N1611 Financial Econometrics Coursework: Analysis and Interpretation

This Financial Econometrics assignment solution addresses two key questions. The first question explores the ARMA model, its evolution from the work of Word (1938), and the Box-Jenkins approach, including the three-stage iterative process of identification, estimation, and model checking. It emphasizes the importance of data stationarity and describes how to identify and transform non-stationary time series data using techniques like the Box-Cox transformation. The second question analyzes Consumer Price Index (CPI) data from 1960 to 2012, discussing its significance in measuring inflation and the cost of living. It includes a graphical representation of the CPI trend and analyzes autocorrelation and partial correlation functions. The assignment uses provided CPI data to demonstrate practical applications of econometric techniques.

Running head: FINANCIAL ECONOMETRICS
Financial Econometrics
Name of the Student:
Name of the University:
Authors Note:

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FINANCIAL ECONOMETRICS
Contents
Question 1:.......................................................................................................................................2
Question 2:.......................................................................................................................................5
References:....................................................................................................................................38

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Question 1:
Word (1938) combined both Auto Regressive (AR) and Moving Average (MA) schemes
to show prove that all stationery items can be modelled using the combined method of ARMA
provided the underlying variables such as number of AR terms and number of MA terms are
correctly specified. With the combined process any series xt can be modelled by using the
following equation:
ARIMA model proposed by Box and Jenkins is also known as Box-Jenkins Approach with “I”
between AR and MA denoting “integration”. The popularity of the model soared during 1970s
with use of empirical study however, it was shattered when the empirical studies later showed
that simple methods were providing more accurate data (Maatallah et. al. 2015).
Preliminary data transformation analysis with an iterative three stage process is described in the
Box Jenkins model. The three stage iterative process includes the following:
I. Identification model.
II. Estimation model.
III. Model checking (Li et. al. 2016).
In the above analysis the assumption is that the realization of stationery process is fundamental
to the time series analysis. In order to describe the DGP in box Jenkins model, ARMA (p, q) is
required to be used for stationery as well as invertible. Hence, at the beginning it is essential to
determine whether the realization of stationery process is about the time series in a fact of case
(Box et. al. 2015). In case it is not then the time series needs to be transformed into stationery

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process. A time series is considered as realization of stationery stochastic process if the
following conditions are satisfied:
I. No systematic change in mean.
II. No systematic change in variance and
III. No periodic verification.
By using graph of the time series values can be obtained to assess the useful indications
concerning the stationery process. Constant variation with constant means in the time series
fluctuation will indicate that the process is stationery otherwise it is non-stationery. The image
below is a typical time series with stationery process (Gairaa et. al. 2016).
A non-stationery process in the time series is represented in the image below which shows the
total number of passengers in international airline.
The above graph shows that the total number of international passengers in airline increases over
time with variance also to increase over a time. Apart from that there will certain peak seasons in
the above graph. As already mentioned in stationery process that the mean and variance shall be
constant. Thus, the above time series is not a stationery realization process as the changes have

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been fluctuating with certain peaks and lean seasons in the above data which is not acceptable in
stationery process (Young, 2015).
Firstly, it is important to stabilize the increasing variability of the data by using Box-Cox
transformation. In order to transform the series to stabilize the data the following transformation
equation of Box-Cox is used:
It is important then to find a transformation g on Xt to ensure that the variance is constant with
g(Xt). In order to achieve that the transformation approximation of Taylor can be used (Taneja
et. al. 2016).
It is important to keep in mind that the above equations are to be used to lay down the procedure
to complete the stationery process of data realization. Finally, after rendering stationery data we
are ready to fir an appropriate model in the data. The following will be the results of ARMA.

