Multiplier analysis refers to the effect, and the timing of the effect, of a change in one variable on the outcome of another variable. The simplest form of multiplier analysis is based on a finite distributed lag model
yt = a + во Xt + PlXt-1 + в2 Xt-2 +——– + в Xt-q + et
The estimated coefficients from this model can be used to produce impact, delay and interim multipliers. The impact multiplier is the impact of a one unit change in xt on the mean of yt. Since X and y are in the same time period the effect is contemporaneous and therefore equal to the initial impact of the change. The s-period delay multiplier is
is the effect of a change in x s-periods in the past on the average value of the dependent variable in the current period. If xt is increased by 1 unit and then maintained at its new level in subsequent periods (t + 1), (t + 2),…, then one can compute the interim multiplier. An interim multiplier simply adds the immediate effect (impact multiplier), в0, to subsequent delay multipliers to measure the cumulative effect. So in period t + 1 the interim effect is в0 + вь In period t + 2, it will be в0 + ві + в2, and so on. The total multiplier is the final effect on y of the sustained increase after q or more periods have elapsed; it is given by ^S=0 es.
The ARDL model adds lagged values of the dependent variable to the AR model,
yt = S + 0iyt-i +—- + Opyt-p + ^xt + 5ixt-i +—- + 5q xt-q + vt (9.21)
and this makes the multiplier analysis a little harder. Basically, this needs to be transformed into an infinite distributed lag model using the properties of the lag operator, L. That is, Lxt = xt-i.
This puts the model into the familiar AR form and the usual definitions of the multipliers can be applied. This is discussed in detail in POE4 and will not be replicated in any detail here.
For the ARDL(1,1) model used to describe Okun’s law we have
Ant = 5 + diAut-i + 5ogt + £1 gt_ i + vt
Written with the lag operator, L
(1 — 9iL) Aut = 5 + (5o + 5iL) gt + vt
Ant = (1 — 6iL)-15 + (1 — 6i L)-1 (5o + 5i L) gt + (1 — 0iL)-ivt
Aut = a + во gt + вт-i + ^2gt_2 + e3gt_3 +———— + et
= a + (eo + eiL + e2L<2 + e3L’3 + • • •) gt + et
This is just an infinite distributed lag model. The coefficients for the multipliers involve the в coefficients, which must be solved for in terms of the estimated parameters of the ARDL. The solutions given in POE4 are
ei =5i + e0$i
вз =ві_i^i for j > 2
The gretl code to accomplish this is simple to construct. In terms of the model of Okun’s Law,
1 open "@gretldirdatapoeokun. gdt"
2 diff u
3 ols d_u(0 to -1) g(0 to -1) const
4 scalar b0 = $coeff(g)
5 scalar b1 = $coeff(d_u_1)*b0+$coeff(g_1)
6 scalar b2 = b1*$coeff(d_u_1)
7 scalar b3 = b2*$coeff(d_u_1)
This can be automated by using a loop to construct the multipliers. Once this is done, it is simple to graph the result up to an arbitrary number of periods.
The script is: 1
7 if І=1
9 elif i=2
15 printf "nThe impact and delay multipliers are n %10.5fn", mult
16 gnuplot 2 1 —matrix=mult —output=display —with-lines —suppress-fitted
Although it took a few lines of code, the results (Figure 9.19 below) look great and the code can easily be reused for other models. It assumes that you have already estimated the ARDL(1,1) for Okun data as done in the previous script. The first thing to do is to decide how many multipliers to compute. I chose 8 and initialized a matrix of zeros that is 8 x 2. We will put lags in the first column and the corresponding multiplier in the second.
The loop begins in line 3 and i will start at 1 and end at 8, with increments of 1. The first two multipliers are computed manually. Then a series of if statements follows. Since there are three forms of the multiplier in equations (9.22) – (9.24), there are 3 if statements. When the index is equal 1, the impact multiplier is placed in m. When the index is equal to 2, the period one delay is placed in m. The last condition fills in m for any i > 2.
Next, we want to be able to plot the multipliers against lag. This is done from a matrix using the —matrix option to gnuplot. Also, using the output=display option sends the plot to the screen which allows for subsequent editing via gretl’s gnuplot interface.
The other option is to convert the matrix to data series and use the regular gretl GUI to make the plots. This requires opening an empty dataset and setting the observations to equal 8. This is done using the nulldata command. The —preserve option is required because without it the matrix containing the multipliers would be cleared from memory. This option preserves the contents of all existing matrices and scalars. The lags are read out of the first column and the multipliers from the second.
1 nulldata 8 —preserve
2 series m = mult[,2]
3 series lag = mult[,1]
4 setinfo m – d "Multipliers" – n "Multiplier"
5 gnuplot m index —with-lines —output=display —suppress-fitted
The edited outcome appears in Figure 9.19 below. The figure shows that an increase in GDP growth leads to an initial reduction in the unemployment rate of about 0.18; the effect diminishes over time and lasts about six or seven quarters.