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### Answer - Introductory Econometrics - Coursework

Introductory Econometrics, EC2017 Coursework Autumn Term, 2016

**Introduction**:

Please make sure you follow the instructions exactly so as to complete this coursework accurately. Therefore, read this document carefully and take the steps in the order they are specified.

**General Instructions**:

1. There are three tasks in this coursework

2. Each task asks you to perform some statistical computing. These will be based on the dataset and samples you take from the population data (see instructions below)

3. You need to work independently and generate your unique dataset by the methods shown below. Please note that failure to work independently will be considered as *academic misconduct*

4. You need to perform the calculations, show all workings, type up and save your work in a single Microsoft Word document and submit it via Moodle before the deadline set. If you use Excel to do any calculation then all such Excel work together with the files should be submitted via Moodle

5. Corresponding to each question within a task, there will be one numeric answer based upon the calculations you have performed above. You will need to enter this answer in an Excel spreadsheet as specified below. Save the Excel file when you have entered all the answers as per instructions and submit it via Moodle before the deadline

6. *Note that it is the answers given in the Excel spreadsheet that will be marked and you will receive the coursework grade based on these answers*

7. *The submitted Word file is for verification purposes only. This means that the examiner must be able to verify how and where you got your numeric answers from. Therefore, you cannot be given any marks if the workings and calculations are not shown in the Word document you submit*

8. You need to run the regression analysis and perform matrix operations using Stata and Stata only. You will not be given any credit if you use any other software to run regressions or perform matrix operations. You must create and submit a Stata .log file detailing all the computations you perform using Stata.

9. At the end, you are required to submit via Moodle: (1) the Excel spreadsheet with your answers; (2) a Word document with your workings and calculations; (3) a STATA log file detailing all your computations. If you do all your calculations in Excel, you are also required to submit (4) an Excel file with your calculations.

** **

**Drawing your sample **

1. First, click open the Excel file ** ec2017cwkdata.xls** uploaded on Moodle.

2. Open the sheet ‘Allocated Number’ and note the number (n) next to your registration/id no.

3. Open the sheet ‘Population’ containing data on demand of a particular product. Here, logq is the logarithm of quantity demanded (q), logy is the logarithm of buyer income (y), logp is the logarithm of the own price of the product (p) and logp2 is the logarithm of the price of a related product (p2).

4. Copy and paste ALL the data on to Stata Data Editor. Choose the option that the first row is treated as the name of a variable.

5. Generate your sample as follows. In the command window (in Stata) type

*sample n, count *

where n is the number you are allocated. For example, to randomly generate a sample of size 35 you should type

*sample 35, count *

Save your sample of n observations as a Stata data file. You will use this sample of observations for all three tasks in the coursework.

**Task 1 (36 marks) **

Task 1 is an exercise on prediction (or forecasting) in the two-variable Classical Linear Regression Model (CLRM)

Consider the following regression model:

*where X is an explanatory variable of your choice*

1. Use STATA to regress the dependent variable log [quantity demanded] expressed as logq on one of the three explanatory variables logy, logp and logp2 (one of your choice). Suppose you try to test how good your regression model is by acquiring two additional observed values of the explanatory variable (values that are in the population data, but not in your sample data). For each observed value, predict the corresponding value of logq and compare it with the actual value of logq as given in the population data.

2. Open the Excel file named ** Answers.xls** and click open the sheet Task1. In cell A2 (colour coded blue), write down the value n (number of observations) allocated to you. Enter, in cells E4, E5, E6 and E8 respectively (all colour coded blue), the estimated values of the constant term, the slope term, the sample mean of the explanatory variable (X) and the standard error of the regression.

3. Enter in cell E10 the value of Explained Sum of Squares (ESS) as given in the Stata output.

4. Randomly choose another observed value of the explanatory variable (a value that is in the population data, but not in your sample data) and the corresponding value of the dependent variable. Enter, in cells E12 and E13 respectively (colour coded blue), the value of the newly acquired observation on the dependent variable and the corresponding value of the explanatory variable.

a. Obtain a point estimate of the predictor and enter this value in cell E15 (colour coded green); (3 marks)

b. Obtain a point estimate of the error of prediction (or forecast error) and enter this value in cell E17 (colour coded green); (3 marks)

c. Estimate the variance of the predictor and enter this value in cell E20 (colour coded green); (6 marks)

d. Estimate the variance of the error of prediction and enter this value in cell E22 (colour coded green); (6 marks)

e. Estimate the 95% confidence interval of and enter the lower and upper limit values of this interval in cells E26 and E27 (both colour coded green), respectively; (6 marks)

f. Randomly choose a second observed value of the explanatory variable (a value that is in the population data, but not in your sample data and that you have not previously chosen) Repeat points (a)-(e) for this new observed value (added results in cells E33, E35, E38, E40, E44 and E45); (8 marks)

g. Describe whether there are any differences in terms of confidence interval and forecasting error in the two cases. What did you learn about extrapolating the regression line to predict values of the dependent variable associated with given values of the independent variable? (add our explanations in the Word document you will submit). (4 marks)

END OF TASK 1

**Task 2 (32 marks) **

Task 2 is an exercise on the CLRM:

*The objectives are (a) construction of the 95% confidence interval of a linear combination of parameters and (b) performing a hypothesis test *

1. Use STATA to regress the dependent variable log[quantity demanded] expressed as logq on the three explanatory variables logy, logp and logp2.

