Time Series Econometric Methods
Step 1
According to Enders, (2004), Purchasing Power Parity (PPP) measures how much money is required to buy same goods or services in between two countries. The PPP is then used to compute the exchange rate between these two countries. Considering the two countries; home country as Switzerland (country A) and foreign country as US (country B), then the Spot Exchange Rate between the two countries will be given by the following relationship;
EMBED Equation.3 :
Where St is the spot exchange between Switzerland and US, Pt is the price level in Switzerland, and Pt* is the price level in US. If the two countries produce tradable goods and there are no trade impediments to international trade, such as tariffs or transaction costs, then the law of one price should hold. Sometimes the PPP is taken as the law of one price if we are dealing with one commodity. This situation at most times is untenable because the possibility of “basket of goods” is almost impossible to get in both countries that are diverse economically. Differences between PPP and the actual rate frequently occur due to miss-measurement of the relevant price indices. If the price indices reflected only traded goods, then the discrepancy would not be so large. Another reason could be because investors have changed their preference from the dollar to non dollar assets (Dickey, & Fuller, 1979).
The PPP should mean that exchange rates should equate the price of goods and services across the countries. For example, 100 Swiss Francs should buy as much as 100 Swiss Francs exchanged into US dollars and used to purchase goods in America. Usually the spot exchange is expressed as EMBED Equation.3 , where lower case letters denotes a variable in logarithms.
Step 2
Using the Consumer Price Index; the prices in the US were slightly higher than Switzerland. Consumers tend to spend more for products and services in the US compared to Switzerland. The trend of the time series shows that there have been gradual increases for the forty observations taken (Wessa, 2010).
Descriptive statistics of the US CPI data
Anderson-Darling A-Squared 1.178
p 0.004
95% Critical Value 0.787
99% Critical Value 1.092
Mean 218.210
Mode #N/A
Standard Deviation 4.598
Variance 21.145
Skewedness 0.561
Kurtosis -0.639
N 48.000
Minimum 211.401
1st Quartile 214.740
Median 217.419
3rd Quartile 220.570
Maximum 227.033
Confidence Interval 1.335
for Mean (Mu) 216.875
0.95 219.546
For Stdev (sigma) 3.828
5.760
for Median 216.476
218.749
Descriptive statistics of the Swiss CPI data
Anderson-Darling A-Squared 1.129
p 0.005
95% Critical Value 0.787
99% Critical Value 1.092
Mean 218.209
Mode #N/A
Standard Deviation 4.687
Variance 21.969
Skewedness 0.438
Kurtosis -0.615
N 48.000
Minimum 210.228
1st Quartile 215.608
Median 217.987
3rd Quartile 220.029
Maximum 226.889
Confidence Interval 1.361
for Mean (Mu) 216.848
0.95 219.570
For Stdev (sigma) 3.902
5.871
for Median 216.177
218.783
F-Test Two-Sample for Variances 0.05 US Switzerland 218.2104 218.2085 21.14503 21.96906 48 48 47 47 0.96 0.448 0.896 Two-tail 1.62 1.78 Two-tail Accept Null Hypothesis because p > 0.05 (Variances are the same)
Accept Null Hypothesis because p > 0.05 (Variances are the same) Multivariate analysis
Step 3
Autocorrelation of the US and Swiss data
The US and Swiss data are autocorelated
US Switzerland 1 0.986169 0.986169 1
Step 4
Regression analysis of the US and Swiss CPI data
SUMMARY OUTPUT Force Constant to Zero FALSE Regression Statistics Multiple R 0.986 R Square 0.973 Goodness of Fit >= 0.80 Adjusted R Square 0.972 Standard Error 0.785 Observations 48 g
ANOVA df SS MS F P-value Regression 1 1004.180 1004.180 1628.486 0.000 Residual 46 28.36518 0.616634 Total 47 1032.545 0.95
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -1.136719056 5.436640586 -0.2090848 0.835 -12.080108 9.80667068
