{"id":46788,"date":"2024-04-26T23:16:02","date_gmt":"2024-04-26T23:16:02","guid":{"rendered":"http:\/\/localhost\/branding\/time-series-econometric-methods\/"},"modified":"2024-04-26T23:16:02","modified_gmt":"2024-04-26T23:16:02","slug":"time-series-econometric-methods","status":"publish","type":"post","link":"https:\/\/sheilathewriter.com\/blog\/time-series-econometric-methods\/","title":{"rendered":"Time Series Econometric Methods"},"content":{"rendered":"

Time Series Econometric Methods<\/p>\n

Step 1<\/p>\n

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;<\/p>\n

EMBED Equation.3 :<\/p>\n

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 \u201cbasket of goods\u201d 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). <\/p>\n

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. <\/p>\n

Step 2<\/p>\n

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).<\/p>\n

Descriptive statistics of the US CPI data<\/p>\n

Anderson-Darling A-Squared 1.178<\/p>\n

p 0.004<\/p>\n

95% Critical Value 0.787<\/p>\n

99% Critical Value 1.092<\/p>\n

Mean 218.210<\/p>\n

Mode #N\/A<\/p>\n

Standard Deviation 4.598<\/p>\n

Variance 21.145<\/p>\n

Skewedness 0.561<\/p>\n

Kurtosis -0.639<\/p>\n

N 48.000<\/p>\n

Minimum 211.401<\/p>\n

1st Quartile 214.740<\/p>\n

Median 217.419<\/p>\n

3rd Quartile 220.570<\/p>\n

Maximum 227.033<\/p>\n

Confidence Interval 1.335<\/p>\n

for Mean (Mu) 216.875<\/p>\n

0.95 219.546<\/p>\n

For Stdev (sigma) 3.828<\/p>\n

5.760<\/p>\n

for Median 216.476<\/p>\n

218.749<\/p>\n

Descriptive statistics of the Swiss CPI data<\/p>\n

Anderson-Darling A-Squared 1.129<\/p>\n

p 0.005<\/p>\n

95% Critical Value 0.787<\/p>\n

99% Critical Value 1.092<\/p>\n

Mean 218.209<\/p>\n

Mode #N\/A<\/p>\n

Standard Deviation 4.687<\/p>\n

Variance 21.969<\/p>\n

Skewedness 0.438<\/p>\n

Kurtosis -0.615<\/p>\n

N 48.000<\/p>\n

Minimum 210.228<\/p>\n

1st Quartile 215.608<\/p>\n

Median 217.987<\/p>\n

3rd Quartile 220.029<\/p>\n

Maximum 226.889<\/p>\n

Confidence Interval 1.361<\/p>\n

for Mean (Mu) 216.848<\/p>\n

0.95 219.570<\/p>\n

For Stdev (sigma) 3.902<\/p>\n

5.871<\/p>\n

for Median 216.177<\/p>\n

218.783<\/p>\n

F-Test Two-Sample for Variances \uf061 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)<\/p>\n

Accept Null Hypothesis because p > 0.05 (Variances are the same) Multivariate analysis<\/p>\n

Step 3<\/p>\n

Autocorrelation of the US and Swiss data<\/p>\n

The US and Swiss data are autocorelated<\/p>\n

\u00a0 US Switzerland 1 0.986169 0.986169 1 <\/p>\n

Step 4<\/p>\n

Regression analysis of the US and Swiss CPI data<\/p>\n

SUMMARY OUTPUT Force Constant to Zero FALSE Regression Statistics \u00a0 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<\/p>\n

ANOVA \u00a0 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 \u00a0 \u00a0 \u00a0 0.95<\/p>\n

