04-12-2017 Groep 1
Drs. N. Bruin 2017-2018
Afstudeerseminaar econometrie Blok 1 en 2
HIDDEN IN THE SHADOWS
An Investigation of Two Prominent Shadow-Rate Term
Structure Models
Author: Gideon Legrand Mentor: Peter Boswijk
Universiteit van Amsterdam Bachelor Thesis, 2017
Statement of Originality
This document is written by Student Gideon Legrand who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Table of Contents:
1. Introduction 1
2. Model Specification 3
2.1 Three-Factor Black Arbitrage-Free
Model 3
2.2 Shadow Rate Term Structure Model 7
3. Methods and Data 8
3.1 Methods 9
3.2 Data 10
4. Results 10
4.1 Parameter Estimation Results 11
4.2 Shadow Rates Compared to the
Effective Federal Funds Rate 11
4.3 Comparison of Fit 13
5. Conclusion 18
1. Introduction
From bond pricing to monetary policy. From pricing financial assets to valuing capital goods. Being able to understand the structure and movement of the yield curve is an essential element for all of these economic tasks. Over the years, constructing models to fit and predict these yield curves has seen an exponential evolution, many steps have been made towards finding a model that is both theoretically compelling and empirically accurate. From just the suggestion of the importance of being able to estimate the shape of the yield curve by Milton Friedman in 1977, to the empirically powerful models that are utilised today such as that of Wu and Xia in 2016; the ‘perfect’ model, seems imminent.
One of these flaws or critiques that earlier researchers failed to account for is the Zero Lower-Bound (ZLB, for short). It was always assumed that the nominal interest rate could never surpass the zero-percent mark and become negative; ‘bounded’ by zero. This due to the fact that currency is (practically) liquid and pure arbitrage would therefore be possible by simply borrowing at a negative interest rate and investing in currency (Buiter, 2009). The ZLB became an even greater issue in the early 2000’s as Japan became stuck in a liquidity trap, their interest rate was stuck at around zero. The models that had previously been successful in estimating the shape of yield curve, were now failing (Uena, Baba & Sakurai, 2006). They believed that this inaccuracy was caused by the flattening out of the yield curve towards the ZLB. Effectively, the ZLB was considered to be an absorbing state, creating a non-linear shape of the data. This non-linearity was later confirmed in, for example, the work of Fernández-Villaverde (2015). Uena, Baba and Sakurai (2006) were one of the first successors in creating a working model to account for this lingering issue by applying the framework of Fischer Black (1995).
In particular, Black (1995) suggested the use of the ‘Shadow-Rate’. The Shadow-Rate is essentially the nominal interest rate in a situation where the ZLB does not exist, hereby allowing a negative interest rate to occur, and Black merely pointed out that the nominal interest rate is simply the option between the shadow interest rate and the zero-value. The interest rate is therefore always non-negative, because the negative values are simply replaced by zero. This idea slowly became the fundamental base of the most popular term structure models used today, especially if the yield curve in question is approaching or stuck at the ZLB.
The model by Uena, Baba and Sakurai (2006), however, only considered one factor in its estimation and could not be expanded into a multifactor model. Kim and Singleton (2012), provided a two-factor Gaussian Shadow Rate model specification, which fit Japanese data even better than previous models. Around that time Krippner (2012) provided a continuous approximation approach in estimating Gaussian Term Structure models. He then continued to also construct a two-factor model that compared considerably to alternative two-factor models based directly on the ‘Black-approach’, such as that of Kim and Singleton (2012).
However, an increasing amount of literature points toward a three-factor model (see: Diebold & Li, 2006) being the optimal choice for term structure models, ergo this is now considered to be the consensus in model specification. This paper will hereby confine itself to the use of three-factor models. In the previous half-decade, there have merely been a couple of models that seem to stand out above the rest; those being that of Wu and Xia (2014) and Christensen and Rudebusch (2015). Christensen, Diebold and Rudebusch (2011) created an affine Gaussian term structure model based on work by Nelson and Siegel (1987) that they concluded to possibly be the best available, due to the fact that their three-factor model was both theoretically accurate and empirically strong. This model did, however, assign positive probabilities to future negative interest rates when interest rates approached the ZLB, as it is linear in Gaussian factors. Thus, they expanded the model, adding the shadow-rate element proposed by Black (1995) and the continuous approximation derived by Krippner (2012) to create a three-factor, shadow rate, arbitrage free, Gaussian term structure model (henceforth, the B-AFNS(3) model). Wu and Xia, on the other hand, derived a method of non-linear approximation to keep the model discrete, as the data is also discrete. This was said to eliminate the error in the numerical approach regarding the continuous approximation as well as being quick in estimation since considerably less numerical approximations are required. The model of Wu and Xia (2014) is also known as the Shadow Rate Term Structure Model (SRTSM), since it was one of the first three factor shadow rate term structure models ever created.
