Standard Error Concept

[danaro e santita]

Just a quick conceptual question about standard error. In the regression outputs displayed throughout the CFAI books, standard error is sometimes calculated based on the actual variation of the observations. Other times it is calculated as 1/sqrt(# observations). For example, the autocorrelations of residuls are calculated using this more static method that doesn’t account for the actual observations. Can anyone explain why there are multiple methods of calculating standard error? Thanks!

 

[S2000magician]

You’re talking about two types of models.
The first is a linear regression; the second is an autoregression.
It’s similar (conceptually, at least) to having different formulae for the volume of a cube and the volume of a sphere: they’re both volumes, but the shapes are completely different.

 

[tickersu]

S2000magician wrote:
You’re talking about two types of models.
The first is a linear regression; the second is an autoregression.
It’s similar (conceptually, at least) to having different formulae for the volume of a cube and the volume of a sphere: they’re both volumes, but the shapes are completely different.

To add to (and really simplify) this– the volume calculation depends on the shape, and the standard error calculation depends on the estimated parameter. Either way, you’re still coming to the same idea.

 

[Harrogath]

danaro e santita wrote:
Just a quick conceptual question about standard error. In the regression outputs displayed throughout the CFAI books, standard error is sometimes calculated based on the actual variation of the observations. Other times it is calculated as 1/sqrt(# observations). For example, the autocorrelations of residuls are calculated using this more static method that doesn’t account for the actual observations. Can anyone explain why there are multiple methods of calculating standard error? Thanks!

Standard error is standard deviation / sqrt(#obs), and you have seen some times just standard deviation. A possible explanation is that calculations made from a population only use standard deviation and those made from samples of a population use the standard error that is that standard deviation but corrected by the squared root of the number of observations of the sample. Compare this adjustment to the sd with the calculation of the sample variance; population variance uses N in the denominator and the sample variance uses N-1. They cannot be the exact same formula.
Note that in some cases time-series data can be a population data, for example: “The following table shows the net income of AAA and BBB companies that started operations on 1,990”. Those time-series of 25 observations are population observations, not samples.
Hope this could help!

 

[tickersu]

Harrogath wrote:
Standard error is standard deviation / sqrt(#obs), and you have seen some times just standard deviation. A possible explanation is that calculations made from a population only use standard deviation and those made from samples of a population use the standard error that is that standard deviation but corrected by the squared root of the number of observations of the sample. Compare this adjustment to the sd with the calculation of the sample variance; population variance uses N in the denominator and the sample variance uses N-1. They cannot be the exact same formula.
Note that in some cases time-series data can be a population data, for example: “The following table shows the net income of AAA and BBB companies that started operations on 1,990”. Those time-series of 25 observations are population observations, not samples.
Hope this could help!

Respectfully, I don’t think this is an accurate explanation. Standard error is not the same as the standard deviation (one deals with variation in the sampling distribution of an estimator and the other is individual variation in a population). You would use the sample standard deviation to describe variation in the individual observations, and the standard error would describe variation in the sample mean from the true mean, for example.
The standard error depends on the estimator you’re working with, and it is considered the standard deviation of the sampling distribution for a given estimator. For example, the standard error of X-bar is the SD divided by the square root of the sample size, but this doesn’t necessarily apply to other estimators. The standard error for a regression slope (say, beta i) is the SER divided by the square root of SST(Xi); SST(Xi) is defined as the total sum of squares for the i-th independent variable (sum squared deviations of Xi about its mean).
It really just boils down to different calculations that depend on the estimator….