\u00a0 Coefficients Standard Error t Stat P-value Lower 95% Upper 95%<\/p>\n

Intercept -1.136719056 5.436640586 -0.2090848 0.835 -12.080108 9.80667068<\/p>\n

US 1.005200879 0.02490926 40.354505 0.000 0.955061 1.055340618<\/p>\n

y = -1.137 +1.005*US Confidence Level<\/p>\n

0.99<\/p>\n

Lower 99% Upper 99%<\/p>\n

-15.745 13.47161<\/p>\n

0.938269 1.072132<\/p>\n

y = -1.137 +1.005*US Observations Predicted Switzerland Residuals Standard Residuals Percentile Switzerland<\/p>\n

1 212.16590 -1.08590 -1.39781 1.04167 210.228<\/p>\n

2 212.59211 -0.89911 -1.15736 3.12500 211.08<\/p>\n

3 213.41436 0.11364 0.14628 5.20833 211.143<\/p>\n

4 213.94712 0.87588 1.12746 7.29167 211.693<\/p>\n

5 215.18854 1.44346 1.85806 9.37500 212.193<\/p>\n

6 217.46432 1.35068 1.73864 11.45833 212.425<\/p>\n

7 219.09274 0.87126 1.12151 13.54167 212.709<\/p>\n

8 218.74997 0.33603 0.43255 15.62500 213.24<\/p>\n

9 218.87361 -0.09061 -0.11663 17.70833 213.528<\/p>\n

10 216.95769 -0.38469 -0.49519 19.79167 213.856<\/p>\n

11 213.04545 -0.62045 -0.79867 21.87500 214.823<\/p>\n

12 211.36375 -1.13575 -1.46197 23.95833 215.351<\/p>\n

13 211.92767 -0.78467 -1.01005 26.04167 215.693<\/p>\n

14 212.79315 -0.60015 -0.77253 28.12500 215.834<\/p>\n

15 212.52978 0.17922 0.23069 30.20833 215.949<\/p>\n

16 212.67453 0.56547 0.72788 32.29167 215.969<\/p>\n

17 212.94795 0.90805 1.16887 34.37500 216.177<\/p>\n

18 214.72414 0.96886 1.24715 36.45833 216.33<\/p>\n

19 214.70604 0.64496 0.83021 38.54167 216.573<\/p>\n

20 215.46296 0.37104 0.47761 40.62500 216.632<\/p>\n

21 215.86906 0.09994 0.12864 42.70833 216.687<\/p>\n

22 216.46515 -0.28815 -0.37091 44.79167 216.741<\/p>\n

23 217.10546 -0.77546 -0.99819 46.87500 217.631<\/p>\n

24 217.32359 -1.37459 -1.76941 48.95833 217.965<\/p>\n

25 217.46331 -0.77631 -0.99929 51.04167 218.009<\/p>\n

26 217.39094 -0.64994 -0.83662 53.12500 218.011<\/p>\n

27 217.43416 0.19684 0.25338 55.20833 218.178<\/p>\n

28 217.36681 0.64219 0.82664 57.29167 218.312<\/p>\n

29 217.17482 1.00318 1.29133 59.37500 218.439<\/p>\n

30 217.19894 0.76606 0.98609 61.45833 218.711<\/p>\n

31 217.64425 0.36675 0.47210 63.54167 218.783<\/p>\n

