Estimation in threshold autoregressive models with a stationary and a unit root regime

This paper treats estimation in a class of new nonlinear threshold autoregressive models with both a stationary and a unit root regime. Existing literature on nonstationary threshold models has basically focused on models where the nonstationarity can be removed by differencing and/or where the threshold variable is stationary. This is not the case for the process we consider, and nonstandard estimation problems are the result.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

A combined random effect and fixed effect forecast for panel data models

When some of the regressors in a panel data model are correlated with the random individual effects, the random effect (RE) estimator becomes inconsistent while the fixed effect (FE) estimator is consistent. Depending on the various degree of such correlation, we can combine the RE estimator and FE estimator to form a combined estimator which can be better than each of the FE and RE estimators. In this paper, we are interested in whether the combined estimator may be used to form a combined forecast to improve upon the RE forecast (forecast made using the RE estimator) and the FE forecast (forecast using the FE estimator) in out-of-sample forecasting. Our simulation experiment shows that the combined forecast does dominate the FE forecast for all degrees of endogeneity in terms of mean squared forecast errors (MSFE), demonstrating that the theoretical results of the risk dominance for the in-sample estimation carry over to the out-of-sample forecasting. It also shows that the combined forecast can reduce MSFE relative to the RE forecast for moderate to large degrees of endogeneity and for large degrees of heterogeneity in individual effects.     

Identification robust confidence set methods for inference on parameter ratios with application to discrete choice models

We study the problem of building confidence sets for ratios of parameters, from an identification robust perspective. In particular, we address the simultaneous confidence set estimation of a finite number of ratios. Results apply to a wide class of models suitable for estimation by consistent asymptotically normal procedures. Conventional methods (e.g. the delta method) derived by excluding the parameter discontinuity regions entailed by the ratio functions and which typically yield bounded confidence limits, break down even if the sample size is large ( Dufour, 1997). One solution to this problem, which we take in this paper, is to use variants of  Fieller’s ( 1940, 1954) method. By inverting a joint test that does not require identifying the ratios, Fieller-based confidence regions are formed for the full set of ratios. Simultaneous confidence sets for individual ratios are then derived by applying projection techniques, which allow for possibly unbounded outcomes. In this paper, we provide simple explicit closed-form analytical solutions for projection-based simultaneous confidence sets, in the case of linear transformations of ratios. Our solution further provides a formal proof for the expressions in Zerbe et al. (1982) pertaining to individual ratios. We apply the geometry of quadrics as introduced by  and , in a different although related context. The confidence sets so obtained are exact if the inverted test statistic admits a tractable exact distribution, for instance in the normal linear regression context. The proposed procedures are applied and assessed via illustrative Monte Carlo and empirical examples, with a focus on discrete choice models estimated by exact or simulation-based maximum likelihood. Our results underscore the superiority of Fieller-based methods.     

Testing for unit roots in the presence of uncertainty over both the trend and initial condition

In this paper we provide a joint treatment of two major problems that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. We suggest decision rules based on the union of rejections of four standard unit root tests (OLS and quasi-differenced demeaned and detrended ADF unit root tests), along with information regarding the magnitude of the trend and initial condition, to allow simultaneously for both trend and initial condition uncertainty.     

On the accuracy of bootstrap confidence intervals for efficiency levels in stochastic frontier models with panel data

We study the construction of confidence intervals for efficiency levels of individual firms in stochastic frontier models with panel data. The focus is on bootstrapping and related methods. We start with a survey of various versions of the bootstrap. We also propose a simple parametric alternative in which one acts as if the␣identity of the best firm is known. Monte Carlo simulations indicate that the parametric method works better than the␣percentile bootstrap, but not as well as bootstrap methods that make bias corrections. All of these methods are valid␣only for large time-series sample size (T), and correspondingly none of the methods yields very accurate confidence intervals except when T is large enough that the identity of the best firm is clear. We also present empirical results for two well-known data sets.      

A note on three-stage confidence intervals for the difference of locations: The exponential case

Fixed-width confidence intervals for the difference of location parameters of two independent negative exponential distributions are constructed via triple sampling when the scale parameters are unknown and unequal. The present three-stage estimation methodology is put forth because (i) it is operationally more convenient than the existing purely sequential counterpart, and (ii) the three-stage and the purely sequential estimation techniques have fairly similar asymptotic second-order characteristics.     

Asymptotic confidence intervals for the effect of closing an area upon the number of birds' nests

Abstract  The methodological discussion in D e R oos -S chaafsma (1981) Section 6 is continued by proposing asymptotic methods as a substitute for laborious confidence interval computations.     

Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model

This paper considers Maximum Likelihood (ML) based estimation and inference procedures for linear dynamic panel data models with fixed effects.     

The effect of uncertainty and instability on dynamic economic models with an application to cournot oligopoly

Consider a dynamic discrete-time economic model in which a state-dependent payoff is earned during each period. We give conditions which ensure that a mean preserving increase in the riskiness of the model's parameters increases (or decreases) the expected payoff earned during n periods.     

