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- United-kingdom
- University of Bristol
- Economics
- Economics 20011
- Gregory Jolivet
- Ch9: T-tests And Confidence Intervals

Harry W.

• 4

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Requirements: t-tests

In order to conduct a test on the assumptions for one parameter of the linear model, we need the following ingredients:

- A null hypothesis H
_{0}and an alternative H_{1} - A test statistic T and its distribution
- A significance level α and a rejection region
- Alternatively, we can compute the p-value.

Method: t-tests

We want to test whether the parameter b is equal or not to a specific numerical value, denoted as b_{(0)}:

_{}

H_{0}: b = b_{(0)} and H_{1}: b ≠ b_{(0)}.

Applied econometrics are often testing whether the effect of a variable on another is significant, in which case the relevant null hypothesis is b = 0.

The test statistic for this test is called the t-statistic:

t = [ b̂ - b_{(0)} ] / σ_{b̂}

where b̂ is the OLS estimate of b and σ_{b̂} is the standard error of b̂.

In a large sample under the null hypothesis, the t-statistic has approximately a standard normal distribution.

For a given significance level α, and given the symmetry of the normal distribution, we can define the rejection region as follows:

Reject H_{0} if |t| > Φ^{-1}(1 - α/2)

Then, defining the rejection region as above means that:

P(t∈ reject. region | H_{0}) = α.

p-values

The p-value of a test is the *lowest significance level* at which we can reject H_{0}. Since the t-statistic t is standard normal, by symmetry the p-value equals:

p-value = 2Φ(-|t|).

We can then reject H_{0} at a significance level α iff p-value ≤ α.

Confidence Intervals

A 95% confidence interval for a coefficient b is an interval that contains the true value of b with probability 0.95. Equivalently, a value b* is the 95% confidence interval if we cannot reject H_{0}: b = b* at the 5% significance level.

We know that we cannot reject H_{0}: b = b* iff:

| (b̂ - b*) / σ̂_{b̂} | < Φ^{-1}(97.5%) ≈ 1.96, equivalently

b* ∈ [ b̂ - 1.96⋅σ̂_{b̂}, b̂ + 1.96⋅σ̂_{b̂}, ].

The 95% confidence interval for b is then:

[ b̂ - 1.96⋅σ̂_{b̂}, b̂ + 1.96⋅σ̂_{b̂}, ].

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