32 218.05939 0.25261 0.32516 65.62500 218.803<\/p>\n

33 218.36297 0.07603 0.09787 67.70833 218.815<\/p>\n

34 219.02238 -0.31138 -0.40081 69.79167 219.086<\/p>\n

35 219.44557 -0.64257 -0.82713 71.87500 219.179<\/p>\n

36 220.42363 -1.24463 -1.60212 73.95833 219.964<\/p>\n

37 221.04886 -0.82586 -1.06307 76.04167 220.223<\/p>\n

38 222.02592 -0.71692 -0.92284 78.12500 221.309<\/p>\n

39 223.21708 0.24992 0.32170 80.20833 223.467<\/p>\n

40 224.05843 0.84757 1.09101 82.29167 224.906<\/p>\n

41 224.66558 1.29842 1.67137 84.37500 225.672<\/p>\n

42 224.86963 0.85237 1.09719 86.45833 225.722<\/p>\n

43 225.55116 0.37084 0.47736 88.54167 225.922<\/p>\n

44 226.30606 0.23894 0.30757 90.62500 225.964<\/p>\n

45 226.91320 -0.02420 -0.03116 92.70833 226.23<\/p>\n

46 226.84686 -0.42586 -0.54818 94.79167 226.421<\/p>\n

47 227.05795 -0.82795 -1.06577 96.87500 226.545<\/p>\n

48 227.07705 -1.40505 -1.80863 98.95833 226.889<\/p>\n

The estimated error correlation (ECM) model will be as bellow\u00a0\u00a0\u00a0<\/p>\n

The process starts by estimating the long run relationship between the variables in the yt\u00a0and\u00a0xt yt\u00a0=\u00a0\u03b20\u00a0+\u00a0\u03b21xt\u00a0+ ut, however, with cointetegration, it is healthy to be confident that the variables in \u03b20\u00a0and\u00a0\u03b21 may never be biased evening when dealing with large samples like 40 in our case. Therefore both \u03b20\u00a0and\u00a0\u03b21\u00a0are 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 <\/p>\n

U (stationary or dynamic)<\/p>\n

HYPERLINK “http:\/\/3.bp.blogspot.com\/-UQ58H-0noqI\/TVaC4Heft2I\/AAAAAAAAAGE\/sWH1DpzZ3eA\/s1600\/cointegration+and+ecm1.JPG” <\/p>\n

INCLUDEPICTURE “http:\/\/upload.wikimedia.org\/wikipedia\/en\/math\/c\/7\/d\/c7dafde3c8b029f02b3192a4acd1cc85.png” * MERGEFORMATINET <\/p>\n

In this equation, the components of the dickey fuller test are as shown below:<\/p>\n

\u03b1\u00a0=is a constant,\u00a0\u03b2=coefficient, p\u00a0=the lag order <\/p>\n

From our original results, we can conclude that the two series have extremely co integrated because the residuals are stationary.<\/p>\n

