Useful Statistical Distributions for Econometrics Econometrics is usually concerned with the estimation of equations of the form: The normal distribution is useful because it is often reasonable to assume that the errors are normally distributed. However, a number of other statistical distributions are also of use. These include the chi-squared, F and t disributions.

The Chi-Squared Distribution The chi-squared distribution is derived from the normal distribution where Z j are i.i.d random variables. V follows a Chi-squared distribution with k degrees of freedom. The Chi-squared distribution is useful in any situation where we are interested in the squared values of a random variable e.g. (1) When examining the variance of the residuals from a regression model. (2) When constructing tests based on the residual sum of squares from a regression model.

For k = 1 or k = 2 the PDF of the chi-squared distribution is downward sloping. Note that the Chi-squared distribution has E(X)=k and V(X) = 2k.

For any value of k greater than 2, the PDF of the chi-squared distribution has the shape illustrated below. The PDF takes the value 0 for x=0, reaches a single peak for some value of x >0 and declines asymptotically to 0 as x becomes large. The distribution is skewed to the left.

As the degrees of freedom increases, the skewness of the PDF becomes less marked and the distribution looks more like the normal

k 1 and k 2 are the degrees of freedom for the F-statistic. Note that the order of these degrees of freedom is important. A variable follows an F-distribution if it is constructed as the ratio of two Chi-squared distributed variables each of which is divided by its degrees of freedom. The F-Distribution

The PDF for the F-distribution has a similar shape to that of the Chi-squared distribution. k 1 = 1 or 2k 1 >2

Student’s t distribution Consider a random variable which is constructed as the ratio of a standard normal random variable to a Chi-squared random variable with degrees of freedom = k. A variable such as this is said to follow Student’s t distribution with k degrees of freedom. The t distribution is useful in many situations in applied econometrics – particularly when we wish to construct hypothesis tests for regression coefficients.

The PDF of the t-distribution has a similar shape to that of the standard normal distribution. The t-distribution has ‘fatter tails’ relative to the normal i.e. larger values of x (in absolute terms) are more probable.