# Why autocorrelation of data lead to the lower estimation of standard error for coefficient? [duplicate]

The problem is from [ISLR][1] Problem 5.R.R2. The data is here.

Consider the linear regression model of $y$ on $X_1$ and $X_2$. Next, plot the data using matplot(Xy,type="l"). Which of the following do you think is most likely given what you see?

Our estimate of $s.e.(\beta^1$) is too low.

The explanation is:

There is very strong autocorrelation between consecutive rows of the data matrix. Roughly speaking, we have about 10-20 repeats of every data point, so the sample size is in effect much smaller than the number of rows (1000 in this case).

But I could not understand it... According to my opinion, , the denominator of this equation (that used to calculate the estimation of standard error for coefficient) indicates no difference whether xi sequence is autocorrelated or not.

• What exactly do you not understand? Commented Mar 24, 2017 at 15:38
• They meant to say "very strong positive autocorrelation." One way to understand it is to contemplate what happens when the autocorrelation equals $1$: there, you clearly have just one piece of information because the repeats don't add anything new.
– whuber
Commented Mar 24, 2017 at 15:46
• Sorry for removing the link of ISLR because the website said that I need 10 reputation to add more than two links.... Commented Mar 25, 2017 at 6:02