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Question 2:
Consumer price index measures the weighted average price of basket of goods in the economy.
The prices changes in each item of goods and services are considered to determine the CPI. In
order to identify the inflation in different periods the consumer price index is used. Also CPI
index is sued to measure the cost of living at a place.
Quarter cpi Log Log Difference
01-01-60 29.4 3.380995 1.419993
01-04-60 29.57 3.38676 1.418978
01-07-60 29.59 3.387436 1.418859
01-10-60 29.78 3.393837 1.417739
01-01-61 29.84 3.39585 1.417388
01-04-61 29.83 3.395515 1.417447
01-07-61 29.95 3.399529 1.416748

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01-10-61 29.99 3.400864 1.416517
01-01-62 30.11 3.404857 1.415825
01-04-62 30.22 3.408504 1.415195
01-07-62 30.31 3.411478 1.414683
01-10-62 30.38 3.413784 1.414287
01-01-63 30.48 3.417071 1.413724
01-04-63 30.53 3.41871 1.413444
01-07-63 30.72 3.424914 1.412386
01-10-63 30.8 3.427515 1.411944
01-01-64 30.93 3.431727 1.41123
01-04-64 30.98 3.433342 1.410957
01-07-64 31.05 3.435599 1.410577
01-10-64 31.19 3.440098 1.40982
01-01-65 31.29 3.443299 1.409283
01-04-65 31.49 3.44967 1.408218
01-07-65 31.58 3.452524 1.407743

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01-10-65 31.75 3.457893 1.406853
01-01-66 32.05 3.467297 1.405302
01-04-66 32.34 3.476305 1.403828
01-07-66 32.62 3.484926 1.402427
01-10-66 32.88 3.492865 1.401145
01-01-67 32.97 3.495598 1.400706
01-04-67 33.17 3.501646 1.399737
01-07-67 33.5 3.511545 1.398161
01-10-67 33.87 3.52253 1.396427
01-01-68 34.2 3.532226 1.39491
01-04-68 34.53 3.541829 1.393418
01-07-68 35 3.555348 1.391336
01-10-68 35.43 3.567559 1.389475
01-01-69 35.87 3.579901 1.387612
01-04-69 36.43 3.595393 1.385298
01-07-69 36.93 3.609024 1.383285
01-10-69 37.5 3.624341 1.381048

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01-01-70 38.1 3.640214 1.378757
01-04-70 38.63 3.654029 1.376786
01-07-70 39.03 3.664331 1.375329
01-10-70 39.6 3.678829 1.373297
01-01-71 39.93 3.687128 1.372145
01-04-71 40.3 3.696351 1.370872
01-07-71 40.7 3.706228 1.369518
01-10-71 41 3.713572 1.368518
01-01-72 41.33 3.721589 1.367432
01-04-72 41.6 3.7281 1.366555
01-07-72 41.93 3.736002 1.365497
01-10-72 42.37 3.746441 1.364108
01-01-73 43.03 3.761898 1.36207
01-04-73 43.93 3.782597 1.359376
01-07-73 44.8 3.802208 1.356861
01-10-73 45.93 3.827118 1.353717
01-01-74 47.3 3.85651 1.350078

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01-04-74 48.57 3.883006 1.34686
01-07-74 49.93 3.910622 1.343569
01-10-74 51.47 3.940999 1.340021
01-01-75 52.57 3.962146 1.337593
01-04-75 53.2 3.974058 1.336241
01-07-75 54.27 3.993972 1.334005
01-10-75 55.27 4.01223 1.33198
01-01-76 55.9 4.023564 1.330735
01-04-76 56.4 4.032469 1.329764
01-07-76 57.3 4.048301 1.328052
01-10-76 58.13 4.062682 1.326511
01-01-77 59.2 4.080922 1.324578
01-04-77 60.23 4.098171 1.322771
01-07-77 61.07 4.112021 1.321335
01-10-77 61.97 4.12665 1.319831
01-01-78 63.03 4.143611 1.318106
01-04-78 64.47 4.1662 1.315836

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01-07-78 65.97 4.1892 1.313558
01-10-78 67.5 4.212128 1.31132
01-01-79 69.2 4.237001 1.308928
01-04-79 71.4 4.268298 1.30597
01-07-79 73.7 4.300003 1.30303
01-10-79 76.03 4.331128 1.300199
01-01-80 79.03 4.369828 1.296751
01-04-80 81.7 4.403054 1.293854
01-07-80 83.23 4.421608 1.29226
01-10-80 85.57 4.449335 1.289911
01-01-81 87.93 4.476541 1.287642
01-04-81 89.77 4.497251 1.285939
01-07-81 92.27 4.524719 1.283711
01-10-81 93.77 4.540845 1.282418
01-01-82 94.6 4.549657 1.281717
01-04-82 95.97 4.564036 1.280581
01-07-82 97.63 4.581185 1.279237

