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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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