Single residual variable output from Multivariate Linear Regression Model

Hi, I'm currently learning a multivariate class.
I know that the traditional multiple linear regression model is the univariate model with a single response variable. Thus, we would also have a correspondent residual.

Now here comes the Multivariate Linear Regression when we want to model with multiple response variables. Is there any method that I could fit the Multivariate Linear Regression, and obtain univariate residuals?

I would like to compare the residuals from the Univariate and Multivariate model.
Thanks!

They should be the same, at least for ordinary least squares:

mv_mod <- lm(cbind(mpg, wt) ~ ., data = mtcars)
mv_mod
#> 
#> Call:
#> lm(formula = cbind(mpg, wt) ~ ., data = mtcars)
#> 
#> Coefficients:
#>              mpg        wt       
#> (Intercept)  15.570618  -0.879401
#> cyl           0.119824  -0.062246
#> disp         -0.013609   0.007252
#> hp           -0.011218  -0.002763
#> drat          1.327261  -0.145385
#> qsec          0.094279   0.195613
#> vs            0.667704  -0.094189
#> am            2.900740  -0.102418
#> gear          1.186497  -0.142945
#> carb         -1.329123   0.304068

resid(mv_mod)
#>                            mpg          wt
#> Mazda RX4           -0.9365932 -0.17842754
#> Mazda RX4 Wag       -0.9893895 -0.03297082
#> Datsun 710          -4.5867058  0.30577894
#> Hornet 4 Drive       1.0366073 -0.23524638
#> Hornet Sportabout    2.3462788 -0.36059306
#> Valiant             -2.4173780  0.03615826
#> Duster 360           1.4214066 -0.40579797
#> Merc 240D            1.5033583  0.10783233
#> Merc 230            -0.3854132 -0.33205269
#> Merc 280            -0.4543157  0.25712184
#> Merc 280C           -1.9108831  0.13975403
#> Merc 450SE           0.3560112  0.50381523
#> Merc 450SL           1.2371554  0.12469263
#> Merc 450SLC         -0.9005562  0.09644743
#> Cadillac Fleetwood  -1.2332895 -0.10785763
#> Lincoln Continental -1.3622341  0.22226885
#> Chrysler Imperial    2.5663091  0.44143401
#> Fiat 128             3.9253258  0.18888586
#> Honda Civic          2.0179626 -0.40769248
#> Toyota Corolla       5.0843265 -0.18752048
#> Toyota Corona       -1.5228350 -0.16694957
#> Dodge Challenger    -1.1739576 -0.07242897
#> AMC Javelin         -2.2226485 -0.08331296
#> Camaro Z28          -0.3643144  0.09643691
#> Pontiac Firebird     3.4806989 -0.26172229
#> Fiat X1-9           -1.1168526  0.03320965
#> Porsche 914-2       -1.0211667  0.23368553
#> Lotus Europa         3.4721306 -0.19067167
#> Ford Pantera L      -3.4758598  0.10922903
#> Ferrari Dino         0.2223466 -0.05818496
#> Maserati Bora        2.0498031 -0.26671344
#> Volvo 142E          -4.6453275  0.45139236

mpg_mod <- lm(mpg ~ . - wt, data = mtcars)
wt_mod  <- lm(wt ~ . - mpg, data = mtcars)

all.equal(
  resid(mv_mod)[, "mpg"],
  resid(mpg_mod)
)
#> [1] TRUE
all.equal(
  resid(mv_mod)[, "wt"],
  resid(wt_mod)
)
#> [1] TRUE

Created on 2020-02-14 by the reprex package (v0.3.0)