Raising the bar

Chapter 9. Hypothesis Testing

Unit 9.1. Concepts of hypothesis testing

Definition 9.1 (Hypotheses). In hypothesis testing we assume two possibilities, called hypotheses, about the state of nature. They must be mutually exclusive and collectively exhaustive events, like "the accused is guilty" and "the accused is innocent". One hypothesis is called a null hypothesis and denoted $latex H_0$ and the other is called an alternative hypothesis and denoted $latex H_a$.

Definition 9.2 (Type I and Type II errors). Rejecting the null, when it is true, we commit an error called Type I error. Rejecting the alternative, when it is true, leads to another error, called Type II error. Equivalently, we commit Type II error if we accept the null when it is wrong.

In practice, the true state of nature is never known, and we have to use statistical evidence to make judgment.

Table 9.1. States of nature and our decisions.

The choice of the hypotheses is up to us and depends on the circumstances. If we seek a strong evidence in favor of a particular outcome, that outcome should be designated as the alternative hypothesis, because, as comparison of Units 9.3 and 9.5 shows, it's easier to control the probability of Type I error.

Table 9.1 hides the probabilistic nature of the decision making process. This video shows what probabilities are hidden behind it.

Example. In a criminal trial the prosecutor wants to prove that the defendant is guilty, and that would be the alternative. For the defendant's lawyer the alternative is that the defendant is innocent.

Definition 9.3 (Cobb-Douglas production function). A Cobb-Douglas production function is defined by $latex f(K, L) = A K^\alpha L^\beta$ where $latex A, \alpha, \beta$ are constants and $latex K, L$ are independent variables (capital and labor). It is homogeneous of degree $latex \alpha + \beta$: $latex f(tK, tL) = A (tK)^\alpha (tL)^\beta = t^{\alpha + \beta} f(K, L)$. We say that this function exhibits increasing, constant or decreasing returns to scale if, respectively, $latex \alpha + \beta > 1$, $latex \alpha + \beta = 1$ or $latex \alpha + \beta < 1$.

The geometry of this function and the way of constructing its surface in Mathematica are explained here.

Example. When estimating the Cobb-Douglas production function in applied research we might want to test

$latex H_0: \alpha + \beta = 1 \textrm{ against } H_a: \alpha + \beta \ne 1. \qquad (\textrm{Eq. 9.1})$

Example. We believe that usually the supply curve is positively sloping. If we model it as a straight line, $latex p = a + bq$, then to prove that it is really positively sloped we test

$latex H_0: b \le 0 \textrm{ against } H_a: b > 0. \qquad (\textrm{Eq. 9.2})$

Example. The hypotheses in equations (9.1) and (9.2) are complementary. In NCT you can see examples of noncomplementary hypotheses, such as

$latex H_0: \alpha + \beta = 1 \textrm{ against } H_a: \alpha + \beta > 1. \qquad (\textrm{Eq. 9.3})$

The same testing procedures are applied in case of noncomplementary hypotheses, just because the statistical science is not perfect.

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