Forecasting
1. Optimal Forecast Criterion - Minimum Mean Square Error Forecast
We have now considered how to determine which ARIMA model we should fit to our
data, we have also examined how to estimate the parameters in the ARIMA model
and how to examine the goodness of fit of our model. So we are finally in a position
to use our model to forecast new values for our time series. Suppose we have
observed a time series up to time T, so that we have known values for 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 ,
and subsequently, we also know 𝑍1 , 𝑍2 , ⋯ , 𝑍𝑇 . This collection of information we
know will be denoted as:
𝐼𝑇 = {𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 ; 𝑍1 , 𝑍2 , ⋯ , 𝑍𝑇 }
Suppose that we want to forecast a value for 𝑋𝑇+𝑘 , 𝑘 time units into the future. We
now introduce the most prevalent optimality criterion for forecasting.
Theorem: The optimal k-step ahead forecast 𝑋𝑇+𝑘,𝑇 (which is a function of 𝐼𝑇 ) that
will minimize the mean square error,
E(𝑋𝑇+𝑘 − 𝑋𝑇+𝑘,𝑇 )
2
is
𝑋𝑇+𝑘,𝑇 = E(𝑋𝑇+𝑘 |𝐼𝑇 )
This optimal forecast is referred to as the Minimum Mean Square Error Forecast.
This optimal forecast is unbiased because
E(𝑋𝑇+𝑘 − 𝑋𝑇+𝑘,𝑇 ) = E [E ((𝑋𝑇+𝑘 − 𝑋𝑇+𝑘,𝑇 )|𝐼𝑇 )] = E[𝑋𝑇+𝑘,𝑇 − 𝑋𝑇+𝑘,𝑇 ] = 0
Lemma: Suppose Z and W are real random variables, then
2
2
min 𝐸 [(𝑍 − ℎ(𝑊)) ] = 𝐸 [(𝑍 − 𝐸(𝑍|𝑊)) ]
ℎ
1
That is, the posterior mean E(Z|W) will minimize the quadratic loss (mean square
error).
Proof:
2
2
𝐸 [(𝑍 − ℎ(𝑊)) ] = 𝐸 [(𝑍 − 𝐸(𝑍|𝑊) + 𝐸(𝑍|𝑊) − ℎ(𝑊)) ]
2
= 𝐸 [(𝑍 − 𝐸(𝑍|𝑊)) ] + 2𝐸[(𝑍 − 𝐸(𝑍|𝑊))(𝐸(𝑍|𝑊) − ℎ(𝑊))]
2
+𝐸 [(𝐸(𝑍|𝑊) − ℎ(𝑊)) ]
Conditioning on W, the cross term is zero. Thus
2
2
2
𝐸 [(𝑍 − ℎ(𝑊)) ] = 𝐸 [(𝑍 − 𝐸(𝑍|𝑊)) ] + 𝐸 [(𝐸(𝑍|𝑊) − ℎ(𝑊)) ]
Note: We usually omit the word ‘optimal’ in the K Periods Ahead (Optimal) Forecast,
because in the time series context, when we refer to ‘forecast’, we mean ‘the optimal
minimum mean square error forecast’, by default.