The CPI Data<\/p>\n

US Data from U.S. Department of Labour: Bureau of Labour Statistics<\/p>\n

Period US Switzerland<\/p>\n

2008-01-01 212.199 211.08 0.99<\/p>\n

2008-02-01 212.623 211.693 1.00<\/p>\n

2008-03-01 213.441 213.528 1.00<\/p>\n

2008-04-01 213.971 214.823 1.00<\/p>\n

2008-05-01 215.206 216.632 1.01<\/p>\n

2008-06-01 217.470 218.815 1.01<\/p>\n

2008-07-01 219.090 219.964 1.00<\/p>\n

2008-08-01 218.749 219.086 1.00<\/p>\n

2008-09-01 218.872 218.783 1.00<\/p>\n

2008-10-01 216.966 216.573 1.00<\/p>\n

2008-11-01 213.074 212.425 1.00<\/p>\n

2008-12-01 211.401 210.228 0.99<\/p>\n

2009-01-01 211.962 211.143 1.00<\/p>\n

2009-02-01 212.823 212.193 1.00<\/p>\n

2009-03-01 212.561 212.709 1.00<\/p>\n

2009-04-01 212.705 213.24 1.00<\/p>\n

2009-05-01 212.977 213.856 1.00<\/p>\n

2009-06-01 214.744 215.693 1.00<\/p>\n

2009-07-01 214.726 215.351 1.00<\/p>\n

2009-08-01 215.479 215.834 1.00<\/p>\n

2009-09-01 215.883 215.969 1.00<\/p>\n

2009-10-01 216.476 216.177 1.00<\/p>\n

2009-11-01 217.113 216.33 1.00<\/p>\n

2009-12-01 217.330 215.949 0.99<\/p>\n

2010-01-01 217.469 216.687 1.00<\/p>\n

2010-02-01 217.397 216.741 1.00<\/p>\n

2010-03-01 217.440 217.631 1.00<\/p>\n

2010-04-01 217.373 218.009 1.00<\/p>\n

2010-05-01 217.182 218.178 1.00<\/p>\n

2010-06-01 217.206 217.965 1.00<\/p>\n

2010-07-01 217.649 218.011 1.00<\/p>\n

2010-08-01 218.062 218.312 1.00<\/p>\n

2010-09-01 218.364 218.439 1.00<\/p>\n

2010-10-01 219.020 218.711 1.00<\/p>\n

2010-11-01 219.441 218.803 1.00<\/p>\n

2010-12-01 220.414 219.179 0.99<\/p>\n

2011-01-01 221.036 220.223 1.00<\/p>\n

2011-02-01 222.008 221.309 1.00<\/p>\n

2011-03-01 223.193 223.467 1.00<\/p>\n

2011-04-01 224.030 224.906 1.00<\/p>\n

2011-05-01 224.634 225.964 1.01<\/p>\n

2011-06-01 224.837 225.722 1.00<\/p>\n

2011-07-01 225.515 225.922 1.00<\/p>\n

2011-08-01 226.266 226.545 1.00<\/p>\n

2011-09-01 226.870 226.889 1.00<\/p>\n

2011-10-01 226.804 226.421 1.00<\/p>\n

2011-11-01 227.014 226.23 1.00<\/p>\n

2011-12-01 227.033 225.672 0.99<\/p>\n

References <\/p>\n

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\/<\/p>\n

Dickey, D. & A. Fuller (1979), \u201cDistribution of the Estimators for Autoregressive Time Series with a Unit Root,\u201d Journal of the American Statistical Association, 74, p. 427\u2013431.<\/p>\n

Enders, W., (2004). Applied Econometric Time Series: Second Edition. John Wiley & Sons: United States.<\/p>\n

Elder, J. & Kennedy, E. (2001)\u00a0\u201cTesting for Unit Roots: What Should Students Be Taught?\u201d.\u00a0Journal of Economic Education, 32(2): 137-146<\/p>\n

