Stock Market Data
by Shirshendu Mukherji
5/22/2020
We first load and look at the data.
## V1 V2 V3 V4 V5
## 1 0.0130338 -0.0078431 -0.0031889 -0.0447693 0.0052151
## 2 0.0084862 0.0166886 -0.0062100 0.0119560 0.0134890
## 3 -0.0179153 -0.0086393 0.0100360 0.0000000 -0.0061428
## 4 0.0215589 -0.0034858 0.0174353 -0.0285917 -0.0069534
## 5 0.0108225 0.0037167 -0.0101345 0.0291900 0.0409751
## 6 0.0101713 -0.0121978 -0.0083768 0.0137083 0.00298950. Contents
This report contains four parts:
1. PC method - We perform factor analysis manually using covariance and correlation matrix and compute loadings and the psi matrix and interpret them.
2. Iterative PC method - We perform factor analysis using “fa” function available in “psych” package which allows the Iterative PC method. Here also we calculate loadings and compute communality and uniqueness with and without varimax rotation. We interpret the results.
3. MLE method - Here we use base R function “factanal” which uses MLE method. We do all the above.
4. Factor scores - We calculate factor scores and check the assumptions.
We begin factor analysis now.
1. PC method
First we perform Factor Analysis using covariance matrix.
Calculating the covariance matrix and its EV-EV pairs.
## V1 V2 V3 V4 V5
## V1 4.332695e-04 0.0002756679 1.590265e-04 6.411929e-05 8.896616e-05
## V2 2.756679e-04 0.0004387172 1.799737e-04 1.814512e-04 1.232623e-04
## V3 1.590265e-04 0.0001799737 2.239722e-04 7.341348e-05 6.054612e-05
## V4 6.411929e-05 0.0001814512 7.341348e-05 7.224964e-04 5.082772e-04
## V5 8.896616e-05 0.0001232623 6.054612e-05 5.082772e-04 7.656742e-04## eigen() decomposition
## $values
## [1] 0.0013676780 0.0007011596 0.0002538024 0.0001426026 0.0001188868
##
## $vectors
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2228228 0.6252260 0.32611218 0.6627590 0.11765952
## [2,] 0.3072900 0.5703900 -0.24959014 -0.4140935 -0.58860803
## [3,] 0.1548103 0.3445049 -0.03763929 -0.4970499 0.78030428
## [4,] 0.6389680 -0.2479475 -0.64249741 0.3088689 0.14845546
## [5,] 0.6509044 -0.3218478 0.64586064 -0.2163758 -0.09371777To check number of factors required we plot the eigenvalues and check screeplot.
It looks like 2 factors is good. We check percentages of variance explained by the five principal components in the vector shown below.
## [1] 52.92607 80.05936 89.88095 95.39935 100.00000So 2 principal components explain 80 percent of the variation.
Next we estimate the loading matrix using the two first two principal components. That is the matrix of eigenvectors scaled by square roots of eigenvalues. This is the 5x2 L matrix.
## [,1] [,2]
## [1,] 0.008240463 0.016555621
## [2,] 0.011364238 0.015103596
## [3,] 0.005725215 0.009122290
## [4,] 0.023630398 -0.006565506
## [5,] 0.024071832 -0.008522342Intepretation
We see the first factor loads on all companies in positive amounts whereas the second factor loads on the first three companies in positive amounts but in negative amounts on the last two companies.
This implies that the first factor could be a general health of economy factor that effects all companies in positive way and the second factor may be an industry specific factor that effects the first three companies negatively and the last two companies positively.
Magnitudes are slightly difficult to interpret because they do not represent correlations as we have used the covariance matrix for factor analysis. But relative magnitudes and signs give some idea
Next we calculate communalities as shown in the vector below.
## [1] 0.0003419938 0.0003572645 0.0001159943 0.0006015016 0.0006520834and the uniquenesses (the psi matrix) shown below.
## [,1] [,2] [,3] [,4] [,5]
## [1,] 9.127563e-05 0.000000e+00 0.0000000000 0.0000000000 0.0000000000
## [2,] 0.000000e+00 8.145269e-05 0.0000000000 0.0000000000 0.0000000000
## [3,] 0.000000e+00 0.000000e+00 0.0001079779 0.0000000000 0.0000000000
## [4,] 0.000000e+00 0.000000e+00 0.0000000000 0.0001209948 0.0000000000
## [5,] 0.000000e+00 0.000000e+00 0.0000000000 0.0000000000 0.0001135907Again magnitudes are slightly difficult to interpret. It will be better to use the correlation matrix for factor analysis.
Now we repeat the process using Correlation matrix
Calculating the correlation matrix and its EV-EV pairs.
