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 unecessarily 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.state-space form, Lyapunov equations, nonnegative and irreducible matrices

Further Model-Based Estimates of U.S. Total Manufacturing Production Capital and Technology, 1949-2005

Production capital and technology (i.e., total factor productivity) in U.S. manufacturing are fundamental for understanding output and productivity growth of the U.S. economy but are unobserved at this level of aggregation and must be estimated before being used in empirical analysis. Previously, we developed a method for estimating production capital and technology based on an estimated dynamic structural economic model and applied the method using annual SIC data for 1947-1997 to estimate production capital and technology in U.S. total manufacturing. In this paper, we update this work by reestimating the model and production capital and technology using annual SIC data for 1949-2001 and partly overlapping NAICS data for 1987-2005.Kalman filter estimation of latent variables

Testing Substitution Bias of the Solow-Residual Measure of Total Factor Productivity Using CES-Class Production Functions

Total factor productivity (TFP) computed as Solow-residuals could be subject to input-substitution bias for two reasons. First, the Cobb-Douglas (CD) production function restricts all input substitutions to one. Second, observed inputs generally differ from optimal inputs, so that inputs observed in a sample tend to move not just due to substitution effects but for other reasons as well. In this paper, we describe using the multi-step perturbation method (MSP) to compute and evaluate total factor productivity (TFP) based on any k+1 times differentiable production function, and we illustrate the method for a CES-class production functions. We test the possible input-substitution bias of the Solow-residual measure of TFP in capital, labor, energy, materials, and services (KLEMS) inputs data obtained from the Bureau of Labor Statistics for U.S. manufacturing from 1949 to 2001. We proceed in three steps: (1) We combine the MSP method with maximum likelihood estimation to determine a best 4th-order approximation of a CES-class production function. The CES class includes not only the standard CES production functions but also the so called tiered CES production functions (TCES), in which the prespecified groups of inputs can have their own input-substitution elasticities and input-cost shares are parameterized (i) tightly as constants, (ii) moderately as smooth functions, and (iii) loosely as successive averages. (2) Based on the best estimated production function, we compute the implied best TFP evaluated at the computed optimal inputs. (3) For the data, we compute Solow-residual TFP and compare it with the best TFP. The preliminary results show that the MSP method can produce almost double precision accuracy, and the results reject a single constant elasticity of substitution among all inputs. For this data, the Solow-residual TFP is on average .1% lower, with a .6% standard error, than the best TFP and, hence, is very slightly downward biased, although the sampling-error uncertainty dominates this conclusion. In further work, we shall attempt to reduce this uncertainty with further testing based on more general CES-class production functions, in which each input has its own elasticity of substitution, and we shall use more finely estimated parametersTaylor-series approximation, model selection, numercial solution, tiered CES production function

Multi-Step Perturbation Solution of Nonlinear Rational Expectations Models

This paper develops and illustrates the multi-step generalization of the standard single-step perturbation (SSP) method or MSP. In SSP, we can think of evaluating at x the computed approximate solution based on x0, as moving from x0 to x in "one big step" along the straight-line vector x-x0. By contrast, in MSP we move from x0 to x along any chosen path, continuous, curved-line or connected-straight-line, in h steps of equal length 1/h. If at each step we apply SSP, Taylor-series theory says that the approximation error per step is 0(e) = h^(-k-1), so that the total approximation error in moving from x0 to x in h steps is 0(e) = h^(-k). Thus, MSP has two major advantages over SSP. First, both SSP and MSP accuracy declines as the approximation point, x, moves from the initial point, x0, although only in MSP can the decline be countered by increasing h. Increasing k is much more costly than increasing h, because increasing k requires new derivations of derivatives, more computer programming, more computer storage, and more computer run time. By contrast, increasing h generally requires only more computer run time and often only slightly more. Second, in SSP the initial point is usually a nonstochastic steady state but can sometimes also be set up in function space as the known exact solution of a close but simpler model. This "closeness" of a related, simpler, and known solution can be exploited much more explicitly by MSP, when moving from x0 to x. In MSP, the state space could include parameters, so that the initial point, x0, would represent the simpler model with the known solution, and the final point, x, would continue to represent the model of interest. Then, as we would move from the initial x0 to the final x in h steps, the state variables and parameters would move together from their initial to final values and the model being solved would vary continuously from the simple model to the model of interest. Both advantages of MSP facilitate repeatedly, accurately, and quickly solving a NLRE model in an econometric analysis, over a range of data values, which could differ enough from nonstochastic steady states of the model of interest to render computed SSP solutions, for a given k, inadequately accurate. In the present paper, we extend the derivation of SSP to MSP for k = 4. As we did before, we use a mixture of gradient and differential-form differentiations to derive the MSP computational equations in conventional linear-algebraic form and illustrate them with a version of the stochastic optimal one-sector growth model.numerical solution of dynamic stochastic equilibrium models

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.Kalman filter estimation of unobserved state variables

Real-Time State Space Method for Computing Smoothed Estimates of Future Revisions of U.S. Monthly Chained CPI

Well known CPI of urban consumers is never revised. Recently initiated chained CPI is initially released every month (ICPI), for that month without delay within BLS and for the previous month with one month delay to the public. Final estimates of chained CPI (FCPI) are released every February for January to December of the calendar year two years before. Every month, simultaneously with the release of ICPI, we would like to have a best estimate, given current information, of FCPI for that month, which will not be released until two calendar years later. ICPI and FCPI data may be indexed in historical time by months of occurrence or in current or real time by months of observation or release. The essence of the solution method is to use data indexed in historical time to estimate models and, then, for an estimated model, to use data indexed in real time to estimate FCPI. We illustrate the method with regression and VARMA models. Using a regression model, estimated FCPI is given directly by an estimated regression line; and, using a VARMA model, estimated FCPI is computed using a Kalman smoother

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

Production capital and total factor productivity or technology are fundamental to understanding output and productivity growth, but are unobserved except at disaggregated levels and must be estimated before being used in empirical analysis. In this paper, we develop estimates of production capital and technology for U.S. total manufacturing based on an estimated dynamic structural economic model. First, using annual U.S. total manufacturing data for 1947-1997, we estimate by maximum likelihood a dynamic structural economic model of a representative production firm. In the estimation, capital and technology are completely unobserved or latent variables. Then, we apply the Kalman filter to the estimated model and the data to compute estimates of model-based capital and technology for the sample. Finally, we describe and evaluate similarities and differences between the model-based and standard estimates of capital and technology reported by the Bureau of Labor Statistics.Kalman filter estimation of latent variables

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.mixed-frequency data, VAR models, maximum-likelihood estimation, Kalman filter

Monte Carlo simulation of baryon and lepton number violating processes at high energies

We report results obtained with the first complete event generator for electroweak baryon and lepton number violating interactions at supercolliders. We find that baryon number violation would be very difficult to establish, but lepton number violation can be seen provided at least a few hundred L violating events are available with good electron or muon identification in the energy range 10 GeV to 1 TeV.Comment: 40 Pages uuencoded LaTeX (20 PostScript figures included), Cavendish-HEP-93/6, CERN-TH.7090/9

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 monthlyquarterly 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 insample 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