To get economic interpretations of the level, slope, and curvature factors I relate them to macroeco-nomic variables since various ecomacroeco-nomic forces change the yield curves. In other words, to determine how factors are driven by changes in macroeconomic variables (that one can directly set) and to determine how those changes propagate to other macro variables through the factors/yield curve.
A total of six macroeconomic variables seem to be non-stationary, and the correlations between estimated factors and these macro variables can be spurious, e.g., CPI only correlate with the level as the level seems to be decreasing. These variables are; Real Personal Income, Real Personal Con-sumption Expenditures, Real Money Stock, Commercial and Industrial Loans, Real Estate Loans, and CPI. All variables seem to fluctuate around a linear trend in the whole period (1997-2017).
Therefore, I detrend these variables and make them stationary, by regressing the variables on the time index and correlate the estimated factors with the residuals from the linear regressions.7
In Figure 7-3, the level factor seems to be lacking strong correlations over the whole period.
6I do not show it in this paper, but ICA produces significantly better forecasts for the one-month-ahead and three-month-ahead forecasts across all maturities. However, there are no significant differences for the six-three-month-ahead and twelve-month-ahead forecasts.
7There are many different ways of making a non-stationary time series stationary, where another approach is to calculate differences between two adjacent time points across the whole dataset.
Table 7.2: Out-of-sample fit in terms of RMSFE (in basis points) and Diebold-Mariano statistics for ICAYM, PCAYM, and DNSM. I report both RMSFE and relative RMSFE for all models. Note that the DNSM is the reference model, and a negative sign for the Diebold-Mariano statistic is in favor of the compared model.
Forecast horizon (months)
h = 1 h = 3 h = 6 h = 12
Model RMSFE D-M RMSFE D-M RMSFE D-M RMSFE D-M
Three-month yield:
ICAYM 2.11 6.89*** 1.25 1.29 0.74 −2.62* 0.77 −3.71***
PCAYM 2.27 7.72*** 1.47 2.18* 0.84 −1.00 0.44 −2.94**
DNSM 25.23 Ref. 34.74 Ref. 49.28 Ref. 59.52 Ref.
Six-month yield:
ICAYM 3.12 7.06*** 1.44 1.79 0.83 −1.52 0.85 −1.52
PCAYM 3.40 7.78*** 1.77 2.67** 1.00 0.01 0.53 −2.17*
DNSM 17.65 Ref. 30.79 Ref. 43.94 Ref. 54.08 Ref.
Nine-month yield:
ICAYM 4.06 7.15*** 1.60 2.05* 0.91 −0.75 0.93 −0.56
PCAYM 4.47 7.85*** 2.03 2.96** 1.15 0.67 0.58 −1.67
DNSM 13.93 Ref. 27.73 Ref. 39.47 Ref. 50.45 Ref.
One-year yield:
ICAYM 4.58 7.21*** 1.71 2.19* 0.96 −0.31 0.99 −0.05
PCAYM 5.09 7.92*** 2.25 3.16** 1.26 1.06 0.60 −1.44
DNSM 12.45 Ref. 25.25 Ref. 35.90 Ref. 48.23 Ref.
Three-year yield:
ICAYM 3.09 7.21*** 1.25 1.18 0.96 −0.25 1.18 0.69
PCAYM 3.55 8.33*** 1.72 2.67** 0.91 −0.40 0.50 −2.07*
DNSM 16.50 Ref. 28.31 Ref. 41.29 Ref. 57.23 Ref.
Five-year yield:
ICAYM 2.43 6.41*** 1.06 0.38 0.95 −0.36 1.13 0.51
PCAYM 2.76 7.46*** 1.29 1.27 0.79 −1.02 0.67 −2.09*
DNSM 20.24 Ref. 36.57 Ref. 53.03 Ref. 68.62 Ref.
Seven-year yield:
ICAYM 2.38 6.25*** 1.05 0.31 0.94 −0.53 1.08 0.35
PCAYM 2.68 7.08*** 1.24 1.13 0.84 −0.94 0.76 −1.70
DNSM 21.41 Ref. 40.52 Ref. 58.99 Ref. 75.60 Ref.
