autocorrelation score of yield data with lag 1. The last column is p-values under 90% confidence. All the parameters are proved to be significant. ARX 𝑐 𝜏 1 − 𝜙𝜏 𝜙𝜏 p-value 3 month 4.88 0.73 0.0000 6 month 4.98 0.75 0.0000 1 year 5.40 0.77 0.0000 2 year 5.67 0.81 0.0000 3 year 5.82 0.83 0.0000 5 year 6.12 0.84 0.0000 7 year 6.36 0.86 0.0000 10 year 6.52 0.87 0.0000 20 year 6.94 0.88 0.0000 30 year 6.89 0.88 0.0000

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17 NFB1 NFB2 SG1 SG2 FG1 FG2 HNO1 HNO2 FB1 FB2 3 month 0.73 -0.07 -0.53 -0.02 -0.44 0.30 -0.38 -0.11 -0.08 0.03 6 month 0.63 -0.06 -0.46 -0.01 -0.40 0.28 -0.41 -0.06 0.00 0.04 1 year 0.54 -0.06 -0.42 -0.01 -0.36 0.27 -0.47 -0.03 0.09 0.05 2 year 0.31 -0.05 -0.37 0.00 -0.19 0.20 -0.49 0.02 0.21 0.05 3 year 0.20 -0.06 -0.35 0.01 -0.12 0.18 -0.44 0.05 0.27 0.02 5 year 0.13 -0.07 -0.18 0.00 -0.23 0.17 -0.38 0.09 0.25 0.01 7 year 0.11 -0.08 -0.08 0.00 -0.28 0.17 -0.33 0.11 0.24 -0.01 10 year 0.09 -0.07 -0.05 0.00 -0.29 0.16 -0.26 0.12 0.22 0.00 20 year 0.08 -0.09 0.01 -0.01 -0.36 0.16 -0.21 0.14 0.21 -0.03 30 year 0.07 -0.08 0.14 -0.01 -0.38 0.14 -0.25 0.11 0.15 -0.03

Table. 3 and Table 4 ARX(1) estimated parameters. The upper table reports the same parameters as AR(1) model. The table below reports the estimated macro variable parameters 𝜑𝜏, from left to right, the columns represent Non-financial business PCA 1&2, State governments PCA 1&2, Federal Government PCA 1&2, Households and Non-profitable organization PCA 1&2 and Financial business PCA 1&2. P-values show that all the parameters under these two tables are significant.

5.3 DNS and DNSX results

The fit of Nelson-Siegel factors is analyzed in section 5.1. The following table compares the variation in estimated yield and variation in actual yield on two different models. DNS seems to have higher quality of approximation with autocorrelation close to 1 while DNSX model has poorer performance. DNSX DNS 3 month 0.88 0.96 6 month 0.87 0.96 1 year 0.86 0.96 2 year 0.82 0.94 3 year 0.79 0.93 5 year 0.75 0.91 7 year 0.72 0.89 10 year 0.69 0.88 20 year 0.65 0.86 30 year 0.66 0.86

Table. 5 Fit goodness of DNSX and DNS model expressed using the correlation between variation in 𝑦̂𝑡𝜏 and variation in 𝑦𝑡𝜏.

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6. Forecasting

As discussed in the previous section, the forecasting sample is comprised of the remaining sample of estimation sample. Only one quarter ahead forecast is carried out in this paper. The following graphs’ data is in percent. 3-month, 2-year, 5-year, 10-year and 30-year maturities are selected in represent of short-term, mid-term and long-term maturities. The predictions of Random walk, AR, VAR, DNS, DNSX and ARX together with actual data are plotted on the same graph for comparison purpose.

Figure. 9 3-month forecast

Figure. 10 2-year forecast -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 20 02/6 /1 20 03/2 /1 20 03/1 0 /1 20 04 /6 /1 20 05/2 /1 20 05/1 0 /1 20 06/6 /1 20 07/2 /1 20 07/1 0 /1 20 08/6 /1 20 09/2 /1 20 09/1 0 /1 20 10 /6 /1 20 11/2 /1 20 11/1 0 /1 20 12/6 /1 20 13/2 /1 20 13/1 0 /1 Yi e ld s (In p e rc e n t) Time Actual data AR RW VAR DNS DNSX ARX -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 20 02/6 /1 20 03/2 /1 20 03/1 0 /1 20 04/6 /1 20 05/2 /1 20 05/1 0 /1 20 06/6 /1 20 07/2 /1 20 07/1 0 /1 20 08/6 /1 20 09/2 /1 20 09/1 0 /1 20 10/6 /1 20 11/2 /1 20 11/1 0 /1 20 12/6 /1 20 13/2 /1 20 13/1 0 /1 Yi e ld s (In p e rc e n t) Time Actual data AR RW VAR DNS DNSX ARX

