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MIEF Quant I (Basic Econometrics)

Practice Final Exam  Answers (Exam date:  August 27, 2019)

Instructions.  This is an open book, open note exam.  Please refer to any books or notes on a computer or notebook that you wish.  Please do watch your time and write as legibly as possible.    For any calculations you are asked to carry out, please use 3 to 4 significant figures (not more, not less).    

1.  The data

What follows is hypothetical (ie, made up) data on 25 SAIS students who graduated last year and is included at Appendix 0.  The variables are:

SAISGPA Final cumulative grade point average (4 point scale)

Econ 1 if an economics major as an undergraduate, 0 otherwise

IR 1 if an international relations major as an undergraduate, 0 otherwise

Math_Science 1 if a math or science major as an undergraduate, 0 otherwise

ColGPA College grade point average (4 point scale)

Jogger 1 if the SAIS student is a jogger, 0 if not a jogger

Internship Number of hours per week on average spent on an internship over the time at SAIS

Appendix 1 uses Stata and sets out to explain SAISGPA in terms  whether the SAIS student was a jogger, several variables that indicate the undergraduate major, College GPA, and time spent on internships

a) For the first function estimated, write out the model including the error term.  What assumptions were made when this regression was carried out?

 

Ie. Linear in the parameters

E[μ│X] = 0

Var(μ) =σ² 

Cov(μᵢμ sub j) = 0 for I ≠ j

μᵢ is normally distributed for all i

Cov(X,μ) = 0  (which is implied by E[μ│X] = 0

b) Put in words each of the slope coefficients on the independent variables; start with a clear description of the constant.  Explain whether the slope coefficients have the sign that you might expect.  

hat tells us that for someone who has a College GPA of zero; no internship hours; is not a jogger and is neither an Econ major, nor an IR nor a math or science major in her undergraduate (ie, some “other” major such as history, psychology, political science, etc), we expect the GPA at SAIS to be 1.2.

hat tells us that one additional point in college GPA is associated with an increase in SAIS GPA of .598 points holding constant Internship, Jogger, and Undergraduate major.

 tells us that spending time on an internship while at SAIS may backfire: each additional hour spent per week on an internship is associated with a GPA lower by 0.01 point. (Luckily, this is not a large effect!), holding constant College GPA, jogger, and undergraduate major.

 tells us that Joggers tend to have a higher GPA: being a jogger increases your GPA by 0.46 points in this sample, holding constant College GPA, Internship hours, and undergraduate major.

 tells us that Econ undergraduate majors have a higher SAIS GPA on average by about .2 points compared to a student with the “other” undergraduate major, holding constant College GPA, Internship Hours, and whether the student is a Jogger

 tells us that  IR undergrad majors perform only slightly  better by .09 points, on average compared to a student with the “other” undergraduate major, holding constant College GPA, Internship Hours, and whether the student is a Jogger

hat tells us that being a math/science major undergrad leads on average to a .238 lower GPA than a student with the “other” major.

c) Based on the p-value computed by Stata, which of these variables are statistically significantly different from zero at the .05 level of significance.  Be clear which numbers you are comparing. What does it mean for a coefficient in multiple regression to be significantly different from zero?

All the variables are statistically significant at the 5 percent level, except IR.   The p value for IR is .179 which is greater than .05.  All the other p values are less than .05.  

Put in words (ie, describe clearly) what the p-value for the IR variable tells us.

The p-value on the IR variable tells us that if the null hypothesis   = 0 is true, the probability is .179 that we could obtain a coefficient that is  .09 or further away from zero.  

Now the second regression estimated drops the undergraduate major variables.  Compute the F statistic to test the null hypothesis that the three undergraduate major variables as a group to do not add anything extra to the explanation of the variability of SAISGPA.   Please compute the F statistic both using the R Squared version and the SSR version.  The two numbers should be close.  Carry out that F test at the .05 level of significance, being clear what degrees of freedom you are using.

