Definition

Let us denote by $R(t,h)$ the interest rate observed at time $t$ on the zero- coupon Treasury bond maturing at date $t+h$. In the analysis, V-lab selects a set of different maturities $\tau =\left\{\mathrm{0.5,1,2,3,5,7,10,20,30}\right\}$ years from the Gurkaynak, Sack and Wright (2006) database. Each element of $\tau$ is denoted by ${\tau }_i$.

We first express the variables as forward rates, denoted by $f_{i,t}$ where:$$\begin{array}{lll}{f}_{1,t}& =& R(t,{\tau }_1)\\ f_{i,t}& =& \frac{{\tau }_{i}R(t,h_i)-{\tau }_{i-1}R(t,{\tau }_{i-1})}{{\tau }_i-{\tau }_{i-1}},\forall i⩾2& \end{array}$$To ensure the zero lower bound, we apply a log-linear transformation to the forward rates whenever they are below the threshold $\overline{f}$: $$\begin{array}{ll}{Y}_{i,t}=[(\overline{f}-c)\log \left(\frac{f_{i,t}-c}{\overline{f}-c}\right)+\overline{f}]\{f_{i,t}<\overline{f}\}+\{f_{i,t}⩾\overline{f}\}{f}_{i,t}& \hfill \text{(1)}\end{array}$$This particular transformation maps the forward rates that take values in ${ℝ}_{+}$ to $ℝ$ without compressing too much volatilities when the forwards are high, and without increasing too much volatilities when the forwards become close to zero. In the application, we choose $\overline{f}=50\text{bps}$ and $c=0$ which is well-fitted to the positive interest rates observed on U.S. data.

Vlab's model is built in two pieces. We compute spreads of long-term transformed forwards against the shortest maturity that we denote by $Z_t$ and model the spreads and the short-rate separately.$$\begin{array}{c}{Z}_t=Y_{2:9,t}-Y_{1,t}\hfill \end{array}$$

Model

The Spreads Model

The dynamics of the transformed spreads are given by the following state-space model:$$\begin{array}{lll}{F}_t& =& A{F}_{t-1}+{\xi }_t\\ Z_t& =& \mu +\varphi F_t+u_t\end{array}$$V-lab's spreads model is inspired from the term structure literature which usually considers that three factors are sufficient to reproduce the bulk of fluctuations in the term structure (see e.g. Nelson and Siegel (1987) or Litterman and Scheinkman (1991)). The factors are computed using Nelson Siegel method, such that:$$\begin{array}{ll}& F_t={(\varphi \text{'}\varphi )}^{-1}\varphi \text{'}(Z_t-\mu ),\text{where}{\varphi }_i=[1,\exp (-\lambda {\tau }_i),\lambda {\tau }_i\exp (-\lambda {\tau }_i)]\end{array}$$We assume that the measurement errors have a diagonal correlation structure, such that$$\begin{array}{l}{u}_t=diag\left(\rho \right)u_{t-1}+{\eta }_t.\end{array}$$Rearranging slightly, and denoting by $H_t^{1/2}{\epsilon }_t={\eta }_t+\varphi {\xi }_t$, we obtain:$$\begin{array}{l}{Z}_t-diag\left(\rho \right)Z_{t-1}={\mu }^{*}+B{F}_{t-1}+H_t^{1/2}{\epsilon }_t\text{where}{H}_t=D_t{R}_t{D}_t\text{and}{\epsilon }_t\sim 𝒩(0,I)\end{array}$$The system is a reduced-rank VAR whenever $\rho =0$. $\rho$ has components constrained in $\left(\mathrm{0,1}\right)$ and is a set of quasi-differencing parameters. $\rho$ is parameterized to control for residual autocorrelation that is not accounted for by the factor structure.