US 1.005200879 0.02490926 40.354505 0.000 0.955061 1.055340618
y = -1.137 +1.005*US Confidence Level
0.99
Lower 99% Upper 99%
-15.745 13.47161
0.938269 1.072132
y = -1.137 +1.005*US Observations Predicted Switzerland Residuals Standard Residuals Percentile Switzerland
1 212.16590 -1.08590 -1.39781 1.04167 210.228
2 212.59211 -0.89911 -1.15736 3.12500 211.08
3 213.41436 0.11364 0.14628 5.20833 211.143
4 213.94712 0.87588 1.12746 7.29167 211.693
5 215.18854 1.44346 1.85806 9.37500 212.193
6 217.46432 1.35068 1.73864 11.45833 212.425
7 219.09274 0.87126 1.12151 13.54167 212.709
8 218.74997 0.33603 0.43255 15.62500 213.24
9 218.87361 -0.09061 -0.11663 17.70833 213.528
10 216.95769 -0.38469 -0.49519 19.79167 213.856
11 213.04545 -0.62045 -0.79867 21.87500 214.823
12 211.36375 -1.13575 -1.46197 23.95833 215.351
13 211.92767 -0.78467 -1.01005 26.04167 215.693
14 212.79315 -0.60015 -0.77253 28.12500 215.834
15 212.52978 0.17922 0.23069 30.20833 215.949
16 212.67453 0.56547 0.72788 32.29167 215.969
17 212.94795 0.90805 1.16887 34.37500 216.177
18 214.72414 0.96886 1.24715 36.45833 216.33
19 214.70604 0.64496 0.83021 38.54167 216.573
20 215.46296 0.37104 0.47761 40.62500 216.632
21 215.86906 0.09994 0.12864 42.70833 216.687
22 216.46515 -0.28815 -0.37091 44.79167 216.741
23 217.10546 -0.77546 -0.99819 46.87500 217.631
24 217.32359 -1.37459 -1.76941 48.95833 217.965
25 217.46331 -0.77631 -0.99929 51.04167 218.009
26 217.39094 -0.64994 -0.83662 53.12500 218.011
27 217.43416 0.19684 0.25338 55.20833 218.178
28 217.36681 0.64219 0.82664 57.29167 218.312
29 217.17482 1.00318 1.29133 59.37500 218.439
30 217.19894 0.76606 0.98609 61.45833 218.711
31 217.64425 0.36675 0.47210 63.54167 218.783
32 218.05939 0.25261 0.32516 65.62500 218.803
33 218.36297 0.07603 0.09787 67.70833 218.815
34 219.02238 -0.31138 -0.40081 69.79167 219.086
35 219.44557 -0.64257 -0.82713 71.87500 219.179
36 220.42363 -1.24463 -1.60212 73.95833 219.964
37 221.04886 -0.82586 -1.06307 76.04167 220.223
38 222.02592 -0.71692 -0.92284 78.12500 221.309
39 223.21708 0.24992 0.32170 80.20833 223.467
40 224.05843 0.84757 1.09101 82.29167 224.906
41 224.66558 1.29842 1.67137 84.37500 225.672
42 224.86963 0.85237 1.09719 86.45833 225.722
43 225.55116 0.37084 0.47736 88.54167 225.922
44 226.30606 0.23894 0.30757 90.62500 225.964
45 226.91320 -0.02420 -0.03116 92.70833 226.23
46 226.84686 -0.42586 -0.54818 94.79167 226.421
47 227.05795 -0.82795 -1.06577 96.87500 226.545
48 227.07705 -1.40505 -1.80863 98.95833 226.889
The estimated error correlation (ECM) model will be as bellow
The process starts by estimating the long run relationship between the variables in the yt and xt yt = β0 + β1xt + ut, however, with cointetegration, it is healthy to be confident that the variables in β0 and β1 may never be biased evening when dealing with large samples like 40 in our case. Therefore both β0 and β1 are extremely consistent. Because the two are also super consistent, it is also healthy to ignore all the dynamic terms and use all the residuals that arise from the co integrating regression in order to test for consitegration. This can provide us with the results for testing variables for the existence of long-term equilibrium associations. The test can use the DF/ADF statistics irrespective of the nature of
U (stationary or dynamic)
HYPERLINK “http://3.bp.blogspot.com/-UQ58H-0noqI/TVaC4Heft2I/AAAAAAAAAGE/sWH1DpzZ3eA/s1600/cointegration+and+ecm1.JPG”
INCLUDEPICTURE “http://upload.wikimedia.org/wikipedia/en/math/c/7/d/c7dafde3c8b029f02b3192a4acd1cc85.png” * MERGEFORMATINET
In this equation, the components of the dickey fuller test are as shown below:
α =is a constant, β=coefficient, p =the lag order
From our original results, we can conclude that the two series have extremely co integrated because the residuals are stationary.
