1
Forecasting and regression
For a simple regression, given
• OLS estimates of the regression coefficients
α
e
,
β
e
• the forecast value of X, X
p
The point forecast is
y
p
=
α
e
+
β
e
X
p
since E[u
p
] = 0
The forecast error 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
The error arises from
uncertainty about u
p
sampling errors in
α
e
,
β
e
The variance of the forecast
The expected forecast error is
E[e
p
]
= E[(
α

α
e
) + (
β

β
e
)X
p
+ u
p
]
= 0
α
e
,
β
e
are unbiased and E[u
p
] = 0
Hence the forecast is unbiased, and Variance = MSE
The forecast variance is
Var[e
p
]
= {Var[
α
e
] + Var[
β
e
](X
p
)
2
+ 2Cov[
α
e
,
β
e
]X
p
} + Var[u
p
]
=
σ
2
[{(1/N)(1 + (X
p
M(X))
2
/Var
N
(X))}+1]
First component (in {}) is uncertainty of the estimated
model: function of sample
disturbances
Second component is ‘true’ uncertainty about
disturbance for forecast
observation
Random sampling ensures that these two components
are independent
Forecast confidence intervals
The forecast is a linear
function of the regression
coefficients and u
p
Hence if coefficients and u
p
are both Normal so is the
forecast error
The 95% confidence interval (with
σ
2
unknown) for the
forecast is
y
p
t
N2
SE[e
p
] < y < y
p
+t
N2
SE[e
p
]
replacing the Normal distribution by the tdistribution
Forecasts based on multiple regressions are also
unbiased and normally distributed, but the variance
formula is more complex
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An example of forecasting: the wage of
a male worker
We forecast the wage of a male worker with 12 years of
schooling and 6 years of work experience
An unconditional
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 Spring '10
 Cowell
 Economics

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