$D_t$ is a diagonal matrix of variance processes which dynamics is given by GJR-GARCH processes: $$\begin{array}{ll}{D}_{i,t}^2={\omega }_i+({\alpha }_i+{\gamma }_i\{{\epsilon }_{i,t-1}<0\}){\left(D_{i,t-1}{\epsilon }_{i,t-1}\right)}^2+{\beta }_i{D}_{i,t-1}^2.& \hfill \text{(2)}\end{array}$$The cross-correlation between the residuals is given by a DCC structure: $$\begin{array}{ll}\{\begin{array}{ccc}{Q}_t& =& \overline{\Omega }(1-a_Q-b_Q)+a_Q\cdot {\epsilon }_{t-1}{\epsilon }_{t-1}^T+b_Q\cdot Q_{t-1}\\ R_t& =& \left[diag{\left(Q_t\right)}^{-1/2}\right]Q_t\left[diag{\left(Q_t\right)}^{-1/2}\right]\end{array}& \hfill \text{(3)}\end{array}$$

The Short-Rate Model

Let $s_t$ be a stationary homogenous Markov chain with 3 states representing monetary policy regimes, upward ($U$), status-quo ($S$), and downward ($D$). The transformed short-rate dynamics are given by: $$\begin{array}{l}{Y}_{1,t}=\{\begin{array}{ccc}b\left(U\right)+Y_{1,t-1}+{\sigma }_t{\zeta }_t& \text{if}& s_t=U\\ b\left(S\right)+a\left(S\right)Y_{1,t-1}+{\sigma }_t{\zeta }_t& \text{if}& s_t=S\\ b\left(D\right)+Y_{1,t-1}+{\sigma }_t{\zeta }_t& \text{if}& s_t=D\end{array}\end{array}$$$$\begin{array}{ll}\text{where}& b\left(U\right)>0,b\left(D\right)<0,b\left(S\right)\in ℝ,\left|a\left(S\right)\right|<1,{\zeta }_t\sim 𝒩\left(\mathrm{0,1}\right).\end{array}$$The variance of the previous equation ${\sigma }_t^2$ is given by a standard GJR-GARCH(1,1).$$\begin{array}{l}{\sigma }_t^2={\omega }^{\left(\sigma \right)}+({\alpha }^{\left(\sigma \right)}+{\gamma }^{\left(\sigma \right)}\{{\zeta }_{t-1}<0\}){\sigma }_{t-1}^2{\zeta }_{t-1}^2+{\beta }^{\left(\sigma \right)}{\sigma }_{t-1}^2.\end{array}$$

Estimation

The mean part of the spreads model is linear and is thus estimated by OLS. The dependent variable is $Z_t-diag\left(\rho \right)Z_{t-1}$ and the independent variable is given by $F_{t-1}$. We parameterize $\rho$ in two steps. We calculate a first set of measurement errors by computing $Z_t-\varphi F_t$, and fit univariate AR(1) models on each of these series. $\rho$ is then set to be the vector gathering all these AR(1) coefficients. Therefore, for each component $Z_{i,t}$, the following regression is then estimated:$$\begin{array}{c}{Z}_{i,t}-{\rho }_i{Z}_{i,t-1}={\mu }_i^{*}+B_i{F}_{t-1}+{\eta }_{i,t}.\hfill \end{array}$$The variance part is estimated in two steps. First, the conditional variance part $D_{i,t}^2$ is estimated by quasi maximum likelihood using the regression residuals ${\eta }_{i,t}$. Stationarity in the variance is easily imposed as ${\alpha }_i+{\beta }_i+{\gamma }_i/2<1$. Second, we obtain the ${\epsilon }_t$ series as ${\epsilon }_t=D_t^{-1}{\eta }_t$. $a_Q$, $b_Q$, and $\overline{\Omega }$ are estimated by quasi maximum likelihood again.

