The use of preliminary data in econometric forecasting: an application with the Bank of Italy Quarterly Model

This paper considers forecasting by econometric and time series models using preliminary (or provisional) data. The standard practice is to ignore the distinction between provisional and final data. We call the forecasts that ignore such a distinction naive forecasts, which are generated as projections from a correctly specified model using the most recent estimates of the unobserved final figures. It is first shown that in dynamic models a multistepahead naive forecast can achieve a lower mean square error than a single-step-ahead one, intuitively because it is less affected by the measurement noise embedded in the preliminary observations. The best forecasts are obtained by combining, in an optimal way, the information provided by the model with the new information contained in the preliminary data. This can be done in the state space framework, as suggested in the literature. Here we consider two simple methods to combine, in general suboptimally, the two sources of information: modifying the forecast initial conditions via standard regressions and using intercept corrections. The issues are explored with reference to the Italian national accounts data and the Bank of Italy Quarterly Econometric Model (BIQM). A series of simulation experiments with the model show that these methods are quite effective in reducing the extra volatility of prediction due to the use of preliminary data.preliminary data, macroeconomic forecasting, Bank of Italy Quarterly Econometric Model

Testing for Stochastic Trends in Series with Structural Breaks

This paper considers the problem of testing for the presence of stochastic trends in multivariate time series with structural breaks. The breakpoints are assumed to be known. The testing framework is the multivariate Locally Best Invariant test and the common trend test of Nyblom and Harvey (2000). The asymptotic distributions of the test statistics are derived under a general specification of the deterministic component, which allows for structural breaks as a particular case. Asymptotic critical values are provided for the case of a single breakpoint. A modified statistic is then proposed, the asymptotic distribution of which is independent of the breakpoint location and belongs to the Cramér-von Mises family. This modification is particularly advantageous in the case of multiple breakpoints. It is also shown that the asymptotic distributions of the test statistics are unchanged when seasonal dummy variables and/or weakly dependent exogenous regressors are included. Finally, as an example, the tests are applied to UK macroeconomic data and to data on road casualties in Great Britain.cointegration, common trends, Cramér-von Mises distribution, locally best invariant test, structural breaks

Bootstrap LR tests of stationarity, common trends and cointegration

The paper considers likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided LM test, that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure -obtained as an extension of the LM test of stationarity- does not possess properties of optimality. Finally, it is showed that the (pseudo) LR tests maintain good size and power properties also for non-Gaussian series. As an empirical illustration, we find evidence of two common stochastic trends in the volatility of the US dollar exchange rate against european and asian/pacific currencies.Kalman filter, state-space models, unit roots

Testing for trend

The paper examines various tests for assessing whether a time series model requires a slope component. We first consider the simple t-test on the mean of first differences and show that it achieves high power against the alternative hypothesis of a stochastic nonstationary slope as well as against a purely deterministic slope. The test may be modified, parametrically or nonparametrically to deal with serial correlation. Using both local limiting power arguments and finite sample Monte Carlo results, we compare the t-test with the nonparametric tests of Vogelsang (1998) and with a modified stationarity test. Overall the t-test seems a good choice, particularly if it is implemented by fitting a parametric model to the data. When standardized by the square root of the sample size, the simple t-statistic, with no correction for serial correlation, has a limiting distribution if the slope is stochastic. We investigate whether it is a viable test for the null hypothesis of a stochastic slope and conclude that its value may be limited by an inability to reject a small deterministic slope. Empirical illustrations are provided using series of relative prices in the euro-area and data on global temperature.Cramér-von Mises distribution, stationarity test, stochastic trend, unit root, unobserved component.

Stationarity Tests for Irregularly Spaced Observations and the Effects of Sampling Frequency on Power

In this paper, starting from continuous-time local level unobserved components models for stock and flow data we derive locally best invariant (LBI) stationarity tests for data available at potentially irregularly spaced points in time. We demonstrate that the form of the LBI test differs between stock and flow variables. In cases where the data are observed at regular intervals throughout the sample we show that the LBI tests for stock and flow data both reduce to the form of the standard stationarity test in the discrete-time local level model. Here we also show that the asymptotic local power of the LBI test increases with the sampling frequency in the case of stock, but not flow, variables. Moreover, for a fixed time span we show that the LBI test for stock (flow) variables is (is not) consistent against a fixed alternative as the sampling frequency increases to infinity. We also consider the case of mixed frequency data in some detail, providing asymptotic critical values for the LBI tests for both stock and flow variables, together with a finite sample power study. Our results suggest that tests which ignore the infra-period aspect of the data involve rather small losses in efficiency relative to the LBI test in the case of flow variables, but can result in significant losses of efficiency when analysing stock variables.Stock and flow variables, local level model, unit root, LBI test, temporal aggregation