## V1 V2 V3 V4 V5
## V1 1.0000000 0.6322878 0.5104973 0.1146019 0.1544628
## V2 0.6322878 1.0000000 0.5741424 0.3222921 0.2126747
## V3 0.5104973 0.5741424 1.0000000 0.1824992 0.1462067
## V4 0.1146019 0.3222921 0.1824992 1.0000000 0.6833777
## V5 0.1544628 0.2126747 0.1462067 0.6833777 1.0000000## eigen() decomposition
## $values
## [1] 2.4372731 1.4070127 0.5005127 0.4000316 0.2551699
##
## $vectors
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4690832 0.3680070 0.60431522 0.3630228 0.38412160
## [2,] 0.5324055 0.2364624 0.13610618 -0.6292079 -0.49618794
## [3,] 0.4651633 0.3151795 -0.77182810 0.2889658 0.07116948
## [4,] 0.3873459 -0.5850373 -0.09336192 -0.3812515 0.59466408
## [5,] 0.3606821 -0.6058463 0.10882629 0.4934145 -0.49755167To check number of factors required we plot the eigenvalues and check screeplot.
It looks like 2 factors is good as only two eigenvalues larger than 1. We check percentages of variance explained by all five principal components in the vector shown below.
## [1] 48.74546 76.88572 86.89597 94.89660 100.00000So 2 principal components explain 76% of the variance.
Next we estimate the loading matrix using the two first two principal components. That is the matrix of eigenvectors scaled by square roots of eigenvalues. This is the 5x2 L matrix.
## [,1] [,2]
## [1,] 0.7323218 0.4365209
## [2,] 0.8311791 0.2804859
## [3,] 0.7262022 0.3738582
## [4,] 0.6047155 -0.6939569
## [5,] 0.5630885 -0.7186401Intepretation
Here the interpretation remains the same as discussed above but now we can make sense of the magnitudes of loadings because they now represent correlations between the factor and the variables. This is due to the fact that we are now using the correlation matrix for factor analysis.
Next, calculating communalities
## [1] 0.7268458 0.7695311 0.6671396 0.8472571 0.8335122and uniqenesses (the psi matrix)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2731542 0.0000000 0.0000000 0.0000000 0.0000000
## [2,] 0.0000000 0.2304689 0.0000000 0.0000000 0.0000000
## [3,] 0.0000000 0.0000000 0.3328604 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000 0.1527429 0.0000000
## [5,] 0.0000000 0.0000000 0.0000000 0.0000000 0.1664878We obtain reasonably high communalities and low uniquenesses suggesting factor model is good.
2. The Iterative PC method
We use the fa function and perform factor analysis using the correlation matrix without rotation first.
## Warning: package 'psych' was built under R version 3.6.3## maximum iteration exceededWe look at loadings first.
##
## Loadings:
## PA1 PA2
## V1 0.625 -0.429
## V2 0.777 -0.342
## V3 0.591 -0.332
## V4 0.704 0.709
## V5 0.509 0.455
##
## PA1 PA2
## SS loadings 2.097 1.121
## Proportion Var 0.419 0.224
## Cumulative Var 0.419 0.644Intepretation
We have similar interpretation as in the basic PC method. The first factor is the general health of the economy factor as it loads moderately posetively on all the companies
The second factor now loads negatively in moderate amounts on the first three companies and positively in moderate amounts on the last two. So the signs have reversed
But separation of company types remains the same.
We also note that two factors explain 64.4% of the variation.
Next we check the uniqueness and communality for each variable.
## V1 V2 V3 V4 V5
## 0.424490154 0.280101509 0.540531664 0.002670401 0.534195813## V1 V2 V3 V4 V5
## 0.5755098 0.7198985 0.4594683 0.9973296 0.4658042Company 4 is well explained by this factor analysis followed by company 2. The others are only moderately well explained by the factor model.
Next we use varimax and check what changes occur.
## maximum iteration exceededWe look at loadings first.
##
## Loadings:
## PA1 PA2
## V1 0.757
## V2 0.821 0.216
## V3 0.669 0.108
## V4 0.110 0.993
## V5 0.115 0.673
##
## PA1 PA2
## SS loadings 1.719 1.499
## Proportion Var 0.344 0.300
## Cumulative Var 0.344 0.644Interpretation
Now there are some changes. We see that all loadings are positive now.
Conclusion here is that again the first factor has positive moderate loading on the first three companies and the second factor has moderate positive loading on the last two companies especialy the fourth company.
So the rotation has removed the overall economy factor and alloted one factor for each company type
The separation of company types is same as before rotation ie the first three are of one type and the last two are of another type.
Next we check uniqueness and communality.
## V1 V2 V3 V4 V5
## 0.424490154 0.280101509 0.540531664 0.002670401 0.534195813## V1 V2 V3 V4 V5
## 0.5755098 0.7198985 0.4594683 0.9973296 0.4658042Similar observation as before rotation. Company 4 is well explained by this factor analysis followed by company 2. The others are only moderately explained by the factor model.
3. The MLE method
We know the number of factors should be 2. We use base R function factanal which uses MLE method. We do factor analysis without rotation and check the loadings.
##
## Loadings:
## Factor1 Factor2
## V1 0.121 0.754
## V2 0.328 0.786
## V3 0.188 0.650
## V4 0.997
## V5 0.685
##
## Factor1 Factor2
## SS loadings 1.622 1.610
## Proportion Var 0.324 0.322
## Cumulative Var 0.324 0.646Interpretation
We get all positive loadings which is different from the PC and iterative PC method without rotation. But similar to iterative PC method after varimax
However separation of company type is same. First three companies are of one type and the last two companies are of another type.