As both the B-AFNS(3) and SRTSM model acquire promising results in times when the short rate approaches or is stuck at zero, it begs the question: which implementation of the shadow rate in term structure models is better in comparison? The goal of this paper is to estimate both models and to test the in-sample fit, comparing the relative accuracy to the actually observed data. The rest of this paper proceeds as follows. In Section (2) both the
B-AFNS(3) and SRTSM model will more meticulously discussed. In (3) the methods and steps taken in order to compare the two models are explained, in (4) the empirical analysis and the results of the comparisons will be displayed and finally in (5); the conclusions will be drawn.
2. Model Specification
In this section, the model from Christensen and Rudebusch (2015) as well as the SRTSM from Wu and Xia (2014) will be specified and discussed. The derivation of both models will be explained, revealing the theoretical differences between the two. Both models have a rich background and are extensions of previously created models. For the complete derivation and for proof of the steps taken towards the final product, it is recommended to read the articles that are called upon. The complete derivation of the models is not the purpose of this paper.
2.1 Three factor Black Arbitrage-Free Nelson-Siegel model
To understand the B-AFNS(3) model, or Black-Arbitrage-Free Nelson-Siegel model, one must go back to the roots; that is, the Nelson-Siegel model. Nelson and Siegel (1987) created a basic model that used three factors, based on the fact that yield curves in the past were either monotonic, humped or S-shaped. These factors were later derived as level, slope and curvature factors (Diebold & Li, 2006). Their formula was parsimonious and relatively simple:
𝑦(𝜏) = 𝛽0+ (𝛽1+ 𝛽2) ∗ [1 − exp (−𝜏𝜆)/(𝜏𝜆)] − 𝛽2∗ exp (−𝜆𝜏)
Here, 𝑦(𝜏) is the yield to maturity on a bill with maturity 𝜏, and 𝜆 a shape parameter. They proved that this model was capable of approximating the shape of the yield curve.
Diebold and Li (2006) further expanded this model in order to enhance its performance in dynamic environments. This model, now known as the Dynamic Nelson-Siegel model (DNS), was a frontrunner in out-of-sample forecasting and is written as follows:
𝑦𝑡(𝜏) = 𝛽1𝑡+ 𝛽2𝑡∗ (1 − exp(−𝜆𝑡𝜏)
𝜆𝑡𝜏 ) + 𝛽3𝑡∗ [
1 − exp(−𝜆𝑡𝜏)
𝜆𝑡𝜏 − exp(−𝜆𝑡𝜏)]
By finding a suitable value for 𝜆𝑡 they restricted the function to the three unknown factors,
which could easily be solved using Ordinary Least Squares (OLS). Thereby, making their model extremely accessible and relatively accurate.
