Working Papers Series

Papers below are in pdf.

J. Isaac Miller

WP 11-04
Cointegrating MiDaS Regressions and a MiDaS Test
J. Isaac Miller

This paper introduces cointegrating mixed data sampling (CoMiDaS) regressions, generalizing nonlinear MiDaS regressions in the extant literature. Under a linear mixed-frequency data-generating process, MiDaS regressions provide a parsimoniously parameterized nonlinear alternative when the linear forecasting model is over-parameterized and may be infeasible. In spite of potential correlation of the error term both serially and with the regressors, I find that nonlinear least squares consistently estimates the minimum mean-squared forecast error parameter vector. The exact asymptotic distribution of the difference may be non-standard. I propose a novel testing strategy for nonlinear MiDaS and CoMiDaS regressions against a general but possibly infeasible linear alternative. An empirical application to nowcasting global real economic activity using monthly covariates illustrates the utility of the approach.

JEL Codes: C12, C13, C22
Keywords: cointegration, mixed-frequency series, mixed data sampling

WP 11-03
Conditionally Efficient Estimation of Long-run Relationships Using Mixed-frequency Time Series
J. Isaac Miller

I analyze efficient estimation of a cointegrating vector when the regressand is observed at a lower frequency than the regressors. Previous authors have examined the effects of specific temporal aggregation or sampling schemes, finding conventionally efficient techniques to be efficient only when both the regressand and the regressors are average sampled. Using an alternative method for analyzing aggregation under more general weighting schemes, I derive an efficiency bound that is conditional on the type of aggregation used on the regressand. This conditional bound differs from the unconditional bound defined by the full-information high-frequency data generating process. I modify a conventionally efficient estimator, canonical cointegrating regression (CCR), to accommodate cases in which the aggregation weights are either unknown or known. In the unknown case, the correlation structure of the error term generally confounds identification of the conditionally efficient weights. In the commonly assumed known case, the correlation structure may be utilized to offset the potential information loss from aggregation, resulting in a conditionally efficient estimator.

JEL Codes: C13, C22
Keywords: cointegration, temporal aggregation, mixed-frequency series, mixed data sampling

WP 10-12
Long-Term Oil Price Forecasts: A New Perspective on Oil and the Macroeconomy
J. Isaac Miller and Shawn Ni

We examine how future real GDP growth relates to changes in the forecasted long-term average of discounted real oil prices and to changes in unanticipated fluctuations of real oil prices around the forecasts. Forecasts are conducted using a state-space oil market model, in which global real economic activity and real oil prices share a common stochastic trend. Changes in unanticipated fluctuations and changes in the forecasted long-term average of discounted real oil prices sum to real oil price changes. We find that these two components have distinctly different relationships with future real GDP growth. Positive and negative changes in the unanticipated fluctuations of real oil prices correlate with asymmetric responses of future real GDP growth. In comparison, changes in the forecasted long-term average are smaller in magnitude but are more influential on real GDP. Persistent upward revisions of forecasts in the 2000s had a substantial negative impact on real GDP growth, according to our estimates.

JEL Codes: E31, E32, Q43
Keywords: oil price and the macroeconomy, oil market fundamental, oil price forecasts, Kalman filter

Revised version published in Macroeconomic Dynamics (2011)

WP 10-01
A Nonlinear IV Likelihood-Based Rank Test for Multivariate Time Series and Long Panels
J. Isaac Miller

A test for the rank of a vector error correction model (VECM) or panel VECM based on the well-known trace test is proposed. The proposed test employs instrumental variables (IV's) generated by a class of nonlinear functions of the estimated stochastic trends of the VECM under the null. The test improves the standard trace test by replacing the non-standard critical values with chi-squared critical values. Extending the result to the panel VECM case, the test is robust to cross-sectional correlation of the disturbances. With this test, I extend earlier research using nonlinear IV's for unit root testing. However, the optimal instrument in the univariate case is not admissable in the more general multivariate case. The chi-squared result suggests that IV tests may be used to replace limits of other standard tests with integrated time series that are given by nonstandard stochastic integrals, even without a panel with which to pool tests.

