Applied Time Series Analysis for the Social Sciences

3Properties of Time Series: Mean and Variance Stationarity

Consider a regression of military expenditures in Jordan, in millions of current US dollars, on the urban population in Fiji, in thousands, during the years 1960–2019.¹ The OLS estimation results are presented in Equation (3.1), with standard errors of the estimates in parentheses.

(3.1)

The estimated coefficient of 4.22 indicates that for each thousand‐person increase in the urban population of Fiji, military expenditure in Jordan increases by $4.22 million. The estimated coefficient is statistically significant at the 0.001 level, and a conventional interpretation of the indicates that 74% of the variance in Jordan’s military expenditures is explained by variation in the size of Fiji’s urban population.

The suggestion of a causal relationship, though, defies credulity. As G. Udny Yule warned in his 1925 Presidential Address to the Royal Statistical Society,

It is fairly familiar knowledge that we sometimes obtain between quantities varying with the time (time‐variables) quite high correlations to which we cannot attach any physical significance whatever, although under the ordinary test the correlation would be held ...