Dow Jones Stock Market Index (2/4): Trade Volume Exploratory Analysis

January 6, 2019
By

(This article was first published on R Programming – DataScience+, and kindly contributed to R-bloggers)

    Categories

    1. Basic Statistics

    Tags

    1. Data Visualisation
    2. Import Data
    3. R Programming

    This is the second part of the 4-series articles about Dow Jones Stock Market. To read the first part go to this link. In this part, I am going to analyze the Dow Jones Industrial Average (DJIA) trade volume.

    Packages

    The packages being used in this post series are herein listed.

    suppressPackageStartupMessages(library(lubridate)) 
    suppressPackageStartupMessages(library(fBasics)) 
    suppressPackageStartupMessages(library(lmtest)) 
    suppressPackageStartupMessages(library(urca)) 
    suppressPackageStartupMessages(library(ggplot2)) 
    suppressPackageStartupMessages(library(quantmod)) 
    suppressPackageStartupMessages(library(PerformanceAnalytics)) 
    suppressPackageStartupMessages(library(rugarch))
    suppressPackageStartupMessages(library(FinTS))
    suppressPackageStartupMessages(library(forecast))
    suppressPackageStartupMessages(library(strucchange))
    suppressPackageStartupMessages(library(TSA))
    

    Getting Data

    We upload the environment status as saved at the end of part 1.

    load(file='DowEnvironment.RData')
    

    Daily Volume Exploratory Analysis

    From the saved environment, we can find back our DJI object. We plot the daily volume.

    dj_vol <- DJI[,"DJI.Volume"]
    plot(dj_vol)
    

    It is remarkable the level jump at the beginning of 2017, something that we will investigate in part 4.

    We transform the volume time series data and timeline index into a dataframe.

    dj_vol_df <- xts_to_dataframe(dj_vol)
    head(dj_vol_df)
    ##   year     value
    ## 1 2007 327200000
    ## 2 2007 259060000
    ## 3 2007 235220000
    ## 4 2007 223500000
    ## 5 2007 225190000
    ## 6 2007 226570000
    
    tail(dj_vol_df)
    ##      year     value
    ## 3015 2018 900510000
    ## 3016 2018 308420000
    ## 3017 2018 433080000
    ## 3018 2018 407940000
    ## 3019 2018 336510000
    ## 3020 2018 288830000
    