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Figure.11 5-year forecast

Figure. 12 10-year forecast

Figure. 13 30-year forecast -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 20 02/6 /1 20 03/2 /1 20 03/1 0 /1 20 04/6 /1 20 05/2 /1 20 05/1 0 /1 20 06/6 /1 20 07/2 /1 20 07/1 0 /1 20 08/6 /1 20 09/2 /1 20 09/1 0 /1 20 10/6 /1 20 11/2 /1 20 11/1 0 /1 20 12/6 /1 20 13/2 /1 20 13/1 0 /1 Yi e ld s (In p e rc e n t) Time Actual data AR RW VAR DNS DNSX ARX -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 20 02/6 /1 20 03/2 /1 20 03/1 0 /1 20 04/6 /1 20 05/2 /1 20 05 /1 0 /1 20 06/6 /1 20 07/2 /1 20 07/1 0 /1 20 08/6 /1 20 09/2 /1 20 09/1 0 /1 20 10 /6 /1 20 11/2 /1 20 11/1 0 /1 20 12/6 /1 20 13/2 /1 20 13/1 0 /1 Yi e ld s (In p e rc e n t) Time Actual data AR RW VAR DNS DNSX ARX -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 20 02/ 6/ 1 20 03/ 2/ 1 20 03/1 0 /1 20 04/ 6/ 1 20 05/ 2/ 1 20 05/1 0 /1 20 06/ 6/ 1 20 07/ 2/ 1 20 07/1 0 /1 20 08/ 6/ 1 20 09/ 2/ 1 20 09/1 0 /1 20 10/ 6/ 1 20 11/ 2/ 1 20 11/1 0 /1 20 12/ 6/ 1 20 13/ 2/ 1 20 13/1 0 /1 Yi e ld s (In p e rc e n t) Time Actual data AR RW VAR DNS DNSX ARX

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In Figure. 9 and Figure. 10, curves without macroeconomic data move in a sneak shape with a sizeable shaking at the beginning of forecast period, then they tend to be stable in the later period. As the maturity going up, curves seem to be more volatile and the forecasting curves are difficult to capture the movements of actual curve. In Figure. 11, Figure. 12 and Figure. 13, non-augmented curves do not have a sizeable hump from the start. But they tend to be more volatile than small maturities in the later period.

From the forecast graphs above, it is not difficult to recognize the exceptional behavior of DNSX and ARX models happens at the end of 2008 till the end of 2009 which is exactly financial crisis time. DNSX model starts its chaotic movement at around middle of 2003 and continues several quarters’ abnormal performance until the later stage of financial crisis – beginning of 2010. Then its predictions are similar to other models’ prediction. Also it is interesting to note that DNSX prediction lines are almost beneath other non-augmented models’ prediction lines under these selected maturities which leads to very high prediction errors over the rest comparative models. This point will elaborate again in evaluation section. ARX moves even wildly than DNSX model. It starts its abnormal movement at the beginning of the prediction period. With early maturities 3-month and 2-year, ARX prediction curves behave similarly as DNSX model. Then as maturity goes up, its movements become more volatile. This might shed a light that macroeconomic data is a considerable impact factor over yield curve models. By analyzing the macro data, it indeed has chaotic dataset starts at the beginning of 2000s. And the lines move dramatically during the crisis period.

Random walk model seems to move lagged behind the actual curves under all selected maturities. This is simply because that this model’s prediction is its latest observation. On 3-month and 2-year forecasts, Random walk, AR, VAR and DNS are moving closely with actual curve. 5-year, 10-year and 30-year forecasts seem to have larger fluctuations.

7. Evaluation

A valuable model of dynamic yield curves should not only fit well in-sample, but also forecast well out-of-sample. The evaluation methods used in this part is Root Mean Squared Prediction Error (RMSPE) and Hit Ratio.