For the short-rate model, we make the regimes observable using the Fed target rate. In particular, we assume that we are in an upward regime at time $t$ if there was an increase in the target rate less than 3 months ago and that the next increase happens less than 3 months after the current date. We are in a downward regime at time $t$ if there was a decrease in the target rate less than 6 months ago and that the next decrease happens less than 6 months after the current date. We assume we are in status quo regime otherwise. Given that the regimes are observable, we estimate the parameters as follows:

• Transition and stationary probabilities of the Markov chain are estimated with their plug-in estimators using the observed regimes.
• $b\left(U\right)$and $b\left(D\right)$are obtained regressing respectively $\Delta Y_{1,t\left(U\right)}$and $\Delta Y_{1,t\left(D\right)}$on a constant, where $t\left(U\right)$and $t\left(D\right)$are the dates where $s_t=U$ and $s_t=D$respectively.
• For the status-quo regime, we parameterize $b\left(S\right)$so that the model-implied mean of $Y_{1,t}$is equal to its in-sample estimate. $b\left(S\right)$is thus expressed as an analytical function of the other parameters and the data. We obtain $a\left(S\right)$by minimizing the squared residuals of the $Y_{1,t\left(S\right)}$by NLS.

The GJR-GARCH parameters are then estimated with QMLE by gathering the 3 series of residuals.

Prediction

Using the multivariate Gaussian assumption on the residuals, it is easy to simulate the model formulated above. The procedure writes, for one path, for each date $t$:

• Simulate 9 Gaussian shocks $({\epsilon }_t\text{'},{\zeta }_t)\text{'}$,
• Compute $H_t$ using the DCC-GARCH equations, and multiply its Cholesky with ${\epsilon }_t$
• Compute the spreads conditional mean ${\mu }^{*}+B{F}_{t-1}+diag\left(\rho \right)Z_{t-1}$and add the simulated shocks $H_t^{1/2}{\epsilon }_t$ to form the simulated$Z_t$
• Update $F_t$ using the simulated value of $Z_t$.
• Compute the GARCH variance of the short-rate shocks and form the reduced-form residuals ${\sigma }_t{\zeta }_t$
• Simulate the regime $s_t$ using the transition matrix probabilities and the last regime value $s_{t-1}$
• Compute the mean part of the short-rate, $b\left(s_t\right)+a\left(s_t\right)Y_{1,t-1}$
• Update the value of the short-rate $Y_{1,t}$ by adding the mean part and the shock,
• Use the simulated short-rate to reconstruct the transformed forwards at other maturities: $Y_{2:9,t}=Z_t+Y_{1,t}$
• Transform $Y_t$ in the yield space:

$$\begin{array}{lll}{f}_{i,t}& =& \overline{f}\exp (\frac{Y_{i,t}}{\overline{f}}-1)\{Y_{i,t}<\overline{f}\}+f_{i,t}\{Y_{i,t}>\overline{f}\}\\ R(t,{\tau }_i)& =& \{\begin{array}{c}{f}_{i,t}\text{if}i=1\hfill \\ (1-\frac{{\tau }_{i-1}}{{\tau }_i})f_{i,t}+\frac{{\tau }_{i-1}}{{\tau }_i}R(t,{\tau }_{i-1}),\forall i⩾2\end{array}\end{array}$$V-lab draws 1,000 paths of simulations with a five year length for each. For each simulation date, quantiles of interest rates are computed to derive the confidence bounds and the median forecasts for each maturity.

Engle, Roussellet and Siriwardane, Scenario generation for long run interest rate risk assessment, Journal of Econometrics 201(2), (2017) https://www.sciencedirect.com/science/article/pii/S0304407617301641

Gurkaynak, Sack and Wright, The U.S. Treasury Yield Curve: 1961 to the Present, Journal of Monetary Economics 54(8), (2007) https://www.federalreserve.gov/Pubs/feds/2006/200628/200628pap.pdf

Nelson and Siegel, Parsimonious Modeling of Yield Curves, The Journal of Business 60(4), (1987) https://www.jstor.org/stable/2352957

Litterman and Scheinkman, Common Factors Affecting Bond Returns, The Journal of Fixed Income 1(1), (1991) https://doi.org/10.3905/jfi.1991.692347