Fisher’s and t-test equation
Summary
The report analyses the relationship between Fisher’s equation and the t-test. It will use data from two countries to try to prove the relationship. The report will use two countries, United State data, and the United Kingdom data as examples. The analysis will be done using various methods as the formulas hold in accordance with Fisher’s equation and t-test. The report will then try to analyze the outcome of the results as it analyzes whether Fisher’s equation holds (Hawawini, 2011, 39). The outcomes of the results will be used to draw conclusions about the interest rates of the countries used in the report. After the conclusion in accordance with the reporting requirement, this will be used to show whether Fisher’s equation holds.
Aim: The aim of this report is to try to prove Fisher’s equation. It intends to look into Fisher’s equation if it holds according to international nominal rates in relation to inflation. The report intends to try to do an analysis of the outcome of the data used as examples (Monro Barbara, 2005, 105). It aims at analyzing the data by applying fisher’s equation and t-test in relation to other methodologies related such as mean and standard deviation.
Objectives: The objectives of the report are to analyze the example data by use of Fisher’s equation. The report also tries to prove Fisher’s equation of the international. Another objective is to determine which country will have the highest interest rate if the international fisher’s equation holds. To determine the t-test of the data recorded. Another is to give the conclusion of the county with a higher interest rate in relation to the country with the low-interest rate and to determine the GPB buying rate in terms of the interest rate.
Data: The data bellow has 60 periods of the exchange rate of the US and the United Kingdom as per day. The data are the records of the respective countries as shown on daily basis excluding Saturday and Sunday
| Jul.Day | YYYY/MM/DD | Wdy | GBP/USD |
| 2456295 | 2013/01/02 | Wed | 0.61523 |
| 2456296 | 2013/01/03 | Thu | 0.61918 |
| 2456297 | 2013/01/04 | Fri | 0.62360 |
| 2456300 | 2013/01/07 | Mon | 0.62192 |
| 2456301 | 2013/01/08 | Tue | 0.62349 |
| 2456302 | 2013/01/09 | Wed | 0.62396 |
| 2456303 | 2013/01/10 | Thu | 0.62020 |
| 2456304 | 2013/01/11 | Fri | 0.62002 |
| 2456307 | 2013/01/14 | Mon | 0.62204 |
| 2456308 | 2013/01/15 | Tue | 0.62158 |
| 2456309 | 2013/01/16 | Wed | 0.62468 |
| 2456310 | 2013/01/17 | Thu | 0.62539 |
| 2456311 | 2013/01/18 | Fri | 0.62992 |
| 2456314 | 2013/01/21 | Mon | 0.63199 |
| 2456315 | 2013/01/22 | Tue | 0.63049 |
| 2456316 | 2013/01/23 | Wed | 0.63103 |
| 2456317 | 2013/01/24 | Thu | 0.63339 |
| 2456318 | 2013/01/25 | Fri | 0.63296 |
| 2456321 | 2013/01/28 | Mon | 0.63756 |
| 2456322 | 2013/01/29 | Tue | 0.63495 |
| 2456323 | 2013/01/30 | Wed | 0.63324 |
| 2456324 | 2013/01/31 | Thu | 0.63073 |
| 2456325 | 2013/02/01 | Fri | 0.63519 |
