1. Classical Regression III
Lecture 11
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2. Today’s plan
Chi-squared test
F test
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3. Total = Explained + Unexplained
More on the ANOVA table
Remember the sum of squares identity:
Total = Explained + Unexplained
The sum of squares identity can also be written as
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4. More on the ANOVA table (2)
The components of the sum of squares identity can be found in the Stata output as the ANOVA table, which looks like this:
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5. Chi-squared test
The next test we’ll look at is the 2 (chi-squared) test
This does not appear on the Stata output
With the 2 test, we are testing the significance of a variance for a particular value
Recall that the Z statistic looks like
The square of this distribution gives the 2 with one degree of freedom
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6. For a 5% significance level
Chi-squared test (2)
With the chi-squared test we want to test:
H0 : sYX2 = 1
For a 5% significance level
The 2 statistic for n-2 degrees of freedom is distributed:
Where 2YX is the hypothesized value given under the null hypothesis
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7. Chi-squared test (3)
The root MSE from the Stata output or Excel output is the square root of
So our 2 statistic will be
Now look at the 2 table:
The format is similar to the F
First column gives degrees of freedom
Across the top row are the areas remaining in the right-hand tail of the curve
The Ho region is the area of the left of the 2 value and the H1 region is the area to the right
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8. Chi-squared test (4)
From the 2 table, we find that for a 95% confidence interval the 2 value is
Since 2 < 228,0, 0.05 we fail to reject the null
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9. F-test
In the second line down in the right hand column of the Stata output we have a F test, which looks at the significance of this regression equation using the sum of squares information from the ANOVA table
Using the sample data, the F test is designed to test the difference of variances between samples drawn from independent populations
The F statistic looks like:
Where m & n are the respective
degrees of freedom
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10. F-test (2) Properties of the F distribution:
1) The F distribution is skewed and positive
2) As the degrees of freedom increases for both samples,
the F distribution will approximate the normal
F(10,2)
F(50,50)
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11. F-test (3)
Last time we looked at the spreadsheet L9.xls and we learned how to use the LINEST function in Excel to run a regression
The LINEST output gives you the model (or explained) sum of squares, the residual sum of squares, and the F statistic
The regression equation we estimated was
Where the numbers in parenthesis are the standard errors for the coefficients
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12. F-test (4) F-test where Ho : b = 0 Our F statistic will be
So from our spreadsheet, we can plug in values to calculate our F statistic
From the LINEST output, we find that the F statistic is [it’s not too far off]
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13. F-test (5)
In the F tables, the first column gives the degrees of freedom for the denominator, or df2
Across the top row are the degrees of freedom for the numerator, or df1
For each pair of degrees of freedom, there are four values for the significance level from 0.25 to 0.01
this refers to the area remaining in the tail of the F distribution
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14. F-test (6) Now let’s look up the F value with df1 = 1 and df2 =28
F value at a 5% significance level is 4.20:
F0.05,1,28 = 4.20
Therefore if F > F0.05,1,28 then we reject the null
reject the null that: b = 0
You will use an F-test when you calculate within sample predictions, or when you do a Chow test
You will also use an F-test when you want to test the significance of a regression when there are multiple independent (or X) variables
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