Ten-year yield:
ICAYM 2.26 6.19*** 1.04 0.26 0.92 −0.85 1.03 0.19
PCAYM 2.53 6.92*** 1.23 1.17 0.89 −0.77 0.82 −1.45
DNSM 23.74 Ref. 45.00 Ref. 65.13 Ref. 82.96 Ref.
***p < 0.001,**p < 0.01,*p < 0.05
Therefore, I construct a rolling window of five-year period8correlations to investigate further whether there are periods of high correlation for specific macroeconomic variables. See the bottom of Figure 7-3. Around the financial crisis in 2008, there was a strong positive correlation between the level factor and the Money Stock. After that, it has been fluctuating around the zero mark. Also, there were strong negative correlations between the level factor and Real Income, Real Consumption Expenditures, and CPI. In 2009, the correlation was decreasing and fluctuate around that point
8I choose a five-year period because that corresponds to 60 observations and is a reasonable length to calculate a correlation between two variables.
until recent years starting to rise back to its previous levels. A remarkable fact is that while both Real Income and Real Consumption Expenditures are going back to these previous levels, CPI moved the other way from negative to a positive correlation.
97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 Time
Level Real Income Real Consumption Expenditures Money Stock CPI ICA: Level factor and Macroeconomic variables
−1.0
Real Income Real Consumption Expenditures Money Stock CPI Rolling correlation window: Level factor and Macroeconomic variables
Figure 7-3: Top: Estimated level factor and Real Income, Real Consumption Expenditures, Money Stock, and CPI. Bottom: Rolling window of five-year period correlations between the estimated level factor and Real Income, Real Consumption Expenditures, Money Stock, and CPI.
As can be seen in Figure 7-4, the slope factor seems to have a strong correlation between both Capacity Utilization and Federal Funds Rate over the whole period. These results are in line with previous literature (e.g., Diebold et al., 2006). Furthermore, the zero lower bound impact from 2009 reflect that the Fed lowered Federal Funds Rate close to zero leaving them with alternative approaches to stimulate the economy because people can always hold cash rather than negative-yielding assets. Thus, the Fed stimulated the economy through quantitative easing, which is an unconventional monetary policy action where they engage in a large-scale asset buying program of government bonds to keep the short-term rates low and push down the long-term rates, which further incentives to borrowing, growth, and spending. Although it is possible to control for unconventional monetary policy (e.g., see Mesters et al., 2014), I do not include such variables.
Also, in the bottom of Figure 7-4, I construct a rolling window of five-year period correlations to investigate further whether there are varying periods of high correlation for these macroeconomic variables. There is a strong positive correlation between the estimated slope factor and both Capacity
Utilization and Federal Funds Rate from 2002 to 2012. After that, Capacity Utilization seems to have a high correlation, whereas Fed Funds Rate appears to go from uncorrelated to highly positive correlate again recovering post-2016, after Fed started increasing the interest rates. These results are in line with the zero lower bound discussion I provide above. I expect the effect to show up in regressions later and a possible way of dealing with it is to include a dummy interaction for a few years. However, I did not have time to do this.
97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18
Time
Slope Capacity Utilization Federal Funds Rate ICA: Slope factor and Macroeconomic variables
−0.75
−0.50
−0.25 0.00 0.25 0.50 0.75 1.00
02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18
Time
Correlation
Capacity Utilization Federal Funds Rate Rolling correlation window: Slope factor and Macroeconomic variables
Figure 7-4: Top: Estimated slope factor and Capacity Utilization and Federal Funds Rate. Bottom:
Rolling window of five-year period correlations between the estimated slope factor and Capacity Utilization and Federal Funds Rate.
In Figure 7-5, there is no strong correlations between the curvature factor and macroeconomic variables over the whole period (1997-2017). Therefore, I construct a rolling window of five-year period correlations to investigate further whether there are periods of high correlation. In fact, Unemployment Rate, Consumer Sentiment Index, and Trade Weighted U.S. Dollar Index fluctuate around the zero mark until recent years with correlation around 0.80 with Unemployment Rate negatively correlated.
