Financial Investment Analysis: Box-Jenkins Methodology, VAR Models, and Impulse Function Graphs

FINANCIAL INVESTMENT ANALYSIS 1
FINANCIAL INVESTMENT ANALYSIS
Course name
Professor’s name
Institution name
City
Date of submission

FINANCIAL INVESTMENT ANALYSIS 2
QUESTION ONE
The box-Jenkins methodology
The Box-Jenkins methodology is usually applied to an ARMA (p, q) model with an aim of
determining the most appropriate values of p and d. It also refers to the method of identifying,
fitting, checking and using ARIMA. The first step in this methodology as suggested by Box-
Jenkins is to check for stationarity by fitting a time series plot, correlogram and partial
correlation plots (Zaharia & Balacesu, 2017).
Time
PX_LAST
0 10 20 30 40 50 60 70
0 2000 6000 10000 14000
Figure 1: Time series plot of bitcoins
The chat above showed that the log returns of bitcoin increase accordingly across a time
period. As shown in the chart above the bitcoin returns rapidly increase during the time
period. There is an evident of an increasing trend in the data. It implies that before analyzing
the data, de-trending methods will be first be applied to the data set in order to remove the

FINANCIAL INVESTMENT ANALYSIS 3
trend component in the data set. One of the methods is using linear regression to model the
bitcoins return data accompanied by linear indices (1,2, 3...n) (Ding et al., 2017). The
resulting model will represent a time series data without trend component. In this case, trend
component will be eliminated from the data. However, in some cases it could still be
presence in the residuals. In order to solve this scenario, some predictors will be added to the
model. Towards the end of the time period there is evident of seasonality component.
0 5 10 15
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
Lag
ACF
Series PX_LAST
Figure 2: Correlogram
Correlogram is usually used to display whether previous lagged values of the time series has
an effect on current state. The chart above show that majority of the autocorrelation crosses
the blue line, implying that these specific lags are significantly associated with the current
bitcoin returns series. This chart also indicates that there is data is non-stationary since its
series is dropping the gradually from the highest to zero (Maravall, et al., 2016). It also

FINANCIAL INVESTMENT ANALYSIS 4
implies that before fitting a model which assume stationarity first the data need to be
transformed.
ARMA ( 1,1 ) =2081.831+0.9394 Xt −0.1024 Xt −1
Order BIC AIC
0,0 1197.34 1369.98
1,1 1386.32 1206.98
2,2 1376.73 1210.69
3,3 1347.89 1230.31
1,0 1367.69 1226.83
0,1 1385.12 1206.99
2,0 1341.87 1250.32
0,2 1364.54 1227.47
3,0 1199.51 1315.95
0,3 1368.02 1226.01
Question 2
Table 1: Continuously compounded rate of returns
Calculations
USDGB
P
EURGB
P
JPYGB
P
GBPUS
D
GBPEU
R
GBPJP
Y
Annual
change 17.27% 27.30% 11.25% -36.05% -48.52% -26.11%
117.27% 127.30% 111.25% 63.95% 51.48% 73.89%
Continuously
compounded
15.93% 24.14% 10.66% -44.70% -66.39% -30.27%

FINANCIAL INVESTMENT ANALYSIS 5
rate of
returns
On the continuously compounded rate of returns, EUROGBP had the highest compounded
rates of return from 2000 to 2018. USDGDP came second at 15.93% followed by JPYGBP at
10.66%. GBPJPY (-30.27%), GBPUSD (-44.70%) and GBPEUR (-66.39%) implicated the
least returns from 2000 to 2018.
a. Best VAR Models
On VAR model selection, an optimal model exists on minimal Akaike Information criterion
(AIC). The lower the AIC, the better the model. According to the analysis, model 1 indicates
the lowest AIC levels (AIC=13,681.83) as compared to model 2 and model 3. This indicates
that the model appropriate in the prediction of interest rates within the selected exchange
rates (Angelidis and Degiannakis, 2018).
Table 2: Regression model 1
Usdgbp Coef Std.
Err t
eurgbp
-
0.10829
94
0.0124
98 -8.67
jpygbp 0.50316
27
0.0111
57 45.1
_cons 0.41105
73
0.0067
76 60.66

FINANCIAL INVESTMENT ANALYSIS 6
Table 3: Regression model 2
usdgbp Coef Std. Err t P>|t| [95 % Conf. Interval]
eurgbp -0.1083 0.012498 -8.67 0 -0.1328 -0.0838
jpygbp 0.503163 0.011157 45.1 0 0.481291 0.52503
_cons 0.411057 0.006776 60.66 0 0.397773 0.424341
On model 2, the AIC=12,484.81 and BIC=-12465.29. The model implies the least in terms of
the VAR measures since it registers the highest AIC scores of the three models. This makes
the models less optimal hence the suitability of model 1 in the forecasting of the exchange
rates presented within the three models.
Table 4: Regression model 3
usdgbp Coef Std. Err t P>|t| [95 % Conf. Interval]
eurgbp 0.578558 0.01282
9 45 0 0.553409 0.60370
8
jpygbp 0.624981 0.01017
2 61.44 0 0.605039 0.64492
4
_cons -0.23705 0.00898
2 -26.39 0 -0.25466 -
0.21944
On model 3, the AIC=-12.8989 and BIC=-12970.32, hence second best optimal model for
selection. In terms of rank, it stands at position 2 as compared to the first model with the least
AIC scores.
b. Final model