Measure of Expectations Heterogeneity

Dear Dynare Community,

I try to develop a measure of heterogeneity of beliefs for the Branch&McGough (2009) model.
A New Keynesian model with heterogeneous expectations.
Journal of Economic Dynamics & Control 33 (2009) pp. 1036-1051.
William A. Branch, Bruce McGough

Motivated by the Di Bartolomeo et al. (2016) paper, more precisely their “consumption inequality” dimension which is driven by diverging expectations, my idea is to augment an ad hoc period loss function by some measure of heterogeneity of beliefs.

The later is given by
\begin{align} \mathrm{var}_i(E_t^i(x_{t+1}))=\alpha(1-\alpha)[E_t(x_{t+1})-\theta^2x_{t-1}]^2\end{align}
which is a cross-sectional variance of one-step ahead forecasts and x is a generic variable.

My question:
Can I implement the measure above into a period loss like
\begin{align} L_t &= \pi_t^2 + \omega_y y_t^2 + \omega_{E\pi} \mathrm{var}_i(E_t^i(\pi_{t+1})) \\&=\pi_t^2 + \omega_y y_t^2 + \omega_{E\pi}\left[ \alpha(1-\alpha)E_t(\pi_{t+1})^2 - 2\alpha(1-\alpha) \theta^2\pi_{t-1}E_t(\pi_{t+1})+ \alpha(1-\alpha)\theta^4\pi_{t-1}^2\right] \end{align}

My problem is the square of the rational expectation.
One may argue in favor of an auxiliary like: E_ppi = ppi(+1); (as additional state variable).
But I am worried about the intertemporal loss J_0=E_0\sum_{k=0}^\infty\beta^k L_{t+k}
since terms like E_0(E_{t+k}(\pi_{t+k+1})^2) show up.

Is the loss still appropriate?

Best,
Max

You can easily deal with nonlinear transformations of expected values using auxiliary variables. See Expected value of a power

Dear Prof. Pfeifer,

If I understand correct one can implement the loss L_t = \omega_{E\pi}\alpha(1-\alpha)[E_t(\pi_{t+1})-\theta^2\pi_{t-1}]^2 + \cdots
as follows.

model(linear);
...
ppi_lag   = ppi(-1);
end;

planner_objective w_Eppi*alp*(1-alp)*(ppi_lead - thet^2*ppi_lag)^2 + .... ;

Yes, indeed. But as you are probably the first person trying this, please report any unusual behavior you encounter.

Can someone tell me how to compute a “targeting rule” under the loss above by hand?

For the sake of simplicity let L_t = \omega_yy_t + \omega_{E\pi} [E_t(\pi_{t+1})-\theta^2\pi_{t-1}]^2 and assume optimal monetary policy under commitment.

The Lagrangian obtains as
L= E_t\sum_{k=0}^\infty \beta^k \{ L_{t+k} + \psi_{t+k}[\pi_{t+k}-\beta\alpha\pi_{t+k+1}-\beta(1-\alpha)\theta^2\pi_{t+k-1} -\kappa y_{t+k}]\} + t.i.p.

How would the FOC w.r.t \pi_{t+k} look like?

What do you mean with “targeting rule”? And where is your problem with the FOC? You would plug in, write out the sum and then take the derivative.

I am worried about the expectation inside the period loss.

If I simply do the job (following your three steps) I obtain the FOC w.r.t \pi_{t+k}
\begin{align}\frac{\partial L}{\partial \pi_{t+k}} =E_t\Big( \beta^k\psi_{t+k} + 2\beta^{k-1}\omega_{E\pi}[E_{t+k-1}(\pi_{t+k})-\theta^2\pi_{t+k-2}] - \beta^{k-1}\beta\alpha\psi_{t+k-1} \\ + 2\beta^{k+1}\omega_{E\pi}[E_{t+k+1}(\pi_{t+k+2})-\theta^2\pi_{t+k}] (-\theta^2) -\beta^{k+1}\beta (1-\alpha)\theta^2 \psi_{t+k+1} \Big) = 0 \end{align}

is this correct?

For k=0 I obtain further:
\begin{align}E_t\Big( \psi_{t} + 2\beta^{-1}\omega_{E\pi}[E_{t-1}(\pi_{t})-\theta^2\pi_{t-2}] - \alpha\psi_{t-1} \\ + 2\beta\omega_{E\pi}[E_{t+1}(\pi_{t+2})-\theta^2\pi_{t}] (-\theta^2) -\beta^{2} (1-\alpha)\theta^2 \psi_{t+1} \Big) = 0 \\ \\ \psi_{t} + 2\beta^{-1}\omega_{E\pi}[E_{t-1}(\pi_{t})-\theta^2\pi_{t-2}] - \alpha\psi_{t-1} \\ - 2\theta^2\beta\omega_{E\pi}[E_{t}(\pi_{t+2})-\theta^2\pi_{t}] -\beta^{2} (1-\alpha)\theta^2 E_t\psi_{t+1} =0 \end{align}

since E_tE_{t-1}=E_{t-1}.

So the targeting rule will involve E_{t-1}.

Targeting rule = reduced form of the FOC / optimal relationship between the target variables (eliminate the lagrange multipliers) (à la Svensson)