I fit a VAR(1) model with estimated ICA factors and a subset of macroeconomic variables. I do not include all variable because I have 246 monthly observations and to use all macro variables leads to few degrees of freedom left since many coefficients have to be estimated. However, it is possible to regularize the VAR(1) model to reduce the parameter space (e.g., see Nicholson et al.,
97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 Time
Curvature Unemployment Rate Consumer Sentiment Index Trade Weighted U.S. Dollar ICA: Curvature factor and Macroeconomic variables
−0.8
Unemployment Rate Consumer Sentiment Index Trade Weighted U.S. Dollar Rolling correlation window: Curvature factor and Macroeconomic variables
Figure 7-5: Top: Estimated curvature factor and Unemployment Rate, Consumer Sentiment Index, and Trade Weighted U.S. Dollar Index. Bottom: Rolling window of five-year period correlations between the estimated curvature factor and Unemployment Rate, Consumer Sentiment Index, and Trade Weighted U.S.
Dollar Index.
2015). Although the regularization approach may be useful, I select variables based on the rolling correlations and previous literature because of the time constraint. Therefore, the variables I choose to include in the VAR(1) model are Capacity Utilization, Federal Funds Rate, CPI, Unemployment Rate, Customer Sentiment Index, and Money Stock.
In Table 7.3, I present parameter estimates for a VAR(1) model with estimated ICA factors and the macro variables. There is a high persistence in each of the factors and variables. Notably, there are two coefficient estimates above 1, i.e., Capacity Utilization and Money Stock with values equal to 1.01. However, they have standard errors equal to 0.02 and 0.01 respectively. A remarkable fact is that the largest significant off-diagonal coefficient is between the slope and curvature factor.
Based on the VAR(1) model, I examine the dynamics of the estimated factors and the macroeco-nomic variables via impulse response functions, which I show (along with 90% confidence intervals) in Figure 7-6 and Figure 7-7. I consider three groups of impulse responses: macro responses to fac-tor shocks, facfac-tor responses to macro shocks, and facfac-tor responses to facfac-tor shocks. In general, the responses for level, slope, and curvature for shocks to Federal Funds Rate and CPI (inflation proxy) are in line with previous literature (e.g., Diebold et al., 2006). However, the shocks to Capacity
Uti-lization is different and this may be due to that they study a different period that does not deal with the same issues as the period I am studying, i.e., with unconventional monetary policy due to the zero lower bound. For example, if inflation is too low or unemployment too high, the central bank may push down short-term rates to increase the incentives to spend. However, too large negative rates are not feasible since people can always hold cash. Therefore, unconventional monetary policy, such as quantitative easing, is a tool that the Fed is using to stimulate the economy. Also, in 2006 all of this was theoretical and zero-lower bound was hardly an issue in the literature. Consequently, it is likely that this tool may affect the results when comparing to literature that looks at other periods.
First, I analyze and discuss the impulses to the six macro vars and the latent factor responses. (i) An increase in Capacity Utilization immediately pushes down the level of the yield curve, whereas it pushes up the slope so that the yield curve is less positively sloped (or more negatively sloped).
The latter is consistent with previous literature (Diebold et al., 2006). However, the former is not, and it is perhaps due to the extra data points as discussed above. Also, the curvature seems to have a negligible response. (ii) An increase in the Federal Funds Rate appears to increase the level of the yield curve, although it is not significant since the zero point is within the confidence bounds. The same goes for the slope factor. An increase in Federal Funds Rate depresses the curvature factor drastically so that the yield curve is less curved. This result is remarkable, since Diebold et al.
(2006) did not find any significant changes and my result may be due to that half the sample has higher short and long-term rate than the mid-term rate (a so-called positive butterfly). Therefore, when short and long end increase more (or decrease less) than the mid-term rates, the convexity of the yield curve increases. (iii) An increase in the CPI, which can be considered to be a proxy for inflation, pushes up the level of the yield curve consistent with long-term inflation expectations not being firmly anchored, and raises the level factor. The slope and curvature factors seem to have negligible responses to a CPI increase. (iv) An increase in the Unemployment Rate depresses the curvature so that the yield curve is less curved. It also seems that the level decrease in the long-run, and that the slope factor is negligible. (v) An increase in Consumer Sentiment Index increase the level marginally in the short-run and also increases the slope marginally with the curvature factor being negligible. (vi) An increase in Money Stock tends to increase the level and curvature. Also, the effect on slope seems to be insignificant.