Shortest confidence intervals and UMVU estimators for families of distributions involving truncation parameters

Summary In this paper, using the pivotal quantity method, new shortest-length confidence intervals and uniformly minimum variance unbiased (UMVU) estimators are constructed, where two independent random samples are available from families of distributions involving truncation parameters. Also, in the case of one sample, we give, for some uniform distributions, confidence intervals which are the shortest among all known confidence intervals.     

The second-order bias and mean squared error of estimators in time-series models

We develop analytical results on the second-order bias and mean squared error of estimators in time-series models. These results provide a unified approach to developing the properties of a large class of estimators in linear and nonlinear time-series models and they are valid for both normal and nonnormal samples of observations, and where the regressors are stochastic. The estimators included are the generalized method of moments, maximum likelihood, least squares, and other extremum estimators. Our general results are applied to four time-series models. We investigate the effects of nonnormality on the second-order bias results for two of these models, while for all four models, the second-order bias and mean squared error results are given under normality. Numerical results for some of these models are also presented.     

The use of dummy variables to compute predictions,prediction errors,and confidence intervals

This paper presents a method for computing predictions, prediction error variances, and confidence intervals, which can be implemented with any regression program. It demonstrates that a regression estimated for an augmented data set, obtained by (1) combining n sample points with r forecast points, and (2) including r dummy variables (each equalling one only for the corresponding forecast point), will yield r dummy variable coefficients and variances which equal the corresponding prediction errors and prediction error variances. Since most programs lack special routines to calculate these magnitudes, while manual computation is cumbersome, the proposed method is of considerable practical value.     

An uncertain suggestion for gold-pricing models: the effect of economic policy uncertainty on gold prices

Gold, whether held in physical form or through financial claims, is of utmost importance to investors, central bankers, and sovereign nations alike. Yet empirically validated explanations of its volatile price remain elusive. Without an ex-post understanding of the determinants of gold prices, ex-ante forecasting is a fruitless endeavor. In this research, an index of US and European economic policy uncertainty is incorporated into a short-run pricing model for gold. The results suggest that in addition to gold being a hedge against inflation, increases in economic policy uncertainty contribute to increases in the price of gold.     

The relationship between power and level for generic unit root tests in finite samples

We show that the maximum power of a generic unit root test against any stationary alternative is equal to the true level of the test. We then use Monte Carlo methods to investigate the implications for several such tests. We show patterns of rejection probabilities over a variety of unit root and stationary processes. We discuss the implications of these results for some of the uses of unit root tests in applied work.     

The effect of data transformation on common cycle,cointegration, and unit root tests: Monte Carlo results and a simple test

Cointegration, common cycle, and related tests statistics are often constructed using logged data, even without clear reason why logs should be used rather than levels. Unfortunately, it is also the case that standard data transformation tests, such as those based on Box–Cox transformations, cannot be shown to be consistent unless assumptions concerning whether variables I(0)$I(0)$ or I(1)$I(1)$ are made. In this paper, we propose a simple randomized procedure for choosing between levels and log-levels specifications in the (possible) presence of deterministic and/or stochastic trends, and discuss the impact of incorrect data transformation on common cycle, cointegration and unit root tests.     

The effect of defense program uncertainty on cost,schedule, and capital investment

Program instability, low levels of capital investment and uneconomical production rates have been recognized as significant problems in defense acquisition programs. In this paper, we use a dynamic programming model under uncertainty to investigate the relations among these problems. We show that capital investment and production rates that appear to be too low may be the result of cost minimizing responses by contractors to program instability. Attempts to change the level of capital investment or production rate on defense programs without regard to the nature of program instability may hinder efficient resource allocation.The refereeing process of this paper was handled through J.F. Muth.     

The effect of temporal aggregation on parameter estimation in distributed lag model

This paper considers the effect of temporal aggregation on parameter estimation in a finite distributed lag model through the least squares procedure. Numerical results are presented through a simple example. It is shown that the loss in efficiency due to aggregation is substantial. Moreover, the loss depends not only on the level of aggregation but also on the nature of the input variable. The loss is more severe if the input variable is negatively autocorrelated than otherwise.     

The comparative forecast performance of univariate and multivariate models: an application to real interest rate forecasting

Does the use of information on the past history of the nominal interest rates and inflation entail improvement in forecasts of the ex ante real interest rate over its forecasts obtained from using just the past history of the realized real interest rates? To answer this question we set up a univariate unobserved components model for the realized real interest rates and a bivariate model for the nominal rate and inflation which imposes cointegration restrictions between them. The two models are estimated under normality with the Kalman filter. It is found that the error-correction model provides more accurate one-period ahead forecasts of the real rate within the estimation sample whereas the unobserved components model yields forecasts with smaller forecast variances. In the post-sample period, the forecasts from the bivariate model are not only more accurate but also have tighter confidence bounds than the forecasts from the unobserved components model.