Uribe (2021) investigates the nature and empirical importance of monetary policy shocks that produce neo-Fisherian dynamics, i.e. move interest rates and inflation in the same direction over the short run. To that end, the author estimates a standard small-scale New-Keynesian model with price stickiness and habit formation, augmented with seven structural shocks.

Three of the shocks are to monetary policy, which is described by the following policy rule:

\[ \frac{1+I_t}{\Gamma_t} = \left[A \left(\frac{1 + \Pi_t}{\Gamma_t} \right)^{\alpha_t} \left(\frac{Y_t}{X_t} \right)^{\alpha_{y}} \right]^{1-\gamma_I} \left(\frac{1 + I_{t-1}}{\Gamma_{t-1}} \right)^{\gamma_I} e^{z_t^{m}}, \]

where \(I_t\) the nominal interest rate, \(Y_t\) is aggregate output, \(\Pi_t\) is the inflation rate, \(\Gamma_t\) is the inflation-target, \(X_t\) is a nonstationary productivity shocks, and \(z_t^{m}\) is a stationary interest-rate shock. The inflation target is defined as

\[ \Gamma_t = X_t^m e^{z_t^{m2}}, \]

where \(X_t^m\) and \(z_t^{m2}\) are permanent and transitory components of the inflation target. It is assumes that \(X_t^m\) and \(X_t\) grow at a rates \(g^m_t\) and \(g_t\), respectively.

There are two preference shocks affecting the lifetime utility function of the representative household, given by

\[ \operatorname{E}_0 \sum_{t=0}^{\infty}\beta^t e^{\xi_t}\Bigg\{ \frac{\left[\left(C_t - \delta \tilde{C}_{t-1}\right) \left(1-e^{\theta_t}h_t \right)^{\chi} \right]^{1-\sigma} - 1}{1-\sigma} \Bigg\}, \]

where \(C_t\) is consumption, \(\tilde{C}_t\) is the cross sectional average of consumption, \(h_t\) is hours worked, \(\xi_t\) is an intertemporal preference shock, and \(\theta_t\) is a shock to labor supply.

In addition to \(X_t\), there is also a stationary productivity shock \(z_t\), which affects the production technology according to

\[ Y_t = e^{z_t}X_t h_t^{\alpha}, \]

The five stationary shocks (\(\xi_t\), \(\theta_t\), \(z_t\), \(z_t^{m}\), and \(z_t^{m2}\) ) and the growth rates of the two non-stationary shocks (\(g_t\) and \(g^m_t\)) are all assumed to follow first-order autoregressive processes.

Uribe (2021) estimates the model using quarterly US data on three variables: per capita output growth (\(\triangle y_t\)), the interest-rate-inflation differential (\(r_t=i_t - \pi_t\)), and the change in the nominal interest rate (\(\triangle i_t\)). All variables are assumed to be observed with measurement errors.