Forecasting for Economics and Business

class: center, middle, inverse, title-slide

.title[
# Forecasting for Economics and Business
]
.subtitle[
## Lecture 7: Vector Autoregression
]
.author[
### David Ubilava
]
.date[
### University of Sydney
]

---

# Economic variables are inter-related

.pull-left[
![](Art/multivariate.png)
]

.pull-right[
Economic variables are often (and usually) interrelated; for example,

- income affects consumption; 
- interest rates impact investment.

The dynamic linkages between two (or more) economic variables can be modeled as a *system of equations*, better known as a vector autoregression, or simply VAR.
]

---

# Vector autoregression

.right-column[
To begin, consider a bivariate first-order VAR.

Let `$\{X_{1t}\}$` and `$\{X_{2t}\}$` be the stationary stochastic processes. A bivariate VAR(1), is then given by:
`$$\begin{aligned}
x_{1t} &= \alpha_1 + \pi_{11}x_{1t-1} + \pi_{12}x_{2t-1} + \varepsilon_{1t} \\
x_{2t} &= \alpha_2 + \pi_{21}x_{1t-1} + \pi_{22}x_{2t-1} + \varepsilon_{2t}
\end{aligned}$$`

where `$\varepsilon_{1,t} \sim iid(0,\sigma_1^2)$` and `$\varepsilon_{2,t} \sim iid(0,\sigma_2^2)$`, and the two can be correlated, i.e., `$Cov(\varepsilon_{1,t},\varepsilon_{2,t}) \neq 0$`.
]

---

# Vector autoregression

.right-column[
An `$n$`-dimensional VAR of order `$p$`, VAR(p), presented in matrix notation: `$$\mathbf{x}_t = \mathbf{\alpha} + \Pi^{(1)} \mathbf{x}_{t-1} + \ldots + \Pi^{(p)} \mathbf{x}_{t-p} + \mathbf{\varepsilon}_t,$$` where `$\mathbf{x}_t = (x_{1,t},\ldots,x_{n,t})'$` is a vector of `$n$` (potentially) related variables; `$\mathbf{\varepsilon}_t = (\varepsilon_{1,t},\ldots,\varepsilon_{n,t})'$` is a vector of error terms, such that `$E\left(\mathbf{\varepsilon}_t\right) = \mathbf{0}$`, `$E\left(\mathbf{\varepsilon}_t^{}\mathbf{\varepsilon}_t^{\prime}\right) = \Sigma_{\mathbf{\varepsilon}}$`, and `$E\left(\mathbf{\varepsilon}_{t}^{}\mathbf{\varepsilon}_{s \neq t}^{\prime}\right) = 0$`.
]
		
---

# A parameter matrix of a vector autoregression

.right-column[
`$\Pi^{(1)},\ldots,\Pi^{(p)}$` are `$n$`-dimensional parameter matrices such that: 
`$$\Pi^{(j)} = 
	    \left[ 
		\begin{array}{cccc} 
		\pi_{11}^{(j)} & \pi_{12}^{(j)} & \cdots &  \pi_{1n}^{(j)} \\ 
		\pi_{21}^{(j)} & \pi_{22}^{(j)} & \cdots &  \pi_{2n}^{(j)} \\  
		\vdots & \vdots & \ddots &  \vdots \\  
		\pi_{n1}^{(j)} & \pi_{n2}^{(j)} & \cdots &  \pi_{nn}^{(j)} 
		\end{array} 
		\right],\;~~j=1,\ldots,p$$`
]
		
---

# Features of vector autoregression

.right-column[
General features of a (reduced-form) vector autoregression are that:
- only the lagged values of the dependent variables are on the right-hand-side of the equations.
  * Although, trends and seasonal variables can also be included.
- Each equation has the same set of right-hand-side variables.
  * However, it is possible to impose different lag structure across the equations, especially when `$p$` is relatively large. This is because the number of parameters increases very quickly with the number of lags or the number of variables in the system.
- The autregressive order, `$p$`, is the largest lag across all equations.
]

---

# Modeling vector autoregression

.right-column[
The autoregressive order, `$p$`, can be determined using system-wide information criteria:

`$$\begin{aligned}
& AIC = \ln\left|\Sigma_{\mathbf{\varepsilon}}\right| + \frac{2}{T}(pn^2+n) \\
& SIC = \ln\left|\Sigma_{\mathbf{\varepsilon}}\right| + \frac{\ln{T}}{T}(pn^2+n)
\end{aligned}$$`

where `$\left|\Sigma_{\mathbf{\varepsilon}}\right|$` is the determinant of the residual covariance matrix; `$n$` is the number of equations, and `$T$` is the total number of observations.
]

---

# Estimating vector autoregression

.right-column[
When each equation of VAR has the same regressors, the OLS can be applied to each equation individually to estimate the regression parameters - i.e., the estimation can be carried out on the equation-by-equation basis.