Recall that we can compute the F statistics in two ways:

 

where q=3 is the number of restriction tested; (n-k-1)=18 is the number of degrees of freedom for the unrestricted model. We will compare this with the critical value of an F with 3 degrees of freedom for the numerator and 18 for the denominator. . Our F-statistics will be:

 

2.   Appendix 2 contains an analysis that utilized a variable which is the product of Age and Height, the two independent variables.  This is a sample of eight young boys aged from 6 to 12.  Height is measured in inches and weight is measured in pounds.

a) Using that regression that contains that product variable (the second regression), write out the relationship that gives the change in Weight as Age changes, holding constant Height, ie, the slope in the Age direction holding constant Height.  Compute the value of the slope for a child with a Height of 40, 50 and 60 inches. 

 

Slope when height = 40 = -6.02+0.0656*40 = -3.396

Slope when height = 50 = -6.02+0.0656*50 = -2.74

Slope when height = 60 = -6.02+0.0656*60 = -2.084

b) What is the standard error of the slope on Age when Age=9 and Height=50?

The coefficient on Age in the second equation that uses the adjusted interactive variable is       -2.74, the same value in a) above. The standard error of that slope on Age is 2.33

3.  Question on Heteroskedasticity

Appendix 3 contains an analysis of hypothetical data for a sample of 33 households where data on savings, income, and family size was obtained.  Page 1 lists the data.

a) State in symbols and words the homoscedasticity assumption.

Homoscedasticity means that the variance of the error term is constant:

The value of the explanatory variables must contain no information about the variance of the unobserved factors.  Alternatively, the variance of the potential u’s for each observation have the same standard deviation σ.  

b) Regression 1 on page 2 sets out to estimate savings as a function of income and family size.  Two regressions are run on page 4 testing whether the assumption discussed in (a) was violated.   State clearly the names of these two tests and the assumptions involved.  From Regression #2, can you tell which variable is causing the problem? What is your conclusion from these two tests? 

Regression #2 is the Breusch-Pagan test, and Regression #3 is the special case of the White test for heteroskedasticity. The Breusch-Pagan tests the null hypothesis of homoskesticity that there is not a linear relationship between the independent variables and the square of the residuals.  The White test assumes a more general functional form that also includes quadratic terms and interaction terms.  Thus the special case of the White test uses yhat and yhat squared to test whether the square of the residuals is related to a linear and non-linear relationship with the independent variables.

It seems that income is causing heteroskedasticity, since when carrying out the Breusch-Pagan test, Income it is much more significant than family size with a p value of .001.

For both the Breusch-Pagan test and the special case of the White test, the F statistics are large with low p-values, thus rejecting the null hypothesis of homoscedasticity and concluding that there is a problem of heteroskedasticity.

 

c) Using Regression #1 and Regression #4 (page 5), write down the function estimated with the standard errors written appropriately below each slope coefficient.   Describe briefly why the standard errors from Regression #4 are more appropriate than the standard errors reported in Regression #1. 

 

(4317.3) (.0232) (1003.8)

[4431.478]  [0.0499] [553.45]

These robust standard errors are more appropriate because they relax the assumption that the error terms have a constant variance which we can reject using the Breusch-Pagan and White Test since they have very large F-values and therefore low p-values on page 4. 

d) Discuss Regression #5 on page 6.  What are the assumptions made with this regression. 

This regression uses weighted least squares to correct for heteroskedasticity. We are assuming that the income variable is the cause of the heteroskedasticity (the variance of each error term is equal to *income).  In order to convert the problem into one with a homeskedastic error term, each variable is divided by 1/sqrt(Income) 

e) Discuss Regression #6 on page 7.  How is that related to Regression #5. 

This is the same regression, using the Stata built-in command for weighted least squared compared with generating the adjusted variables directly.  Note that the weights used in the weighted least squares is  1/Income. 

f) On page 7 there is also a scatter plot that was produced following Regression #1.  Describe briefly what information you see in this scatter 

We see that the residuals increase as income increases. Clearly, there is some relationship between income and the size of the residuals. 

g) On page 8 there are two functions estimated, Regression #7 and Regression #8.  Explain the model being estimated here.  How is this different from Regression #5

o This is a feasible generalized least squares model that generates the hhat directly from the data where we are assuming that the Heteroskedasticity is  of unknown form. We use data to estimate the form of heteroskedasticity. (Estimate the weights. Then, compute WLS).