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01-10-82 97.93 4.584253 1.278998
01-01-83 98 4.584967 1.278943
01-04-83 99.13 4.596432 1.278053
01-07-83 100.1 4.60617 1.277303
01-10-83 101.1 4.61611 1.27654
01-01-84 102.53 4.630155 1.27547
01-04-84 103.5 4.639572 1.274758
01-07-84 104.4 4.64823 1.274106
01-10-84 105.3 4.656813 1.273462
01-01-85 106.27 4.665983 1.272778
01-04-85 107.23 4.674976 1.272111
01-07-85 107.9 4.681205 1.27165
01-10-85 109 4.691348 1.270904
01-01-86 109.57 4.696564 1.270522
01-04-86 109.03 4.691623 1.270884
01-07-86 109.7 4.697749 1.270435
01-10-86 110.47 4.704744 1.269924

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01-01-87 111.8 4.716712 1.269055
01-04-87 113.07 4.728007 1.26824
01-07-87 114.27 4.738564 1.267482
01-10-87 115.33 4.747798 1.266823
01-01-88 116.23 4.755571 1.266271
01-04-88 117.57 4.767034 1.265461
01-07-88 119 4.779123 1.264612
01-10-88 120.3 4.789989 1.263853
01-01-89 121.67 4.801312 1.263067
01-04-89 123.63 4.817293 1.261966
01-07-89 124.6 4.825109 1.26143
01-10-89 125.87 4.83525 1.260739
01-01-90 128.03 4.852265 1.259588
01-04-90 129.3 4.862135 1.258924
01-07-90 131.53 4.879235 1.257783
01-10-90 133.77 4.896122 1.256665
01-01-91 134.77 4.90357 1.256176

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FINANCIAL ECONOMETRICS
01-04-91 135.57 4.909488 1.255788
01-07-91 136.6 4.917057 1.255294
01-10-91 137.73 4.925295 1.254758
01-01-92 138.67 4.932097 1.254317
01-04-92 139.73 4.939712 1.253826
01-07-92 140.8 4.94734 1.253335
01-10-92 142.03 4.956038 1.252778
01-01-93 143.07 4.963334 1.252313
01-04-93 144.1 4.970508 1.251857
01-07-93 144.77 4.975146 1.251563
01-10-93 145.97 4.983401 1.251042
01-01-94 146.7 4.98839 1.250728
01-04-94 147.53 4.994032 1.250374
01-07-94 148.9 5.003275 1.249795
01-10-94 149.77 5.009101 1.249432
01-01-95 150.87 5.016419 1.248978
01-04-95 152.1 5.024538 1.248476

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01-07-95 152.87 5.029588 1.248164
01-10-95 153.7 5.035003 1.247831
01-01-96 155.07 5.043877 1.247287
01-04-96 156.4 5.052417 1.246766
01-07-96 157.3 5.058155 1.246417
01-10-96 158.67 5.066827 1.245892
01-01-97 159.63 5.072859 1.245528
01-04-97 160 5.075174 1.245388
01-07-97 160.8 5.080161 1.245088
01-10-97 161.67 5.085557 1.244765
01-01-98 162 5.087596 1.244643
01-04-98 162.53 5.090863 1.244447
01-07-98 163.37 5.096018 1.24414
01-10-98 164.13 5.100659 1.243863
01-01-99 164.73 5.104308 1.243646
01-04-99 165.97 5.111807 1.243202
01-07-99 167.2 5.119191 1.242766