Appendix<\/p>\n

Period US<\/p>\n

2008-01-01 212.199<\/p>\n

2008-02-01 212.623<\/p>\n

2008-03-01 213.441<\/p>\n

2008-04-01 213.971<\/p>\n

2008-05-01 215.206<\/p>\n

2008-06-01 217.470<\/p>\n

2008-07-01 219.090<\/p>\n

2008-08-01 218.749<\/p>\n

2008-09-01 218.872<\/p>\n

2008-10-01 216.966<\/p>\n

2008-11-01 213.074<\/p>\n

2008-12-01 211.401<\/p>\n

2009-01-01 211.962<\/p>\n

2009-02-01 212.823<\/p>\n

2009-03-01 212.561<\/p>\n

2009-04-01 212.705<\/p>\n

2009-05-01 212.977<\/p>\n

2009-06-01 214.744<\/p>\n

2009-07-01 214.726<\/p>\n

2009-08-01 215.479<\/p>\n

2009-09-01 215.883<\/p>\n

2009-10-01 216.476<\/p>\n

2009-11-01 217.113<\/p>\n

2009-12-01 217.330<\/p>\n

2010-01-01 217.469<\/p>\n

2010-02-01 217.397<\/p>\n

2010-03-01 217.440<\/p>\n

2010-04-01 217.373<\/p>\n

2010-05-01 217.182<\/p>\n

2010-06-01 217.206<\/p>\n

2010-07-01 217.649<\/p>\n

2010-08-01 218.062<\/p>\n

2010-09-01 218.364<\/p>\n

2010-10-01 219.020<\/p>\n

2010-11-01 219.441<\/p>\n

2010-12-01 220.414<\/p>\n

2011-01-01 221.036<\/p>\n

2011-02-01 222.008<\/p>\n

2011-03-01 223.193<\/p>\n

2011-04-01 224.030<\/p>\n

2011-05-01 224.634<\/p>\n

2011-06-01 224.837<\/p>\n

2011-07-01 225.515<\/p>\n

2011-08-01 226.266<\/p>\n

2011-09-01 226.870<\/p>\n

2011-10-01 226.804<\/p>\n

2011-11-01 227.014<\/p>\n

2011-12-01 227.033<\/p>\n

Swiss CPI data<\/p>\n

Period Swiss CPI data<\/p>\n

2008-01-01 211.08<\/p>\n

2008-02-01 211.693<\/p>\n

2008-03-01 213.528<\/p>\n

2008-04-01 214.823<\/p>\n

2008-05-01 216.632<\/p>\n

2008-06-01 218.815<\/p>\n

2008-07-01 219.964<\/p>\n

2008-08-01 219.086<\/p>\n

2008-09-01 218.783<\/p>\n

2008-10-01 216.573<\/p>\n

2008-11-01 212.425<\/p>\n

2008-12-01 210.228<\/p>\n

2009-01-01 211.143<\/p>\n

2009-02-01 212.193<\/p>\n

2009-03-01 212.709<\/p>\n

2009-04-01 213.24<\/p>\n

2009-05-01 213.856<\/p>\n

2009-06-01 215.693<\/p>\n

2009-07-01 215.351<\/p>\n

2009-08-01 215.834<\/p>\n

2009-09-01 215.969<\/p>\n

2009-10-01 216.177<\/p>\n

2009-11-01 216.33<\/p>\n

2009-12-01 215.949<\/p>\n

2010-01-01 216.687<\/p>\n

2010-02-01 216.741<\/p>\n

2010-03-01 217.631<\/p>\n

2010-04-01 218.009<\/p>\n

2010-05-01 218.178<\/p>\n

2010-06-01 217.965<\/p>\n

2010-07-01 218.011<\/p>\n

2010-08-01 218.312<\/p>\n

2010-09-01 218.439<\/p>\n

2010-10-01 218.711<\/p>\n

2010-11-01 218.803<\/p>\n

2010-12-01 219.179<\/p>\n

2011-01-01 220.223<\/p>\n

2011-02-01 221.309<\/p>\n

2011-03-01 223.467<\/p>\n

2011-04-01 224.906<\/p>\n

2011-05-01 225.964<\/p>\n

2011-06-01 225.722<\/p>\n

2011-07-01 225.922<\/p>\n

2011-08-01 226.545<\/p>\n

2011-09-01 226.889<\/p>\n

2011-10-01 226.421<\/p>\n

2011-11-01 226.23<\/p>\n

2011-12-01 225.672<\/p>\n","protected":false},"excerpt":{"rendered":"

Time Series Econometric Methods Step 1 According to Enders, (2004), Purchasing Power Parity (PPP) measures how much money is required<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-46788","post","type-post","status-publish","format-standard","hentry"],"yoast_head":"\nTime Series Econometric Methods - sheilathewriter<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/sheilathewriter.com\/blog\/time-series-econometric-methods\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Time Series Econometric Methods - sheilathewriter\" \/>\n<meta property=\"og:description\" content=\"Time Series Econometric Methods Step 1 According to Enders, (2004), Purchasing Power Parity (PPP) measures how much money is required\" \/>\n<meta property=\"og:url\" content=\"https:\/\/sheilathewriter.com\/blog\/time-series-econometric-methods\/\" \/>\n<meta property=\"og:site_name\" content=\"sheilathewriter\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-26T23:16:02+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/sheilathewriter.com\/blog\/time-series-econometric-methods\/\",\"url\":\"https:\/\/sheilathewriter.com\/blog\/time-series-econometric-methods\/\",\"name\":\"Time Series Econometric Methods - 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