When processes are covariance-stationarity, conventional t-tests and F-tests are applicable for hypotheses testing.
]

---

# U.S. Interest Rates and Inflation

.right-column[
![](07-Multivariate_files/figure-html/unnamed-chunk-2-1.png)
]

---

# U.S. Interest Rates and Inflation: VAR(3)

.pull-left[
<table class="myTable">
 <thead>
  <tr>
   <th style="text-align:left;"> p </th>
   <th style="text-align:left;"> AIC </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> -2.892 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> -3.189 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> -3.244 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:left;"> -3.208 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<table class="myTable table" style="font-size: 22px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> `$\alpha_i$` </th>
   <th style="text-align:right;"> `$\pi_{i1}^{(1)}$` </th>
   <th style="text-align:right;"> `$\pi_{i1}^{(2)}$` </th>
   <th style="text-align:right;"> `$\pi_{i1}^{(3)}$` </th>
   <th style="text-align:right;"> `$\pi_{i2}^{(1)}$` </th>
   <th style="text-align:right;"> `$\pi_{i2}^{(2)}$` </th>
   <th style="text-align:right;"> `$\pi_{i2}^{(3)}$` </th>
  </tr>
 </thead>
<tbody>
  <tr grouplength="2"><td colspan="8" style="border-bottom: 1px solid;"><strong>Interest Rate (i=1)</strong></td></tr>
<tr>
   <td style="text-align:left;padding-left: 2em;" indentlevel="1"> estimate </td>
   <td style="text-align:right;"> 0.193 </td>
   <td style="text-align:right;"> 1.429 </td>
   <td style="text-align:right;"> -0.651 </td>
   <td style="text-align:right;"> 0.174 </td>
   <td style="text-align:right;"> 0.199 </td>
   <td style="text-align:right;"> -0.311 </td>
   <td style="text-align:right;"> 0.147 </td>
  </tr>
  <tr>
   <td style="text-align:left;padding-left: 2em;" indentlevel="1"> s.e. </td>
   <td style="text-align:right;"> 0.082 </td>
   <td style="text-align:right;"> 0.053 </td>
   <td style="text-align:right;"> 0.086 </td>
   <td style="text-align:right;"> 0.053 </td>
   <td style="text-align:right;"> 0.103 </td>
   <td style="text-align:right;"> 0.167 </td>
   <td style="text-align:right;"> 0.102 </td>
  </tr>
  <tr grouplength="2"><td colspan="8" style="border-bottom: 1px solid;"><strong>Inflation Rate (i=2)</strong></td></tr>
<tr>
   <td style="text-align:left;padding-left: 2em;" indentlevel="1"> estimate </td>
   <td style="text-align:right;"> 0.047 </td>
   <td style="text-align:right;"> 0.044 </td>
   <td style="text-align:right;"> 0.027 </td>
   <td style="text-align:right;"> -0.070 </td>
   <td style="text-align:right;"> 1.250 </td>
   <td style="text-align:right;"> -0.097 </td>
   <td style="text-align:right;"> -0.164 </td>
  </tr>
  <tr>
   <td style="text-align:left;padding-left: 2em;" indentlevel="1"> s.e. </td>
   <td style="text-align:right;"> 0.042 </td>
   <td style="text-align:right;"> 0.027 </td>
   <td style="text-align:right;"> 0.043 </td>
   <td style="text-align:right;"> 0.027 </td>
   <td style="text-align:right;"> 0.052 </td>
   <td style="text-align:right;"> 0.085 </td>
   <td style="text-align:right;"> 0.052 </td>
  </tr>
</tbody>
</table>
]

---

# Testing in-sample Granger causality

.right-column[
Consider a bivariate VAR(p): 
`$$\begin{aligned}
x_{1t} &= \alpha_1 + \pi_{11}^{(1)} x_{1t-1} + \cdots + \pi_{11}^{(p)} x_{1t-p} \\
&+ \pi_{12}^{(1)} x_{2t-1} + \cdots + \pi_{12}^{(p)} x_{2t-p} +\varepsilon_{1t}  \\
x_{2t} &= \alpha_1 + \pi_{21}^{(1)} x_{1t-1} + \cdots + \pi_{21}^{(p)} x_{1t-p} \\
&+ \pi_{22}^{(1)} x_{2t-1} + \cdots + \pi_{22}^{(p)} x_{2t-p} +\varepsilon_{2t} 
\end{aligned}$$`