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01-10-99 168.43 5.12652 1.242335
01-01-00 170.1 5.136386 1.241757
01-04-00 171.43 5.144175 1.241303
01-07-00 173 5.153292 1.240773
01-10-00 174.23 5.160376 1.240363
01-01-01 175.9 5.169916 1.239813
01-04-01 177.13 5.176884 1.239413
01-07-01 177.63 5.179703 1.239251
01-10-01 177.5 5.178971 1.239293
01-01-02 178.07 5.182177 1.23911
01-04-02 179.47 5.190008 1.238663
01-07-02 180.43 5.195343 1.23836
01-10-02 181.5 5.201256 1.238024
01-01-03 183.37 5.211506 1.237445
01-04-03 183.07 5.209869 1.237537
01-07-03 184.43 5.21727 1.23712
01-10-03 185.13 5.221058 1.236907

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01-01-04 186.7 5.229503 1.236434
01-04-04 188.17 5.237346 1.235997
01-07-04 189.37 5.243703 1.235643
01-10-04 191.4 5.254365 1.235053
01-01-05 192.37 5.259421 1.234774
01-04-05 193.67 5.266156 1.234403
01-07-05 196.6 5.281171 1.233581
01-10-05 198.43 5.290436 1.233077
01-01-06 199.47 5.295664 1.232793
01-04-06 201.27 5.304647 1.232307
01-07-06 203.17 5.314043 1.231801
01-10-06 202.33 5.3099 1.232024
01-01-07 204.32 5.319687 1.231498
01-04-07 206.63 5.33093 1.230897
01-07-07 207.94 5.33725 1.230561
01-10-07 210.49 5.349438 1.229915
01-01-08 212.77 5.360212 1.229347

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01-04-08 215.54 5.373147 1.228668
01-07-08 218.86 5.388432 1.227872
01-10-08 213.85 5.365275 1.229081
01-01-09 212.43 5.358613 1.229431
01-04-09 213.48 5.363543 1.229172
01-07-09 215.35 5.372265 1.228714
01-10-09 217 5.379897 1.228316
01-01-10 217.4 5.381739 1.22822
01-04-10 217.28 5.381187 1.228249
01-07-10 218.01 5.384541 1.228074
01-10-10 219.64 5.39199 1.227687
01-01-11 222.03 5.402813 1.227128
01-04-11 224.57 5.414187 1.226542
01-07-11 226.18 5.421331 1.226176
01-10-11 226.97 5.424818 1.225998
01-01-12 228.27 5.430529 1.225707
01-04-12 228.84 5.433023 1.22558

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01-07-12 230.03 5.43821 1.225316
01-10-12 231.28 5.443629 1.225041 (Wang
and Ding, 2015)
The above table contains the CPI from 1960 to 2012 shows how the general price level in the
country have changed over the course of the above period. It is an effective economic indicator
often used by the economists to measure the general inflation level in the economy apart from
the cost of living and living standard in a country. One of the most crucial and important thing
that can e understood from the above data is how constantly the price level in the country have
increased without even single period where the general price level the country declining. Thus,
the increasing trend is a constant in CPI.
Part a:
The graph below is the graph prepared to reflect the movement of CPI in the country from 1960
to 2012. Preliminary look at the graph below clearly outlines the movement in CPI in the country
with constant increase in the general price level. The slope in the line below is upward since its
beginning.

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1/1/1960
4/1/1962
7/1/1964
10/1/1966
1/1/1969
4/1/1971
7/1/1973
10/1/1975
1/1/1978
4/1/1980
7/1/1982
10/1/1984
1/1/1987
4/1/1989
7/1/1991
10/1/1993
1/1/1996
4/1/1998
7/1/2000
10/1/2002
1/1/2005
4/1/2007
7/1/2009
10/1/2011
0
50
100
150
200
250
cpi
The series is not stationery as can be seen there has been fluctuation in the above graph.
However, the series shows one thing very clearly and that it has grown continuously. Thus,
throughout the period the CPI has increased on a regular basis. Thus, a non-stationery time series
the above graph is indicative of uneven fluctuation in the consumer prices.
Part b:
Correlation of a signal with delayed copy of itself with the function of delay is known as
Autocorrelation function. Partial autocorrelation function is a conditional correlation. Generally
the fluctuating shape is taken in stationery autoregressive process (Siti Roslindar et. al. 2015).
Part c:
Autocorrelation and partial correlation function are provided in the table below:
Autocorrelation Partial correlation
0.972352 0.648235

