This paper deals with the analysis of the temperatures in a group of 29 stations located in twelve European countries by looking at the coefficients in a linear time trend regression model and allowing for long memory patterns in the error term. The results indicate that long memory is present in practically all cases, and the time trend coefficients are statistically significant in the majority of the cases implying evidence of increasing warming trends. This pattern is particularly noticeable in the case of several stations located across Italy and France, which might be related with micro climates affecting these regions.
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I(1) denotes integration of order 1, which means that the series needs to be first differenced to render it stationary I(0). This latter concept (I(0)) is considered as a minimal requirement in time series to make statistical inference. Fractional integration or I(d) takes place when the number of differences required in the series is d and d is a fractional value.
Long memory is a feature of the I(d, d > 0) models that is characterized by the large degree of association between observations which are far distant in time. Mean reversion means that shocks will be transitory disappearing in the long run and takes place when d < 1.
A description of the functional form of this method can be found in Gil-Alana and Robinson (1997).
The length of the series varies from 67 to 359 observations. Nevertheless, the method employed in the paper, in spite of its asymptotic nature, it performs well even in small samples as it was shown in Gil-Alana and Robinson (1997) and other papers.
In other words, if you do a 30 hypothesis tests at P[Type-1 error] = 10%, you will end up rejecting about 3 of the hypothesis where the null is actually true, simply because the test is set up that way. The same situation holds for confidence intervals: the 95% confidence intervals will fail to contain the true value in 5 of every 100 cases just because such intervals are designed for a single case, not multiple times usage. Thus, the 95% intervals reported across Tables 1 and 2 contain a non-ignorable number of false positives.
The spectral density function is basically the frequency domain representation of the autocovariances based on Fourier transforms.
Once more note that the results for Navacerrada (NAVA), Salamanca (SALA) and Cadiz (CADI) should be taken with caution.
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Comments from the Editor and three anonymous reviewers are gratefully acknowledged.
Prof. Luis A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Economía y Competitividad (ECO2017-85503-R).
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Gil-Alana, L.A., Sauci, L. Temperatures across Europe: evidence of time trends. Climatic Change 157, 355–364 (2019). https://doi.org/10.1007/s10584-019-02568-6
- European temperatures
- Long memory
- Time trends