One thing two note is that the factors have reversed roles. Factor 2 now affects the first three companies and Factor 1 on the last two companies.
64 percent of variation explained by 2 factors.
We check uniqueness (1 - communality)
## V1 V2 V3 V4 V5
## 0.4165374 0.2746902 0.5420233 0.0050000 0.5298429We see that they indicate the same thing as in the case of iterative PC.
Next we perform FA with rotation varimax and check the loadings.
##
## Loadings:
## Factor1 Factor2
## V1 0.763
## V2 0.819 0.232
## V3 0.668 0.108
## V4 0.113 0.991
## V5 0.108 0.677
##
## Factor1 Factor2
## SS loadings 1.725 1.507
## Proportion Var 0.345 0.301
## Cumulative Var 0.345 0.646Interpretation
Same result (separation of company types) as before varimax. But factors have switched roles again.
64 percent is explained by the two factors as before varimax
Uniqueness (and communality) also remains identical as before varimax as shown below.
## V1 V2 V3 V4 V5
## 0.4165374 0.2746902 0.5420233 0.0050000 0.52984294. Factor scores
In this section we estimate factor scores by two methods - regression and WLS - after performing factor analyis by MLE method using factanal function.
First we do it by regression method and display some of the scores.
## Factor1 Factor2
## [1,] 0.16535864 -1.8342740
## [2,] 0.36753184 0.2555092
## [3,] -0.39519052 -0.1079285
## [4,] 0.62520403 -1.2878906
## [5,] -0.06003873 0.9494523
## [6,] -0.37607023 0.3996291To check assumptions first we check bivariate normality of the factor scores. It passes the three tests - mardia, hz and royston - available in the MVN package via MVN function.
## Warning: package 'MVN' was built under R version 3.6.3## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2## sROC 0.1-2 loaded## Test Statistic p value Result
## 1 Mardia Skewness 0.745285405187422 0.945629116871027 YES
## 2 Mardia Kurtosis -0.0196531043319442 0.98432010086735 YES
## 3 MVN <NA> <NA> YES## Test HZ p value MVN
## 1 Henze-Zirkler 0.4749564 0.6592693 YES## Test H p value MVN
## 1 Royston 0.8014243 0.6698429 YESNext we calculate the sample mean of scores of both factors and check the mean vector. Both are very nearly zero as seen below.
## [1] 2.967346e-18 2.554559e-17Next we calculate the sample covariance matrix which is reasonably close to the 2x2 Identity matrix. So they are uncorrelated.
## Factor1 Factor2
## Factor1 0.81757553 0.02145502
## Factor2 0.02145502 0.99244004Below we present normal qqplots of both factors individually and they are close to standard normal. They also pass the Shapiro Wilk test individually. Therefore the two factors can be said to be indeed independent as they are uncorrelated and bivariate normal.
##
## Shapiro-Wilk normality test
##
## data: fa.ml$scores[, 1]
## W = 0.99118, p-value = 0.7423
##
## Shapiro-Wilk normality test
##
## data: fa.ml$scores[, 2]
## W = 0.98649, p-value = 0.3831Next we repeat the above on factor scores by WLS method and see some of the scores.
## Factor1 Factor2
## [1,] 0.25089936 -1.8536707
## [2,] 0.44303382 0.2478778
## [3,] -0.48078772 -0.0983568
## [4,] 0.79921292 -1.3149790
## [5,] -0.09859658 0.9588163
## [6,] -0.47081640 0.4128516To check assumptions first we check bivariate normality of the factor scores. It passes the three tests - mardia, hz and royston - available in the MVN package via MVN function.
## Test Statistic p value Result
## 1 Mardia Skewness 0.745285405187427 0.945629116871026 YES
## 2 Mardia Kurtosis -0.0196531043319431 0.984320100867351 YES
## 3 MVN <NA> <NA> YES## Test HZ p value MVN
## 1 Henze-Zirkler 0.4749564 0.6592693 YES## Test H p value MVN
## 1 Royston 0.9058111 0.6357782 YESNext we calculate the sample mean of scores of both factors. Both are very nearly zero as seen below.
## [1] 3.938180e-17 2.824978e-19Next we calculate the sample covariance matrix which is reasonably close to the 2x2 Identity matrix. So they are uncorrelated.
## Factor1 Factor2
## Factor1 1.22382288 -0.02645716
## Factor2 -0.02645716 1.00818951Below we present normal qqplots of both factors individually and they are close to standard normal. They also pass the Shapiro Wilk test individually. Therefore the two factors can be said to be indeed independent as they are uncorrelated and bivariate normal.
##
## Shapiro-Wilk normality test
##
## data: fa.ml$scores[, 1]
## W = 0.99163, p-value = 0.7779
##
## Shapiro-Wilk normality test
##
## data: fa.ml$scores[, 2]
## W = 0.98594, p-value = 0.3497