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Econometrics and the t Distribution

The t distribution is used quite a bit in econometrics. You probably used the t distribution extensively when dealing with means in your statistics class, but in econometrics you also use it for regression coefficients. Before you find out how that works, you should know how the t distribution is derived and its basic properties.

The t distribution is derived from a ratio of a standard normal random variable and the square root of a chi-squared random variable. It’s bell-shaped, symmetrical around zero, and approaches a normal distribution, as the degrees of freedom (number of observations) increases.

The figure shows how the t distribution changes with degrees of freedom. The df1, df2, and df3 indicate increasing degrees of freedom (or observations). As the sample size approaches the population size, the t distribution approaches the standard normal.

image0.jpg

If you have a normally distributed sample mean, such as

image1.jpg

then you can convert it to a standard normal by

image2.jpg

Similarly, if you have a squared normal, such as the sample variance

image3.jpg

you can convert it to a chi-squared by

image4.png

When you take the ratio of the standard normal to the square root of your chi-squared distribution, you end up with a t distribution:

image5.jpg
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