2
2. Forecast in MA(1)
The MA(1) model is:
𝑋𝑡 = 𝑍𝑡 + 𝜃𝑍𝑡−1 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
Given the observed data {𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 }, the white noise term 𝑍𝑡 is not directly
observed, however, we can perform an Recursive Estimation of the White Noise as
follows:
Given the initial condition 𝑍0 (not important), we have:
𝑍1 = 𝑋1 − 𝜃𝑍0
𝑍2 = 𝑋2 − 𝜃𝑍1
⋯
𝑍𝑇 = 𝑋𝑇 − 𝜃𝑍𝑇−1
Therefore, given {𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 }, we can also obtain {𝑍1 , 𝑍2 , ⋯ , 𝑍𝑇 }, the entire
information known can thus be written as: 𝑰𝑻 = {𝑿𝟏 , 𝑿𝟐 , ⋯ , 𝑿𝑻 ; 𝒁𝟏 , 𝒁𝟐 , ⋯ , 𝒁𝑻 }
Since 𝑋𝑇+1 = 𝑍𝑇+1 + 𝜃𝑍𝑇 , the one period ahead (optimal) forecast is
𝑋𝑇+1,𝑇 = 𝐸(𝑋𝑇+1|𝐼𝑇 ) = 𝐸(𝑍𝑇+1 + 𝜃𝑍𝑇 |𝐼𝑇 )
= 𝐸(𝑍𝑇+1 |𝐼𝑇 ) + 𝜃𝐸(𝑍𝑇 |𝐼𝑇 )
= 𝐸(𝑍𝑇+1 ) + 𝜃𝐸(𝑍𝑇 |𝑍𝑇 )
= 0 + 𝜃𝑍𝑇 = 𝜃𝑍𝑇
Note: Per convention in time series literature, we wrote:
𝐸(𝑍𝑇 |𝑍𝑇 ) = 𝑍𝑇
Although one can write this more rigorously as:
𝐸(𝑍𝑇 |𝑍𝑇 = 𝑧𝑇 ) = 𝑧𝑇
Similarly, we wrote
𝐸(𝑋𝑇 |𝑋𝑇 ) = 𝑋𝑇 ,
which can be written more rigorously as:
𝐸(𝑋𝑇 |𝑋𝑇 = 𝑥𝑇 ) = 𝑥𝑇
The forecast error is
𝑒𝑇+1,𝑇 = 𝑋𝑇+1 − 𝑋𝑇+1,𝑇 = 𝑍𝑇+1
Since the forecast is unbiased, the minimum mean square error is equal to the
forecast error variance:
2
E(𝑋𝑇+1 − 𝑋𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑋𝑇+1 − 𝑋𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑍𝑇+1 ) = 𝜎 2
Given 𝐼𝑇 = {𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 ; 𝑍1 , 𝑍2 , ⋯ , 𝑍𝑇 }, and 𝑋𝑇+2 = 𝑍𝑇+2 + 𝜃𝑍𝑇+1 ,
the two periods ahead (optimal) forecast is
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2|𝐼𝑇 ) = 𝐸(𝑍𝑇+2 + 𝜃𝑍𝑇+1|𝐼𝑇 )
= 𝐸(𝑍𝑇+2 |𝐼𝑇 ) + 𝜃𝐸(𝑍𝑇+1|𝐼𝑇 )
= 0+𝜃∗0 =0
3
The forecast error is
𝑒𝑇+2,𝑇 = 𝑋𝑇+2 − 𝑋𝑇+2,𝑇 = 𝑍𝑇+2 + 𝜃𝑍𝑇+1
Since the forecast is unbiased, the minimum mean square error is equal to the
forecast error variance:
2
E(𝑋𝑇+2 − 𝑋𝑇+2,𝑇 ) = 𝑉𝑎𝑟(𝑋𝑇+2 − 𝑋𝑇+2,𝑇 ) = 𝑉𝑎𝑟(𝑍𝑇+2 + 𝜃𝑍𝑇+1 ) = (1 + 𝜃 2 )𝜎 2
In summary, for more than one period ahead, just like the two-periods ahead
forecast, the forecast is zero, the unconditional mean. That is,
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2|𝐼𝑇 ) = 0
𝑋𝑇+3,𝑇 = 𝐸(𝑋𝑇+3|𝐼𝑇 ) = 0
⋯
The MA(1) process is not forecastable for more than one period ahead (apart from
the unconditional mean).
Forecast for MA(1) with Intercept (non-zero mean)
If the MA(1) model includes an intercept
𝑋𝑡 = 𝜇+𝑍𝑡 + 𝜃𝑍𝑡−1 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
We can perform forecasting using the same approach.