Christensen, Diebold and Rudebusch (2011), however, displayed some scepticism on the DNS model because of the fact that it “does not impose the restrictions necessary to eliminate opportunities for riskless arbitrage”; which was, for the authors, theoretically incorrect. They perceived that only two types of models were commonly used in practice, one that was either theoretically lacking, but empirically powerful (DNS), or vice versa (see: Duffee & Kan, 1996). Their goal in 2011 was to create a model that was both empirically accurate and theoretically sound. In other words, expand the accurate and reliable dynamic Nelson-Siegel model with no-arbitrage restrictions. This goal was well achieved, as two types of Arbitrage-Free Nelson-Siegel (AFNS) models were presented, one with and one without correlated factors. The general form of both is shown below:
𝑦𝑡(𝜏) = 𝑋𝑡1 + 𝑋𝑡2 ∗ (1−exp(−𝜆𝜏)𝜆𝜏 ) + 𝑋𝑡3 ∗ (1−exp(−𝜆𝜏)𝜆𝜏 − exp(−𝜆𝜏)) −𝐴(𝜏)𝜏 (1)
This model takes the form of the previously mentioned DNS model, with an adjustment caused by the arbitrage-free restriction. The vector 𝑋𝑡= (𝑋𝑡1 𝑋𝑡2 𝑋𝑡3)′ still represents the
level, slope and curvature, respectively1. The shadow short-rate in this model was defined as:
𝑠𝑡 = 𝑋𝑡1 + 𝑋
𝑡2 (2)
The dynamics of 𝑋𝑡 under the ℚ-measure being defined as:
( 𝑑𝑋𝑡1 𝑑𝑋𝑡2 𝑑𝑋𝑡3 ) = − (0 00 𝜆 −𝜆0 0 0 𝜆 ) ( 𝑋𝑡1 𝑋𝑡2 𝑋𝑡3 ) 𝑑𝑡 + ( 𝜎11 0 0 𝜎21 𝜎22 0 𝜎31 𝜎32 𝜎33) ( 𝑑𝑊𝑡1,𝑄 𝑑𝑊𝑡2,𝑄 𝑑𝑊𝑡3,𝑄 ) (3)
1 For the exact notation for the yield adjustment term (the arbitrage-free term), see the work by Christensen, Diebold and Rudebusch
And the dynamics of 𝑋𝑡 under the ℙ-measure is given by the following set of differential equations: ( 𝑑𝑋𝑡1 𝑑𝑋𝑡2 𝑑𝑋𝑇3 ) = ( 10−7 0 0 𝜅21ℙ 𝜅 22ℙ 𝜅23ℙ 0 0 𝜅33ℙ ) (( 0 𝜃2ℙ 𝜃3ℙ) − ( 𝑋𝑡1 𝑋𝑡2 𝑋𝑡3 )) 𝑑𝑡 + 𝛴 ( 𝑑𝑊𝑡𝑋1,ℙ 𝑑𝑊𝑡𝑋2,ℙ 𝑑𝑊𝑡𝑋3,ℙ ) (4)
The matrix 𝐾ℙ here is the simplified version of the maximally fully flexible defined 𝐾ℙ
matrix. This simplification was justified in their paper (Christensen, Diebold & Rudebusch, 2011).
Their conclusions are that after testing this model, their correlated factor model considerably outperforms previous arbitrage-free models and performs well due to the Nelson-Siegel approach, in regards to in-sample fitting. Out-of-sample forecasting by the independent-factor AFNS model, however, provides strong results relative to the correlated-factor AFNS model, but it still outperforms other no-arbitrage alternatives.
Although the results provide compelling conclusions, the researchers failed to test their model in a period of near zero interest rates, the Zero Lower Bound. However, Christensen and Rudebusch tackled this obstacle again in 2015, extending their model to the now frequently implemented B-AFNS(3) model.
As the name implies, the model is based on the work of Fischer Black (1995). It implements the ‘call-option’ when describing the instantaneous, risk-free rate 𝑟𝑡.
𝑟𝑡 = max{0, 𝑠𝑡} (4)
𝑠𝑡 is now considered to be the ‘shadow’-rate. As can be seen, the shadow rate is the risk-free interest rate so long as this rate is above the zero percent mark. Numerous models were constructed based on this approach prior to Christensen and Rudebusch (2015), but these mainly were constricted to one or two factors, due to the numerical difficultly stemming from higher dimensions (see: Bomfim, 2003; Kim and Singleton, 2012). Christensen and Rudebusch decided to construct a three-factor model based on the Black approach, but using a continuous approximation invented by Krippner (2012). The idea of this approximation stems
from the fact that due to the availability of currency, if the time between buying and selling a zero-coupon bond that pays $1 at maturity becomes increasingly small (say one day, or 𝛿), the price of the bond will be stuck at 1, since the buyer also has the option to hold on to currency and prices will never be above par. As a result, this price (𝑃(𝑡, 𝑡 + 𝛿)) has a maximum value of 1 and will otherwise equal the shadow (risk free) bond price. Krippner (2012) uses this idea and the availability of the values of European call options (American call options are difficult to analytically value) to create a continuous approximation of the yield to maturity. The equation is shown here:
𝑦𝑡(𝑇) = 𝑦𝑡(𝑇) + ∫ lim 𝛿→0[ 𝜕 𝜕𝛿( 𝐶𝐸(𝑡,𝑠,𝑠+𝛿;1) 𝑃(𝑡,𝑠) )] 𝑑𝑠 𝑇 𝑡 (5)
Here, 𝐶𝐸(𝑡, 𝑠, 𝑠 + 𝛿; 1) is the European call option value at time t and maturity s, strike price
set at 1, using a shadow discount bond maturing at time 𝑠 + 𝛿. 𝑃(𝑡, 𝑠) is the price of a shadow rate zero-coupon bond in a situation without an existence of currency, 𝑦(𝑡, 𝑇) is the unconstrained shadow-bond yield.
