Introduction to Econometrics
81
Technical Notes
Introduction to Econometrics
Lecture 8: Forecasts and Structural Breaks
Forecasting and Forecast Confidence Intervals
In a simple regression, if we know the forecast value of the regressor X
p
, we can forecast Y conditional on
X using
Y
p
=
α
e
+
β
e
X
p
since E[u
p
] = 0. This forecast should be contrasted with the unconditional forecast of Y given by
Y
u
=
μ
e
where
μ
e
is the estimated sample mean of Y
1
.
The forecast error e
p
is
e
p
= Y  Y
p
= (
α
+
β
X
p
+ u
p
)  (
α
e
+
β
e
X
p
)
=
α

α
e
+ (
β

β
e
)X
p
+ u
p
with expectation
E[e
p
]
= {E[
α

α
e
] + X
p
E[
β

β
e
]} + E[u
p
] = 0
because the OLS estimates of the parameters are unbiased. Therefore the forecast is unbiased, but forecast
error arises (even if we know X
p
) both because we cannot forecast the random variable u
p
, and also because
our estimates of
α
and
β
are subject to sampling error. This sampling error component (the term in braces)
depends on the sample
disturbances, but not on u
p
, so the two types of error are independent. The two
components of the forecasting error are both Normally distributed, so their sum is also Normally
distributed: this result can be generalised easily to multiple regression. It is also possible to show that
forecasts based on OLS regression coefficients have lower variance than other linear unbiased forecasts:
this is an analogue of the GaussMarkov theorem that the OLS parameter estimators are BLUE.
Using Forecasts  Prediction
There are two main ways of using these results on the distribution of a forecast. The first of these
corresponds to the conventional use of the word ‘forecast’: an attempt to predict the outcome for Y for an
observation where we know (or can predict) the explanatory variables. Although very often the ‘users’ of
such predictions are only interested in the single central value Y
p
, it is also important to provide a measure
of forecast precision in the form of a confidence interval.
To compute the forecast
confidence interval for a simple regression, compute
Var[e
p
] = E[{e
p
 E[e
p
]}
2
]
= E[{(
α

α
e
) + (
β

β
e
)X
p
+ u
p
}
2
]
= {Var(
α
e
) + Var(
β
e
)(X
p
)
2
+ 2Cov(
α
e
,
β
e
)X
p
} + Var(u
p
)
1
In this discussion of forecasts we concentrate on the case where the data is generated by random sampling, so
forecasting amounts to predicting the value of Y for another draw from the underlying population. The problems
involved in forecasting the value of a timeseries variable are discussed later in the course. However it is helpful to
note here that for time series the distinction between unconditional and conditional forecasts is less useful than
that between forecasts which use only the past history of Y (‘univariate forecasts’) and those which use other
variables (‘multivariate forecasts’).
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82
Technical Notes
since E[(
α

α
e
)u
p
] = E[(
β

β
e
)X
p
u
p
] = 0, because u
p
is independent of the sample
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 Spring '10
 Cowell
 Economics, Econometrics, Regression Analysis, Chow, Structural Breaks, predictive failure

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