8
One-Sided Alternatives (cont)
Having picked a significance level, a, we look up the (1 – a)th percentile in a t distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null
Under the CLM assumptions, conditional on the sample values of the independent variable s
bˆ j ~ Normal b j ,Var bˆ j , so that
bˆ j b j sd bˆ j ~ Normal 0,1
7
t Test: One-Sided Alternatives
Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided
because we have to estimate s 2by sˆ 2
Note the degrees of freedom : n k 1
5
The t Test (cont)
Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis
9
One-Sided Alternatives (cont)
yi = b0 + b1xi1 + … + bkxik + ui
H0: bj = 0
H1ห้องสมุดไป่ตู้ bj > 0
Fail to reject
H1: bj > 0 and H1: bj < 0 are one-sided H1: bj 0 is a two-sided alternative
If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%
"the"t statistic for bˆj :tbˆ j bˆ j se bˆ j
We will then useour t statistic along with a rejection rule to determine whether ot accept thenull hypothesis, H0
mean and variance s2: u ~ Normal(0,s2)
1
CLM Assumptions (cont)
Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows
bˆj is distributed normally becauseit
is a linear combinatio n of the errors
4
The t Test
Under the CLM assumptions
bˆ j b j se bˆ j ~ tnk1
Note this is a t distributi on (vs normal)
For example, H0: bj=0
If accept null, then accept that xj has no effect on y, controlling for other x’s
6
The t Test (cont)
To performour test we first need to form
2
The homoskedastic normal distribution with a single explanatory variable
y
f(y|x)
.
.
E(y|x) = b0 + b1x
Normal distributions
x1
x2
3
Normal Sampling Distributions
Assumptions of the Classical Linear Model (CLM)
So far, we know that given the GaussMarkov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2,…, xk and u is normally distributed with zero
y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality