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3.4: Forecasting

Suppose that the variables \(X_1,\ldots,X_n\) of a weakly stationary time series \((X_t\colon t\in\mathbb{Z})\) have been observed with the goal  to predict or forecast the future values of \(X_{n+1},X_{n+2},\ldots\). The focus is here on so-called one-step best linear predictors (BLP). These are, by definition, linear combinations

   \[\hat{X}_{n+1}=\phi_{n0}+\phi_{n1}X_n+\ldots+\phi_{nn}X_1 \tag{3.4.1}\]

of the observed variables \(X_1,\ldots,X_n\) that minimize the mean-squared error

\[ E\left[\{X_{n+1}-g(X_1,\ldots,X_n)\}^2\right] \]

for functions g of \(X_1,\ldots,X_n\). Straightforward generalizations yield definitions for the m-step best linear predictors \(\hat{X}_{n+m}\) of \(X_{n+m}\) for arbitrary \(m\in\mathbb{N}\) in the same fashion. Using Hilbert space theory, one can prove the following theorem which will be the starting point for our considerations.

Theorem 3.4.1 Best linear prediction
Let \((X_t\colon t\in\mathbb{Z})\) be a weakly stationary stochastic process of which \(X_1,\ldots,X_n\) are observed. Then, the one-step BLP \(\hat{X}_{n+1}\) of \(X_{n+1}\) is determined by    the equations    

\[ E\left[(X_{n+1}-\hat{X}_{n+1})X_{n+1-j}\right]=0 \]    

for all \(j=1,\ldots,n+1\), where \(X_0=1\).

The equations specified in Theorem (3.4.1.) can be used to calculate the coefficients \(\phi_{n0},\ldots,\phi_{nn}\) in (3.4.1.). It suffices to focus on mean zero processes \((X_t\colon t\in\mathbb{Z})\) and thus to set \(\phi_{n0}=0\) as the following calculations show. Assume that \(E[X_t]=\mu\) for all \(t\in\mathbb{Z}\). Then, Theorem 3.4.1 gives that \(E[\hat{X}_{n+1}]=E[X_{n+1}]=\mu\) (using the equation with \(j=n+1\). Consequently, it holds that

\[ \mu=E[\hat{X}_{n+1}] =E\left[\phi_{n0}+\sum_{\ell=1}^n\phi_{n\ell}X_{n+1-\ell}\right] =\phi_{n0}+\sum_{\ell=1}^n\phi_{n\ell}\mu. \]

Using now that \(\phi_{n0}=\mu(1-\phi_{n1}-\ldots-\phi_{nn})\), equation (3.4.1.) can be rewritten as

\[ \hat{Y}_{n+1}=\phi_{n1}Y_n+\ldots+\phi_{nn}Y_1, \]

where \(\hat{Y}_{n+1}=\hat{X}_{n+1}-\mu\) has mean zero.

With the ACVF \(\gamma\) of \((X_t\colon t\in\mathbb{Z})\), the equations in Theorem 3.4.1. can be expressed as

\[\sum_{\ell=1}^n\phi_{n\ell}\gamma(j-\ell)=\gamma(j),\qquad j=1,\ldots,n. \tag{3.4.2}\]

Note that due to the convention \(\phi_{n0}=0\), the last equation in Theorem \ref{th:3.4.1} (for which \(j=n+1\)) is omitted. More conveniently, this is restated in matrix notation. To this end, let \(\Gamma_n=(\gamma(j-\ell))_{j,\ell=1,\ldots,n}\), \(\phi_n=(\phi_{n1},\ldots,\phi_{nn})^T\) and \(\gamma_n=(\gamma(1),\ldots,\gamma(n))^T\), where \(^T\) denotes the transpose. With these notations, (3.4.2.) becomes

   \[\Gamma_n\phi_n=\gamma_n \qquad\Longleftrightarrow\qquad \phi_n=\Gamma_n^{-1}\gamma_n, \tag{3.4.3}\]

provided that \(\Gamma_n\) is nonsingular.

The determination of the coefficients \(\phi_{n\ell}\) has thus been reduced to solving a linear equation system and depends only on second-order properties of \((X_t\colon t\in\mathbb{Z})\) which are given by the ACVF \(\gamma\).