RMSPE is a single statistic summing the individual forecasting errors over an entire forecast period. It is a popular statistic in forecast evaluation. The model-𝑚, time-𝑇 RMSPE for a 𝜏-month maturity is given by

𝑅𝑀𝑆𝑃𝐸𝑚𝑇 = √_𝑇1∑𝑇𝑡=1(𝑦̂𝑡+1|𝑡,𝑚𝑇 − 𝑦𝑡+1𝑇 )2 (7.1) where 𝑦𝑡+1𝑇 is the yield for a 𝜏-month maturity observed at time 𝑡 + 1, while 𝑦̂𝑡+1|𝑡,𝑚𝑇 is its

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model- 𝑚 forecast, made at time 𝑡. Since Random walk model is set as benchmark model, the RMSPEs of the remaining models on each maturities divide the Random walk RMSPE to get a comparative ratio. The ratio larger than one implies that the model under performs Random walk model while the ratio smaller than one implies an out Random walk model performance.

3 month 1 year 5 year 10 year 30 year

RW 1 1 1 1 1 AR 1.08 1.06 1.05 1.03 1.02 VAR 1.18 1.22 1.14 1.11 1.07 DNS 1.14 1.08 1.11 1.11 1.01 DNSX 2.11 2.27 1.77 1.73 1.84 ARX 2.10 2.18 2.37 2.68 2.94

Table. 6 RMSPE results over Random walk, AR, VAR, DNS, DNSX and ARX models on selected maturities from second quarter 2002 to first quarter 2014

RMSPE results are not positive in regards of ARX and DNSX models. As reported in Table. 6, the RMSPEs of ARX and DNSX model on 3-month maturity are twice worse than Random walk model. DNSX model’s RMSPE has the highest ratio on 1-year maturity. And the situation eases on mid-term maturities, then goes worse again at the long-end of the yield curve. ARX is the worst, the 30-year RMSPE is even three times worse than Random walk model. ARX and DNSX model have the most number of parameters over these models. It has 100 × 1 dimension for constant parameter, 3 × 3 for second parameter – Nelson-Siegel factors and 3 × 10 dimension for macro variables. Those variables might cause too much unwanted noises in estimation procedure. As can be seen from the figures in forecasting part, DNSX model seems to amplify the fluctuation of the actual yield curve. With the same downwards trend in yield data and macro data during financial crisis period (2008-2009), the predicted yields in this period plummet sharply to -4.00%. On the other hand, AR model is the best forecast model under this text statistic. RMSPE ratios under all maturities are very close to Random walk model. A possible explanation is that Random walk’s prediction is simply the last observation and AR(1) model specifies that the regressor depends linearly on its one period lagged values, which means they all captures how closely are past observations and the latest observation relates to each other. DNS model cannot outperform AR model, but it has smaller errors than VAR model. Hence, DNS model can be considered a predictive model. It provides good estimates of the term structure of interest rate by taking account of yield curve’s shape. VAR model is not prominent under this test statistic. The enlarged parameter matrix may cause more noises than AR model.

The Hit ratio, however, not compares the value of errors between sample and predictions, but investigates whether the move directions of the sample and predictions are the same. The Hit ratio can be calculated with the following equations.

𝑣𝑡,𝜏,1≡ 1𝑦_𝑡+1𝜏 _−𝑦

𝑡𝜏>0 (7.2)

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if the rate goes up and zero when it goes down.

𝑣̂𝑡,𝜏,1𝑚 = 1𝑦̂_{𝑡+1|𝑡,𝑚}𝜏 _−𝑦

𝑡𝜏>0 (7.3)

(7.3) is an indicator function which specifies the direction of change of predicted interest rates is one if the predicted rate goes up and zero when it goes down.

𝐻𝑡,𝜏,1𝑚 =_𝑇1∑ (1 𝑣̂_𝜏,1𝑚_=𝑣 𝜏,1

𝑇

𝑡=1 ) (7.4)

(7.4) is the expression of the Hit ratio. It obtained by comparing the results of (7.2) and (7.3) and computes the average times they agree with each other. The larger the Hit ratio is, the better the direction prediction of the model. It can be deduced that the Random walk model has Hit ratio value 0.5. The following table is the Hit ratios of various models.