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The autocorrelation function and partial correlation function is quite significant in data analysis
as both shows the correlation between data to consider in the decision making process. Partial
correlation is again in line with autocorrelation as it is lower than the value of autocorrelation.
The autocorrelation of 0.972 (approx.) of data shows the significant correlation amongst the data
to suggest that the data are interrelated. The tentative model that has been identified is provided
below in the form of a graph below:
1/1/1960
4/1/1962
7/1/1964
10/1/1966
1/1/1969
4/1/1971
7/1/1973
10/1/1975
1/1/1978
4/1/1980
7/1/1982
10/1/1984
1/1/1987
4/1/1989
7/1/1991
10/1/1993
1/1/1996
4/1/1998
7/1/2000
10/1/2002
1/1/2005
4/1/2007
7/1/2009
10/1/2011
0
50
100
150
200
250
cpi
The graph below shows how the cpi has continuously moved upward with each quarter. The fact
that the consumer price index has increase with each quarter is clear from the above fact. From
the above model it is clearly understandable that the cpi shows upward movement trend
consistently.
Part d:
quarter cpi MA (4) Residuals

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01-01-60 29.4 29.585
01-04-60 29.57 29.695
01-07-60 29.59 29.76
01-10-60 29.78 29.585 29.85
01-01-61 29.84 29.695 29.9025
01-04-61 29.83 29.76 29.97
01-07-61 29.95 29.85 30.0675
01-10-61 29.99 29.9025 30.1575
01-01-62 30.11 29.97 30.255
01-04-62 30.22 30.0675 30.3475
01-07-62 30.31 30.1575 30.425
01-10-62 30.38 30.255 30.5275
01-01-63 30.48 30.3475 30.6325
01-04-63 30.53 30.425 30.745
01-07-63 30.72 30.5275 30.8575
01-10-63 30.8 30.6325 30.94

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01-01-64 30.93 30.745 31.0375
01-04-64 30.98 30.8575 31.1275
01-07-64 31.05 30.94 31.255
01-10-64 31.19 31.0375 31.3875
01-01-65 31.29 31.1275 31.5275
01-04-65 31.49 31.255 31.7175
01-07-65 31.58 31.3875 31.93
01-10-65 31.75 31.5275 32.19
01-01-66 32.05 31.7175 32.4725
01-04-66 32.34 31.93 32.7025
01-07-66 32.62 32.19 32.91
01-10-66 32.88 32.4725 33.13
01-01-67 32.97 32.7025 33.3775
01-04-67 33.17 32.91 33.685
01-07-67 33.5 33.13 34.025
01-10-67 33.87 33.3775 34.4
01-01-68 34.2 33.685 34.79

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FINANCIAL ECONOMETRICS
01-04-68 34.53 34.025 35.2075
01-07-68 35 34.4 35.6825
01-10-68 35.43 34.79 36.165
01-01-69 35.87 35.2075 36.6825
01-04-69 36.43 35.6825 37.24
01-07-69 36.93 36.165 37.79
01-10-69 37.5 36.6825 38.315
01-01-70 38.1 37.24 38.84
01-04-70 38.63 37.79 39.2975
01-07-70 39.03 38.315 39.715
01-10-70 39.6 38.84 40.1325
01-01-71 39.93 39.2975 40.4825
01-04-71 40.3 39.715 40.8325
01-07-71 40.7 40.1325 41.1575
01-10-71 41 40.4825 41.465
01-01-72 41.33 40.8325 41.8075
01-04-72 41.6 41.1575 42.2325

