Some analysts at the onset of 2008 have noted that a bad first day leads to a bullish year, others have said that the first five days of trading are an indication of the year ahead.
The scatter plot below shows the percent change in the Dow Jones Industrial Average over the first five days of trading versus one year performance dating back to 1915. As shown, the trend line is sloping up indicating some correlation, but the R squared value of that trend line is 0.04 and can hardly be deemed significant.
Looking also at the first day of trading versus the rest of the year, we find that there is no correlation whatsoever. As shown below, the R squared value is zero. (Note that the 80% gain from the chart above is not shown in the chart below, the change on 1/2/1915 was only more recently available.)