| 2456328 | 2013/02/04 | Mon | 0.63548 |
| 2456329 | 2013/02/05 | Tue | 0.63894 |
| 2456330 | 2013/02/06 | Wed | 0.63883 |
| 2456331 | 2013/02/07 | Thu | 0.63650 |
| 2456332 | 2013/02/08 | Fri | 0.63242 |
| 2456335 | 2013/02/11 | Mon | 0.63828 |
| 2456336 | 2013/02/12 | Tue | 0.63887 |
| 2456337 | 2013/02/13 | Wed | 0.64338 |
| 2456338 | 2013/02/14 | Thu | 0.64509 |
| 2456339 | 2013/02/15 | Fri | 0.64444 |
| 2456343 | 2013/02/19 | Tue | 0.64805 |
| 2456344 | 2013/02/20 | Wed | 0.65361 |
| 2456345 | 2013/02/21 | Thu | 0.65524 |
| 2456346 | 2013/02/22 | Fri | 0.65551 |
| 2456349 | 2013/02/25 | Mon | 0.66179 |
| 2456350 | 2013/02/26 | Tue | 0.66118 |
| 2456351 | 2013/02/27 | Wed | 0.66078 |
| 2456352 | 2013/02/28 | Thu | 0.65832 |
| 2456353 | 2013/03/01 | Fri | 0.66527 |
| 2456356 | 2013/03/04 | Mon | 0.66302 |
| 2456357 | 2013/03/05 | Tue | 0.66242 |
| 2456358 | 2013/03/06 | Wed | 0.66508 |
| 2456359 | 2013/03/07 | Thu | 0.66529 |
| 2456360 | 2013/03/08 | Fri | 0.66999 |
| 2456363 | 2013/03/11 | Mon | 0.67129 |
| 2456364 | 2013/03/12 | Tue | 0.67182 |
| 2456365 | 2013/03/13 | Wed | 0.67004 |
| 2456366 | 2013/03/14 | Thu | 0.66440 |
| 2456367 | 2013/03/15 | Fri | 0.66077 |
| 2456370 | 2013/03/18 | Mon | 0.66189 |
| 2456371 | 2013/03/19 | Tue | 0.66147 |
| 2456372 | 2013/03/20 | Wed | 0.66083 |
| 2456373 | 2013/03/21 | Thu | 0.65880 |
| 2456374 | 2013/03/22 | Fri | 0.65625 |
| 2456377 | 2013/03/25 | Mon | 0.65859 |
| 2456378 | 2013/03/26 | Tue | 0.65983 |
| 2456379 | 2013/03/27 | Wed | 0.66181 |
| Jul.Day | YYYY/MM/DD | Wdy | USD/GBP |
| 2456295 | 2013/01/02 | Wed | 1.6254 |
| 2456296 | 2013/01/03 | Thu | 1.6150 |
| 2456297 | 2013/01/04 | Fri | 1.6036 |
| 2456300 | 2013/01/07 | Mon | 1.6079 |
| 2456301 | 2013/01/08 | Tue | 1.6039 |
| 2456302 | 2013/01/09 | Wed | 1.6027 |
| 2456303 | 2013/01/10 | Thu | 1.6124 |
| 2456304 | 2013/01/11 | Fri | 1.6128 |
| 2456307 | 2013/01/14 | Mon | 1.6076 |
| 2456308 | 2013/01/15 | Tue | 1.6088 |
| 2456309 | 2013/01/16 | Wed | 1.6008 |
| 2456310 | 2013/01/17 | Thu | 1.5990 |
| 2456311 | 2013/01/18 | Fri | 1.5875 |
| 2456314 | 2013/01/21 | Mon | 1.5823 |
| 2456315 | 2013/01/22 | Tue | 1.5861 |
| 2456316 | 2013/01/23 | Wed | 1.5847 |
| 2456317 | 2013/01/24 | Thu | 1.5788 |
| 2456318 | 2013/01/25 | Fri | 1.5799 |
| 2456321 | 2013/01/28 | Mon | 1.5685 |
| 2456322 | 2013/01/29 | Tue | 1.5749 |
| 2456323 | 2013/01/30 | Wed | 1.5792 |
| 2456324 | 2013/01/31 | Thu | 1.5855 |
| 2456325 | 2013/02/01 | Fri | 1.5743 |
| 2456328 | 2013/02/04 | Mon | 1.5736 |
| 2456329 | 2013/02/05 | Tue | 1.5651 |
| 2456330 | 2013/02/06 | Wed | 1.5654 |
| 2456331 | 2013/02/07 | Thu | 1.5711 |
| 2456332 | 2013/02/08 | Fri | 1.5812 |
| 2456335 | 2013/02/11 | Mon | 1.5667 |
| 2456336 | 2013/02/12 | Tue | 1.5653 |
| 2456337 | 2013/02/13 | Wed | 1.5543 |
| 2456338 | 2013/02/14 | Thu | 1.5502 |
| 2456339 | 2013/02/15 | Fri | 1.5517 |
| 2456343 | 2013/02/19 | Tue | 1.5431 |
| 2456344 | 2013/02/20 | Wed | 1.5300 |
