Data Analysis Project – Professor Simonoff
The S&P 500 and the World’s MarketsPage 1 of 15
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The S&P 500 and the World’s Markets
Project
With the rapid globalization of the world’s markets and with the increased trade activity between the developed worlds, one would expect that their stock indexes are related. For my project I will be looking at the change in the Standard & Poor’s 500 (S&P) in comparison with the major world markets to see if this is the case. By looking at the bi-monthly returns on these indexes, I will be seeking to build a regression model to investigate their relationship.
Data
The data for my project came from and covers the period for May 1994 through December 1995. I will utilize information about the changes in the NASDAQ Composite (NASDAQ), Dow Jones Industrial Average (DJIA), Nikkei 225, Hang Seng, Australian market, Singaporean market, Paris CAC-40, London Financial Times 100 and German DAX-30 to build my regression of the Standard & Poor’s 500
Missing Data
In building my regression, it would be helpful to have details on the change of exchange rates over these periods rate to see these affect the returns on the indexes (because they partially reflect movements of capital). Unfortunately, these details are not considered. In addition, it would be more helpful if I had a longer period of time included in my data but, once again, I will make due with what I have.
Looking at the Data
Well, before any further delay, let’s take a preliminary look at my data.
Variable N N* Mean Median TrMean StDev
S&P chan 40 0 0.00795 0.00902 0.00861 0.01583
DJIA Cha 40 0 0.00875 0.00699 0.00946 0.02004
NASDAQ c 40 0 0.01102 0.01396 0.01174 0.03556
Nikkei c 40 0 0.00084 0.00056 -0.00115 0.03721
Hang Sen 40 0 0.00374 0.00457 0.00490 0.04534
Australi 40 0 0.00188 0.00110 0.00165 0.02399
Singapor 40 0 0.00001 -0.00038 0.00047 0.02602
French c 40 0 -0.00311 -0.01009 -0.00342 0.03313
British 40 0 0.00444 0.00982 0.00500 0.02401
German c 39 1 0.00090 0.00448 0.00127 0.03074
Variable SE Mean Minimum Maximum Q1 Q3
S&P chan 0.00250 -0.03547 0.03821 0.00057 0.01808
DJIA Cha 0.00317 -0.04365 0.04072 -0.00161 0.02689
NASDAQ c 0.00562 -0.09985 0.09631 -0.01156 0.03351
Nikkei c 0.00588 -0.07062 0.13779 -0.02832 0.02582
Hang Sen 0.00717 -0.11493 0.10357 -0.02217 0.02779
Australi 0.00379 -0.05057 0.05974 -0.01319 0.01707
Singapor 0.00411 -0.07092 0.05243 -0.01440 0.01728
French c 0.00524 -0.07183 0.06953 -0.03173 0.02414
British 0.00380 -0.04767 0.05330 -0.01459 0.02525
German c 0.00492 -0.05802 0.05629 -0.02379 0.02592
From here we can see that the average bi-monthly change for the indexes is quite diverse ranging from -.311% to 1.102% with the S&P returning .795% (as discussed, looking at the means for markets is a tricky notion though) Now let’s look at the markets in comparison with the S&P:
Most of these show at least some relation to the S&P although some are much stronger than others. For example, the DJIA is fairly strongly related while the French CAC-40 is fairly weak.
Of course some of our data is going to have collinearity as is evidenced by the below:
Unfortunately, our different predictors are not distinct.
Before moving on let’s take a quick look at a couple of histograms of our data. I won’t include them all but here’s two:
The data looks relatively normal so we will pass on logging the data or adjusting it any other way.
For completeness, we will also look at some box plots, once again, I will only include a couple for your review:
Interestingly the outlier (period 4) shown on the S&P boxplot does not show up as an outlier for any on any of the other boxplots.