The impulses to the three latent factors and the macro vars responses show that shocks to the level and curvature factor seem to have no response in the macro variables. However, an increase in the slope factor depresses Capacity Utilization immediately. The same goes for CPI, although it seems to be only in the short run and later on appears to increase CPI. The inverse relationship is for Money Stock, where it first increase in the short term and after that decrease rapidly. An increase in the slope factor pushes the Federal Funds Rate, and the same goes for Unemployment
Rate and Consumer Sentiment Index.
Finally, most off-diagonal responses of the level, slope, and curvature own-dynamics are insignif-icant except the shock to slope factor which pushes up the curvature factor drastically making the yield curve more curved in the short term and making the yield curve less curved in the long term.
Table 7.3: Parameter estimates for VAR(1) model with estimated ICA factors and a subset of macroeco-nomic variables. The macro variables are Capacity Utilization (CU), Unemployment Rate (UR), Consumer Sentiment Index (CSI), Money Stock (MS), Federal Funds Rate (FFR), and CPI. Standard errors appear in parentheses.
f1,t−1 f2,t−1 f3,t−1 CUt-1 URt-1 CSIt-1 MSt-1 FFRt-1 CPIt-1
f1,t 0.90*** 0.01 −0.14* 0.06*** −0.02** 0.05 −0.04*** −0.01 0.08**
(0.04) (0.01) (0.06) (0.01) (0.01) (0.03) (0.01) (0.01) (0.02)
f2,t 0.05 0.88*** 0.72** −0.26*** 0.13*** 0.01 0.16** 0.05 −0.44***
(0.14) (0.05) (0.23) (0.05) (0.04) (0.12) (0.05) (0.02) (0.09)
f3,t −0.10*** 0.03*** 0.73*** 0.03** −0.01* 0.05* −0.04*** 0.04*** 0.06**
(0.03) (0.01) (0.04) (0.01) (0.01) (0.02) (0.01) (0.00) (0.02)
CUt −0.17*** 0.01 0.06 1.01*** −0.06*** 0.18*** −0.01 0.05*** 0.04
(0.04) (0.01) (0.07) (0.02) (0.01) (0.04) (0.01) (0.01) (0.03)
URt −0.03 0.00 −0.07 0.06*** 0.95*** 0.08* −0.03* 0.04*** 0.02
(0.04) (0.01) (0.07) (0.01) (0.01) (0.04) (0.01) (0.01) (0.03)
CSIt 0.12 0.04 −0.05 0.05* −0.02 0.63*** 0.03 0.00 −0.07
(0.06) (0.02) (0.10) (0.02) (0.01) (0.05) (0.02) (0.01) (0.04)
MSt 0.01 −0.02 0.00 −0.03 0.01 0.16*** 1.01*** −0.03*** −0.06*
(0.04) (0.01) (0.07) (0.02) (0.01) (0.04) (0.01) (0.01) (0.03)
FFRt −0.04 0.09 −0.84** 0.25*** −0.11** 0.13 −0.21*** 0.94*** 0.51***
(0.15) (0.05) (0.25) (0.06) (0.04) (0.14) (0.05) (0.03) (0.10)
CPIt 0.03 0.00 −0.16* −0.02 0.01 −0.10** 0.03* −0.04*** 0.91***
(0.04) (0.01) (0.07) (0.02) (0.01) (0.04) (0.01) (0.01) (0.03)
***p < 0.001,**p < 0.01,*p < 0.05
0 10 25
Figure 7-6: Impulse response functions for the estimated factors (level, slope, and curvature) and a subset of macroeconomic variables based on the VAR(1) model in Table 7.3. The macro variables are Capacity Utilization (CU), Federal Funds Rate (FFR), CPI, Unemployment Rate (UR), Consumer Sentiment Index (CSI), and Money Stock (MS).
0 10 25
−1.00.5
Shock to Level
Level response 0 10 25−1.00.5
Shock to Slope
Level response 0 10 25−1.00.5
Shock to Curv.
Level response
0 10 25
−1.00.5
Shock to Level
Slope response 0 10 25−1.00.5
Shock to Slope
Slope response 0 10 25−1.00.5
Shock to Curv.
Slope response
0 10 25
−1.00.5
Shock to Level
Curvature response 0 10 25−1.00.5
Shock to Slope
Curvature response 0 10 25−1.00.5
Shock to Curv.
