Hypothesis Testing Using One-Sample t-Tests

You've collected your data, you've got your model, you've run your regression and you've got your results. Now what do you do with your results?

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

The theory behind Okun's Law was described in the article: "Instant Econometrics Project 1 - Okun's Law":

Okun's law is an empirical relationship between the change in the unemployment rate and the percentage growth in real output, as measured by GNP. Arthur Okun estimated the following relationship between the two:

Y_t = - 0.4 (X_t - 2.5 )

This can also be expressed as a more traditional linear regression as:

Y_t = 1 - 0.4 X_t

Where:
Y_t is the change in the unemployment rate in percentage points.
X_t is the percentage growth rate in real output, as measured by real GNP.

So our theory is that the values of our parameters are B₁ = 1 for the slope parameter and B₂ = -0.4 for the intercept parameter.

We used American data to see how well the data matched the theory. From "How to Do a Painless Econometrics Project" we saw that we needed to estimate the model:

Y_t = b₁ + b₂ X_t

Where:
Y_t is the change in the unemployment rate in percentage points.
X_t is the change in the percentage growth rate in real output, as measured by real GNP.
b₁ and b₂ are the estimated values of our parameters. Our hypothesized values for these parameters are denoted B₁ and B₂.

Using Microsoft Excel, we calculated the parameters b₁ and b₂. Now we need to see if those parameters match our theory, which was that B₁ = 1 and B₂ = -0.4. Before we can do that, we need to jot down some figures that Excel gave us. If you look at the results screenshot you'll notice that the values are missing. That was intentional, as I want you to calculate the values on your own. For the purposes of this article, I will make up some values and show you in what cells you can find the real values. Before we begin our hypothesis testing, we need to jot down the following values:

Observations

Number of Observations (Cell B8) Obs = 219

Intercept

Coefficient (Cell B17) b₁ = 0.47 (appears on chart as "AAA")
Standard Error (Cell C17) se₁ = 0.23 (appears on chart as "CCC")
t Stat (Cell D17) t₁ = 2.0435 (appears on chart as "x")
P-value (Cell E17) p₁ = 0.0422 (appears on chart as "x")

X Variable

Coefficient (Cell B18) b₂ = - 0.31 (appears on chart as "BBB")
Standard Error (Cell C18) se₂ = 0.03 (appears on chart as "DDD")
t Stat (Cell D18) t₂ = 10.333 (appears on chart as "x")
P-value (Cell E18) p₂ = 0.0001 (appears on chart as "x")

If you did the regression, you'll have different values than these. These values are just used for demonstration purposes, so make sure to substitute your values in for mine when you do your analysis.

In the next section we'll look at hypothesis testing and we'll see if our data matches our theory.

Be Sure to Continue to Page 2 of "Hypothesis Testing Using One-Sample t-Tests".