Let \(X_n=(X_n,X_{n-1},\ldots,X_1)^T\). Then, \(\hat{X}_{n+1}=\phi_n^TX_n\). To assess the quality of the prediction, one computes the mean-squared error with the help of (3.4.3.) as follows:

   \[P_{n+1} =E\left[(X_{n+1}-\hat{X}_{n+1})^2\right] \nonumber\]
          \[=E\left[(X_{n+1}-\phi_n^T X_n)^2\right] \nonumber\]
          \[=E\left[(X_{n+1}-\gamma_n^T\Gamma_n^{-1} X_n)^2\right]\nonumber\]
          \[=E\left[X_{n+1}^2-2\gamma_n^T\Gamma_n^{-1} X_nX_{n+1} +\gamma_n^T\Gamma_n^{-1} X_n X_n^{T}\Gamma_n^{-1}\gamma_n\right]\nonumber\]
          \[=\gamma(0)-2\gamma_n^T\Gamma_n^{-1}\gamma_n +\gamma_n^T\Gamma_n^{-1}\Gamma_n\Gamma_n^{-1}\gamma_n\nonumber\]
          \[=\gamma(0)-\gamma_n^T\Gamma_n^{-1}\gamma_n. \tag{3.4.4}\]

As an initial example, we explain the prediction procedure for an autoregressive process of order 2.

Example 3.4.2(Prediction of an AR(2) Process). Let \((X_t\colon t\in\mathbb{Z})\) be the causal AR(2) process \(X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+Z_t\). Suppose that only an observation of \(X_1\) is available to forecast the value of \(X_2\). In this simplified case, the single prediction equation (3.4.2.) is

 \[ \phi_{11}\gamma(0)=\gamma(1), \]

so that \(\phi_{11}=\rho(1)\) and \(\hat{X}_{1+1}=\rho(1)X_1\).

In the next step, assume that observed values of \(X_1\) and \(X_2\) are at hand to forecast the value of \(X_3\). Then, one similarly obtains from (3.4.2.) that the predictor can be computed from

\[\hat{X}_{2+1} =\phi_{21}X_{2}+\phi_{22}X_1 =\phi_2^T X_2=(\Gamma_2^{-1}\gamma_2)^T X_2\]

 \[=(\gamma(1),\gamma(2))\left(\begin{array}{c@{\quad}c} \gamma(0) & \gamma(1) \\ \gamma(1) & \gamma(0) \end{array}\right)^{-1} \left(\begin{array}{c} X_2 \\ X_1 \end{array}\right).\]

However, applying the arguments leading to the definition of the PAC in Section 3.3.3., one finds that

 \[E\left[\{X_3-(\phi_1X_2+\phi_2X_1)\}X_1\right]=E[Z_3X_1]=0,\]
      \[E\left[\{X_3-(\phi_1X_2+\phi_2X_1)\}X_2\right]=E[Z_3X_2]=0.\]
   Hence, \(\hat{X}_{2+1}=\phi_1X_2+\phi_2X_1\) and even \(\hat{X}_{n+1}=\phi_1X_n+\phi_2X_{n-1}\) for all \(n\geq 2\), exploiting the particular autoregressive structure.
Since similar results can be proved for general causal AR(p) processes, the one-step predictors have the form

\[ \hat{X}_{n+1}=\phi_1X_n+\ldots+\phi_pX_{n-p+1} \]

whenever the number of observed variables n is at least p.

The major drawback of this approach is immediately apparent from the previous example: For larger sample sizes n, the prediction procedure requires the calculation of the inverse matrix \(\Gamma_n^{-1}\) which is computationally expensive. In the
remainder of this section, two recursive prediction methods are introduced that bypass the inversion altogether. They are known as
Durbin-Levinson algorithm and innovations algorithm. Finally, predictors based on the {\it infinite past}\/ are introduced which are often easily applicable for the class of causal and invertible ARMA processes. 

Method 1 (The Durbin-Levinson algorithm) If \((X_t\colon t\in\mathbb{Z})\) is a zero mean weakly stationary process with ACVF \(\gamma\) such that \(\gamma(0)>0\) and \(\gamma(h)\to 0\) as \(h\to\infty\), then the coefficients \(\phi_{n\ell}\) in (3.4.2.) and the mean squared errors \(P_n\) in (3.4.4.) satisfy the recursions
\[ \phi_{11}=\frac{\gamma(1)}{\gamma(0)},\qquad P_0=\gamma(0), \]
and, for \(n\geq 1\),
\[ \phi_{nn}=\frac{1}{P_{n-1}} \left(\gamma(n)-\sum_{\ell=1}^{n-1}\phi_{n-1,\ell}\gamma(n-\ell)\right), \]
\[ \left(\begin{array}{l}\phi_{n1} \\ {~}\vdots \\ \phi_{n,n-1}\end{array}\right) =\left(\begin{array}{l} \phi_{n-1,1} \\ {~}\vdots \\ \phi_{n-1,n-1}\end{array}\right) -\phi_{nn}\left(\begin{array}{l} \phi_{n-1,n-1} \\ {~}\vdots \\ \phi_{n-1,1}\end{array}\right) \]
and
\[ P_{n}=P_{n-1}(1-\phi_{nn}^2). \]
It can be shown that under the assumptions made on the process \((X_t\colon t\in\mathbb{Z})\), it holds indeed that \(\phi_{nn}\) is equal to the value of the PACF of \((X_t\colon t\in\mathbb{Z})\) at lag n. The result is formulated as Corollary 5.2.1 in Brockwell and Davis (1991). This fact is highlighted in an example.

