Equally Weighted Moving Average (EqWMA) of the pth. exponentiated values

The function eqwma returns an Equally Weighted Moving Average (EqWMA) of the pth. exponentiated values lagged k times (the default of k is 1). Optionally, the absolute values are computed before averaging if abs=TRUE, and the natural log of the values is returned if log=TRUE. The function leqwma is a wrapper to eqwma with abs=TRUE and log=TRUE.

If x is financial return (possibly mean-corrected) and p=2, then this gives the socalled 'historical' model, also known as an integrated ARCH model where the ARCH coefficients all have the same value with sum equal to one. In the log-variance specification the lag of log(EqWMA) is thus a financial volatility proxy. It may be an imperfect proxy compared with high-frequency data (which can also be included as regressors), but - in contrast to high-frequency data - is always available and easy to compute.

Usage

eqwma(x, length=5, k=1, p=1, abs=FALSE, log=FALSE,  as.vector=FALSE,
  lag=NULL, start=NULL)
leqwma(x, length=5, k=1, p=2, as.vector=FALSE, lag=NULL, start=NULL)

Arguments

x: numeric vector, time-series or zoo object
length: integer or vector of integers each equal to or greater than 1. The length or lengths of the moving window or windows of averages
k: integer that determines how many periods the term(s) should be lagged. If 0 (or smaller), then the moving averages are not lagged
p: numeric value. The exponent p in x^p when abs=FALSE, and in abs(x)^p when abs=TRUE
log: logical with default FALSE. If TRUE, then the logarithm of the moving average is returned
abs: logical with default FALSE. If TRUE, then x is transformed to absolute values before x is exponentiated
as.vector: logical with default FALSE. If TRUE, and if length(length)==1, then the result is returned as a vector. Otherwise the returned value is always a matrix
lag: deprecated
start: deprecated

Details

The intended primary use of eqwma is to construct mixed frequency regressors for the mean specification of an arx model.

The intended primary use of leqwma is to construct volatility proxies for the log-variance specification in an arx model. In the latter case, the default is the lagged log of an equally weighted moving average of the squared residuals, where each average is made up of m observations. This is equivalent to an integrated ARCH(p) model where the p coefficients are all equal. For further details on the use of log(EqWMA) as a volatility proxy, see Sucarrat and Escribano (2012).

Value

numeric matrix, vector or zoo object

References

Genaro Sucarrat and Alvaro Escribano (2012): 'Automated Financial Model Selection: General-to-Specific Modelling of the Mean and Volatility Specifications', Oxford Bulletin of Economics and Statistics 74, Issue no. 5 (October), pp. 716-735

Pretis, Felix, Reade, James and Sucarrat, Genaro (2018): 'Automated General-to-Specific (GETS) Regression Modeling and Indicator Saturation for Outliers and Structural Breaks'. Journal of Statistical Software 86, Number 3, pp. 1-44

Author

Genaro Sucarrat, http://www.sucarrat.net/

Examples

##generate an iid normal series:
set.seed(123)
x <- rnorm(100)

