Financial and Statistical Modelling

Financial and Statistical Modelling

The report:

  1. Find sources of real-life time series data and select two series of interest, one without seasonal effect and the other with seasonality.

Average monthly temperatures across the world

For seasonal effect, I have chosen the data of “Average monthly temperatures in Edgbaston” for the year range “1963 – 1969” because this as shown in the graph portrays seasonality. Seasonality is visible in the graph as you can see that the trend is recurrent at given months.

 

Figure 1: Average Monthly Temperature across the world from 1963 to 1969

For the non-seasonal effect, I have chosen the data of “Exchange rates monthly data – Pounds vs. Euro” for the year range “2008-2014” as it shows the non-seasonal effect as seen in the graph provided.

 

 

 

 

 

Exchange Rates in Euro and Pound

 

 

Figure 2: Euro/GB Pound Exchange Rate

The rates were colected for every beginning of the quarters. For every year there are 4 rates. From 2008 to 2014, there are 72 months or 24 quarters.  The minimum rate was experienced in May 2008 and in 2014, being 1.128.

Trace the two series back in order to collect sufficient observations to perform a meaningful analysis. (This will require a minimum of 24 observations for data collected quarterly and a minimum of 72 for monthly data). The source of data is www.themoneyconverter.com.

 

Figure 3: Average Monthly Temperature across the world

What is this graph here for?

  1. 2.       Investigate the background for each of the two series: what it represents, how the data are collected, if there have been changes in the way in which it was collected, what factors may affect the series, etc.

In the scenario above, the projected data on Edgbaston monthly moving average Temperature (oC) demonstrates seasonality.  The first series represents the current and the past average temperatures based on available data in the global scene First series represents current and past average temperatures? How?. The second series is a projection of the future values future values??? based on the factors that surround the temperatures. The data presented by the two graphs shows that there has been a trend of uniform fluctuation in the temperatures. The data is a representation of the real events around the variation in the global temperatures Which data? Be precise. Data collection in this project was done from the secondary information in the online libraries. There was no data change made to the data since they all belong to different seasons What does this sentence even mean?. The major factors that can change the Euro to pounds rate is the inflation rate and the global risk exposure to foreign currency exchange losses Good. The second possible factor is the global economic inflation rateClimatic changes and human activities tend to affect futuristic trends when it comes to monthly moving temperatures Such as what human activities, which climatic changes?. The exchange rate on the other hand is affected by the market forces, demand and supply.  The exchange rate has distinct and significant effect on growth Is there not anything else you can add onto this.

 

Figure 4: Projected Average Monthly Temperature across the world

  1. If you think that other factors affect the series, make sure that you collect data for those factors so you can include them later in the analysis as explanatory variables.

Average Temperatures

For the variation in the average temperatures, the data was collected for climatic changes in the aspect of Seasons. For example, during winter seasons and autumn season, the average temperatures are extremely low, falling below 0oC in winter. On the contrary, during spring and summer, temperatures grow extremely high, ranging between 38 and 40oC.

The data is as follows:

Season Temperature
su1

37

su2

38

su3

39

su4

40

su5

38

su6

37

su7

36

a1

34

a2

30

a3

25

a4

22

a5

15

a6

7

a7

4

w1

2

w2

0

w3

-2

w4

-4

w5

-5

w6

-4

w7

-2

sp1

0

sp2

4

sp3

10

sp4

25

sp5

30

sp6

36

sp7

37

 

 

Figure 5: Temperature variation with seasons

  1. 4.       For each of the series analyse the data and forecast the next three time points using a variety of forecasting techniques (variety of forecasting techniques it says? What have you used? Explain which techniques you use please stop just answering it in your way instead of properly saying which techniques are being used) you are meant to use a variety. I only see one thing you have done. How can you think this work would be this easy? This question on the question document clearly said See below for the forecasting techniques and I have already explained this. You are meant to find the next three time points using techniques such as moving averages etc can you please refer to the other document I sent to you explaining how I wanted this question to be answered once again!!!! Can you please read it thoroughly as I cannot keep having to type it out.

Forecasting in this sense involve finding the three next points by finding the average value of the three last values. The variation between the three numbers from the average provides the range between the next three figures. In question one, the average rate of exchange is:

1/3 (1.36 +1.36 +1.42) = (5.14) / 3 = 1.713.

The range is 1.713 – 1.42 = 0.293

The nest three numbers are

1.42 + 0.293 = 1.713

1.713 + 0.293 = 2.006

2.006 + 0.293 = 2.299

 

  1. 5.       Comment critically on your findings

The projection gives the next three numbers as 1.713, 2.006 and 2.299. Scores of time series statistics follow recurring seasonal trends. For instance, yearly exchange rate will are highest during the winter that is December and Jan. In 2008, the Exchange Rate Euro vs. Pound was at 1.33879 while the temperature was at 4 degrees centigrade. During the summer holiday of 2008, the Exchange Rate Euro vs. Pound was at 1.26166 and 1.26096 during the month of July and August respectively, while the temperatures were all time high at 17 degree centigrade’s.