Curvature response
Figure 7-7: Impulse response functions for the estimated factors (level, slope, and curvature) based on the VAR(1) model in Table 7.3.
Section 8
Conclusion
I find that the loadings from ICA give an interpretation of ICA factors as level, slope, and curvature.
They also have similar shapes as the loadings in PCA and DNSM. The estimated ICA factors have highly persistent dynamics and correlate with estimates from PCA and DNSM (especially with PCA).
All three-factor models fit the yield curve well and produce good forecasts. ICAYM and PCAYM forecast short-term yields with a long forecast horizon significantly better than DNSM, across most forecast horizons. Concerning the yield curves longer maturities, there are no statistically significant differences between ICAYM and DNSM. However, DNSM is significantly better to forecast future yields at a short forecast horizon. Now, since ICAYM has a good forecast performance for the short-term maturities and multiple horizons, one can investigate if it is possible to improve long-short-term forecasts to estimate the bond risk premia and apply it to forecasted short-term yields to obtain mid to long-term yields. To estimate the bond risk premia, one may use NY FED ACM model (Adrian et al., 2013, 2014) or estimate how, e.g., the 7y forward 3m rate differ from the 10y forward 3m rate.
This whole approach relies on the assumption that people do not have strong expectations about interest rates that far out and that the bond risk premium is a significant yield curve influencer.
As pointed out in section 4, I assume that the factors distribution are a reciprocal cosh density.
However, whether this is the best density remains to be answered. Furthermore, extensions can be considered, such as estimating ICA factor loadings jointly in a state-space model (similar to DNSM). However, this is likely to be computationally cumbersome, whereas this paper’s proposed ICA-based model is fast concerning computational time. Another possible extension may be time-varying loadings, or to use a rolling window approach (e.g., Shah, 2013). An application related to the rolling window approach can be to study whether there are evident dynamic patterns regarding a coupling metric (Shah and Roberts, 2013), i.e., a dependency measure, and can be used to study how factors interact across different maturities in the yield curve for trading strategies, risk management
purposes, or as an economic indicator. Also, all these approaches can be combined with wavelets to study time-scale dependencies (Shah and Roberts, 2008).
Traders may be interested in a high-frequency setting, where it may be easier to obtain indepen-dent factors because there are more observations than in the monthly case. However, one has to be cautious in selecting the period of interest because of the changes in the economic environment.
The literature for term structure models is vast, where a significant part belongs to the affine no-arbitrage models. Although ICAYM is not arbitrage-free, it gives interpretable results, good fit, and produces good forecasts. One future extension is to investigate feasibility to impose absence of arbitrage in ICAYM, similar as in the DNSM extension proposed by Christensen et al. (2011), where they add a yield-adjustment term.
The estimated factors have periods of high correlation with some of the macroeconomic vari-ables. In particular, the correlation between the level factor and Real Income, Real Consumption Expenditures, and CPI seem to be strongly negative around financial crisis in 2008, and in recent years (except CPI with opposite sign). The slope factor appears to have a high correlation with Capacity Utilization and Federal Funds Rate over the whole period. Finally, the correlation between the curvature factor and Unemployment Rate, Consumer Sentiment Index, and Trade Weighted U.S.
Dollar Index, seems to be high in recent years.
An increase in the Fed Funds Rate appears to increase the level and slope of the yield curve, whereas it depresses the curvature factor drastically so that the yield curve is less curved. An increase in the CPI pushes up the level of the yield curve, whereas the slope and curvature factors seem to have negligible responses. An increase in the Unemployment Rate decrease the level in the long-run and also depresses the curvature so that the yield curve is less curved.
In conclusion, today the most popular yield curve models are either based on parametric loading shapes or loadings for uncorrelated factors that explain the maximal variance in the yields. Here, I propose a dynamic ICA-based yield curve model that use independent factors. It fits the yield curve well and produces better forecasts for the short-term maturities and a long forecast horizon than the commonly used Diebold and Li (2006) model, with similar results for the long-term maturities. I show that the estimated factors from the proposed model relate to macroeconomic variables. Finally, I have demonstrated that ICA has a place in yield curve modeling, and many future extensions can further increase our understanding of the characteristics and dynamics of the yield curve, leading to more accurate asset pricing, portfolio allocation, risk management, and monetary policy.
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