Higher-Dimensional Centering and Linear Prediction

fhdbetween is a generalization of fbetween to efficiently predict with multiple factors and linear models (i.e. predict with vectors/factors, matrices, or data frames/lists where the latter may contain multiple factor variables). Similarly, fhdwithin is a generalization of fwithin to center on multiple factors and partial-out linear models.

The corresponding operators HDB and HDW additionally allow to predict / partial out full lm() formulas with interactions between variables.

Usage

fhdbetween(x, ...)
 fhdwithin(x, ...)
       HDB(x, ...)
       HDW(x, ...)

# Default S3 method
fhdbetween(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)
# Default S3 method
fhdwithin(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)
# Default S3 method
HDB(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)
# Default S3 method
HDW(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)

# S3 method for class 'matrix'
fhdbetween(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)
# S3 method for class 'matrix'
fhdwithin(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, lm.method = "qr", ...)
# S3 method for class 'matrix'
HDB(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, stub = .op[["stub"]],
    lm.method = "qr", ...)
# S3 method for class 'matrix'
HDW(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE, stub = .op[["stub"]],
    lm.method = "qr", ...)

# S3 method for class 'data.frame'
fhdbetween(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE,
           variable.wise = FALSE, lm.method = "qr", ...)
# S3 method for class 'data.frame'
fhdwithin(x, fl, w = NULL, na.rm = .op[["na.rm"]], fill = FALSE,
          variable.wise = FALSE, lm.method = "qr", ...)
# S3 method for class 'data.frame'
HDB(x, fl, w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]], fill = FALSE,
    variable.wise = FALSE, stub = .op[["stub"]], lm.method = "qr", ...)
# S3 method for class 'data.frame'
HDW(x, fl, w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]], fill = FALSE,
    variable.wise = FALSE, stub = .op[["stub"]], lm.method = "qr", ...)

# Methods for indexed data / compatibility with plm:

# S3 method for class 'pseries'
fhdbetween(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE, ...)
# S3 method for class 'pseries'
fhdwithin(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE, ...)
# S3 method for class 'pseries'
HDB(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE, ...)
# S3 method for class 'pseries'
HDW(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE, ...)

# S3 method for class 'pdata.frame'
fhdbetween(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE,
           variable.wise = TRUE, ...)
# S3 method for class 'pdata.frame'
fhdwithin(x, effect = "all", w = NULL, na.rm = .op[["na.rm"]], fill = TRUE,
          variable.wise = TRUE, ...)
# S3 method for class 'pdata.frame'
HDB(x, effect = "all", w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]],
    fill = TRUE, variable.wise = TRUE, stub = .op[["stub"]], ...)
# S3 method for class 'pdata.frame'
HDW(x, effect = "all", w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]],
    fill = TRUE, variable.wise = TRUE, stub = .op[["stub"]], ...)

Arguments

x

a numeric vector, matrix, data frame, 'indexed_series' ('pseries') or 'indexed_frame' ('pdata.frame').

fl

a numeric vector, factor, matrix, data frame or list (which may or may not contain factors). In the HDW/HDB data frame method fl can also be a one-or two sided lm() formula with variables contained in x. Interactions (:) and full interactions (*) are supported. See Examples and the Note.

w

a vector of (non-negative) weights.

cols

data.frame methods: Select columns to center (partial-out) or predict using column names, indices, a logical vector or a function. Unless specified otherwise all numeric columns are selected. If NULL, all columns are selected.

na.rm

remove missing values from both x and fl. by default rows with missing values in x or fl are removed. In that case an attribute "na.rm" is attached containing the rows removed.

fill

If na.rm = TRUE, fill = TRUE will not remove rows with missing values in x or fl, but fill them with NA's.

variable.wise

(p)data.frame methods: Setting variable.wise = TRUE will process each column individually i.e. use all non-missing cases in each column and in fl (fl is only checked for missing values if na.rm = TRUE). This is a lot less efficient but uses all data available in each column.

effect

plm methods: Select which panel identifiers should be used for centering. 1L takes the first variable in the index, 2L the second etc.. Index variables can also be called by name using a character vector. The keyword "all" uses all identifiers.

stub

character. A prefix/stub to add to the names of all transformed columns. TRUE (default) uses "HDW."/"HDB.", FALSE will not rename columns.

lm.method

character. The linear fitting method. Supported are "chol" and "qr". See flm.

...

further arguments passed to fixest::demean (other than notes and im_confident) and chol / qr. Possible choices are tol to set a uniform numerical tolerance for the entire fitting process, or nthreads and iter to govern the higher-order centering process.

Details

fhdbetween/HDB and fhdwithin/HDW are powerful functions for high-dimensional linear prediction problems involving large factors and datasets, but can just as well handle ordinary regression problems. They are implemented as efficient wrappers around fbetween / fwithin, flm and some C++ code from the fixest package that is imported for higher-order centering tasks (thus fixest needs to be installed for problems involving more than one factor).