 

o We use the exponential function because variance has to be positive

 

o Procedure:

1) Estimate the original model (Regression #1 )  and obtain the residuals .  Generate the square of  called residualsSquared.  Take the log of residualsSquared calling it logresidualsquared on page 8.

2) Run the regression  and obtain the fitted values ,  on page 8 with Regression #7

3) Take the exponential of the fitted values  also on page 8, called hhat.

4) Next generate the adjusted variables for weighted least squares by dividing each variable by the square root of hhat.

5) Regression #8 on page 8 estimated the weighted least squares regression using these adjusted variables.  This is a feasible generalized least squares estimates that are biased but consistent.  The actual data used is listed on the top of page 9.

6) These differ from Regression #5 as the hhat for regression #5 was assumed to be the Income variable.  Here we let the data determine hhat which likely included some of the impact of family size. 

h) Explain the connection between Regression #9 is on page 9 and Regression #8 on page 8.

Regression #9 on page 9 runs the weighted least squared directly with Stata using the hhat determined in the feasible generalized least squared process as the weights.   This is accomplished by using the [aweight = 1/hhat] option in Stata.   The results are the same as Regression #8.  The F statistic in Regression #9 is the same as recomputed after Regression #8. 

4.  Pooled Cross Section (plus review of dummy variables; quadratics; and logs)

The analysis in Appendix 4 using hypothetical (made up) data on SAIS students drawn from a sample in 2008 and another sample in 2015.  The data is listed on pp 1-4 and are the same as question 1 with fellowship=1 if a student has a fellowship.

a. Regression #1 on Page 4 sets out to explain SAISGPA in terms of undergraduate GPA (ColGPA), time spent studying (StudyHrs), time spent on internships (InternshipHrs), and a special fellowship that was enhanced in 2012.  

1) Write out the function estimated with standard errors and t-values in parentheses.   

 

(0.585) (0.75) (0.0063) (0.0297) (0.0045) (0.085) (0.077) (0.114)

(8.41) (-3.41) (2.02) (-1.97) (2.61) (1.41) (1.77) (0.62)

2) Put in words the constant and each of the coefficients on Y2015, Fellowship, and Y15Fellowship.

A person with a college GPA of zero, no internship hours, who studies zero hours per week, and does not have a fellowship is predicted to have a GPA of 0.36 in 2008.

Comparing two students with the same undergraduate GPA, time spent on internships, study hours, and fellowship status, the student in 2015 would have a SAISGPA .109 higher.

Comparing two students with the same undergraduate GPA, time spent on internships, study hours, the student with a fellowship in 2008 is predicted to have a GPA 0.223 points higher than a student who does not have a fellowship in 2008.

The coefficient of Y15Fellowship of .201 indicates that the value of a fellowship in 2015 is .201 higher than in 2008 (ie, the value of a fellowship in 2015 is .223+.201 = .424 compared to someone without a fellowship in 2015 (the value in the constant).  

3) Put in words the coefficients on StudyHrs and StudyHrsSq.  What is the impact on SAISGPA when StudyHrs=20 and when StudyHrs=40. 

The effect of hours of study on GPA can be obtained as:

 

The negative coefficient on the quadratic term suggests that there is a diminishing marginal return to to studying. For each additional hour studied per week (holding constant undergraduate GPA, time spent on internship and whether a fellowship was held), a student’s GPA is predicted to rise by , so the final effect actually depends on how many hours one individual is already allocating to study. A student who studies 33 hours and increases their studying by one more hour will actually see his or her GPA fall.

When StudyHrs = 20, the effect of one additional hour studied on predicted GPA is
 0.06-2*0.00088*20=0.0248

When StudyHrs = 40, the effect of one additional hour studied on predicted GPA is
 0.06-2*0.00088*40=-0.0108. 

