Squared-error Measure Research Articles

Stock return predictability: Using the cyclical component of the price ratio

We examine whether the cyclical component of the log dividend-price and price-earnings ratios contain forecast power for stock returns. While the levels of these series contain slow moving information for predicting long horizon returns, they typically provide poor short horizon forecasts. Using three approaches to extract the ratios cyclical component, we conduct several in- and out-of-sample tests. In-sample estimation using the cyclical component leads to economically sensible values, as well as an improved fit compared to the ratio level results. Out-of-sample evidence reveals forecast improvement over a historical mean model, although this varies according to the metric used. The historical mean model is preferred using mean absolute and squared error measures, but the predictive models perform better using Mincer-Zarnowitz and encompassing regressions. Using economic based forecast evaluation, the predictive models are clearly preferred, with a stronger ability to predict the future direction of return movements and with higher trading returns. A further examination of the results reveal that this greater performance largely arises from a superior ability to predict future negative returns.

Exchange rate prediction redux: New models, new data, new currencies

Previous assessments of nominal exchange rate determination, following Meese and Rogoff (1983) have focused upon a narrow set of models. Cheung et al. (2005) augmented the usual suspects with productivity based models, and “behavioral equilibrium exchange rate” models, and assessed performance at horizons of up to 5 years. In this paper, we further expand the set of models to include Taylor rule fundamentals, yield curve factors, and incorporate shadow rates and risk and liquidity factors. The performance of these models is compared against the random walk benchmark. The models are estimated in error correction and first-difference specifications. We examine model performance at various forecast horizons (1 quarter, 4 quarters, 20 quarters) using differing metrics (mean squared error, direction of change), as well as the “consistency” test of Cheung and Chinn (1998). The purchasing power parity estimated in the error correction form delivers the best performance at long horizons by a mean squared error measure. Moreover, along a direction-of-change dimension, certain structural models do outperform a random walk with statistical significance. While one finds that these forecasts are cointegrated with the actual values of exchange rates, in most cases, the elasticity of the forecasts with respect to the actual values is different from unity. Overall, model/specification/currency combinations that work well in one period and one performance metric will not necessarily work well in another period and alternative performance metric.

Exchange Rate Prediction Redux: New Models, New Data, New Currencies

Previous assessments of nominal exchange determination, following Meese and Rogoff (1983) have focused upon a narrow set of models. Cheung et al. (2005) augmented the usual suspects with productivity based models, and behavioral equilibrium exchange rate models, and assessed performance at horizons of up to 5 years. In this paper, we further expand the set of models to include Taylor rule fundamentals, yield curve factors, and incorporate shadow rates and risk and liquidity factors. The performance of these models is compared against the random walk benchmark. The models are estimated in error correction and first-difference specifications. We examine model performance at various forecast horizons (1 quarter, 4 quarters, 20 quarters) using differing metrics (mean squared error, direction of change), as well as the “consistency” test of Cheung and Chinn (1998). No model consistently outperforms a random walk, by a mean squared error measure, although purchasing power parity does fairly well. Moreover, along a direction-of-change dimension, certain structural models do outperform a random walk with statistical significance. While one finds that these forecasts are cointegrated with the actual values of exchange rates, in most cases, the elasticity of the forecasts with respect to the actual values is different from unity. Overall, model/specification/currency combinations that work well in one period will not necessarily work well in another period

