### A.5 Proof of Corollary 6

From the definition of the first-order autocorrelation, we calculate $ \rho_{\Delta c}^{HY} = \text{Cov}\left(\Delta c_{t+1}^{HY}, \Delta c_{t}^{HY}\right) / \text{Var}\left(\Delta c_{t}^{HY}\right) $ . Changes in consumption under the hybrid model are expressed as $ \Delta C_{t+1}^{HY} = (1 - \lambda) \Delta C_{t+1} + \lambda \Delta X_{t+1} $ , which we normalize by $ X_{t} $ to obtain $ \Delta c_{t+1}^{HY} = (1 - \lambda) \Delta c_{t+1} + \lambda g_{t} $ .

First, $ Cov\left(\Delta c_{t+1}^{HY},\Delta c_{t}^{HY}\right) $ , the numerator of $ \rho_{\Delta c}^{HY} $ , is given by

$$ \begin{align*}Cov\left(\Delta c_{t+1}^{HY},\Delta c_{t}^{HY}\right)\ =&\ \left(1-\lambda\right)^{2}Cov\left(\Delta c_{t+1},\Delta c_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t+1},\Delta c_{t}\right)\\&+\lambda\left(1-\lambda\right)Cov\left(\Delta c_{t+1},g_{t}\right)+\lambda^{2}Cov\left(g_{t+1},g_{t}\right).\end{align*} $$

Because both $ \Delta c_{t} $ and $ g_{t} $ follow the AR(1) process (see Proposition 2), $ Cov\left(\Delta c_{t+1}, \Delta c_{t}\right) = \omega^{*}Var\left(\Delta c_{t}\right) $ , $ Cov\left(g_{t+1}, \Delta c_{t}\right) = \omega^{*}Cov\left(g_{t}, \Delta c_{t}\right) $ , $ Cov\left(\Delta c_{t+1}, g_{t}\right) = \phi Cov\left(g_{t}, \Delta c_{t}\right) $ , and $ Cov\left(g_{t+1}, g_{t}\right) = \phi Var\left(g_{t}\right) $ . Substituting these expressions into the above equation yields

$$ \begin{align*}Cov\left(\Delta c_{t+1}^{HY},\Delta c_{t}^{HY}\right)=&\ \left[\left(1-\lambda\right)^{2}Var\left(\Delta c_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)\right]\omega^{*}\\&+\left[\lambda^{2}Var\left(g_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)\right]\phi.\end{align*} $$

Next, the variance $ Var\left(\Delta c_{t}^{HY}\right) $ , the denominator of $ \rho_{\Delta c}^{HY} $ , can similarly be calculated. The resulting expression is

$$ \begin{aligned}Var\left(\Delta c_{t}^{HY}\right)\ =&\ \left(1-\lambda\right)^{2}Var\left(\Delta c_{t}\right)+2\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)\\&\ +\lambda^{2}Var\left(g_{t}\right).\end{aligned} $$

Finally, taking the ratio of $ (50) $ to $ (51) $ yields

$$ \begin{array}{rcl}\rho_{\Delta c}^{HY}&=&\displaystyle\frac{\left(1-\lambda\right)^{2}Var\left(\Delta c_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)}{Var\left(\Delta c_{t}^{HY}\right)}\omega^{*}+\frac{\lambda^{2}Var\left(\hat{g}_{t}\right)+\lambda\left(1-\lambda\right)Cov\left(g_{t},\Delta c_{t}\right)}{Var\left(\Delta c_{t}^{HY}\right)}\phi\\&=&\left(1-\Lambda\right)\omega^{*}+\Lambda\phi,\end{array} $$

which is (20) in the main text. We note that $ \Lambda $ is a function not only of $ \lambda $ but also of $ \operatorname{Var}\left(\Delta c_{t}\right) $ , $ \operatorname{Cov}\left(g_{t},\Delta c_{t}\right) $ , and $ \operatorname{Var}\left(g_{t}\right) $ :

$$ \begin{align*}Var\left(\Delta c_{t}\right)&=\frac{1}{1-(\omega^{*})^{2}}\left[\frac{\left(1-\omega\right)\left(1+r\right)}{1+r-\phi}\right]^{2}\sigma^{2},\\Cov\left(g_{t},\Delta c_{t}\right)&=\frac{1}{1-\omega^{*}\phi}\frac{\left(1-\omega\right)\left(1+r\right)}{1+r-\phi}\sigma^{2},\\Var\left(g_{t}\right)&=\frac{1}{1-\phi^{2}}\sigma^{2},\end{align*} $$

where all of these moments are a function of $ \phi $ , and $ \omega $ . In this proof, we used the assumption that $ \mu = 0 $ . In the case of $ \mu \neq 0 $ , we can similarly calculate $ \rho_{\Delta c}^{HY} $ , using the stochastic process of $ \Delta c_{t+1} $