A distance between pairs of sets of autoregressive moving average (ARMA) processes is proposed. Its main properties are discussed. The paper also shows how the proposed distance finds application in time series analysis. In particular it can be used to evaluate the distance between portfolios of ARMA models or the distance between vector autoregressive (VAR) models.

Investigating the relationship between Gross Domestic Product and unemployment is one of the most important challenges in macroeconomics. In this paper, we compare French and German economies in terms of the dynamic linkage between these variables. In particular, we use an empirical methodology to investigate how much the relationship between Gross Domestic Product and unemployment growth rates are dynamically different in the two major European economies over the period 2003–2019. To this aim, a Vector Autoregressive model is specified for each country to jointly model the growth rate of the two variables. Then a new statistical test is proposed to assess the distance between the two estimated models. Results indicate that the dynamic linkage between Gross Domestic Product and unemployment is very similar in the two countries. This empirical evidence does not imply identical product and labor markets in France and Germany, but it ensures that in these markets there are common dynamics. This could favor the process of economic convergence between the two countries.

The aim of this paper is to investigate the relationships among Interhemispheric Temperature Asymmetry (ITA) and the principal modes of natural variability: the Atlantic Multidecadal Oscillation (AMO), the Southern Oscillation Index (SOI), and the Pacific Decadal Oscillation (PDO). In particular, Granger causality tests are used to capture the linkages among these variables. Our analysis provides strong evidence that AMO causes ITA, the causal role of PDO is weak, and SOI seems to have no causal influence.

The contribution of this paper is to investigate a particular form of lack of invariance of causality statements to changes in the conditioning information sets. Consider a discrete-time three-dimensional stochastic process $z={(x,y_1,y_2)}^{\prime }$ . We want to study causality relationships between the variables in $y={(y_1,y_2)}^{\prime }$ and x. Suppose that in a bivariate framework, we find that $y_1$ Granger causes x and $y_2$ Granger causes x, but these relationships vanish when the analysis is conducted in a trivariate framework. Thus, the causal links, established in a bivariate setting, seem to be spurious. Is this conclusion always correct? In this note, we show that the causal links, in the bivariate framework, might well not be ‘genuinely’ spurious: they could be reflecting causality from the vector y to x. Paradoxically, in this case, it is the non-causality in trivariate system that is misleading.

It is well known the effect on uncentered $$R^2$$R 2 stemming from adding a constant to dependent variable in a linear regression model with intercept. In this paper, we investigate the effect of adding a constant to variables on the uncentered $$R^2$$R 2 when a linear regression through the origin is used. In particular, we consider two cases. First, a constant $$c \in \mathbb {R}$$c ∈ R is added to all observations of the dependent variable. Second, a constant $$c \in \mathbb {R}$$c ∈ R is added to all the observations of both the dependent variable and at least an independent variable. We show that in both cases there is an artificial variation of the uncentered $$R^2$$R 2 . This quantity is not invariant under location change.

Estimating the magnitude of the Arctic equilibrium climate sensitivity is key for understanding the climate dynamics of this polar region. Using a linear regression model, we find that the Arctic equilibrium climate sensitivity is 3.61°C. To the best of our knowledge, this is the first time that an estimate of the Arctic equilibrium climate sensitivity is provided. The obtained results confirm that this region is more sensitive to climate change than the rest of the world. The Arctic sensitivity is almost one and a half times that of the global sensitivity. In this paper, we have used the estimated Arctic equilibrium climate sensitivity to obtain projections of the Arctic temperature under different forcing scenarios.