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15.3 Dynamic Multipliers and Cumulative Dynamic Multipliers

The following terminology regarding the coefficients in the distributed lag model (15.2) is useful for upcoming applications:

  • The dynamic causal effect is also called the dynamic multiplier. \(\beta_{h+1}\) in (15.2) is the \(h\)-period dynamic multiplier.

  • The contemporaneous effect of \(X\) on \(Y\), \(\beta_1\), is termed the impact effect.

  • The \(h\)-period cumulative dynamic multiplier of a unit change in \(X\) and \(Y\) is defined as the cumulative sum of the dynamic multipliers. In particular, \(\beta_1\) is the zero-period cumulative dynamic multiplier, \(\beta_1 + \beta_2\) is the one-period cumulative dynamic multiplier and so forth.

    The cumulative dynamic multipliers of the distributed lag model (15.2) are the coefficients \(\delta_1,\delta_2,\dots,\delta_r,\delta_{r+1}\) in the modified regression \[\begin{align} Y_t =& \, \delta_0 + \delta_1 \Delta X_t + \delta_2 \Delta X_{t-1} + \dots + \delta_r \Delta X_{t-r+1} + \delta_{r+1} X_{t-r} + u_t \tag{15.3} \end{align}\] and thus can be directly estimated using OLS which makes it convenient to compute their HAC standard errors. \(\delta_{r+1}\) is called the long-run cumulative dynamic multiplier.

It is straightforward to compute the cumulative dynamic multipliers for (15.1), the estimated distributed lag regression of changes in orange juice concentrate prices on freezing degree days, using the corresponding model object orange_DLM and the function cumsum().

# compute cumulative multipliers
cum_mult <-cumsum(orange_DLM$coefficients[-1])

# rename entries
names(cum_mult) <- paste(0:6, sep = "-", "period CDM")

cum_mult
## 0-period CDM 1-period CDM 2-period CDM 3-period CDM 4-period CDM 
##    0.4714329    0.6164542    0.6748177    0.7489835    0.7852874 
## 5-period CDM 6-period CDM 
##    0.8340436    0.8842895

Translating the distributed lag model with six lags of \(FDD\) to (15.3), we see that the OLS coefficient estimates in this model coincide with the multipliers stored in cum_mult.

# estimate cumulative dynamic multipliers using the modified regression
cum_mult_reg <-dynlm(FOJC_pctc ~ d(FDD) + d(L(FDD,1:5)) + L(FDD,6))
coef(cum_mult_reg)[-1]
##          d(FDD) d(L(FDD, 1:5))1 d(L(FDD, 1:5))2 d(L(FDD, 1:5))3 
##       0.4714329       0.6164542       0.6748177       0.7489835 
## d(L(FDD, 1:5))4 d(L(FDD, 1:5))5       L(FDD, 6) 
##       0.7852874       0.8340436       0.8842895

As noted above, using a model specification as in (15.3) allows to easily obtain standard errors for the estimated dynamic cumulative multipliers.

# obtain coefficient summary that reports HAC standard errors
coeftest(cum_mult_reg, vcov. = vcovHAC)
## 
## t test of coefficients:
## 
##                 Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)     -0.69296    0.23668 -2.9278 0.0035431 ** 
## d(FDD)           0.47143    0.13583  3.4709 0.0005562 ***
## d(L(FDD, 1:5))1  0.61645    0.13145  4.6896 3.395e-06 ***
## d(L(FDD, 1:5))2  0.67482    0.16009  4.2151 2.882e-05 ***
## d(L(FDD, 1:5))3  0.74898    0.17263  4.3387 1.682e-05 ***
## d(L(FDD, 1:5))4  0.78529    0.17351  4.5260 7.255e-06 ***
## d(L(FDD, 1:5))5  0.83404    0.18236  4.5737 5.827e-06 ***
## L(FDD, 6)        0.88429    0.19303  4.5810 5.634e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1