# Introduction

## The negative comovement problem

One implication of simple RBC-like models with steady state growth is that a persistent increase in the growth rate of productivity can trigger a decline in output. More generally, positive changes in productivity growth can have negative impact on hours worked, output or investment. For instance (Viard 1993), (Carroll 1994) showed that a decline in the productivity growth rate should elicit an immediate rise in the saving rate, (Campbell 1994) showed that in a real business cycle model, a persistent decline in the productivity growth rate yielded the “perverse” effect of a rise in employment and output. In both (Edge 2007) and (Boz 2011), a representative agent reduces her labor supply in response to a positive and persistent trend growth shock, due to the wealth effect. When the persistence of the shock is higher than a threshold (around 0.2 in (Boz 2011) calibration) the decline in labor supply leads to a fall in output even after that capital starts to accumulate (Butler 2005)

(Edge 2007) argues that this negative co-movement property is consequence of assuming full information, to the extent that it can be mitigated or eliminated, for sensible parameter values, if we introduce imperfect information about productivity shocks. The reason is simple: negative co-movement appears with permanent changes in growth but not with transitory ones. If the agent is unsure about the nature of the shock, permanent ones will never hit with the same force, because now agents believe that the shock is transitory with some probability. Hence, income effects are mitigated and negative co-movements disappear. Driving those negative co-movements are large income effect generated by permanent or very persistent changes in productivity growth, overcoming substitution effects, inducing a decline in consumption and leisure. Any modification that atone the influence or the perceived magnitude of this income effect, will work against those negative co-movements. In this paper I’d like to explore how much we gain if we assume that the agent, on top of imperfect information about the nature of the shock, it doesn’t fully trust the model for productivity evolution and seeks to limit the losses induced by model misspecification.

To include fear of model misspecification I’ll use the robust control formulation advocated in economics by Sargent, Hansen and several coauthors, specially the ones dealing with hidden states in Markov environments 1. I’ll say more about imperfect information and fear of misspecification in the next subsections

## Imperfect information

Since this negative co-movement problem would not arise in these models if the shocks to growth were purely transitory, making the agent doubt whether he is facing a transitory or permanent shock to growth weakens the effects of truly permanent shocks and the may lose their ability to induce negative co-movements. Now the agent has to solve a filtering problem to imagine the future of productivity levels and growth rates. Examples of this specific strand of the literature are (Boz 2011) and (Edge 2007). There are other recent works that inquire on the modifications brought about by the imperfect knowledge of some key aspect of the economy, see for instance (Eusepi 2011) for a case when agent do not known the exact mapping from individual decisions to equilibrium prices or the work of (Milani 2011) (Milani 2007) for a least-square type of adaptive learning and it consequences for the business cycle.

Since equations for the evolution of growth rates are linear and the the shocks are gaussian, the filtering problem, or Bayesian updating, is solved by the Kalman filter. However, little attention has been given to non-stationary instances of learning, this is, before the covariance of the hidden stated has converged, although (Boz 2011) and (Edge 2007) do some comparative static of two economies with different degrees of noise which is closely related to examine two different covariance matrices of the hidden state. Under stationary learning the long run constant mean of the productivity growth is effectively pinned down by the agent, eliminating a potentially important source of variation in their responses to productivity shocks. In particular, is the agent it’s also her estimate of the long run growth, then is not only pondering whether it will take a instant or a decade to go back to the long run growth rate, but on top of that she may think that the long run growth rate best estimate is not longer, say, 2 percent but 1.7 or 2.3. This paper make the case for assessing the importance of such variations before settling with constant-gain learning version of the Kalman filter. 2

## Fear of misspecification

If the agent fears that a part of her model is misspecified, she may want to protect herself by making decisions that are robust to errors of specification. In this paper she is worried that the statistical model she uses to learn about the true evolution of the productivity is only an approximation, with productivity actually evolving in a way she can’t or won’t specify.

The robust solution she’ll find can be interpreted as the outcome of a bayesian agent who is using a new, optimally distorted predictive conditional distribution of productivity growth. By distorting the beliefs about the hidden state, it will change both the innovation and the gain used to update her beliefs, reinforcing or depressing the wealth effect induced by a positive shock to growth. Thus, is crucial to examine the sign of changes in this distributions relatives to those used in the partial information case.

When solving for optimally distorted beliefs, the dynamic and stochastic features of the system and preferences of the agent determine the features of the resulting distribution. For instance, if in the approximating the model conditional innovations are gaussian, the optimally distorted distribution is also gaussian. If the model is linear quadratic, then no approximation is needed (beyond numerically solving the Riccati equation in the quadratic value function) and the resulting distribution for the hidden state (and therefore to the signal distribution) has a constant adjustment to the mean and a time-varying, but deterministic adjustment to the covariance matrix. In non linear quadratic models, a first order Taylor perturbation approximation also gives a constant adjustment to the mean. This will amount to a constant difference in the income effects found in the partial information model. Interestingly, using a different route to find approximated solutions, Hansen and Borovicka show that a first order representation of the robust optimal policy includes a belief distortion that is state dependent and hence non constant, implying in for our case that how different are the income effects in the partial information model and the robust control model, depends on how fast productivity is growing.

