01:220:322 Chap.6 Inference with t Tests, F Tests and Intervals
Inference with t Tests, F Tests and Intervals
One coefficient at a time
The t statistic is the estimate minus the hypothesised value divided by the standard error. Under MLR.1 to MLR.4, with robust standard errors, it is approximately standard normal in large samples, so the course uses normal critical values: 1.645, 1.96 and 2.58 are the workhorses.
Hypotheses are always statements about population parameters.
p-values and intervals
The p-value is the smallest significance level at which the null is rejected. The confidence interval is the estimate plus or minus a critical value times the standard error; a 95% interval contains every value a two-sided 5% test would not reject.
The 95% describes the procedure over repeated samples, not a probability about one computed interval.
Significance versus importance
A tiny effect can be statistically significant in a huge sample, and a large effect can be insignificant when it is noisy.
Good answers report size, sign and precision together, and say fail to reject rather than accept.
Linear combinations
Testing whether two coefficients are equal needs the standard error of their difference, which includes their covariance.
Stata's lincom command reports it directly.
Joint tests
When a claim sets several coefficients to zero, for example that a variable entering with its square has no effect, separate t tests are not valid. The course uses the software-computed robust F statistic from the test command and compares it with an F critical value that depends on the number of restrictions.
Coefficients that are individually insignificant can be jointly significant when the variables move together.
Reading the Stata table
The column labelled with the absolute t shows the two-sided p-value for the null that each coefficient is zero, and the last two columns give the 95% interval.
For a non-zero null, such as an elasticity equal to one, compute the t statistic yourself or use the test or lincom commands after the regression.
Joint tests you will meet
A variable that enters with its square, a dummy that enters on its own and in an interaction, or a block of related variables that move together all call for a joint test.
Write the null as every relevant coefficient equal to zero and the alternative as the null being false.
What this chapter covers
- 01
Critical values for t statistics
- 02
One-sided and two-sided alternatives
- 03
The p-value
- 04
Confidence intervals and their interpretation
- 05
Testing a difference with lincom
- 06
Joint exclusion restrictions and the F test
Worked example · free
Three ways to the same decision
- 2t = 0.084/0.030 = 2.80, which exceeds 1.96, so reject.
- 2Two-sided p = 2P(Z > 2.80) = 0.005, below 0.05, so reject.
- 295% interval: 0.084 plus or minus 0.059 gives 0.025 to 0.143, which excludes zero, so reject.
Key terms
- T Statistic
- The estimate minus its hypothesised value, divided by its standard error.
- Confidence Interval
- A range built from a procedure that covers the true parameter in a stated share of repeated samples.
- Joint Hypothesis
- A null that restricts several parameters at the same time.
- F Statistic
- The test statistic for several restrictions at once, compared with an F critical value.
- Economic Significance
- Whether an estimated effect is large enough to matter in practice, judged from its size.
- Exclusion Restriction
- A null hypothesis that a set of regressors has zero coefficients and can be dropped.
Inference with t Tests, F Tests and Intervals FAQ
Why can I not test two coefficients with two separate t tests?
Each t test holds the other coefficient unrestricted, so together they do not test the joint claim, and correlated regressors can make both look insignificant. The F test evaluates the restrictions at once.
What does a 95% confidence interval really mean?
The method produces intervals that contain the true coefficient in 95% of repeated samples. Any single interval either contains the fixed parameter or does not, so avoid probability language about it.
When is a one-sided test appropriate?
When the question itself predicts a direction, such as a claim that an effect is positive. Halve the two-sided p-value only if the estimate has the predicted sign.
What does fail to reject mean in practice?
The data are consistent with the null, but they are also consistent with other values nearby. A wide interval that includes zero says the effect is imprecisely estimated, not that it is zero.
When is F equal to t squared?
When the F test has a single restriction. Testing one coefficient equal to zero with an F statistic gives exactly the square of its t statistic, and the decisions from the two-sided t test and the F test agree.
Exam move
Write the three decision routes for every coefficient you test, then practise writing joint nulls for models with quadratics and interactions.
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