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Catalano and Bruckner conclude in their letter that there is no association between the Great Recession and life expectancy at birth (LEB) in the USA. What they are actually trying to do is to refute that recessions and expansions are associated with changes in mortality. That association has been shown by a number of authors in different ways, often employing the unemployment rate as economic indicator. A standard approach to demonstrate a potentially causal link between two time series is to ‘prewhiten’ them and then to examine the cross-correlation function to look for significant crosscorrelations. To prewhiten a series means to transform it so that the resulting series has negligible autocorrelation. This usually means removing lowfrequency components of the series, and prewhitening is therefore closely related to detrending. To transform the series into first differences—i.e. annual variation if data are annual (Figure 1, panels A and D) is a common method to detrend a series that may also prewhiten it. For the period 1948–2013, the annual change in unemployment has an autocorrelation of 0.10, whereas that of LEB is 0.02. With autocorrelations as close to zero as these, most algorithms indicate that these two series in first differences are adequately prewhitened. Considering the sample, 1948–2013 unemployment and LEB in first differences cross-correlate 0.42 at lag 0 (Table 1), a highly significant correlation revealing that the annual changes of both variables are substantially synchronized in the 65-year sample (Figure 1, panels D and G). Note for instance how in the recessions of the mid-1970s and 2008–09, large annual increases in unemployment coincide with large annual gains in LEB (Figure 1, panel D). Many procedures can be used to detrend a series. Common methods are subtracting a non-linear trend like the Hodrick-Prescott (HP) filter or a linear trend, i.e. a straight line (Figure 1, panels A to C). The autocorrelation of the transformed series indicates how good is the prewhitening. The series of LEB and unemployment rates linearly detrended have respective autocorrelations of 0.88 and 0.73, whereas the autocorrelations of the series detrended with the HP filter (using a smoothing parameter 6.25 which is recommended for annual data) are, respectively, 0.01 and 0.22. Thus the prewhitening is poor with linear detrending and much better with the HP filter. The cross-correlations at lag 0 (Table 1) are 0.46 for the linearly detrended series and 0.45 for the HP-detrended series of LEB and unemployment, both highly significant. Scatterplots of annual variations or deviation from HP or linear trend (Figure 1, panels G to I) indicate that the statistically positive cross-correlations at lag 0 are not determined by outliers. A more sophisticated method for prewhitening a series is obtaining the residuals from fitting an ARIMA (p,d,q) model. This requires choosing a value p for the autoregressive (AR) order, a value d for the integrating (I) order and a value q for the moving average (MA) order. Choosing an

Statistical evidence shows that mortality tends to fall during recessions: a rebuttal to Catalano and Bruckner