Forecasting macroeconomic fundamentals in economic crises

Abstract

The paper studies the way economic turmoils influence the lay agents’ predictions of macroeconomic fundamentals. The recent economic crises have, in fact, led several authors to challenge the standard macroeconomic view that all agents are Muth-rational, hence omniscient and homogeneous, forecasters. In this paper lay agents are assumed to be heterogeneous in their predictive ability. Heterogeneity is modeled by assuming that people have equal loss functions, but different asymmetry parameters. The adopted methodological tools are grounded in the standard operational research theory. Specifically, we develop a dynamic stochastic optimization problem, which is solved by performing extensive Monte Carlo simulations. Results show that the less sophisticated forecasters in our setting—the medians—never perform as muthians and that second best (SB) agents do that only occasionally. This regardless the size of the crisis. Thus, as in the real world, in our artificial economy heterogeneity is a structural trait. More intriguingly, simulations also show that the medians’ behavior tend to be relatively smoother than that of SB agents, and that the difference between them widens in the case very serious crises. In particular, great recessions make SB agents’ predictions relatively more biased. An explanation is that dramatic crises extend the available information set (e.g., due to greater mass media coverage), and this leads SB agents, who are more attentive to revise their forecasts than medians. The point is that more information does not necessarily mean better forecasting performances. All considered, thus, our simulations suggest a rewording of Ackoff’s famous phrase: it is not silly to not look for an optimal solution to a mess.

Appendix

We present the numerical procedures in a block-wise form.

First block

The first building block of consists of the generation of the dynamics of agents’ GDP forecast $\{\tilde{\mathbf {f}}_t\}_t$ in (1).

• (A.1) set $j=1$;

• (A.2) set $t=1$;

• (A.3) set $\tilde{f}_{j,t}=\tilde{f}_0$;

• (A.4) set $t=t+1$ and go to step (A.5);

• (A.5) if $t=T+1$, go to step (A.7). Otherwise, go to step (A.6);

• (A.6) set $\tilde{f}_{j,t}=\tilde{f}_{j,t-1}+\rho /(1+\theta (j,t))*\tilde{f}_{j,t-1}$ and go to step (A.4);

• (A.7) set $j=j+1$ and go to step (A.8);

• (A.8) if $j=J+1$, stop. Otherwise, go to step (A.2).

Indices $t=1, \ldots , T$ and $j=1, \ldots , J$ represent time and the components of the vectors $\theta _t$, for each $t$.

Second block

In the second building block of the application, the stochastic dynamics of the true GDP $\{f_t\}_t$ is constructed. In particular, $H=1000$ scenarios are simulated by extracting $H=1000$ times the standard normal variable $\epsilon$ in Eq. (3). The values of $\alpha$ are also considered.

• (B.1) set $a=1$;

• (B.2) set $h=1$;

• (B.3) set $t=1$;

• (B.4) set ${f}_{t,h,a}=f_0$;

• (B.5) set $t=t+1$ and go to step (B.6);

• (B.6) if $t=T+1$, go to step (B.8). Otherwise, go to step (B.7);

• (B.7) set $f_{t,h,a}=f_{t-1,h,a}+\rho *f_{t-1,h,a}+\sigma (a)*f_{t-1,h,a}*N_h(0,1)$ and go to step (B.5);

• (B.8) set $h=h+1$ and go to step (B.9);

• (B.9) if $h=H+1$, go to step (B.10). Otherwise, go to step (B.3);

• (B.10) set $a=a+1$ and go to step (B.11);

• (B.11) if $a=3$, stop. Otherwise, go to step (B.2).

Indices $h=1, \ldots , H$ and $a=1,2$ point to the random scenarios and to the intensity of the crisis, respectively. In particular, $a=1,2$ stand for $\alpha =\alpha _m,\alpha _d$, respectively.

In accord to the setting of the problem, function $\sigma (a)$ introduced in step (B.7) is such that $\sigma (1)=0.01$, $\sigma (2)=0.4$.

$N_h(0,1)$ is the adopted notation for the standard normal variable, which is extracted in correspondence of the values of index $h$.