Christensen and Rudebusch use Equation (1) and compute the instantaneous forward rate:
𝑓(𝑡, 𝑇) = −𝜕𝑇𝜕 ln(𝑃(𝑡, 𝑇)) = 𝑋𝑡1 + 𝑒−𝜆(𝑇−𝑡)∗ 𝑋
𝑡2+ 𝜆(𝑇 − 𝑡)𝑒−𝜆(𝑡−𝑇)∗ 𝑋𝑡3+𝜕(𝐴(𝑡,𝑇))𝜕𝑇 (6)
They then proceed to derive an analytical formula for valuing the European call option 𝐶𝐸(𝑡, 𝑠, 𝑠 + 𝛿; 1) from (3). The calculated instantaneous shadow rate, (6), is then plugged into
an approximation derived by Krippner (2012) to obtain the ZLB instantaneous forward rate:
𝑓(𝑡, 𝑇) = 𝑓(𝑡, 𝑇) ∗ 𝛷 (𝜔(𝑡,𝑇)𝑓(𝑡,𝑇)) + 𝜔(𝑡, 𝑇) ∗2𝜋1 ∗ exp (−12[𝜔(𝑡,𝑇)𝑓(𝑡,𝑇)]2) (7)
𝜔(𝑡, 𝑇) is a time-based function of the variance estimates of 𝑋𝑡. This result, combined with
the analytical formula for the European call option, are plugged into Equation (3) for the resulting B-AFNS(3) model:
𝑦(𝑡, 𝑇) =𝑇−𝑡1 ∫ [𝑓(𝑡, 𝑠)Φ (𝜔(𝑡,𝑠)𝑓(𝑡,𝑠)) + 𝜔(𝑡, 𝑠) ( 1 √2𝜋) exp (− 1 2[ 𝑓(𝑡,𝑠) 𝜔(𝑡,𝑠)] 2 )] 𝑑𝑠 𝑇 𝑡 (8)
With 𝑦(𝑡, 𝑇) the zero-coupon bond yields that account for the ZLB. 𝜔(𝑡, 𝑠) stems from the shadow price bond formula and is a function of 𝜎 and 𝜆. The calculation of this function can be found in the work of Christensen and Rudebusch (2015). In order to apply their model empirically, the ℙ-measure dynamics of the historical data, to connect the dynamics under the ℚ-measure, were applied from the work of Cheridito, Filipovic and Kimmel (2007).
2.2 Shadow Rate Term Structure Model
In the work of Wu and Xia (2014), they also acknowledge the performance superiority of Gaussian Affine Term Structure Models, such as the one proposed by Christensen, Diebold and Rudebusch (2011), but also provide the same critique as Christensen and Rudebusch (2015) mentioned in the foregoing section. Since the models are linear in Gaussian factors, the models function poorly in non-linear environments, ergo, in the liquidity trap. Wu and Xia, however, provide a different approximation of the forward rate; a discrete variant. This discrete aspect, alongside the fact that they managed to do this whilst maintaining the three-factor structure, is what makes their model so popular to this day. Due to the discreteness of interest data, it can be applied directly, avoiding numerical approximation error, making their model extremely tractable.