Apparent mass of small children: Modelling

Mass-spring-damper models are widely available for quantifying the whole-body vibration characteristics of primates, human adolescents and human adults, but no models have previously been developed for small children. In this study a single degree of freedom, linear, mass-spring-damper with base support model was determined from the seated vertical apparent mass modulus function of each of eight small children of less than 18 kg in mass. A differential evolution optimisation algorithm was used in conjunction with a mean squared error measure and penalty functions to identify the optimal child model parameter values. The eight child models were characterised by a mean moving mass m 1 of 8.5 kg, a mean body stiffness k 1 of 21,131 N/m and a mean damping coefficient c 1 of 329 Ns/m. Comparison to the parameter values of similar models reported in the literature for Rhesus monkeys, Baboons, large children and adults suggests that the values obtained in the current study for small children are intermediate between the smaller primates and the larger humans. A regression analysis of the model parameters was performed as a function of subject mass for a data set consisting of the eight child models, 12 similar models for primates, and 60 similar models for large children and adults. The moving mass m 1 of the group of models grew with a power exponent of approximately unity, the body stiffness k 1 grew with a power exponent of approximately +1/2, the damping coefficient c 1 grew with a power exponent of approximately +3/4 and the dimensionless damping ratio was independent of subject mass. The natural frequency of the models grew with a power exponent of approximately −1/4.

Empirical Exchange Rate Models of the Nineties: Are Any Fit to Survive?

Previous assessments of nominal exchange rate determination have focused upon a narrow set of models typically of the 1970's vintage. The canonical papers in this literature are by Meese and Rogoff (1983, 1988), who examined monetary and portfolio balance models. Succeeding works by Mark (1995) and Chinn and Meese (1995) focused on similar models. In this paper we re-assess exchange rate prediction using a wider set of models that have been proposed in the last decade: interest rate parity, productivity based models, and behavioral equilibrium exchange rate' models. The performance of these models is compared against a benchmark model the Dornbusch-Frankel sticky price monetary model. The models are estimated in error correction and first-difference specifications. Rather than estimating the cointegrating vector over the entire sample and treating it as part of the ex ante information set as is commonly done in the literature, we recursively update the cointegrating vector, thereby generating true ex ante forecasts. We examine model performance at various forecast horizons (1 quarter, 4 quarters, 20 quarters) using differing metrics (mean squared error, direction of change), as well as the consistency' test of Cheung and Chinn (1998). No model consistently outperforms a random walk, by a mean squared error measure; however, along a direction-of-change dimension, certain structural models do outperform a random walk with statistical significance. Moreover, one finds that these forecasts are cointegrated with the actual values of exchange rates, although in a large number of cases, the elasticity of the forecasts with respect to the actual values is different from unity. Overall, model/specification/currency combinations that work well in one period will not necessarily work well in another period.

Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models

Nine macroeconomic variables are forecast in a real-time scenario using a variety of flexible specification, fixed specification, linear, and nonlinear econometric models. All models are allowed to evolve through time, and our analysis focuses on model selection and performance. In the context of real-time forecasts, flexible specification models (including linear autoregressive models with exogenous variables and nonlinear artificial neural networks) appear to offer a useful and viable alternative to less flexible fixed specification linear models for a subset of the economic variables which we examine, particularly at forecast horizons greater than 1-step ahead. We speculate that one reason for this result is that the economy is evolving (rather slowly) over time. This feature cannot easily be captured by fixed specification linear models, however, and manifests itself in the form of evolving coefficient estimates. We also provide additional evidence supporting the claim that models which ‘win’ based on one model selection criterion (say a squared error measure) do not necessarily win when an alternative selection criterion is used (say a confusion rate measure), thus highlighting the importance of the particular cost function which is used by forecasters and ‘end-users’ to evaluate their models. A wide variety of different model selection criteria and statistical tests are used to illustrate our findings.