    Basic statistics summary

    (dj_stats <- dataframe_basicstats(dj_vol_df))
    ##                     2007         2008         2009         2010
    ## nobs        2.510000e+02 2.530000e+02 2.520000e+02 2.520000e+02
    ## NAs         0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
    ## Minimum     8.640000e+07 6.693000e+07 5.267000e+07 6.840000e+07
    ## Maximum     4.571500e+08 6.749200e+08 6.729500e+08 4.598900e+08
    ## 1. Quartile 2.063000e+08 2.132100e+08 1.961850e+08 1.633400e+08
    ## 3. Quartile 2.727400e+08 3.210100e+08 3.353625e+08 2.219025e+08
    ## Mean        2.449575e+08 2.767164e+08 2.800537e+08 2.017934e+08
    ## Median      2.350900e+08 2.569700e+08 2.443200e+08 1.905050e+08
    ## Sum         6.148432e+10 7.000924e+10 7.057354e+10 5.085193e+10
    ## SE Mean     3.842261e+06 5.965786e+06 7.289666e+06 3.950031e+06
    ## LCL Mean    2.373901e+08 2.649672e+08 2.656970e+08 1.940139e+08
    ## UCL Mean    2.525248e+08 2.884655e+08 2.944104e+08 2.095728e+08
    ## Variance    3.705505e+15 9.004422e+15 1.339109e+16 3.931891e+15
    ## Stdev       6.087286e+07 9.489163e+07 1.157199e+08 6.270480e+07
    ## Skewness    9.422400e-01 1.203283e+00 1.037015e+00 1.452082e+00
    ## Kurtosis    1.482540e+00 2.064821e+00 6.584810e-01 3.214065e+00
    ##                     2011         2012         2013         2014
    ## nobs        2.520000e+02 2.500000e+02 2.520000e+02 2.520000e+02
    ## NAs         0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
    ## Minimum     8.410000e+06 4.771000e+07 3.364000e+07 4.287000e+07
    ## Maximum     4.799800e+08 4.296100e+08 4.200800e+08 6.554500e+08
    ## 1. Quartile 1.458775e+08 1.107150e+08 9.488000e+07 7.283000e+07
    ## 3. Quartile 1.932400e+08 1.421775e+08 1.297575e+08 9.928000e+07
    ## Mean        1.804133e+08 1.312606e+08 1.184434e+08 9.288516e+07
    ## Median      1.671250e+08 1.251950e+08 1.109250e+08 8.144500e+07
    ## Sum         4.546415e+10 3.281515e+10 2.984773e+10 2.340706e+10
    ## SE Mean     3.897738e+06 2.796503e+06 2.809128e+06 3.282643e+06
    ## LCL Mean    1.727369e+08 1.257528e+08 1.129109e+08 8.642012e+07
    ## UCL Mean    1.880897e+08 1.367684e+08 1.239758e+08 9.935019e+07
    ## Variance    3.828475e+15 1.955108e+15 1.988583e+15 2.715488e+15
    ## Stdev       6.187468e+07 4.421660e+07 4.459353e+07 5.211034e+07
    ## Skewness    1.878239e+00 3.454971e+00 3.551752e+00 6.619268e+00
    ## Kurtosis    5.631080e+00 1.852581e+01 1.900989e+01 5.856136e+01
    ##                     2015         2016         2017         2018
    ## nobs        2.520000e+02 2.520000e+02 2.510000e+02 2.510000e+02
    ## NAs         0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
    ## Minimum     4.035000e+07 4.589000e+07 1.186100e+08 1.559400e+08
    ## Maximum     3.445600e+08 5.734700e+08 6.357400e+08 9.005100e+08
    ## 1. Quartile 8.775250e+07 8.224250e+07 2.695850e+08 2.819550e+08
    ## 3. Quartile 1.192150e+08 1.203550e+08 3.389950e+08 4.179200e+08
    ## Mean        1.093957e+08 1.172089e+08 3.112396e+08 3.593710e+08
    ## Median      1.021000e+08 9.410500e+07 2.996700e+08 3.414700e+08
    ## Sum         2.756772e+10 2.953664e+10 7.812114e+10 9.020213e+10
    ## SE Mean     2.433611e+06 4.331290e+06 4.376432e+06 6.984484e+06
    ## LCL Mean    1.046028e+08 1.086786e+08 3.026202e+08 3.456151e+08
    ## UCL Mean    1.141886e+08 1.257392e+08 3.198590e+08 3.731270e+08
    ## Variance    1.492461e+15 4.727538e+15 4.807442e+15 1.224454e+16
    ## Stdev       3.863238e+07 6.875709e+07 6.933572e+07 1.106550e+08
    ## Skewness    3.420032e+00 3.046742e+00 1.478708e+00 1.363823e+00
    ## Kurtosis    1.612326e+01 1.122161e+01 3.848619e+00 3.277164e+00
    

    In the following, we make specific comments to some relevant above shown metrics.

    Mean

    Years when Dow Jones daily volume has positive mean are:

    filter_dj_stats(dj_stats, "Mean", 0)
    ##  [1] "2007" "2008" "2009" "2010" "2011" "2012" "2013" "2014" "2015" "2016"
    ## [11] "2017" "2018"
    

    All Dow Jones daily volume mean values in ascending order.

    dj_stats["Mean",order(dj_stats["Mean",,])]
    ##          2014      2015      2016      2013      2012      2011      2010
    ## Mean 92885159 109395714 117208889 118443373 131260600 180413294 201793373
    ##           2007      2008      2009      2017      2018
    ## Mean 244957450 276716364 280053730 311239602 359371036
    

    Median

    Years when Dow Jones daily volume has positive median are:

    filter_dj_stats(dj_stats, "Median", 0)
    ##  [1] "2007" "2008" "2009" "2010" "2011" "2012" "2013" "2014" "2015" "2016"
    ## [11] "2017" "2018"
    

    All Dow Jones daily volume median values in ascending order.

    dj_stats["Median",order(dj_stats["Median",,])]
    ##            2014     2016      2015      2013      2012      2011      2010
    ## Median 81445000 94105000 102100000 110925000 125195000 167125000 190505000
    ##             2007      2009      2008      2017      2018
    ## Median 235090000 244320000 256970000 299670000 341470000
    