Wu and Xia also began with the framework provided by Black (1995), equation (4). Although the authors assumed a lower bound of 𝑟 = 0,25%, not exactly the Zero Lower Bound. This decision was based on the level of interest that the Federal Reserve of America maintained since 2008. Their shadow rate was also a differently defined function of the state variables, namely: 𝑠𝑡 = 𝛿0+ 𝛿1′𝑋 𝑡 , 𝛿1 = ( 1 1 0) (9)
The ℙ-measure dynamics also differ, as Wu and Xia define their state variables 𝑋𝑡 as VAR(1)
variables. Their ℚ-measure dynamics differ as well, as it is based on the work by Duffee (2002), implying that they are also defined as a VAR(1) process. Under the standard definition of forward rates, they derived a formula to estimate the forward rate of the SRTSM, namely:
𝑓𝑛,𝑛+1,𝑡𝑆𝑅𝑇𝑆𝑀 = 𝑟 + 𝜎
𝑛ℚ[𝑧Φ(𝑧) + 𝜙(𝑧)], 𝑧 =𝑎𝑛+𝑏𝑛
′𝑋 𝑡−𝑟
𝜎𝑛ℚ (10)
With Φ(. ) being the CDF of the standard-Normal distribution and 𝜙(. ) the pdf of that same distribution. 𝜎𝑛ℚ is defined as the variance of 𝑠𝑡+𝑛 (the shadow rate at time 𝑡 + 𝑛) under the
ℚ-measure. This approximation by Wu and Xia was derived from the truncated Normal distribution and is later proved to be accurate to only a few basis points. 𝑎𝑛 and 𝑏𝑛 are defined
as follows: 𝑎𝑛 ≡ 𝛿0+ 𝛿1′[ ∑(𝜌ℚ)𝑗 𝑛−1 𝑗=0 ] 𝜇ℚ ⇒ 𝑎 𝑛 ≡ 𝑎𝑛 − 1 2𝛿1′[∑(𝜌ℚ)𝑗 𝑛−1 𝑗=0 ] 𝛴𝛴′[∑(𝜌ℚ)𝑗 𝑛−1 𝑗=0 ] ′ 𝛿1 𝑏𝑛′ ≡ 𝛿 1′(𝜌ℚ)𝑛
𝜇ℚ and 𝜌ℚ are parameters from the VAR(1)-processes of the state variables under the ℚ- and
ℙ-measures, shown below:
𝑋𝑡+1= 𝜇ℙ+ 𝜌ℙ𝑋
𝑡+ 𝛴𝜖𝑡+1ℙ , 𝜖𝑡+1~𝑁(0, 𝐼)
𝑋𝑡+1= 𝜇ℚ+ 𝜌ℚ𝑋
𝑡+ 𝛴𝜖𝑡+1ℚ , 𝜖𝑡+1ℚ ~𝑁(0, 𝐼)
While the 𝛿 parameters are the same as those defined in (9).
Both of these models have proven to fare well empirically, but both models are based on different approximations and function under different ℚ- and ℙ-measures. Which still begs the question; which implementation of the shadow rate fares best?
3. Methods and Data
In this section, the methods are described in order to test the relative superiority of the previously mentioned B-AFNS(3) model from Christensen and Rudebusch (2015) and that of the SRTSM from Wu and Xia (2014). Following the methods explanation, the data will be presented, along with its source and relevance.
3.1 Methods
The first step was to program both models based on codes written by the authors themselves, made available to the public23. The models were adjusted to fit the data and
subsequently estimated, providing estimates used to graph and test both models. On account of the non-linearity the pair of models appreciate, an extended version of the Kalman-filter will be used to maximize the likelihood of the estimates. Christensen and Rudebusch (2015) show that the standard extended Kalman-filter provides near identical results compared to the unscented extended Kalman-filter used in other studies (see: Kim and Priebsch, 2012), but is simpler to implement. Therefore, the standard version shall also be utilised in this research study.
Since the purpose of this paper is to investigate the implementation of the shadow-rate models’ approximations of the forward shadow-rates, two adjustments will be made to the B-AFNS model in order to make it more comparable and to isolate the shadow-rate implementation better. Firstly, equation (8) will be disregarded, as Wu and Xia merely provided us with their approximation of the shadow forward-rates and not the shadow yields. Therefore, the comparisons of the models will be performed on the solely the shadow forward-rates and not on the approximation of the zero-coupon bond yields. Considering (8) is an integral of the shadow forward rates, this adjustment should not have any repercussions for the found results shown in the upcoming section. Secondly, the B-AFNS model will firstly be estimated in full in order to acquire the parameter estimates, the initial forward rates from (6) are then replaced with the forward rates that were used in the SRTSM-model. This also should have no repercussions, as both forward rates are derived from the same data set. It
2 Link to B-AFNS code: http://cepr.org/sites/default/files/events/Extended_Kalman_filter_B_CR_shadow_rate_model.r 3 Link to SRTSM code: https://sites.google.com/site/jingcynthiawu/
merely eliminates the possible error that could result if either the parameter estimates or factor estimates of the B-AFNS model prove to be inaccurate. Additionally, since the forward-rates are identical, this actually isolates the implementation of the shadow forward-rate approximation, as the difference in the models is now merely the steps taken to derive equation (7) and (10). Also, the lower bound of the SRTSM is set to zero (𝑟 = 0), in order to minimise the difference between the models even further.