4) Put in words the R squared. 

The R squared tells us that about 82% of the variation in SAISGPA within our sample is explained by the model (ie. by the linear relationship with ColGPA, InternshipHrs, StudyHrs, StudyHrsSq, Fellowship, Y2015,  Y15*Fellowship.) 

5) What is the standard error of the estimate.  What is it an estimate of? 

The standard error of estimate is the Root Mean Square Error (RMSE) which is equal to 0.176. This is an estimate of the standard deviation of the errors (the Us). The RMSE is defined as:

  

6) Stata produces a 95 percent confidence interval for the population coefficient on ColGPA.  Compute an 80 percent confidence interval.  Explain why it is narrower or wider. 

There are n-k-1 = 47-7-1 = 39 degrees of freedom, so the critical t-value for an 80% confidence interval is 1.303. This means that the confidence interval will be:

0.63+/-1.303*0.075=0.63+/-0.098=[0.532, 0.728].

This is narrower than the 95% interval produced by Stata because we are requiring less confidence.

7) What hypothesis is being tested with the F statistic displayed.  Show the hypothesis test with a picture and critical F value.

 

H sub 1:  The alternative hypothesis is that at least one of the slope coefficients is not equal to zero.

 

When we test the joint significance of all variables in the model, our F statistics reduces to

 

b.   On page 5 are two regressions run for the separate time periods.  Also on page 7 is the same regression run with the pooled data.   Test using the SSR form of the F test whether there was a structural change in the function between the two time periods.  What is this test called?

This is a Chow Test, which allows us to understand whether there has been a structural change between the two time periods considered.

The F statistic for this test is constructed as:

 

In our case this will be:

 

we will need to compare this with the relevant critical value for an F with 9 degrees of freedom at the numerator and 29 at the denominator. Let’s choose a significance level of 0.01. Then:

 

Notice that the Null Hypothesis in a Chow test is that the coefficients are stable over time. If we reject it, it means that there has indeed been a structural break between 2008 and 2015.

c. On page 6 (Regression #4) and page 7 (Regression #5) are regressions that can be used to test if there was a structural change in the function between the two time periods.  Use the R squared form of this F test.  What is this test called?  Compare the result with that from part b.

This is again a Chow test. The Chow test can be implemented in two ways: either running separate regressions on separate tie period plus a pooled regression – as we did in the previous question; or running a pooled regression plus a regression that includes a dummy variable for one of the time periods, as well as interactions of each independent variable with that dummy – which is what we have in this case. We can interpret this as a case of restricted vs. unrestricted model, so our F statistics in this case will be:

 

As it is always the case for F-statistics, we can re-write this in term of the the R-squared of the two regressions. Notice that the regression including the dummy and all the interaction terms will be the unrestricted model; while the other regression will be the restricted one:

 

(Unsurprisingly – since we are doing exactly the same thing as in the previous question, but in a different way, we reach the same conclusion)

d. Use Regression #5 on page 7 and Regression #6 on page 7 to test if undergraduate major is significant in determining SAISGPA.   

Again this is an F test:

 

This is larger than the critical value at 1% (which is 4.31), so we would reject at 1%.

e. Regression #8 on page 8 and Regression #9 on page 9 examine a policy change for SAIS students with fellowships (this is apocryphal) in that they are offered a special quiet place to study to see if it would improve their grades.  Develop a Differences of Differences analysis for these two outputs.  Show that you obtain the same results from Regression #10 on page 9.  It is important that a difference –in- differences (the average treatment effect) is statistically significant.  Test if the average treatment effect is statistically significant.  Please show a picture of the distribution, and the critical t-value for α=.01.  Is your conclusion consistent with the p-value displayed?  Put in words what this p-value tells you.

In this question we will look at a policy that was only aimed at Fellowship student. So Fellowship students will constitute our Treatment group (T), whereas students without fellowship are the Control group (C). Also, year 2015 will be our second period (post) and year 2008 will be the pre-treatment period.