For example, since 𝑋𝑇+1 = 𝜇+𝑍𝑇+1 + 𝜃𝑍𝑇 , the one period ahead (optimal)
forecast is
𝑋𝑇+1,𝑇 = 𝐸(𝑋𝑇+1 |𝐼𝑇 ) = 𝐸(𝜇 + 𝑍𝑇+1 + 𝜃𝑍𝑇 |𝐼𝑇 )
= 𝜇 + 𝐸(𝑍𝑇+1 |𝐼𝑇 ) + 𝜃𝐸(𝑍𝑇 |𝐼𝑇 )
= 𝜇 + 0 + 𝜃𝑍𝑇 = 𝜇 + 𝜃𝑍𝑇
The forecast error is
𝑒𝑇+1,𝑇 = 𝑋𝑇+1 − 𝑋𝑇+1,𝑇 = 𝑍𝑇+1
Since the forecast is unbiased, the minimum mean square error is equal to the
forecast error variance:
2
E(𝑋𝑇+1 − 𝑋𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑋𝑇+1 − 𝑋𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑍𝑇+1 ) = 𝜎 2
3. Generalization to MA(q):
For a MA(q) process, we can forecast up to q out-of-sample periods. Can you write
down the (optimal) point forecast for each period, the forecast error, and the
forecast error variance? Please practice before the exam.
4. Forecast in AR(1)
For the AR(1) model,
X𝑡 = ∅X𝑡−1 +𝑍𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
Given 𝐼𝑇 = {𝑋1 , 𝑋2 , ⋯ , 𝑋𝑇 ; 𝑍1 , 𝑍2 , ⋯ , 𝑍𝑇 };
The One Period Ahead (Optimal) Forecast can be computed as follows:
4
X 𝑇+1 = ∅X 𝑇 +𝑍𝑇+1
So, given data 𝐼𝑇 , the one period ahead forecast is
𝑋𝑇+1,𝑇 = 𝐸(X 𝑇+1|𝐼𝑇 ) = 𝐸(∅X 𝑇 +𝑍𝑇+1 |𝐼𝑇 )
= 𝐸(∅X 𝑇 |𝐼𝑇 ) + 𝐸(𝑍𝑇+1|𝐼𝑇 )
= ∅X 𝑇 + 0
= ∅X 𝑇
The forecast error is
𝑒𝑇+1,𝑇 = 𝑋𝑇+1 − 𝑋𝑇+1,𝑇 = 𝑍𝑇+1
The forecast error variance:
𝑉𝑎𝑟(𝑒𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑍𝑇+1 ) = 𝜎 2
Two Periods Ahead Forecast
By back-substitution
X 𝑇+2 = ∅X 𝑇+1 +𝑍𝑇+2
X 𝑇+2 = ∅(∅X 𝑇 +𝑍𝑇+1 )+𝑍𝑇+2
X 𝑇+2 = ∅2 X 𝑇 +∅𝑍𝑇+1 +𝑍𝑇+2
So, given data 𝐼𝑇 , the two periods ahead (optimal) forecast is
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2 |𝐼𝑇 ) = 𝐸(∅2 X 𝑇 +∅𝑍𝑇+1 +𝑍𝑇+2 |𝐼𝑇 )
= ∅2 X 𝑇 + ∅0 + 0
= ∅2 X 𝑇
That is, the 2 periods ahead forecast is also a linear function of the final observed
value, but with the coefficient ∅2 .
The forecast error is
𝑒𝑇+2,𝑇 = 𝑋𝑇+2 − 𝑋𝑇+2,𝑇 = ∅𝑍𝑇+1 +𝑍𝑇+2
The forecast error variance is:
Var(𝑒𝑇+2,𝑇 ) = 𝑉𝑎𝑟(∅𝑍𝑇+1 +𝑍𝑇+2 ) = (1 + ∅2 )𝜎 2
K Periods Ahead Forecast
Similarly
X 𝑇+𝑘 = ∅𝑘 X 𝑇 +∅𝑘−1 𝑍𝑇+1 + ⋯ + ∅𝑍𝑇+𝑘−1 + 𝑍𝑇+𝑘
So, given I𝑇 , the k periods ahead optimal forecast is
𝑋𝑇+𝑘,𝑇 = 𝐸(X 𝑇+𝑘 |I𝑇 ) = ∅𝑘 X 𝑇
AR(1) with Intercept
If the AR(1) model includes an intercept
X𝑡 = 𝛼 + ∅X𝑡−1 + 𝑍𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
Then the one period ahead forecast is
𝑋𝑇+1,𝑇 = 𝐸(𝑋𝑇+1|𝐼𝑇 )
= 𝛼 + 𝐸(∅𝑋𝑇 |𝐼𝑇 ) + 𝐸(𝑍𝑇+1|𝐼𝑇 )
5
= 𝛼 + ∅𝑋𝑇
What is the two periods ahead forecast?