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FINANCIAL ECONOMETRICS
01-07-72 41.93 41.465 42.815
01-10-72 42.37 41.8075 43.5325
01-01-73 43.03 42.2325 44.4225
01-04-73 43.93 42.815 45.49
01-07-73 44.8 43.5325 46.65
01-10-73 45.93 44.4225 47.9325
01-01-74 47.3 45.49 49.3175
01-04-74 48.57 46.65 50.635
01-07-74 49.93 47.9325 51.7925
01-10-74 51.47 49.3175 52.8775
01-01-75 52.57 50.635 53.8275
01-04-75 53.2 51.7925 54.66
01-07-75 54.27 52.8775 55.46
01-10-75 55.27 53.8275 56.2175
01-01-76 55.9 54.66 56.9325
01-04-76 56.4 55.46 57.7575
01-07-76 57.3 56.2175 58.715

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FINANCIAL ECONOMETRICS
01-10-76 58.13 56.9325 59.6575
01-01-77 59.2 57.7575 60.6175
01-04-77 60.23 58.715 61.575
01-07-77 61.07 59.6575 62.635
01-10-77 61.97 60.6175 63.86
01-01-78 63.03 61.575 65.2425
01-04-78 64.47 62.635 66.785
01-07-78 65.97 63.86 68.5175
01-10-78 67.5 65.2425 70.45
01-01-79 69.2 66.785 72.5825
01-04-79 71.4 68.5175 75.04
01-07-79 73.7 70.45 77.615
01-10-79 76.03 72.5825 79.9975
01-01-80 79.03 75.04 82.3825
01-04-80 81.7 77.615 84.6075
01-07-80 83.23 79.9975 86.625
01-10-80 85.57 82.3825 88.885

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FINANCIAL ECONOMETRICS
01-01-81 87.93 84.6075 90.935
01-04-81 89.77 86.625 92.6025
01-07-81 92.27 88.885 94.1525
01-10-81 93.77 90.935 95.4925
01-01-82 94.6 92.6025 96.5325
01-04-82 95.97 94.1525 97.3825
01-07-82 97.63 95.4925 98.1725
01-10-82 97.93 96.5325 98.79
01-01-83 98 97.3825 99.5825
01-04-83 99.13 98.1725 100.715
01-07-83 100.1 98.79 101.8075
01-10-83 101.1 99.5825 102.8825
01-01-84 102.53 100.715 103.9325
01-04-84 103.5 101.8075 104.8675
01-07-84 104.4 102.8825 105.8
01-10-84 105.3 103.9325 106.675
01-01-85 106.27 104.8675 107.6

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FINANCIAL ECONOMETRICS
01-04-85 107.23 105.8 108.425
01-07-85 107.9 106.675 108.875
01-10-85 109 107.6 109.325
01-01-86 109.57 108.425 109.6925
01-04-86 109.03 108.875 110.25
01-07-86 109.7 109.325 111.26
01-10-86 110.47 109.6925 112.4025
01-01-87 111.8 110.25 113.6175
01-04-87 113.07 111.26 114.725
01-07-87 114.27 112.4025 115.85
01-10-87 115.33 113.6175 117.0325
01-01-88 116.23 114.725 118.275
01-04-88 117.57 115.85 119.635
01-07-88 119 117.0325 121.15
01-10-88 120.3 118.275 122.55
01-01-89 121.67 119.635 123.9425
01-04-89 123.63 121.15 125.5325

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FINANCIAL ECONOMETRICS
01-07-89 124.6 122.55 126.95
01-10-89 125.87 123.9425 128.6825
01-01-90 128.03 125.5325 130.6575
01-04-90 129.3 126.95 132.3425
01-07-90 131.53 128.6825 133.91
01-10-90 133.77 130.6575 135.1775
01-01-91 134.77 132.3425 136.1675
01-04-91 135.57 133.91 137.1425
01-07-91 136.6 135.1775 138.1825
01-10-91 137.73 136.1675 139.2325
01-01-92 138.67 137.1425 140.3075
01-04-92 139.73 138.1825 141.4075
01-07-92 140.8 139.2325 142.5
01-10-92 142.03 140.3075 143.4925
01-01-93 143.07 141.4075 144.4775
01-04-93 144.1 142.5 145.385
01-07-93 144.77 143.4925 146.2425

