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.
|Year||10 year||2 year||Yield Curve||Yield Curve Forecast||Net Bias Based on Yield Curve|
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.
|Year||10 year||2 year||Upper Bias as per Prediction Intervals||Lower Bias as per Prediction Intervals||Net Bias Based on Prediction Limits|
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.