In a sample of size 25, you calculate a sample autocorrelation to be 0.35. You need to know if this autocorrelation is statistically significant at the 5% level of significance. Is it? (Keep in mind that the standard error of a sample autocorrelation is 1/(N^0.5), where N is the sample size.

Question

In a sample of size 25, you calculate a sample autocorrelation to be 0.35. You need to know if this autocorrelation is statistically significant at the 5% level of significance. Is it?  (Keep in mind that the standard error of a sample autocorrelation is 1/(N^0.5), where N is the sample size.

BlackTom AI · Accepted Answer

First, identify the given values: a sample size N = 25 and a sample autocorrelation r = 0.35. The standard error of the sample autocorrelation is provided as SE = 1 / sqrt(N) = 1 / 5 = 0.2. Next, compute the test statistic as t = r / SE = 0.35 / 0.2 = 1.75.

Option A: 'Yes, the autocorrelation is statistically significant at the 5% level (t-stat must be 1.96)'. This statement would be true only if the calculated t-statistic met or exceeded 1.96 (for a two-tailed test at 5% significance) or the appropriate critical value for the chosen test. Since 1.75 < 1.96, the t-statistic does not reach the 5% threshold. Therefore, this option incorrectly asserts significance at the 5% level.

Option B: 'No, the autocorrelation is significant at the 10% level (t-stat must be 1.645), but not at the 5% level'. Here, the key is the comparison to the critical values. A t-statistic of 1.75 exceeds 1.645, which means the result would be significant at the 10% level. However, it does not reach 1.96, so it is not significant at the 5% level. This option correctly describes the level at which the result is significant and the level at which it is not.

To summarize the logic in a different framing: compare the computed t = 1.75 to the standard critical thresholds 1.645 (10% level) and 1.96 (5% level). Since 1.645 < 1.75 < 1.96, the outcome sits between those cutoffs. That means it passes the 10% test but fails the 5% test. The conclusion about which option is supported by these comparisons is that the result is not significant at 5% but is significant at 10%.

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