##compute lag of EqWMA(20) for x^2:
eqwma(x, p=2)
#>          EqWMA(5)
#>   [1,]         NA
#>   [2,]         NA
#>   [3,]         NA
#>   [4,]         NA
#>   [5,]         NA
#>   [6,] 0.56367460
#>   [7,] 1.08913759
#>   [8,] 1.12103000
#>   [9,] 0.95519167
#>  [10,] 1.04855075
#>  [11,] 1.08493060
#>  [12,] 0.79631627
#>  [13,] 0.77972072
#>  [14,] 0.49176828
#>  [15,] 0.39986505
#>  [16,] 0.42193400
#>  [17,] 0.76087047
#>  [18,] 0.78454829
#>  [19,] 1.52594135
#>  [20,] 1.62187123
#>  [21,] 1.60478570
#>  [22,] 1.19422348
#>  [23,] 1.15415508
#>  [24,] 0.59117549
#>  [25,] 0.59905196
#>  [26,] 0.63248043
#>  [27,] 0.97341781
#>  [28,] 1.10429262
#>  [29,] 0.89846026
#>  [30,] 1.05127491
#>  [31,] 1.28755046
#>  [32,] 0.75493794
#>  [33,] 0.63197395
#>  [34,] 0.78751928
#>  [35,] 0.68267183
#>  [36,] 0.50326055
#>  [37,] 0.56173128
#>  [38,] 0.60568280
#>  [39,] 0.44619942
#>  [40,] 0.31069837
#>  [41,] 0.20465091
#>  [42,] 0.20632939
#>  [43,] 0.15361036
#>  [44,] 0.47308933
#>  [45,] 1.39524069
#>  [46,] 1.65812350
#>  [47,] 1.81387452
#>  [48,] 1.83769184
#>  [49,] 1.56099969
#>  [50,] 0.74179481
#>  [51,] 0.45135046
#>  [52,] 0.21190993
#>  [53,] 0.17960968
#>  [54,] 0.13642381
#>  [55,] 0.38936914
#>  [56,] 0.39817356
#>  [57,] 0.84527613
#>  [58,] 1.32484019
#>  [59,] 1.39282727
#>  [60,] 1.02128080
#>  [61,] 1.02041244
#>  [62,] 0.58930105
#>  [63,] 0.16003977
#>  [64,] 0.11389056
#>  [65,] 0.31832174
#>  [66,] 0.53874288
#>  [67,] 0.52834358
#>  [68,] 0.51805621
#>  [69,] 0.49641267
#>  [70,] 0.45902896
#>  [71,] 1.06985112
#>  [72,] 1.09964751
#>  [73,] 2.12592129
#>  [74,] 2.32766140
#>  [75,] 2.25813908
#>  [76,] 1.51224081
#>  [77,] 1.67437782
#>  [78,] 0.62414477
#>  [79,] 0.71987312
#>  [80,] 0.62585417
#>  [81,] 0.53504116
#>  [82,] 0.32468848
#>  [83,] 0.33815754
#>  [84,] 0.06760497
#>  [85,] 0.14407500
#>  [86,] 0.14993971
#>  [87,] 0.17194892
#>  [88,] 0.38287188
#>  [89,] 0.39327070
#>  [90,] 0.33147275
#>  [91,] 0.58570167
#>  [92,] 0.76109580
#>  [93,] 0.58063248
#>  [94,] 0.55415446
#>  [95,] 0.61176139
#>  [96,] 0.71808462
#>  [97,] 0.59273695
#>  [98,] 1.48947423
#>  [99,] 1.94785473
#> [100,] 1.88011245

