ECON0022
Econometrics for Macro and Finance
Problem Set III
Dennis Kristensen
October 22, 2021
1 Practice Problems
1. Consider iterative forecasts over the horizon T + 1; :::; T + h using an AR(1)
model, Yt = + Yt1 + "t, given data up to time T . We ignore parameter
uncertainty and treat the parameters and as known such that the iterative
forecast takes the form
YT+kjT = + YT+k1jT ; k = 1; 2; :::; h;
where YT jT = YT . We wish to analyze the behaviour of the forecasts in the
stationary (jj < 1) and non-stationary ( = 1) case.
(a) Show that the …nal h-step ahead forecast, YT+hjT , can be expressed as
YT+hjT = hYT +
n
1 + + 2 + :::+ h1
o
:
[hint: Use backward recursion starting at time T +h and stopping at time
T ]
(b) Use the above expression to argue that for long-horizon forecasts (h very
large):
If jj < 1 : YT+hjT

1 = Y ;
If = 1 : YT+hjT ! 1;
where YT+hjT ! 1 should be interpreted as YT+hjT exploding (increas-
ing in absolute value without any bound) as h!1. Interpret the above
results.
(c) Let eT+h be h-step ahead forecast error de…ned as:
eT+h = YT+h YT+hjT :
Expresss eT+h as a function of "T+1; :::; "T+h. [Hint: Use backward recur-
sion to show that YT+h = YT+hjT +
Ph1
i=0
i"T+hi]
1
(d) Show that
E [eT+h] = 0; Var (eT+h) =
2
"
n
1 + 2 + ::::2(h1)
o
:
What is the h-step MSFE for the iterative procedure using an AR(1)
model? What happens to the MSFE as h grows large?
(e) Explain how you could use the results in question (d) to construct a 95%
forecast interval for your h-period ahead point forecast.
2. Suppose we have obtained the following estimates of an ADL model for annual
UK unemployment (unemp) using in‡ation (infl) as additional predictor,
udnempt = 1:304 + :647unempt1 + :184inflt1; ^" = :883: (1)
(:490) (:084) ((:041)
(a) Suppose that unemp2012 = 7:8 and infl2012 = 1:3. What is your forecast
of unemployment in 2013?
(b) Suppose the errors in the ADL model are normally distributed. Compute
a 95% and 67% forecast interval.
(c) Can you use the above ADL model to forecast employment for 2014?
Explain.
(d) You decide to build a model for in‡ation in addition to the above model
for unemployment. You …nd that an AR(1) model provides an adequate
description of in‡ation,
icnflt = 1:277 + :665inflt1; ^" = :653: (2)
(:558) (:107)
Based on (1)-(2), is there empirical support for inflt being stationary and
mixing? What about unempt?
(e) Use eqs. (1)-(2) to compute a forecast of in‡ation and unemployment for
2014.
Note: the joint model for infl and unemp as given in eqs. (1)-(2) is
a particular example of so-called Vector Autoregressive (VAR) models.
These allow for joint modelling of multiple economic time series.