Convergences of prices and rates of inflation

We consider how unit root and stationarity tests can be used to study the convergence properties of prices and rates of inflation. Special attention is paid to the issue of whether a mean should be extracted in carrying out unit root and stationarity tests and whether there is an advantage to adopting a new (Dickey-Fuller) unit root test based on deviations from the last observation. The asymptotic distribution of the new test statistic is given and Monte Carlo simulation experiments show that the test yields considerable power gains for highly persistent autoregressive processes with relatively large initial conditions, the case of primary interest for analysing convergence. We argue that the joint use of unit root and stationarity tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whether convergence occurs in relative prices or just in rates of inflation. The tests are applied to the monthly series of the Consumer Price Index in the Italian regional capitals over the period 1970-2003. It is found that all pairwise contrasts of inflation rates have converged or are in the process of converging. Only 24% of price level contrasts appear to be converging, but a multivariate test provides strong evidence of overall convergence.Dickey-Fuller test, initial condition, law of one price, stationarity test

Testing against stochastic trend and seasonality in the presence of unattended breaks and unit roots

This paper considers the problem of testing against stochastic trend and seasonality in the presence of structural breaks and unit roots at frequencies other than those directly under test, which we term unattended breaks and unattended unit roots respectively. We show that under unattended breaks the true size of the Kwiatkowski et. al. (1992) [KPSS] test at frequency zero and the Canova and Hansen (1995) [CH] test at the seasonal frequencies fall well below the nominal level under the null with an associated, often very dramatic, loss of power under the alternative. We demonstrate that a simple modification of the statistics can recover the usual limiting distribution appropriate to the case where there are no breaks, provided unit roots do not exist at any of the unattended frequencies. Where unattended unit roots occur we show that the above statistics converge in probability to zero under the null. However, computing the KPSS and CH statistics after pre-filtering the data is simultaneously efficacious against both unattended breaks and unattended unit roots, in the sense that the statistics retain their usual pivotal limiting null distributions appropriate to the case where neither occurs. The case where breaks may potentially occur at all frequencies is also discussed. The practical relevance of the theoretical contribution of the paper is illustrated through a number of empirical examples.stationarity tests, structural breaks, pre-filtering, unattended unit roots

Testing for the Presence of a Random Walk in Series with Structural Breaks - (Now published in Journal of Time Series Analysis, 22 (2001), pp.127-150.)

The paper considers tests for the presence of a random walk component in a stationary or trend stationary time series and extends them to series which contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the breakpoint and its form is that of the generalised Cram?r-von Mises distribution, with degrees of freedom depending on the number of breakpoints. The performance of this modified test is shown, via some simulation experiments, to be comparable to that of the LBI test. An unconditional test, based on the assymption that there is a single break at an unknown point is also examined. The use of the tests is illustrated with data on the flow of the Nile and US Gross National Product.Brownian bridge, Cram?r-von Mises distribution, intervention analysis, locally best invariant test, structural time series model, unobserved components.

Inflation convergence and divergence within the European Monetary Union

We study the convergence properties of inflation rates among the countries of the European Monetary Union over the period 1980-2004. Given the Maastricht agreements and the adoption of the single currency, the sample can be naturally split into two parts, before and after the birth of the euro. We study convergence in the first sub-sample by means of univariate and multivariate unit root tests on inflation differentials, arguing that the power of the tests is considerably increased if the Dickey-Fuller regressions are run without an intercept term. Overall, we are able to accept the convergence hypothesis over the period 1980-1997. We then investigate whether the second sub-sample is characterized by stable inflation rates across the European countries. Using stationarity tests on inflation differentials, we find evidence of diverging behaviour. In particular, we can statistically detect two separate clusters, or or convergence clubs: a lower inflation group that comprises Germany, France, Belgium, Austria, Finland and a higher inflation one with Spain, Netherlands, Greece, Portugal and Ireland. Italy appears to form a cluster of its own, standing in between the other two. JEL Classification: C12, C22, C32, E31Absolute Convergence, inflation differentials, stability, Unit Root Tests