ECON 61001: Hypothesis Testing
Alastair R. Hall
The University of Manchester
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Examples of parameter restrictions in econometric models
assetreturns:R−Rf =β0(Rm−Rf)+error
β0 < 0 ⇒ stock is inversely related to market index;
β0 = 1 ⇒ stock moves in line with market index. returns to education (dropping exp to simplify and
reparameterizing):
ln(w) = β0,1 + β0,2 ∗ ed + β0,3 ∗ D + β0,4 ∗ (D ∗ ed) + error
D = 1 is female and zero else
β0,3 = 0, β0,4 = 0 ⇒ no difference between men and women.
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Examples of parameter restrictions in econometric models
aggregate production function:
lnQ = β0,1 + β0,2 ∗ln(L) + β0,3 ∗ln(K) + error
< β0,2 + β0,3 =
>
1 ⇒
diminishing
constant returns to scale
increasing
Alastair R. Hall
ECON 61001: Hypothesis Testing
3 / 13
Hypothesis testing
A statistical hypothesis is a conjecture about the distribution of one or more random variables.
The classical theory of hypothesis testing provides a framework for deciding whether a particular hypothesis is correct.
Within this framework, there are only two possible decisions: the hypothesis is true or it is not. A decision procedure for such a problem is called a test.
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Basic framework
Assume: our hypothesis involves θ, the parameter vector indexing distribution of V , and Θ denotes the parameter space with
Θ ⊂ Rp.
Divide Θ into two mutually exclusive and exhaustive parts:
Θ0 = {θ : such that the hypothesis is true},
Θ1 = {θ : such that the hypothesis is false}.
Using this partition, we can state the object as being to test the
null hypothesis,
against the alternative hypothesis,
H0 : θ∈Θ0 H1 : θ ∈ Θ1.
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Basic framework
To facilitate the choice between H0 and H1, collect sample of T observations.
Base inference on some function of this sample, known as a test statistic; denoted by ST .
In a test procedure, divide sample space of ST into two mutually exclusive and exhaustive regions, R0 and R1, such that
ST ∈ R0 ⇒ H0 is accepted or rather not rejected ST ∈ R1 ⇒ H0 is rejected in favour of H1
where
R0 is known as the acceptance region.
R1 is known as the rejection region or the critical region.
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Basic framework
Choice of R0 and R1 based on possible outcomes of test.
Decision may be correct, but can also make error:
a Type I error, in which H0 is rejected when it is true;
a Type II error, in which H0 is not rejected when its false.
Ideally we would make both errors as small as possible but there is a tension between them.
So limit P(Type I error) to be no larger than: 0.1, 0.05 or 0.01.
To do this, we need to know the distribution of the test statistic under H0.
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Size and significance level of a test
Let Pθ ( · ) denote the probability of the event in parentheses if the parameter vector takes the value θ.
Define α(θ) = Pθ(R1) that is, α(θ) describes the probability of a type I error for values of θ that satisfy H0.
The quantity supθ∈Θ0 α(θ) is known as the size of the test, and equals the maximal probability of a type I error.
To implement test, specify α such that α(θ) ≤ α for all θ ∈ Θ0.
α is an upper bound on the probability of a type one error. 100α% is known as the significance level of the test.
In general, the size and α coincide (but they need not.)
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Example 2.8 in Lecture Notes
Suppose that:
vt ∼ IN(θ,σ2), t = 1,2,…T; assume σ2 known.
wish to test H0 : θ = 0 versus H1 : θ ̸= 0.
can show v ̄T ∼ N(θ,σ2/T), where v ̄T = T−1 Tt=1 vt.
→ test statistic is:
τT = σ2/T
UnderH0,τT ∼N(0,1).
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v ̄T
Example 2.8 in Lecture Notes
Reject H0 if τT is sufficiently far away from zero.
So the decision rule takes form: reject H0 if |τT | > c for some
constant c.
Choose c to control P(Type I error).
Θ0 ={0}andsosupθ∈Θ0α(θ)=P(|τT| > cθ=0). ⇒ c = z1−α/2.
⇒(forexample): rejectH0 : θ=0infavourofH1 : θ̸=0atthe 5% significance level if |τT | > 1.96.
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Example 2.8 in Lecture Notes: Acceptance and rejection regions
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Observed significance level aka the p-value
The observed significance level or p-value is the significance level for which the test statistic lies on the boundary of the acceptance and rejection regions.
In our example: p-value = δ such that |τT | = z1−δ/2. Interpretation of p-value: we reject H0 at all significance levels
100α% for which α > p − value.
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Further discussion of the topics in this podcast can be found in Section 2.8.1 of the Lecture Notes.
Same source discusses application of framework to hypotheses of the form:
H0 : θ ≤ 0 versus H1 : θ > 0 H0 : θ ≥ 0 versus H1 : θ < 0.
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