Term Structure forecast and gradient analysis

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interest rate hike

Using Multiple Regression Analysis with prediction intervals for forecasting:

A multiple regression analysis is an equation for estimating relationship amongst dependent and independent variables.  Specifically, this means how the value of the dependent variable changes as the value of the independent variable is varied.  The multiple regression equation tries to predict the average value of the dependent variable when the independent variables are fixed.  Thus this methodology estimates the conditional expectations of the average value of the dependent variable when the independent variables are fixed.  Adding predictive inferences to these equations adds to their functionality as prediction intervals are estimations of an interval in which the future observations will fall in light of the historical data with a certain degree of probability.  Thus prediction intervals present a forecast with a random variable that is yet to be observed.  This differs from the confidence interval which is computed from the data which is assumed to be non-random and unknown.

The multiple regression equation may be used independently or in conjunction with prediction intervals to ascertain bias.  For example, the below stated data calculates the term structure of bonds.  If the term structure is downward sloping, meaning that the yield on long term bonds (10 years) is decreasing at a faster rate than short term bonds (2-year treasury yields), would manifest in a negative slope and investors would be deemed to be risk averse.  Converse would also be true.

We would like to check what is the term structure bias, or it future tendency.  An upward tendency would indicate to a positive sentiment in the outlook of the market whereas a negative tendency would negate that and, as has been in the past, be a precursor to economic slowdown.  Two biases or upper and lower limit of the prediction interval may be calculated for the regression equation in which the dependent variables are the 2 year and the 10-year bond yield and the term structure is the dependent variable.  We witness, that there is an upper and a lower bias for each bond yield for a given year.  For example, for year 2016, the upper bias is 0.175 and the lower bias is 0.191.  Looking at the magnitudes of these numbers we see that there is a residual tendency for the term structure to point downwards, thus the net bias based on prediction limits is negative 0.015 (- 0.015), which exhibits a net downward pressure on the term structure.  We witness bias on a positive slope from 2009 – 2012, when a maximum momentum of 0.339 was reached and generally the economy was in a faster rate of expansion.  After this point bias changed as indicated by a steadily reducing number from 2012-2015 before turning negative in 2016.  From a historical perspective what seems apparent is that this bias should continue, barring any uncharacteristic conditions suddenly reversing it.

Term Structure Analysis as per prediction intervals

 

Year 10 year 2 year Yield Curve Yield Curve Forecast Net Bias Based on Yield Curve
2016 1.691 0.734 0.434 0.431 -0.003
2015 2.035 0.645 0.317 0.369 0.052
2014 2.495 0.583 0.234 0.298 0.064
2013 2.615 0.321 0.123 0.226 0.103
2012 1.633 0.234 0.143 0.331 0.188
2011 1.917 0.247 0.129 0.298 0.169
2010 2.512 0.426 0.170 0.262 0.092
2009 3.305 0.948 0.287 0.274 -0.013
2008 3.829 1.968 0.514 0.427 -0.087
2007 4.594 3.988 0.868 0.764 -0.104
2006 4.633 4.691 1.013 0.910 -0.102
2005 4.332 4.173 0.963 0.837 -0.126
2004 4.123 2.593 0.629 0.524 -0.105
2003 3.939 1.464 0.372 0.305 -0.067
2002 3.6 1.691 0.470 0.396 -0.073
2001 4.59 2.851 0.621 0.521 -0.100
2000 5.808 6.004 1.034 1.044 0.011
1999 5.881 5.591 0.951 0.946 -0.004
1998 4.429 4.302 0.971 0.853 -0.119
1997 6.103 5.783 0.948 0.960 0.012
1996 6.7 6.101 0.911 0.953 0.042
1995 6.182 5.868 0.949 0.968 0.019
1994 7.608 6.585 0.866 0.943 0.077
1993 5.374 3.875 0.721 0.642 -0.079
1992 6.354 3.803 0.599 0.503 -0.095
1991 7.452 5.973 0.802 0.831 0.029
1990 8.81 8.191 0.930 1.136 0.206

 

A similar bias can be estimate via a yield curve forecast which is calculated based on the same set of independent and dependent variables.  We see that the forecasted term structure is given by the equation:

Term Structure = 0.4916 + 0.2232 2 year – 0.1331 10 year

The term structure forecast is an estimation based on this equation.  If we calculate this forecast by deducting from it the actual term structure, then naturally a negative bias would be foretelling of a negative bias and a positive notation would speak of a positive bias in the term structure.

Term Structure Analysis as per multiple regression equation

 

Year 10 year 2 year Upper Bias as per Prediction Intervals Lower Bias as per Prediction Intervals Net Bias Based on Prediction Limits
2016 1.691 0.734 0.175 0.191 -0.015
2015 2.035 0.645 0.247 0.118 0.128
2014 2.495 0.583 0.269 0.096 0.174
2013 2.615 0.321 0.330 0.036 0.294
2012 1.633 0.234 0.352 0.013 0.339
2011 1.917 0.247 0.242 0.123 0.119
2010 2.512 0.426 0.192 0.173 0.019
2009 3.305 0.948 0.169 0.196 -0.027
2008 3.829 1.968 0.080 0.284 -0.205
2007 4.594 3.988 0.085 0.280 -0.195
2006 4.633 4.691 0.091 0.274 -0.183
2005 4.332 4.173 0.057 0.308 -0.251
2004 4.123 2.593 0.092 0.273 -0.181
2003 3.939 1.464 0.111 0.254 -0.142
2002 3.6 1.691 0.089 0.276 -0.187
2001 4.59 2.851 0.115 0.249 -0.134
2000 5.808 6.004 0.223 0.143 0.080
1999 5.881 5.591 0.134 0.231 -0.097
1998 4.429 4.302 0.155 0.210 -0.055
1997 6.103 5.783 0.213 0.152 0.061
1996 6.7 6.101 0.200 0.165 0.035
1995 6.182 5.868 0.258 0.107 0.151
1994 7.608 6.585 0.176 0.189 -0.013
1993 5.374 3.875 0.110 0.256 -0.146
1992 6.354 3.803 0.144 0.223 -0.079
1991 7.452 5.973 0.284 0.082 0.201
1990 8.81 8.191 0.416 -0.049 0.464

 

Both the above stated procedures would give a similar result and the shape of the graphs is similar.  Also statistically we witness that both sets of data are showing a high degree of positive correlation computed at 0.88, which is almost linear, thus both approaches work from a simplistic standpoint.

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