So what we want to assess is the change in the SAISGPA, for fellowship students compared to non-fellowship students, from before to after the policy change. This will be:

)-) 

=3.043+0.5544=3.597

=3.043

=3.173-0.112=3.061

=3.173 

 

We can see that this is exactly the effect that we find in regression 10 for the Y2015*Fellowship variable, which in fact gives us the average treatment effect. In regression 10 we have

 

where:

· The constant (3.173) gives us the SAISGPA for the Control group, before the policy change

· The coefficient on Fellowship (-.0112) gives us the pre-existing difference between the control and treatment group

· The coefficient on Y2015 (-.1296) gives us the pure time effect

· The coefficient on Y2015*Fellowship (0.666) is the average treatment effect

f. Regression #11 (page 10) changes the functional form by having the dependent variable (SAISGDP) logged as well as ColGDP.  State clearly in words the meaning of the slope coefficient on ColGDP.  State clearly in words the meaning the slope coefficient on InternshipHrs.  State clearly the meaning of the slope coefficient on Fellowship (compute in two different ways).

A 1% increase in college GPA will increase SAIS GPA by 0.63%, holding constant internship hours, study hours, whether a fellowship was obtained, and undergraduate major

A one-hour increase in internship hours per week will decrease SAIS GPA by 0.84%, holding constant CollegeGPA,  hours of study,  whether a fellowship was obtained, and undergraduate major

Having the fellowship will increase [exp(0.071) – 1]*100 = 7.36%, holding constant College GPA, study hours, internship hours and undergraduate major

g. The calculations on page 11 are used to compare Regression #11 with Regression #5.  Discuss this comparison with some mention of the sample correlation between a dependent variable and the predicted value of the dependent variable.

The r-squared tells us what proportion of the variation in y is predicted by our model. Also, it is the square of the correlation between y and y-hat. When our y-variable is logged, we cannot compare the fit of that model with the fit of a model where the y-variable is not logged. Thus, we have to calculate the adjusted predicted values of y (where the adjustment factor is ), find their correlation with the actual values of y, and square it to get the comparable r-squared. Once we do this, we find that the “linearized” R-square for the log regression is .818 which is  higher than the one we found in regression 5 of .810.

h. Regression #12 on page 12 is designed to examine the multicollinearity issues in Regression #5.  Compute the VIF (Variance Inflation Factor) for the StudyHrs Squared.  What impact do you think this VIF has on the standard errors (and therefore the t-values) for the StudyHrs and StudyHrsSq variables in Regression #5 on page.7.

The variance inflation factor is equal to   , where  is the R-squared from regressing independent variable j on all the other independent variables. This is one part of the equation for the variance of beta-hat j:

 

 

 

We know that the variance of beta-hat will increase when Xj is highly correlated with the other independent variables. It will also increase as the standard deviation of the u-hats increases, and will fall as the variable Xj is more varied.

 In this case, the  is .9861 so that the VIF = 1/(1-.9864) = 1/.0136 = 73.5.  

Thus the variance of the StudyHrs^2 variable is 73.5 times higher than it would be if that variable had a zero with the other independent variables.  Thus with the higher standard error, the t value will be lower

5.   Appendix 5 provides an analysis of two period panel data.

a. The data is listed on page 2 and 3.  Describe how the data is included in the data set.

This is panel data, with two rows of data for each city. The data include the population, number of crimes, crime rate, and unemployment rate for each city in 1982 and 1987. There is a dummy variable for whether the year is 1987. The data also include information on the change in the crime rate and unemployment rate from 1982 to 1987.

b. The first regression on page 1 treats the data as pooled cross section data.  Why would you think that the slope coefficient on unemployment rate (to explain the dependent variable crime rate) is statistically insignificant?

We have not accounted for any unobservable factors within the cities that relate to crime rate which means coefficient is likely to be biased. We have left out important independent variables that are likely correlated with the unemployment rate.

c. The second regression on page 1 takes the first difference of the two variables and we do end up with a statistically significant slope coefficient on the unemployment rate (change in the unemployment rate).  Discuss this in terms of how the fixed effects are handled.