Finally, the in-sample fit will be tested for different maturity levels using the Mean Squared Errors (MSE’s) across the time-horizon Likewise, the fit will also be tested for the different maturity levels with MSE’s, at specific points in time. Hereby testing their accuracy for short and long-term time periods at different maturity levels. The time-horizon in question is specified in the forthcoming paragraph.
3.2 Data
The data that will be used is the same data Krippner used in his research in 2013 and that Wu and Xia used in their paper in 2014. The data in question are Treasury quotes by the Federal Reserve Bank of New York for 1, 2, 5 and 10-year rates from a paper of Gurkaynak, Sack and Wright (2007, henceforth: GSW), ranging from January 2008 till October 2017. It was then enriched by adding 3 and 6 month forward rates based on treasury bills, provided by the Board of Governors of the Federal Reserve System. This provides easier computations, as the data has already been cleaned and is user-ready. For further explanation on the method of the calculation of the forward rates and the reasoning behind it, please see GSW (2007). The data of the Effective Federal Funds Rate used in the next section, was retrieved from FRED, the Federal Reserve Bank of St. Louis (Board of Governors of the Federal Reserve System (US), 2017).
This paper confined the data to the time period of when the forward rates were approaching the Zero Lower Bound and climbing away from the ZLB. Along these lines the models are tested and compared in the period of time for which these models were structured. Since the models are similarly Gaussian in the foundation, they should also converge to similar models away from the ZLB. By confining them, their relative fitting-performance is at its maximum.
4. Results
In this section, the test results and comparisons will be displayed and discussed, both empirically and graphically. First, the absorbing properties of the ZLB are displayed, showing the necessity of the non-linear models along with a visual representation of the option-based model (i.e. the shadow rate) introduced by Black (1995). Next, the two models are compared graphically and the resulting MSE’s are calculated.
4.1 Parameter Estimation Results
For both models, the estimators found are significant from zero (see Table 1), which is actually quite surprising giving the results found in the next paragraph. There it will be shown that the factors estimated (𝑋𝑡) seem to be inaccurate, which can be confirmed by these results. The
estimated 𝜅22ℙ is significantly negative, which is odd, since positive shocks to 𝑋𝑡2 should
B-AFNS(3) SRTSM
Parameter Parameter Estimation
Standard Error Parameter Parameter Estimation Standard Error 𝜿𝟐𝟏ℙ 0.5896 0.0437 𝜌 11ℚ 0.9937 0.0007 𝜿𝟐𝟐ℙ 0.0325 0.0239 𝜌22ℚ 0.9730 0.0008 𝜿𝟐𝟑ℙ -0.4560 0.1845 𝜌 33 ℚ 0.9730 0.0008 𝜿𝟑𝟑ℙ 0.4410 0.1367 𝜎11 1.2407 0.2330 𝜽𝟐ℙ 0.0920 0.0292 𝜎21 1.1946 0.2082 𝜽𝟑ℙ 0.4195 0.0291 𝜎22 0.2814 0.0353 𝝈𝟏𝟏 0.3218 0.0305 𝜎31 -0.0401 0.0085 𝝈𝟐𝟐 0.3076 0.0268 𝜎32 -0.0156 0.0068 𝝈𝟑𝟑 0.04897 0.0036 𝜎33 0.0344 0.0025 𝝀 0.5351 0.0081 𝛿0 3.2288 0.4356 Log(L) 4162.131 - Log(L) 467.556 Table 1: Parameter estimation results
positively influence 𝑋𝑡+12 and vice versa, but this is a mere assumption as no evidence for this
remark could be discovered. Hereby, the found results are assumed correct.