Forecast AR(1) recursively.
Alternatively, rather than using the back-substitution method directly as qwe have
shown above, we can forecast the AR(1) recursively as follows:
X 𝑇+2 = ∅X 𝑇+1 +𝑍𝑇+2
So, given data 𝐼𝑇 , the two periods ahead (optimal) forecast is
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2 |𝐼𝑇 ) = 𝐸(∅X 𝑇+1 +𝑍𝑇+2 |𝐼𝑇 ) = ∅𝐸(X 𝑇+1|𝐼𝑇 ) = ∅𝑋𝑇+1,𝑇
Similarly, one can obtain the general recursive forecast relationship as:
𝑋𝑇+𝑘,𝑇 = 𝐸(𝑋𝑇+𝑘 |𝐼𝑇 ) = 𝐸(∅X 𝑇+𝑘−1 +𝑍𝑇+𝑘 |𝐼𝑇 )
= ∅𝐸(X 𝑇+𝑘−1 |𝐼𝑇 ) = ∅𝑋𝑇+𝑘−1,𝑇
= ⋯ = ∅𝑘−1 𝑋𝑇+1,𝑇
This way, once you have computed the one-step ahead forecast as:
𝑋𝑇+1,𝑇 = 𝐸(X 𝑇+1|𝐼𝑇 ) = 𝐸(∅X 𝑇 +𝑍𝑇+1 |𝐼𝑇 )
= 𝐸(∅X 𝑇 |𝐼𝑇 ) + 𝐸(𝑍𝑇+1|𝐼𝑇 )
= ∅X 𝑇 + 0
= ∅X 𝑇
You can obtain the rest of the forecast easily.
Note: As mentioned in class, I prefer the back-substitution method (introduced first)
in general because it can provide you with the forecast error directly as well.
5. AR(p) Process
Consider an AR(2) model with intercept
X𝑡 = 𝛼 + ∅1 X𝑡−1 + ∅2 X𝑡−2 + 𝑍𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
We have
X 𝑇+1 = 𝛼 + ∅1 X 𝑇 + ∅2 X 𝑇−1 + 𝑍𝑇+1
Then the one period ahead forecast is
𝑋𝑇+1,𝑇 = 𝐸(𝑋𝑇+1|𝐼𝑇 )
= 𝛼 + 𝐸(∅1 𝑋𝑇 |𝐼𝑇 ) + 𝐸(∅2 𝑋𝑇−1 |𝐼𝑇 ) + 𝐸(𝑍𝑇+1|𝐼𝑇 )
= 𝛼 + ∅1 𝑋𝑇 + ∅2 X 𝑇−1
The two periods ahead forecast (by the recursive foreecasting method) is
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2|𝐼𝑇 )
= 𝛼 + 𝐸(∅1 𝑋𝑇+1 |𝐼𝑇 ) + 𝐸(∅2 𝑋𝑇 |𝐼𝑇 ) + 𝐸(𝑍𝑇+2|𝐼𝑇 )
= 𝛼 + ∅1 𝑋𝑇+1,𝑇 + ∅2 X 𝑇
= 𝛼 + ∅1 (𝛼 + ∅1 𝑋𝑇 + ∅2 X 𝑇−1 ) + ∅2 X 𝑇
= (1 + ∅1 )𝛼 + (∅1 2 + ∅2 )X 𝑇 + ∅1 ∅2 X 𝑇−1
6
Alternatively, the two periods ahead forecast (by the back-substitution method) can
be derived as:
X 𝑇+2 = 𝛼 + ∅1 X 𝑇+1 + ∅2 X 𝑇 + 𝑍𝑇+2
= 𝛼 + ∅1 (𝛼 + ∅1 X 𝑇 + ∅2 X 𝑇−1 + 𝑍𝑇+1 ) + ∅2 X 𝑇 + 𝑍𝑇+2
= (1 + ∅1 )𝛼 + (∅1 2 + ∅2 )X 𝑇 + ∅1 ∅2 X 𝑇−1 + ∅1 𝑍𝑇+1 + 𝑍𝑇+2
Therefore
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2|𝐼𝑇 )
= (1 + ∅1 )𝛼 + (∅1 2 + ∅2 )X 𝑇 + ∅1 ∅2 X 𝑇−1
The forecast error is
𝑒𝑇+2,𝑇 = 𝑋𝑇+2 − 𝑋𝑇+2,𝑇 = ∅1 𝑍𝑇+1 + 𝑍𝑇+2
The forecast error variance is:
Var(𝑒𝑇+2,𝑇 ) = 𝑉𝑎𝑟(∅1 𝑍𝑇+1 + 𝑍𝑇+2 ) = (1 + ∅12 )𝜎 2
What are the forecasts for more future periods?
