Kent D. Wall

Kalman filtering estimation of unobserved rational expectations with an application to the German hyperinflation

Abstract The assumption that rational expectations always lie on a convergent path is subject to an empirical test using the German hyperinflation data. The estimation technique employs a Kalman filtering algorithm. After presenting a brief background for the convergent expectations problem and a derivation of the various model specifications, a generalized expectations model and its attendant Kalman filtering estimation technique are discussed. Additional estimation details and empirical results are then presented. Based on an assumption of normally distributed errors, the null hypothesis of convergent paths is rejected in all situations involving a deterministic specification of the evolution of the unobserved parameter which characterizes the convergent path. The same null hypothesis is rejected in four of the six cases corresponding to a stochastic specification of the evolution of the unobserved parameter which characterizes the convergent path. A discussion of these findings, their economic significance, and suggestions for further research concludes the paper.

Estimation of Unobserved Expected Monthly Inflation Using Kalman Filtering

Hamilton developed a technique for estimating financial market expectations of inflation based on the observed time-series properties of interest rates and inflation. The technique is based on a state-space representation derived from an underlying vector autoregressive process of the expected real interest rate and the expected inflation rate on lagged expectations and lagged values of the observed Treasury bill rate and the actual inflation rate. This article extends this work in two ways. First, we use monthly data, since the quarterly data used by Hamilton may obscure many interesting movements, especially for determining the role of inflationary expectations in stock price movements, and this is one of our primary interests. Second, we employ an alternative method developed by Burmeister and Wall for estimating the parameters of the model, and this method leads to a different identification proof. Both approaches share the use of the Kalman filter to estimate the unobserved variables, in this case, e...

Two estimators for the apt model when factors are measured

Abstract Non-linear SUR and ITSUR techniques are proposed for the estimation of the APT and the CAPM when the factors are observed. These techniques estimate all of the parameters of the model simultaneously and directly impose the models non-linear parameter restrictions.

Asymptotic distribution theory for the kalman filter state estimator

An asymptotic distribution theory for the state estimate from a Kalman filter in the absence of the usual Gaussian assumption is presented. It is found that the stability properties of the state transition matrix playa key role in the distribution theory. Specifically, when the state equation is neutrally stable (i.e., borderline stable-unstable) the state estimate is asymptotically normal when the random terms in the model have arbitrary distributions. This case includes the popular random walk state equation. However, when the state equation is either stable or unstable, at least some of the random terms in the model must be normally distributed beyond some finite time before the state estimate is asymptotically normal.

Unobserved Rational Expectations and the German Hyperinflation with Endogenous Money Supply

A Kalman filtering technique is employed to test the convergent expectation hypothesis in the great German hyperinflation when the money supply is endogenously determined. After converting the model structure to a state space form, the parameters are estimated via minimization of a Gaussian likelihood function. The result is then used in a filter- smoother combination to yield smoothed estimates of the state variable associated with the arbitrary constant of the rational expectations solution. Bo th para-metric and nonparametric tests lead us to reject the null hypothesis of convergent expectations. Copyright 1987 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

A State Space Approach to Bootstrapping Conditional Forecasts in Arma Models

A bootstrap approach to evaluating conditional forecast errors in ARMA models is presented. The key to this method is the derivation of a reverse-time state space model for generating conditional data sets that capture the salient stochastic properties of the observed data series. We demonstrate the utility of the method using several simulation experiments for the MA(q) and ARMA(p,q) models. Using the state space form, we are able to investigate conditional forecast errors in these models quite easily whereas the existing literature has only addressed conditional forecast error assessment in the pure AR(p) form. Our experiments use short data sets and non-Gaussian, as well as Gaussian, disturbances. The bootstrap is found to provide useful information on error distributions in all cases and serves as a broadly applicable alternative to the asymptotic Gaussian theory.

State Space and Unobserved Component Models: Resampling in state space models

Resampling the innovations sequence of state space models has proved to be a useful tool in many respects. For example, while under general conditions, the Gaussian MLEs of the parameters of a state space model are asymptotically normal, several researchers have found that samples must be fairly large before asymptotic results are applicable. Moreover, problems occur if the any of parameters are near the boundary of the parameter space. In such situations, the bootstrap applied to the innovation sequence can provide an accurate assessment of the sampling distributions of the parameter estimates. We have also found that a resampling procedure can provide insight into the validity of the model. In addition, the bootstrap can be used to evaluate conditional forecast errors of state space models. The key to this method is the derivation of a reverse-time innovations form of the state space model for generating conditional data sets. We will provide some theoretical insight into our procedures that show why resampling works in these situations, and we provide simulations and data examples that demonstrate our claims.

A model of decision making under bounded rationality

Abstract A model of decision making under bounded rationality is presented that combines satisficing behavior with learning and adaptation through environmental feedback. The aspirations, or goals, of the decision maker dynamically adjust in response to the observed sequence of past decisions and their corresponding effects on the decison makers objective function. A simple linear response model is employed to represent the beliefs of the decison maker concerning the causal connection between his/her decisions and the resulting objective function value. The combination of these simple elements yields a decision process model rich in dynamic behavior; it can exhibit optimizing behavior in the long-run and chaotic pseudo-random search in the short-run. As such, the model bridges the gap between substantive rationality and procedural rationality.

Generalized expectations modeling in macroeconometrics

Abstract The Generalized Expectations Model (GEM) representation is introduced as a medium for the development of econometric models involving expectational variables. Motivation is provided by viewing the various expectation formation processes of economics as special cases of a more general representation. This representation is shown to result in a form of state space model where the states relate to the expectations while the output equations represent the behavioral relations. When coupled with Kalman filtering techniques, a powerful empirical tool results which possesses the ability to simultaneously estimate: (1) the structural model; (2) the expectation formation mechanism; and (3) the time series of expectations. Both unstable expectations and non-stationary stochastic shocks are permitted. An example using a monetary model of hyperinflation illustrates the salient features of modeling with the GEM representation.

Are Errors in Official U.S. Budget Receipts Forecasts Just Noise

Existing evidence suggests that U.S. Government budget receipts forecasts are unbiased and efficient. Our study is an attempt to examine the veracity of these findings. The time series framework employed in this study is distinguished from previous work in three ways. First, we build a model that explicitly admits serial correlation in the residuals by allowing for autoregressive, moving-average, serial correlation. Second, we employ the nonparametric Monte-Carlo bootstrap to free ourselves from reliance on asymptotic distribution theory which is suspect given the short data series available for this study. Third, we control for errors in the macroeconomic and financial assumptions used to produce the U.S. Governments budget forecasts. We find that the U.S. Governments annual, one-year ahead, budget receipts forecasts for fiscal years 1963 through 2003 are biased and inefficient. In addition, we find that these forecasts exhibit serial correlation in their errors and thus do not efficiently exploit all available information. Finally, we find evidence that is consistent with strategic bias that may reflect the political goals of the Administration in power.