| 2456345 | 2013/02/21 | Thu | 1.5262 |
| 2456346 | 2013/02/22 | Fri | 1.5255 |
| 2456349 | 2013/02/25 | Mon | 1.5111 |
| 2456350 | 2013/02/26 | Tue | 1.5125 |
| 2456351 | 2013/02/27 | Wed | 1.5134 |
| 2456352 | 2013/02/28 | Thu | 1.5190 |
| 2456353 | 2013/03/01 | Fri | 1.5032 |
| 2456356 | 2013/03/04 | Mon | 1.5083 |
| 2456357 | 2013/03/05 | Tue | 1.5096 |
| 2456358 | 2013/03/06 | Wed | 1.5036 |
| 2456359 | 2013/03/07 | Thu | 1.5031 |
| 2456360 | 2013/03/08 | Fri | 1.4926 |
| 2456363 | 2013/03/11 | Mon | 1.4897 |
| 2456364 | 2013/03/12 | Tue | 1.4885 |
| 2456365 | 2013/03/13 | Wed | 1.4925 |
| 2456366 | 2013/03/14 | Thu | 1.5051 |
| 2456367 | 2013/03/15 | Fri | 1.5134 |
| 2456370 | 2013/03/18 | Mon | 1.5108 |
| 2456371 | 2013/03/19 | Tue | 1.5118 |
| 2456372 | 2013/03/20 | Wed | 1.5132 |
| 2456373 | 2013/03/21 | Thu | 1.5179 |
| 2456374 | 2013/03/22 | Fri | 1.5238 |
| 2456377 | 2013/03/25 | Mon | 1.5184 |
| 2456378 | 2013/03/26 | Tue | 1.5155 |
| 2456379 | 2013/03/27 | Wed | 1.5110 |
Methodology: Methodology is the method or methods that apply in computing the data (Hawawini, 2011, 39). This is actually to give meaning to the data that has been collected and arranged in a particular manner. This report will apply several methodologies to analyze the data by at the end to use t – test and finally international Fisher’s equation. The analysis will involve the calculation of mean, variance, standard deviation, which enables arriving at the t – test. The analysis of the data is as follows:
Taking the data to be unpaired and running the t – test, the results are as follows (Hawawini, 2011, 39)
t-test (Z)
Z =
t= -159.
sdev= 0.313E-01
degrees of freedom =118
The probability of this result, assuming the null hypothesis, is less than .0001
Group A: Number of items (n) = 60
0.615 0.619 0.620 0.620 0.622 0.622 0.622 0.623 0.624 0.624 0.625 0.625 0.630 0.630 0.631 0.631 0.632 0.632 0.633 0.633 0.633 0.635 0.635 0.635 0.636 0.638 0.638 0.639 0.639 0.639 0.643 0.644 0.645 0.648 0.654 0.655 0.656 0.656 0.658 0.659 0.659 0.660 0.661 0.661 0.661 0.661 0.661 0.662 0.662 0.662 0.662 0.663 0.664 0.665 0.665 0.665 0.670 0.670 0.671 0.672
Mean () = summation of data/data size = 38.67831/60 = 0.645
Mean () = 0.645
95% confidence interval for Mean: 0.6366 thru 0.6526
Variance of the data = 0.000287
Standard Deviation = √variance
Standard Deviation = 1.694E-02
High = 0.672 Low = 0.615
Median = 0.641
Average Absolute Deviation from Median = 1.524E-02
Group B: Number of items (n) = 60
1.49 1.49 1.49 1.49 1.50 1.50 1.50 1.51 1.51 1.51 1.51 1.51 1.51 1.51 1.51 1.51 1.51 1.51 1.52 1.52 1.52 1.52 1.52 1.53 1.53 1.53 1.54 1.55 1.55 1.55 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.58 1.58 1.58 1.58 1.58 1.58 1.59 1.59 1.59 1.60 1.60 1.60 1.60 1.60 1.61 1.61 1.61 1.61 1.61 1.61 1.63
Mean () = summation of data/data size = 93.139 /60 =1.55
Mean = 1.55
This mean is on the assumption that the used data sampled randomly and independently.