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FINANCIAL ECONOMETRICS
01-10-93 145.97 144.4775 147.275
01-01-94 146.7 145.385 148.225
01-04-94 147.53 146.2425 149.2675
01-07-94 148.9 147.275 150.41
01-10-94 149.77 148.225 151.4025
01-01-95 150.87 149.2675 152.385
01-04-95 152.1 150.41 153.435
01-07-95 152.87 151.4025 154.51
01-10-95 153.7 152.385 155.6175
01-01-96 155.07 153.435 156.86
01-04-96 156.4 154.51 158
01-07-96 157.3 155.6175 158.9
01-10-96 158.67 156.86 159.775
01-01-97 159.63 158 160.525
01-04-97 160 158.9 161.1175
01-07-97 160.8 159.775 161.75
01-10-97 161.67 160.525 162.3925

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FINANCIAL ECONOMETRICS
01-01-98 162 161.1175 163.0075
01-04-98 162.53 161.75 163.69
01-07-98 163.37 162.3925 164.55
01-10-98 164.13 163.0075 165.5075
01-01-99 164.73 163.69 166.5825
01-04-99 165.97 164.55 167.925
01-07-99 167.2 165.5075 169.29
01-10-99 168.43 166.5825 170.74
01-01-00 170.1 167.925 172.19
01-04-00 171.43 169.29 173.64
01-07-00 173 170.74 175.065
01-10-00 174.23 172.19 176.2225
01-01-01 175.9 173.64 177.04
01-04-01 177.13 175.065 177.5825
01-07-01 177.63 176.2225 178.1675
01-10-01 177.5 177.04 178.8675
01-01-02 178.07 177.5825 179.8675

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01-04-02 179.47 178.1675 181.1925
01-07-02 180.43 178.8675 182.0925
01-10-02 181.5 179.8675 183.0925
01-01-03 183.37 181.1925 184
01-04-03 183.07 182.0925 184.8325
01-07-03 184.43 183.0925 186.1075
01-10-03 185.13 184 187.3425
01-01-04 186.7 184.8325 188.91
01-04-04 188.17 186.1075 190.3275
01-07-04 189.37 187.3425 191.7025
01-10-04 191.4 188.91 193.51
01-01-05 192.37 190.3275 195.2675
01-04-05 193.67 191.7025 197.0425
01-07-05 196.6 193.51 198.9425
01-10-05 198.43 195.2675 200.585
01-01-06 199.47 197.0425 201.56
01-04-06 201.27 198.9425 202.7725

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01-07-06 203.17 200.585 204.1125
01-10-06 202.33 201.56 205.305
01-01-07 204.32 202.7725 207.345
01-04-07 206.63 204.1125 209.4575
01-07-07 207.94 205.305 211.685
01-10-07 210.49 207.345 214.415
01-01-08 212.77 209.4575 215.255
01-04-08 215.54 211.685 215.17
01-07-08 218.86 214.415 214.655
01-10-08 213.85 215.255 213.7775
01-01-09 212.43 215.17 214.565
01-04-09 213.48 214.655 215.8075
01-07-09 215.35 213.7775 216.7575
01-10-09 217 214.565 217.4225
01-01-10 217.4 215.8075 218.0825
01-04-10 217.28 216.7575 219.24
01-07-10 218.01 217.4225 221.0625

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01-10-10 219.64 218.0825 223.105
01-01-11 222.03 219.24 224.9375
01-04-11 224.57 221.0625 226.4975
01-07-11 226.18 223.105 227.565
01-10-11 226.97 224.9375 228.5275
01-01-12 228.27 226.4975 229.605
01-04-12 228.84 227.565
01-07-12 230.03 228.5275
01-10-12 231.28 229.605
Estimation of ARMA models from order (0, 0) to (4, 4) for a sample period from first quarter of
1960 to last quarter of 2009. From the above data it is clear that the preferred model is the
ARIMA model with “I” standing for integrated. The reason that ARIMA is the preferred model
is because AR and MA is integrated under the model. As per the above table AIC is 1.2 with
SBIC 1.7.