Third block

The third block aims at constructing the dynamics of the errors $\{x_t\}_t$ as in formula (4).

• (C.1) set $j=1$;

• (C.2) set $a=1$;

• (C.3) set $h=1$;

• (C.4) set $t=1$;

• (C.5) set ${x}_{j,t,h,a}=0$;

• (C.6) set $t=t+1$ and go to step (C.7);

• (C.7) if $t=T+1$, go to step (C.9). Otherwise, go to step (C.8);

• (C.8) set $x_{j,t,h,a}=f_{t,h,a}-\tilde{f}(j,t)$ and go to step (C.6);

• (C.9) set $h=h+1$ and go to step (C.10);

• (C.10) if $h=H+1$, go to step (C.11). Otherwise, go to step (C.4);

• (C.11) set $a=a+1$ and go to step (C.12);

• (C.12) if $a=3$, go to step (C.13). Otherwise, go to step (C.3);

• (C.13) set $j=j+1$ and go to step (C.14);

• (C.14) if $j=J+1$, stop. Otherwise, go to step (C.2).

By using the definition of the loss function $L$ in (5) and the quantities $\theta (j,t)$ and $x_{j,t,h,a}$ introduced in the previous steps of the numerical procedure, we have obtained a $J\times T\times H\times 2$ matrix $\tilde{\mathbf {L}}=(\tilde{L}_{j,t,h,a})_{j,t,h,a}$ as follows:

$$\begin{aligned} L_{j,t,h,a}=\Big |\frac{1}{\theta (j,t)^2}*\left[ exp((\theta (j,t))* x_{j,t,h,a})-\theta (j,t)* x_{j,t,h,a}-1\right] -(x_{j,t,h,a})^2\Big |. \end{aligned}$$

Recalling that the index $h$ is associated to the simulated scenarios, we have then taken the average on $h$ and obtain the matrix $\mathbf {L}=(L_{j,t,a})_{j,t,a}$:

$$\begin{aligned} L_{j,t,a}=\frac{1}{H}\sum _{h=1}^H \tilde{L}_{j,t,h,a}. \end{aligned}$$

The cases $a=1$ and $a=2$ have been distinguished. Fixed $a=\bar{a}$, the median and the second best forecaster have been identified as follows:

• Median forecaster

For each $\bar{t}=1,2, \ldots , T$, we have taken the median of $(L_{j,\bar{t},\bar{a}})_j$, say for $j=j^\star _{\bar{t}}$, and identified the corresponding $\theta (j^\star _{\bar{t}},\bar{t})$. When the number of the data is pair, the median do not correspond to any $\theta$. In this case, we have sorted $(L_{j,\bar{t},\bar{a}})_j$ in increasing order and found the index $\bar{j}_{\bar{t}}$ such that the median of $(L_{j,\bar{t},\bar{a}})_j$ is $1/2(L_{\bar{j}_{\bar{t}}-1,\bar{t},\bar{a}}+L_{\bar{j}_{\bar{t}},\bar{t},\bar{a}})$. We have then set $j^\star _{\bar{t}}=\bar{j}_{\bar{t}}$, hence leading to $\theta (j^\star _{\bar{t}},\bar{t})=\theta (\bar{j}_{\bar{t}},\bar{t})$. The trajectory of the $\theta$ related to the median forecaster is

$$\begin{aligned} \theta _{median}=(\theta (j^\star _1,1), \theta (j^\star _2,2), \ldots , \theta (j^\star _T,T)). \end{aligned}$$

• Second best forecaster

For each $\bar{t}=1,2, \ldots , T$, we have taken the minimum over $j$ of $(L_{j,\bar{t},\bar{a}})_j$, which is assumed to be attained for $j=j^\bullet _{\bar{t}}$, and identified the corresponding $\theta (j^\bullet _{\bar{t}},\bar{t})$.

The trajectory of the $\theta$ related to the second best forecaster is

$$\begin{aligned} \theta _{sb}=(\theta (j^\bullet _1,1), \theta (j^\bullet _2,2), \ldots , \theta (j^\bullet _T,T)). \end{aligned}$$