12.1 Simple versus Joint Hypothesis Tests

We have already considered all there is to know about simple hypothesis tests.

\[H_0: \beta = 0 \quad \text{versus} \quad H_1: \beta \neq 0\]

With the established (one-sided or two-sided) hypotheses, we were able to calculate a test statistic given a nonarbitrary value of \(\beta\), calculate a p-value, and conclude. There is nothing more to it than that.

A simple hypothesis test follows the same constraints as how we interpret single coefficients: all else equal. In particular, when we conduct a simple hypothesis test, we must calculate a test statistic under the null while assuming that all other coefficients are unchanged. This might be fine under some circumstances, but what if we want to test the population values of multiple regression coefficients at the same time? Doing this requires going from simple hypothesis tests to joint hypothesis tests.

Joint hypothesis tests consider a stated null involving multiple PRF coefficients simultaneously. Consider the following general PRF:

\[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3 X_{3i} + \varepsilon_i\]

A simple hypothesis test such as

\[H_0: \beta_1 = 0 \quad \text{versus} \quad H_1: \beta_1 \neq 0\]

is conducted under the assumption that \(\beta_2\) and \(\beta_3\) are left to be whatever the data says they should be. In other words, a simple hypothesis test can only address a value for one coefficient at a time while being silent on all others.

A joint hypothesis states a null hypothesis that considers multiple PRF coefficients simultaneously. The statement in the null hypothesis can become quite sophisticated and test some very interesting statements.

For example, we can test if all population coefficients are equal to zero - which explicitly states that none of the independent variables are important simultaneously.

\[H_0: \beta_1 = \beta_2 = \beta_3 = 0 \quad \text{versus} \quad H_1: \beta_1 \neq 0,\; \beta_2 \neq 0,\; \text{or} \; \beta_3 \neq 0\]

We don’t have to be so extreme and test that just two of the three coefficients are simultaneously zero.

\[H_0: \beta_1 = \beta_3 = 0 \quad \text{versus} \quad H_1: \beta_1 \neq 0\; \text{or} \; \beta_3 \neq 0\]

If we have a specific theory in mind, we could also test if PRF coefficients are simultaneously equal to specific (nonzero) numbers.

\[H_0: \beta_1 = 1 \; \text{or} \; \beta_3 = 4 \quad \text{versus} \quad H_1: \beta_1 \neq 1\; \text{or} \; \beta_3 \neq 4\]

Finally, we can test if PRF coefficients behave according to some relative measures. Instead of stating in the null that coefficients are equal to some specific number, we can state that they are equal (or opposite) to each other or they behave according to some linear mathematical condition.

\[H_0: \beta_1 = -\beta_3 \quad \text{versus} \quad H_1: \beta_1 \neq -\beta_3\]

\[H_0: \beta_1 + \beta_3 = 1 \quad \text{versus} \quad H_1: \beta_1 + \beta_3 \neq 1\]

\[H_0: \beta_1 + 5\beta_3 = 3 \quad \text{versus} \quad H_1: \beta_1 + 5\beta_3 \neq 3\]

As long as you can state a hypothesis involving multiple PRF coefficients in a linear expression, then we can test the hypothesis using a joint test. There are an infinite number of possibilities, so it is best to give you a couple of concrete examples to establish just how powerful these tests can be.

Application

One chapter of my PhD dissertation concluded with a single joint hypothesis test. The topic I was researching was the Bank-Lending Channel of Monetary Policy Transmission, which is a bunch of jargon dealing with how commercial banks respond to changes in monetary policy established by the Federal Reserve. A 1992 paper written by Ben Bernanke and Alan Blinder established that aggregate bank lending volume responded to changes in monetary policy (identified as movements in the Federal Funds Rate).⁴⁶ A simplified version of their model (below) considers the movement in commercial bank lending as the dependent variable and the movement in the Fed Funds Rate (FFR) as the independent variable.

\[L_i = \beta_0 + \beta_1 FFR_i + \varepsilon_i\]

While this is a simplification of the model actually estimated, you can see that \(\beta_1\) will concisely capture the change in bank lending given an increase in the Fed Funds Rate.

\[\beta_1 = \frac{\Delta L_i}{\Delta FFR_i}\]

Since an increase in the Federal Funds Rate indicates a tightening of monetary policy, the authors proposed a simple hypothesis test to show that an increase in the FFR delivers a decrease in bank lending.

\[H_0:\beta_1 \geq 0 \quad \text{versus} \quad H_1:\beta_1 < 0\]

Their 1992 paper rejects the null hypothesis above, which gave them empirical evidence that bank lending responds to monetary policy changes. The bank lending channel was established!

My dissertation tested an implicit assumption of their model: symmetry.

\[\beta_1 = \frac{\Delta L_i}{\Delta FFR_i}\]

The interpretation of the slope of this regression works for both increases and decreases in the Fed Funds Rate. Assuming that \(\beta_1 <0\), a one-unit increase in the FFR will deliver an expected decline of \(\beta_1\) units of lending on average. However, it also states that a one-unit decrease in the FFR will deliver an expected increase of \(\beta_1\) units of lending on average. This symmetry is baked into the model. The only way we can explicitly test this assumption is to extend the model and perform a joint hypothesis test.

Suppose we separated the FFR variable into increases in the interest rate and decreases in the interest rate.

\[FFR_i^+ = FFR_i >0 \quad \text{(zero otherwise)}\] \[FFR_i^- = FFR_i <0 \quad \text{(zero otherwise)}\]

If we were to put both of these variables into a similar regression, then we could separate the change in lending from increases and decreases in the interest rate.

\[L_i = \beta_0 + \beta_1 FFR_i^+ + \beta_2 FFR_i^- + \varepsilon_i\]

\[\beta_1 = \frac{\Delta L_i}{\Delta FFR_i^+}, \quad \beta_2 = \frac{\Delta L_i}{\Delta FFR_i^-}\]

Notice that both \(\beta_1\) and \(\beta_2\) are still hypothesized to be negative numbers. However, the first model imposed the assumption that they were the same negative number while this model allows them to be different. We can therefore test the hypothesis that they are the same number by performing the following joint hypothesis:

\[H_0: \beta_1=\beta_2 \quad \text{versus} \quad H_1: \beta_1 \neq \beta_2\]

In case you were curious, the null hypothesis gets rejected and this provides evidence that the bank lending channel is indeed asymmetric. This implies that banks respond more to monetary tightenings than monetary expansions, which should make sense given all of the low amounts of bank lending in the post-global recession of 2008 despite interest rates being at all time lows.

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46. Bernanke, B., & Blinder, A. (1992). The Federal Funds Rate and the Channels of Monetary Transmission. The American Economic Review, 82(4), 901-921.↩︎