Example 3.4.2 (The PACF of an AR(2) process).  Let \((X_t\colon t\in\mathbb{Z})\) be a causal AR(2) process. Then, \(\rho(1)=\phi_1/(1-\phi_2)\) and all other values can be computed recursively from
   \[ \rho(h)-\phi_1\rho(h-1)-\phi_2\rho(h-2)=0,\qquad h\geq 2. \]
   Note that the ACVF \(\gamma\) satisfies a difference equation with the same coefficients, which is seen by multiplying the
   latter equation with \(\gamma(0)\). Applying the Durbin-Levinson algorithm gives first that
   \[ \phi_{11}=\frac{\gamma(1)}{\gamma(0)}=\rho(1) \qquad\mbox{and}\qquad P_1=P_0(1-\phi_{11}^2)=\gamma(0)(1-\rho(1)^2). \]
   Ignoring the recursion for the error terms \(P_n\) in the following, the next \(\phi_{n\ell}\) values are obtained as
 
      \[\phi_{22} =\frac{1}{P_1}\left[\gamma(2)-\phi_{11}\gamma(1)\right] =\frac{1}{1-\rho(1)^2}\left[\rho(2)-\rho(1)^2\right]\]
               \[=\frac{\phi_1^2(1-\phi_2)^{-1}+\phi_2-[\phi_1(1-\phi_2)^{-1}]^2} {1-[\phi_1(1-\phi_2)^{-1}]^2}=\phi_2,\]
      \[\phi_{21} =\phi_{11}-\phi_{22}\phi_{11}=\rho(1)(1-\phi_2)=\phi_1, \]
      \[\phi_{33} =\frac{1}{P_2}\left[\gamma(3)-\phi_{21}\gamma(2)-\phi_{22}\gamma(1)\right] =\frac{1}{P_2}\left[\gamma(3)-\phi_1\gamma(2)-\phi_2\gamma(2)\right]=0.\]
   
   Now, referring to the remarks after Example 3.3.7., no further computations are necessary to determine the PACF because \(\phi_{nn}=0\) for all \(n>p=2\).

Method 2 (The innovations algorithm) In contrast to the Durbin-Levinson algorithm, this method can also be applied to nonstationary processes. It should thus, in general, be preferred over Method 1. The innovations algorithm gets its name from the fact that one directly uses the form of the prediction equations in Theorem 3.4.1. which are stated in terms of the innovations \((X_{t+1}-\hat{X}_{t+1})_{t\in\mathbb{Z}}\). Observe that the sequence consists of uncorrelated random variables.

The one-step predictors \(\hat{X}_{n+1}\) can be calculated from the recursions
\[ \hat{X}_{0+1}=0,\qquad P_1=\gamma(0) \]
and, for \(n\geq 1\),
   

\[\hat{X}_{n+1} =\sum_{\ell=1}^n\theta_{n\ell}(X_{n+1-\ell}-\hat{X}_{n+1-\ell})\]
\[P_{n+1} =\gamma(0)-\sum_{\ell=0}^{n-1}\theta_{n,n-\ell}^2P_{\ell+1},\]
where the coefficients are obtained from the equations
\[ \theta_{n,n-\ell}=\frac{1}{P_{\ell+1}} \left[\gamma(n-\ell)-\sum_{i=0}^{\ell-1}\theta_{\ell,\ell-i}\theta_{n,n-i}P_{i+1}\right], \qquad\ell=0,1,\ldots,n-1. \]
As example we show how the innovations algorithm is applied to a
moving average time series of order 1.

Example 3.4.3 (Prediction of an MA(1) Process).  Let \((X_t\colon t\in\mathbb{Z})\) be the MA(1) process \(X_t=Z_t+\theta
   Z_{t-1}\). Note that
   \[ \gamma(0)=(1+\theta^2)\sigma^2,\qquad\gamma(1)=\theta\sigma^2
      \qquad\mbox{and}\qquad\gamma(h)=0\quad(h\geq 2). \]
   Using the innovations algorithm, one can compute the one-step
   predictor from the values
   \begin{align*}
      \theta_{n1}=\frac{\theta\sigma^2}{P_n},\qquad
      \theta_{n\ell}=0 \quad(\ell=2,\ldots,n-1),
   \end{align*}
   and
   \begin{align*}
      P_1    &=(1+\theta^2)\sigma^2,\\[.2cm]
      P_{n+1}&=(1+\theta^2-\theta\theta_{n1})\sigma^2
   \end{align*}
   as
   \[ \hat{X}_{n+1}=\frac{\theta\sigma^2}{P_n}(X_n-\hat{X}_{n}).\]

