Chapter 15
Forecasting
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All rights reserved.
Slides by Niels-Hugo Blunch
Washington and Lee University
What Is Forecasting?
• In general, forecasting is the act of predicting the future
• In econometrics, forecasting is the estimation of the expected value of
a dependent variable for observations that are not part of the same
data set
• In most forecasts, the values being predicted are for time periods in
the future, but cross-sectional predictions of values for countries or
people not in the sample are also common
• To simplify terminology, the words prediction and forecast will be used
interchangeably in this chapter
– Some authors limit the use of the word forecast to out-of-sample prediction
for a time series
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What Is Forecasting? (cont.)
• Econometric forecasting generally uses a single linear
equation to predict or forecast
• Our use of such an equation to make a forecast can be
summarized into two steps:
1. Specify and estimate an equation that has as its
dependent variable the item that we wish to forecast:
(15.2)
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What Is Forecasting? (cont.)
2. Obtain values for each of the independent
variables for the observations for which we
want a forecast and substitute them into our
forecasting equation:
(15.3)
•
Figure 15.1 illustrates two examples
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Figure 15.1a
Forecasting Examples
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Figure 15.1b
Forecasting Examples
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More Complex
Forecasting Problems
• The forecasts generated in the previous section are quite simple,
however, and most actual forecasting involves one or more
additional questions—for example:
1. Unknown Xs: It is unrealistic to expect to know the
values for the independent variables outside the sample
• What happens when we don’t know the values of the
independent variables for the forecast period?
2. Serial Correlation: If there is serial correlation involved,
the forecasting equation may be estimated with GLS
• How should predictions be adjusted when forecasting equations
are estimated with GLS?
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More Complex Forecasting
Problems (cont.)
3. Confidence Intervals: All the previous forecasts were
single values, but such single values are almost never
exactly right, so maybe it would be more helpful if we
forecasted a confidence interval instead
• How can we develop these confidence intervals?
4. Simultaneous Equations Models: As we saw in
Chapter 14, many economic and business equations are
part of simultaneous models
• How can we use an independent variable to forecast a
dependent variable when we know that a change in value of the
dependent variable will change, in turn, the value of the
independent variable that we used to make the forecast?
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Conditional Forecasting (Unknown X
Values for the Forecast Period)
• Unconditional forecast: all values of the independent
variables are known with certainty
– This is rare in practice
• Conditional forecast: actual values of one or more of
the independent variables are not known
– This is the more common type of forecast
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Conditional Forecasting (Unknown X
Values for the Forecast Period) (cont.)
• The careful selection of independent variables can
sometimes help avoid the need for conditional
forecasting
• This opportunity can arise when the dependent variable
can be expressed as a function of leading indicators:
– A leading indicator is an independent variable the movements
of which anticipate movements in the dependent variable
– The best known leading indicator, the Index of Leading
Economic Indicators, is produced each month
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Forecasting with Serially
Correlated Error Terms
• Recall from Chapter 9 that when serial correlation is severe, one
remedy is to run Generalized Least Squares (GLS) as noted in
Equation 9.18:
(9.18)
• If Equation 9.18 is estimated, the dependent variable will be:
(15.7)
• Thus, if a GLS equation is used for forecasting, it will produce
predictions of Y*T + 1 rather than of YT+1
• Such predictions thus will be of the wrong variable!
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Forecasting with Serially
Correlated Error Terms (cont.)
• If forecasts are to be made with a GLS equation, Equation 9.18
should first be solved for YT before forecasting is attempted:
(15.8)
• Next, substitute T+1 for t (to forecast time period T+1) and insert
estimates for the coefficients, ρs and Xs into the equation to get:
(15.9)
• Equation 15.9 thus should be used for forecasting when an
equation has been estimated with GLS to correct for serial
correlation
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Forecasting Confidence
Intervals
• The techniques we use to test hypotheses can also be
adapted to create forecasting confidence intervals
• Given a point forecast,
all we need to generate a
confidence interval around that forecast are tc, the critical
t-value (for the desired level of confidence), and SF, the
estimated standard error of the forecast:
(15.11)
• The critical t-value, tc, can be found in Statistical Table B1 (for a two-tailed test with T-K-1 degrees of freedom)
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Forecasting Confidence
Intervals (cont.)