##compute lag of EqWMA(5) and lag of EqWMA(10) for x:
eqwma(x, length=c(5,10))
#>             EqWMA(5)    EqWMA(10)
#>   [1,]            NA           NA
#>   [2,]            NA           NA
#>   [3,]            NA           NA
#>   [4,]            NA           NA
#>   [5,]            NA           NA
#>   [6,]  0.1935702609           NA
#>   [7,]  0.6486783876           NA
#>   [8,]  0.7868971267           NA
#>   [9,]  0.2221432170           NA
#>  [10,]  0.0706709683           NA
#>  [11,] -0.0443189727  0.074625644
#>  [12,] -0.1425156106  0.253081388
#>  [13,] -0.1627360864  0.312080520
#>  [14,]  0.1704304506  0.196286834
#>  [15,]  0.3299375642  0.200304266
#>  [16,]  0.3079017313  0.131791379
#>  [17,]  0.4204679991  0.138976194
#>  [18,]  0.4480753294  0.142669621
#>  [19,] -0.0254023921  0.072514029
#>  [20,]  0.0927322450  0.211334905
#>  [21,]  0.1093421904  0.208621961
#>  [22,] -0.4616051781 -0.020568589
#>  [23,] -0.6047702567 -0.078347464
#>  [24,] -0.4166477150 -0.221025054
#>  [25,] -0.7026971412 -0.304982448
#>  [26,] -0.7331467132 -0.311902261
#>  [27,] -0.8569206342 -0.659262906
#>  [28,] -0.6457682423 -0.625269250
#>  [29,] -0.4098927291 -0.413270222
#>  [30,] -0.4917418707 -0.597219506
#>  [31,] -0.1159710329 -0.424558873
#>  [32,]  0.3066604736 -0.275130080
#>  [33,]  0.0800887681 -0.282839737
#>  [34,]  0.2284392767 -0.090726726
#>  [35,]  0.6316933616  0.069975745
#>  [36,]  0.5452465937  0.214637780
#>  [37,]  0.5976818003  0.452171137
#>  [38,]  0.7674796276  0.423784198
#>  [39,]  0.5760721532  0.402255715
#>  [40,]  0.3392529230  0.485473142
#>  [41,]  0.0988425065  0.322044550
#>  [42,] -0.1778269401  0.209927430
#>  [43,] -0.3301939265  0.218642851
#>  [44,] -0.5708908547  0.002590649
#>  [45,] -0.0759071288  0.131672897
#>  [46,]  0.2417794710  0.170310989
#>  [47,]  0.1560991502 -0.010863895
#>  [48,]  0.1171056387 -0.106544144
#>  [49,]  0.2768538383 -0.147018508
#>  [50,] -0.0009443311 -0.038425730
#>  [51,] -0.2592105441 -0.008715537
#>  [52,]  0.0160748754  0.086087013
#>  [53,]  0.0909424914  0.104024065
#>  [54,]  0.1756994706  0.226276654
#>  [55,]  0.2934269038  0.146241286
#>  [56,]  0.2649465199  0.002867988
#>  [57,]  0.5175769380  0.266825907
#>  [58,]  0.2135357283  0.152239110
#>  [59,]  0.3390325696  0.257366020
#>  [60,]  0.0900829616  0.191754933
#>  [61,]  0.1784254725  0.221685996
#>  [62,] -0.0489407518  0.234318093
#>  [63,]  0.1603451184  0.186940423
#>  [64,] -0.0232191083  0.157906731
#>  [65,] -0.2517050337 -0.080811036
#>  [66,] -0.5092515927 -0.165413060
#>  [67,] -0.5244737610 -0.286707256
#>  [68,] -0.3343671146 -0.087010998
#>  [69,] -0.2571247926 -0.140171950
#>  [70,]  0.1310437776 -0.060330628
#>  [71,]  0.7554189601  0.123083684
#>  [72,]  0.5965069986  0.036016619
#>  [73,]  0.0450312677 -0.144667923
#>  [74,]  0.2355781273 -0.010773333
#>  [75,] -0.0907155188  0.020164129
#>  [76,] -0.6383341793  0.058542390
#>  [77,] -0.3350136721  0.130746663
#>  [78,]  0.0698655016  0.057448385
#>  [79,] -0.3754257457 -0.069923809
#>  [80,] -0.1973248973 -0.144020208
#>  [81,] -0.0875014465 -0.362917813
#>  [82,] -0.2914628832 -0.313238278
#>  [83,] -0.1574522016 -0.043793350
#>  [84,]  0.0125593345 -0.181433206
#>  [85,]  0.1051739483 -0.046075475
#>  [86,]  0.0888549084  0.000676731
#>  [87,]  0.1540584640 -0.068702210
#>  [88,]  0.2963701864  0.069458992
#>  [89,]  0.4575384909  0.235048913
#>  [90,]  0.2634768641  0.184325406
#>  [91,]  0.5373357002  0.313095304
#>  [92,]  0.6696800786  0.411869271
#>  [93,]  0.5599916678  0.428180927
#>  [94,]  0.5207017167  0.489120104
#>  [95,]  0.4603068186  0.361891841
#>  [96,]  0.5026757846  0.520005742
#>  [97,]  0.1839230960  0.426801587
#>  [98,]  0.5117103027  0.535850985
#>  [99,]  0.7704860809  0.645593899
#> [100,]  0.8489272243  0.654617021

