Problem Set #4                                                                                    Economics 43

Due Friday, November 14th at the start of class                             Prof. Stephen Schmidt

 

This problem set gives you a chance to practice the techniques that econometricians use when the assumptions of the Gauss-Markov theorem do not hold. The first problem deals with serial correlation; the second with heteroskedasticity.

 

1. The Federal Reserve Board exercises control over the macroeconomy by altering the money supply to meet economic demand for various forms of money. In order to do this, they need accurate models of money demand; such models are estimated on time series data and are often subject to the problem of autocorrelation.

In Eviews, load the datafile pset4.wf1 from the Courses/Econ/Eco043 folder. This datafile contains quarterly data from 1950 to 2002 on the following variables (not all of which have data over the full sample):

 

M1                               - The money supply, measured by M1, in current (nominal) dollars

PCHANGEM1            - The percent increase (or decrease) in M1 from previous quarter

GOVEXP                   - Total Federal expenditure (billions of current dollars)

PCHANGEEXP            - The percent increase in GOVEXP from previous quarter

LABORFORCE             - Size of the labor force (millions of persons)

PARTICIPATION            - Labor force participation rate (percentage points)

UNEMPLOYMENT            - The unemployment rate (percentage points)

a) Plot a line graph of M1. Does it appear to be growing smoothly, or does its growth rate vary over time? In what time period did it grow fastest? In what years has it shrunk?

b) Use this data to estimate the following regression (print the results and turn them in along with your answers):

            M1 = b1 + b2* PCHANGEEXP + b3*UNEMPLOYMENT + b4*LABORFORCE + e

 

What is the Durbin-Watson statistic for this regression? What are the lower and upper critical values for the Durbin-Watson test for this regression? Does this regression have autocorrelation, or not? If it does, re-estimate the model using an AR(1) correction. What is the estimated value of r? What problem does this suggest?

c) To solve this problem, we’ll estimate the regression expressing some of the variables in percentage growth terms instead of levels. We’ll drop LABORFORCE and replace it with PARTICIPATION (because labor force participation is a percentage) and we won’t change UNEMPLOYMENT because it is already a percentage. Estimate the regression:

PCHANGEM1 = b1 + b2*PCHANGEEXP + b3*UNEMPLOYMENT +
                        b5* PARTICIPATION + e

 

This regression is based on a model first proposed by Robert Barro, an early monetarist, who believed that money demand and labor supply, including unemployment, were closely linked. (The model was proposed in the paper “Unanticipated Money Growth and Unemployment in the United States”, American Economic Review, in 1977. You may consult this paper if you would like to know more about this theory, although I strongly suggest you not try to read the econometric sections of that paper, which are very complex.) Are any of the variables insignificant? If so, which? (You do not need to print this regression.)

c) Try dropping variables until you reach a model in which all the variables included are significant. Do not drop any variables that are already significant, and make sure by appropriate T-tests, or F-tests, that any variables you drop are definitely irrelevant to the model. Show the results of your T-test or F-test in your solutions. When you have a model in which all independent variables are significant, print that regression and turn it in along with your answers. Then, in the regression window, press the View button, select Actual-Fitted-Residual, and select Graph. Based on this graph, do the errors look autocorrelated or not? How do you know?

d) What is the Durbin-Watson statistic for this model? What are the critical values for the Durbin-Watson test for this data? Do you have autocorrelation, no autocorrelation, or can you not tell? If you have autocorrelation, is it positive or negative?

e) Rerun the regression including an AR(1) term in the model to correct for autocorrelation. Print the results of this model and turn them in with your answers. Have the estimates of the parameters changed very much? Are the standard errors larger or smaller after the AR(1) term is included? How have the t-statistics changed? Is the AR(1) term significant or not? Does it take the sign you expected based on your answer to part d? Does the Durbin-Watson statistic look acceptable now, or not?

 

2. In this problem, we’ll examine the productivity of American manufacturing industries, checking to see whether the data is heteroskedastic, and whether the heteroskedasticity affects the results of the regression if so. Load the datafile pset4a.wf1 from the Courses/Econ/Eco043 folder. This data set (which we have also used in class) contains the following observations on 435 industries from 1994:

SIC                  - Standard Industry Code number for this industry

SHIPMENTS  - Total value of output shipments by this industry in 1994 (thousands $)

MANAGERS  - Total number of managers employed by this industry

WORKERS  - Total number of production workers employed by this industry

MATERIALS - Total dollar value of materials used by this industry (thousands $)

NEWCAP       - Total expenditure on new capital by this industry (thousands $)

INVENTORY - Total value of inventories held by firms in this industry (thousands $)

a) Do a Histogram and Stats on SHIPMENTS. Are these industries all about the same size, or is there a lot of variation in the value of shipments across industries? How many times bigger are the shipments of the largest industry than the shipments of the smallest industry?

b) Run the following regression:

 

SHIPMENTS = b1 + b2*MANAGERS + b3*WORKERS + b4*MATERIALS +

b5*NEWCAP + b5*INVENTORY + e

 

How many variables are significant? What is the Durbin-Watson statistic for this regression? Does that surprise you or not surprise you, and why?

c) In the regression window, go to the View button, select Residual Tests, and select White test (no cross terms). Is this regression heteroskedastic or not?

d) First we’ll try to ignore the heteroskedasticity, and just turn on the Robust Standard Errors option. In the regression window, click the Estimate button, then click the Options button, and turn on the Heteroskedasticity-Corrected Standard Errors option. Then rerun the regression. Have the parameter estimates changed? Have the standard errors changed? Do things seem to be more precisely or less precisely estimated than they were before?

e) Now we’ll try to fix the heteroskedasticity with a weighting series. Let’s assume that the standard deviation of the errors is proportional to MATERIALS (this turns out to be a reasonably good assumption). We’ll need to generate a weighting series for the regression; go to the Eviews command line and type genr weight = 1/materials. Then go back to the regression window; click the Estimate button, then click the Options button, and turn on the Weighted Least Squares option. Type weight for the name of the weighting series. (Turn off the Heteroskedasticity-Corrected Standard Errors option if it’s still on.) Then click OK, and rerun the regression. Now have the parameter estimates changed? Are things estimated more precisely or less precisely then they appeared to be before? Which estimated standard errors are the correct ones?

f) In the corrected regression, again do the White test (no cross terms). Has the heteroskedasticity been eliminated or not? Do you think the weighting series helped or not? Why?
g) Last, we’ll try changing the functional form to see if we can get rid of the heteroskedasticity that way; we’ll convert to a log-linear functional form, corresponding to a Cobb-Douglass production function for each industry. Type the following commands to convert each variable into log format:

 

genr logships = log(shipments)

genr logmangs = log(managers)

genr logworks = log(workers)

genr logmats = log(materials)

genr lognewcap = log(newcap)

genr loginv = log(inventory)

 

Then estimate the following regression without using weighted least squares (click on the Options button before estimating the regression and make sure the Weighted option is turned off):

 

LOGSHIPS = b1 + b2*LOGMANGS + b3*LOGWORKS + b4*LOGMATS +

                                    b5*LOGNEWCAP + b5*LOGINV + e

 

Have any of the coefficients changed signs? Do any of these results look surprising? Which of the two functional forms do you prefer, or does it not seem to make much difference? Perform the White test again. Has the change in functional form gotten rid of the heteroskedasticity? If not, are you closer to passing the test than you were before?