Sung K. Ahn

Nested Reduced-Rank Autoregressive Models for Multiple Time Series

Abstract The nested reduced-rank autoregressive (AR) model is considered in order to simplify and provide a more detailed description of the structure of the multivariate time series and to reduce the number of parameters in the time series modeling. The multivariate AR model is Yt = Σ p j=1 Φ j Y t-j + et , where Yt is m × 1, and the structure of the model considered is such that the rj = rank(Φ j ) are nonincreasing as the lag j increases, so the Φ j have the factorization Φ j = AjBj and range(Aj ) ⊃ range(A j+1). Specification of the coefficient rank structures for such models through the use of canonical correlation analysis between Yk,t = [Y′t, …, Y′t-k ]′ and Y k,t-1 is discussed. A canonical variable transformation that produces simpler structure in the model and explicitly illustrates how different components of the vector series depend on past lagged values to differing degrees is also examined. A Gaussian parameter-estimation procedure is described and asymptotic properties of the Gaussian estim...

Estimation of partially nonstationary vector autoregressive models with seasonal behavior

Abstract In this paper the concept of multivariate partial nonstationarity considered by Ahn and Reinsel (1990) is extended to nonstationary seasonal models. The relationship between the partially non- stationary vector autoregressive model with seasonal behavior and seasonal cointegration and the error correction model is presented. The Gaussian reduced rank estimation procedure along with asymptotic properties of its estimator are studied. This estimation procedure yields an estimated model in which the same nonstationary characteristics are imposed as those possessed by the underlying process. Therefore, we can achieve better understanding of the nature of the process, improved forecasts, and better seasonal adjustment. We also study a two-step reduced rank estimation procedure that yields an estimator which is asymptotically as efficient as the Gaussian reduced rank estimator. Asymptotic properties of the Gaussian reduced rank estimator are also studied when deterministic seasonal components are included in the model. The finite sample properties of the estimators are briefly examined through a simulation. An example is presented to illustrate the methods and concepts.

Inference of Vector Autoregressive Models with Cointegration and Scalar Components

Abstract For the partially nonstationary vector autoregressive model of Ann and Reinsel, I further assume that the first differenced series has scalar components of lower order and study estimation of these models along with asymptotic properties of the estimators. It is shown that Gaussian reduced rank estimation can be easily carried out by simple modification of the Ahn and Reinsels method. The asymptotic distribution for the estimator of the nonstationary parameter is a locally asymptotically mixed normal, and for that of the stationary parameter is asymptotically a normal. Testing hypothesis of the assumed structure of scalar components, including serial correlation common feature, is briefly discussed. A numerical example is provided to illustrate the methods.

Additional sources of bias in half-life estimation

Recently, an increasing amount of attention is being paid to biases in the measurement of time series dynamics based on calculations of half-life. In particular, this issue amplifies the controversy surrounding the purchasing power parity doctrine. Cross-sectional and temporal aggregations, along with mis-specified models, were previously identified as sources of this bias. We identified several other sources of bias, namely, sampling error, incorrect approximations, and structural breaks in time series. These sources should also receive sufficient attention for a sound measurement of half-life.

UNIT ROOT TESTS WITH INFINITE VARIANCE ERRORS

This paper considers the asymptotic properties of some unit root test statistics with the errors belonging to the domain of attraction of a symmetric α-stable law with 0 < α < 2. The results obtained can be viewed as a parallel extension of the asymptotic results for the finite-variance case. The test statistics considered are the Dickey-Fuller, the Lagrange multiplier, the Durbin-Watson and Phillips-type modified. Their asymptotic distributions are expressed as functionals of a standard symmetric α-stable Lévy motion. Percentiles of these test statistics are obtained by computer simulation. Asymptotic distributions of sample moments that are part of the test statistics are found to have explicit densities. A small Monte Carlo simulation study is performed to assess small-sample performance of these test statistics for heavy-tailed errors.

F-Probability plot and its application to multivariate normality

We introduce the squared jackknife distance of a multivariate observation and derive its finite sampling distribution when a random sample is taken from a multivariate normal distribution. Based on the finite sampling distribution of the squared jackknife distance we recommend the use of an F-probability plot for checking multivariate normality and the F-probability plot correlation coefficient and the F-probability plot intercept as numerical measures of detecting deviation from normality. We report empirical percentiles of the F-probability plot correlation coefficient and the F-probability plot intercept. We compare the powers of tests for multivariate normality based on the F-probability plot and based on the “usual” chi-square probability plot and conduct extensive power studies of the proposed numerical measures.

Estimation of Vector Error Correction Models with Mixed‐Frequency Data

Vector autoregressive (VAR) models with error‐correction structures (VECMs) that account for cointegrated variables have been studied extensively and used for further analyses such as forecasting, but only with single‐frequency data. Both unstructured and structured VAR models have been estimated and used with mixed‐frequency data. However, VECMs have not been studied or used with mixed‐frequency data. The article aims partly to fill this gap by estimating a VECM using the expectation‐maximization (EM) algorithm and US data on four monthly coincident indicators and quarterly real GDP and, then, using the estimated model to compute in‐sample monthly smoothed estimates and out‐of‐sample monthly forecasts of GDP. Because the model is treated as operating at the highest monthly frequency and the monthly‐quarterly data are used as given (neither interpolated to all‐monthly data, nor aggregated to all‐quarterly data), the application is expected to be unbiased and efficient. A Monte Carlo analysis compares the accuracy of VECMs estimated with the given mixed‐frequency data vs. with their single‐frequency temporal aggregate.

Some tests for unit roots in seasonal time series with deterministic trends

Using the Lagrange multiplier principle, we develop test statistics for testing seasonal unit roots in a time series with possible deterministic trends. The asymptotic distributions of the test statistics are derived: they are functionals of stochastic integrals of standard Brownian bridges. Empirical percentiles of the test statistics for selected seasonal periods are provided.

On the predictive power of monetary exchange rate model: the case of the Malaysian ringgit/US dollar rate

The predictive power of the monetary model for the Malaysian ringgit/US dollar (RM/USD) rate is analysed using quarterly data ending in 2006:Q3. We find compelling evidence of a long-run relationship between exchange rates and the economic fundamental determinant. Macroeconomic factors systematically affect the long-run movement of the RM/USD rate. Additionally, the RM/USD rate was overvalued by about 10% several quarters before the 1997 crisis; after the crisis, rates fluctuated close to the equilibrium value. The out-of-sample forecasts demonstrate that the monetary model outperforms the naïve random walk model. The monetary and Purchasing Power Parity (PPP) models do well at the four to eight quarters horizon.

Maximum Eigenvalue Test for Seasonal Cointegrating Ranks

The maximum eigenvalue (ME) test for seasonal cointegrating ranks is presented using the approach of Cubadda [Oxford Bulletin of Economics and Statistics (2001), Vol. 63, pp. 497–511], which is computationally more efficient than that of Johansen and Schaumburg [Journal of Econometrics (1999), Vol. 88, pp. 301–339]. The asymptotic distributions of the ME test statistics are obtained for several cases that depend on the nature of deterministic terms. Monte Carlo experiments are conducted to evaluate the relative performances of the proposed ME test and the trace test, and we illustrate these tests using a monthly time series.