    Skewness

    Years when Dow Jones daily volume has positive skewness are:

    filter_dj_stats(dj_stats, "Skewness", 0)
    ##  [1] "2007" "2008" "2009" "2010" "2011" "2012" "2013" "2014" "2015" "2016"
    ## [11] "2017" "2018"
    

    All Dow Jones daily volume skewness values in ascending order.

    dj_stats["Skewness",order(dj_stats["Skewness",,])]
    ##             2007     2009     2008     2018     2010     2017     2011
    ## Skewness 0.94224 1.037015 1.203283 1.363823 1.452082 1.478708 1.878239
    ##              2016     2015     2012     2013     2014
    ## Skewness 3.046742 3.420032 3.454971 3.551752 6.619268
    

    Excess Kurtosis

    Years when Dow Jones daily volume has positive excess kurtosis are:

    filter_dj_stats(dj_stats, "Kurtosis", 0)
    ##  [1] "2007" "2008" "2009" "2010" "2011" "2012" "2013" "2014" "2015" "2016"
    ## [11] "2017" "2018"
    

    All Dow Jones daily volume excess kurtosis values in ascending order.

    dj_stats["Kurtosis",order(dj_stats["Kurtosis",,])]
    ##              2009    2007     2008     2010     2018     2017    2011
    ## Kurtosis 0.658481 1.48254 2.064821 3.214065 3.277164 3.848619 5.63108
    ##              2016     2015     2012     2013     2014
    ## Kurtosis 11.22161 16.12326 18.52581 19.00989 58.56136
    

    Box-plots

    dataframe_boxplot(dj_vol_df, "DJIA daily volume box plots 2007-2018")
    

    The trade volume starts to decrease from 2010 and on 2017 a remarkable increase occurred. Year 2018 volume has been even larger than 2017 and other years as well.

    Density plots

    dataframe_densityplot(dj_vol_df, "DJIA daily volume density plots 2007-2018")
    

    Shapiro Tests

    dataframe_shapirotest(dj_vol_df)
    ##            result
    ## 2007 6.608332e-09
    ## 2008 3.555102e-10
    ## 2009 1.023147e-10
    ## 2010 9.890576e-13
    ## 2011 2.681476e-16
    ## 2012 1.866544e-20
    ## 2013 6.906596e-21
    ## 2014 5.304227e-27
    ## 2015 2.739912e-21
    ## 2016 6.640215e-23
    ## 2017 4.543843e-12
    ## 2018 9.288371e-11
    

    The null hypothesis of normality is rejected for all years.

    QQ plots

    dataframe_qqplot(dj_vol_df, "DJIA daily volume QQ plots 2007-2018")
    

    QQplots visually confirm the non-normality of daily trade volume distribution.

    Daily volume log-ratio Exploratory Analysis

    Similarly to log-returns, we can define the trade volume log ratio as.

    \[
    v_{t}\ := ln \frac{V_{t}}{V_{t-1}}
    \]
    We can compute it by CalculateReturns within the PerformanceAnalytics package and plot it.

    dj_vol_log_ratio <- CalculateReturns(dj_vol, "log")
    dj_vol_log_ratio <- na.omit(dj_vol_log_ratio)
    plot(dj_vol_log_ratio)
    

    Mapping the trade volume log-ratio time series data and timeline index into a dataframe.

    dj_vol_df <- xts_to_dataframe(dj_vol_log_ratio)
    head(dj_vol_df)
    ##   year        value
    ## 1 2007 -0.233511910
    ## 2 2007 -0.096538449
    ## 3 2007 -0.051109832
    ## 4 2007  0.007533076
    ## 5 2007  0.006109458
    ## 6 2007  0.144221282
    
    tail(dj_vol_df)
    ##      year       value
    ## 3014 2018  0.44563907
    ## 3015 2018 -1.07149878
    ## 3016 2018  0.33945998
    ## 3017 2018 -0.05980236
    ## 3018 2018 -0.19249224
    ## 3019 2018 -0.15278959
    

