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Regularized Solutions to Linear Rational Expectations Models
\newtheorem{lem}{Lemma} \newtheorem{thm}{Theorem} \newtheorem{cor}{Corollary} \newtheorem{prop}{Proposition} \newtheorem{exmp}{Example} \newtheorem{defn}{Definition} \newtheorem{claim}{Claim} \newtheorem{alg}{Algorithm} \newtheorem{rmk}{Remark} \newtheorem{res}{Result}
\abstract{ This paper proposes an algorithm for computing regularized solutions to linear rational expectations models. The algorithm allows for regularization cross-sectionally as well as across frequencies. A variety of numerical examples illustrate the advantage of regularization.
JEL Classification: C62, C63, E00.
Keywords: Linear rational expectations model, regularization, indeterminacy, computational methods.}
Recently, spectral showed that non-unique solutions to multivariate linear rational expectations models (LREMs) are not generally continuous with respect to their parameters, invalidating crucial assumptions for both frequentis and Bayesian methods. For frequentist analyses, the objective function (e.g.\ the likelihood function) has to be at least continuous. For Bayesian analysis, the posterior cannot have atoms at unknown locations. spectral demonstrated that these two conditions are not guaranteed under current methodology and proposed a regularization solution.
Regularization is a method for selecting from among infinitely many solutions to an LREM a unique solution that accords with prior information that the researcher may have about what a solution should look like (e.g. that its spectral density should concentrate in the range of observed business cycles). spectral provided a theoretical analysis of regularizaiton. The aim of this paper is to provide an algorithm for computing such solutions based on the sims framework.
This work is related to several more recent works. The main result of this paper builds on sunspots and linsys. farmeretal and bianchinicolo provide alternative parametrizations of solutions to LREMs to sunspots. funovits counts the dimension of the solution space to a given LREM. This paper can also be seen as part of the recent interest in frequency domain analysis of LREMs as exemplified by onatski, tanwalker, and tan. Such methods have found important applications in addressing the identification problem for LREMs as seen in kn, tq17, kk18, and ident.
This paper is organized as follows. Section (ref) reviews results of sims and sunspots. Section (ref) shows how regularization can be achieved and provides the main result of this paper. Section (ref) provides illustrative examples of how regularization works. Section (ref) concludes. The Matlab code for reproducing the computations presented in this paper can be found in the accompanying file, regular.zip.
We begin by reviewing results developed by sims and sunspots. This is necessary in order to set the notation and obtain the basic ingredients that we will need. Because regularization is only defined in a stationary context, we will restrict attention to covariance stationary solutions.
Assuming, as sims does, that $\det(\Gamma_0+\Gamma_1 x)$ is not identically zero (i.e.\ it is impossible to cancel out $y$ by elementary algebraic operations), then by Theorem VI.1.9 and Exercise VI.1.3 of stewartsun, there are orthogonal matrices $Q,Z\in\mathbb{R}^{n\times n}$ such that $Q\Gamma_0 Z$ and $Q\Gamma_1 Z$ are block upper triangular with either $1\times1$ or $2\times2$ blocks on the diagonal. Under the stronger assumption that $\det(\Gamma_0+\Gamma_1 x)\neq0$ for all $x\in\mathbb{C}$ with $|x|=1$ (i.e.\ the aforementioned cancellation is impossible and there are no unit roots in the system), then these matrices can be partitioned conformably as
where the polynomial $\det(\Lambda_{11}+\Omega_{11}x)$ has all its zeros outside the unit circle (this implies that $\Lambda_{11}$ is non-singular), and the polynomial $\det(\Lambda_{22}+\Omega_{22}x)$ has all its zeros inside the unit circle (this implies that $\Omega_{22}$ is non-singular). As shown in the online appendix to linsys, this step is an implicit Wiener-Hopf factorization. Note that sims and sunspots use the complex QZ decomposition but never explain how the final answer is real; using the real QZ decomposition obviates any need for such a discussion.
