Temporal Terms (Lag/Lead)

When working with time series data it is common to want to access past or future values of your predictors. These are called lagged (past) or lead (future) variables.

StatsModels supports basic lead and lag functionality:

• lag(x, n) accesses data for variable x from n rows (time steps) ago.
• lead(x, n) accesses data for variable x from n rows (time steps) ahead.

In both cases, n can be omitted, and it defaults to 1 row. missing is used for any entries that are lagged or lead out of the table.

Note that this is a purely structural lead/lag term: it is unaware of any time index of the data. It is up to the user to ensure the data is sorted, and following a regular time interval, which may require inserting additional rows containing missings to fill in gaps in irregular data.

Below is a simple example:

julia> using StatsModels, DataFrames

julia> df = DataFrame(y=1:5, x=2:2:10)
5×2 DataFrame
 Row │ y      x
     │ Int64  Int64
─────┼──────────────
   1 │     1      2
   2 │     2      4
   3 │     3      6
   4 │     4      8
   5 │     5     10

julia> f = @formula(y ~ x + lag(x, 2) + lead(x, 2))
FormulaTerm
Response:
  y(unknown)
Predictors:
  x(unknown)
  (x)->lag(x, 2)
  (x)->lead(x, 2)

julia> f = apply_schema(f, schema(f, df))
FormulaTerm
Response:
  y(continuous)
Predictors:
  x(continuous)
  lag(x, 2)
  lead(x, 2)

julia> modelmatrix(f, df)
5×3 reshape(::Matrix{Union{Missing, Int64}}, 5, 3) with eltype Union{Missing, Int64}:
  2   missing   6
  4   missing   8
  6  2         10
  8  4           missing
 10  6           missing

Programmatic construction of lead and lag terms

StatsModels.jl provides methods for lead and lag that allow LeadLagTerms to be constructed programmatically (at run time). See the section on Constructing a formula programmatically for more information. For a short example, you can produce the same formula as above without the @formula macro like this:

julia> y, x = term(:y), term(:x);

julia> f2 = y ~ x + lag(x, 2) + lead(x, 2)
FormulaTerm
Response:
  y(unknown)
Predictors:
  x(unknown)
  lag(x, 2)
  lead(x, 2)

julia> f2 = apply_schema(f2, schema(f2, df))
FormulaTerm
Response:
  y(continuous)
Predictors:
  x(continuous)
  lag(x, 2)
  lead(x, 2)

julia> modelmatrix(f2, df)
5×3 reshape(::Matrix{Union{Missing, Int64}}, 5, 3) with eltype Union{Missing, Int64}:
  2   missing   6
  4   missing   8
  6  2         10
  8  4           missing
 10  6           missing