• Lastly, the standard error of the forecast, SF, for an equation with just
one independent variable, equals the square root of the forecast error
variance:
(15.13)
where:
s2
= the estimated variance of the error term
T
= the number of observations in the sample
XT+1 = the forecasted value of the single independent variable
X
= the arithmetic mean of the observed Xs in the sample
• Figure 15.2 illustrates an example of a forecast confidence interval
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Figure 15.2
A Confidence Interval for
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Forecasting with Simultaneous
Equations Systems
• How should forecasting be done in the context of a
simultaneous model?
• There are two approaches to answering this question,
depending on whether there are lagged endogenous
variables on the right-hand side of any of the equations in
the system:
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Forecasting with Simultaneous
Equations Systems (cont.)
1.
No lagged endogenous variables in the system:
• the reduced-form equation for the particular endogenous variable can
be used for forecasting because it represents the simultaneous solution
of the system for the endogenous variable being forecasted
2.
Lagged endogenous variables in the system:
• then the approach must be altered to take into account the dynamic
interaction caused by the lagged endogenous variables
• For simple models, this sometimes can be done by substituting
for the lagged endogenous variables where they appear in the
reduced-form equations
• If such a manipulation is difficult, however, then a technique called
simulation analysis can be used
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ARIMA Models
• ARIMA is a highly refined curve-fitting device that uses current and
past values of the dependent variable to produce often accurate
short-term forecasts of that variable
– Examples of such forecasts are stock market price predictions created by
brokerage analysts (called “chartists” or “technicians”) based entirely on
past patterns of movement of the stock prices
• If ARIMA models thus essentially ignores economic theory (by
ignoring “traditional” explanatory variables), why use them?
• The use of ARIMA is appropriate when:
– little or nothing is known about the dependent variable being forecasted,
– the independent variables known to be important cannot be forecasted
effectively
– all that is needed is a one or two-period forecast
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ARIMA Models (cont.)
• The ARIMA approach combines two different
specifications (called processes) into one equation:
1. An autoregressive process (AR):
• expresses a dependent variable as a function of past values of the
dependent variable
• This is similar to the serial correlation error term function of
Chapter 9 and to the dynamic model of Chapter 12
2. a moving average process (MA):
• expresses a dependent variable as a function of past values of the
error term
• Such a function is a moving average of past error term observations
that can be added to the mean of Y to obtain a moving average of past
values of Y
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ARIMA Models (cont.)
• To create an ARIMA model, we begin with an econometric equation with
no independent variables:
• and then add to it both the autoregressive and moving-average
processes:
(15.17)
where the θs and the φs are the coefficients of the autoregressive and
moving-average processes, respectively, and p and q are the number
of past values used of Y and ε, respectively
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ARIMA Models (cont.)
• Before this equation can be applied to a time series, however, it
must be ensured that the time series is stationary, as defined in
Section 12.4
• For example, a non-stationary series can often be converted
into a stationary one by taking the first difference:
(15.18)
• If the first differences do not produce a stationary series, then
first differences of this first-differenced series can be taken—i.e.
a second-difference transformation:
(15.19)
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ARIMA Models (cont.)
• If a forecast of Y* or Y** is made, then it must be converted back into
Y terms
• For example, if d = 1 (where d is the number of differences taken to
make Y stationary), then:
(15.20)
• This conversion process is similar to integration in mathematics, so
the “I” in ARIMA stands for “integrated”
• ARIMA thus stands for Auto-Regressive Integrated Moving Average
– An ARIMA model with p, d, and q specified is usually denoted as ARIMA
(p,d,q) with the specific integers chosen inserted for p, d, and q
– If the original series is stationary and d therefore equals 0, this is
sometimes shortened to ARMA
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Key Terms from Chapter 15
• Unconditional forecast
• Conditional forecast
• Leading indicator
• Confidence interval (of forecast)
• Autoregressive process
• Moving-average process
• ARIMA(p,d,q)
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