Please practice before the exam.
6. ARMA Process
Consider an ARMA(1,1) model
X𝑡 = ∅X𝑡−1 + 𝑍𝑡 + 𝜃𝑍𝑡−1 , 𝑤ℎ𝑒𝑟𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎 2 )
Again we can use the back-substitution method to compute point forecast.
X 𝑇+1 = ∅X 𝑇 + 𝑍𝑇+1 + 𝜃𝑍𝑇
X 𝑇+2 = ∅X 𝑇+1 + 𝑍𝑇+2 + 𝜃𝑍𝑇+1
= ∅(∅X 𝑇 + 𝑍𝑇+1 + 𝜃𝑍𝑇 ) + 𝑍𝑇+2 + 𝜃𝑍𝑇+1
One-step ahead forecast:
𝑋𝑇+1,𝑇 = 𝐸(𝑋𝑇+1|𝐼𝑇 )
= 𝐸(∅𝑋𝑇 |𝐼𝑇 ) + 𝐸(𝑍𝑇+1 |𝐼𝑇 ) + 𝐸(𝜃𝑍𝑇 |𝐼𝑇 )
= ∅𝑋𝑇 + 𝜃𝑍𝑇
Two-step ahead forecast:
𝑋𝑇+2,𝑇 = 𝐸(𝑋𝑇+2|𝐼𝑇 )
= 𝐸(∅𝑋𝑇+1 |𝐼𝑇 ) + 𝐸(𝑍𝑇+2 |𝐼𝑇 ) + 𝐸(𝜃𝑍𝑇+1|𝐼𝑇 )
= ∅𝐸(𝑋𝑇+1 |𝐼𝑇 ) = ∅(∅𝑋𝑇 + 𝜃𝑍𝑇 )
The two-step ahead forecast error is
𝑒𝑇+2,𝑇 = 𝑋𝑇+2 − 𝑋𝑇+2,𝑇 = ∅𝑍𝑇+1 + 𝑍𝑇+2 + 𝜃𝑍𝑇+1 = (∅ + 𝜃)𝑍𝑇+1 + 𝑍𝑇+2
Again, since the forecast is unbiased, the minimum mean square error is equal to the
forecast error variance:
2
E(𝑋𝑇+2 − 𝑋𝑇+2,𝑇 ) = 𝑉𝑎𝑟(𝑋𝑇+2 − 𝑋𝑇+2,𝑇 ) = 𝑉𝑎𝑟[(∅ + 𝜃)𝑍𝑇+1 + 𝑍𝑇+2 ]
= [(∅ + 𝜃)2 + 1]𝜎 2
7
7. ARIMA Process
Suppose 𝑋𝑡 follows the ARIMA(1,1,0) model. That means its first difference
𝑌𝑡 = ∇𝑋𝑡 = 𝑋𝑡 − 𝑋𝑡−1
follows the ARMA(1,0) model, which is simply the AR(1) model as follows:
𝑌𝑡 = 𝜙𝑌𝑡−1 + 𝑍𝑡
𝐻𝑒𝑟𝑒 𝑤𝑒 𝑎𝑠𝑠𝑢𝑚𝑒 𝑍𝑡 ~𝑊𝑁(0, 𝜎𝑍2 )
𝐍𝐨𝐭𝐞: we used σ2Z instead of σ2 so you can get familiar with different notations.
