Stock Returns vs BAA

Continuing research of the four bond rates, we model normalized total returns of S&P (nominal or real) as a regression upon the BAA rates at the end of last year, and the change in BAA rates throughout this year. See a GitHub repository.

Description: Let $Q(t)$ be nominal or real total returns during year $t.$ Let $V(t)$ be the annual volatility during year $t.$ Let $R(t)$ be the BAA rate at end of year $t.$

We consider two models, both for the nominal and the real versions of returns.

Model 1. $\frac{Q(t)}{V(t)} = a + bS(t-1) + c(S(t) - S(t-1)) + \delta(t).$

Model 2. $\frac{Q(t)}{V(t)} = \frac{a}{V(t)} + b\frac{S(t-1)}{V(t)} + c\frac{S(t) - S(t-1)}{V(t)} + k + \delta(t).$

Results: In each of the four models, residuals $\delta(t)$ are IID Gaussian, judging by the normality tests and the autocorrelation function plots.

But what is the goodness of fit? We get $R^2 = 15\%$ for nominal Model 1 and $R^2 = 19\%$ for real Model 1. But $R^2 = 44\%$ for nominal Model 2, and $R^2 = 42\%$ for real Model 2.

Regression results are: The coefficient $b$ is insignificant judging by the Student T-test, but $c < 0$ is significant. In both versions of Model 2, $k = -0.0103,$ significantly different from zero. For Model 2, actually $p = 38\%$ for the nominal version and $p = 7\%$ for the real version.