Interest Rates & Bond Supply Study
Thomas Vermaelen
6/27/2019
Summary
In this study, we attempt to find an Instrument Variable (IV) to study the relationship between price informativeness and US interest rates. The ratio of US publicly owned debt to GDP is used as an IV.
As an initial step, we regress Annual US Treasury rates (treasury_yield) on Bond supply (Debt/GDP). We then run the diagnostics for the regression and fit the appropriate transformations to the data.
We then look at the significance of adding a linear time trend. The trend variable ends up being significant without affecting the significance of our IV, log(Debt/GDP)
In order to capture the autocorrelation of the residuals from our initial (transformed) regression we evaluate two different techniques:
The Feasible Generalized Least Squares (FGLS) regression (Pairs-Winsten), where the independent variable log(Debt/GDP) is assumed to be exogenous
OLS using serial correlation-robust standard error of the coefficient on log(Debt/GDP) (Newey-West’s HAC)
In both cases, we find log(Debt/GDP) to be significant
Variables
All variables are stored in the table \(bond.data\):
\(Fiscal Year\), US fiscal Year from 1919 to 2012, which is assumed to run from June 30 of the previous year up to September 30 of the given year (e.g. 2012 FY runs from June 30, 2011 to September 30, 2012). This is according to Krishnamurthy & Vissing-Jorgensen (2012), Appendix B
\(bond\)_\(supply\) (Debt/GDP) is the supply of Treasuries scaled by GDP, at market value, calculated as: (Debt/GDP)*(Total mkt value of T-Bills/total face value of T-Bills), for the given fiscal year. The series comes from Henning Bohn and has been annualized.
\(treasury\)_\(yield\) is the 10-Year Treasury Constant Maturity yield on September 30 (end of FY in the US)
Methodology
Perform Regression analysis with US interested rates as a dependent variable and US bond supply as an independent variable; check OLS assumptions
To make sure the relationship between interest rates and bond supply is not spurious, check for a time trend in the data
Control for autocorrelations in bond supply by estimating model parameters using FBLS and adjusting the standard errors using Newey-West’s HAC
Regression Analysis
1. a) Regressing treasury_yield on Debt/GDP
##
## Call:
## lm(formula = treasury_yield ~ bond_supply, data = bond.data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.365 -1.387 -0.544 0.883 6.827
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.548 0.594 12.707 < 2e-16 ***
## bond_supply -5.756 1.270 -4.531 1.76e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.3 on 92 degrees of freedom
## Multiple R-squared: 0.1824, Adjusted R-squared: 0.1735
## F-statistic: 20.53 on 1 and 92 DF, p-value: 1.761e-05Regression analysis yields a significant coefficient of -5.756 on Debt/GDP implying that an increase in the ratio of Debt/GDP by 1 leads to a decrease in the interest rate by 5.76 percentage points. This interpretation is not very intuitive, therefore, it would make more sense to take logs of the variables
1. b) Regression Diagnostics for treasury_yield ~ Debt/GDP
Linearity
Non-linearity is present. This suggests that a non-linear transformation of the independent variable is more appropriate
Homoskedasticity
Variance increases for larger values of Y implying that a log-transformation of the dependent variable is needed, which we will perform next
Normality of Residuals
Residuals seem to be non-normally distributed
2. a) Regressing treasury_yield on log(Debt/GDP)
##
## Call:
## lm(formula = treasury_yield ~ log_bond_supply, data = bond.data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9413 -1.3596 -0.6160 0.8711 6.8283
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.9293 0.6132 4.777 6.69e-06 ***
## log_bond_supply -2.3124 0.6048 -3.824 0.000239 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.363 on 92 degrees of freedom
## Multiple R-squared: 0.1371, Adjusted R-squared: 0.1277
## F-statistic: 14.62 on 1 and 92 DF, p-value: 0.0002391Regression reveals a coefficient of -2.312 for log(bond supply) that is significant at a 5% level signifying that a 1% change in Debt/GDP leads to a decrease in the interest rate by aproximately 0.02 percentage points. R- squared = 0.137 (slightly better than above)
2. b) Regression diagnostics for treasury_yield ~ log(Debt/GDP)
Homoskedasticity
Variance seems to increase at larger values of log(bond supply) still suggesting a log-transformation of treasury yield
Normality of Residuals
Residuals do not seem to be normally distributed
3 . a) Regressing Log(treasury_yield) ~ Log(Debt/GDP)
##
## Call:
## lm(formula = log_treasury_yield ~ log_bond_supply, data = bond.data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.81145 -0.29314 -0.02922 0.24238 0.89214
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.0256 0.1091 9.403 4.11e-15 ***
## log_bond_supply -0.5248 0.1076 -4.879 4.46e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4203 on 92 degrees of freedom
## Multiple R-squared: 0.2055, Adjusted R-squared: 0.1969
## F-statistic: 23.8 on 1 and 92 DF, p-value: 4.461e-06Coefficient on log(bond supply) is -0.525 and significant. A 1% change in Debt/GDP leads to a 0.53% decrease in interest rates. R-squared is 0.201, which is 46.72% improvement on the previous model.