Substituting 𝑌𝑡 = ∇𝑋𝑡 = 𝑋𝑡 − 𝑋𝑡−1 , and 𝑌𝑡−1 = ∇𝑋𝑡−1 = 𝑋𝑡−1 − 𝑋𝑡−2 we can write
the original ARIMA(1,1,0) model as:
𝑋𝑡 − 𝑋𝑡−1 = 𝜙(𝑋𝑡−1 − 𝑋𝑡−2 ) + 𝑍𝑡
That is:
𝑋𝑡 = (1 + 𝜙)𝑋𝑡−1 − 𝜙𝑋𝑡−2 + 𝑍𝑡
One-step ahead forecast:
𝑋𝑇+1,𝑇 = 𝐸[𝑋𝑇+1 |𝐼𝑇 ] = 𝐸[(1 + 𝜙)𝑋𝑇 − 𝜙𝑋𝑇−1 + 𝑍𝑇+1 |𝐼𝑇 ]
= (1 + 𝜙)𝑋𝑇 − 𝜙𝑋𝑇−1
Thus the forecast error
𝑒𝑇+1,𝑇 = 𝑋𝑇+1 − 𝑋𝑇+1,𝑇
= (1 + 𝜙)𝑋𝑇 − 𝜙𝑋𝑇−1 + 𝑍𝑇+1 − [(1 + 𝜙)𝑋𝑇 − 𝜙𝑋𝑇−1 ]
= 𝑍𝑇+1
And the variance of the forecast error
𝑉𝑎𝑟(𝑒𝑇+1,𝑇 ) = 𝑉𝑎𝑟(𝑍𝑇+1 ) = 𝜎𝑍2
Two-step ahead forecast:
𝑋𝑇+2 = (1 + 𝜙)𝑋𝑇+1 − 𝜙𝑋𝑇 + 𝑍𝑇+2
= (1 + 𝜙)[(1 + 𝜙)𝑋𝑇 − 𝜙𝑋𝑇−1 + 𝑍𝑇+1 ] − 𝜙𝑋𝑇 + 𝑍𝑇+2
= (1 + 𝜙 + 𝜙 2 )𝑋𝑇 − (𝜙 + 𝜙 2 )𝑋𝑇−1 + 𝑍𝑇+2 + (1 + 𝜙)𝑍𝑇+1
Therefore
𝑋𝑇+2,𝑇 = 𝐸[𝑋𝑇+2 |𝐼𝑇 ]
2 )𝑋
2
= 𝐸[(1 + 𝜙 + 𝜙
𝑇 − (𝜙 + 𝜙 )𝑋𝑇−1 + 𝑍𝑇+2 + (1 + 𝜙)𝑍𝑇+1 |𝐼𝑇 ]
2
= (1 + 𝜙 + 𝜙 )𝑋𝑇 − (𝜙 + 𝜙 2 )𝑋𝑇−1
Thus the forecast error
𝑒𝑇+2,𝑇 = 𝑋𝑇+2 − 𝑋𝑇+2,𝑇 = 𝑍𝑇+2 + (1 + 𝜙)𝑍𝑇+1
And the variance of the forecast error
𝑉𝑎𝑟(𝑒𝑇+2,𝑇 ) = (2 + 2𝜙 + 𝜙 2 )𝜎𝑍2
8. Prediction (Forecast) Error, and Prediction Interval
Recall the k-step ahead prediction error is defined as
e + , = X + − E(X + |I )
By the law of total variance:
Var( ) = E[Var( |X)] + Var[E( |X)]
8
We can derive the variance of the prediction error:
Var(e
+ ,
) = Var(X
+
|I )
Based on the predictor and its error variance, we may construct the prediction
intervals. For example, if the X ’s are normally distributed, we can consider the 95%
prediction interval
E(X
+
|I )
0.02
√Var(X
+
|I )
9