Intended areas of use are to efficiently obtain residuals and predicted values from data, and to prepare data for complex linear models involving multiple levels of fixed effects. Such models can now be fitted using (g)lm() on data prepared with fhdwithin / HDW (relying on bootstrapped SE's for inference, or implementing the appropriate corrections). See Examples.

If fl is a vector or matrix, the result are identical to lm i.e. fhdbetween / HDB returns fitted(lm(x ~ fl)) and fhdwithin / HDW residuals(lm(x ~ fl)). If fl is a list containing factors, all variables in x and non-factor variables in fl are centered on these factors using either fbetween / fwithin for a single factor or fixest C++ code for multiple factors. Afterwards the centered data is regressed on the centered predictors. If fl is just a list of factors, fhdwithin/HDW returns the centered data and fhdbetween/HDB the corresponding means. Take as a most general example a list fl = list(fct1, fct2, ..., var1, var2, ...) where fcti are factors and vari are continuous variables. The output of fhdwithin/HDW | fhdbetween/HDB will then be identical to calling resid | fitted on lm(x ~ fct1 + fct2 + ... + var1 + var2 + ...). The computations performed by fhdwithin/HDW and fhdbetween/HDB are however much faster and more memory efficient than lm because factors are not passed to model.matrix and expanded to matrices of dummies but projected out beforehand.

The formula interface to the data.frame method (only supported by the operators HDW | HDB) provides ease of use and allows for additional modeling complexity. For example it is possible to project out formulas like HDW(data, ~ fct1*var1 + fct2:fct3 + var2:fct2:fct3 + var2:var3 + poly(var5,3)*fct5) containing simple (:) or full (*) interactions of factors with continuous variables or polynomials of continuous variables, and two-or three-way interactions of factors and continuous variables. If the formula is one-sided as in the example above (the space left of (~) is left empty), the formula is applied to all variables selected through cols. The specification provided in cols (default: all numeric variables not used in the formula) can be overridden by supplying one-or more dependent variables. For example HDW(data, var1 + var2 ~ fct1 + fct2) will return a data.frame with var1 and var2 centered on fct1 and fct2.

The special methods for 'indexed_series' (plm::pseries) and 'indexed_frame's (plm::pdata.frame) center a panel series or variables in a panel data frame on all panel-identifiers. By default in these methods fill = TRUE and variable.wise = TRUE, so missing values are kept. This change in the default arguments was done to ensure a coherent framework of functions and operators applied to plm panel data classes.

Note

On the differences between fhdwithin/HDW... and fwithin/W...:

• fhdwithin/HDW can center data on multiple factors and also partial out continuous variables and factor-continuous interactions while fwithin/W only centers on one factor or the interaction of a set of factors, and does that very efficiently.

• HDW(data, ~ qF(group1) + qF(group2)) simultaneously centers numeric variables in data on group1 and group2, while W(data, ~ group1 + group2) centers data on the interaction of group1 and group2. The equivalent operation in HDW would be: HDW(data, ~ qF(group1):qF(group2)).

• W always does computations on the variable-wise complete observations (in both matrices and data frames), whereas by default HDW removes all cases missing in either x or fl. In short, W(data, ~ group1 + group2) is actually equivalent to HDW(data, ~ qF(group1):qF(group2), variable.wise = TRUE). HDW(data, ~ qF(group1):qF(group2)) would remove any missing cases.

• fbetween/B and fwithin/W have options to fill missing cases using group-averages and to add the overall mean back to group-demeaned data. These options are not available in fhdbetween/HDB and fhdwithin/HDW. Since HDB and HDW by default remove missing cases, they also don't have options to keep grouping-columns as in B and W.

Value

HDB returns fitted values of regressing x on fl. HDW returns residuals. See Details and Examples.

See also

fbetween, fwithin, fscale, TRA, flm, fFtest, Data Transformations, Collapse Overview

