Forecasting I
(some general forecasting issues)
“There are two kind of forecasters: those who don´t know and those who don´t know they don´t know”
John Kenneth Galbraith (1993)
Gloria González-Rivera
University of California, Riverside
and
Jesús Gonzalo U. Carlos III de Madrid
Spring 2002
Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but
please if you improve this material send us your new version.
Forecasting in Action
Forecasts are made to guide decisions in a variety of fields.
•
Operations planning and Control: Firms use forecasts to decide what to
produce, when to produce and where to produce.
•
Marketing: Pricing decisions, distribution path decisions, and advertising
expenditure decisions all rely heavily on forecasts of responses of sales to
different marketing schemes.
•
Economics: The forecast of the major economic variables, such as GDP,
unemployment, consumption, investment, the price level, and interest rates are
used for governments to guide monetary and fiscal policy. Private firms use
them for strategic planning, because economy-wide economic fluctuations
typically have industry-level and firm-level effects.
•
Financial speculation: Speculators in asset markets have an interest in
forecasting asset returns (stock returns, interest rates, exchange rates, ...). Such
forecasts are made routinely. Are these forecasts successful???
Forecasting in Action (cont)
•
Financial risk management: Volatility forecasts are crucial for evaluating
and insuring risks associated with asset portfolios. Volatility forecasts are also
crucial for firms and investors who need to price assets such options and other
derivatives.
•
Capacity planning: Capacity planning decisions rely heavily on a variety of
forecasts related both to product demand and supply.
•
Business and government planning: Business and governments of all sorts
must constantly plan and justify their expenditures. A major component of the
budgeting process is the revenue forecast.
•
Demography: Population forecasts are crucial for planning government
expenditure on health care, infrastructure, social insurance, antipoverty
programs, and so forth.
Basic Elements of Any Forecast
Think on any economic variable you want to forecast. What do you
need?
•Information: Univariate or Multivariate
•A Model: Univariate or Multivariate
Once you have done your forecast, someone else can come with
another forecast of the same variable. How do you compare these
forecasts?
• Forecast Evaluation: Different measures of the forecast
errors.
Forecasting with Regression Models:
The regression model is an explicitly multivariate model, in which
variables are explained and forecast on the basis of their own
history and the histories of other, related variables. You have
already studied regression models in your Econometric course, and
very likely you have covered the forecasting issue. In the next
slides we will review it.
y t   0  1x t   t
 t is WN(0,  2 )
(a) Conditional Forecasting Models
A conditional forecasting model is one that can be used to
produce forecasts for a variable of interest, conditional upon
assumptions about other variables. With the regression model,
y t   0  1x t   t
 t is N(0,  2 )
our h-step ahead conditional forecast for y, given that the h-step
value of x is x *T  h is
y T  h , T | x *T  h   0  1x *T  h
Assuming normality, we use the conditional density forecast
N( y T  h, T | x *T  h ), and from it we get conditional interval forecasts.
We make the procedure operational by replacing unknown
parameters with estimates.
Conditional Forecasting Models (cont)
Forecasts are subjetc to error. There are at least three sources of
such error:
•Specification uncertainty: All models are wrong!!!!
•Innovation uncertainty: Future innovations are not known when
the forecast is made.
•Parameter uncertainty: The coefficients that we use to produce
forecasts are, of course, just estimates, and the estimates are
subject to sampling variability.
Q1: Which type of uncertainty is less important??
Conditional Forecasting Models (cont)
When using a conditional forecasting model, simple calculation allow
us to quantify both innovation and parameter uncertainty. Consider the
following simple example:
y t  x t   t
Suppose we want to predict yT+h at xT+h= x*T+h . Then
y T  h   x *T  h   T  h
Thus
ŷ T  h , T | x *T  h  ˆ x *T  h
with corresponding error
ˆ T  h, T  y T  h  ŷ T  h, T  (  ˆ ) x *T  h   T  h
Thus,
var( ˆ T  h, T )  ( x *T  h ) 2 var( ˆ )   2
Conditional Forecasting Models (cont)
In the latter expression, the first term accounts for parameter
uncertainty, while the second accounts for the usual innovation
uncertainty. Taken together we get an operational density forecast that
accounts for parameter uncertainty:
N(ˆ x *T  h , Var (ˆ T  h , T ))
from which interval forecasts may be constructed as well.
(b) Unconditional Forecasting Models
Often we do not want to make forecasts of y conditional upon
assumptions about x, rather, we just want the best possible forecast of
y-an unconditional forecast. To get an unconditional forecast from a
regression model, we often encounter the forecasting the right-handside variables problem. That is, to get an optimal unconditional point
forecast for y , we cannot insert an arbitrary value for future x, rather,
we need to insert the optimal point forecast, xT+h,T ,which yields the
unconditional forecast
y T  h, T  0  1x T  h, T
We usually don`t have such a forecast for x and the regression model
at hand doesn’t help us. Assuming this variable follows and ARIMA
representation, you will learn how to produce these forecasts in the
next set of slides: FORECASTING II
Evaluation of Forecasts
There are many ways of making forecasts, but all of them need the
following common ingredients in order for success:
(i) that there are regularities to capture
(ii) that such regularities are informative about the future
(iii) they are encapsulated in the selected forecasting method, and
(iv) non-regularities and excluded.
The main alternatives are (for some of them see Reading I):
1) Guessing
2) Extrapolation
3) Leading Indicators
4) Surveys
5) Time-Series Models
6) Econometric Models
Evaluation of Forecasts (cont)
The most common overall accuracy measures are:
T
mean squared error:
1
2
MSE 
T
root mean squared error
RMSE 
mean absolute error

