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

First we'll consider our hypothesis that the intercept variable equals one. The idea behind this is explained quite well in Gujarati's Essentials of Econometrics. On page 105 Gujarati describes hypothesis testing:

"[S]uppose we hypothesize that the true B₁ takes a particular numerical value, e.g., B₁ = 1. Our task now is to "test" this hypothesis."

"In the language of hypothesis testing a hypothesis such as B₁ = 1 is called the null hypothesis and is generally denoted by the symbol H₀. Thus H₀: B₁ = 1. The null hypothesis is usually tested against an alternative hypothesis, denoted by the symbol H₁. The alternative hypothesis can take one of three forms:

H₁: B₁ > 1, which is called a one-sided alternative hypothesis, or
H₁: B₁ < 1, also a one-sided alternative hypothesis, or
H₁: B₁ not equal 1, which is called a two-sided alternative hypothesis. That is the true value is either greater or less than 1."

In the above I've substituted in our hypothesis for Gujarati's to make it easier to follow. In our case we want a two-sided alternative hypothesis, as we're interested in knowing if B₁ is equal to 1 or not equal to 1.

The first thing we need to do to test our hypothesis is to calculate at t-Test statistic. The theory behind the statistic is beyond the scope of this article. Essentially what we are doing is calculating a statistic which can be tested against a t distribution to determine how probable it is that the true value of the coefficient is equal to some hypothesized value. When our hypothesis is B₁ = 1 we denote our t-Statistic as t₁(B₁=1) and it can be calculated by the formula:

t₁(B₁=1) = (b₁ - B₁ / se₁)

Let's try this for our first variable, the marginal propensity to consume. Recall we had the following data:

X Variable 1

b₂ = 0.9370
se₂ = 0.0019

Our t-Statistic for the hypothesis that B₂ = 1 is simply:

t₂(B₂=1) = (0.9370 - 1) / 0.0019 = -33.157

So t₂(B₂=1) is -33.157. We can also calculate our t-test for the hypothesis that our second X variable is 0:

X Variable 2

b₃ = -13.7194
se₃ = 1.4186

Our t-Statistic for the hypothesis that B₃ = 0 is simply:

t₃(B₃= 0) = ((-13.7194) - 0) / 1.4186 = -9.617

So t₃(B₃= 0) is -9.617. Next we have to convert these into p-values. The p-value "may be defined as the lowest significance level at which a null hypothesis can be rejected...As a rule, the smaller the p value, the stronger is the evidence against the null hypothesis." (Gujarati, 113) As a standard rule of thumb, if the p-value is lower than 0.05, we reject the null hypothesis and accept the alternative hypothesis. This means that if the p-value associated with the test t₁(B₁=1) is less than 0.05 we reject the hypothesis that B₁=1 and accept the hypothesis that B₁ not equal to 1. If the associated p-value is equal to or greater than 0.05, we do just the opposite, that is we accept the null hypothesis that B₁=1.

Calculating the p-value

Unfortunately, you cannot calculate the p-value. To obtain a p-value, you generally have to look it up in a chart. Most standard statistics and econometrics books contain a p-value chart in the back of the book. Fortunately with the advent of the internet, there's a much simpler way of obtaining p-values. The site Graphpad Quickcalcs: One sample t test allows you to quickly and easily obtain p-values. Using this site, here's how you obtain a p-value for each test.

Steps Needed to Estimate a p-value for B₂=1

• Click on the radio box containing "Enter mean, SEM and N." Mean is the parameter value we estimated, SEM is the standard error, and N is the number of observations.
• Enter 0.9370 in the box labelled "Mean:".
• Enter 0.0019 in the box labelled "SEM:"
• Enter 179 in the box labelled "N:", as this is the number of observations we had.
• Under " 3. Specify the hypothetical mean value" click on the radio button beside the blank box. In that box enter 1, as that is our hypothesis.
• Click "Calculate Now"

You should get an output page. On the top of the output page you should see the following information:

P value and statistical significance:
The two-tailed P value is less than 0.0001
By conventional criteria, this difference is considered to be extremely statistically significant.

So our p-value is 0.0001 which is less than 0.05. In this case we reject our null hypothesis and accept our alternative hypothesis. In our words, for this parameter, our theory did not match the data.

Be Sure to Continue to Page 3 of "Hypothesis Testing With Multivariate Regressions Using One-Sample t-Tests".