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FINANCIAL ECONOMETRICS
1/1/1960
2/1/1962
3/1/1964
4/1/1966
5/1/1968
6/1/1970
7/1/1972
8/1/1974
9/1/1976
10/1/1978
11/1/1980
12/1/1982
1/1/1985
2/1/1987
3/1/1989
4/1/1991
5/1/1993
6/1/1995
7/1/1997
8/1/1999
9/1/2001
10/1/2003
11/1/2005
12/1/2007
0
50
100
150
200
250
300
350
400
450
500
Chart Title
Log Difference MA (4)
Part e:
The re-estimate of preferred models using the data from first quarter of 1960 to the last quarter of
2009 the following graph has been created. As can be seen in the graph below that re-estimation
has resulted in significant difference as another parallel line has appeared in the graph below.
1/1/1960
2/1/1962
3/1/1964
4/1/1966
5/1/1968
6/1/1970
7/1/1972
8/1/1974
9/1/1976
10/1/1978
11/1/1980
12/1/1982
1/1/1985
2/1/1987
3/1/1989
4/1/1991
5/1/1993
6/1/1995
7/1/1997
8/1/1999
9/1/2001
10/1/2003
11/1/2005
12/1/2007
0
100
200
300
400
500
600
700
Chart Title
Series1 Series2 Series3

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FINANCIAL ECONOMETRICS
The residual checks and estimated models fit the data well as is clear from the above graphical
representation.
Part f:
The static forecast graph is presented below.
1/1/2010
3/1/2010
5/1/2010
7/1/2010
9/1/2010
11/1/2010
1/1/2011
3/1/2011
5/1/2011
7/1/2011
9/1/2011
11/1/2011
1/1/2012
3/1/2012
5/1/2012
7/1/2012
9/1/2012
210
215
220
225
230
235
Chart Title
The actual graph of cpi series is presented below by using the data in the sample period.
1/1/2010
3/1/2010
5/1/2010
7/1/2010
9/1/2010
11/1/2010
1/1/2011
3/1/2011
5/1/2011
7/1/2011
9/1/2011
11/1/2011
1/1/2012
3/1/2012
5/1/2012
7/1/2012
9/1/2012
0
50
100
150
200
250
300
350
400
450
500
Chart Title
Series1 Series2

36
FINANCIAL ECONOMETRICS

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References:
Box, G.E., Jenkins, G.M., Reinsel, G.C. and Ljung, G.M., 2015. Time series analysis:
forecasting and control. John Wiley & Sons.
Gairaa, K., Khellaf, A., Messlem, Y. and Chellali, F., 2016. Estimation of the daily global solar
radiation based on Box–Jenkins and ANN models: A combined approach. Renewable and
Sustainable Energy Reviews, 57, pp.238-249.
Li, K., Zhu, Y., Yang, J. and Jiang, J., 2016. Video super-resolution using an adaptive
superpixel-guided auto-regressive model. Pattern Recognition, 51, pp.59-71.
Maatallah, O.A., Achuthan, A., Janoyan, K. and Marzocca, P., 2015. Recursive wind speed
forecasting based on Hammerstein Auto-Regressive model. Applied Energy, 145, pp.191-197.
Siti Roslindar, Y., Noor Azlinna, A., Maizah Hura, A., Roslinazairimah, Z., Agrawal, M. and
Boland, J., 2015. Preliminary Analysis on Hybrid Box-Jenkins-GARCH Modeling In
Forecasting Gold Price.
Taneja, K., Ahmad, S., Ahmad, K. and Attri, S.D., 2016. Time series analysis of aerosol optical
depth over New Delhi using Box–Jenkins ARIMA modeling approach. Atmospheric Pollution
Research, 7(4), pp.585-596.
Wang, X. and Ding, F., 2015. Convergence of the auxiliary model-based multi-innovation
generalized extended stochastic gradient algorithm for Box–Jenkins systems. Nonlinear
Dynamics, 82(1-2), pp.269-280.
Young, P.C., 2015. Refined instrumental variable estimation: maximum likelihood optimization
of a unified Box–Jenkins model. Automatica, 52, pp.35-46.

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