The behaviour of inconsistent instrumental variables estimators in dynamic systems with autocorrelated errors

To appropriately interpret time-series evidence when empirical relationships are incorrectly formulated, a general mis-specification framework is required. A linear, stationary, dynamic, simultaneous system with autoregressive errors is postulated to investigate instrumental variables ables estimators when the instruments are unknowingly correlated with the equation errors. The approach uses control variates (Hendry and Harrison, Journal of Econometrics, July 1974) to develop asymptotic distributions and exact moments for approximations to the econometric estimators. The accuracy of the asymptotic results for finite sample moments is corroborated by simulation. The analysis highlights the need for care in interpreting estimated equations and tests for predictive failure.     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

Unit root tests for panel MTAR model with cross-sectionally dependent error

Unit root tests are constructed for dynamic panels whose component series are momentum threshold autoregressive processes. Gaussian null asymptotics are established for the proposed tests. A Monte–Carlo experiment is conducted to compare finite sample properties of the proposed tests. The tests are illustrated by a real data set.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

Several efficient two-step estimators for the dynamic simultaneous equations model with autoregressive disturbances

Several asymptotically efficient methods are suggested on both the full and the limited information approach to estimate the simultaneous equations model in which the lagged endogenous variables and the autoregressive disturbances coexist. They are two-step procedures and do not involve iterations. A method is suggested also for the case where any portion of the autoregressive parameter matrix is specified to be zero. Since the consistency and efficiency depend upon the asymptotic, local identifiability, the necessary and sufficient condition is derived for it. It does not depend on the exclusion of the lagged endogenous variables.     

On the Applicability of the Sieve Bootstrap in Time Series Panels*

In this article, we investigate the validity of the univariate autoregressive sieve bootstrap applied to time series panels characterized by general forms of cross‐sectional dependence, including but not restricted to cointegration. Using the final equations approach we show that while it is possible to write such a panel as a collection of infinite order autoregressive equations, the innovations of these equations are not vector white noise. This causes the univariate autoregressive sieve bootstrap to be invalid in such panels. We illustrate this result with a small numerical example using a simple DGP for which the sieve bootstrap is invalid, and show that the extent of the invalidity depends on the value of specific parameters. We also show that Monte Carlo simulations in small samples can be misleading about the validity of the univariate autoregressive sieve bootstrap. The results in this article serve as a warning about the practical use of the autoregressive sieve bootstrap in panels where cross‐sectional dependence of general form may be present.     

Relative efficiencies of some simple Bayes estimators of coefficients in dynamic models — I

In this paper we consider some approximations to Bayes estimators of coefficients in simple autoregressive models and give an example of a Monte Carlo experiment where these approximate Bayes estimators yield a substantial improvement over the usual sampling theory or quasi-Bayesian estimators. The practical situation is represented by the case where the coefficient vector is known to lie in or on a hypersphere of radius r with center at 0. We show that arbitrariness in the choice of the value of r is often not catastrophic if r is sufficiently large, but finite.     

Improved likelihood ratio tests for cointegration rank in the VAR model

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but as usual the asymptotic results do not require normally distributed innovations. Our tests differ from existing tests in two respects. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott et al. (1996). Second, our tests incorporate a “sign” restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.     

Content horizons for univariate time-series forecasts

This paper investigates the maximum horizon at which conditioning information can be shown to have value for univariate time series forecasts. In particular, we consider the problem of determining the horizon beyond which forecasts from univariate time series models of stationary processes add nothing to the forecast implicit in the unconditional mean. We refer to this as the content horizon for forecasts, and provide a formal definition of the corresponding forecast content function at horizons s=1,… S. This function depends upon parameter estimation uncertainty as well as on autocorrelation structure of the process. We show that for autoregressive processes it is possible to give an asymptotic expression for the forecast content function, and show by simulation that the expression gives a good approximation even at modest sample sizes. The results are applied to the growth rate of GDP and to inflation, using US and Canadian data.     

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

Prewhitening Bias in HAC Estimation*

Heteroskedasticity and autocorrelation consistent (HAC) estimation commonly involves the use of prewhitening filters based on simple autoregressive models. In such applications, small sample bias in the estimation of autoregressive coefficients is transmitted to the recolouring filter, leading to HAC variance estimates that can be badly biased. The present paper provides an analysis of these issues using asymptotic expansions and simulations. The approach we recommend involves the use of recursive demeaning procedures that mitigate the effects of small‐sample autoregressive bias. Moreover, a commonly used restriction rule on the prewhitening estimates (that first‐order autoregressive coefficient estimates, or largest eigenvalues, >0.97 be replaced by 0.97) adversely interferes with the power of unit‐root and [ Kwiatkowski, Phillips, Schmidt and Shin (1992) Journal of Econometrics, Vol. 54, pp. 159–178] (KPSS) tests. We provide a new boundary condition rule that improves the size and power properties of these tests. Some illustrations of the effects of these adjustments on the size and power of KPSS testing are given. Using prewhitened HAC estimates and the new boundary condition rule, the KPSS test is consistent, in contrast to KPSS testing that uses conventional prewhitened HAC estimates [ Lee, J. S. (1996) Economics Letters, Vol. 51, pp. 131–137].     