Robust control jointly with estimation of a hidden state’s distribution has been used, among other, by (Hansen 2010) to produce state dependent enhanced risk premia, in (Cogley 2008) to study incentives to actively experiment when undecided between a Phillip’s curve and a neoclassical setup for inflation, and by (Hansen 1999) where their agent lives in an endowment economy close to Hall’s permanent income model but with consumption habit. A production economy with stochastic growth, but without labor supply decision and written in continuous time, can be found in (Cagetti 2002), where the agent doesn’t observe the exact decomposition of productivity growth between small frequent shocks and large but infrequent ones, modeled as diffusion and jumps. (Galí 2010)

It’s common to this literature to put greater emphasis in the pricing rather than allocative implications of fear of misspecification and to use model that allow for closed form solutions. This paper, however is primarily concerned with relations between quantities, but more than dynamics is about the margin along which substitution effects and income effect can cancel each other and how this margin is modified by the presence of fear of misspecification, and those are the same forces behind assets pricing behavior: changes in the envisioned stream of possible incomes. A further difference is that by moving into typical RBC-type of models, closed form solution are no longer available and concern for robustness is partly uncover at each degree of approximation. Regarding this problem, this paper uses extensively the series expansion approach advocated and developed for the robust control case by (Borovicka 2013).

1. See (Hansen 2007) for a general treatment of such models

2. A related paper is (Moore 2002) that uses learning about persistent and transitory changes in productivity to produce booms and bust in aggregate investment.

# A simple model with only neutral productivity

If we fixed the intertemporal elasticity of substitution (IES) for consumption and leisure and consider only one type of non-stationary productivity, labor of TFP, we get a model like the one in this section. Later, I’ll allow for different IESs and the presence of a possibly cointegrated investment-specific productivity process, but this simple case delivers the main message and results can be compared with those in (Edge 2007), who use essentially the same model when producing their results 1. (Hansen 2010a)

## Preferences

Since productivity has a stochastic trend some variables will not be stationary. Indeed, in equilibrium all variables, excepting leisure –and therefore labor–, will grow at constant expected rate, defining a Balanced Growth Path. In what follows, variables with tilde (i.e. $$\tilde{C}, \tilde{Y}$$, etc.) are non-stationary.

A social planner tries to maximize

$\label{eq:toy_preferences_u} E_0 \sum_{t=0}^{\infty} \beta^t \tilde{u}_t$

$\label{eq:toy_preferences_logs} \tilde{u}_t := \nu \ln \tilde{C}_t + (1-\nu) \ln (1- L_t)$

Let $$A$$ denote the level of total factor productivity in this economy. The following, transformed, variables are stationary: $\label{eq:toy_stationary_variables} C_t := \frac{\tilde{C}_t}{A_{t-1} }, \quad K_t := \frac{\tilde{K}_t}{A_{t-1} }, \quad I_t := \frac{\tilde{I}_t}{A_{t-1} }$

We can re-state the Planner’s objective using stationary variables: $\label{eq:toy_preferences_u_stationary} E_0 \sum_{t=0}^{\infty} \beta^t u_t$ $\label{eq:toy_preferences_logs_stationary} u_t := \nu \ln C_t + (1-\nu) \ln (1- L_t)$

## Production

There is a single good, produced with Cobb-Douglas technology. The way we have written the production function reminds us that in this case the logarithm of TFP is equal to the logarithm of labor productivity times the labor share.

$\label{eq:Fisher_prod_tilde} {\tilde{Y}}_{t} = A_t^{1-\alpha} {\tilde{K}}^{\alpha}_{t} L^{1-\alpha}_{t}$

Output is used for consumption and investment $\label{eq:toy_CIYtilde} {\tilde{C}}_t + {\tilde{I}}_t \leq {\tilde{Y}}_{t}$

There is no investment-specific productivity yet: $\label{eq:toy_Kevol_tilde} {\tilde{K}}_{t+1} = I_t + (1-\delta) {\tilde{K}}_{t}$

implying $\label{eq:toy_CKYtilde} {\tilde{C}}_t + {\tilde{K}}_{t+1} - (1-\delta){\tilde{K}}_{t} \leq A_t^{1-\alpha} {\tilde{K}}^{\alpha}_{t} L^{1-\alpha}_{t}$

Stationary versions of \eqref{eq:toy_CIYtilde}, \eqref{eq:toy_Kevol_tilde} and \eqref{eq:toy_CKYtilde} are goven by

## Productivity evolution

Evolution of productivity is given by \begin{aligned} A_{t+1} = & A_{t}~ e^{a_{t+1}}, \quad A_0 ~\text{given} \end{aligned}

\begin{aligned} a_{t} & = \mu_{a} + \zeta_{a,t} + \sigma_{a} w_{a,t} , \quad w_{\zeta_a,t+1} \overset{iid}{\sim} N(0,1) \\ \zeta_{a,t} & = \rho_a \zeta_{a,t-1} + \sigma_{\zeta_a} w_{\zeta_a, t} , \quad w_{a,t} \overset{iid}{\sim} N(0,1) \end{aligned}

This is the same processes for productivity specified in (Edge 2007), you can obtain their system of equations by defining 2

$a_t := p_{t}-p_{t-1}, \, \zeta_{a,t-1} := g_{p,t} - \bar{g}, \, \mu_a := \bar{g}, \, w_{a,t} := \frac{\epsilon_t}{\sigma_{\epsilon}}, \, w_{\zeta_a, t} := \frac{v_t}{\sigma_{v}}, \, \sigma_{a} := \sigma_{\epsilon}, \, \sigma_{\zeta_a} := \sigma_{v}$

1. They specify a general CRRA utility, but the calibration actually used in the paper sets IES=1, their two productivity processes are independent of each other, effects of both are decoupled and show us what happen on the face of changes of TFP growth rate

2. I’ve chosen this formulation because make comparisons to (Edge 2007) easy. An alternative formulation that is more natural in a state space formulation, and that allows easily to consider correlation between tfp