2. Open the Excel file named ** Answers.xls** and click open the sheet Task2.

3. In cell A2 (colour coded blue), enter the value n (number of observations) allocated to you. Enter, in cells D10, D11, D12 and D13 respectively (all colour coded blue), the estimated values of the coefficients for logy, logp, logp2 and the constant term, as reported in the Stata output.

4. Copy the elements of the estimated variance-covariance matrix (e(V)) exactly as given in the Stata output in the 4 X 4 array of cells A5-D8, (colour coded blue).

**Note**: You must run the regression so that the Stata output e(V) shows the exact order of the explanatory variables as follows (STATA command is *mat list e(V)*):

symmetric e(V)[4,4]

logy logp logp2 _cons

logy .0005018

logp -.000007132 .00057139

logp2 -.00002487 .000003610 .00045251

_cons -.04728342 -.05761722 -.04269444 14.945162

Thus, given that my e(V) output is as above, I should enter the following numbers in the array A5-D8 as shown below:

.0005018 |
-.00000713 |
-.00002487 |
-.04728342 |

-.00000713 |
.00057139 |
.000003610 |
-.05761722 |

-.00002487 |
.000003610 |
.00045251 |
-.04269444 |

-.04728342 |
-.05761722 |
-.04269444 |
14.945162 |

*Failure to follow the above steps will cost you marks *

a. Calculate the 95% confidence interval for the parameter value : enter the values of the lower and upper limit of this interval in cells E16 and E17, respectively (both colour coded green); (8 marks)

b. Test the hypothesis that the estimated demand function is homogeneous of degree 0 in income y and the two prices p and p2:

- Enter the value of the t-statistic as per your calculations in cell E21 (colour coded green); (6 marks)
- Enter the number 0 in cell E24 (colour coded green) if you reject the null hypothesis – otherwise enter the number 1; (6 marks)
- Enter the number 0 in cell E27 (colour coded green) if your test is significant – otherwise enter the number 1; (6 marks)
- Enter the number 0 in cell E30 (colour coded green) if there is enough evidence to suggest that demand is not homogeneous of degree 0 – otherwise enter the number 1. (6 marks)

END OF TASK 2

**Task 3 (32 marks) **

Task 3 is an exercise on the CLRM:

*The objective is to test whether or not a ‘structural break’ occurs as new data is acquired *

Use the Stata multiple regression output you obtained in Task 2 above as the estimated regression output based on original data. Next, randomly select data on *m* more observations (different from those you had selected earlier) where *m* is an integer of your choice satisfying. You hypothesise that the additional data fits into the original model. You will now perform a Chow test at the 5% level significance whether or not your hypothesis is correct.

1. Open the Excel file named ** Answers.xls** and click open the sheet Task3.

2. In cell A2 (colour coded blue), write down the value n (number of observations) allocated to you.

3. Enter the value of RSSR and RSSU in cell D2 and E2 (colour coded blue), respectively

4. In cell A4 (colour coded blue), write down the value of *m* you have selected.

- Write down the null hypothesis and the alternative hypothesis for this Chow Test (in the Word document); (3 marks)
- Enter the value of the F-statistic as per your calculations in cell E7 (colour coded green); (5 marks)
- Enter the critical F value for this test in cell E9 (colour coded green); (4 marks)
- Enter the number 1 in cell E12 (colour coded green) if you reject the null hypothesis based on your analysis – otherwise enter the number 0. (4 marks)
- Enter the number 1 in cell E15 (colour coded green) if your test is significant – otherwise enter the number 0. (4 marks)
- Enter the number 1 in cell E18 (colour coded green) if your conclusion is that there is a structural break – otherwise enter the number 0. (4 marks)
- Consider now that your sample data have a time dimension. Add a variable called
*year*to your data starting in the past and ending in 2016 (your*m additional observations are the most recent ones*). Use this time variable and explain how you will conduct a dummy variable alternative to the Chow Test.- Create the dummy variable
- Specify the multiple regression model
- Write the mean demand functions for the different time periods
- Report the differential intercept and the differential slope coefficients (in cells E21 and E22)

(8 marks)

END OF TASK 3

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