SCHOOL OF ECONOMICS

University College Cork - National University of Ireland, Cork

MSc Finance (Banking and Risk Management)

EC6063 Applied Time Series Analysis

2021/22 21/03/22

EC6063 Assignment 2

Question 1:

Clearly explain the differences between an Autoregressive (AR) time series process and a Moving Average (MA) time series process, including examples of specifications of AR and MA processes.

What is the Autocorrelation Function? What is the Partial Autocorrelation Function? Show with the aid of diagrams, and clearly explain, the theoretical patterns of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for AR(1) and MA(1) processes and more generally for AR(p) and MA(q) processes.

Table 1 below presents results for a number of alternative ARIMA models. Which specification appears to be the most appropriate ARIMA model? Clearly explain your reasoning.

The following is an estimation using quarterly data on US inflation (INF) measured as the percentage change in the CPI and US wage growth (WGWTH) measured as the percentage change in average hourly earnings from 1971Q2 to 2010Q1.

INFt = -0.062 + 0.433INFt-1 + 0.062INFt-3 + 0.410WGWTHt

(t=-0.66) (t=6.35) (t=3.11) (t=3.74)

Given the information presented in Table 2 and that WGWTH2010Q2 = 0.6, WGWTH2010Q3 = 0.5 and WGWTH2010Q4 = 0.7, forecast inflation in 2010Q2, 2010Q3 and 2010Q4. Clearly show all of your workings.

Given that the Root Mean Square Error (RMSE) = 0.51264 from the above model find 95% forecast intervals for the inflation forecasts. The following information is also provided to you:

se(u_1 )=s

se(u_2 )=sv(1+?_1^2 )

se(u_3 )=sv(1+?_1^2+?_1^4 )

where se(ut) is the standard error of the forecast errors u1, u2, u3,…ut

Table 1: Alternative ARIMA models

p = 1-7

q = 0 p = 1-2

q = 0 p = 1, 2, 7

q = 0 p = 1

q = 1 p = 1, 2

q = 1 p = 1,2

q = 1,7

?1 1.177

(t=15.84) 1.108

(t=15.39) 1.088

(t=14.77) 0.817

(t=17.44) 0.444

(t=2.93) 0.312

(t=3.02)

?2 -0.466

(t=-4.17) -0.0244

(t=3.39) -0.210

(t=-2.73) 0.350

(t=2.42) 0.488

(t=4.82)

?3 0.386

(t=3.37)

?4 -0.339

(t=-2.93)

?5 0.319

(t=2.78)

?6 -0.379

(t=2.78)

?7 -0.150

(t=2.02) -0.045

(t=-1.22)

?1 0.368

(t=4.88) 0.725

(t=6.20) 0.862

(t=13.80)

?7 -0.144

(t=-3.77)

AIC 738.906 748.664 749.139 745.608 742.389 734.553

SC 764.712 758.341 762.042 755.285 755.292 750.682

Q(4) 0.230 6.874 6.246 6.598 1.422 1.355

Q(8) 4.501 21.286** 21.320** 20.481** 12.509 3.107

Q(12) 13.072 29.559** 28.081** 27.579** 21.090* 12.231

Notes:

1. p = number of AR terms, q = number of MA terms

2. t = t-values in parentheses

3. AIC = Akaike’s information criterion, SC = Schwarz’s information criterion

4. Q(n) = Ljung-Box (Q) test for white noise in the residuals where n represents the degrees of freedom.

** Reject H0 at 5% level of sig, * Reject H0 at 10% level of sig.

Table 2: US Inflation-US wage growth data set (selected dates)

US inflation

(INF) US wage growth (WGWTH)

2009q1 -0.56 0.78

2009q2 0.46 0.42

2009q3 0.91 0.66

2009q4 0.65 0.73

2010q1 0.38 0.51

Question 2:

Outline the main differences, including advantages and disadvantages, between VAR models and single equation time series models, with appropriate examples to illustrate.

