Every row twice
You fit a simple linear regression by ordinary least squares on observations, and compute the usual -statistic of the slope . Then, by mistake, every row of the data set is duplicated ( rows) and the regression is run again.
By what factor is the -statistic of the slope multiplied?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The estimates do not change. What happens to , to the residual sum of squares, and to the residual degrees of freedom?
with .
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Duplicating rows adds no information, but the formula for the standard error believes it does: it sees twice as many observations and shrinks the standard error by about .
Step 1: what doubles. With every row twice, the least-squares problem is the original one multiplied by , so and are unchanged. The residuals are the same, each counted twice: . Likewise . The residual degrees of freedom go from to .
Step 2: the standard error.
Step 3: the -statistic.
The lesson. For large the factor tends to . Duplicated or overlapping observations (for example overlapping return windows) make -statistics look much stronger than they are. The standard errors assume independent rows.