    Basic statistics summary

    (dj_stats <- dataframe_basicstats(dj_vol_df))
    ##                   2007       2008       2009       2010       2011
    ## nobs        250.000000 253.000000 252.000000 252.000000 252.000000
    ## NAs           0.000000   0.000000   0.000000   0.000000   0.000000
    ## Minimum      -1.606192  -1.122526  -1.071225  -1.050181  -2.301514
    ## Maximum       0.775961   0.724762   0.881352   1.041216   2.441882
    ## 1. Quartile  -0.123124  -0.128815  -0.162191  -0.170486  -0.157758
    ## 3. Quartile   0.130056   0.145512   0.169233   0.179903   0.137108
    ## Mean         -0.002685   0.001203  -0.001973  -0.001550   0.000140
    ## Median       -0.010972   0.002222  -0.031748  -0.004217  -0.012839
    ## Sum          -0.671142   0.304462  -0.497073  -0.390677   0.035162
    ## SE Mean       0.016984   0.016196   0.017618   0.019318   0.026038
    ## LCL Mean     -0.036135  -0.030693  -0.036670  -0.039596  -0.051141
    ## UCL Mean      0.030766   0.033100   0.032725   0.036495   0.051420
    ## Variance      0.072112   0.066364   0.078219   0.094041   0.170850
    ## Stdev         0.268536   0.257612   0.279677   0.306661   0.413341
    ## Skewness     -0.802037  -0.632586   0.066535  -0.150523   0.407226
    ## Kurtosis      5.345212   2.616615   1.500979   1.353797  14.554642
    ##                   2012       2013       2014       2015       2016
    ## nobs        250.000000 252.000000 252.000000 252.000000 252.000000
    ## NAs           0.000000   0.000000   0.000000   0.000000   0.000000
    ## Minimum      -2.158960  -1.386215  -2.110572  -1.326016  -1.336471
    ## Maximum       1.292956   1.245202   2.008667   1.130289   1.319713
    ## 1. Quartile  -0.152899  -0.145444  -0.144280  -0.143969  -0.134011
    ## 3. Quartile   0.144257   0.149787   0.134198   0.150003   0.141287
    ## Mean          0.001642  -0.002442   0.000200   0.000488   0.004228
    ## Median       -0.000010  -0.004922   0.013460   0.004112  -0.002044
    ## Sum           0.410521  -0.615419   0.050506   0.123080   1.065480
    ## SE Mean       0.021293   0.019799   0.023514   0.019010   0.019089
    ## LCL Mean     -0.040295  -0.041435  -0.046110  -0.036952  -0.033367
    ## UCL Mean      0.043579   0.036551   0.046510   0.037929   0.041823
    ## Variance      0.113345   0.098784   0.139334   0.091071   0.091826
    ## Stdev         0.336667   0.314299   0.373274   0.301780   0.303028
    ## Skewness     -0.878227  -0.297951  -0.209417  -0.285918   0.083826
    ## Kurtosis      8.115847   4.681120   9.850061   4.754926   4.647785
    ##                   2017       2018
    ## nobs        251.000000 251.000000
    ## NAs           0.000000   0.000000
    ## Minimum      -0.817978  -1.071499
    ## Maximum       0.915599   0.926101
    ## 1. Quartile  -0.112190  -0.119086
    ## 3. Quartile   0.110989   0.112424
    ## Mean         -0.000017   0.000257
    ## Median       -0.006322   0.003987
    ## Sum          -0.004238   0.064605
    ## SE Mean       0.013446   0.014180
    ## LCL Mean     -0.026500  -0.027671
    ## UCL Mean      0.026466   0.028185
    ## Variance      0.045383   0.050471
    ## Stdev         0.213032   0.224658
    ## Skewness      0.088511  -0.281007
    ## Kurtosis      3.411036   4.335748
    

    In the following, we make specific comments to some relevant above-shown metrics.

    Mean

    Years when Dow Jones daily volume log-ratio has positive mean are:

    filter_dj_stats(dj_stats, "Mean", 0)
    ## [1] "2008" "2011" "2012" "2014" "2015" "2016" "2018"
    

    All Dow Jones daily volume log-ratio mean values in ascending order.

    dj_stats["Mean",order(dj_stats["Mean",,])]
    ##           2007      2013      2009     2010     2017    2011  2014
    ## Mean -0.002685 -0.002442 -0.001973 -0.00155 -1.7e-05 0.00014 2e-04
    ##          2018     2015     2008     2012     2016
    ## Mean 0.000257 0.000488 0.001203 0.001642 0.004228
    