This means that we have controlled for anything that does not change within a city over this time period. We controlled for these fixed effects by using first differences which would have these fixed effects drop out.

6.   Appendix 6 uses data that is not seasonally adjusted.  

The variable Women is the number of female wage and salary workers, 25 years old and over, who work part-time.  The variable Men is the number of male wage and salary workers, 25 years old and over, who work part-time.  The data is listed on Page 1 and top of Page 2

a) Write out Regression 1 in the usual fashion with standard errors and t statistics in parentheses.  Put in words the constant and each of the slope coefficients.  

 

             (95.81)                    (1.884)             (101.55)              (101.56)                (103.09)

             (98.68)                    (15.46)             (5.59)                       (3.93)                (4.63)

· The constant is the average number of women who work part-time in Q3 over the entire period

· The coefficient for time tells us that the number of women working part-time have increased by 29.1 every quarter from 2000 through the first half of 2016.

· The coefficient on Q1 tells us that on average from 2000 to the first half of 2016,  we can expect 567.97 more women who work part-time in quarter 1 than in quarter 3

· The coefficient on Q2 tells us that on average from 2000 to the first half of 2016,  we can expect 399.37 more women who work part-time in quarter 2 than in quarter 3

· The coefficient on Q4 tells us that on average from 2000 to the first half of 2016, we can expect 476.81 more women who work part-time in quarter 4 than in quarter 3

b) Briefly compare the difference between Regression 2 and Regression 1 in terms of the values of the coefficients.

Regression 2 does the same thing for men. Broadly, the coefficients have the same signs. The constant is lower for men, and the time trend is bigger. The coefficients on the seasonal dummies are all positive and again the coefficient for Q1 is the largest and the coefficient for Q2 is the smallest.

c) In Regression 3 and Regression 4 on Page 3, state clearly the meaning of the coefficient on Time and Q4.

Regression 3: the coefficient on time tells us that the number of women working part-time has increase by 0.273% every quarter from 2000 through the first half of 2016. The coefficient on Q4 tells that on average from 2000 thought the first half of 2016, approximately 4.5% more women work part-time in Q4 than in Q3 or (exp(.04549)-1)* 100  = 4.65% more women work part-time in Q4 than in Q3.

Regression 4: the coefficient on time tells us that the number of men working part-time has increased by 0.530% every quarter. The coefficient on Q4 tells that within each year, approximately 4.7% more men work part-time in Q4 than in Q3 or (exp(.04726)-1)*100 = 4.84% more men work part-time in Q4 than in Q3.

d) On Page 4 and top of Page 5 are four regressions carried out.  Discuss the R Squared value.  What would happen if you tried to “detrend” Q1?

The R-square is close to zero in all cases, which is not surprising since we are simply regressing an indicator of seasonality on a time trend .  There is no trend at all in seasonal dummy variables. If we tried to “detrend” on of the seasonal dummies, there would be no change.

e) Regression 9 is found on Page 5 with a follow up regression at the bottom of the page.  Discuss how the regression at the bottom of the page is related to Regression 1 on Page 2.

They are essentially doing the same thing. In regression 1 we are regressing women on a time trend and 3 seasonal dummies, here we are first de-trending women and then regressing the de-trended version on the seasonal dummies. But the result is the same because by including the time trend in regression 1 we are de fact de-trending the dependent variable.

7. Appendix 7 contains some U.S. National Income data quarterly seasonally adjusted (billions of U.S. $) from 2005 to early 2016.  The regressions on Page 2 set out to explain consumption of durable goods in terms of GDP and either investment in fixed assets or investment in residential structures.

a) For Regression 1 on Page 2, put in words the coefficient on GDP

For every additional billion of GDP, consumption of durables goods increases by 0.026 billion, controlling for (holding constant) fixed investment  at the current and two prior periods.

b) Draw a diagram of the lag distribution for Regression 1.  Put into words each of these coefficients.  Briefly state if this is what you expected.

