Illustrative Worksheets: Boston, New York, F Test

(The attached PDF file has better formatting.)

The project template explains the F test in detail and give many suggestions for its use.

The illustrative worksheets for Boston and New York show that the optimal regression equation has two past years for Boston and one past year for New York. We ask: "How different are Boston and New York?"

Your student project should explain what you are testing.

The null hypothesis assumes the same regression equation for both teams.

Jacob: The two teams clearly have different regression equations. The optimal equation uses two past years for Boston and one past year for New York. The ordinary least squares estimators differ. Why assume the equations are the same?

Rachel: We use 50 data points for each team. Losing percentages are stochastic. Perhaps the two teams have the same regression equation, and the differences stem from random fluctuation. We examine two possibilities:

For scenario #1, neither α nor β differ for the two teams. For scenario #2, we normalize each team’s long-run won-loss record to 50% before applying the F test.

We can use either one or two past years for the F test. The illustrative worksheet uses one past year. It includes the data for the second past year, and you are encouraged to run an F test with two past years.

Your student project may have three or four past years for the F test.

Copy the Boston and New York data for the current year and one past year (or two past years, depending on the F test) into a new worksheet.

Take heed: The instructions here delete some rows to simplify the worksheet. If year T+1 is copied from Year T and you delete Year T, year T+1 is missing a value.

The comments above may not make sense until you do the work. As you do the work, check for Excel error warnings (values! names!).

Use two regression equations:

The dummy variable is 0 for New York and 1 for Boston.

If the first letter of the team name (Column C) is "N" (for New York), the dummy variable = 0; otherwise, the dummy variable = 1.

Take heed: You don’t have to use an IF function. You can enter 0 in Cells H11:H60 and 1 in H61:H110.

Take heed: Instead of an F test with dummy variables, you can use the version of the F test with αN and βN.

Take heed: We have not normalized the won-loss records to 50%. New York has a better average won-loss record over the fifty years (a lower losing percentage), so the same regression equation is unlikely to work for both teams. We redo the analysis below after normalizing the won-loss records to 50%.

The restricted equation uses one past year and no dummy variable. We are tempted to reason as follows:

The restricted equation has a high R² (52.84%) vs 42.89% for New York alone and 44.60% for Boston alone. We explain the rationale.

The two teams have similar slope coefficients, but New York has a lower losing percentage (on average) in 1901-1960. We want to use the past experience. To make the intuition clear, we give an extreme illustration

Illustration: Suppose Boston had an 80% losing percentage (on average) and New York had a 20% losing percentage, and last year’s losing percentage does not help predict the current year’s losing percentage if each team is examined alone.

The optimal regression equation for each team alone has α = 20% for New York and 80% for Boston, β = 0, and R² . 0.

If we examine the teams together, we place great weight on last year’s losing percentage.

The optimal regression depends on the stochasticity. As σ ÿ 0, β ÿ 1 and R² ÿ 100%.

Take heed: The discussion forum postings for the project templates give the rationale for the statistical results. As you work through your student project, jot down the implications of the statistical results and include these in your write-up.

Take heed: Explaining the intuition for your results is a good habit, helping you understand the regression and avoid errors.

The unrestricted equation uses three (2k + 1) explanatory variables and one intercept.

See the general discussion of restricted vs unrestricted equations.

The R² always increases and the ESS always decreases. The F test says if the increase in R² and the decrease in the ESS reflects a better fit or simply more explanatory variables.

We determine the degrees of freedom for the numerator and denominator.

Numerator: The number of constraints is two (not one): the same α and same β. That is, the coefficient of D (the dummy variable) is zero and the coefficient of D × Year -1 is zero.

Take heed: The number of constraints is the number of independent variables + 1.

Denominator: 50 years × 2 teams = 100 data points. The unrestricted equation has four explanatory variables (including the intercept). The degrees of freedom for the denominator is 100 – 4 = 96.

Take heed: The write-up to your student project should state the degrees of freedom for the numerator and denominator.