##compute lag of log(EqWMA(20)) for x^2:
leqwma(x)
#>        logEqWMA(5)
#>   [1,]          NA
#>   [2,]          NA
#>   [3,]          NA
#>   [4,]          NA
#>   [5,]          NA
#>   [6,] -0.57327815
#>   [7,]  0.08538618
#>   [8,]  0.11424791
#>   [9,] -0.04584326
#>  [10,]  0.04740897
#>  [11,]  0.08151602
#>  [12,] -0.22775885
#>  [13,] -0.24881948
#>  [14,] -0.70974764
#>  [15,] -0.91662816
#>  [16,] -0.86290636
#>  [17,] -0.27329215
#>  [18,] -0.24264716
#>  [19,]  0.42261150
#>  [20,]  0.48358056
#>  [21,]  0.47299023
#>  [22,]  0.17749617
#>  [23,]  0.14336854
#>  [24,] -0.52564236
#>  [25,] -0.51240694
#>  [26,] -0.45810599
#>  [27,] -0.02694189
#>  [28,]  0.09920497
#>  [29,] -0.10707281
#>  [30,]  0.05000363
#>  [31,]  0.25274155
#>  [32,] -0.28111973
#>  [33,] -0.45890710
#>  [34,] -0.23886742
#>  [35,] -0.38174102
#>  [36,] -0.68664725
#>  [37,] -0.57673168
#>  [38,] -0.50139886
#>  [39,] -0.80698929
#>  [40,] -1.16893271
#>  [41,] -1.58644962
#>  [42,] -1.57828141
#>  [43,] -1.87333604
#>  [44,] -0.74847105
#>  [45,]  0.33306694
#>  [46,]  0.50568654
#>  [47,]  0.59546517
#>  [48,]  0.60851035
#>  [49,]  0.44532645
#>  [50,] -0.29868260
#>  [51,] -0.79551117
#>  [52,] -1.55159394
#>  [53,] -1.71696923
#>  [54,] -1.99198899
#>  [55,] -0.94322745
#>  [56,] -0.92086728
#>  [57,] -0.16809193
#>  [58,]  0.28129184
#>  [59,]  0.33133569
#>  [60,]  0.02105752
#>  [61,]  0.02020690
#>  [62,] -0.52881810
#>  [63,] -1.83233292
#>  [64,] -2.17251732
#>  [65,] -1.14469263
#>  [66,] -0.61851686
#>  [67,] -0.63800849
#>  [68,] -0.65767153
#>  [69,] -0.70034771
#>  [70,] -0.77864197
#>  [71,]  0.06751950
#>  [72,]  0.09498968
#>  [73,]  0.75420526
#>  [74,]  0.84486407
#>  [75,]  0.81454106
#>  [76,]  0.41359253
#>  [77,]  0.51544164
#>  [78,] -0.47137293
#>  [79,] -0.32868030
#>  [80,] -0.46863789
#>  [81,] -0.62541160
#>  [82,] -1.12488909
#>  [83,] -1.08424339
#>  [84,] -2.69407382
#>  [85,] -1.93742125
#>  [86,] -1.89752202
#>  [87,] -1.76055785
#>  [88,] -0.96005486
#>  [89,] -0.93325711
#>  [90,] -1.10420968
#>  [91,] -0.53494471
#>  [92,] -0.27299604
#>  [93,] -0.54363728
#>  [94,] -0.59031181
#>  [95,] -0.49141295
#>  [96,] -0.33116786
#>  [97,] -0.52300457
#>  [98,]  0.39842319
#>  [99,]  0.66672863
#> [100,]  0.63133159