Preliminary Regression
Let’s run a quick regression to see where we are:
The regression equation is
S&P change = 0.00167 + 0.617 DJIA Change + 0.0133 NASDAQ change - 0.0049 Nikkei change + 0.0694 Hang Seng change + 0.0469 Australian change - 0.0378 Singapore change - 0.0661 French change + 0.0525 British change + 0.0116 German change
39 cases used 1 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.001675 0.001166 1.44 0.162
DJIA Cha 0.61745 0.07511 8.22 0.000 2.3
NASDAQ c 0.01329 0.03438 0.39 0.702 1.5
Nikkei c -0.00487 0.03575 -0.14 0.893 1.8
Hang Sen 0.06935 0.03809 1.82 0.079 3.0
Australi 0.04692 0.05699 0.82 0.417 1.9
Singapor -0.03777 0.06240 -0.61 0.550 2.6
French c -0.06607 0.04215 -1.57 0.128 2.0
British 0.05252 0.06993 0.75 0.459 2.9
German c 0.01156 0.05186 0.22 0.825 2.5
S = 0.006198 R-Sq = 88.5% R-Sq(adj) = 84.9%
Analysis of Variance
Source DF SS MS F P
Regression 9 0.00857907 0.00095323 24.82 0.000
Residual Error 29 0.00111387 0.00003841
Total 38 0.00969294
Source DF Seq SS
DJIA Cha 1 0.00819907
NASDAQ c 1 0.00004158
Nikkei c 1 0.00000001
Hang Sen 1 0.00020926
Australi 1 0.00003024
Singapor 1 0.00000188
French c 1 0.00006384
British 1 0.00003128
German c 1 0.00000191
Unusual Observations
Obs DJIA Cha S&P chan Fit StDev Fit Residual St Resid
1 -0.0060 -0.015014 -0.001571 0.002520 -0.013443 -2.37R
18 -0.0165 0.009550 -0.007594 0.002028 0.017144 2.93R
R denotes an observation with a large standardized residual
There are some P values that are fairly high so we will get rid of some variables to check how it effects our calculation.
A Second Try at Regression
Let’s remove the NASDAQ, Nikkei, Singapore and Frankfurt DAX-30 to see the effect:
The regression equation is
S&P change = 0.00168 + 0.640 DJIA Change + 0.0583 Hang Seng change + 0.0408 Australian change - 0.0581 French change + 0.0433 British change
Predictor Coef StDev T P VIF
Constant 0.001679 0.001074 1.56 0.127
DJIA Cha 0.64005 0.06651 9.62 0.000 2.0
Hang Sen 0.05835 0.02459 2.37 0.023 1.4
Australi 0.04081 0.04856 0.84 0.407 1.6
French c -0.05813 0.03642 -1.60 0.120 1.7
British 0.04326 0.05630 0.77 0.448 2.1
S = 0.005836 R-Sq = 88.1% R-Sq(adj) = 86.4%
Analysis of Variance
Source DF SS MS F P
Regression 5 0.0086109 0.0017222 50.56 0.000
Residual Error 34 0.0011581 0.0000341
Total 39 0.0097690
Source DF Seq SS
DJIA Cha 1 0.0082697
Hang Sen 1 0.0002277
Australi 1 0.0000263
French c 1 0.0000671
British 1 0.0000201
Unusual Observations
Obs DJIA Cha S&P chan Fit StDev Fit Residual St Resid
1 -0.0060 -0.015014 -0.001622 0.001603 -0.013392 -2.39R
18 -0.0165 0.009550 -0.007839 0.001823 0.017389 3.14R
R denotes an observation with a large standardized residual
Our R- Squared did not change much at all and our model is much simpler. There still may be room for improving our model though.
A Little Simpler
Now let’s remove the Australia and London FT 100 (probably should have gotten rid of the London earlier but I thought I would try leaving it in):
The regression equation is
S&P change = 0.00162 + 0.680 DJIA Change + 0.0648 Hang Seng change - 0.0432 French change
Predictor Coef StDev T P VIF
Constant 0.001620 0.001041 1.56 0.128
DJIA Cha 0.68012 0.05606 12.13 0.000 1.5
Hang Sen 0.06475 0.02375 2.73 0.010 1.4
French c -0.04319 0.03036 -1.42 0.163 1.2
S = 0.005783 R-Sq = 87.7% R-Sq(adj) = 86.6%
Analysis of Variance
Source DF SS MS F P
Regression 3 0.0085651 0.0028550 85.37 0.000
Residual Error 36 0.0012039 0.0000334
Total 39 0.0097690
Source DF Seq SS
DJIA Cha 1 0.0082697
Hang Sen 1 0.0002277
French c 1 0.0000677
Unusual Observations
Obs DJIA Cha S&P chan Fit StDev Fit Residual St Resid
1 -0.0060 -0.015014 -0.001648 0.001494 -0.013366 -2.39R
9 0.0103 -0.001430 0.010132 0.001583 -0.011562 -2.08R
17 0.0193 0.014588 0.007952 0.003264 0.006637 1.39 X
18 -0.0165 0.009550 -0.007509 0.001780 0.017059 3.10R
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence.
