Fox Module 11 Units of Measurement Practice problems


Fox Module 11 Units of Measurement Practice problems

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NEAS
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Fox Module 11: Statistical inference for simple linear regression

 

(The attached PDF file has better formatting.)

 

Regression Analysis Units of Measurement Practice problems

 

Know how ordinary least squares estimators, their standard errors, t-values, and p-values depend on the units of measurement and displacement from the origin. The principles are

 


           Multiplying the explanatory variable by k multiplies its â by 1/k.

           Multiplying the response variable by k multiplies all the â’s by k.

           Displacements of explanatory variables and the response variable from the origin changes á, not the ß’s.


 

 

Intuition: â is in units of response variable / explanatory variable.

 

Illustration: Suppose claim frequency = á + â × kilometers driven.

 


 

            á is in units of claim frequency.

            â is in units of claim frequency / kilometers driven


 

 

If we write the regression equation as claim frequency = á + (â/1,000) × meters driven.

 


 

            á is in units of claim frequency.

            â is in units of claim frequency / kilometers driven


 

 

Intuition: The ß’s depend on the deviations of the values from their means. A constant displacement of all the values doesn’t affect the deviations. But a constant displacement of k raises the response variable Y by k × â. á has the same displacement as the response variable, so it also rises by k × â.

 


 

Elasticities, standardized coefficients, and t-values are unit-less.

 


 

           Elasticities are percentage changes:  Y/Y /  X/X.

           The change in a value has the same units as the value itself.

           If X is kilometers driven, then  X is also measured in kilometers driven.

           If Y is claim frequency, then  Y is also measured in claim frequency.


 

 

Standardized coefficients are â × óx / óy.

 


 

            â is in units of Y / X.

            óx is in units of X.

            óy. Is in units of Y.


 

 

The standardized coefficient is unit-less.

 

Measures of significance are not affected by units of measurement.

 


 

           The t-value is the ordinary least squares estimator divided by its standard deviation. 

           The estimator and its standard deviation have the same units, so the t-value is unit-less.


 

 

The correlation between two random variables is unrelated to units of measurement, so the R2 statistic is also unit-less.

 


 

*Question 11.1: Goodness-of-fit and Units of Measurement

 

We use least squares regression with N pairs of observations (Xi, Yi) to estimate average annual claims cost in dollars per average miles driven each week, giving Y= 50 + 40X + å.

 

If we change the parameters to annual claims costs in Euros and kilometers driven per week, which of the following is true?

 


 

A.    The R2 increases and the t value for kilometers driven increases

B.     The R2 increases and the t value for kilometers driven decreases

C.     The R2 decreases and the t value for kilometers driven increases

D.    The R2 decreases and the t value for kilometers driven decreases

E.     The R2 stays the same and the t value for kilometers driven stays the same

 

Answer 11.1: E

 

The R2 and the t statistic are both unit-less.

 


 

           The R2 is a proportion.  If we double the units of Y, the TSS, RegSS, and RSS all increase by a factor of 22 = 4.  The R2 doesn’t change.

           The t statistic is the ordinary least squares estimator divided by its standard deviation.  If we double the units of X, both the estimator and its standard deviation decrease by 50%.


 

 

 


 

*Question 11.2: Miles Driven and Annual Claim Costs

 

We use least squares regression with N pairs of observations (Xi, Yi) to estimate average annual claims cost in dollars per average miles driven per day, giving Y= 50 + 40X + å.  For instance, a policyholder who drives an average of 25 miles a day has average claim costs of 50 + 40 × 25 = 1,050 dollars a year.

 

If we change the parameters to annual claims costs in Euros and kilometers driven a day, what is the revised regression equation?  For this problem, assume €1.00 = $1.25 and 1 kilometer = ⅝ mile (five eighths of a mile).