JEL Codes: C12, C32 and C33
Keywords: VECM, panel VECM, cointegrating rank, trace test, nonlinear instruments

Published in Journal of Time Series Econometrics (2010)

WP 08-10
Crude Oil and Stock Markets: Stability, Instability, and Bubbles
J. Isaac Miller & Ronald A. Ratti

We analyze the long-run relationship between the world price of crude oil and international stock markets over 1971:1-2008:3 using a cointegrated vector error correction model with additional regressors. Allowing for endogenously identified breaks in the cointegrating and error correction matrices, we find evidence for breaks after 1980:5, 1988:1, and 1999:9. We find a clear long-run relationship between these series for six OECD countries for 1971:1-1980.5 and 1988:2-1999.9, suggesting that stock market indices respond negatively to increases in the oil price in the long run. During 1980.6-1988.1, we find relationships that are not statistically significantly different from either zero or from the relationships of the previous period. The expected negative long-run relationship appears to disintegrate after 1999.9. This finding supports a conjecture of change in the relationship between real oil price and real stock prices in the last decade compared to earlier years, which may suggest the presence of several stock market bubbles and/or oil price bubbles since the turn of the century.

JEL Codes: C13, C32, Q43
Keywords: crude oil, stock market prices, cointegrated VECM, structural stability, stock market bubble, oil price bubble

Published in Energy Economics (2009)

WP 08-03
Testing the Bounds: Empirical Behavior of Target Zone Fundamentals
J. Isaac Miller

Standard target zone exchange rate models are based on nonlinear functions of unobserved economic fundamentals, which are assumed to be bounded, similarly to the target zone exchange rates themselves. Using a novel estimation and testing strategy, I show how this key but often overlooked assumption may be tested. Empirical results cast doubt on its validity in practice, providing a reason for well-documented empirical difficulties of these models in the literature.

JEL Codes: F3, C22, C52
Keywords: target zone exchange rates, economic fundamental, unscented Kalman filter, rescaled range statistic

Published in Economic Modelling (2011)

WP 08-01
Nonlinearity, Nonstationarity, and Thick Tails: How They Interact to Generate Persistency in Memory
J. Isaac Miller and Joon Y. Park

We consider nonlinear transformations of random walks driven by thick-tailed innovations that may have infinite means or variances. These three nonstandard characteristics: nonlinearity, nonstationarity, and thick tails interact to generate a spectrum of asymptotic autocorrelation patterns consistent with long-memory processes. Such autocorrelations may decay very slowly as the number of lags increases or may not decay at all and remain constant at all lags. Depending upon the type of transformation considered and how the model error is speci- fied, the autocorrelation functions are given by random constants, deterministic functions that decay slowly at hyperbolic rates, or mixtures of the two. Such patterns, along with other sample characteristics of the transformed time series, such as jumps in the sample path, excessive volatility, and leptokurtosis, suggest the possibility that these three ingredients are involved in the data generating processes of many actual economic and financial time series data. In addition to time series characteristics, we explore nonlinear regression asymptotics when the regressor is observable and an alternative regression technique when it is unobservable. To illustrate, we examine two empirical applications: wholesale electricity price spikes driven by capacity shortfalls and exchange rates governed by a target zone.

Revised version published in Journal of Econometrics (2010)

JEL Codes: C22, C16
Keywords: persistency in memory, nonlinear transformations, random walks, thick tails, stable distributions, wholesale electricity prices, target zone exchange rates

WP 07-22
Cointegrating Regressions with Messy Regressors: Missingness, Mixed Frequency, and Measurement Error
J. Isaac Miller

We consider a cointegrating regression in which the integrated regressors are messy in the sense that they contain data that may be mismeasured, missing, observed at mixed frequencies, or have other irregularities that cause the econometrician to observe them with mildly nonstationary noise. Least squares estimation of the cointegrating vector is consistent. Existing prototypical variance-based estimation techniques, such as canonical cointegrating regression (CCR), are both consistent and asymptotically mixed normal. This result is robust to weakly dependent but possibly nonstationary disturbances.