- `$\{X_2\}$` does not Granger cause `$\{X_1\}$` if `$\pi_{12}^{(1)}=\cdots=\pi_{12}^{(p)}=0$`
- `$\{X_1\}$` does not Granger cause `$\{X_2\}$` if `$\pi_{21}^{(1)}=\cdots=\pi_{21}^{(p)}=0$`
]

---

# Testing in-sample Granger causality

.right-column[
<table class="myTable">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> F </th>
   <th style="text-align:left;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Inflation Rate GC Interest Rate </td>
   <td style="text-align:left;"> 2.825 </td>
   <td style="text-align:left;"> 0.039 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Interest Rate GC Inflation Rate </td>
   <td style="text-align:left;"> 5.223 </td>
   <td style="text-align:left;"> 0.002 </td>
  </tr>
</tbody>
</table>
]

---

# One-step-ahead forecasts

.right-column[
Consider realizations of dependent variables from a bivariate VAR(1): 
`$$\begin{aligned}
x_{1t+1} &= \alpha_1 + \pi_{11} x_{1t} + \pi_{12} x_{2t} + \varepsilon_{1t+1} \\
x_{2t+1} &= \alpha_2 + \pi_{21} x_{1t} + \pi_{22} x_{2t} + \varepsilon_{2t+1}
\end{aligned}$$`

Point forecasts are:
`$$\begin{aligned}
x_{1t+1|t} &= E(x_{1t+1}|\Omega_t) = \alpha_1 + \pi_{11} x_{1t} + \pi_{12} x_{2t} \\
x_{2t+1|t} &= E(x_{2t+1}|\Omega_t) = \alpha_2 + \pi_{21} x_{1t} + \pi_{22} x_{2t}
\end{aligned}$$`
]

---

# One-step-ahead forecasts

.right-column[
Forecast errors are: 
`$$\begin{aligned}
e_{1t+1} &= x_{1t+1} - x_{1t+1|t} = \varepsilon_{1t+1} \\
e_{2t+1} &= x_{2t+1} - x_{2t+1|t} = \varepsilon_{2t+1}
\end{aligned}$$`

Forecast variances are:
`$$\begin{aligned}
\sigma_{1t+1}^2 &= E(x_{1t+1} - x_{1t+1|t}|\Omega_t)^2 = E(\varepsilon_{1t+1}^2) = \sigma_{1}^2 \\
\sigma_{2t+1}^2 &= E(x_{2t+1} - x_{2t+1|t}|\Omega_t)^2 = E(\varepsilon_{2t+1}^2) = \sigma_{2}^2
\end{aligned}$$`
]

---

# Multi-step-ahead forecasts

.right-column[
Realizations of dependent variables in period `$t+h$`: 
`$$\begin{aligned}
x_{1t+h} &= \alpha_1 + \pi_{11} x_{1t+h-1} + \pi_{12} x_{2t+h-1} + \varepsilon_{1t+h} \\
x_{2t+h} &= \alpha_2 + \pi_{21} x_{1t+h-1} + \pi_{22} x_{2t+h-1} + \varepsilon_{2t+h}
\end{aligned}$$`

Point forecasts are:
`$$\begin{aligned}
x_{1t+h|t} &= E(x_{1t+1}|\Omega_t) = \alpha_1 + \pi_{11} x_{1t+h-1|t} + \pi_{12} x_{2t+h-1|t} \\
x_{2t+h|t} &= E(x_{2t+1}|\Omega_t) = \alpha_2 + \pi_{21} x_{1t+h-1|t} + \pi_{22} x_{2t+h-1|t}
\end{aligned}$$`
]

---

# Multi-step-ahead forecasts

.right-column[
Forecast errors are: 
`$$\begin{aligned}
e_{1t+h} &= x_{1t+h} - x_{1t+h|t} = \pi_{11} e_{1t+h-1} + \pi_{12} e_{2t+h-1} + \varepsilon_{1t+h} \\
e_{2t+h} &= x_{2t+h} - x_{2t+h|t} = \pi_{21} e_{1t+h-1} + \pi_{22} e_{2t+h-1} + \varepsilon_{2t+h}
\end{aligned}$$`

Forecast variances are the functions of error variances and covariances, and the model parameters.
]

---

# Forecasting interest rate

.right-column[
![](07-Multivariate_files/figure-html/unnamed-chunk-6-1.png)
]

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# Forecasting inflation rate

.right-column[
![](07-Multivariate_files/figure-html/unnamed-chunk-7-1.png)
]

---