Clearly explain the methodology underlying the Granger causality test.

The data file usdata.xlsx contains annual data for US real GDP growth (growth) and US inflation (inflation) from 1960 to 2019. Assume both variables are integrated or order 0 i.e. I(0).

Copy the data into STATA and create a time variable called ‘date’ in STATA and use this new variable to declare the data set to be time-series data.

Estimate two VAR models with (i) 1 extra lag of growth and inflation and (ii) 2 extra lags of growth and inflation. Which model is preferred? Explain your answer.

Carry out Granger causality test on your preferred model to see if growth ‘granger causes’ inflation and vice versa. Based on the results of the granger causality tests, do you think a VAR model was appropriate in this instance? Explain your answer.

Generate an impulse response function graph for your preferred model and comment on the results.

Question 3:

Clearly explain the terms auto-regressive conditional heteroscedasticity, with reference to the specification of an ARCH(1) model.

Clearly explain the differences between the ARCH and GARCH models. In an ARCH(1) model how would you interpret the coefficients? In a GARCH(1, 1) model how would you interpret the coefficients?

The data file nasdaq.xlsx contains monthly returns data for the Nasdaq stock market from 1988m1 to 2010m7.

Copy the data into STATA and create a time variable called ‘date’ in STATA and use this new variable to declare the data set to be time-series data. Plot the nasdaq data over time and comment on the extent of volatility in the data.

Estimate a GARCH(1, 1) model using the Nasdaq returns as the dependent variable in the mean equation with a lag of the dependent variable included as an independent variable. Interpret the results of the model.

Estimate a T-GARCH(1, 1) model and two GARCH (1, 1) in mean models, one using the variance to capture risk in the mean equation and one using the standard deviation to capture risk in the mean equation. Interpret the results of each of these models. Which of the four specifications is preferred? Clearly explain your answer.

Question 4:

You have panel data but decide to estimate a pooled least squares model. Outline two unrealistic assumptions underlying the pooled least squares model.

Compare and contrast the fixed effects and random effects models for estimating relationships using panel data. Include a description of the specification of each model, the advantages and disadvantages of each model and the essential differences between the two models.

Assignment Layout:

The project should contain the following parts:

Report ? This provides the answers to each of the questions above. When discussing work carried out in Stata, you should show the relevant Stata output in this report. You should copy and paste the Stata output from your log (see below), change the font to Courier New and the font size to 8. Always proofread your work before submission.

Appendix: Stata Log ? you create a log at the start of your session. You can add to your own comments to the log if you desire. Please ensure that you save the log using your own student number i.e. *********.log

Submission Details:

This is an individual assignment. The deadline is Tuesday April 19th 2022. The assignment is to be uploaded to Canvas by 6pm on that day. You are allowed unlimited submissions before the deadline, Canvas keeps the latest submission.

Answers should be typed. Only Microsoft Word documents (.doc or .docx) can be submitted to Canvas. A front page should be included containing the following information: Essay title, name and student identification number, word count and submission date. The level of professional presentation, which includes writing style, layout and presentation will be considered when deciding on a grade.

When submitting your Assignment online through Canvas, a Plagiarism review will automatically be carried out using Turnitin. This will provide students with a similarity report – where elements of the student’s work is similar to material found via the internet and other student submissions. I will set up the online submission in such a way that you can submit multiple times until the due date and therefore use Turnitin to identify large segments of unoriginal material before the final submission date.

Please ensure that you have a backup copy of your assignment, in case material is lost. Those who submit their assignments late, citing this excuse (or similar excuses) will not be accepted!

Canvas will accept submissions after the deadline but they will be marked late and a late penalty will apply. Penalties: Where work is submitted up to and including 7 days late, 10% of the total marks available shall be deducted from the mark achieved. Where work is submitted up to and including 14 days late, 20% of the total marks available shall be deducted from the mark achieved. Work submitted 15 days late or more shall be assigned a mark of zero.

This assignment counts for 50% of your final mark in EC6063.

