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# Hypothesis Testing With Multivariate Regressions Using One-Sample t-Tests

## Hypothesis Testing With Multivariate Regressions Using One-Sample t-Tests

In this article we consider the Okun's Law model and results from the article "How to Do a Painless Multivariate Econometrics Project". One sample t-tests will be introduced and used in order to see if the theory matches the data. In that article we developed our theory:

For this econometrics project, I'm going to calculate the marginal propensity to consume (MPC) in the United States. (If you're more interested in doing a simpler, univariate econometrics project, please see "How to Do a Painless Econometrics Project" The marginal propensity to consume is defined as how much an agent spends when given an extra dollar from an additional dollar's personal disposable income. My theory is that consumers keep a set amount of money aside for investment and emergency, and spend the rest of their disposable income on consumption goods. Therefore my null hypothesis is that MPC = 1.

I'm also interested in seeing how changes in the prime rate influence consumption habits. Many believe that when the interest rate rises, people save more and spend less. If this is true, we should expect that there is a negative relationship between interest rates such as the prime rate, and consumption. My theory, however, is that there is no link between the two, so all else being equal, we should see no change in the level of the propensity to consume as the prime rate changes.

In order to test my hypotheses, I need to create an econometric model. First we'll define our variables:

Yt is the nominal personal consumption expenditure (PCE) in the United States.
X2t is the nominal disposable after-tax income in the United States. X3t is the prime rate in the U.S.

Our model is then:

### Yt = b1 + b2*X2t + b3*X3t

Where b1, b2, and b3 are the parameters we will be estimating via linear regression. These parameters represent the following:

• b1 is the level of PCE when nominal disposable after-tax income (X2t) and the prime rate (X3t) are both zero. We do not have a theory about what the "true" value of this parameter should be, as it holds little interest to us.

• b2 represents the amount PCE rises when the nominal disposable after-tax income in the United States rises by a dollar. Note that this is the definition of the marginal propensity to consume (MPC), so b2 is simply the MPC. Our theory is that MPC = 1, so our null hypothesis for this parameter is b2 = 1.

• b3 represents the amount PCE rises when the prime rate increases by a full percent (say from 4% to 5% or from 8% to 9%). Our theory is that changes in the prime rate do not influence consumption habits, so our null hypothesis for this parameter is b2 = 0.
So we will be comparing the results of our model:

### Yt = b1 + b2*X2t + b3*X3t

to the hypothesized relationship:

### Yt = b1 + 1*X2t + 0*X3t

where b1 is a value that does not particularly interest us.
Using Microsoft Excel, we calculated the parameters b1, b2, b3. Now we need to see if those parameters match our theory, which was that b2 = 1 and b3 = 0. We are not interested in the value of b1 so we will not be running any tests on it. Before we can do that, we need to jot down some figures that Excel gave us. After running the regression, you should have gotten the following values:

### Observations

Number of Observations = 179

### X Variable 1

Coefficient b2 = 0.9370
Standard Error se2 = 0.0019
t Stat t2 = 488.1184
P-value p2 = 0.0000

### X Variable 2

Coefficient b3 = -13.7194
Standard Error se3 = 1.4186
t Stat t3 = -9.6708
P-value p3 = 0.0000
In the next section we'll look at hypothesis testing and we'll see if our data matches our theory.