S&P Returns vs 3 Spreads with Volatility

This is the continuation of the research in this main post and addendum post. We remove earnings yield from regression for stock index returns. The rates-only.py code is from GitHub/asarantsev repository 3spreads-CAPE-simulator

Consider annual S&P returns $Q(t)$ (price/total, nominal/real)

$Q(t) = a + \sum_{i=1}^3b_iS_i(t) + cV(t) + V(t)Z(t).$

Standard analysis of residuals $Z(t)$ explained in this main post shows they are well modeled as independent identically distributed Gaussian. All $b_1, b_2, b_3$ are not significant: Student T-test has p-values greater than 5%. The most significant (having the smallest p-values) is $b_1$ corresponding to the BAA-AAA spread. Exceptions: for Total Nominal Returns, $b_1$ has $p = 3.3\%.$ The coefficient $c < 0$ is very statistically significant with $p < 0.15\%.$ See below the graphs for simulated volatility and spreads.

We added to this GitHub repository the entire simulator for the rates-only model (with annual volatility, of course). It is done in Python file rates-only-sim.py in the same repository. See below the graph.

Below we show one simulated path of prices and wealth. This simulation is for the case of real (inflation-adjusted) version and 20-year horizon.

Finally, pick $Z(t)$ for the case of total nominal returns. Below we see the autocorrelation plot for $Z(t)$ , for $|Z(t)|$ , and the quantile-quantile plot for $Z(t)$ versus the Gaussian distribution.

As before, we have this strange value at lag 4 for autocorrelation. This presents no problem, since other values are low.

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S&P Returns vs 3 Spreads with Volatility

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