Appunti Econometria

NULL HYPOTHESIS H0: This is typically the hypothesis that there is no eMect or no

relationship. For example, you might hypothesize that a parameter (such as beta1 in a

regression model) is equal to zero, implying that the predictor has no eMect on the outcome

variable.

ALTERNATIVE HYPOTHESIS H1: This represents the hypothesis that contradicts the null

hypothesis. The exact form of H1 depends on the direction or nature of the eMect you are

testing for.

Two-Tailed Test (Non-Directional Alternative):

à

If we are interested in whether the parameter beta1 is diMerent from zero in either direction

(i.e., it could be either positive or negative), we would define the alternative hypothesis as:

This is a two-tailed test because we are testing for the possibility that beta1 is either less

than zero or greater than zero. This test is used when we do not have a strong directional

expectation about the sign of the eMect, but we are interested in whether it is diMerent from

zero at all.

One-Tailed Test (Directional Alternative):

à

If we have a specific hypothesis about the direction of the eMect (e.g., we expect the

parameter to be positive or negative), then the alternative hypothesis would reflect that.

If you hypothesize that beta1 is greater than zero, the alternative hypothesis would be:

If you hypothesize that beta1 is less than zero, the alternative hypothesis would be:

OUR GENERAL EXAMPLE: Testing Alpha

Null Hypothesis:

Alternative Hypothesis:

Thus, ALPHA CAN BE NEGATIVE, and there’s no reason to automatically assume that the only

alternative is that alpha is positive unless our research or theory specifically suggests that.

If there’s no specific expectation about the direction of alpha, the two-tailed test would be the

correct choice.

OUR IN-CLASS EXAMPLE:

B) When the matrices are ORTHOGONAL:

Another case where a relation like XY = Y’X’ might hold is when the matrices are orthogonal.

Two orthogonal matrices X and Y satisfy the condition:

X’X = I and Y’Y = I

where I is the identity matrix.

When cannot we do XY = Y’X’?

In general, XY ≠ Y’X’ when:

If the matrices are not symmetric, the relation XY = Y’X’ will generally not hold.

If the matrices X and Y are not square or do not have the same dimensions, the multiplication

XY and Y’X’ might not even be defined.

Matrix Multiplication is not commutative, meaning that in general XY ≠ YX, so we can’t do

XY = Y’X’ without specific conditions.

THIS IS WHAT WE NEED: