Date        V1              V2
1 1996-01-04 0.04383562 days 0.1203920
2 1996-01-04 0.12054795 days 0.1094760
..............
3 1996-02-01 0.04383562 days 0.1081815
4 1996-02-01 0.12054795 days 0.1092450
..............
5 1996-03-01 0.04109589 days 0.1553875
6 1996-03-01 0.13687215 days 0.1469690

IV<-data %>% group_by(Date) %>% approx(V1,V2,xout=0.08)

Error in approx(., V1, V2, xout = 0.08) :
  invalid interpolation method
In addition: Warning message:
In if (is.na(method)) stop("invalid interpolation method") :
  the condition has length > 1 and only the first element will be used

Results<-unsplit(lapply(split(data,data$Date),function(x){m<-lm(V2~V1,x)
                                                       cbind(x,predict(m,0.08))}),data$Date)

Error in model.frame.default(formula = x[, 3] ~ x[, 2], data = x, drop.unused.levels = TRUE) :
  invalid type (list) for variable 'x[, 3]'

IV<-data %>% group_by(Date) %>% predict(lm(V2~V1,data=data,0.08)

Error in UseMethod("predict") :
  no applicable method for 'predict' applied to an object of class "c('grouped_df', 'tbl_df', 'tbl', 'data.frame')"

library(data.table)
#created as df instead of dt for use in dplyr solution later
df <- data.frame(grp=sample(letters[1:2],10,T),
             v1=rnorm(10),
             v2=rnorm(10))

dt <- data.table(df)

dt[, approx(v1,v2,xout=.08), by=grp]

#output
   grp    x          y
1:   b 0.08 -0.5112237
2:   a 0.08 -1.4228923

library(dplyr)

df %>%
    group_by(grp) %>%
    summarise(out=list(approx(v1,v2,xout=.08))) %>%
    ungroup() %>%
    mutate(x=purrr::map_dbl(out,'x'),
           y=purrr::map_dbl(out,'y')) %>%
    select(-out)

#output
# A tibble: 2 × 3
     grp     x          y
  <fctr> <dbl>      <dbl>
1      a  0.08 -1.4228923
2      b  0.08 -0.5112237