4.2 The Shadow Rates Compared to the Effective Federal Funds Rate
The Effective Federal Funds Rate is per definition: The interest rate that federal funds are traded at by depository institutions overnight (Board of Governors of the Federal Reserve System (US), 2017). In Wu and Xia (2014), they provided evidence that the Effective Federal Funds Rate is similarly correlated to the state factors 𝑋𝑡 as that of the Shadow Rate. The main
difference is the ability the Shadow Rate has to dip below the zero-percent mark. As the factor dynamics differ, so do their respective shadow rates; equation (2) and (9). Wu and Xia (2014) use VAR(1) processes for their dynamics, due to the discreteness of their model, while Christensen and Rudebusch (2015), use differential equations for their continuous time model. Their respective shadow rates and the EFFR can be seen in Figure 1, with the green, dotted-line the EFFR and the red and blue curves the estimated shadow rates for the C&R-model and the SRTSM, respectively. The grey shaded area represents the time of the liquidity trap. Here, the shadow rate outside of the ZLB period of the SRTSM seems to match the EFFR, as it did in the paper of Wu and Xia as well. The shadow rate of the B-AFNS model, however, fails to resemble the EFFR during this same period. The factor dynamics of the AFNS-model were proven to be accurate in the work of Christensen, Rudebusch and Diebold (2011) and their parameters were significantly estimated according to their values and standard deviations (Table 1), but Figure 1 suggests that the model has trouble finding factor values that still fit the data well. Most likely due to the majority of the data being observed during a liquidity trap, which resulted in an oddly-estimated negative 𝜅22ℙ . This could be solved by
simply expanding the time-horizon, to include more observations away from the Zero Lower Bound and thereby making the estimates resemble more of those of the original AFNS-model. Furthermore, these results also provide additional reason to use the forward rates that were calculated by Wu and Xia, because using the estimated factors of the B-AFNS model to then calculate their shadow forward rates, would only provide false or inaccurate results. Only the 𝜔𝑡 function (based on the significant variance parameters of the factors, the time 𝑇 − 𝑡 and 𝜆) is necessary.
The EFFR is a major tool in measuring the effect of monetary policy on the economy (Wu & Xia, 2014). A trending topic of discussion is how this effect can properly be measured when the EFFR is stuck at the ZLB. Bullard (2012) attempted to study the monetary stance using an estimated shadow rate, this was later debunked by Christensen and Bauer (2013), stating that the level of the shadow rate was model related and therefore could vary, dependent on the model in use. Wu and Xia (2014) then countered Christensen and Bauer stating that although the height of the shadow rate differs per model, the conclusions that can possibly be drawn remain the same. This discussion has yet to finalise, further emphasising the importance of shadow rate research. However, this topic is beyond the scope of this paper. We therefore continue with the comparison of the accuracy of the SRTSM and B-AFNS models, for various maturity lengths.
4.3 Comparison of fit
Although the previous paragraph suggests that the factor estimations of the B-AFNS model are not accurate when only considering a ZLB state, does the shadow forward-approximation still fare well when disregarding the factor estimates? In figure 2 (a),(b) and (c), both the B-AFNS model and SRTSM are plotted for various maturity levels (green and blue lines, respectively), at different points in time. The estimated forward rates were averaged to obtain the average forward rate of the given year. The red circles are the averaged, actually observed forward rates in those specific time frames. The three years can be viewed as such: In 2008; the crisis began and the short-term forward rates headed towards the ZLB, in 2012; the short-term forwards were stuck at the ZLB and in 2016; the forward rates began ascending away from the zero percent mark.
In these figures, the previous suggestion is confirmed. The Christensen and Rudebusch implied model shows too much curvature towards the short-end in all three situations, when the data advocate that the rates actually flatten due to the absorbing attribute of the ZLB. This ‘flattening’ can explicitly be seen when comparing Figure 2(a) with 2(b). Maturities of the shorter length seem to be drawn towards zero and the larger maturity lengths also portray this affect, with the SRTSM almost becoming linear. However, let it be mentioned that the B-AFNS model displays incredible relative accuracy when comparing the fitting of the larger maturities. On the other hand, the model from Wu and Xia does tend to fit the short end well, with the tails flattening to match the data. The problem that comes with this flattening however, is that the curve slopes upwards too soon and has to overcompensate when trying to fit the 12 and 24-month data. Hereby, completely overshooting the 120-month maturity level. This same phenomenon occurs in all three situations, meaning that in practice; one should adjust the results to account for this over-compensating trait for the longer maturities.