The CPI Data
US Data from U.S. Department of Labour: Bureau of Labour Statistics
Period US Switzerland
2008-01-01 212.199 211.08 0.99
2008-02-01 212.623 211.693 1.00
2008-03-01 213.441 213.528 1.00
2008-04-01 213.971 214.823 1.00
2008-05-01 215.206 216.632 1.01
2008-06-01 217.470 218.815 1.01
2008-07-01 219.090 219.964 1.00
2008-08-01 218.749 219.086 1.00
2008-09-01 218.872 218.783 1.00
2008-10-01 216.966 216.573 1.00
2008-11-01 213.074 212.425 1.00
2008-12-01 211.401 210.228 0.99
2009-01-01 211.962 211.143 1.00
2009-02-01 212.823 212.193 1.00
2009-03-01 212.561 212.709 1.00
2009-04-01 212.705 213.24 1.00
2009-05-01 212.977 213.856 1.00
2009-06-01 214.744 215.693 1.00
2009-07-01 214.726 215.351 1.00
2009-08-01 215.479 215.834 1.00
2009-09-01 215.883 215.969 1.00
2009-10-01 216.476 216.177 1.00
2009-11-01 217.113 216.33 1.00
2009-12-01 217.330 215.949 0.99
2010-01-01 217.469 216.687 1.00
2010-02-01 217.397 216.741 1.00
2010-03-01 217.440 217.631 1.00
2010-04-01 217.373 218.009 1.00
2010-05-01 217.182 218.178 1.00
2010-06-01 217.206 217.965 1.00
2010-07-01 217.649 218.011 1.00
2010-08-01 218.062 218.312 1.00
2010-09-01 218.364 218.439 1.00
2010-10-01 219.020 218.711 1.00
2010-11-01 219.441 218.803 1.00
2010-12-01 220.414 219.179 0.99
2011-01-01 221.036 220.223 1.00
2011-02-01 222.008 221.309 1.00
2011-03-01 223.193 223.467 1.00
2011-04-01 224.030 224.906 1.00
2011-05-01 224.634 225.964 1.01
2011-06-01 224.837 225.722 1.00
2011-07-01 225.515 225.922 1.00
2011-08-01 226.266 226.545 1.00
2011-09-01 226.870 226.889 1.00
2011-10-01 226.804 226.421 1.00
2011-11-01 227.014 226.23 1.00
2011-12-01 227.033 225.672 0.99
References
Wessa P., (2010). Autocorrelation Function (v1.0.9) in Free Statistics Software (v1.1.23-r7), Office for Research Development and Education, URL http://www.wessa.net/rwasp_autocorrelation.wasp/
Dickey, D. & A. Fuller (1979), “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” Journal of the American Statistical Association, 74, p. 427–431.
Enders, W., (2004). Applied Econometric Time Series: Second Edition. John Wiley & Sons: United States.
Elder, J. & Kennedy, E. (2001) “Testing for Unit Roots: What Should Students Be Taught?”. Journal of Economic Education, 32(2): 137-146
Appendix
Period US
2008-01-01 212.199
2008-02-01 212.623
2008-03-01 213.441
2008-04-01 213.971
2008-05-01 215.206
2008-06-01 217.470
2008-07-01 219.090
2008-08-01 218.749
2008-09-01 218.872
2008-10-01 216.966
2008-11-01 213.074
2008-12-01 211.401
2009-01-01 211.962
2009-02-01 212.823
2009-03-01 212.561
2009-04-01 212.705
2009-05-01 212.977
2009-06-01 214.744
2009-07-01 214.726
2009-08-01 215.479
2009-09-01 215.883
2009-10-01 216.476
2009-11-01 217.113
2009-12-01 217.330
2010-01-01 217.469
2010-02-01 217.397
2010-03-01 217.440
2010-04-01 217.373
2010-05-01 217.182
2010-06-01 217.206
2010-07-01 217.649
2010-08-01 218.062
2010-09-01 218.364
2010-10-01 219.020
2010-11-01 219.441
2010-12-01 220.414
2011-01-01 221.036
2011-02-01 222.008
2011-03-01 223.193
2011-04-01 224.030
2011-05-01 224.634
2011-06-01 224.837
2011-07-01 225.515
2011-08-01 226.266
2011-09-01 226.870
2011-10-01 226.804
2011-11-01 227.014
2011-12-01 227.033
Swiss CPI data
Period Swiss CPI data
2008-01-01 211.08
2008-02-01 211.693
2008-03-01 213.528
2008-04-01 214.823
2008-05-01 216.632
2008-06-01 218.815
2008-07-01 219.964
2008-08-01 219.086
2008-09-01 218.783
2008-10-01 216.573
2008-11-01 212.425
2008-12-01 210.228
2009-01-01 211.143
2009-02-01 212.193
2009-03-01 212.709
2009-04-01 213.24
2009-05-01 213.856
2009-06-01 215.693
2009-07-01 215.351
2009-08-01 215.834
2009-09-01 215.969
2009-10-01 216.177
2009-11-01 216.33
2009-12-01 215.949
2010-01-01 216.687
2010-02-01 216.741
2010-03-01 217.631
2010-04-01 218.009
2010-05-01 218.178
2010-06-01 217.965
2010-07-01 218.011
2010-08-01 218.312
2010-09-01 218.439
2010-10-01 218.711
2010-11-01 218.803
2010-12-01 219.179
2011-01-01 220.223
2011-02-01 221.309
2011-03-01 223.467
2011-04-01 224.906
2011-05-01 225.964
2011-06-01 225.722
2011-07-01 225.922
2011-08-01 226.545
2011-09-01 226.889
2011-10-01 226.421
2011-11-01 226.23
2011-12-01 225.672