University College Cork - National University of Ireland, Cork

MSc Finance (Banking and Risk Management)

EC6063 Applied Time Series Analysis

2021/22 21/03/22

EC6063 Assignment 2

Question 1:

Clearly explain the differences between an Autoregressive (AR) time series process and a Moving Average (MA) time series process, including examples of specifications of AR and MA processes.

What is the Autocorrelation Function? What is the Partial Autocorrelation Function? Show with the aid of diagrams, and clearly explain, the theoretical patterns of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for AR(1) and MA(1) processes and more generally for AR(p) and MA(q) processes.

Table 1 below presents results for a number of alternative ARIMA models. Which specification appears to be the most appropriate ARIMA model? Clearly explain your reasoning.

The following is an estimation using quarterly data on US inflation (INF) measured as the percentage change in the CPI and US wage growth (WGWTH) measured as the percentage change in average hourly earnings from 1971Q2 to 2010Q1.

INFt = -0.062 + 0.433INFt-1 + 0.062INFt-3 + 0.410WGWTHt

(t=-0.66) (t=6.35) (t=3.11) (t=3.74)

Given the information presented in Table 2 and that WGWTH2010Q2 = 0.6, WGWTH2010Q3 = 0.5 and WGWTH2010Q4 = 0.7, forecast inflation in 2010Q2, 2010Q3 and 2010Q4. Clearly show all of your workings.

Given that the Root Mean Square Error (RMSE) = 0.51264 from the above model find 95% forecast intervals for the inflation forecasts. The following information is also provided to you:

se(u_1 )=s

se(u_2 )=sv(1+?_1^2 )

se(u_3 )=sv(1+?_1^2+?_1^4 )

where se(ut) is the standard error of the forecast errors u1, u2, u3,…ut

Table 1: Alternative ARIMA models

p = 1-7

q = 0 p = 1-2

q = 0 p = 1, 2, 7

q = 0 p = 1

q = 1 p = 1, 2

q = 1 p = 1,2

q = 1,7

?1 1.177

(t=15.84) 1.108

(t=15.39) 1.088

(t=14.77) 0.817

(t=17.44) 0.444

(t=2.93) 0.312

(t=3.02)

?2 -0.466

(t=-4.17) -0.0244

(t=3.39) -0.210

(t=-2.73) 0.350

(t=2.42) 0.488

(t=4.82)

?3 0.386

(t=3.37)

?4 -0.339

(t=-2.93)

?5 0.319

(t=2.78)

?6 -0.379

(t=2.78)

?7 -0.150

(t=2.02) -0.045

(t=-1.22)

?1 0.368

(t=4.88) 0.725

(t=6.20) 0.862

(t=13.80)

?7 -0.144

(t=-3.77)

AIC 738.906 748.664 749.139 745.608 742.389 734.553

SC 764.712 758.341 762.042 755.285 755.292 750.682

Q(4) 0.230 6.874 6.246 6.598 1.422 1.355

Q(8) 4.501 21.286** 21.320** 20.481** 12.509 3.107

Q(12) 13.072 29.559** 28.081** 27.579** 21.090* 12.231

Notes:

1. p = number of AR terms, q = number of MA terms

2. t = t-values in parentheses

3. AIC = Akaike’s information criterion, SC = Schwarz’s information criterion

4. Q(n) = Ljung-Box (Q) test for white noise in the residuals where n represents the degrees of freedom.

** Reject H0 at 5% level of sig, * Reject H0 at 10% level of sig.

Table 2: US Inflation-US wage growth data set (selected dates)

US inflation

(INF) US wage growth (WGWTH)

2009q1 -0.56 0.78

2009q2 0.46 0.42

2009q3 0.91 0.66

2009q4 0.65 0.73

2010q1 0.38 0.51

Question 2:

Outline the main differences, including advantages and disadvantages, between VAR models and single equation time series models, with appropriate examples to illustrate.

Clearly explain the methodology underlying the Granger causality test.

