date	CPI
2010-01-01	2.51
2010-02-01	2.12
2010-03-01	2.43
2010-04-01	2.67
2010-05-01	2.45
2010-06-01	2.38
2010-07-01	2.27
2010-08-01	2.38
2010-09-01	2.33
2010-10-01	2.48
2010-11-01	2.60
2010-12-01	3.04
2011-01-01	3.36
2011-02-01	3.64
2011-03-01	3.47
2011-04-01	3.78
2011-05-01	3.81
2011-06-01	3.62
2011-07-01	3.76
2011-08-01	3.89
2011-09-01	4.44
2011-10-01	4.30
2011-11-01	4.07
2011-12-01	3.69
2012-01-01	3.12
2012-02-01	3.07
2012-03-01	3.06
2012-04-01	2.76
2012-05-01	2.53
2012-06-01	2.27
2012-07-01	2.36
2012-08-01	2.27
2012-09-01	2.08
2012-10-01	2.42
2012-11-01	2.44
2012-12-01	2.43
2013-01-01	2.42
2013-02-01	2.48
2013-03-01	2.52
2013-04-01	2.20
2013-05-01	2.41
2013-06-01	2.60
2013-07-01	2.47
2013-08-01	2.40
2013-09-01	2.36
2013-10-01	1.99
2013-11-01	1.92
2013-12-01	1.89
2014-01-01	1.79
2014-02-01	1.65
2014-03-01	1.52
2014-04-01	1.68
2014-05-01	1.46
2014-06-01	1.80
2014-07-01	1.57
2014-08-01	1.55
2014-09-01	1.30
2014-10-01	1.35
2014-11-01	1.09
2014-12-01	0.75
2015-01-01	0.54
2015-02-01	0.35
2015-03-01	0.33
2015-04-01	0.26
2015-05-01	0.45
2015-06-01	0.34
2015-07-01	0.47
2015-08-01	0.37
2015-09-01	0.25
2015-10-01	0.25
2015-11-01	0.44
2015-12-01	0.52
2016-01-01	0.64
2016-02-01	0.62
2016-03-01	0.76
2016-04-01	0.70
2016-05-01	0.73
2016-06-01	0.84
2016-07-01	0.90
2016-08-01	0.96
2016-09-01	1.26
2016-10-01	1.28
2016-11-01	1.47
2016-12-01	1.79
2017-01-01	1.93
2017-02-01	2.33
2017-03-01	2.35
2017-04-01	2.65
2017-05-01	2.73
2017-06-01	2.57
2017-07-01	2.55
2017-08-01	2.74
2017-09-01	2.81
2017-10-01	2.78
2017-11-01	2.85
2017-12-01	2.72
2018-01-01	2.71
2018-02-01	2.45
2018-03-01	2.29
2018-04-01	2.20
2018-05-01	2.30
2018-06-01	2.30
2018-07-01	2.29
2018-08-01	2.39
2018-09-01	2.20
2018-10-01	2.23
2018-11-01	2.15
2018-12-01	2.01
2019-01-01	1.77
2019-02-01	1.81
2019-03-01	1.84
2019-04-01	2.00
2019-05-01	1.96
2019-06-01	1.94
2019-07-01	1.99
2019-08-01	1.70
2019-09-01	1.69
2019-10-01	1.50
2019-11-01	1.50
2019-12-01	1.37
2020-01-01	1.75
2020-02-01	1.73
2020-03-01	1.55
2020-04-01	0.92
2020-05-01	0.68
2020-06-01	0.80
2020-07-01	1.15
2020-08-01	0.51
2020-09-01	0.75
2020-10-01	0.89
2020-11-01	0.58
2020-12-01	0.82
2021-01-01	0.93
2021-02-01	0.73
2021-03-01	0.96
2021-04-01	1.64
2021-05-01	2.12
2021-06-01	2.45
2021-07-01	2.09
2021-08-01	3.03
2021-09-01	2.92
2021-10-01	3.83
2021-11-01	4.58
2021-12-01	4.84
2022-01-01	4.90
2022-02-01	5.48
2022-03-01	6.22
2022-04-01	7.78
2022-05-01	7.87
2022-06-01	8.18
2022-07-01	8.76
2022-08-01	8.61
2022-09-01	8.81
2022-10-01	9.59
2022-11-01	9.35
2022-12-01	9.24
2023-01-01	8.84
2023-02-01	9.16
2023-03-01	8.87
2023-04-01	7.85
2023-05-01	7.89
2023-06-01	7.31
2023-07-01	6.39
2023-08-01	6.30

You are given monthly data for the UK Consumer Price Index (CPI) over the period 2010M1 to 2023M8. The data file name is "CPI.xls", which is uploaded to Canvas along with this file. First calculate the UK inflation rate, i.e., cpit=cpitcpit1, where cpit is the natural logarithm of the CPI at time t and is the first difference operator. Then: [10\%] d) Identify the appropriate models that you would estimate for the cpit series based on a sample period from 2010M1 to 2021M12 (note that this is not the full sample) by: i) Obtaining the autocorrelation function (ACF) and the partial autocorrelation function (PACF) for the cpir series (specify the number of lags as 6). [10%] ii) Estimating all ARMA models from order (0,0) to (6,6) for the cpit series. (You would also need to report all relevant information for the models that you estimate, including the value of the AIC and SBIC and other relevant required criteria in a table). [20%] e) Re-estimate only the appropriate model(s) identified in question (d). Again, use only the 2010M1 to 2021M12 sample. Report and comment on the results. Carry out diagnostic checks on the residuals from these estimated model(s). Do the model(s) fit the data well? [10%] f) Use the model(s) estimated in question (e) to generate one step ahead (static) forecasts for the period 2022M1 - 2023M8. Plot a graph of the actual cpit series and the forecasts that you have generated over the specified out-of-sample period. Comment on the results. [15%] Conduct all your statistical tests at the 5% level