ESS Version of F Test

The numerator is the difference in the ESS divided by its degrees of freedom:

(0.47639 – 0.455791) / 2 = 0.01030

The denominator is ESS for the unrestricted equation divided by its degrees of freedom.

0.455791 / 96 = 0.00475

The F statistic is 0.01030 / 0.00475 = 2.16931

Use Excel’s fdist built-in function to determine the p value for an F statistic of 2.16931 with (2, 96) degrees of freedom. Typing =FDIST(0.216931,2,96) gives 0.1198255.

You can also interpolate with the table of critical values for the F test in the textbook.

R² Version of F Test

The numerator is the difference in the R² divided by its degrees of freedom:

(54.88% – 52.84%) / 2 = 1.02%

The denominator is the R² for the unrestricted equation divided by its degrees of freedom.

(1 – 54.88%) / 96 = 0.00470

The F statistic is 0.01020 / 0.00470 = 2.170213

The regression output for the unrestricted equation confirms the results of the F test.

The dummy variable has a coefficient of 0.024993, a standard error of 0.071397, a t value of 0.350057, and a p value of 0.727063.

The (dummy variable × Year -1) has a coefficient of 0.015221, standard error of 0.156349, t value of 0.09735, and p value of 0.922651.

The unrestricted equation says that the slope coefficient for Boston alone is 0.65651, and the slope coefficient for New York alone is 0.65651 + 0.015221 = 0.671731. This difference is small A it reflects random fluctuations in the data sample of 50 years for each team.

Take heed: Confirm your F test results by examining the regression equations for each team (or league or conference or division) separately.

The F test above considers two possibilities: New York and Boston are the same or they have different intercepts and slopes. We might also consider a third possibility:

We might surmise that Boston and New York have different average won-loss records, but the past year’s losing percentage has the same effect on the current year’s for both teams.

The average losing percentage differs materially for the two teams. In 1911-1960:

The slope coefficient for the two teams is almost the same. Boston’s is higher by

0.015221 / 0.65651 = 2.32%.

Normalization does not affect the slope coefficient, as shown in the unrestricted equation.

Jacob: Don’t we check if the long-run average losing percentages are the same by the standard deviations and the means?

Rachel: Your procedure is correct if losing percentages are uncorrelated from year to year. But suppose β = 99%, the true mean is 45.5% for both teams, and the losing percentage the first year is 40% for New York and 51% for Boston. With a 10% standard deviation for the losing percentage each year, this is a likely scenario: each team’s observed value is within half a standard deviation of the mean. With β = 99%, each team’s expected losing percentage is about the same from year to year, with a slow reversion to the mean.

We ask: "If we normalize both teams to an average won-loss record of 50%, how would the F test change?"

Method: Let BN be the average losing percentage for Boston in 1910-1960 and NN be the average losing percentage for New York in 1910-1960. The illustrative worksheet shows these averages at the bottom of the column labeled forecast. Give these averages names: Nyaverage and Boaverage, using Excel’s insert A name A define.

Copy Columns C, D, and E to Columns P, Q, and R.

Normalize the losing percentage figures to a 50% average won-loss record for both teams.

NYaverage is a name, so it always refers to the same figure (41.0%).

Do the same for the Boston figures.

We multiply the Boston figures by 50% / BN and the New York figures by 50% / NN. Re-compute the averages to make sure they are both 50% (in column S).

For the restricted equation, the R² and slope coefficient decrease with normalization, and the intercept increases. See the discussion above for the rationale.

The degrees of freedom for the F test do not change. The difference in the ESS (or in the R²) decreases almost to zero. The F statistic decreases almost to zero. We surely do not reject the null hypothesis that the regression equations for the two teams are the same except for the difference in the average won-loss record.

Take heed: A league (or division or conference) has an average won-loss record of 50%. If your student project compares two leagues, you do not have to normalize.

Take heed: The R² for a regression equation does not change if the figures are multiplied by a constant. The ESS changes by the square of the normalizing constant.

Jacob: Do we expect the new F statistic to be higher or lower?

Rachel: If teams differ in average quality (won-loss records), we expect lower F statistics.