e
t  h, t
t 1
T
1
T

e2
t  h, t
t 1
T
1
MAE 
T

t 1
where et+h,t=yt+h-yt+h,t are the forecast errors.
| e t  h, t |
(a) Comparing Forecast Accuracy
Suppose two competing forecasting procedures produce errors et(1) and
et(2) for t=1, ..., T. Then if expected squared error is to be the criterion,
the procedure yielding the lower MSE over the sample period will be
judged superior.
How can we test MSE(1) = MSE(2) versus the opposite?
Assume that the individual forecast errors are unbiased and not
autocorrelated. Consider, now, the pair of random variables
et(1)+ et(2) and et(1)- et(2) . Now
E[(e (1)  e (2) )(e (1)  e (2) )]   2   2
1
2
so the two expected expected squared errors, will be equal iff this pair
of random variables is uncorrelated.
Q2: Find an easy way of testing this hypothesis (Hint: use regression
analysis).
Forecast combination
Let ft(1) and ft(2) be two forecasts of yt with errors
( j)
( j)
e t  y t  f t for j  1,2, and such that
( j)
( j)2
E(e t )  0, E(e t )   2 , and
j
(1) (2)
E(e t e t )  1 2
Consider now a combined forecast, taken to be a weighted average of
the two individual forecasts,
(1)
(2)
C t  kf t  (1  k)f t
The forecast error is
(c)
(1)
(2)
e t  y t  C t  ke t  (1  k)e t
Forecast combination (cont)
Hence the error variance is
 c2  k 2  2  (1  k ) 2  2  2k (1  k )1 2
1
2
This expression is minimized for the value of k given by
ko 
2
2  1 2
12   2
2  21 2
and substituting in the top expression, the minimum achievable error
variance is
2  2 (1   2 )

1 2
2 
c,0
12   2
2  21 2
1
2


or

 c2,0  min( 12 ,  2
)
, unless
2
2
1
Note that
. If either
equality holds, then the variance of the combined forecast is equal to
the smaller of the two error variances.
Problems on Forecast combination
P1: Show that
2
k o   0 if and only if
 
1
P 2: Explain what happens with  c2, o as  approaches to –1 or
+1.