Now suppose $(y,\eta)$ is a solution to (ref), define $w(t)=Z'y(t)$, and rewrite the system as
If we partition
conformably with (ref), then
where
is partitioned conformably with (ref). Applying the conditional expectation $E_{t-1}$ we obtain
This implies that
Therefore,
where we have used the fact that $E\|E_tw_2(s)\|^2\leq E(E_t\|w_2(s)\|^2)=E\|w_2(s)\|^2$ williams. The covariance stationarity of $y$ implies that $E\|w_2(t)\|^2=E\|w_2(s)\|^2$. Since our choice of QZ decomposition ensures that the eigenvalues of $\Omega_{22}^{-1}\Lambda_{22}$ are inside the unit circle, $\|(\Omega_{22}^{-1}\Lambda_{22})^{s-t}\|<1$ for large enough $s-t$ and then it must be the case that $E\|w_2(t)\|^2=E\|w_2(s)\|^2=0$. Therefore,
Now plugging this back into (ref) we have that
Multiplying on the right by $z'(t)$, taking expectations, and utilizing the joint covariance stationarity of $\eta$ and $z$, we arrive at
But since $E(z(0)z'(0))$ is invertible by assumption, a necessary condition for existence is
It also follows that
where $(Q_{2\cdot}\Pi)^\dag$ is the Moore-Penrose generalized inverse of $Q_{2\cdot}\Pi$, and
Thus, for a given matrix $K$ whose columns form a basis for $\ker(Q_{2\cdot}\Pi)$ there is a martingale difference sequence with respect to $z$, denoted by $\nu$, such that
Every solution is therefore representable as
with
Note that $\nu$ inters into the system along $\mathrm{rank}(Q_{1\cdot}\Pi K)$ independent directions, what funovits calls the dimension of indeterminacy.
In fact, (ref) is not only necessary but also sufficient for existence. To see this, simply construct the pair $(y,\eta)$ from (ref) with $\nu$ set to the zero process; it is easily checked that this pair is a solution to (ref).
Turning now to uniqueness, we see that the arbitrary $\nu$ plays no role in the solution if and only if $\Theta_\nu=0$ or, equivalently, if and only if $Q_{\cdot1}\Pi K=0$, which can be expressed as
Since (ref), generated with $\nu(t)=Az(t)$ for $t\in\mathbb{Z}$ defines a solution for any matrix $A$, it must be that (ref) is necessary and sufficient for uniqueness. Note that sims expresses (ref) equivalently in terms of the row spaces of $Q_{1\cdot}\Pi$ and $Q_{2\cdot}\Pi$.
To summarize, we have proven the following.
Current methodology utilizes (ref) or variants thereof. spectral has demonstrated that these solutions can be discontinuous (as we will see shortly) and proposed using regularized solutions to ensure continuity. We now turn to the problem of computing such solutions.
We begin with the basic setting. Suppose a symmetric positive semi-definite matrix $W\in\mathbb{R}^{n\times n}$ is given and we are interested in selecting among all solutions to (ref), one that minimizes
If the solution to (ref) is unique, there is nothing to solve for. If not, it will be convenient in the subsequent computations to introduce the martingale difference sequence, $\zeta$, defined as
The process $\zeta$ is the residual from regressing $\nu(t)$ on $z(t)$. This implies that
where $CC'=E(\zeta(0)\zeta'(0))$ and $\Sigma_{zz}=E(z(0)z'(0))$. Thus, finding a regularized solution is equivalent to minimizing
with respect to $B$ and $C$. Using the properties of the trace of a produce of matrices,
where
Note that $\Xi$ is the unique solution to the Lyapunov equation
See Section B.1.8 of lp. Taking the gradient of $\mathscr{L}$, we obtain the following first order conditions
If $\Theta_\nu'\Xi\Theta_\nu$ is invertible, there exists a unique regularized solution determined by
If $\Theta_\nu'\Xi\Theta_\nu$ is not invertible, there are infinitely many regularized solutions determined by
for arbitrary $X$ and $Y$ of the appropriate sizes such that $\mathrm{im}(X),\mathrm{im}(Y)\subseteq\ker(\Theta_\nu'\Xi\Theta_\nu)$.