Once again not much change in our R-squared but our regression is much simpler and therefore, much better.
Even Simpler
Now let’s try getting rid of the French also to see how it strengthens our model:
The regression equation is
S&P change = 0.00197 + 0.0617 Hang Seng change + 0.656 DJIA Change
Predictor Coef StDev T P VIF
Constant 0.001973 0.001025 1.93 0.062
Hang Sen 0.06171 0.02397 2.57 0.014 1.3
DJIA Cha 0.65635 0.05425 12.10 0.000 1.3
S = 0.005862 R-Sq = 87.0% R-Sq(adj) = 86.3%
Analysis of Variance
Source DF SS MS F P
Regression 2 0.0084974 0.0042487 123.62 0.000
Residual Error 37 0.0012716 0.0000344
Total 39 0.0097690
Source DF Seq SS
Hang Sen 1 0.0034672
DJIA Cha 1 0.0050302
Unusual Observations
Obs Hang Sen S&P chan Fit StDev Fit Residual St Resid
1 0.019 -0.015014 -0.000790 0.001386 -0.014224 -2.50R
17 -0.115 0.014588 0.007571 0.003298 0.007018 1.45 X
18 0.012 0.009550 -0.008107 0.001753 0.017657 3.16R
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence.
Once again little change in the R-squared and a much stronger regression than we started out with. The VIFs are fairly low so there isn’t too much collinearity. In addition the F statistic for each is high which means low values. Both indexes are therefore significant predictors and the Hang Seng shows a real “Asian effect’. The Hang Seng only shows minimal (0.0617) effect though so let’s see what happens if remove it.
Sweet and Simple
The simple regression is:
The regression equation is
S&P change = 0.00159 + 0.727 DJIA Change
Predictor Coef StDev T P
Constant 0.001588 0.001086 1.46 0.152
DJIA Cha 0.72677 0.05020 14.48 0.000
S = 0.006281 R-Sq = 84.7% R-Sq(adj) = 84.2%
Analysis of Variance
Source DF SS MS F P
Regression 1 0.0082697 0.0082697 209.59 0.000
Residual Error 38 0.0014994 0.0000395
Total 39 0.0097690
Unusual Observations
Obs DJIA Cha S&P chan Fit StDev Fit Residual St Resid
4 -0.0436 -0.035475 -0.030136 0.002812 -0.005339 -0.95 X
18 -0.0165 0.009550 -0.010425 0.001612 0.019975 3.29R
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence.
This is actually quite a good regression. As the R-squared doesn’t go down too much (87% down to 84.7%), we will use this for the moment.
Checking Assumptions
Let’s check our assumptions:
From our data, it does not appear that there is any structure related to the order of the data. In addition, they appear to be displaying homoscedasticity in that the variability is random and not stronger for any specific members of the population.
Unfortunately, however, as this is time data, we must consider autocorrelation. As this has to do with the stock market, the geometric random walk says there should be no autocorrelation. By looking at Residual Versus the Order of the Data graph, this appears to be the case. Unfortunately, any further research is beyond the scope of our class (the semester is a bit too short )
Checking for Potential Outliers
From the above, there is clearly at least one point (#18) that needs to be addressed. This point may be an influential point. I tried to check what was happening at this time to see if anything could explain this unusual activity but, unfortunately, I was unsuccessful in finding anything. Omitting the point does not change things that much though:
The regression equation is
S&P change =0.000751 + 0.761 DJIA Change
Predictor Coef StDev T P
Constant 0.0007507 0.0009559 0.79 0.437
DJIA Cha 0.76129 0.04395 17.32 0.000
S = 0.005383 R-Sq = 89.0% R-Sq(adj) = 88.7%
Analysis of Variance
Source DF SS MS F P
Regression 1 0.0086942 0.0086942 300.01 0.000
Residual Error 37 0.0010723 0.0000290
Total 38 0.0097664
In addition, the leverage value of 2.5 * (p+1)/N is .14 and the value for observation 18 (0.065823) is well below this. In addition the Cook value (0.381355) is well below 1.
