7 statistical tests after OLS regression
vad indikerar en small SE
att estimator is precise and we should believe that the causal effect is pretty close to the true value
beskriv de olika typerna av errors vid hypotestesting
type 1: rejecting H0 when it is true
type 2: not reject when H0 is false
type 1 är värst att göra. om vi gör typ-2 fel så lär vi oss inget och det är inte så farligt som att göra typ-1 fel
vad äsger vi när vi inte can reject?
we dont have enough evidence to reject H0
sant eller falskt:
we can only learn from rejecting in statistical testing
sant
vad säger P(reject| H0)
probability of rejecting when H0 is true, alltså probability of type 1 error.
detta kallas också: size of the test
size of the test kan delas in i empirical och nominal, förklara dessa
nominal: is the intended size of the test. om ett test has a nominal size of 10%, then it is constructed in a way that is supposed to guarantee that it has an empirical size of 10%. but this may not work put in practice då some tests only work well under really large samples
=> smaller sample size can result in different levels of empirical and nominal
empirical: is the size of the test that we experiance when we apply the test in real life.
so a nominal size of 10% is the same as saying that the peobability of doing a type 1 error is 10%. if we have a larger size (empirical) if the sample size is smaller, we also have a higher probability of making a type 1 error
vad säger P(not reject| H1)
the peobability of not rejecting when H0 is false. it is the probability of making a type 2 error. om denna är låg så är det mer sannolikt att vi reject H0 when its false, which we want. "we are more likely to find evidence that they are cheating, than we are to find insufficient evidence." detta kallas för ett POWERFULL test when we are good at detecting a false H0.
1 - power of test = P(not reject| H1) = P(type 2 error)
vad säger H0: B1 = 0
there is no causal effect of x1 on Y
what is the formula for test-statistic and what defines a large vs. small t-value?
T = (B1^ - B1H0) / SE(B1^)
if we are estimating a large value for B1^ thab we hav hypothesized B1H0, then T will be large. Small if it is tje other way around
vad är ca? vad är formeln i stata?
critical value. det vi jämför emot ifall vi ska reject or not with our observed t-value.
we dont want ca to be too small, since we will reject to often and invreases the risk of type 1 error.
not too large either då vi inte kommer kunns rejecta och inte lära oss något.
invnormal(0,974) om a= 0,05
what is the formula for probability with a two-sided t-test?
1 - a/2
what is the risk if we choose a large alfa for the nominal size of test?
making a type 1 error, since we reject when p-value is lower that alfa. but i we choose a very low alfa, then there is an incressef risk of type 2 error and we have a less powerfull test.
sant eller falskt:
när the power of a test is strong, there is less probability of making a type 2 error
sant, we are more likely to reject when H0 is false, which is correct.
how can we increase the power of a test?
decrease the sd(B1^), whixh will make B1^ a more precise estimate which is closer to the true value => more likely to reject H0 when it is false and less probability of making a type 2 error.
how di we decrease the standard error?
by looking at the formuka fåor variance. incresing n will decrase var which is the same as decreasing sd
choosing a size of a test equal to 10%, we are accepting..
the probability of type 1 error of 10%
how do we test a joint test?
with an F-test with F-distribution
what says a large F?
indicates that our estimated betad are very different from what we have hypothesized
when do we reject in an F-test?
when p-value is smaller than alfa.
when F-value is smaller than ca
how do we calculate an F-test? (F-distribution) is we have B1=B2=0 and n = 40
F(df1, df2)
df1 is the degrees of freedom. it is the number of restrictions
df2 is the number of observations minus the number of B in our model.
F(2, 37)
what is the fomtula in stata for the critacal value in an F-test?
invF(df1,df1,0,95) if we have alfa=0,05
how many restictions?
H0: B1 = B2 = 0
H0: B1-B2=0 dvs B1=B2
2
1 det går inte att sätta lika med 0 på höger sidan eftersom B1 och B2 kanske har värdet 3 och inte 0