Aaron D. Smallwood

Joint Tests for Non-linearity and Long Memory: The Case of Purchasing Power Parity

A pervasive finding of unit roots in macroeconomic data often runs counter to intuition regarding the stochastic nature of the process under consideration. Two econometric techniques have been utilized in an attempt to resolve the finding of unit roots, namely long memory and models that depart from linearity. While the use of long memory and stochastic regime switching models have developed almost independently of each other, it is now clear that the two modeling techniques can be intimately linked. In particular, both modeling techniques have been used in isolation to study the dynamics of the real exchange rate. To determine the importance of each technique in this context, I employ a testing and estimation procedure that allows one to jointly test for long memory and non-linearity (regime switching behavior) of the STAR variety. I find that there is substantial evidence of non-linear behavior for the real exchange rate for many developing and European countries, with little evidence for ESTAR non-linearity for countries outside the European continent including Japan and Canada. In cases where non-linearity is found, I also find significant evidence of long memory for the majority of the countries in my sample. Thus, long memory and non-linearity can also be viewed as compliments rather than substitutes. The linear model in isolation appears to be inadequate for breaking down the paradox known as the PPP puzzle. On the other hand, a combination of long memory and non-linearity may be a promising research avenue for pursuing an answer to the paradox.

Long Memory Regressors and Predictive Testing: A Two-stage Rebalancing Approach

Predictability tests with long memory regressors may entail both size distortion and incompatibility between the orders of integration of the dependent and independent variables. Addressing both problems simultaneously, this paper proposes a two-step procedure that rebalances the predictive regression by fractionally differencing the predictor based on a first-stage estimation of the memory parameter. Extensive simulations indicate that our procedure has good size, is robust to estimation error in the first stage, and can yield improved power over cases in which an integer order is assumed for the regressor. We also extend our approach beyond the standard predictive regression context to cases in which the dependent variable is also fractionally integrated, but not cointegrated with the regressor. We use our procedure to provide a valid test of forward rate unbiasedness that allows for a long memory forward premium.

An Encompassing Test of Real Interest Rate Equalization

Much confusion about the real interest rate connection amongst different countries may result from a narrow approach to analyzing the data. Using an encompassing methodology that accommodates many different types of times-series processes, we find that real interest rates are mean-reverting long-memory variables. We show that cointegration methodology can often fail in this environment. Using a more general approach, we detect a limited connection between real interest rates across countries. In particular, Germany is connected with several European countries, but the US is connected only with Canada and possibly the United Kingdom. Copyright

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching, Structural Break, and Nonlinear Data

The potential observational equivalence between various types of nonlinearity and long memory has been recognized by the econometrics community since at least the contribution of Diebold and Inoue (2001). A large literature has developed in an attempt to ascertain whether or not the long memory finding in many economic series is spurious. Yet to date, no study has analyzed the consequences of using long memory methods to test for unit roots when the “truth” derives from regime switching, structural breaks, or other types of mean reverting nonlinearity. In this article, I conduct a comprehensive Monte Carlo analysis to investigate the consequences of using tests designed to have power against fractional integration when the actual data generating process is unknown. I additionally consider the use of tests designed to have power against breaks and threshold nonlinearity. The findings are compelling and demonstrate that the use of long memory as an approximation to nonlinearity yields tests with relatively high power. In contrast, misspecification has severe consequences for tests designed to have power against threshold nonlinearity, and especially for tests designed to have power against breaks.

Estimating cointegrating vectors using near unit root variables

This paper argues that the predominant method of estimating equilibrium relationships in macroeconometric models, namely the VECM system of Johansen, is severely flawed if the underlying variables are distributed as near unit root processes. Researchers may apply cointegration techniques to these processes, as the power of rejecting near unit roots using standard unit root tests is extremely low. Using Monte Carlo analysis, problematic behaviour of cointegration analysis is found in detecting the true underlying form of the connection between the near unit root processes. Furthermore the connecting vector is imprecisely estimated, resulting in problematic inference for error correction models.

Mean Reversion in the Real Interest Rate and the Effects of Calculating Expected Inflation

We investigate the measured persistence in the real interest rate using a variety of methods to annualize inflation and calculate the real rate. Results from a battery of conventional unit root tests yield conflicting conclusions for the various real rates, adding to an existing confusion regarding mean reversion. Both long memory and exponential smooth-transition autoregressive models (ESTAR) nonlinearity are considered as possible alternatives, and in contrast to the unit root test results, we find highly robust evidence against the unit root null. Based on the empirical results, Monte Carlo analysis is performed to study the disparate results obtained using fractional integration and unit root tests.

Generalized long memory and mean reversion of the real exchange rate

Definitive evidence regarding a rapid mean reversion of the real exchange rate is not present when using standard linear methodology, including unit root tests and fractional integration. To consider the robustness of these results, we use an encompassing model, the Gegenbauer AutoRegressive Moving Average (GARMA) model, which nests as special cases the existing linear methods. The GARMA model accommodates a complete notion of persistence and allows shocks to dissipate slowly in a cyclical manner. We find evidence supporting a weak version of purchasing power parity, where equilibrium errors are long memory with strongly persistent cycles. However, this new form of cyclical mean reversion is likely too slow to be economically meaningful. The inability to find a strong equilibrium attractor process, using a very general encompassing linear methodology provides support for the recent models that allow for a nonlinear attraction process and for shifting real exchange rate equilibria.