Unformatted text preview: Var bi s N (0, 1) dβ
we do not know Var bi . We replace Var bi with Var bi .
β
β
r
dβ
β
Note that Stata reports se bi = Var bi so we compute
b
β
βi
ri
dβ
Var bi Lecture 4 (econometrics and simple linear regression) = b
βi βi se bi
β EMET2007/6007 s tn 2 13 th March 2013 12 / 50 Under assumptions SLR 1  SLR 6:
b s N β , Var b
βi
βi
i
b
βi βi β
se bi s N (0, 1) The estimators are normally distributed around the true parameters
with the variance that was derived earlier
The standardized estimators follow a standard normal distribution Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 13 / 50 Testing hypotheses about a population parameter
Under assumptions SLR 1  SLR 6: (tdistribution for standardized
estimators)
b
βi βi
s tn 2
β
se bi
If the standardization is done using the estimated standard deviation
(se bi = standard error), the normal distribution is replaced by a
β
tdistribution Note: The tdistribution is close to the standard normal distribution if
n 2 is large. Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 14 / 50 Null hypothesis (for more general hypotheses, see below)
H0 : β 1 = 0 The population parameter is equal to zero, i.e. after controlling for the
other independent variables, there is no e¤ect of x on y
t statistic (or t ratio)
tb =
β
1 b
βi se bi
β The tstatistic will be used to test the above null hypothesis. The farther
the estimated coe¢ cient is away from zero, the less likely it is that the null
hypothesis holds true. But what does “far” away from zero mean? Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 15 / 50 Distribution of the tstatistic if the null hypothesis is true
tb =
β
1 b
βi
β
se bi s tn 2 Goal: De…ne a rejection rule so that, if it is true, H0 is rejected only
with a small probability (= signi…cance level, e.g. 5%) Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 16 / 50 Testing against onesided alternatives (H0 : β1 0, H1 : β1 > 0) Reject the null hypothesis in favour of the alternative hypothesis if...
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 Two '13
 strachan
 Normal Distribution, Regression Analysis, Statistical hypothesis testing, Tn

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