Smoothing models Why are you explaining what a smoothing model is when you haven’t even said where you have done one. Please re-do question 4 and then do this according to the VARIETY of forecasting techniques you have to use demonstrates a linear upwards pattern of exchange rate over the time and a recurring trend or season in a given year. (For instance, exchange rate is highest during summer and the low season is represented. The report adopts a seasonal decomposition to separate those elements, hence cluster the series into the pattern effect and seasonal effects. The regression model presents one of the finest forecasts of the dependent variable (Y), based on the independent variable (X). Nevertheless, issues to with weather can be uncertain to forecast, hence demonstrating a significant variation of the visible points around the fitted regression line.

i)                    Moving average and decomposition

 

 

 

The moving average of exchange rate in Euros v. Pounds demonstrates a non-seasonality trend owing to a trend that is not repetitive. The last three points shows a reducing trend in the rates of exchange while the previous three points before them indicates a gradual increase. The forecast is therefore showing trends that are contrary to the current situation.

 

Edgbaston monthly averages temperature demonstrates a seasonal behavior owing to a repetitive trend. The trend has been visible during certain months. March of 2008 and 2009 exhibited a similar temperature. The same pattern is visible between august 2008 and 2009, December 2008 and 2009, Jan 2009 and 2010, March 2009 and 2010 and so forth. This trend is largely impacted by climatic variations, which is normally tied to annual cycles. The seasonality in this case is a regular fluctuation that is repeated year after year with a similar timing and intensity.

 

Regression Analysis of Edgbaston average temperatures

 

 

Regression Analysis of exchange rates for Euro and Pound

Method of Data Collection and Analysis

For the exchange rates between Euro and pound, the data collection was based on secondary soutces, from the online data market website. Regression Analysis was done in SPSS, using bivariate linear regression models. The data shows 72 observations of the exchange rates , inclusinmg its forecast for the next three months.

For Edgbaston average temperatures, the data collection took place in a similar manner as that of the exchange rates.  Data analysis took place in SPSS, in form of two variables Linear correlation analysis.

ii)                  Regression analysis (including GAMLSS)

 

Descriptive Statistics

 

Mean

Std. Deviation

N

Month

6.50

3.476

72

Edgbaston Monthly Average Temperatures 0C

9.81

4.895

72

 

 

 

 

Correlations

   

Month

Edgbaston Monthly Average Temperatures 0C

Pearson Correlation Month

1.000

.214

Edgbaston Monthly Average Temperatures 0C

.214

1.000

Sig. (1-tailed) Month

.

.035

Edgbaston Monthly Average Temperatures 0C

.035

.

N Month

72

72

Edgbaston Monthly Average Temperatures 0C

72

72

 

 

Model Summary

 
Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Change Statistics

Durbin-Watson

 

R Square Change

F Change

df1

df2

Sig. F Change

 
1

.214a

.046

.032

3.420

.046

3.372

1

70

.071

.814

 
a. Predictors: (Constant), Edgbaston Monthly Average Temperatures 0C  
b. Dependent Variable: Month  
 

Coefficientsa

  Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

Collinearity Statistics

 

B

Std. Error

Beta

Tolerance

VIF

  1 (Constant)

5.007

.907

5.518

.000

  Edgbaston Monthly Average Temperatures 0C

.152

.083

.214

1.836

.071

1.000

1.000

  a. Dependent Variable: Month

 

The analysis shows the expected probability (cumulative) against the observed cumulative probability. Apart from the scattered dot lines, the analysis shows a linear regression line of best fit for the Edgbaston average temperatures. This measures the accuracy of prediction. It represents the variation between the real probability observed and the projected values.

Descriptive Statistics

 

Mean

Std. Deviation

N

Month

6.50

3.476

72

Exchange Rate

1.1850941

.05707804

72

 

 

Correlations

   

Month

Exchange Rate

Pearson Correlation Month

1.000

-.022

Exchange Rate

-.022

1.000

Sig. (1-tailed) Month

.

.426

Exchange Rate

.426

.

N Month

72

72

Exchange Rate

72

72

 

 

Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Change Statistics

Durbin-Watson

R Square Change

F Change

df1

df2

Sig. F Change

1

.022a

.001

-.014

3.500

.001

.035

1

70

.852

.782

a. Predictors: (Constant), Exchange Rate
b. Dependent Variable: Month

 

 

Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

Collinearity Statistics

B

Std. Error

Beta

Tolerance

VIF

1 (Constant)

8.120

8.634

.940

.350

Exchange Rate

-1.367

7.278

-.022

-.188

.852

1.000

1.000

a. Dependent Variable: Month

 

 

This was another linear regression analysis involving the expected probabilities against the actual probabilities for the Exchange rates. The analysis shows the expected probability (cumulative) against the observed cumulative probability. Apart from the scattered dot lines, the analysis shows a linear regression line of best fit. This measures the accuracy of prediction. It represents the variation between the real probability observed and the projected values.  The regression analysis ensures that unconditional volatility and the velocity of mean restrictions are constant.   Additionally, large data are required to accurately classify different elements. Nevertheless, the slow moving pattern is mean reversion to a constant value and the notion that volatility procedure ultimately changes to a fixed level.               

 

iii)                ARIMA models

 

 

 

Fig 1: Showing

order