 

Holding constant GDP, if Fixed investment increases by 1 billion $ in a given quarter, this function predicts that Durable Goods expenditure will increase by $.500 billion that quarter, fall by  $ .293 billion the following quarter, and then increase by $.001 billion the subsequent quarter.

c) For Regression 2 on Page 2, answer the same questions as in a) and b) above.

a) For every additional billion of GDP, consumption of durables goods increases by 0.063 billion, controlling for (holding constant) residential investment  at the current and two prior periods.

 

Holding constant GDP, if Residential investment increases by 1 billion $ in a given quarter, this function predicts that Durable Goods expenditure will increase by $.491 billion that quarter, fall by   $ .387 billion the following quarter, and then increase by $.238 billion the subsequent quarter.

8.  Appendix 8 uses the sample of eight young boys ages 6 to 12 with Height measured in inches and weight measured in pounds.  The regression at the top of Page 1 (not identified with a number) is the basic regression that sets out to explain the Weight of these children as a function of the Age and Height.

Regression 1 toward the bottom of Page 1 uses several additional variables.  What is being tested with this function?  What is the name of this test?  What is the conclusion?

This is a RESET test, i.e. a specification test. We want to assess whether our liner model misses any relevant non-linear predictor of the dependent variable.   We add the square and cube of the predicted value of the dependent variable and then rerun the regression adding these two variables.  If we can reject the null hypothesis that both coefficients are equal to zero using an F test, we would conclude that we have some non-linarites  that were not included.  In this case we end up with a very low F value resulting in a very high p value so we conclude that the linear model was correctly specified.

Regression 2 on Page 2 uses a different functional form and then predicts based on that new functional form.  Regression 3 at the bottom of Page 2 uses that prediction as an additional independent variable.  What is being tested here?  What is the name of the test?  What is the conclusion?

This is the Davidson-McKinnon test. The main idea behind the Davidson-MacKinnon Test is that if a particular specification is appropriate, then the fitted values of some alternative specification should not be significant predictors of the outcome variable. Indeed, the coefficient on the fitted values in regression 3 is not significant, suggesting the regression that uses the independent variables in level form is correctly specified.

Discuss Regression 4 on Page 3 and the tests that follow.  What is the name of this test?  What are the conclusions?

This is the Mizon and Richard approach to testing non-nested alternatives. It is a variation on the Davidson McKinnon test, and we reach the same conclusion as far as the logged variables are concerned. However, the test also fails to reject the null for the level variables, so it is inconclusive.

9   Appendix 9 lists the data now for 9 children except that Child 9 appears not to be a child

a) Regression 1 runs the regression that sets out to predict Weight as a function of Age and Height.  It generated two residuals.  Explain each of the residuals.

One is the regular OLS residual, the other one is the studentized residual, i.e. the residual standardized by its standard deviation which is computed using the standard error of estimate from the regression which excludes Child 9.

b) Regression 2 on Page 2 runs the regression again adding the variable Outlier.  Briefly describe the Outlier variable.  The Outlier variable has a coefficient, standard error, and t value.  Indicate where two of these numbers show up elsewhere on Appendix 9.

The outlier dummy is equal to 1 for observation 9, i.e. a child that exhibits an especially large weight compared to the rest of the sample. The coefficient will tell us the difference between the observed value of the outcome for the outlier and the predicted value of the Outlier variable from the regression line obtained without using the outlier observation.  Those calculations come at the end of the appendix leading to the number 94.624, the same as the coefficient on Outlier.   The t-value for that coefficient is 5.55, the same value as the studentized r residual from Regression 1.  

c) Regression 3 on the bottom of Page 2 and the calculations on Page 3 work to produce the final number on Page 3.  Discuss briefly what is being carried out with the regression and the calculations.

We are computing the residual for the outlier, based on a regression that excludes the outlier, and showing that it is equal to the coefficient that we found in the previous regression that did include the outlier but with a dummy variable for the Outlier included.