## 4.11 Inference / Hypothesis Testing

SW 3.2, 3.3

Often in statistics/econometrics, we have some theory that we would like to test. Pretty soon, we will be interested in testing a theory like: some economic policy had no effect on some outcome of interest.

In this section, we’ll focus on the relatively simple case of conducting inference on $$\mathbb{E}[Y]$$, but very similar arguments will apply when we try to start estimating more complicated things soon. Because we’re just focusing on $$\mathbb{E}[Y]$$, the examples in this section may be a somewhat trivial/uninteresting, but I want us to learn some mechanics, and then we’ll be able to apply these in more complicated situations.

Let’s start with defining some terms.

Null Hypothesis This is the hypothesis (or theory) that we want to test. We’ll often write it in the following way

$H_0 : \mathbb{E}[Y] = \mu_0$ where $$\mu_0$$ is some actual number (e.g., 0 or 10 or just whatever coincides with the theory you want to test).

Alternative Hypothesis This is what is true if $$H_0$$ is not. There are other possibilities, but I think the only alternative hypothesis that we will consider this semester is

$H_1 : \mathbb{E}[Y] \neq \mu_0$ i.e., that $$\mathbb{E}[Y]$$ is not equal to the particular value $$\mu_0$$.

The key conceptual issue is that, even if the null hypothesis is true, because we estimate $$\mathbb{E}[Y]$$ with a sample, it will generally be the case that $$\bar{Y} \neq \mu_0$$. This is just the nature of trying to estimate things with a sample.

What we are going to go for is essentially trying to tell the difference (or at least be able to weigh the evidence) regarding whether the difference between $$\bar{Y}$$ and $$\mu_0$$ can be fully explained by sampling variation or that the difference is “too big” to be explained by sampling variation. Things will start to get “mathy” in this section, but I think it is helpful to just hold this high-level idea in your head as we go along.

Next, let’s define the standard error of an estimator. Suppose that we know that our estimator is asymptotically normal so that

$\sqrt{n}(\hat{\theta} - \theta) \rightarrow N(0,V) \quad \textrm{as } n \rightarrow \infty$ Then, we define the standard error of $$\hat{\theta}$$ as

$\textrm{s.e.}(\hat{\theta}) := \frac{\sqrt{\hat{V}}}{\sqrt{n}}$ which is just the square root of the estimate of the asymptotic variance $$V$$ divided by the square root of the sample size. For example, in the case where we are trying to estimate $$\mathbb{E}[Y]$$, recall that, by the CLT, $$\sqrt{n}(\bar{Y} - \mathbb{E}[Y]) \rightarrow N(0,V)$$ where $$V=\mathrm{var}(Y)$$, so that

$\textrm{s.e.}(\bar{Y}) = \frac{\sqrt{\widehat{\mathrm{var}}(Y)}}{\sqrt{n}}$ where $$\widehat{\mathrm{var}}(Y)$$ is just an estimate of the variance of $$Y$$, i.e., just run var(Y) in R.

Over the next few sections, we are going to consider several different way to conduct inference (i.e., weigh the evidence) about some theory (i.e., the null hypothesis) using the data that we have. For all of the approaches that we consider below, the key ingredients are going to an estimate of the parameter of interest (e.g., $$\bar{Y}$$), the value of $$\mu_0$$ coming from the null hypothesis, and the standard error of the estimator.