#compute lag of log(EqWMA(5)) and lag of log(EqWMA(8))
#for abs(x)^2:
leqwma(x, length=c(4,8))
#>         logEqWMA(4) logEqWMA(8)
#>   [1,]           NA          NA
#>   [2,]           NA          NA
#>   [3,]           NA          NA
#>   [4,]           NA          NA
#>   [5,] -0.356083095          NA
#>   [6,] -0.468309051          NA
#>   [7,]  0.298752989          NA
#>   [8,] -0.230804569          NA
#>   [9,]  0.176258820 -0.05489919
#>  [10,]  0.267359159 -0.03429666
#>  [11,] -0.476744253 -0.01563263
#>  [12,] -0.059448138 -0.14146046
#>  [13,] -0.554157855 -0.12369273
#>  [14,] -0.699630862 -0.10354092
#>  [15,] -0.798112965 -0.62457405
#>  [16,] -1.878471984 -0.60229433
#>  [17,] -0.084772156 -0.29217381
#>  [18,] -0.061310753 -0.33038113
#>  [19,]  0.644148103  0.16319891
#>  [20,]  0.667880243  0.05018093
#>  [21,]  0.188732205  0.06130160
#>  [22,]  0.358244618  0.17031076
#>  [23,] -0.742761703  0.17402142
#>  [24,] -0.484517498  0.24923754
#>  [25,] -0.366823379 -0.05095151
#>  [26,] -0.682130734 -0.03234022
#>  [27,]  0.186391589 -0.17394152
#>  [28,]  0.110820625 -0.14318421
#>  [29,] -0.009793082 -0.17245832
#>  [30,]  0.195916323 -0.14968180
#>  [31,] -0.107357511  0.05026455
#>  [32,] -0.263704320 -0.05900976
#>  [33,] -0.243235788 -0.11971791
#>  [34,] -0.414667047 -0.06348026
#>  [35,] -0.775818562 -0.38674298
#>  [36,] -0.538526172 -0.39170393
#>  [37,] -0.385078370 -0.31164427
#>  [38,] -0.585565306 -0.49646983
#>  [39,] -1.007941011 -0.88515975
#>  [40,] -1.515837855 -0.91225906
#>  [41,] -1.985898083 -0.89446246
#>  [42,] -1.708123283 -0.99696330
#>  [43,] -1.655195615 -1.28009036
#>  [44,] -0.565707151 -0.93193437
#>  [45,]  0.535241899 -0.08060362
#>  [46,]  0.668854448  0.06447858
#>  [47,]  0.813830782  0.20195686
#>  [48,]  0.640172374  0.20894977
#>  [49,] -0.254689532  0.21632775
#>  [50,] -0.575452200  0.22890707
#>  [51,] -1.390925672  0.22529348
#>  [52,] -1.494733527  0.05879527
#>  [53,] -1.771543436 -0.74947691
#>  [54,] -3.993050538 -1.23633444
#>  [55,] -0.723660372 -1.00264051
#>  [56,] -0.730486941 -1.04131165
#>  [57,]  0.054858789 -0.48901380
#>  [58,]  0.504157908 -0.17791081
#>  [59,]  0.241192508 -0.12911837
#>  [60,]  0.234168839 -0.13608766
#>  [61,] -0.355825619 -0.12954722
#>  [62,] -1.988011593 -0.10950338
#>  [63,] -2.166251579 -0.36573556
#>  [64,] -1.976681165 -0.35497204
#>  [65,] -0.951284716 -0.60987345
#>  [66,] -0.450362483 -0.94885968
#>  [67,] -0.515256659 -1.03288963
#>  [68,] -0.478336699 -0.96976839
#>  [69,] -1.018484376 -0.98432018
#>  [70,] -1.249660461 -0.77219130
#>  [71,]  0.273290120 -0.04519155
#>  [72,]  0.280911495 -0.02832239
#>  [73,]  0.977084469  0.41139454
#>  [74,]  0.992114988  0.39996150
#>  [75,]  0.572087436  0.43380749
#>  [76,]  0.604328597  0.45563828
#>  [77,] -0.274559158  0.53550057
#>  [78,] -0.639979053  0.47754881
#>  [79,] -0.256054251  0.24140068
#>  [80,] -0.409505139  0.22074141
#>  [81,] -0.901766003 -0.53977451
#>  [82,] -0.952989260 -0.78428690
#>  [83,] -2.992557440 -0.88641770
#>  [84,] -2.573233456 -0.99389123
#>  [85,] -1.741421634 -1.23593943
#>  [86,] -1.674422791 -1.25001149
#>  [87,] -1.726950310 -2.17162310
#>  [88,] -0.811384384 -1.34605281
#>  [89,] -0.947309151 -1.26752667
#>  [90,] -0.910837241 -1.22145187
#>  [91,] -0.350114690 -0.81821940
#>  [92,] -0.429851356 -0.60253124
#>  [93,] -0.387952056 -0.62902003
#>  [94,] -0.406262580 -0.62705744
#>  [95,] -0.832956513 -0.56267234
#>  [96,] -0.429486375 -0.42966885
#>  [97,] -0.406861505 -0.39736209
#>  [98,]  0.613884566  0.22862045
#>  [99,]  0.848547894  0.32606610
#> [100,]  0.635145539  0.23828439