The data file usdata.xlsx contains annual data for US real GDP growth (growth) and US inflation (inflation) from 1960 to 2019. Assume both variables are integrated or order 0 i.e. I(0).

Copy the data into STATA and create a time variable called ‘date’ in STATA and use this new variable to declare the data set to be time-series data.

Estimate two VAR models with (i) 1 extra lag of growth and inflation and (ii) 2 extra lags of growth and inflation. Which model is preferred? Explain your answer.

Carry out Granger causality test on your preferred model to see if growth ‘granger causes’ inflation and vice versa. Based on the results of the granger causality tests, do you think a VAR model was appropriate in this instance? Explain your answer.

Generate an impulse response function graph for your preferred model and comment on the results.

Question 3:

Clearly explain the terms auto-regressive conditional heteroscedasticity, with reference to the specification of an ARCH(1) model.

Clearly explain the differences between the ARCH and GARCH models. In an ARCH(1) model how would you interpret the coefficients? In a GARCH(1, 1) model how would you interpret the coefficients?

The data file nasdaq.xlsx contains monthly returns data for the Nasdaq stock market from 1988m1 to 2010m7.

Copy the data into STATA and create a time variable called ‘date’ in STATA and use this new variable to declare the data set to be time-series data. Plot the nasdaq data over time and comment on the extent of volatility in the data.

Estimate a GARCH(1, 1) model using the Nasdaq returns as the dependent variable in the mean equation with a lag of the dependent variable included as an independent variable. Interpret the results of the model.

Estimate a T-GARCH(1, 1) model and two GARCH (1, 1) in mean models, one using the variance to capture risk in the mean equation and one using the standard deviation to capture risk in the mean equation. Interpret the results of each of these models. Which of the four specifications is preferred? Clearly explain your answer.

Question 4:

You have panel data but decide to estimate a pooled least squares model. Outline two unrealistic assumptions underlying the pooled least squares model.

Compare and contrast the fixed effects and random effects models for estimating relationships using panel data. Include a description of the specification of each model, the advantages and disadvantages of each model and the essential differences between the two models.

Assignment Layout:

The project should contain the following parts:

Report ? This provides the answers to each of the questions above. When discussing work carried out in Stata, you should show the relevant Stata output in this report. You should copy and paste the Stata output from your log (see below), change the font to Courier New and the font size to 8. Always proofread your work before submission.

Appendix: Stata Log ? you create a log at the start of your session. You can add to your own comments to the log if you desire. Please ensure that you save the log using your own student number i.e. *********.log

Submission Details:

This is an individual assignment. The deadline is Tuesday April 19th 2022. The assignment is to be uploaded to Canvas by 6pm on that day. You are allowed unlimited submissions before the deadline, Canvas keeps the latest submission.

Answers should be typed. Only Microsoft Word documents (.doc or .docx) can be submitted to Canvas. A front page should be included containing the following information: Essay title, name and student identification number, word count and submission date. The level of professional presentation, which includes writing style, layout and presentation will be considered when deciding on a grade.

When submitting your Assignment online through Canvas, a Plagiarism review will automatically be carried out using Turnitin. This will provide students with a similarity report – where elements of the student’s work is similar to material found via the internet and other student submissions. I will set up the online submission in such a way that you can submit multiple times until the due date and therefore use Turnitin to identify large segments of unoriginal material before the final submission date.

Please ensure that you have a backup copy of your assignment, in case material is lost. Those who submit their assignments late, citing this excuse (or similar excuses) will not be accepted!

Canvas will accept submissions after the deadline but they will be marked late and a late penalty will apply. Penalties: Where work is submitted up to and including 7 days late, 10% of the total marks available shall be deducted from the mark achieved. Where work is submitted up to and including 14 days late, 20% of the total marks available shall be deducted from the mark achieved. Work submitted 15 days late or more shall be assigned a mark of zero.

This assignment counts for 50% of your final mark in EC6063.

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