Examples

HDW(mtcars$mpg, mtcars$carb)                   # Simple regression problems
#>  [1]  3.3505410  3.3505410 -1.0166151 -2.4166151 -3.0608964 -5.7166151
#>  [7] -3.3494590  2.6391036  1.0391036  1.5505410  0.1505410 -3.3051777
#> [13] -2.4051777 -4.5051777 -7.2494590 -7.2494590 -2.9494590  8.5833849
#> [19]  8.6391036 10.0833849 -2.3166151 -6.2608964 -6.5608964 -4.3494590
#> [25] -2.5608964  3.4833849  4.2391036  8.6391036 -1.8494590  6.1619784
#> [31]  5.5734158 -0.3608964
HDW(mtcars$mpg, mtcars[-1])
#>  [1] -1.599505761 -1.111886079 -3.450644085  0.162595453  1.006565971
#>  [6] -2.283039036 -0.086256253  1.903988115 -1.619089898  0.500970058
#> [11] -1.391654392  2.227837890  1.700426404 -0.542224699 -1.634013415
#> [16] -0.536437711  4.206370638  4.627094192  0.503261089  4.387630904
#> [21] -2.143103442 -1.443053221 -2.532181498 -0.006021976  2.508321011
#> [26] -0.993468693 -0.152953961  2.763727417 -3.070040803  0.006171846
#> [31]  1.058881618 -2.968267683
HDW(mtcars$mpg, qM(mtcars[-1]))
#>  [1] -1.599505761 -1.111886079 -3.450644085  0.162595453  1.006565971
#>  [6] -2.283039036 -0.086256253  1.903988115 -1.619089898  0.500970058
#> [11] -1.391654392  2.227837890  1.700426404 -0.542224699 -1.634013415
#> [16] -0.536437711  4.206370638  4.627094192  0.503261089  4.387630904
#> [21] -2.143103442 -1.443053221 -2.532181498 -0.006021976  2.508321011
#> [26] -0.993468693 -0.152953961  2.763727417 -3.070040803  0.006171846
#> [31]  1.058881618 -2.968267683
head(HDW(qM(mtcars[3:4]), mtcars[1:2]))
#>                     HDW.disp     HDW.hp
#> Mazda RX4         -56.791929 -29.668202
#> Mazda RX4 Wag     -56.791929 -29.668202
#> Datsun 710         -7.001038   6.283636
#> Hornet 4 Drive     43.577448 -28.558294
#> Hornet Sportabout  38.455459 -19.007424
#> Valiant            -8.969914 -42.715033
head(HDW(iris[1:2], iris[3:4]))                # Partialling columns 3 and 4 out of columns 1 and 2
#>   HDW.Sepal.Length HDW.Sepal.Width
#> 1       0.21483967       0.2001352
#> 2       0.01483967      -0.2998648
#> 3      -0.13098262      -0.1255786
#> 4      -0.33933805      -0.1741510
#> 5       0.11483967       0.3001352
#> 6       0.41621663       0.6044681
head(HDW(iris[1:2], iris[3:5]))                # Adding the Species factor -> fixed effect
#>   HDW.Sepal.Length HDW.Sepal.Width
#> 1       0.14989286       0.1102684
#> 2      -0.05010714      -0.3897316
#> 3      -0.15951256      -0.1742640
#> 4      -0.44070173      -0.3051992
#> 5       0.04989286       0.2102684
#> 6       0.17930818       0.3391766

head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c + qF(year))) # Partialling out 2 fixed effects
#>   HDW.PCGDP HDW.LIFEEX
#> 1 1578.6211 -1.3980224
#> 2 1412.8849 -1.1838196
#> 3  917.2033 -1.0547978
#> 4  627.8605 -0.8296048
#> 5  168.0458 -0.6683027
#> 6 -234.9535 -0.4708428
head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c + qF(year), variable.wise = TRUE)) # Variable-wise
#>   HDW.PCGDP HDW.LIFEEX
#> 1        NA  -6.706423
#> 2        NA  -6.688440
#> 3        NA  -6.562210
#> 4        NA  -6.472079
#> 5        NA  -6.445378
#> 6        NA  -6.367659
head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c + qF(year) + ODA)) # Adding ODA as a continuous regressor
#>   HDW.PCGDP HDW.LIFEEX
#> 1 -324.3991 -1.1765307
#> 2 -439.5404 -0.9751559
#> 3 -598.9266 -0.7835446
#> 4  100.2175 -0.6186010
#> 5  -70.7664 -0.4966332
#> 6  330.3561 -0.2257800
head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c:qF(decade) + qF(year) + ODA)) # Country-decade and year FE's
#>    HDW.PCGDP  HDW.LIFEEX
#> 1  411.79228 -0.55122290
#> 2  231.95880 -0.36367639
#> 3  -73.18195 -0.20459213
#> 4   43.93176 -0.05394933
#> 5 -136.49858  0.06637048
#> 6 -151.30884  0.24440305

head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c*year))          # Country specific time trends
#>    HDW.PCGDP    HDW.LIFEEX
#> 1  -3.035801 -0.1540994153
#> 2  -7.963841 -0.1544275887
#> 3 -35.533424 -0.1407557620
#> 4 -29.220766 -0.1100839354
#> 5 -38.876368 -0.0614121088
#> 6 -16.317261  0.0002597179
head(HDW(wlddev, PCGDP + LIFEEX ~ iso3c*poly(year, 3))) # Country specific cubic trends
#>    HDW.PCGDP   HDW.LIFEEX
#> 1   8.885334  0.023614035
#> 2  13.685446  0.006724458
#> 3 -10.597857 -0.011405315
#> 4  -6.279492 -0.023928535
#> 5 -22.048660 -0.025998452
#> 6  -8.561088 -0.018768318

