# DS 809 BUA The PACF Plots of The Series for Future Forecasting Question

DS 809 – Final ExamStatistical inference purposes you can use an 𝜶 (significance) level of 0.05. For each case, please

clearly state your hypotheses, rejection criteria, and conclusion when needed.

Please make sure to submit your R Code along with your detailed write-up on canvas.

1) Consider the data given in the file Q1data.csv where you have information about the average

daily fluctuations in the price of a product. Use the first 190 observations as your training set and

the rest as your test sample.

a) Obtain the time series, ACF, and the PACF plots of the series. What do you observe?

b) Find a suitable/best fit ARIMA/SARIMA model that best describes the data. Briefly

explain your reasoning. Write down your estimated model. Discuss the significance of

your estimated parameters. Obtain a fit plot (training sample data versus your fitted

model).

c) Investigate if the residuals from your best fit model are white noise.

d) Investigate if an ARCH/GARCH model is suitable for your model from part b. If needed,

re-estimate the model with the respective ARCH/GARCH structures. Report the new

model and discuss the significance of the parameters.

e) Using MAPE estimates, compare the predictive performance of the following two

approaches using the test sample:

1. 10-step ahead predictions using your best fit model.

2. Sequential predictions using your best fit model updated sequentially. For

instance, in predicting data point no: 191 use all the previous data (1:190), in

predicting point no:192 use all the previous data (1:191), etc. For each case, reestimate the respective model using the past data and obtain the prediction for the

next time period.

3. Which approach yields a better predictive performance.

2) Consider the monthly data given in the file unemployment.txt where the only variable is the

monthly unemployment rate in the United States for twenty-five years. Data starts in January

1980. Use the first 280 observations as your training sample and the rest as your test sample.

a) Obtain the time series and ACF plots of the series. What do you observe? Is the series

stationary? If needed, show how you can induce stationarity. Obtain the PACF of the

stationary series.

b) Based on your answer in part b, find a suitable ARIMA/SARIMA model that best

describes the unemployment data. Write down your estimated model. Discuss the

significance of your estimated parameters. Obtain a fit plot (training sample data

versus your fitted model).

c) Investigate if the residuals from your best fit model from part b are white noise.

d) Estimate a seasonal (monthly) indicator model with a trend component.

e) Obtain the 20-step ahead predictions for the test sample for your models from parts b

and d. Compare the predictive performances using the MSE and MAPE measures.

Which model provides a better predictive performance? Plot your predictions against

your test data.

f) Investigate if an ARCH/GARCH model is suitable for your model from part b. If

needed, re-estimate the model with the respective ARCH/GARCH structures. Report

the new model and discuss the significance of the parameters.

3) Q3_Mortgage.csv data contains monthly mortgage default counts of 4 different mortgage

pools initiated during years 1994, 1995, 1996, and 1997 during the same months (variables are

denoted as coh94, coh95, coh96, coh96 in the file). The first set of observations start in January

of 1997 for each cohort.

VARMA Model

a) Obtain the time series plots of all four series. What do you observe? Do you think there is

strong co-movement?

b) Obtain the ACF and CCF plots. What do you observe? Are all four series stationary? Can

you induce stationarity?

c) Using stationary transformations for all four series, investigate if you can find a suitable

VARMA model. You can identify several suitable VARMA models and compare their fit

performance. Which one provides the best fit? Write down your estimated model for the

best fit model. Discuss the significance of the model parameters. Briefly explain the

implications of your estimated model.

d) Check if the residuals of your estimated VARMA model exhibit white noise behavior.

e) Using your best fit model, obtain the plot of the actual data and your fitted model.

f) Using your best fit model, obtain predictions for the next 5 months and plot them.

Transfer Function Model

Consider using the 1994 cohort data (coh94) as your output variable and 1995 cohort data as

your input variable (coh95).

a) Investigate if you need to pre-whiten your input variable.

b) Obtain the CCF of your output and input (pre-whitened if needed) variables to identify a

suitable transfer function (TF) model.

c) Estimate a TF model and discuss the significance of the model parameters. Briefly

explain the implications of your estimated model. Hint: A simple low order model would

suffice in this case.

d) Check if the residuals are white noise.

e) Estimate a corrected TF model (TF-Noise Model) and write down your estimated model.

Show that the residuals from your corrected model are white noise. Compare the AIC

estimates for your TF model from part c and your corrected TF model.

f) Plot the actual data versus your fitted models.

UNEMPLOYMENT

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