Calculate a data frame of total effects, resulting from a pulse experiment
(i.e., an exogenous and temporary change in a single variable in time t=0) or
a press experiment (i.e., an exogenous and permanent change in a single variable
starting in time t=0 and continuing for n_lags times), representing the
estimated effect of a change in any variable on every other variable and any time-lag
from 0 (simultaneous effects) to a user-specified maximum lag.
Usage
total_effect(object, n_lags = 4, type = c("pulse", "press"))Arguments
- object
Output from
dsem- n_lags
Number of lags over which to calculate total effects
- type
Whether a pulse or press experiment is intended. A pulse experiment answers the question:
What happens if a variable is changed for only a single time-interval?" A press experiment answers the question:What happens if a variable is permanently changed starting in a given time-interval?
Value
A data frame listing the time-lag (lag), variable that is undergoing some exogenous change (from), and the variable being impacted (to), along with the total effect (total_effect) including direct and indirect pathways, and the partial "direct" effect (direct_effect)
Details
Total effects are taken from the Leontief matrix \(\mathbf{(I-P)^{-1}}\),
where \(\mathbf{P}\) is the path matrix across variables and times.
\(\mathbf{(I-P)}^{-1} \mathbf{\delta} \)
calculates the effect of a perturbation represented by vector \(\mathbf{\delta}\)
with length \(n_{\mathrm{lags}} \times n_{\mathrm{J}}\) where \(n_{\mathrm{J}}\) is the number of variables.
\(\mathbf{(I-P)}^{-1} \mathbf{\delta} \) calculates the total effect of
a given variable (from)
upon any other variable (to) either in the same time (\(t=0\)), or subsequent times
(\(t \geq 1\)), where \(\mathbf{\delta} = \mathbf{i}_{\mathrm{T}} \otimes \mathbf{i}_{\mathrm{J}}\),
where \(\mathbf{i}_{\mathrm{J}}\) is one for the from variable and zero otherwise.
For a pulse experiment, \(\mathbf{i}_{\mathrm{T}}\) is one at \(t=0\) and zero for other times.
For a press experiment, \(\mathbf{i}_{\mathrm{T}}\) is one for all times.
We compute and list the total effect at each time from time t=0
to t=n_lags-1. For press experiments, this includes transient values as the the total effect
approaches its asymptotic value (if this exists) as \(t\) approaches infinity.
If the analyst wants an asymptotic effect from a press experiment, we recommend
using a high lag (e.g., n_lags = 100) and then confirming that it has
reached it's asymptote (i.e., the total effect is almost identical for the last
and next-to-last lag), and then reporting the value for that last lag.
Examples
### EXAMPLE 1
# Define linear model with slope of 0.5
sem = "
# from, to, lag, name, starting_value
x -> y, 0, slope, 0.5
"
# Build DSEM with specified value for path coefficients
mod = dsem(
sem = sem,
tsdata = ts(data.frame(x=rep(0,20),y=rep(0,20))),
control = dsem_control( run_model = FALSE )
)
#> List of estimated fixed and random effects:
#> Coefficient_name Number_of_coefficients Type
#> 1 beta_z 3 Fixed
#> 2 mu_j 2 Random
# Show that total effect of X on Y from pulse experiment is 0.5 but does not propagate over time
pulse = total_effect(mod, n_lags = 2, type = "pulse")
subset( pulse, from=="x" & to=="y")
#> lag to from total_effect direct_effect
#> 3 0 y x 0.5 0.5
#> 4 1 y x 0.0 0.0
### EXAMPLE 2
# Define linear model with slope of 0.5 and autocorrelated response
sem = "
x -> y, 0, slope, 0.5
y -> y, 1, ar_y, 0.8
"
mod = dsem(
sem = sem,
tsdata = ts(data.frame(x=rep(0,20),y=rep(0,20))),
control = dsem_control( run_model = FALSE )
)
#> List of estimated fixed and random effects:
#> Coefficient_name Number_of_coefficients Type
#> 1 beta_z 4 Fixed
#> 2 mu_j 2 Random
# Show that total effect of X on Y from pulse experiment is 0.5 with decay of 0.8 for each time
pulse = total_effect(mod, n_lags = 4, type = "pulse")
subset( pulse, from=="x" & to=="y")
#> lag to from total_effect direct_effect
#> 5 0 y x 0.500 0.5
#> 6 1 y x 0.400 0.0
#> 7 2 y x 0.320 0.0
#> 8 3 y x 0.256 0.0
# Show that total effect of X on Y from press experiment asymptotes at 2.5
press = total_effect(mod, n_lags = 50, type = "press")
subset( press, from=="x" & to=="y")
#> lag to from total_effect direct_effect
#> 51 0 y x 0.500000 0.5
#> 52 1 y x 0.900000 0.5
#> 53 2 y x 1.220000 0.5
#> 54 3 y x 1.476000 0.5
#> 55 4 y x 1.680800 0.5
#> 56 5 y x 1.844640 0.5
#> 57 6 y x 1.975712 0.5
#> 58 7 y x 2.080570 0.5
#> 59 8 y x 2.164456 0.5
#> 60 9 y x 2.231565 0.5
#> 61 10 y x 2.285252 0.5
#> 62 11 y x 2.328201 0.5
#> 63 12 y x 2.362561 0.5
#> 64 13 y x 2.390049 0.5
#> 65 14 y x 2.412039 0.5
#> 66 15 y x 2.429631 0.5
#> 67 16 y x 2.443705 0.5
#> 68 17 y x 2.454964 0.5
#> 69 18 y x 2.463971 0.5
#> 70 19 y x 2.471177 0.5
#> 71 20 y x 2.476942 0.5
#> 72 21 y x 2.481553 0.5
#> 73 22 y x 2.485243 0.5
#> 74 23 y x 2.488194 0.5
#> 75 24 y x 2.490555 0.5
#> 76 25 y x 2.492444 0.5
#> 77 26 y x 2.493955 0.5
#> 78 27 y x 2.495164 0.5
#> 79 28 y x 2.496131 0.5
#> 80 29 y x 2.496905 0.5
#> 81 30 y x 2.497524 0.5
#> 82 31 y x 2.498019 0.5
#> 83 32 y x 2.498415 0.5
#> 84 33 y x 2.498732 0.5
#> 85 34 y x 2.498986 0.5
#> 86 35 y x 2.499189 0.5
#> 87 36 y x 2.499351 0.5
#> 88 37 y x 2.499481 0.5
#> 89 38 y x 2.499585 0.5
#> 90 39 y x 2.499668 0.5
#> 91 40 y x 2.499734 0.5
#> 92 41 y x 2.499787 0.5
#> 93 42 y x 2.499830 0.5
#> 94 43 y x 2.499864 0.5
#> 95 44 y x 2.499891 0.5
#> 96 45 y x 2.499913 0.5
#> 97 46 y x 2.499930 0.5
#> 98 47 y x 2.499944 0.5
#> 99 48 y x 2.499955 0.5
#> 100 49 y x 2.499964 0.5