Autoregressive models are widely used throughout econometrics when dealing with stochastic

processes whose future values are related to their past values plus a random disturbance term. In

general, the current value of an AR(????) process is given by a linear combination of its previous ???? values

(plus a random disturbance term)

???????? = ????0 + ????1?????????1 + ????2?????????2 + ? + ????????????????????? + ???????? (1)

In the simple case of an AR(0) process, the value of ???????? is not dependent upon the history of the process

in any way, and is determined solely by the random disturbance term. . For an AR(1) process, if ????1 =

1???????????? ????0 = 0 then the process is known as a random walk. Normally we assume that the disturbance

term is randomly drawn from some distribution which is constant throughout time. However, in this

assignment we will examine a first order autoregressive model with specific disturbances that are given

in the question. It can be viewed as the simplest form of a simulation.

Use a spreadsheet program (e.g., Excel) to plot the following processes. You may write the discussions

(legibly) by hand or use a word processor, but the text and the plots should be integrated so that the

assignment is a well thought through document that communicates the ideas in a clear and succinct

manner.

Consider the AR(1) process

???????? = ????0 + ????1?????????1 + ???????? (2)

In all cases below assume y0 = 0 unless otherwise specified. Suppose there is an isolated shock of

magnitude 3 in period 7. That is to say that ????7 = 3, and ???????? = 0 for all ???? ? 7.

a) On each of three separate graphs, plot ???????? as a function of ????, for ???? = 0, … , 15, for the following five cases

(that is, 3 plots with 5 lines per plot):

i) ????0 = 0, ????1 = {0 , 0.5 , 0.75 , 1 , 1.1}

ii) ????0 = 0.5, ????1 = {0 , 0.5 , 0.75 , 1 , 1.1}

iii) ????0 = {0 , 0.5 , 0.75 , 1 , 1.1}, ????1 = 0.5

Where there is a set of values for one of the parameters, this implies that you should plot each value in

turn on a single graph. In light of these plots, discuss the concept of stationary and non-stationary time

series processes. What are the respective roles of ????0 and ????1?

b) Building on the plots from part (a), make three new (but similar) plots with different error terms. Set

????7 = 3 and ????12 = ?1, with ???????? = 0 for all other ????. Discuss the differences in the evolution of ???????? both

among these new plots and in comparison to the plots form (a). What can be said about the long term

trends of these processes?

c) For ????0 = 0 and ????1 = 1, show that

???????? = ????0 +?????????

????

????=1

(3)

BONUS: Equation (3) can be generalized to arbitrary values of ????0 and ????1. Derive this equation.

d) Plot ???????? as a function of ????, with ????0 = 0, ????1 = 1.5, with ????1 = 1 and ???????? = 0 for all ???? ? 1. What would

happen if this system never experienced a shock? What would happen if ????0 = ? 1

3

instead of 0? Is this

result dependant on the values of the disturbance terms?

e) Redo part (a), but change the initial condition. Assume that y0 = 3 instead of y0=0.

f) Discuss the concept of a unit-root with respect to parts a) thru e).

Aside: In the above we have restricted all of the examples to cases where the coefficients are positive.

You may restrict your discussion to this subset of cases. Allowing a negative slope coefficient is a bit

trickier than focussing on positive possibilities, although there is no theoretical reason to restrict the

coefficients.