DateS&P changeDJIA ChangeNASDAQ changeNikkei changeHang Seng changeAustralian changeSingapore changeFrench changeBritish changeGerman changeHI1COOK1
5/13/94-0.0150141-0.0059782-0.02258540.0276550.0188100.0018876-0.00438550.0097139-0.00195180.00568570.038861 0.080084
5/31/940.02782910.02696680.03845730.0346730.0458510.0057005-0.0019465-0.0718336-0.0476725-0.05801880.046188 0.028391
6/15/940.00900330.0085250-0.00388020.014750-0.042292-0.00355460.0021213-0.03243510.0253493-0.02490950.025003 0.000496
6/30/94-0.0354747-0.0436496-0.0452594-0.030025-0.042747-0.0411203-0.0269406-0.0366893-0.0415654-0.02379140.200393 0.113219
7/15/940.02226120.03554520.01926170.0061140.0409450.0346388-0.01195100.04365220.05330230.03370790.070846 0.027681
7/29/940.00902770.00284780.0079512-0.0154430.0401220.00170070.00368460.05084600.00253680.02532950.027228 0.010516
8/15/940.0064810-0.00111830.01869460.0086530.000350-0.00276500.0432057-0.03279050.0193668-0.00363360.031224 0.013728
8/31/940.03091730.04072290.05521370.0001070.0467270.03225020.00475720.03095740.03468800.03460290.090279 0.000098
9/15/94-0.00143010.01033880.0231214-0.034377-0.006722-0.0335988-0.0137761-0.0443579-0.0426291-0.04467990.025161 0.037212
9/30/94-0.0255260-0.0279953-0.0325473-0.017850-0.034615-0.01077630.0227065-0.0495878-0.0277573-0.04835900.111255 0.081746
10/14/940.01385380.01750630.01256820.0207260.003118-0.01118940.01915010.02861250.02656710.04671550.029893 0.000084
10/31/940.0069282-0.00060100.03573220.0010170.0099800.01934200.0007277-0.0141385-0.0029935-0.01619390.030589 0.013768
11/15/94-0.0154970-0.0209205-0.0021305-0.029912-0.008355-0.0505673-0.01590980.02562850.01226840.01888370.081241 0.004311
11/30/94-0.0243855-0.0227710-0.0178324-0.016299-0.114932-0.0261152-0.04232480.0109540-0.0172227-0.02960560.088473 0.119834
12/15/940.00363680.0070175-0.02746900.002385-0.0244150.0022743-0.0275903-0.0226930-0.03504900.00211400.025192 0.003127
12/30/940.00863090.01831640.02684790.031480-0.0082960.00934040.0272127-0.02586610.03097460.02630340.030841 0.016349
1/13/950.01458840.01930400.0153610-0.019870-0.114601-0.0281801-0.0709202-0.0144433-0.0056108-0.02418610.032110 0.000460
1/31/950.0095500-0.0165282-0.0125463-0.0352460.012453-0.01517110.0012640-0.0302484-0.0186005-0.01671510.065823 0.381355
2/15/950.03001570.03702270.0670071-0.0353260.1035720.00671910.01167830.03559710.02784460.05628640.076042 0.002613
2/28/950.00588190.00624160.0000231-0.0521130.0276910.04183620.0090146-0.0456791-0.0213340-0.01539080.025403 0.000020
3/15/950.00921230.00681120.0255954-0.0226700.004526-0.0081771-0.0235907-0.02152690.0125278-0.04378310.025241 0.002408
3/31/950.01795150.02954660.0080709-0.0316120.0265780.00115530.00799410.06952720.0298326-0.04355420.052615 0.019393
4/13/950.01701580.01214380.02634460.0185160.0086700.0597430-0.01036730.01179360.02259470.03321560.025734 0.014978
4/28/950.01076130.02687390.02316260.022384-0.0347450.0147496-0.00007240.01969270.00246200.01484560.045972 0.068660
5/15/950.02531520.02689020.0357356-0.0117240.099122-0.00068290.05243160.03808780.02909810.03512010.046010 0.011219
5/31/950.01072500.00623550.0036189-0.0706160.023677-0.0135689-0.0006927-0.02188270.00274900.00260210.025405 0.007190
6/15/950.00697410.00697180.0489449-0.036894-0.015454-0.0296388-0.0112791-0.01398910.01536420.01713530.025203 0.000034
6/30/950.01420540.01330660.0508604-0.023532-0.0059880.0285044-0.0281272-0.0322174-0.0165559-0.02071880.026324 0.003056
7/14/950.02779260.03351990.09631430.1377860.0565900.04804160.04544690.04814880.03457430.04845170.064177 0.003157