95% confidence interval for Mean: 1.544 thru 1.560, as calculated
Variance = 0.001666, as calculated
Standard Deviation = √variance
Standard Deviation = 4.082E-02, as calculated
High = 1.63 Low = 1.49
Median = 1.56
Average Absolute Deviation from Median = 3.668E-02
International Fisher’s Equation
E(e) = (i$ – ic)/(1 + ic) = i$ – ic
Rearranging this equation gives
E(e) = (1 + i$)/(1 + ic) – 1 = i$ – ic
E(e) – the expected rate of change in the exchange rate
i$ – nominal interest rate
ic – real interest rate
Spot exchange between United States and United Kingdom is 1.55 USD/GPB, as calculated
US interest rate as per now = 4, as calculated
U.K interest rate as per now = 6%, as calculated
From the given data we can estimate the expected spot change rate in 12 months from now according to International Fisher’s effect (Wang Peijie, 2009, 321)
From the equation: E(e) = (1 + i$)/(1 + ic) = (1 + 4%)/(1 + 6%) = 0.981
1.55* 0.981 = 1.521
Rearranging the equation and performing the calculations
E(e) = (4% – 6%)/(1 + 6%) = -0.017632
E(e) = ((1 + 4%)/(1 + 6%)) – 1 = -0.01762
From the two calculations, the percentage is -1.7632%. (Wang Peijie, 2009, 276)
Analysis: From the methodologies, calculations of means are on the assumption that the used data are sampled randomly and independently. This assumption gives the confident interval for the mean of 95% by many sampled data. The calculations are done on the unpaired groups of data. The difference in their means is then taken into account on how far apart are the differences. The confident interval is 95% as in the methodology, which shows that the assumptions are true, and gives how far the means of the data are.
From the international Fisher’s equation, the expected percentage rate of change is a depreciation of 1.7632% as it gives a negative result from the calculation. That is, the expected rate of change yields the same values of -1.7632% (Wang Peijie, 2009, 213). This depreciation is for the GBP of the country with higher interest rate as compared to the other country (Wang Peijie, 2009, 213). It shows that the cost of purchasing $ per 1GBP is now only 1.521 rather than 1.55 as it was calculated before. This shows that the value of the currency of the country that has higher interest rate will depreciate. This shows that International fisher’s equation holds (Wang Peijie, 2009, 418).
Conclusion
From the analysis, it is true that International Fisher’s equation holds. This shows that the country with the high interest rate will actually have the depreciation. The report has proved this from the analysis of the ample data. These results may change if there are some changes within the data. The methodologies hold as long the there are consistent in how the data occurs. These changes may be because of the factors that come with immediate changes may be in economy. This leads to immediate change in the interest rate, which will affect the entire calculation. Otherwise, the prediction of the calculations as shown holds for any data collected. The analysis of the data using statistical methods gives the way to enable arriving at the right conclusion over the particular data.
Therefore, from the calculations, the GBP of the country has been seen to go higher depending on the interest rate. The higher interest rate therefore needs to be controlled to prevent the country from form having inflation. Inflation in this case, will mean that the county’s economy is down.
Works Cited
Hawawini, Gabriel A, and Claude Viallet. Finance for Executives: Managing for Value
Creation. Mason, Ohio: South-Western Cengage Learning, 2011. Print.
Munro, Barbara H. Statistical Methods for Health Care Research. Philadelphia: Lippincott
Williams & Wilkins, 2005. Print.
Wang, Peijie. The Economics of Foreign Exchange and Global Finance. Berlin: Springer-
Verlag, 2009. Internet resource.
Appendices
Unpaired data: data, which are not corresponding in accordance to collection
Period: the number of times the data occurs
Data: raw facts that has not been analyzed to give the information
Methodology: the method applied to analyze the data
Depreciation: the decrease of something downwards not as it is expected
Equation: an expression that is used to analyze certain problem mathematically
Hypothesis: answers of the objectives of what is intended to be achieved
Null hypothesis: hypothesis, which is not positive