Long memory dynamics in a discrete-time real business cycle DSGE model and a continuous-time macro-financial model

Abstract

Long memory refers to a property of a stationary stochastic process or a time series. More specifically, a stationary time series is called a long memory process if its autocorrelation function (ACF) decays very slowly to zero. Indeed, the convergence is so slow that the sum of the ACF’s absolute values diverges. In contrast, traditional time series models such as ARMA processes are so-called short memory processes as their ACF decays rapidly, such that these processes permit only a limited dependency structure. This dissertation is motivated by the following observation. In the 1960s, Mandelbrot initiated research on long memory processes. After the work of Hosking, Granger, and Joyeux in the early 1980s, who developed a class of long memory processes (the so-called ARFIMA processes), there is increasing empirical evidence that many macroeconomic time series can be well-described by long memory processes. Moreover, some theoretical explanations exist for the presence of long memory in (macro)economic time series. For instance, the aggregation of microdata can induce long memory in macro data. On the other hand, stochastic models build a cornerstone in modern macroeconomics to explain macroeconomic relationships, analyze counterfactual scenarios, or make forecasts. Two representative types of stochastic models are the discrete-time dynamic stochastic general equilibrium (DSGE) models and the continuous-time macro-financial models. Both types of these models use exogenous stochastic processes to describe the dynamics of the model’s variables. However, the exogenous stochastic processes often assumed for modeling are predominantly short memory processes. This becomes evident for DSGE models, in which technology shocks, monetary policy shocks, preference shocks, etc., are described by first-order autoregressive processes (AR(1) processes). However, since DSGE models are typically estimated with macro data, it may be appropriate to use a long memory process instead of a short memory process in a DSGE model.This dissertation aims to contribute to the integration of these two strands of the literature by introducing long memory dynamics in a DSGE and a macro-financial model. Before Chapter 4 and Chapter 5 introduce long memory into these two types of models, Chapter 2 introduces the mathematical framework and the discrete-time and continuous-time long memory processes that will later be used for modeling purposes. Chapter 3 gives an overview of long memory in economic and econometric research and underlines the relevance of long memory. Chapter 4 considers a real business cycle (RBC) model extended by long memory in the exogenous technology shock. In order to ensure that this is a true generalization of the existing model, the class of so-called ARFIMA processes is used. More precisely, the assumption of an exogenous AR(1) technology shock is replaced with an exogenous long memory ARFIMA(1, d, 0) process. Compared to the former, the latter has an additional parameter d that specifies the ACF’s decay rate and controls the strength of the long memory in the process. Setting this parameter to zero returns the well-known standard model (the AR(1) process) as a special case. However, the derivation of the solution of such a model is not trivial. If one considers a higher-order ARMA process instead of an AR(1) process, this can be done quite easily by expanding the model’s state space. For ARFIMA processes, this procedure does not work since they do not have such a finite-dimensional state space representation. Thus, Chapter 4 focuses on the solvability of such a long memory DSGE model. It turns out that the solution method of Klein (2000) can be extended to such models. This opens the possibility of analyzing the responses of the model’s variables to different specifications of the exogenous shocks using impulse-response functions (IRF). In addition, besides pure short and long memory processes, mixed processes, as well as so-called trend shocks with a permanent character, are considered and contrasted with each other. It turns out that the model’s responses to pure long memory shocks do not differ qualitatively from pure short memory AR(1) shocks. At first glance, this seems surprising from a model perspective. One might have expected an infinitely-lived representative agent with rational expectations to account for the long-lasting shock effects in his intertemporal consumption and labor supply decision. That this is not the case can be explained by the fact that the household in the model discounts its expected utility at an exponential rate. Thus, the shock effects in later periods have little impact on his consumption and labor supply decisions immediately after the time of shock occurrence. However, it is also shown that the model’s responses in the mixed short and long memory cases differ significantly from the responses in the corresponding pure cases. It is shown that the effect of the shock in the period after its occurrence is equal to the sum of the two memory parameters. Thus, long memory not only affects the model in the long term but can also affect model dynamics in the short term. Finally, the model responses are compared with those of a permanent growth shock. It is illustrated that the model’s responses to shocks with high short and long memory parameters are similar to those of a trend shock in the short run. In the long run, the economy reaches a new balanced growth path in response to the growth shock, whereas, in the mixed short and long memory case, it slowly returns to its old steady state. In Chapter 5, a continuous-time macro-financial model is extended to allow for long memory in the economy’s aggregate output growth rates. For modeling purposes, the Brownian motion assumed in the reference model is replaced with an approximation of a fractional Brownian motion. This replacement allows the exogenous shock to be split into a drift and volatility effect and the model to be solved using existing solution methods. It turns out that the evolution of the wealth distribution between the two agents in the model, which serves as a state variable, depends only on the volatility effect. In particular, the presence of long memory slows down the convergence of the state variable toward its steady state value. Moreover, the evolution of the aggregate wealth can be decoupled to some extent from the evolution of the wealth distribution and, thus, from the evolution of the state variable. While both models considered in this thesis can be solved given the more general long memory dynamics, the price for introducing long memory this way seems high. For example, estimating a long memory DSGE model is difficult because the associated DSGE model no longer has a finite-dimensional state space representation, which is typically used for estimating DSGE models. In the continuous-time model, the outlined decoupling of an economy’s total wealth from wealth distribution allows for more sophisticated modeling. However, this feature seems to contradict the general model structure of this kind of models. Generally, in this model category, all variables can be expressed as functions of the underlying state variables. This no longer holds true in the model under consideration, so generalizations to more complex models appear difficult.