Testing for ARCH in the presence of additive outliers

In this paper we investigate the properties of the Lagrange Multiplier [LM] test for autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH (GARCH) in the presence of additive outliers (AOs). We show analytically that both the asymptotic size and power are adversely affected if AOs are neglected: the test rejects the null hypothesis of homoscedasticity too often when it is in fact true, while the test has difficulty detecting genuine GARCH effects. Several Monte Carlo experiments show that these phenomena occur in small samples as well. We design and implement a robust test, which has better size and power properties than the conventional test in the presence of AOs. We apply the tests to a number of US macroeconomic time series, which illustrates the dangers involved when nonrobust tests for ARCH are routinely applied as diagnostic tests for misspecification. Copyright © 1999 John Wiley & Sons, Ltd.     

Modeling and forecasting of stock index volatility with APARCH models under ordered restriction

This article examines volatility models for modeling and forecasting the Standard & Poor 500 (S&P 500) daily stock index returns, including the autoregressive moving average, the Taylor and Schwert generalized autoregressive conditional heteroscedasticity (GARCH), the Glosten, Jagannathan and Runkle GARCH and asymmetric power ARCH (APARCH) with the following conditional distributions: normal, Student's t and skewed Student's t‐distributions. In addition, we undertake unit root (augmented Dickey–Fuller and Phillip–Perron) tests, co‐integration test and error correction model. We study the stationary APARCH (p) model with parameters, and the uniform convergence, strong consistency and asymptotic normality are prove under simple ordered restriction. In fitting these models to S&P 500 daily stock index return data over the period 1 January 2002 to 31 December 2012, we found that the APARCH model using a skewed Student's t‐distribution is the most effective and successful for modeling and forecasting the daily stock index returns series. The results of this study would be of great value to policy makers and investors in managing risk in stock markets trading.     

Instrumental variables estimation of a nearly nonstationary,heterogeneous error component model

This paper studies instrumental variables (IV) estimation for an error component model with stationary and nearly nonstationary regressors. It is assumed that the numbers of cross section and time series observations are infinite. Furthermore, autoregressive disturbances are assumed for the error component model, the structure of which may vary with individuals. The estimators considered are the Within-IV-OLS, Within-IV-GLS and IV-GLS estimators. The GLS estimators use Gohberg's formula, which is particularly useful when autoregressive structures are imposed on the disturbance terms. Sequential limit theories for the estimators are derived, and it is shown that all of the estimators have normal distributions in the limit. Additionally, Wald tests for coefficient vectors are shown to have chi-square distributions in the limit. Simulation results regarding the estimator efficiency and the size of the Wald tests are also reported. The results show that the Within-IV-GLS and IV-GLS estimators are more efficient than the Within-IV-OLS estimator in most cases and that the Wald tests keep nominal size reasonably well. The relation between the trade and budget deficits of 23 OECD nations is examined using the panel IV estimators. The empirical results support the view that the budget and trade deficits move in the same direction.     

Inference in a nearly integrated autoregressive model with nonnormal innovations

Robust tests and estimators based on nonnormal quasi-likelihood functions are developed for autoregressive models with near unit root. Asymptotic power functions and power envelopes are derived for point-optimal tests of a unit root when the likelihood is correctly specified. The shapes of these power functions are found to be sensitive to the extent of nonnormality in the innovations. Power loss resulting from using least-squares unit-root tests in the presence of thick-tailed innovations appears to be greater than in stationary models.     

Generalizations of the KPSS-test for stationarity

We propose automatic generalizations of the KPSS-test for the null hypothesis of stationarity of a univariate time series. We can use these tests for the null hypotheses of trend stationarity, level stationarity and zero mean stationarity. We introduce the asymptotic null distributions and we determine consistency against relevant nonstationary alternatives. We compare the properties of the tests with those of other proposed tests for stationarity. Monte Carlo simulations support the relevance of the tests when an autoregressive process with large positive autocorrelations is likely under the null hypothesis.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

Markov switching causality and the money–output relationship

The causal link between monetary variables and output is one of the most studied issues in macroeconomics. One puzzle from this literature is that the results of causality tests appear to be sensitive with respect to the sample period that one considers. As a way of overcoming this difficulty, we propose a method for analysing Granger causality which is based on a vector autoregressive model with time‐varying parameters. We model parameter time‐variation so as to reflect changes in Granger causality, and assume that these changes are stochastic and governed by an unobservable Markov chain. When applied to US data, our methodology allows us to reconcile previous puzzling differences in the outcome of conventional tests for money–output causality. Copyright © 2005 John Wiley & Sons, Ltd.     

Testing for a unit root in a random coefficient panel data model

This paper proposes new unit root tests in the context of a random autoregressive coefficient panel data model, in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance. The asymptotic distributions of the test statistics are derived and simulation results are provided to suggest that they perform very well in small samples.     

The exact moments of the least squares estimator for the autoregressive model

Exact mean and variance of the least squares estimate of the stationary first-order autoregressive coefficient, i.e., β in y_t=α+βx_t+u_t are evaluated algebraically as well as numerically. It turns out that the least squares estimate is seriously biased for the sample of two-digits sizes typically dealt with in econometrics if the mean of the process is unknown, i.e., if the equation has a non-zero intercept (α≠0). Kendall's approximation to the mean and Barlett's approximation to the variance are shown to be fairly good. Also, our numerical results confirm Orcutt and Winokur's (Econometrica, Vol. 37) based on Monte Carlo experiments.