Source coding and vector quantization with codebook-excited neural networks

The quantization property of layered neural networks is studied in this paper. We first review the layered neural network-based coders or quantizers developed in recent years and show that their poor performances is due to their independent training schemes for each component in the coder. Then an alternative model named a codebook-excited neural network is proposed, where an encoded vector is approximated by the output of the network driven by one vector selected from an excitation codebook. The network and the excitation codebook are jointly trained with the error back-propagation algorithm. Simulations with a Gauss-Markov source demonstrate that the quantization performance of the codebook-excited feedforward neural network is not worse than that of the connectionist vector quantizer formed by a set of single-layer neural units which satisfied the optimal quantization conditions, and that the performance of the codebook-excited recurrent neural network is very close to the asymptotic performance bound of block quantizers. The codebook-excited neural network is applicable with any distortion measure. For a zero-mean, unit variance, memory-less Gaussian source and a squared-error measure, a 1 bit/sample two-dimensional quantizer with a codebook-excited feedforward neural network is found always to escape from the local minima and converge to the best one of the three local minima which are known to exist in the vector quantizer designed using the LBG algorithm. Moreover, due to its conformal mapping characteristic, the codebook-excited neural network can be applied to designing the vector quantizer with any required structural form on its codevectors.

KERNEL REGRESSION SMOOTHING OF TIME SERIES

Abstract. A class of non‐parametric regression smoothers for times series is defined by the kernel method. The kernel approach allows flexible modelling of a time series without reference to a specific parametric class. The technique is applicable to detection of non‐linear dependences in time series and to prediction in smooth regression models with serially correlated observations.In practice these estimators are to be tuned by a smoothing parameter. A data‐driven selector for this smoothing parameter is presented that asymptotically minimizes a squared error measure. We prove asymptotic optimality of this selector. We illustrate the technique with a simulated example and by constructing a smooth prediction curve for the variation of gold prices. In both cases the non‐parametric method proves to be useful in uncovering non‐linear structure.

Improved estimation under collinearity and squared error loss

This paper examines the performance of several biased, Stein-like and empirical Bayes estimators for the general linear statistical model under conditions of collinearity. A new criterion for deleting principal components, based on an unbiased estimator of risk, is introduced. Using a squared error measure and Monte Carlo sampling experiments, the resulting estimator's performance is evaluated and compared with other traditional and non-traditional estimators.

Connectionist nonparametric regression: Multilayer feedforward networks can learn arbitrary mappings

It has been recently shown (e.g., Hornik, Stinchcombe & White, 1989, 1990) that sufficiently complex multilayer feedforward networks are capable of representing arbitrarily accurate approximations to arbitrary mappings. We show here that these approximations are learnable by proving the consistency of a class of connectionist nonparametric regression estimators for arbitrary (square integrable) regression functions. The consistency property ensures that as network “experience” accumulates (as indexed by the size of the training set), the probability of network approximation error exceeding any specified level tends to zero. A key feature of the demonstration of consistency is the proper control of the growth of network complexity as a function of network experience. We give specific growth rates for network complexity compatible with consistency. We also consider automatic and semi-automatic data-driven methods for determining network complexity in applications, based on minimization of a cross-validated average squared error measure of network performance. We recommend cross-validated average squared error as a generally applicable criterion for comparing relative performance of differing network architectures and configurations.

Forecasting temporally aggregated vector ARMA processes

AbstractIf interest centres on forecasting a temporally aggregated multiple time series and the generation process of the disaggregate series is a known vector ARMA (autoregressive moving average) process then forecasting the disaggregate series and temporally aggregating the forecasts is at least as efficient, under a mean squared error measure, as forecasting the aggregated series directly. Necessary and sufficient conditions for equality of the two forecasts are given. In practice the data generation process is usually unknown and has to be determined from the available data. Using asymptotic theory it is shown that also in this case aggregated forecasts from the disaggregate process will usually be superior to forecasts obtained from the aggregated process.

Forecasting Contemporaneously Aggregated Vector ARMA Processes

Given a multiple time series that is generated by a multivariate ARMA process and assuming the objective is to forecast a weighted sum of the individual variables, then under a mean squared error measure of forecasting precision, it is preferable to forecast the disaggregated multiple time series and aggregate the forecasts, rather than forecast the aggregated series directly, if the involved processes are known. This result fails to hold if the processes used for forecasting are estimated from a given set of time series data. The implications of these results for empirical research are investigated using different sets of economic data.