The resulting MSE’s in Table 2, display a strong bias towards the model implied by Christensen and Rudebusch. Logically, the larger differences towards the far-end maturity approximations tilt the stakes in the B-AFNS’s favour. When only measuring the MSE’s at the 3, 6 and 12-month marks, the MSE of the SRTSM is approximately two times smaller than that of the B-AFNS model for the same three years demonstrated below. This provides some scepticism to the otherwise obvious dominant performance of the B-AFNS model. This
scepticism is further taken under a microscope when the MSE’s are calculated for the separate maturities during the entire time-period of this dataset.
Figure 2: Averaged, fitted forward rates for both the SRTSM (blue line) and B-AFNS model (green line), for a given year. With
the averaged, observed forward rates in the given year marked by the red circles. (a): Year 2008
(c): Year 2016 (b): Year 2012
Year MSE B-AFNS MSE SRTSM
2008 0.0912 0.6318
2012 0.0482 0.2353
2016 0.1363 0.5758
In Figure 3; the B-AFNS and SRTSM approximated shadow forward rates (green and blue line, respectively) and the observed forward rates (red line) for 3, 24 and 120-month maturity levels were plotted in (a),(b) and (c), respectively. The corresponding MSE’s for each maturity level are displayed in Table 3.
What was suspect with regards to the previous test, is further confirmed here. In Figure 3(a) one can see that the SRTSM outperforms the B-AFNS when judging the fit of both curves. Even in pre- and post-liquidity trap periods. The extra curvature that the B-AFNS model displayed at the short-end in the foregoing section provides a constant over-estimation of the forward rate at short maturity lengths. In 3(b) the difference between both models becomes relatively small, conclusive with the previous results. The fact that the ratio between the MSE’s with a maturity of 3M is just as large as that of the reverse of 24M, even though the curves display relatively minor deviation, is due in part to the larger
percentage points of the average forward rate with a larger maturity. However, the lesser MSE’s do still represent the more precise model. The conclusion of the over-estimation of the SRTSM for larger maturity lengths is visually defended in Figure 3(c), with the calculated MSE’s empirically confirming that assumption.
Maturity (Months) MSE B-AFNS(3) MSE SRTSM
3 8.3626 3.7681
24 3.8413 8.0018
120 8.3728 32.7997
Table 3: MSE of fit against time, for different lengths of maturities
Figure 3: The observed forward rates (red line) are plotted against the fitted shadow forward rates of the
B-AFNS model and SRTSM (green and red, respectively) at 3, 24 and 120-month maturity levels over the entire time period of the dataset.
(a): 3 month maturity level
(b): 24 month maturity level
5. Conclusion
In conclusion, the better implementation of the shadow rate in term structure models, remains inconclusive. Both the B-AFNS(3) model from Christensen and Rudebusch (2015) and the SRTSM from Wu and Xia (2014) provide positive, as well as negative, results. Although the C&R-model is theoretically strong with the addition of the no-arbitrage term, their dynamics showed little to know resemblance to the dynamics of the EFFR; especially when compared to that of Wu and Xia. When comparing the fit of the forward rates during a ZLB timeframe, however, the more dominant performer remained to be conclusive. The SRTSM accounted better for the ZLB in terms of forward rates with smaller maturities, as their yield function flattened out towards the shorter end, resembling the absorbing effect of the Zero Lower Bound. It failed marginally though, when trying to fit forwards with larger maturities relative to its B-AFNS counterpart.
An argument can be made that the SRTSM should be the preferred model of choice, due to the tiebreaking superiority of its fit to the EFFR, suggesting the VAR(1) dynamics hereby display real-world properties. But this could already be solved by including more historical observations, since the AFNS model has proved to fit the data well in the past (Christensen, Diebold & Rudebusch, 2011). Hereby, the EFFR during a liquidity trap is
remains stuck around zero and how the shadow rate can be interpreted (and which shadow rate can be interpreted better) remains to be discovered. Therefore, an exact conclusion on which shadow rate implementation fares best, remains hidden in the shadows.
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