    Median

    Years when Dow Jones daily volume log-ratio has positive median are:

    filter_dj_stats(dj_stats, "Median", 0)
    ## [1] "2008" "2014" "2015" "2018"
    

    All Dow Jones daily volume log-ratio median values in ascending order.

    dj_stats["Median",order(dj_stats["Median",,])]
    ##             2009      2011      2007      2017      2013      2010
    ## Median -0.031748 -0.012839 -0.010972 -0.006322 -0.004922 -0.004217
    ##             2016   2012     2008     2018     2015    2014
    ## Median -0.002044 -1e-05 0.002222 0.003987 0.004112 0.01346
    

    Skewness

    Years when Dow Jones daily volume log-ratio has positive skewness are:

    filter_dj_stats(dj_stats, "Skewness", 0)
    ## [1] "2009" "2011" "2016" "2017"
    

    All Dow Jones daily volume log-ratio mean values in ascending order.

    dj_stats["Skewness",order(dj_stats["Skewness",,])]
    ##               2012      2007      2008      2013      2015      2018
    ## Skewness -0.878227 -0.802037 -0.632586 -0.297951 -0.285918 -0.281007
    ##               2014      2010     2009     2016     2017     2011
    ## Skewness -0.209417 -0.150523 0.066535 0.083826 0.088511 0.407226
    

    Excess Kurtosis

    Years when Dow Jones daily volume has positive excess kurtosis are:

    filter_dj_stats(dj_stats, "Kurtosis", 0)
    ##  [1] "2007" "2008" "2009" "2010" "2011" "2012" "2013" "2014" "2015" "2016"
    ## [11] "2017" "2018"
    

    All Dow Jones daily volume log-ratio excess kurtosis values in ascending order.

    dj_stats["Kurtosis",order(dj_stats["Kurtosis",,])]
    ##              2010     2009     2008     2017     2018     2016    2013
    ## Kurtosis 1.353797 1.500979 2.616615 3.411036 4.335748 4.647785 4.68112
    ##              2015     2007     2012     2014     2011
    ## Kurtosis 4.754926 5.345212 8.115847 9.850061 14.55464
    

    Box-plots

    dataframe_boxplot(dj_vol_df, "DJIA daily volume box plots 2007-2018")
    

    The most positive extreme values can be spotted on years 2011, 2014 and 2016. The most negative extreme values, on years 2007, 2011, 2012, 2014.

    Density plots

    dataframe_densityplot(dj_vol_df, "DJIA daily volume density plots 2007-2018")
    

    Shapiro Tests

    dataframe_shapirotest(dj_vol_df)
    ##            result
    ## 2007 3.695053e-09
    ## 2008 6.160136e-07
    ## 2009 2.083475e-04
    ## 2010 1.500060e-03
    ## 2011 3.434415e-18
    ## 2012 8.417627e-12
    ## 2013 1.165184e-10
    ## 2014 1.954662e-16
    ## 2015 5.261037e-11
    ## 2016 7.144940e-11
    ## 2017 1.551041e-08
    ## 2018 3.069196e-09
    

    Based on reported p-values, for all we can reject the null hypothesis of normal distribution.

    QQ plots

    dataframe_qqplot(dj_vol_df, "DJIA daily volume QQ plots 2007-2018")
    

    Departure from normality can be spotted for all reported years.

    Saving the current enviroment for further analysis.

    save.image(file='DowEnvironment.RData')
    

    If you have any questions, please feel free to comment below.

    References

    1. Dow Jones Industrial Average
    2. Skewness
    3. Kurtosis
    4. An introduction to analysis of financial data with R, Wiley, Ruey S. Tsay
    5. Time series analysis and its applications, Springer ed., R.H. Shumway, D.S. Stoffer
    6. Applied Econometric Time Series, Wiley, W. Enders, 4th ed.
    7. Forecasting – Principle and Practice, Texts, R.J. Hyndman
    8. Options, Futures and other Derivatives, Pearson ed., J.C. Hull
    9. An introduction to rugarch package
    10. Applied Econometrics with R, Achim Zeileis, Christian Kleiber – Springer Ed.
    11. GARCH modeling: diagnostic tests

    Disclaimer

    Any securities or databases referred in this post are solely for illustration purposes, and under no regard should the findings presented here be interpreted as investment advice or promotion of any particular security or source.

    Related Post

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