7/31/950.0038758-0.0000743-0.03555140.009676-0.0281800.0010407-0.0146092-0.01451510.00994400.01548810.029977 0.002215
8/15/95-0.0062093-0.01436350.02666640.046481-0.0552300.0079864-0.04350830.0136299-0.00545720.00399780.059132 0.005909
8/31/950.0059258-0.0065247-0.01246470.0380740.0278290.00000000.0407408-0.03227810.00969690.00480340.039909 0.045236
9/15/950.03821100.04056120.02122160.0353990.0672710.0153305-0.0032191-0.00618570.02495830.03516050.089620 0.069954
9/29/950.0018171-0.0017696-0.0066722-0.045072-0.015418-0.0138523-0.0090715-0.0445521-0.0158223-0.05609380.032073 0.000997
10/13/950.00015400.0009814-0.0257743-0.0017990.024615-0.01718410.00414100.01603710.01704580.00447640.028858 0.001787
10/31/95-0.0051326-0.00798950.0504912-0.012650-0.010258-0.0120534-0.0101736-0.0016621-0.0109025-0.01316440.042904 0.000495
11/15/950.02142730.0351742-0.00885130.001592-0.0358820.0176496-0.01908530.03372090.01198610.00842290.069585 0.033369
11/30/950.01921000.03082420.00040440.0600400.0404990.02298250.0294849-0.02501090.02601220.02591750.056113 0.018232
12/15/950.01812110.0201478-0.09984500.0321230.0046170.02871970.03524250.0169394-0.00592200.01869960.033293 0.001614
12/29/95-0.00066520.00007530.05650770.0269610.014152-0.00801510.02879110.00684150.0128205*0.029810 0.002137
This is clearly a unique observation though so I will look at the regression to see if it changes if I take it out. Of course, I will start at the beginning and then see if I still work down to the simple regression[1]:
The regression equation is
S&P change = 0.00088 + 0.652 DJIA Change + 0.0169 NASDAQ change + 0.0027 Nikkei change + 0.0557 Hang Seng change + 0.0453 Australian change - 0.0358 Singapore change - 0.0605 French change + 0.0590 British change + 0.0081 German change
From here we will decide to take out the Dow, Hang Seng, French and British:
The regression equation is
S&P change =0.000737 + 0.702 DJIA Change + 0.0483 Hang Seng change - 0.0549 French change + 0.0543 British change
Getting just down to the DJIA gives us:
The regression equation is
S&P change =0.000751 + 0.761 DJIA Change
Predictor Coef StDev T P
Constant 0.0007507 0.0009559 0.79 0.437
DJIA Cha 0.76129 0.04395 17.32 0.000
S = 0.005383 R-Sq = 89.0% R-Sq(adj) = 88.7%
Analysis of Variance
Source DF SS MS F P
Regression 1 0.0086942 0.0086942 300.01 0.000
Residual Error 37 0.0010723 0.0000290
Total 38 0.0097664
Not too much of a change but quite a bit if you consider that we only removed one observation. I will not consider it enough to effect my final conclusion though and will use the previous simple regression.
Conclusion
Our calculation is:
yi = o + 1x + iS&P change = 0.00159 + 0.727 DJIA Change + error
It is interesting to note that on the news when they report the DJIA one would be tempted to think that it is irrelevant as it only includes 30 stocks. As we can see from this calculation though, they clearly also generally reflect the movement of a larger number of stocks.
In addition, we may want to use the calculation that includes the Heng Seng. It offers significant predictive power although it makes our calculation slightly less simple. In that case our solution would be:
yi = o + 1x +2x + iS&P change = 0.00197 + 0.0617 Hang Seng change + 0.656 DJIA Change+ error
Although we may want to keep the other variables in to show that we accounted for them, the above is the simplest and most efficient regression.
One Step Further
From the above, we were not able to generate any earth moving conclusions. I will, therefore, remove the other US stock indexes (DJIA and NASDAQ) and just check to see how the S&P relates to the foreign markets.