# More complex examples
lm(HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~ factor(cyl)*carb + vs + wt:gear + wt:gear:carb))
#> 
#> Call:
#> lm(formula = HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~factor(cyl) * 
#>     carb + vs + wt:gear + wt:gear:carb))
#> 
#> Coefficients:
#> (Intercept)       HDW.hp  
#>  -2.038e-16   -3.265e-02  
#> 
lm(mpg ~ hp + factor(cyl)*carb + vs + wt:gear + wt:gear:carb, data = mtcars)
#> 
#> Call:
#> lm(formula = mpg ~ hp + factor(cyl) * carb + vs + wt:gear + wt:gear:carb, 
#>     data = mtcars)
#> 
#> Coefficients:
#>       (Intercept)                 hp       factor(cyl)6       factor(cyl)8  
#>          42.11872           -0.02366           -3.70912           -3.80071  
#>              carb                 vs  factor(cyl)6:carb  factor(cyl)8:carb  
#>          -1.64558           -0.81529           -0.82919           -1.56964  
#>           wt:gear       carb:wt:gear  
#>          -1.52766            0.26438  
#> 

lm(HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~ factor(cyl)*carb + vs + wt:gear))
#> 
#> Call:
#> lm(formula = HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~factor(cyl) * 
#>     carb + vs + wt:gear))
#> 
#> Coefficients:
#> (Intercept)       HDW.hp  
#>  -2.038e-16   -3.265e-02  
#> 
lm(mpg ~ hp + factor(cyl)*carb + vs + wt:gear, data = mtcars)
#> 
#> Call:
#> lm(formula = mpg ~ hp + factor(cyl) * carb + vs + wt:gear, data = mtcars)
#> 
#> Coefficients:
#>       (Intercept)                 hp       factor(cyl)6       factor(cyl)8  
#>           36.4543            -0.0274            -6.2463            -9.4541  
#>              carb                 vs  factor(cyl)6:carb  factor(cyl)8:carb  
#>            0.2508            -0.3227             0.5897             1.0374  
#>           wt:gear  
#>           -0.8238  
#> 

lm(HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~ cyl*carb + vs + wt:gear))
#> 
#> Call:
#> lm(formula = HDW.mpg ~ HDW.hp, data = HDW(mtcars, ~cyl * carb + 
#>     vs + wt:gear))
#> 
#> Coefficients:
#> (Intercept)       HDW.hp  
#>  -3.898e-17   -2.151e-02  
#> 
lm(mpg ~ hp + cyl*carb + vs + wt:gear, data = mtcars)
#> 
#> Call:
#> lm(formula = mpg ~ hp + cyl * carb + vs + wt:gear, data = mtcars)
#> 
#> Coefficients:
#> (Intercept)           hp          cyl         carb           vs     cyl:carb  
#>    48.42617     -0.02151     -2.80751     -1.72418     -1.03254      0.36051  
#>     wt:gear  
#>    -0.81296  
#> 

lm(HDW.mpg ~ HDW.hp, data = HDW(mtcars, mpg + hp ~ cyl*carb + factor(cyl)*poly(drat,2)))
#> 
#> Call:
#> lm(formula = HDW.mpg ~ HDW.hp, data = HDW(mtcars, mpg + hp ~ 
#>     cyl * carb + factor(cyl) * poly(drat, 2)))
#> 
#> Coefficients:
#> (Intercept)       HDW.hp  
#>  -9.917e-16   -4.725e-02  
#> 
lm(mpg ~ hp + cyl*carb + factor(cyl)*poly(drat,2), data = mtcars)
#> 
#> Call:
#> lm(formula = mpg ~ hp + cyl * carb + factor(cyl) * poly(drat, 
#>     2), data = mtcars)
#> 
#> Coefficients:
#>                 (Intercept)                           hp  
#>                    29.87184                     -0.06227  
#>                         cyl                         carb  
#>                    -0.32237                     -2.19559  
#>                factor(cyl)6                 factor(cyl)8  
#>                    -1.60109                           NA  
#>              poly(drat, 2)1               poly(drat, 2)2  
#>                    27.84148                     -8.41291  
#>                    cyl:carb  factor(cyl)6:poly(drat, 2)1  
#>                     0.35323                    -49.59226  
#> factor(cyl)8:poly(drat, 2)1  factor(cyl)6:poly(drat, 2)2  
#>                   -18.35266                    -18.70972  
#> factor(cyl)8:poly(drat, 2)2  
#>                    -0.56842  
#>