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## 4.2 One Sided Tests

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4.2 One Sided Tests

-Before we construct a rule for rejecting H0, we need to pick an ALTERNATE HYPOTHESIS

-an example of a ONE SIDED ALTERNATIVE would be:

-Which technically expands the null hypothesis to

-Which means we don’t care about negative values of Bj

-This can be due to introspection or economic theory

4.2 One Sided Tests

-If we pick an α (level of significance) of 5%, we are willing to reject H0 when it is true 5% of the time

-in order to reject H0, we need a “sufficiently large” positive t value

-a one sided test with α=0.05 would leave 5% in the right tail with n-k-1 degrees of freedom

-our rejection rule becomes reject H0 if:

-where t* is our CRITICAL VALUE

4.2 One Sided Example

-Take the following regression where we are interested in testing whether Pepsi consumption has a +’ve effect on coolness:

-We therefore have the following hypotheses:

4.2 One Sided Example

-We then construct our test statistic:

-With degrees of freedom=43-3=40 and a 1% significance level, from a t table we find that our critical t, t*=2.423

-We therefore do not reject H0 at a 1% level of significance; Pepsi has no positive effect on coolness at the 1% significance level in our study

4.2 One Sided Tests

-From looking at a t table, we see that as the significance level falls, t* increases

-We therefore need a bigger test t statistic in order to reject H0 (the hypothesis that a variable is not significant)

-as degrees of freedom increase, the t distribution approximates the normal distribution

-after df=120, one can in practice use normal critical values

4.2 One Sided Tests

-The other one-sided test we can conduct is:

-Which technically expands the null hypothesis to

-Here we don’t care about positive values of Bj

-We now reject H0 if:

4.2 Two Sided Tests

-It is important to decide the nature of our one-sided test BEFORE running our regression

-It would be improper to base our alternative on whether Bjhat is positive or negative

-A way to avoid this and a more general test is a two-tailed (or two sided) test

-Two sided tests work well when a variable’s sign isn’t determined by theory or common sense

-Our alternate hypothesis now becomes:

4.2 Two Sided Tests

-For a two sided test, we reject H0 if:

-In finding our t*, since we now have two rejection regions, α/2 will fit into each tail

-For example, if α=0.05, we will have 2.5% in each tail

-When we reject H0, we say that “xj is statistically significant at the ()% level”

-When we do not reject H0, we say that “xj is statistically insignificant at the ()% level”

4.2 Two Sided Example

-Going back to our Pepsi example, we instead ask if Pepsi has ANY effect (positive or negative) on coolness:

-We therefore have the following hypotheses:

4.2 Two Sided Example

-We then construct our same test statistic as before:

-With degrees of freedom=43-3=40 and a 1% significance level, from a t table we find that our critical t, t*=2.704 (bigger than before)

-We therefore reject H0 at a 1% level of significance; Pepsi has an effect on coolness at the 1% significance level in our study

4.2 Other Simple Tests

-We sometimes want to test whether Bj is equal to a certain number, such as:

-Which makes the alternate hypothesis:

-Which changes our test t statistic to (t* is found the same from tables):

4.2 Another Pepsi Example

-Foolishly, we forget that coolness is a log-log model (see GH 2009), making each slope parameter the partial elasticity:

-wanting to see if Pepsi has a unit partial elasticity, we have the following hypotheses:

4.2 Two Sided Example

-We then construct our new test statistic:

-With degrees of freedom=43-3=40 and a 1% significance level, from a t table we find that our critical t, t*=2.704 (same as 2-tailed)

-Therefore don’t reject H0 at a 1% level of significance; Pepsi may have unit partial elasticity at the 1% significance level

4.2 p-values

-So far we have taken a CLASSICAL approach to hypothesis tests

-choosing an α ahead of time can skew our results

-if a variable is insignificant at 1%, but significant at 5%, it is still highly significant!

-we can instead ask: “given the observed value of the t statistic, what is the SMALLEST significance level at which the null hypothesis would be rejected? This level is known as the P-VALUE.”

4.2 p-values

-P-VALUES relate to probabilities and are therefore always between zero and 1

-regression packages (such as Shazam) usually report p-values for the null hypothesis Bj=0

-testing commands can give other p-values of the form:

-ie: P-values are the areas in the tails

4.2 p-values

-a small p-value argues for rejecting the null hypothesis

-a large p-value argues for not rejecting the null hypothesis

-once a level of significance (α) has been chosen, reject H0 if:

-regression packages generally list the p-value for a two-tailed test.

-for a one-tailed test, simply use p/2

4.2 Statistical Mumbo-Jumbo

-If we reject H0, we can state that “Ho is rejected at a ()% level of significance’

-If we do not reject H0, we CANNOT say that “H0 is accepted at a ()% level of significance”

-while a null hypothesis of H0:Bj=2 may be not rejected, a similar H0:Bj=2.2 may also not be rejected

-Bj cannot equal both 2 and 2.2

-we can conclude a certain number ISN’T valid, but we can’t conclude on ONE valid number

4.2 Economic and Statistical Significance

-STATISTICAL significance depends on the value of t

-ECONOMIC significance depends upon the size of Bj

-since we know that t depends on the size and standard error of Bj:

-a coefficient may test significant due to a very small se(Bj); a STATISTICALLY significant coefficient may be too small to be economically significant

4.2 Insignificant Example

-Theoretically, World Peace (WP) can only be achieved if House (H) episodes resume and people eat more chicken (C):

-although both House and Chicken would test as being significant variables (their standard errors are very small compared to their values), B3 is so small chicken has a very small impact

-you’d have to eat so much chicken to cause world peace it’s ECONOMICALLY insignificant

4.2 Significance and Large Samples

-As sample size increases, standard errors also tend to increase

-coefficients tend to be more statistically significant in large samples

-some researchers argue for smaller significance levels in large samples and larger significance levels in small samples

-this can often be due to an agenda

-in large samples, it is important to examine the MAGNITUDE of any statistically significant variables.

4.2 Multicollinearity Strikes Back

-Recall that large standard errors can also be caused by Multicollinearity

-This can cause small t stats and insignificance

-This can be fought by

- Collecting more data
- Dropping or combining (preferred) independent variables

4.2 3 Easy (honest) steps for tests

When testing, follow these 3 easy steps:

- If a variable is significant, examine its coefficient’s magnitude and explain its impact (this may be complicated if not linear)
- If a variable is insignificant at usual levels, check it’s p-value to see if some case for significance can be made
- If a variable has the “wrong” sign, ask why – are there omitted variables or other issues?

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