Graham Elliott

Efficient Tests for an Autoregressive Unit Root

This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.

Inference in Models with Nearly Integrated Regressors

This paper examines regression tests of whether x forecasts y when the largest autoregressive root of the regressor is unknown. It is shown that previously proposed two-step procedures, with first stages that consistently classify x as I(1) or I(0), exhibit large size distortions when regressors have local-to-unit roots, because of asymptotic dependence on a nuisance parameter that cannot be estimated consistently. Several alternative procedures, based on Bonferroni and Scheffe methods, are therefore proposed and investigated. For many parameter values, the power loss from using these conservative tests is small.

Efficient Tests for a Unit Root When the Initial Observation Is Drawn from Its Unconditional Distribution

Researchers desire powerful tests for unit roots. This paper derives the family of asymptotically most powerful tests for unit roots when the initial condition is drawn from its unconditional distribution under the alternative. This enables both the examination of previously proposed statistics and the construction of powerful tests against this alternative model. Copyright 1999 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Tests for Unit Roots and the Initial Condition

The paper analyzes the impact of the initial condition on the problem of testing for unit roots. To this end, we derive a family of optimal tests that maximize a weighted average power criterion with respect to the initial condition. We then investigate the relationship of this optimal family to popular tests. We find that many unit root tests are closely related to specific members of the optimal family, but the corresponding members employ very different weightings for the initial condition. The popular Dickey-Fuller tests, for instance, put a large weight on extreme deviations of the initial observation from the deterministic component, whereas other popular tests put more weight on moderate deviations. Since the power of unit root tests varies dramatically with the initial condition, this paper explains the results of comparative power studies of unit root tests. The results allow a much deeper understanding of the merits of particular tests in specific circumstances, and a guide to choosing which statistics to use in practice.

Optimal Forecast Combinations Under General Loss Functions and Forecast Error Distributions

Existing results on the properties and performance of forecast combinations have been derived in the context of mean squared error loss. Under this loss function empirical studies have generally found that estimates of optimal forecast combination weights lead to higher losses than equally-weighted combined forecasts which in turn outperform the best individual predictions. We show that this and other results can be overturned when asymmetries are introduced in the loss function and the forecast error distribution is skewed. We characterize the optimal combination weights for the most commonly used alternatives to mean squared error loss and demonstrate how the degree of asymmetry in the loss function and skews in the underlying forecast error distribution can significantly change the optimal combination weights. We also propose estimation methods and investigate their small sample properties in simulations and in an inflation forecasting exercise.

Heterogeneous Expectations and Tests of Efficiency in the Yen/Dollar Forward Exchange Rate Market

This paper examines the efficiency of the forward yen/dollar market using micro survey data. We first argue that the conventional tests of efficiency (unbiasedness) of the forward rate or of the survey forecasts do not correspond directly to the zero-profit condition. Instead, we use the survey data to calculate directly potential profits of individual forecasters based on a natural trading rule. We find that although the survey data are not the best predictor of future spot rate in terms of typical mean square forecast error criteria, the survey data can be used to obtain on average positive profits. However, these profits are small and highly variable. We also examine profits generated by a trading rule using regression forecasts, where forward premium is an explanatory variable. These profits are also small and highly variable.

Testing for unit roots with stationary covariates

We derive the family of tests for a unit root with maximal power against a point alternative when an arbitrary number of stationary covariates are modeled with the potentially integrated series. We show that very large power gains are available when such covariates are available. We then derive tests which are simple to construct (involving the running of vector autoregressions) and achieve at a point the power envelopes derived under very general conditions. These tests have excellent properties in small samples. We also show that these are obvious and internally consistent tests to run when identifying structural VARs using long run restrictions.

Confidence intervals for autoregressive coefficients near one

Often we are interested in the largest root of an autoregressive process. Available methods rely on inverting t-tests to obtain confidence intervals. However, for large autoregressive roots, t-tests do not approximate asymptotically uniformly most powerful tests and do not have optimality properties when inverted for confidence intervals. We exploit the relationship between the power of tests and accuracy of confidence intervals, and suggest methods which are asymptotically more accurate than available interval construction methods. One interval, based on inverting the P(T) or Q(T) statistic, has good asymptotic accuracy and is easy to compute.

On the Failure of Purchasing Power Parity for Bilateral Exchange Rates after 1973

Point estimates suggest mean reversion in real exchange rates; however, it still remains uncomfortable that models without any mean reversion are often compatible with data from the floating period. Studies with data over longer periods find mean reversion, but at the cost of mixing in data from earlier exchange rate arrangements. Pooling the floating period data potentially mixes country pairs with and without mean reversion.We examine tests for mean reversion for individual country pairs where greater power against close alternatives is gained through modeling other economic variables with the real exchange rate. By increasing the power of the tests we find strong evidence of mean reversion.

Optimal Power for Testing Potential Cointegrating Vectors With Known Parameters for Nonstationarity

Theory often specifies a particular cointegrating vector among integrated variables, and testing for a unit root in the known cointegrating vector is often required. Although it is common to simply use a univariate test for a unit root for this test, it is known that this does not take into account all available information. We show here that in such testing situations, a family of tests with optimality properties exists. We use this to characterize the extent of the loss in power from using popular methods, as well as to derive a test that works well in practice. We also characterize the extent of the losses of not imposing the cointegrating vector in the testing procedure. We apply various tests to the hypothesis positing that price forecasts from the Livingston data survey are cointegrated with prices, and find that although most tests fail to reject the presence of a unit root in forecast errors, the tests presented here strongly reject this (implausible) hypothesis.