Peter A. Zadrozny

Forecasting Quarterly German GDP at Monthly Intervals Using Monthly IFO Business Conditions Data

The paper illustrates and evaluates a Kalman filtering method for forecasting German real GDP at monthly intervals. German real GDP is produced at quarterly intervals but analysts and decision makers often want monthly GDP forecasts. Quarterly GDP could be regressed on monthly indicators, which would pick up monthly feedbacks from the indicators to GDP, but would not pick up implicit monthly feedbacks from GDP onto itself or the indicators. An efficient forecasting model which aims to incorporate all significant correlations in monthly-quarterly data should include all significant monthly feedbacks. We do this with estimated VAR(2) models of quarterly GDP and up to three monthly indicator variables, estimated using a Kalman-filtering-based maximum-likelihood estimation method. Following the method, we estimate monthly and quarterly VAR(2) models of quarterly GDP, monthly industrial production, and monthly, current and expected, business conditions. The business conditions variables are produced by the Ifo Institute from its own surveys. We use early in-sample data to estimate models and later out-of-sample data to produce and evaluate forecasts. The monthly maximum-likelihood-estimated models produce monthly GDP forecasts. The Kalman filter is used to compute the likelihood in estimation and to produce forecasts. Generally, the monthly German GDP forecasts from 3 to 24 months ahead are competitive with quarterly German GDP forecasts for the same time-span ahead, produced using the same method and the same data in purely quarterly form. However, the present mixed-frequency method produces monthly GDP forecasts for the first two months of a quarter ahead which are more accurate than one-quarter-ahead GDP forecasts based on the purely-quarterly data. Moreover, quarterly models based on purely-quarterly data generally cannot be transformed into monthly models which produce equally accurate intra-quarterly monthly forecasts.

Asymptotic Distributions of Impulse Responses, Step Responses, and Variance Decompositions of Estimated Linear Dynamic Models

Formulas are derived for computing asymptotic covariance matrices of sets of impulse responses, step responses, or variance decompositions of estimated dynamic simultaneous-equations models in vector autoregressive moving-average (VARMA) form. Computed covariances would be used to test linear restrictions on sets of impulse responses, step responses, or variance decompositions. The results unify and extend previous formulas to handle any model in VARMA form, provide accurate computations based on analytic derivates, and provide insights into the structures of the asymptotic covariances. Copyright 1993 by The Econometric Society.

Necessary and Sufficient Restrictions for Existence of a Unique Fourth Moment of a Univariate GARCH(p,q) Process

A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m, eigenvalue restrictions have been proposed as necessary and sufficient restrictions for existence of a unique mth moment of the output of a univariate GARCH process or, equivalently, the 2mth moment of the underlying variable. However, proofs in the literature that an eigenvalue restriction is necessary and sufficient for existence of unique 4th or higher even moments of the underlying variable, are either incorrect, incomplete, or unnecessarily long. Thus, the paper contains a short and general proof that an eigenvalue restriction is necessary and sufficient for existence of a unique 4th moment of the underlying variable of a univariate GARCH process. The paper also derives an expression for computing the 4th moment in terms of the GARCH parameters, which immediately implies a necessary and sufficient inequality restriction for existence of the 4th moment. Because the inequality restriction is easily computed in a finite number of basic arithmetic operations on the GARCH parameters and does not require computing eigenvalues, it provides an easy means for computing “by hand” the 4th moment and for checking its existence for low-dimensional GARCH processes. Finally, the paper illustrates the computations with some GARCH(1,1) processes reported in the literature.

Estimated U.S. Manufacturing Production Capital and Technology Based on an Estimated Dynamic Economic Model

Production capital and technology, fundamental to understanding output and productivity growth, are unobserved except at disaggregated levels and must be estimated prior to being used in empirical analysis. We develop and apply a new estimation method, based on advances in economics, statistics, and applied mathematics, which involves estimating a structural dynamic economic model of a representative production firm and using the estimated model to compute Kalman-filtered estimates of capital and technology for the sample period. We apply the method to annual data from 1947-97 for U.S. total manufacturing and compare the estimates with those reported by the Bureau of Labor Statistics.

Estimation of Vector Error Correction Models with Mixed‐Frequency Data

Vector autoregressive (VAR) models with error‐correction structures (VECMs) that account for cointegrated variables have been studied extensively and used for further analyses such as forecasting, but only with single‐frequency data. Both unstructured and structured VAR models have been estimated and used with mixed‐frequency data. However, VECMs have not been studied or used with mixed‐frequency data. The article aims partly to fill this gap by estimating a VECM using the expectation‐maximization (EM) algorithm and US data on four monthly coincident indicators and quarterly real GDP and, then, using the estimated model to compute in‐sample monthly smoothed estimates and out‐of‐sample monthly forecasts of GDP. Because the model is treated as operating at the highest monthly frequency and the monthly‐quarterly data are used as given (neither interpolated to all‐monthly data, nor aggregated to all‐quarterly data), the application is expected to be unbiased and efficient. A Monte Carlo analysis compares the accuracy of VECMs estimated with the given mixed‐frequency data vs. with their single‐frequency temporal aggregate.