3. b) Regression diagnostics for Log(treasury_yield) ~ Log(Debt/GDP)
Homoskedasticity
Not sure if Homoskedacity seems to improve
Normality of Residuals
Normality of residuals looks better
PACF suggests largelly significant lag 1 after controlling for other lags
Time series plot of residuals
Fitting Time Trend
Graph suggests there is long-term linear trend perhaps even a quadratic one
Fit linear time trend
##
## Call:
## lm(formula = bond.data$log_treasury_yield ~ bond.data$trend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.03016 -0.30879 -0.01081 0.29615 0.89239
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.054188 0.081053 13.006 < 2e-16 ***
## bond.data$trend 0.009678 0.001482 6.532 3.52e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3898 on 92 degrees of freedom
## Multiple R-squared: 0.3168, Adjusted R-squared: 0.3094
## F-statistic: 42.67 on 1 and 92 DF, p-value: 3.524e-09Model suggests the presence of a linear time trend . R-squared = 0.309
Fit quadratic time trend
##
## Call:
## lm(formula = bond.data$log_treasury_yield ~ bond.data$trend +
## bond.data$trend_sqrd)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9223 -0.2954 0.0001 0.2802 0.8549
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.392e-01 1.229e-01 7.644 2.07e-11 ***
## bond.data$trend 1.687e-02 5.969e-03 2.825 0.0058 **
## bond.data$trend_sqrd -7.567e-05 6.088e-05 -1.243 0.2171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3886 on 91 degrees of freedom
## Multiple R-squared: 0.3282, Adjusted R-squared: 0.3135
## F-statistic: 22.23 on 2 and 91 DF, p-value: 1.375e-08Model suggests no presence of a qudratic time trend
Adding back log(Debt/GDP) to the regression: log(treasury_yield) ~ Trend
##
## Call:
## lm(formula = bond.data$log_treasury_yield ~ bond.data$trend +
## bond.data$log_bond_supply)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.70786 -0.18194 -0.04292 0.18898 0.67506
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.413584 0.100698 4.107 8.74e-05 ***
## bond.data$trend 0.010934 0.001147 9.533 2.42e-15 ***
## bond.data$log_bond_supply -0.624457 0.077221 -8.087 2.53e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.299 on 91 degrees of freedom
## Multiple R-squared: 0.6025, Adjusted R-squared: 0.5938
## F-statistic: 68.96 on 2 and 91 DF, p-value: < 2.2e-16Coefficient of log(bond supply) is still significant after adding linear trend, it seems as if the relationship between log(treasury yield) and log(Debt/GDP) is not spurious. R-squared = 0.594
Linear trend model diagnostics : log(treasury_yield) ~ log(Debt/GDP) + Trend
Homoskedasticity
Homoskedasdicity seems to improve compared to the model without a linear trend.
Normality
Normality still looks somewhat good
Autocorrelation of residuals
PACF suggests largelly significant lag 1 after controlling for other lags
Time series plot of residuals
Accounting for Autocorrelation
1. Feasible Generalized Least Squares (Pairs-Winsten), assuming strict exogeneity of X
Method:
- Run OLS on log(treasury_yield) and log(Debt/GDP) and obtain residuals
- Run AR(1) on residuals, save coefficient \(\rho\):
- Run OLS on the following (provided that the intercept coefficient is divided by \(1 - \rho\)):
- \(\log(treasury \_ yield_t) - \rho log(treasury\_ yield_{t-1})\)
- \(\log(bond \_ supply_t) - \rho \log(bond\_ supply_{t-1})\)
##
## Call:
## lm(formula = FGLS$y.tilde ~ FGLS$x.tilde)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.33429 -0.06396 -0.00546 0.06210 0.23613
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.028475 0.009877 2.883 0.00491 **
## FGLS$x.tilde -0.244530 0.092014 -2.658 0.00930 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09417 on 91 degrees of freedom
## Multiple R-squared: 0.07202, Adjusted R-squared: 0.06182
## F-statistic: 7.063 on 1 and 91 DF, p-value: 0.009297Using FGLS (Pairs-Winsten), the coefficient on log(Debt/GDP) is -0.245 and signficant, suggesting that a 1% increase in bond supply leads to a 0.25% decrease in interest rates. However, R-squared is very low with a value of = 0.072
2. Diagnostics of FBLS
Normality of residuals improves slightly
Homoskedasticity looks good for larger fitted values.
Autocorrelation has been accounted for
Using OLS and Newey-West Heteroskedastic and Autocorrelation Consistent (HAC) standard errors
Method:
- run OLS on Log(treasury_yield) ~ log(Debt/GDP)
- compute the correction factor that adjusts for serially correlated errors using the estimated autocorrelation coefficient and a chosen truncation parameter
- derive the Newey-West variance estimator using correction factor and use it to compute the HAC standard error for the estimated coefficient of log(Debt/GDP)
##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.02563 0.16758 6.1202 2.26e-08 ***
## bond.data$log_bond_supply -0.52482 0.19233 -2.7288 0.007616 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1OLS regression with serial correlation-robust standard errors yields a significant coefficient of -0.525 for log(Debt/GDP)
Sources
Regression Model Diagnostics: http://www.sthda.com/english/articles/39-regression-model-diagnostics/161-linear-regression-assumptions-and-diagnostics-in-r-essentials/#linearity-of-the-data
OLS assumptions: https://statisticsbyjim.com/regression/ols-linear-regression-assumptions/
GLS and autocorrelation : https://pdfs.semanticscholar.org/7c1b/5ba81a4fcde9abe7efe49228244a98d5ce11.pdf
FBLS regression: https://www.youtube.com/watch?v=GsLacO_ZJ-s
Newey-West HAC : https://www.econometrics-with-r.org/15-4-hac-standard-errors.html