We have established the following. First, a regularized solution to (ref) exists if and only if solutions to (ref) exist. Second, the regularized solution is unique if and only if either the solution to (ref) is unique, in which case the regularized solution is the unique solution,
or $\Theta_\nu'\Xi\Theta_\nu$ is invertible, in which case the regularized solution has the representation,
where
The intuition of this result is quite simple. Write
Now if $W^{1/2}\Theta_\nu$ is of full column rank, then the regularized solution is unique. That is, if $W$ attaches non-trivial weight to every contemporaneous instance of indeterminacy, then regularization eliminates indeterminacy. More generally, we have proven that regularization eliminates indeterminacy if and only if $W$ attaches non-trivial weight to every instance of indeterminacy whether contemporaneous or lagged. From a linear systems point of view, regularization leads to uniqueness if and only if the triple $(\Theta_1,\Theta_\nu,W^{1/2})$ is input observable sm, which is to say, again, that the weight matrix detects all of the indeterminacy in the system.
The analysis above suggests a generalization of the basic setting. We have constructed an algorithm for minimizing
where $f$ is the spectral density of $y$. The above expression allows us to choose different weights along the cross-section of $y$. More generally, we may consider choosing weights on frequencies of oscillation of $y$. In particular, we may consider minimizing
where $W$ is a bounded measurable function, with $W(\omega)$ Hermitian positive semi definite and $W(\omega)^\ast=W(-\omega)'$ for all $\omega\in(-\pi,\pi]$. If, for example, we like to impose that the solution should display the frequency characteristics of the business cycle, we could choose
which penalizes oscillations of period smaller than a year and greater than eight years in quarterly data. To that end, we first note that
This implies that
where
It is easily checked that $\Xi$ is a real symmetric positive semi definite matrix and that it reduces to our previous expression when $W(\omega)$ is constant. Following the same line of argument as above, we arrive finally at the main result of the paper.
Consider first, the Cagan model with mean zero, independent, and identically distributed shocks
There are infinitely many solutions to this system. To compute the regularized solution minimizing $EX_t^2$, we reformulate this model as
with
This implies that
Solving for the first element, we obtain $0.5\left(\frac{0.5-L}{1-0.5L}\right)\varepsilon_t$, which was obtained analytically in spectral. This regularized solution is actually a white noise process and therefore has a flat spectral density. We may instead impose that the solution avoid empirically unlikely frequencies. If we use the weight matrix
we obtain a different regularized solution with the spectral density plotted in the Figure (ref).
Consider next the New Keynesian model of ls04.
The model is calibrated using Lubik & Schorfheide's estimates reported in their Table 3 in the column titled “Pre-Volcker (Prior 1)”. Figure (ref) plots the impulse responses of the first three variables to the three shocks. The impulse responses are generated from the non-regularized solution, the solution regularized with constant weight matrix with equal weights on the first three variables, and the solution regularized with a variable weight matrix emphasizing business cycle frequencies in the first three variables.
Clearly, regularization produces more stable dynamics. Unlike the case in Figure (ref), however, regularizing by constant and variable weight matrices did not produce dramatically different results.
Consider now the system
The shocks are again zero mean, independent, and identically distributed. This system also has infinitely many solutions. Although it is simple, it concretely illustrates the failure of current methodology to account for discontinuity of solutions to LREMs. spectral demonstrates its discontinuity analytically and studies its Gaussian likelihood function. We will now demonstrate its discontinuity numerically.
In order to reformulate this system into the form (ref), we use the second equation to obtain
and then combine this equation with the first equation of the original system to obtain
This system is equivalent to the original one, provided $\theta\neq0$. We can now set
with
The weight matrix is
For $\theta=10^{-6}$, the first three impulse responses of the non-regularized solution are
Clearly, these are quite far from the impulse responses of the $\theta=0$ model, which ought to be
On the other hand, the first three impulse responses of the regularized solution are
The continuity of regularized solutions is proven in Theorem 6 of spectral.
This paper has provided an algorithm for computing regularized solutions to LREMs. This work suggests at least three venues for further investigation. First, it is likely that regularization helps resolve identifiability issues in LREMs due to its imposition of uniqueness but since it is strictly more general than the class of solutions considered in ident its identifiability requires separate examination. Second, the algorithm presented here is given without any claim to efficiency; it would be helpful to consider other methods of obtaining regularized solutions and compare their accuracy and speed. Finally, recent work has sought to relax the assumption that the information set includes all exogenous variables (e.g.\ ht, rondina, angeletos, and han); regularization in that context would be a fruitful venue for follow up work.
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