Click on the chart to enlarge.

During the current recession, the moving average saw a peak on the week that ended on April 4, 2009. That would put the end of the recession somewhere between April and May (zero to eight weeks later). And if the past is any indication, most likely in May.
The usual caveat applies: the pattern-recognition method is based on only six episodes of recession, and six is a very small number to make any claim on the significance of this finding. Moreover, each episode is spaced by several years from each other, so the hypothetical relationship between UI claims and the business cycle may have changed since last time.
I can’t help, nonetheless, to find some solace in that chart.
*Here I take the NBER dating of the business cycles. The NBER, however, provides the month in which a business cycle trough occurs, not the week. I take the liberty of dating the end of each recession on the last week of the month when the NBER identifies a business cycle trough.
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