 


 

A.    Y = 40 + 40X + å

B.     Y = 40 + 20X + å

C.     Y = 40 + 64X + å

D.    Y = 62.5 + 25X + å

E.     Y = 62.5  + 64X + å

 


 

Answer 11.2: B

 

The estimate of â is the covariance ñ(x,y) divided by the variance of X. 

 


 

           Using euros multiplies each Y value by 1.00 / 1.25 = 0.80.

           Using kilometer multiplies each X value by 8/5 = 1.60.


 

 

Illustration: $10.00 = 10 × 0.80 = €8.00, and 10 miles = 10 × 1.60 = 16 kilometers.

 

Multiplying the Y values by 0.80 and the X values by 1.60

 


 

           Multiplies the covariance by 0.80 × 1.60 = 1.280

           Multiplies the variance of X by 1.602 = 2.560


 

 

This multiplies â by 1.280 / 2.560 = 0.500.

 

á is not affected by the units of X, since the product â × X is not affected by the units of X.  But á varies directly with the units of Y: if Y is multiplied by 0.80, á is multiplied by 0.80.

 

Jacob: Is the product â × X unit-less?

 

Rachel: No; the product is in the units of Y.

 

We can check our result numerically:

 


 

           Before the change, if X = 0 miles, Y = $50.  Now X = 0 gives Y = €40, so á is 40.

           Before the change, if X = 5 miles, Y = $250.  Now X = 8 kilometers gives Y = $250 × 0.8 = €200.  Since á = 40, â is (200 – 40) / 8 = 20.


 

 

 

 


 

*Question 11.3: Displacement

 

We regress Y on X with a two-variable regression model .

 


 

           X is the number of hours studied as a deviation from its mean.

           Y is the exam score as a deviation from its mean.


 

 

We change the values of X and Y to

 


 

           X is the actual number of hours studied (mean = 80 hours)

           Y is the actual exam score (mean score = 80)


 

 

Which of the following is true?

 


 

A.    The R2 increases and the adjusted R2 increases

B.     The R2 increases and the adjusted R2 stays the same

C.     The R2 decreases and the adjusted R2 increases

D.    The R2 decreases and the adjusted R2 stays the same

E.     The R2 stays the same and the adjusted R2 stays the same

 

Answer 11.3: E

 

The displacement of X and Y does not affect the correlation between the random variables, so it does not affect the R2 or the adjusted R2.

 


 

*Question 11.4: Displacement

 

We regress Y on X with a two-variable regression model . Which of the following is true?

 


 

A.    If we double each X value and decrease each Y value by 1, á increases.

B.     If we double each X value but don’t change the Y values, á decreases.

C.     If we double each X value and increase each Y value by 1, á decreases.

D.    If we double each X value and increase each Y value by 1, á increases.

E.     If we double each X value and decrease each Y value by 1, á stays the same.

 

Answer 11.4: D

 


 

           Doubling each X value reduces â by 50% but does not change á.

           Increasing each Y value by 1 increases á by 1 but does not change â.


 

 

 


 

*Question 11.5: Standardized Coefficients and Elasticities

 

We regress the average auto insurance loss costs in dollars (the Y dependent variable) on the number of hours the auto is driven each week (the X independent variable).  We estimate the ordinary least squares estimator , the standardized coefficient *, and the elasticity ç.

 

If we use Euros for the loss costs instead of dollars, which of the following is true?  Assume that one Euro is 1.25 dollars.

 


 

A.                 increases; * and ç stay the same.

B.      decreases; * and ç stay the same.

C.      and * stay the same; and ç increases.

D.     and * stay the same; and ç decreases.

E.      and ç stay the same, and * increases.


 

Answer 11.5: B

 

If an hour of driving each week increases loss costs by $10, it increases loss costs by €8, so â decreases.

 

The standardized coefficient and elasticity are unit-less, so they are not affected by a change in the units of measurement.

 

 


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Is it a typo on Page 1?

Should it be "claim frequency = α + β × meters driven/ 1000" instead of "claim frequency = α + 1000β × meters driven" ?

[NEAS: Thank you; typo is now corrected.]


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