JEL Codes: C13, C14, C32
Keywords: cointegration, canonical cointegrating regression, near-epoch dependence, messy data, missing data, mixed-frequency data, measurement error, interpolation

Published as "Cointegrating Regressions with Messy Regressors and an Application to Mixed-frequency Series" in Journal of Time Series Analysis, 2010

WP 06-09
A Random Coefficients Autoregressive Model with Exogenously-Driven Stochastic Unit Roots
J. Isaac Miller

We develop a random coefficients autoregressive (RCA) model with time-varying coefficients generated by a bounded nonlinear function of an exogenous time series that may be a mixingale or integrated. Moreover, we allow for exogenously-driven heteroskedasticity in the error term. By restricting the range of the function essentially to the unit interval, we show that the two series of autoregressive coefficients and variances of such a model are covariance stationary, even though these series may be nonergodic. Time series driven by such a data generating process are stationary, but may have (stochastic) unit or near-unit roots over periods of time. Under appropriate assumptions, we show that maximum likelihood estimation yields asymptotically normal or mixed normal parameter estimates. A data generating process of this form may engender commonly observed time series characteristics that defy the simple I(0)-I(1) dichotomy, but is more structural in nature than statistical I(d) models. Moreover, this approach provides a nonspurious way to model relationships between a nonstationary and a stationary time series. The utility of the proposed econometric model is demonstrated with an empirical application, in which inflation drives the autoregressive coefficient of interest rate volatility.

JEL Codes: C13, C22, C32
Keywords: random coefficients autoregressive models, stochastic unit roots, nonlinear transformations, mixingales, near-epoch dependent processes, integrated processes, interest rate volatility

WP 06-04
Testing for Purchasing Power Parity Under a Target Zone Exchange Rate Regime
J. Isaac Miller

Using a simple flexible price monetary model combined with a standard target zone exchange rate model, we show theoretically that when the latent economic fundamental is integrated (as is generally assumed) the real exchange rate (RER) between currencies whose nominal exchange rates are governed by a target zone regime generally cannot exhibit short memory (mean reversion). Typical cointegration-based tests for long-run purchasing power parity (PPP) are therefore inherently misspecified. We explain how the fundamental in such models may be estimated using nonlinear filtering techniques, such as the extended Kalman filter (EKF). We then use the EKF to estimate the latent fundamental for Denmark and the Euro area. Finding empirical evidence to support the integratedness of this fundamental, our theoretical results thus imply that long-run PPP between Denmark and the Euro area cannot hold, and empirical tests on this RER support this result.

JEL Codes: C13, C22, C32
Keywords: target zone exchange rates, purchasing power parity, nonlinear functions of integrated time series, nonlinear filtering, extended Kalman filter

WP 05-07
Extracting a Common Stochastic Trend: Theory with Some Applications
Yoosoon Chang, J. Isaac Miller and Joon Y. Park

This paper investigates the statistical properties of estimators of the parameters and unobserved series for state space models with integrated time series. In particular, we derive the full asymptotic results for maximum likelihood estimation using the Kalman filter for a prototypical class of such models -- those with a single latent common stochastic trend. Indeed, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator and show that the conventional method of inference is valid for this class of models. The models we explicitly consider comprise a special -- yet useful -- class of models that may be employed to extract the common stochastic trend from multiple integrated time series. Such models can be very useful to obtain indices that represent fluctuations of various markets or common latent factors that affect a set of economic and financial variables simultaneously. Moreover, our derivation of the asymptotics of this class makes it clear that the asymptotic Gaussianity and the validity of the conventional inference for the maximum likelihood procedure extends to a larger class of more general state space models involving integrated time series. Finally, we demonstrate the utility of this class of models extracting a common stochastic trend from three sets of time series involving short- and long-term interest rates, stock return volatility and trading volume, and Dow Jones stock prices.

JEL Codes: C13, C32
Keywords: state space model, Kalman filter, common stochastic trend, maximum likelihood estimation, permanent-transitory decomposition, interest rates, volume and volatility, stock price index

Published in Journal of Econometrics (2009)