UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
Regression Analysis Practice--Solutions
1) The results of an OLS regression analysis include: n = 62, k = 8, RSS = 200, ESS = 800, and TSS = 1000.
Calculate Ftest.
Ftest = [RSS/(k-1)] / [ESS/(n-k)] = [200/(8-1)] / [800/(62-8)] = 1.929
What is the hypothesis that is tested using Ftest?
H0 = all B’s are zero
H1 = one or more of the B’s is not equal to zero
What are the values of d.f.numerator and d.f.denominator for this F-test?
d.f. numerator = k - 1 = 7. d.f. denominator = n - k = 54.
For alpha = 5% level of significance, use the F-table to find Fcrit.
d.f. numerator = k - 1 = 7. d.f. denominator = n - k = 54. alpha = 5%.
So, Fcritical = between 2.10 and 2.25 (approximately)
Does the regression(as a whole) explain a statistically significant (at the alpha = 5% level of significance)
percentage of the variation in the dependent variable? Very briefly, why or why not?
Since Ftest < Fcritical, we accept H0.
Thus, the regression (as a whole) does not explain a statistically significant percentage of variation in the
dependent variable at the alpha = 5% level of significance.
Calculate R2.
R2 = RSS / TSS, however, because the regression equation is not statistically significant (based on the Ftest), R2 is not valid and should not be calculated.
Calculate Rbar2.
Rbar2 = 1-(1-R2)((n-1)/(n-k)), but, because the regression equation is not statistically significant (based on
the F-test), R-bar2 is not valid and should not be calculated.
If the regression is/were statistically significant, which should be used for this particular regression, R2 or
Rbar2?
Because the regression equation is not statistically significant (based on the F-test), neither R2 nor R-bar2 is
valid, and neither should be used.
Briefly, what does R2 (or Rbar2) tell us?
R2 (or Rbar2) tells us the percentage of the variation in Y that is explained by the regression model, but
neither should be used in this example, as the regression equation is not statistically significant (based on
the F-test).
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UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
2) Suppose you do some consulting work for a client who is interested in the potential relationships between the price of ear
buds (Pbuds) and the prices of two of the materials used in ear bud production, the price of resin (Presin) and the price of
copper wire (Pwire). You decide to run the following OLS regression analysis in SAS:
proc reg data=dataset01;
model Pbuds = Presin Pwire;
run;
The results of the analysis are shown below.
The REG Procedure
Model: MODEL1
Dependent Variable: Pbuds
Number of Observations Read
Number of Observations Used
26
26
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
23
25
1551.00585
509.95569
2060.96154
775.50293
22.17199
Root MSE
Dependent Mean
Coeff Var
4.70871
14.96154
31.47212
R-Square
Adj R-Sq
F Value
Pr > F
34.98
<.0001
0.7526
0.7310
Parameter Estimates
Variable
Label
Intercept
Presin
Pwire
Intercept
Presin
Pwire
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
1
1
1
7.70949
0.90064
0.03093
1.46982
0.14345
0.05922
5.25
6.28
0.52
<.0001
<.0001
0.6064
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UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
Referring to the OLS regression analysis results on the previous page, answer the following questions,
assuming α = 0.05:
What was the dependent variable in the regression analysis, and what were the independent variables?
The dependent variables is Pbuds. The independent variables are Presin and Pwire.
What was the sample size (n) used in the analysis?
n = 26
What does the F-Value tell you about the regression model?
The F-Value tells us whether the regression model, as a whole, explains a statistically-significant percentage of the
variation in the Y variable.
What are the values of d.f.numerator and d.f.denominator for use in finding Fcritical from the F-table?
d.f.numerator = k – 1 = 3 – 1 = 2
d.f.denominator = n – k = 26 – 3 = 23
What is the value of Fcritical from the F-table (using α = 0.05)?
Fcritical = 3.49 approximately
Is the F-Value significant? Briefly, how can you tell?
From the regression output, Ftest = 34.98. Since Ftest > Fcritical, the F-Value is significant, which means that we reject
H0 = all β’s are zero, and we accept H1 = one or more of the β’s is not equal to zero.
What do R-Square and Adj R-Sq tell you? For this regression, which should you look at, and (very briefly) why?
R-Square and Adj R-Sq tell us the percentage of the variation in Y that is explained by the regression model. For
this regression, we should look at Adj R-Sq, because the regression model has more than one X variable.
In SAS, the SER is called “Root MSE?” Briefly, what does it tell us?
SER tells us the average distance of a data point from the regression line/curve.
Briefly, what do the numbers in the “Parameter Estimate” column tell us?
The numbers in the “Parameter Estimate” column tell us the values of the 𝛽̂ ’s. The Parameter Estimate for the
“Intercept” is 𝛽̂ 0. The Parameter Estimate for Presin is the 𝛽̂ 1 in the regression equation. The Parameter Estimate
for Pwire is the 𝛽̂ 2 in the regression equation.
Briefly, what do the numbers given in the “t Value” column tell us?
The numbers in the “t Value” column tell us the ttest values for the t-tests of whether each 𝛽̂ parameter is equal to
zero. We can compare these ttest values with tcritical numbers from a t-table to test H0: 𝛽̂ = 0 vs. H1: 𝛽̂ ≠ 0. (Since
this is a two-sided test, use α/2 when retrieving tcritical from the t-table.)
Briefly, what do the numbers in the “Pr > |t| column tell us?
The numbers in the “Pr > |t|” column tell us the p-values for the t-tests of the 𝛽̂ parameters. We can compare these
p-values to the α/2-value in order to test H0: 𝛽̂ = 0 vs. H1: 𝛽̂ ≠ 0.
Is the effect of Presin on Pbuds statistically significant (is the 𝛽̂ for Presin different from zero)? If so, what is the sign
and magnitude of the effect of Presin on Pbuds?
 The effect of Presin on Pbuds is statistically significant at the α/2 significance level, because the t test value for
Presin is 6.28, which is greater than tcritical = 2.069 from the t-table (two-sided, α/2 = 0.025, d.f. = n-k = 26-3 =
23).
 Parameter estimate 𝛽̂ 1 gives the effect of Presin (variable X1) on Pbuds (the Y variable). The value of 𝛽̂ 1 is
0.90064. This means that a one unit increase in Presin (variable X1) results in a 0.90064 unit increase in Pbuds
(the Y variable). We know that the effect of X1 on Y is positive because 𝛽̂ 1 is positive.
Is the effect of Pwire on Pbuds statistically significant (is the 𝛽̂ for Pwire different from zero)? If so, what is the sign
and magnitude of the effect of Pwire on Pbuds?
 The effect of Pwire on Pbuds is not statistically significant at the α/2 significance level, because the ttest value
for Pwire is 0.52, which is less than tcritical = 2.069 from the t-table (two-sided, α/2 = 0.025, d.f. = n-k = 26-3 =
23).
 Because the effect of Pwire on Pbuds is not statistically significant, the effect of Pwire on Pbuds is zero.
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