*
* Sample graphs from pages 4-15
*
all 20
*
* Dynamic multipliers from page 4
*
set r1 = .8**(t-1)
set r2 = (-.8)**(t-1)
set r3 = 1.1**(t-1)
set r4 = (-1.1)**(t-1)
*
* This sets the horizontal axis label to the symbol font. The "phi" is
* represented by an f character in that font. The spgraph allows the
* graphs to be placed in a 2x2 "matrix" of graphs. By default, the
* cells are filled going down columns, which is why the "r3" graph is
* done before r2.
*
grparm(font=symbol) hlabel 36
spgraph(hfields=2,vfields=2,header='Figure 1.1. Dynamic Multipliers For First-Order Difference Equations')
graph(number=0,style=bar,hlabel='f=0.8',min=-1.0,max=1.0)
# r1
graph(number=0,style=bar,hlabel='f=1.1',min=-7.0,max=7.0)
# r3
graph(number=0,style=bar,hlabel='f=-0.8',min=-1.0,max=1.0)
# r2
graph(number=0,style=bar,hlabel='f=-1.1',min=-7.0,max=7.0)
# r4
spgraph(done)
*
* Impulse response graphs from page 5. In the first case,
* the w is a spike when t is 6. This is done by using the
* == operator. t==6 is 1 when t is 6 and 0 otherwise. The
* set instruction uses the "first=0" option to give the
* series a computable value when t is 1. If you don't put
* this on, y will be all missing values because y(1)=phi*y(0)+w(1),
* but y(0) doesn't exist, so y(1) is missing; which forces
* y(2) to be missing, etc. You have to be careful when
* generated recursively defined data.
*
compute phi=.8
set w = (t==6)
set(first=0.0) y = phi*y{1}+w
grparm hlabel 24
spgraph(vfields=2,header='Figure 1.2 Paths of Input (W) and Output (Y) Variables')
graph(number=0,style=bar,hlabel='W',noticks)
# w
graph(number=0,style=bar,hlabel='Y',noticks)
# y
spgraph(done)
*
* In this example, the w shocks are 1 from period 6 on. The >= operator is used
* to create this dummy variable.
*
set w = (t>=6)
set(first=0.0) y = phi*y{1}+w
grparm hlabel 24
spgraph(vfields=2,header='Figure 1.3 Paths of Input (W) and Output (Y) Variables for Long-Run Effect')
graph(number=0,style=bar,hlabel='W',noticks,max=6.0)
# w
graph(number=0,style=bar,hlabel='Y',noticks)
# y
spgraph(done)
*
* Brute force calculations of the second order difference equations
* This uses the polynomial root finder, which returns complex numbers.
* The coefficients are put in from the 0 power up, so for the difference
* equation y(t)=phi1*y(t-1)+phi2*y(t-2)+w, the polynomial is x**2-phi1*x-phi2,
* which is input as coefficients -phi2,-phi1,1.0. The %real and %imag functions
* return the real and complex parts of a complex number, %cabs gets the absolute
* value, and %arg the argument (the "theta").
*
* Note that the formulas in the text are based at j=0, where RATS series start
* with T=1, which is why (T-1) is used everywhere in the formulas.
*
compute lambda=%polycxroots(||-.2,-.6,1.0||)
compute l1=%real(lambda(1))
compute l2=%real(lambda(2))
compute c1=l1/(l1-l2)
compute c2=l2/(l2-l1)
set r1 = c1*l1**(t-1)+c2*l2**(t-1)
*
compute lambda=%polycxroots(||.8,-.5,1.0||)
compute r=%cabs(lambda(1))
compute theta=%arg(lambda(1))
compute c1i = lambda(1)/(lambda(1)-lambda(2))
compute c2i = lambda(2)/(lambda(2)-lambda(1))
compute alpha=%real(c1i),beta=%imag(c1i)
*
set r2 = 2*alpha*r**(t-1)*cos(theta*(t-1))-2*beta*r**(t-1)*sin(theta*(t-1))
*
grparm(font=symbol) hlabel 24
spgraph(vfields=2,header='Figure 1.4 Dynamic Multiplier for Second Order Difference Equations')
graph(number=0,style=bar,hlabel='f1=0.6,f2=0.2')
# r1
graph(number=0,style=bar,hlabel='f1=0.5,f2=-0.8')
# r2
spgraph(done)
*
* Same thing, but using RATS instructions. The IMPULSE instruction is
* designed precisely to compute these impulse response functions. All
* it needs is an equation which describes the difference equation. This
* is done with the EQUATION instruction. The syntax of EQUATION as used
* here is
*
* EQUATION(COEFFS=vector of coefficients,VARIANCE=residual variance) equation_name  dependent_variable
* # list of explanatory variables
*
* The explanatory variables here are the first and second lags of the dependent variable. We give
* a variance of 1 because the IMPULSE instruction, by default, gives the equation a one standard
* deviation shock, which we want to be 1.
*
equation(coeffs=||.6,.2||,variance=1.0) diffeq2 y
# y{1 2}
impulse(noprint) 1 20
# diffeq2 r1
equation(coeffs=||.5,-.8||,variance=1.0) diffeq2 y
# y{1 2}
impulse(noprint) 1 20
# diffeq2 r2
*
* The graph is the same as above
*
grparm(font=symbol) hlabel 24
spgraph(vfields=2,header='Figure 1.4 Dynamic Multiplier for Second Order Difference Equations')
graph(number=0,style=bar,hlabel='f1=0.6,f2=0.2')
# r1
graph(number=0,style=bar,hlabel='f1=0.5,f2=-0.8')
# r2
spgraph(done)