We will start with a quick regression:
S&P change = 0.00564 + 0.0306 Nikkei change + 0.188 Hang Seng change + 0.225 Australian change - 0.179 Singapore change - 0.0576 French change + 0.260 British change - 0.0329 German change
39 cases used 1 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.005639 0.001911 2.95 0.006
Nikkei c 0.03061 0.06292 0.49 0.630 1.7
Hang Sen 0.18785 0.05982 3.14 0.004 2.3
Australi 0.22479 0.09480 2.37 0.024 1.6
Singapor -0.1793 0.1024 -1.75 0.090 2.2
French c -0.05762 0.07524 -0.77 0.450 2.0
British 0.2598 0.1150 2.26 0.031 2.4
German c -0.03294 0.09235 -0.36 0.724 2.5
S = 0.01113 R-Sq = 60.4% R-Sq(adj) = 51.5%
Analysis of Variance
Source DF SS MS F P
Regression 7 0.0058556 0.0008365 6.76 0.000
Residual Error 31 0.0038373 0.0001238
Total 38 0.0096929
Source DF Seq SS
Nikkei c 1 0.0008523
Hang Sen 1 0.0030776
Australi 1 0.0009944
Singapor 1 0.0002322
French c 1 0.0000375
British 1 0.0006460
German c 1 0.0000158
Unusual Observations
Obs Nikkei c S&P chan Fit StDev Fit Residual St Resid
1 0.028 -0.01501 0.00998 0.00322 -0.02499 -2.35R
2 0.035 0.02783 0.01061 0.00750 0.01722 2.10R
17 -0.020 0.01459 -0.00994 0.00592 0.02453 2.60R
R denotes an observation with a large standardized residual
We can see immediately that our R-squared is much lower now. We will now try to simplify by taking out the Nikkei, French and German indexes:
The regression equation is
S&P change = 0.00601 + 0.171 Hang Seng change + 0.245 Australian change - 0.153 Singapore change + 0.189 British change
Predictor Coef StDev T P VIF
Constant 0.006009 0.001729 3.48 0.001
Hang Sen 0.17067 0.05251 3.25 0.003 1.9
Australi 0.24530 0.08026 3.06 0.004 1.3
Singapor -0.15250 0.08784 -1.74 0.091 1.8
British 0.18931 0.08151 2.32 0.026 1.3
S = 0.01069 R-Sq = 59.0% R-Sq(adj) = 54.4%
Analysis of Variance
Source DF SS MS F P
Regression 4 0.0057673 0.0014418 12.61 0.000
Residual Error 35 0.0040017 0.0001143
Total 39 0.0097690
Source DF Seq SS
Hang Sen 1 0.0034672
Australi 1 0.0014703
Singapor 1 0.0002130
British 1 0.0006167
Unusual Observations
Obs Hang Sen S&P chan Fit StDev Fit Residual St Resid
1 0.019 -0.01501 0.00998 0.00208 -0.02500 -2.38R
2 0.046 0.02783 0.00651 0.00542 0.02132 2.31R
4 -0.043 -0.03547 -0.01513 0.00423 -0.02034 -2.07R
17 -0.115 0.01459 -0.01071 0.00560 0.02530 2.78R
R denotes an observation with a large standardized residual
This seems quite strong. Playing further with the data doesn’t come up with a better regression though. The VIFs are quite low also.
A look at the conclusions gives us the shows us some outliers again and brings the same conclusions regarding the assumptions.
DateHI1COOK1
5/13/940.0377040.044501
5/31/940.2567580.369692
6/15/940.1169380.011457
6/30/940.1563960.159059
7/15/940.2164200.076599
7/29/940.0530880.001796
8/15/940.1609800.006442
8/31/940.1014120.002223
9/15/940.1640100.025742
9/30/940.1860610.099990
10/14/940.0873700.011592
10/31/940.0481230.002187
11/15/940.1963260.082083
11/30/940.2023450.031855
12/15/940.1153320.003461
12/30/940.1220440.000000
1/13/950.2743850.583391
1/31/950.0694700.010994
2/15/950.2095610.000832
2/28/950.1834700.045314
3/15/950.0785190.000380
3/31/950.0606820.000873
4/13/950.2155670.073966
4/28/950.0787090.007038
5/15/950.1905590.012941
5/31/950.0540690.001205
6/15/950.0977880.008194
6/30/950.1395080.000378
7/14/950.1643340.000217
7/31/950.0481360.000265
8/15/950.1210660.029385
8/31/950.1036860.000047
9/15/950.1304970.041644
9/29/950.0470330.001078
10/13/950.0716040.010343
10/31/950.0398030.002648
11/15/950.0861140.026116
11/30/950.0696490.000007
12/15/950.1663960.048627
12/29/950.0780880.004279
In addition, the leverage value of 2.5 * (p+1)/N is .25 and the value for observations 1 & 4 are below this while 2 & 17 are above it. The Cook value for each unusual observation are well below 1. Therefore, these observations are leverage values but not influentials.