Identifiability of regular and singular multivariate autoregressive models from mixed frequency data

This paper is concerned with identifiability of an underlying high frequency multivariate AR system from mixed frequency observations. Such problems arise for instance in economics when some variables are observed monthly whereas others are observed quarterly. If we have identifiability, the system and noise parameters and thus all second moments of the output process can be estimated consistently from mixed frequency data. Then linear least squares methods for forecasting and interpolating nonobserved output variables can be applied. Two ways for guaranteeing generic identifiability are discussed.

Analytic Derivatives for Estimation of Discrete-Time

ESTIMATION OF PARAMETERS in discrete-time, linear-quadratic, infinite-horizon, dynamic, optimization models (Hansen and Sargent, 1980, 1981) is a nonlinear estimation problem. An impediment to effectively computing parameter estimates and their covariances in these models with gradient algorithms (Kennedy and Gentle, 1980, pp. 425-512) has been the absence of expressions for computing analytic VoP, the gradient matrix of first-partial derivatives of the optimal policy matrix P with respect to parameters of the optimization problem collected in vector 0. Derivatives can always be approximated numerically, but gradient algorithms perform more reliably, accurately, and quickly when analytic derivatives are used. In this paper we derive linear systems whose solution yields analytic V P. We also express Vo P in a closed form, which, while not computationally recommended because it involves sparse, Kronecker-product, matrices, may be analytically useful. We consider a higher-order problem put into first-order, state-space, form. Present results can be used with Euler-equation solution methods after making the appropriate translation (Hansen and Sargent, 1981, pp. 134-135).

Kalman-filtering methods for computing information matrices for time-invariant, periodic, and generally time-varying VARMA models and samples

Abstract Under general conditions, the inverse sample information matrix can be used to establish a Cramer-Rao lower bound of the covariance matrix of parameter estimates of a model, and the inverse asymptotic information matrix is the asymptotic covariance matrix of the parameter estimates. The paper does two things. First, it derives a recursive Kalman-filtering method for computing exact sample and asymptotic information matrices for time-invariant, periodic, or generally time-varying Gaussian vector autoregressive moving-average (VARMA) models and samples. Second, it specializes the recursive method to a nonrecursive method for computing exact asymptotic information matrices for time-invariant or periodic VARMA models and samples

Analytic derivatives of the matrix exponential for estimation of linear continuous-time models1

Abstract Linear-in-variables continuous-time processes are estimated nonlinearly, because the coefficients of the implied linear-in-variables discrete-time estimating equations are the exponential of a matrix formed with the continuous-time parameters. Even with sampling complications such as irregular intervals, mixed frequencies, and stock and flow variables, using Van loans (1978) results, the mapping from continuous- to discrete-time parameters and its derivatives can be expressed as the submatrix of a matrix exponential. For quicker estimation and more accurate hypothesis testing or sensitivity analysis, it is often better to compute analytically the first-order derivatives of the mapping. This paper explains how to compute efficiently the continuous- to discrete-time parameter mapping and its derivatives, without computing an eigenvalue decomposition, the common way of doing this. By linking present results with previous ones, a complete chain rule is obtained for computing the Gaussian likelihood function and its derivatives with respect to the continuous-time parameters.

Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem

The paper obtains two principal results. First, using a new definition ofhigher-order (>2) matrix derivatives, the paper derives a recursion forcomputing any Gaussian multivariate moment. Second, the paper uses this resultin a perturbation method to derive equations for computing the 4th-orderTaylor-series approximation of the objective function of the linear-quadraticexponential Gaussian (LQEG) optimal control problem. Previously, Karp (1985)formulated the 4th multivariate Gaussian moment in terms of MacRaesdefinition of a matrix derivative. His approach extends with difficulty to anyhigher (>4) multivariate Gaussian moment. The present recursionstraightforwardly computes any multivariate Gaussian moment. Karp used hisformulation of the Gaussian 4th moment to compute a 2nd-order approximationof the finite-horizon LQEG objective function. Using the simpler formulation,the present paper applies the perturbation method to derive equations forcomputing a 4th-order approximation of the infinite-horizon LQEG objectivefunction. By illustrating a convenient definition of matrix derivatives in thenumerical solution of the LQEG problem with the perturbation method, the papercontributes to the computational economists toolbox for solving stochasticnonlinear dynamic optimization problems.