Hit ratio 3 month 1 year 3 year 5 year 10 year 30 year

AR 0.66 0.53 0.51 0.47 0.47 0.47

VAR 0.53 0.57 0.47 0.51 0.43 0.49

DNS 0.60 0.52 0.50 0.58 0.48 0.50

DNSX 0.53 0.53 0.57 0.51 0.43 0.47

ARX 0.49 0.51 0.55 0.57 0.55 0.54

Table. 7 Hit ratio results over AR, VAR, DNS and DNSX models on selected maturities from second quarter 2002 to first quarter 2014

The results are, however, positive for augmented models. The outstanding result of DNSX model occurs at 3-year maturity with the leading 0.57. This cannot be considered as a prominent advantage although. The remaining ratios of DNSX model are worse than other models, especially at the long-end of the yield curve. One possible explanation is that incorporating macro data introduces more uncertainty into the model, thus, more noises. The same as DNSX model, ARX model has bad ratios with smaller maturities, but it performs very well in the larger maturities. This gives some hope on incorporating macroeconomic data into yield curve model. They may fail to provide accurate predictions, but they can come up with better direction prediction. It will be much helpful than having a precise forecast, because it is indeed very difficult to have an accurate yield prediction. Thus, it might be better to consider the direction of interest rate instead.

AR model, again, is the second best model on average under this evaluation ratio. The reason is that AR model follows closely with actual data which leads to smaller errors or higher Hit ratios. The Dynamic Nelson-Siegel model performs well under the Hit ratio. Except 3-month’s score, the rest maturities’ scores are very close to AR model, some are even a little better than AR model. This is might be another proof of DNS model successfully captures the shape of yield curve. VAR model has the similar Hit ratio to DNSX model which can be considered not well performed under this test ratio.

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8. Further recommendations

Further studies are recommended on financial crisis period. In this paper, most errors and unproductive model results might because of the chaotic crisis period. Financial sector is no doubt suffered heavy losses over the crisis. As the important economic indicator and manipulate tool, interest rate reacts intimately to financial events. That’s the reason why the models discussed in this paper lost its prediction power over that period. More elaborated model is needed in specific for crisis period analysis. Alternatively, data can be divided into two periods – before and during financial crisis. Then one can compare the errors of the same model on the same maturity to see if financial crisis is indeed the impact factor of the model. If so, models or methods of eliminating the crisis effects should be carried out. A powerful forecasting model is more valuable in crisis period than non-crisis period. Moreover, other economic variables can be examined in further studies. This paper chooses four sectors in macroeconomic scope – central bank, government, business and household. It might be interesting to consider international trade into the model, for instance. Cash flows from all over the world do have their power over interest rate. As pointed out in literature, the root cause of financial crisis is financial imbalance. The blending of current account deficits and surpluses which entailed massive gross international flows before crisis built the bobble in U.S. financial market.

The research on decay parameter – 𝜆 is also intriguing. Recall that 𝜆 specifies the maturity at which the loading on the mid-term, factor achieves it maximum (Diebold and Li, 2005). 𝜆 is determined at 0.0609 in order to maximize the 30-month factor loading. In this paper, mid-term maturity is no longer 30-month. Instead, according to the yield data, 3-year and 5-year are the mid-term maturities. So, pick a middle point, 48-month is the medium maturity. Therefore, the use of 𝜆 = 0.0609 is inappropriate and might cause bias in DNS and DNSX models. There are already some studies of choosing the 𝜆 values on Nelson-Siegel model. It would be better to incorporate macro factor data and financial crisis issue together with appropriate 𝜆 value into an elaborated model. The results of that model will be intriguing.

9. Conclusion

Dynamic Nelson-Siegel model and Augmented Dynamic Nelson-Siegel model are examined in this paper and the out-of-sample yield curve forecasting performance is analyzed. Four major macro-economic factors are included in Augmented Dynamic Nelson-Siegel model – government, central bank, business and household. Cash flows of these factors are extracted from Federal Reserve’s official database. The fit of Nelson-Siegel model is good. It provides proof that Nelson-Siegel factors well capture the shape of the yield curve. Although, there are no promising results on Augmented Dynamic Nelson-Siegel model over Root Mean Squared Prediction Error and Hit ratio, the estimation and forecasting results on Dynamic Nelson-Siegel model are

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satisfying. The well performed Hit ratio test of ARX model proves macroeconomic data has its power on predicting the direction of yield curve. This is saying that macroeconomic data does benefit on yield curve forecasting. Theoretically and practically macroeconomics indeed has important predictive information about interest rates. The failure of exploring its power over interest rate prediction is partially linked to the recent financial crisis when the macro data moves chaotically.

From the perspective of interest rate forecasting power, DNS and DNSX models are failure to give accurate out-of-sample forecast. On the other hand, parsimonious models such as Autoregressive model, Vector Autoregressive model and Random walk model provide better forecast than macroeconomic models, but they failed to catch the direction of yield curves. Further studies are recommended on choosing appropriate 𝜆 value and specific models or analysis are advised over financial crisis period. More economic variables are encouraged to consider when building models for forecasting interest rate.

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