T-Tests: Hypothesis Testing and Interval Estimation

The t-test is a common tool in OLS for hypothesis testing, particularly useful when dealing with a normal random variable with an estimated standard error.
The rule of thumb: if the absolute value of the t-statistic is greater than 2, reject the null hypothesis.
- |t| > 2 \implies \text{Reject null hypothesis}
(se(B_1)) denotes the standard error of the coefficient estimate.

We calculate $B_1$ and assess its surprisingness if the null hypothesis were true.
The scale of $B1$ is crucial. For example, in a presidential election model, if $B1 = 2.29$ , its significance depends on its standard error.
A large $B_1$ is unlikely if its standard error is small (e.g., 0.52), but plausible if the standard error is large (e.g., 2.0).

The key is the size of the $B_1$ coefficient relative to its standard error.
We are unlikely to observe a $B_1$ coefficient much larger than its standard error.
Test statistic: The estimated coefficient divided by the estimated standard deviation:
- $\text{Test Statistic} = \frac{B1}{se(B1)}$
The test statistic indicates how many standard errors the estimated coefficient is from zero. For instance, if $B1 = 6$ and $se(B1) = 2$ , the test statistic is 3.

Dividing $B1$ by its standard error addresses the scale issue but introduces a new challenge: understanding the distribution of $\frac{B1}{se(B_1)}$ .
While $B1$ is normally distributed, $se(B1)$ is also a random variable dependent on the estimated $B_1$ .
The distribution of $\frac{B1}{se(B1)}$ follows a t-distribution, as discovered by the Guinness Brewery in the early 20th century.

The t-distribution is bell-shaped, like a normal distribution but with “fatter tails”.
Fatter tails mean that extreme values have higher probabilities compared to the normal distribution.
The extent of these tails depends on the sample size; larger samples result in tails that resemble the normal distribution more closely.
The t-distribution addresses the caution needed when $se(B_1)$ might be underestimated, especially with small datasets.
The specific shape depends on the degrees of freedom ([df]), calculated as the sample size minus the number of estimated parameters.
- For a bivariate OLS model (estimating $B0$ and $B1$ ), with a sample size of 50, the degrees of freedom are $50 - 2 = 48$ .

With lower degrees of freedom (e.g., 2), the t-distribution has heavier tails than the normal distribution.
As degrees of freedom increase (e.g., 5), the t-distribution starts to resemble the normal distribution more closely.
At high degrees of freedom (e.g., 50), the t-distribution is virtually indistinguishable from the normal distribution.

A critical value is a threshold; if the test statistic exceeds it, we reject the null hypothesis.
The decision rule depends on the alternative hypothesis.
- Two-sided alternative: Reject the null if the absolute value of the t-statistic exceeds the critical value.
 - HA: B1 \neq 0 \implies \text{Reject } H0 \text{ if } |\frac{B1}{se(B1)}| > \text{critical value}
- One-sided alternative ((B_1 > 0)): Reject the null if the t-statistic exceeds the critical value.
 - HA: B1 > 0 \implies \text{Reject } H0 \text{ if } \frac{B1}{se(B1)} > \text{critical value}
- One-sided alternative ((B_1 < 0)): Reject the null if the t-statistic is less than -1 times the critical value.
 - HA: B1 < 0 \implies \text{Reject } H0 \text{ if } \frac{B1}{se(B1)} < -1 \times \text{critical value}

Example using adult height as a coefficient.
T statistic from a homoscedastic model: 4.225 ((B_1) is 4.225 standard deviations from zero).
T statistic from a heteroscedastic model: 4.325 (essentially the same).
To determine statistical significance, compare the t-statistic to a critical value.
For a two-sided test with $\alpha = 0.05$ and degrees of freedom = 1,908, the critical value is approximately 1.96.
Since 4.225 > 1.96, we reject the null hypothesis.

T-tests can be extended to null hypotheses where the value is not zero.
If the null hypothesis is $H0: B1 = 7$ versus $HA: B1 \neq 7$ , we check how many standard deviations $B_1$ is from 7.
- $\text{Test} = \frac{B1 - 7}{se(B1)}$
More generally, to test $H0: B1 = B_{Null}$ , use:
- $\text{Test} = \frac{B1 - B{Null}}{se(B_1)}$

William Sealy Gosset (pen name "Student") developed the t-test in 1908 while working for Guinness Brewery in Dublin.
The standard error of $B_1$ follows a $\chi^2$ distribution, and the ratio of a normally distributed random variable and a $\chi^2$ random variable follows a t-distribution.