Method 3 (Prediction based on the infinite past) Suppose that a causal and invertible ARMA(p,q) process is analyzed. Assume
further that (unrealistically) the complete history of the process can be stored and that thus all past variables
\((X_t\colon t\leq n)\) can be accessed. Define then
\[ \tilde{X}_{n+m}=E[X_{n+m}|X_n,X_{n-1},\ldots], \]
as the m-step ahead predictor based on the infinite past. It can be shown that, for large sample sizes n, the difference between the values of \(\hat{X}_{n+m}\) and \(\tilde{X}_{n+m}\) vanishes at an exponential rate. Exploiting causality and invertibility of the ARMA process, one can transform the predictor \(\tilde{X}_{n+m}\) so that it is in a computationally more feasible form. To do so, note that by causality

   \[\tilde{X}_{n+m} =E[X_{n+m}|X_n,X_{n-1},\ldots]\nonumber\]
   \[=E\left[\sum_{j=0}^\infty\psi_jZ_{n+m-j}\Big|X_n,X_{n-1},\ldots\right]\nonumber\]
   \[=\sum_{j=m}^\infty\psi_jZ_{n+m-j} \tag{3.4.5}\]


because \(E[Z_t|X_n,X_{n-1},\ldots]\) equals zero if t>n and equals Z_t if \(t\leq n\) (due to invertibility!). The representation in
(3.4.5.) can be used to compute the mean squared prediction error \(\tilde{P}_{n+m}\). It follows from causality that

   \[ \tilde{P}_{n+m}=E[(X_{n+m}-\tilde{X}_{n+m})^2] =E\left[\left(\sum_{j=0}^{m-1}\psi_jZ_{n+m-j}\right)^2\right] =\sigma^2\sum_{j=0}^{m-1}\psi_j^2. \tag{3.4.6}\]

On the other hand, (3.4.5.) does not allow to directly calculate the forecasts because \(\tilde{X}_{n+m}\) is given in terms
of the noise variables \(Z_{n+m-j}\). Instead invertibility will be utilized. Observe first that
\[ E[X_{n+m-j}|X_n,X_{n-1},\ldots]=\left\{\begin{array}{c@{\quad}l} \tilde{X}_{n+m-j}, & j<m.\\[.2cm] X_{n+m-j}, & j\geq m. \end{array}\right. \]
By invertibility (the ``0='' part follows again from causality),

   \[0=E[Z_{n+m}|X_n,X_{n-1},\ldots]\]
    \[=E\left[\sum_{j=0}^\infty\pi_jX_{n+m-j}\Big|X_n,X_{n-1},\ldots\right]\]
    \[=\sum_{j=0}^\infty\pi_jE[X_{n+m-j}|X_n,X_{n-1},\ldots].\]

Combining the previous two statements, yields

   \[\tilde{X}_{n+m}=-\sum_{j=1}^{m-1}\pi_j\tilde{X}_{n+m-j} -\sum_{j=m}^\infty\pi_jX_{n+m-j}. \tag{3.4.7}\]

The equations can now be solved recursively for \(m=1,2,\ldots\) Note, however, that for any \(m\geq 1\) the sequence
\((X_{n+m+t}-\tilde{X}_{n+m+t}\colon t\in\mathbb{Z})\) does not consist of uncorrelated random variables. In fact, if \(h\in\mathbb{N}_0\), it holds that


   \[E[(X_{n+m}-\tilde{X}_{n+m})(X_{n+m+h}-\tilde{X}_{n+m+h})]\]
    \[=E\left[\sum_{j=0}^{m-1}\psi_jZ_{n+m-j}\sum_{i=0}^{m+h-1}\psi_iZ_{n+m+h-i}\right]\]
    \[=\sigma^2\sum_{j=0}^{m-1}\psi_j\psi_{j+h}.\]

Finally, for practical purposes the given forecast needs to be truncated. This is accomplished by setting
\[ \sum_{j=n+m}^\infty\pi_jX_{n+m-j}=0. \]
The resulting equations (see (3.4.7.) for comparison) yield recursively the truncated m-step predictors \(X_{n+m}^*\):

\[X_{n+m}^*=-\sum_{j=1}^{m-1}\pi_jX_{n+m-j}^*-\sum_{j=m}^{n+m-1}\pi_jX_{n+m-j}. \tag{3.4.8}\]