Time Series Econometric Methods

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