I will remove all four observations and see what it does to our regression calculation:
The regression equation is
S&P change = 0.00613 + 0.186 Hang Seng change + 0.201 Australian change - 0.0993 Singapore change + 0.170 British change + 0.0098 Nikkei change - 0.0147 French change + 0.0251 German change
35 cases used 1 cases contain missing values
Predictor Coef StDev T P VIF
Constant 0.006130 0.001492 4.11 0.000
Hang Sen 0.18586 0.05164 3.60 0.001 2.6
Australi 0.20141 0.07117 2.83 0.009 1.5
Singapor -0.09932 0.08042 -1.23 0.228 2.0
British 0.16988 0.08960 1.90 0.069 2.2
Nikkei c 0.00975 0.05255 0.19 0.854 2.2
French c -0.01472 0.05474 -0.27 0.790 1.7
German c 0.02515 0.07060 0.36 0.724 2.4
S = 0.007968 R-Sq = 74.7% R-Sq(adj) = 68.1%
Once again I will remove the Nikkei, French and German
The regression equation is
S&P change = 0.00590 + 0.187 Hang Seng change + 0.219 Australian change - 0.101 Singapore change + 0.178 British change
Predictor Coef StDev T P VIF
Constant 0.005904 0.001341 4.40 0.000
Hang Sen 0.18654 0.04053 4.60 0.000 1.8
Australi 0.21919 0.05836 3.76 0.001 1.2
Singapor -0.10097 0.06511 -1.55 0.131 1.5
British 0.17783 0.06842 2.60 0.014 1.4
S = 0.007580 R-Sq = 74.1% R-Sq(adj) = 70.7%
Analysis of Variance
Source DF SS MS F P
Regression 4 0.0050913 0.0012728 22.15 0.000
Residual Error 31 0.0017813 0.0000575
Total 35 0.0068727
Source DF Seq SS
Hang Sen 1 0.0035651
Australi 1 0.0010547
Singapor 1 0.0000833
British 1 0.0003882
Taking out the Singaporean index helps more this time.
The regression equation is
S&P change = 0.00592 + 0.157 Hang Seng change + 0.220 Australian change
+ 0.163 British change
Predictor Coef StDev T P VIF
Constant 0.005915 0.001370 4.32 0.000
Hang Sen 0.15740 0.03669 4.29 0.000 1.4
Australi 0.22004 0.05963 3.69 0.001 1.2
British 0.16321 0.06924 2.36 0.025 1.4
S = 0.007745 R-Sq = 72.1% R-Sq(adj) = 69.5%
Analysis of Variance
Source DF SS MS F P
Regression 3 0.0049531 0.0016510 27.52 0.000
Residual Error 32 0.0019195 0.0000600
Total 35 0.0068727
Source DF Seq SS
Hang Sen 1 0.0035651
Australi 1 0.0010547
British 1 0.0003334
Our calculation is:
yi = o + 1x + iS&P change = 0.00592 + 0.157 Hang Seng change + 0.220 Australian change + 0.163 British change + error
The result of our research is that the S&P is definitely strongly related to the DJIA. We also know with almost complete certainty that the S&P is related to the Hang Seng, Australian Index and British London FT100 and moves in relation to them. This is reflected by the fact that the F statistic is 27.52! and therefore the P value is extremely low. Of course, we should also note that the predictive power of the regression with the foreign markets (about 70%) is not as strong as the one with all the markets (r-squared about 85%).
[1] I of course did all the interim steps on Minitab and check all the variables. I have omitted them here for the sake of (relative) brevity.