Hypothesis Testing With One-Sample t-Tests

Hypothesis Testing With One-Sample t-Tests Youve collected your data, youve got your model, youve run your regression and youve got your results. Now what do you do with your results? In this article we consider the Okuns 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 Okuns Law was described in the article: Instant Econometrics Project 1 - Okuns Law: Okuns 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: Yt - 0.4 (Xt - 2.5 ) This can also be expressed as a more traditional linear regression as: Yt 1 - 0.4 Xt Where:Yt is the change in the unemployment rate in percentage points.Xt 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: Yt = b1 + b2 Xt Yt Xt b1 b2 B1 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 youll 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) In the next section well look at hypothesis testing and well see if our data matches our theory. Be Sure to Continue to Page 2 of Hypothesis Testing 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: â€Å"[S]uppose we hypothesize that the true B1 takes a particular numerical value, e.g., B1 1. Our task now is to â€Å"test† this hypothesis.†Ã¢â‚¬Å"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, orH1: B1 1, also a one-sided alternative hypothesis, orH1: 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(B11) and it can be calculated by the formula: t1(B11) (b1 - B1 / se1) Let’s try this for our intercept data. Recall we had the following data: Intercept b1 0.47se1 0.23 Our t-Statistic for the hypothesis that B1 1 is simply: t1(B11) (0.47 – 1) / 0.23 2.0435 So t1(B11) is 2.0435. We can also calculate our t-test for the hypothesis that the slope variable is equal to -0.4: X Variable b2 -0.31se2 0.03 Our t-Statistic for the hypothesis that B2 -0.4 is simply: t2(B2 -0.4) ((-0.31) – (-0.4)) / 0.23 3.0000 So t2(B2 -0.4) is 3.0000. 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(B11) is less than 0.05 we reject the hypothesis that B11 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 B11. 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 B11 Click on the radio box containing â€Å"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 0.47 in the box labelled â€Å"Mean:†.Enter 0.23 in the box labelled â€Å"SEM:†Enter 219 in the box labelled â€Å"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 â€Å"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 equals 0.0221By conventional criteria, this difference is considered to be statistically significant. So our p-value is 0.0221 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 Using One-Sample t-Tests. Again using site Graphpad Quickcalcs: One sample t test we can quickly obtain the p-value for our second hypothesis test: Steps Needed to Estimate a p-value for B2 -0.4 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 -0.31 in the box labelled “Mean:”. Enter 0.03 in the box labelled “SEM:” Enter 219 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 -0.4, as that is our hypothesis. Click “Calculate Now” P value and statistical significance: The two-tailed P value equals 0.0030By conventional criteria, this difference is considered to be statistically significant. We used U.S. data to estimate the Okuns Law model. Using that data we found that both the intercept and slope parameters are statistically significantly different than those in Okuns Law. Therefore we can conclude that in the United States Okuns Law does not hold. Now youve seen how to calculate and use one-sample t-tests, you will be able to interpret the numbers youve calculated in your regression. 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